Plane Thermoelastic Waves in Infinite Half-Space Caused FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 13, N o 2, 2015, pp. 181 - 191 1NON-LINEAR MATHEMATICAL MODELS IN THE THEORY OF EXPERIMENTAL DESIGN: APPLICATION IN THE MANUFACTURING PROCESSES UDC 621.7+519.2 Anđela Lazarević, Dragoljub Lazarević Faculty of Mechanical Engineering, University of Niš, Serbia Abstract. Since real processes usually are not linear, different non-linear mathematical models are usually applied for studying their behavior: high order polynomials, power, exponential and other functions. In this paper, the application of multiple power functions is presented, with and without interactions between the influential factors, in modelling various technological problems. The modeling results show that the proper mathematical model is selected, one that guarantees high accuracy in the entire experimental space. Furthermore, an important conclusion is reached, stating that interactions between the influential factors are not of importance in such mathematical models; thus they can be neglected. Therefore, it is shown that it is more practical to use the basic mathematical model (without interactions) than the expanded mathematical model (with interactions) Key Words: Experimental Design, Non-linear Mathematical Models, Plasma Cutting, Face Milling 1. INTRODUCTION The theory of experimental design represents a qualitatively new approach to the theoretical-experimental analysis and optimization of complex processes/systems, with universal application and range of advantages in comparison to the concept and practice of the one-factor-at-a-time method [1-2]. The theory of experimental design addresses management of an experiment, i.e., its preparation and physical realization as well as processing and analysis of the experimental results according to the previously determined plan, which enables the variations of the influential factors simultaneously on various levels, in each of the following series of trials. The experimental design was proven as successful method in various fields, which is especially evident in complex research objects with a large number of influential factors. Received June 4, 2015 / Accepted July 20, 2015 Corresponding author: Anđela Lazarević Affiliation: Faculty of Mechanical Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš E-mail: andjela.lazarevic@gmail.com Original scientific paper 182 A. LAZAREVIĆ, D. LAZAREVIĆ The basic characteristics and advantages of the theory of experimental design are as follows [1,5,6]: number of trials, successive experiment management, i.e., in stages (step by step), from simpler to more complex designs, a simple statistical (regression and dispersion) analysis of experimental results, a possibility of qualitative and quantitative assessment of effects of each of the influential factors and their interactions on the target function, easy optimization of a process/system which is the subject of research, on the basis of obtained empirical (regression) model of the target function that encompasses the complete experimental space, minimum time and material losses (expenses) for experiment realization, elimination of the subjective influence of the researcher, etc. [7]. The outline of recommended procedure in the theory of experimental design, as a statistical approach in designing and analysing an experiment, include [1-4]: recognition and statement of the problem, choice of factors, levels, and range, selection of the response variable, choice of experimental design, performing the experiment, statistical analysis of the data, and conclusions and recommendations. The selection of mathematical model (regression equation), which is used to establish a connection between influential factors and target function, depends on the research objective, the complexity of the phenomenon under study, the selected experimental design, as well as quantity and quality of available information. Both theory and practice have shown that in most cases the best choice is a mathematical model in the form of polynomials (linear, quasi-linear, square, etc.). More complex mathematical models ensure higher accuracy in the prediction of researched system behaviour in the selected hyperspace. However, such models also imply a more complex experimental procedures, higher expenses, time-consuming analysis and interpretation of experimental results etc. On the other hand, the selection of the simplest linear mathematical model does not enable an analysis of the factor interactions’ effects; various examples undoubtedly show that the factor interactions’ effect on the target function can be more significant than that of separate factors. It should also be emphasized that it is the case of lower-order factor interactions, since the higher-order factor interactions can be neglected. Namely, interactions of many factors, as a rule, do not influence the accuracy of the selected mathematical model. Various experiments and practical knowledge point to the fact that the mathematical models used for the analysis and optimisation of complex processes are those in the form of multiple power, exponential, or some other function instead of polynomials. Such mathematical models are easily transformed into linear functions (first order polynomials). Therefore, the synergy of a complex non-linear mathematical model and simple processing and analysis of experimental data is established [8]. Upon linearization of the aforementioned functions, a basic mathematical model without interactions is obtained. As it was mentioned, the influence of interactions should not be rejected a priori. The choice of appropriate criterion for building the mathematical model is not often obvious [9-10]. The accuracy of basic non-linear models without interactions and expanded non-linear mathematical models with interactions are comparatively analyzed on examples given in this paper. Non-linear Mathematical Models in the Theory of Experimental Design... 183 2. COMPLEX POWER FUNCTIONS AS MATHEMATICAL MODELS Numerous experiments have shown that a successful approximation can be made for modelling most diverse technological processes by using a complex power equation [11- 16] in the following form: ki p k p i ppp c XXXXXCF  321 321 (1) where Xi (i=1,2,3…k) - influential factors (natural coordinates), C, pi - constants to be determined, k - number of factors. By using logarithmic form for the Eq. (1) it could be rewritten in a reduced linear form: kkii xbxbxbby  110 (2) where iiiic XxpbCbFy ln,,ln,ln 0  Eq. (2) is represented in coded coordinates. The connection between natural and coded coordinates is established through the following transformation equations: ki XX XX x ii ii i ....3,2,1; lnln lnln 21 minmax max     (3) According to Eq. (3), coded coordinates take integer values (xi=+1  0  -1). For the purposes of dispersion analysis it is necessary to repeat trials at certain points in the experimental hyperspace. The systems of trial repetition are as follows: 1. Repetition for n0 times only in the central point of the experimental design (xi=0); 2. Uniform repetition for n times in each vertex of an experimental hypercube (xi=1); 3. Non-uniform repetition for nu times in certain vertices of an experimental hypercube, or possibly, only in one point. It should be emphasised that for such mathematical model the determination of basic levels of influential factors is conducted by applying the following relations: minmax 2 0 iii XXX  (4) After determining the coefficients of linear regression Eq. (2), the unknown constants of target function Eq. (1) are calculated using the following formulas: )ln(exp; )/ln( 2 max 10minmax i k i i k i i ii i i XpbC XX b p    (5) The mathematical model can be expanded by introducing interactions of influential factors, when Eq. (2) turns into the quasi-linear form. By introducing transformation equations into this equation, one gets the expanded form of power function Eq. (1) in the form: )ln...lnln...lnlnln ... 3 ln 2 ln 1 ln 123 ... 2 ln 1 ln 12 (exp...2 2 1 1 k X j X i X kijlm p l X j X i X mijl p XXXpXXp p X p XCK      (6) 184 A. LAZAREVIĆ, D. LAZAREVIĆ 2.1 Selected examples of mathematical models application Example I: Modelling of the kerf in plasma arc cutting The objective of the plasma cutting process is to concentrate a large amount of energy on a small surface of a workpiece which leads to intensive heating of the material surface. The source of energy is high temperature and high speed ionized gas. The gas is ionized by means of a direct current passing between the cathode (inside the nozzle) and the anode (workpiece). The plasma jet cuts the material by releasing the energy spent for the plasma gas ionization upon hitting the workpiece surface. The removal of the melted material from the cutting zone is done by the plasma jet kinetic energy. Even though it is the case of a complex process, which is characterised by a large number of influential factors, the previous analysis has shown that this number can be reduced to three influential factors (input values, independent variables): cutting current (I), cutting speed (V) and material thickness (s) (Table 1). As the target function (output value, dependent variable) one of the basic characteristics of the cutting quality is selected – kerf (W). Influential factors were varied on two levels. Table 1 Cutting factors and their levels Cutting factor Symbol Unit Factor levels Level 1 (Low) Level 2 (High) Cutting current I A 45 80 Material thickness s mm 4 6 Cutting speed V m/min 0.9 1.3 For the purpose of this research, a number of experiments were done [17]. The entire experiment consists of 25 trials, while 8 measurement results are used for regression analysis (Table 2). The main experimental matrix (used for all the examples in this paper) is represented by the full factorial design of type 2 3 . Such a design enables the application of mathematical models including independent assessment of the main effects, as well as all influential factor interactions. Straight-line cuts are made in the performed experiment by varying input process factors, according to Table 1. Test samples are made of stainless steel X10CrNiMn-16- 10-2 EN designation (EN10025). The experimental research of plasma cutting process is conducted on the CNC machine for plasma cutting, type HPm Steel Max 6.25. The given data on the kerf represent mean values from obtained 3 measurements. For the purposes of using Eqs. (1-6), the following substitution of variables was introduced: Fc=W; X1=I; X2=V; X3=s. The need to determine the unknown constant C and exponents p1; p2; p3 in this case arises from the basic Eq. (1). The regression analysis is conducted, coefficients of the linear mathematical model are generated, and thus the regression equation in coded coordinates obtains the following form: 1 2 3 0.62338 0.03463 0.04263 0.008125y x x x    (7a) Non-linear Mathematical Models in the Theory of Experimental Design... 185 This equation is easily transformed, in the afore-described manner, into non-linear regression equation in natural coordinates: 04008.023186.012038.0 08073.1 sVIW   (7b) The kerf increases with the increase in cutting current and material thickness since the exponents I and s are positive, whereas the exponent of cutting speed is negative, indicating decreasing tendency of kerf, with increasing cutting speed. Eq. (7b) is suitable for application in engineering practice. By introducing interactions into the Eq. (7a), the quasi-linear regression equation is obtained in the form: 3213231 21321 008125.0006125.0007375.0 001125.0008125.004263.003463.062338.0 xxxxxxx xxxxxy   (8a) In the natural coordinates, this regression equation has the form: )lnlnln7577.0lnln266585.3lnln185933.0ln ln225267.1(exp75.3 73409.050963.517675.0 1 sVIsVsIV IsVIW    (8b) For the purposes of verification of the generated mathematical models, the results from 17 trials are used, otherwise not used for the purposes of regression analysis, as shown in Table 3. This experiment confirmation, which contains a great number of trails, should contribute to higher reliability of conclusions from the following analysis. Table 2 Main experiment I V s Wexp Wcal   1 80 (+1) 1.3 (+1) 6 (+1) 1.88 1.8776 1.8655 0.1277- 0.7713- 0.1277 0.7713 2 45 (-1) 1.3 (+1) 6 (+1) 1.75 1.7507 1.7406 0.0400 0.5371- 0.0400 0.5371 3 80 (+1) 0.9 (-1) 6 (+1) 2.05 2.0483 2.0315 0.0829- -0.9024 0.0829 0.9024 4 45 (-1) 0.9 (-1) 6 (+1) 1.86 8571.1 8955.1 0.1559- 1.9086 0.1559 1.9086 5 80 (+1) 1.3 (+1) 4 (-1) 1.83 1.8276 1.8354 0.1311- 0.2951 0.1311 0.2951 6 45 (-1) 1.3 (+1) 4 (-1) 1.70 1.6989 1.7126 0.0647- 0.7412 0.0647 0.7412 7 80 (+1) 0.9 (-1) 4 (-1) 1.98 1.9779 1.9987 0.1061- 0.9444 0.1061 0.9444 8 45 (-1) 0.9 (-1) 4 (-1) 1.91 1.9079 1,8650 0.1099- -2.3560 0.1099- -2.3560 Note: Results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model Into Table 2 and Table 3, the calculation results obtained by applying Eq. (7b) and Eq. (8b) are also inserted, as well as corresponding errors. 186 A. LAZAREVIĆ, D. LAZAREVIĆ The following relations were used for the calculation of errors:  For the percentage error () and mean percentage error (  ), ny yy cal       ;(%)100 exp exp (9a)  For absolute percentage error () and mean absolute percentage error ( Δ ), ; n        (9b) where n - the number of trials in the experiment. Table 3 Confirmation experiment I V s Wexp Wcal   80 1.025 6 2.02 1.9862 1.9711 1.6733- -2.4208 1.6733 2.4208 80 1.075 6 1.96 1.9640 1.9495 0.2041 -0.5357 0.2041 0.5357 80 1.125 6 1.98 1.9430 1.9291 1.8687- -2.5707 1.8687 2.5707 80 1.175 6 1.91 1.9231 1.9097 0.6859 -0.0157 0.6859 0.0157 80 1.225 6 1.93 1.9042 1.8913 1.3368- -2.0052 1.3368 2.0052 80 1.275 6 1.92 1.8863 1.8739 1.7552- -2.4010 7552.1 2.4010 80 1.0 4 1.83 1.9336 1.9505 5.6612 6.5847 5.6612 6.5847 80 1.2 4 1.82 1.8593 1.8698 2.1593 2.7363 2.1593 2.7363 45 1.0 6 1.81 1.8259 1.8498 0.8785 2.1989 0.8785 2.1989 45 1.1 6 1.84 1.7982 1.8094 2.2717- -1.6630 2.2717 1.6630 45 0.95 4 1.80 1.8756 1.8418 4.2000 2.3222 4.2000 2.3222 45 1.0 4 1.80 1.8455 1.8200 2.5278 1.1111 2.5278 1.1111 45 1.05 4 1.72 1.8173 1.7995 5.6570 4.6221 5.6570 4.6221 45 1.1 4 1.75 1.7909 1.7802 2.3371 1.7257 2.3371 1.7257 45 1.15 4 1.67 1.7659 1.7619 5.7425 5.5030 5.7425 5.5030 45 1.20 4 1.63 1.7424 1.7446 6.8957 7.0307 6.8957 7.0307 45 1.25 4 1.62 1.7201 1.7282 6.1790 6.6790 6.1790 6.6790 Note: Results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model Based on the data from Table 2, one can conclude that the calculation errors are negligible, regardless whether the basic regression Eq. (7b) or the expanded regression Eq. (8b) is applied. As it could have been expected, calculation errors are larger in relation to confirmation experiment results (Table 3). Non-linear Mathematical Models in the Theory of Experimental Design... 187 Fig. 1 Correlations between the experimental results and results predicted by the generated model for the kerf calculation: a.) for basic model, b) for expanded model The conformance assessment of the mathematical model prediction with the experimental results can also be conducted using the correlation coefficient (R). Based on the Fig. 1 it could be concluded that both mathematical models ensure high levels of correlation, which are very similar, although not identical. The perfect prediction implies that all points should lie on a straight line passing through the origin and inclined at 45º. From Fig. 1 one can observe that there is a relatively small deviation of the line of regression (which represents the best linear approximation of the data) from the ideal line. Table 4 Adequacy assessment criteria of the mathematical models related for the plasma cutting process Experiment Basic mathematical model Expanded mathematical model max Δ R max Δ R Main experiment 2.36 1.06 0.911 0.16 0.1 0.925 Confirm.experiment 7.03 3.07 6.9 3.06 Taking three criteria (max,  , R) into consideration results in the fact that an expanded mathematical model cannot guarantee better prediction for randomly selected points within the chosen experimental space (Table 4). Example II: Modelling of the thermal stress in the face milling Face milling process has periodical characteristics, since the number of teeth in contact with material change periodically. Periodical heating and cooling of the teeth is very unfavourable, since it itself represents a thermal process. In this investigation three influential factors are chosen for modelling the face milling process: cutting speed (V), unit feed rate (fz) and depth of cut (a) (Table 5). 188 A. LAZAREVIĆ, D. LAZAREVIĆ Table 5 Cutting factors and their levels Cutting factor Symbol Unit Factor levels Level 1 (Low) Level 2 (High) Cutting speed V m/s 2.32 3.67 Unit feed rate fz mm/tooth 0.178 0.280 Depth of cut a mm 1.00 2.25 Thermo-electric current (U) is chosen as the target function. Influential factors are varied on two levels. A sample of 16 trials is extracted from the literature [18]. One half of trials are used for conducting the regression analysis (Table 6), while another half is used for the selected mathematical models verification (Table 7). Table 6 Main experiment V fz a Uexp Ucal   3.67 (+1) 0.28 (+1) 2.25 (+1) 14.8 8000.14 8188.14 0000.0 1270.0 0000.0 1270.0 2.32 (-1) 0.28 (+1) 2.25 (+1) 13.4 3998.13 3733.13 0015.0 1993.0   0015.0 1993.0 3.67 (+1) 0.18 (-1) 2.25 (+1) 14.0 0000.14 0413.14 0000.0 2950.0 0000.0 2950.0 2.32 (-1) 0.18 (-1) 2.25 (+1) 12.7 6998.12 6716.12 0016.0 2236.0   0016.0 2236.0 3.67 (+1) 0.28 (+1) 1.00 (-1) 14.2 2000.14 1413.14 0000.0 4134.0 0000.0 4134.0 2.32 (-1) 0.28 (+1) 1.00 (-1) 12.7 7000.12 7618.12 0000.0 4866.0 0000.0 4866.0 3.67 (+1) 0.18 (-1) 1.00 (-1) 13.4 4001.13 3993.13 0007.0 0052.0 0007.0 0052.0 2.32 (-1) 0.18 (-1) 1.00 (-1) 12.1 1000.12 0922.12 0000.0 0645.0 0000.0 0645.0 Note: Results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model Test samples are made of steel C 60 DIN designation. The experiment is done on the milling machine FS-GVK-3 (Prvomajska). Cutting head JAL G-750 ( 125 mm) with hard material inserts SPAN 12 03 ER is used. The metering of the thermo-electric current (in mV) is done using thermocouples and heatmeter (Digital Multi-thermometer). The metering is done in a few seconds after the beginning of the cutting process, namely in the stationary phase of the process. Non-linear Mathematical Models in the Theory of Experimental Design... 189 It should be mentioned that in the confirmation experiment the first two trials are related to the central point of the experimental design. The other trails are related to the experimental points located on the coordinate axes of the experimental design, therefore, both models generate the same results. Table 7 Confirmation experiment V fz a Uexp Ucal   2.95 0.223 1.0 13.4 13.3863 13.3863 0.1022- -0.1022 0.1022 0.1022 2.95 0.223 1.0 13.7 13.3863 13.3863 2.2898- -2.2898 2.2898 2.2898 1.83 0.223 1.5 12.1 12.0805 12.0805 0.1612- -0.1612 0.1612 0.1612 4.65 0.223 1.5 14.9 14.8332 14.8332 0.4483- -0.4483 0.4483 0.4483 2.95 0.142 1.5 12.6 12.6839 12.6839 0.6659 0.6659 0.6659 0.6659 2.95 0.351 1.5 14.1 14.1276 14.1276 0.1957 0.1957 0.1957 0.1957 2.95 0.223 0.67 12.8 12.7742 12.7742 0.2016- -0.2016 0.2016 0.2016 2.95 0.223 3.37 15.0 14.0276 14.0276 6.4827- -6.4827 6.4827 6.4827 Note: Results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model Applying the same methodology as in the previous examples, the following regresion equations are generated:  for the mathematical model without interactions: 321 02340.002695.005132.059423.2 xxxy  (10a) 05771.011898.022380.029976.12 afVU z (10b)  for the mathematical model with interactions: 3213231 21321 000959.0000356.0002106.0 001439.002340.002695.005132.059423.2 xxxxxxx xxxxxy   (11a) )lnlnln06834.0lnln06930.0 lnln07980.0(exp1562.12 04102.011741.023298.0 afVaf aVafVU zz z    (11b) From Eqs. (10) and (11) it is obvious that the thermo-electric current increases with the increase of cutting speed, unit feed rate and depth of cut, since exponents V, fz and a are positive. The regression equation in the form of Eq. (10b) is suitable for application in engineering practice. Based on Fig. 2 (which relates to the entire experiment) one can conclude that both mathematical models ensure high levels of correlations, which are almost identical. 190 A. LAZAREVIĆ, D. LAZAREVIĆ Fig. 2 Correlations between the experimental results and results predicted by the generated model for the face milling process: a) for basic model, b) for expanded model Taking all criteria into consideration results in the fact that an expanded mathematical model cannot guarantee a better prediction for randomly selected points within the entire experimental hyperspace (Table 8). Table 8 Adequacy assessment criteria of the mathematical models related to the face milling process Experiment Basic mathematical model Expanded mathematical model max Δ R max Δ R Main experiment 0.49 0.23 0.965 0.02 0.005 0.966 Confirm. experiment 6.48 1.32 6.48 1.32 3. CONCLUSIONS This paper presents the application of non-linear mathematical models for different technological process modelling. It is possible to draw the following conclusions from the analysis of described example: 1. The classical theory of experimental design shows that non-linear mathematical models in the form of higher-order polynomials ensure a higher accuracy than the linear ones. These models, especially models including factors’ interactions, make intricate the analysis and interpretation of modeling results. However, non-linear mathematical models in the form of complex power functions with interactions do not guarantee higher accuracy than the same models without interactions. 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