13240 FACTA UNIVERSITATIS Series: Mechanical Engineering https://doi.org/10.22190/FUME241129010S © 2025 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper A NOVEL APPROACH TO PREDICTING THE CUTTING FORCE IN TURNING USING DIMENSIONAL ANALYSIS Jelena Stanojković1, Miloš Madić2, Milan Trifunović2, Predrag Janković2, Dušan Petković2 1University of Priština, Faculty of Technical Sciences, Serbia 2University of Niš, Faculty of Mechanical Engineering in Niš, Serbia Abstract. Cutting forces are a critical indicator of the machining process, and their modeling is important for a variety of reasons, including tool life assessment, chatter prediction, tool condition monitoring, assessment of machining strategies and machining process optimization and control. This paper presents a new principle of modeling the main cutting force using dimensional analysis (DA). Dry longitudinal single-pass turning of two different steels (20MnCrS5 and S235JRG2) with two different cutting inserts, was considered. Taguchi's 33×21 design with 6 trials was applied to arrange seven parameters: depth of cut, feed rate, cutting speed, feed velocity, rake angle, cutting edge angle, and workpiece material parameter, i.e., tensile strength. The obtained results, including additional validation tests, showed a very good prediction capacity of the DA- based model in estimating the cutting force during the turning process. The analysis includes the influence of parameters on the cutting force as well as examination of 3D surface diagrams and correlation coefficients. The chip slenderness ratio proved to be the most important dimensionless group for cutting force prediction. By performing additional experimental trials, the correction coefficient for the tool nose radius was estimated and extended models were developed. The well-known Victor-Kienzle model can be used to predict the cutting force if the exact values of mc and kc1.1 coefficients. The proposed DA-based models proved to be valid for predicting all three cutting force components with high accuracy. Key words: Turning, Cutting force, Dimensional Analysis, Modeling, Taguchi’s Design Received: November 29, 2024 / Accepted March 01, 2025 Corresponding author: Jelena Stanojković Faculty of Technical Sciences, University of Priština in Kosovska Mitrovica, Knjaza Miloša 7, 38220 Kosovska Mitrovica, Serbia E-mail: jelena.stanojkovic@pr.ac.rs 2 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ 1. INTRODUCTION The turning process is one of the most commonly used in the metal processing industry's production operations [1]. As one of the oldest production technologies, it still plays an important role in modern industry due to favorable cutting mechanics for a wide range of materials, and can ensure high productivity at acceptable costs [2]. Nonetheless, as well as other traditional and non-traditional technologies [3-5], turning is a multi-input process characterized by the existence of multiple machinability and output performances. In addition, these performances for a particular application can be of different significance, and as a rule, are mutually opposed [5]. Cutting forces are one of the most important machinability indicators for understanding the turning process. The cutting forces generated during the turning process have a direct impact on heat generation, which in turn affects tool life and machining accuracy. They also correlate with other factors, such as the quality of the machined surface, self- excitation, and forced vibrations [6-8]. Knowing the cutting forces for different cutting parameters helps the designer-manufacturer increase the machine tool's efficiency. In addition, it enables the optimization of processing parameters, which can lead to improved productivity, reduced production costs, and increased process efficiency [9-11]. In order to achieve a complete match between the predicted and real responses, it is necessary to capture all aspects of the interactions of the cutting tool, workpiece, machine tool, and environment. However, in practice, this is often difficult because the actual states of some aspects and physical phenomena are unknown or partially known in a given process or even certain parameter space. These aspects also include environmental influences, homogeneity of the workpiece material chemical composition and mechanical properties, variable lubrication and chip removal conditions, random formation of deposits and their removal, etc. [12]. Therefore, general modeling approaches are based on simplifying the real process. Various influencing factors, including processing parameters, workpiece material properties, the kinematics and dynamics of the cutting process, cutting tool geometry, and lubrication conditions, have led to the establishment of numerous cutting force prediction models today. Various methods exist for modeling the cutting force, typically categorized as analytical, empirical, and numerical methods [13]. In practice, empirical models, with the Victor-Kienzle model being one of the most frequently applied approaches, are used in determining cutting forces [14]. Empirical models are based on experimental determination of the model coefficients, whose form has been usually determined in advance, and typically relate cutting force as the dependent variable to different independent variables such as cutting speed, depth of cut, feed rate, material type, cutting tool geometry, etc. These empirical models are often simple to implement and provide acceptable results in a wide range of industrial applications. Different researchers have previously used different approaches to predict the cutting force during the turning process, Table 1. A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 3 Table 1 Review prediction of the cutting force during the turning process Ref. Tool Material workpiece Process parameters Experimental design (trials) Model type [15] coated carbide AISI 4340 steel vc, f, ap Taguchi (27) linear [16] ceramic with reference AISI D3 steel vc, f, ap, CT Taguchi (18) quasi-linear ANN [17] coated carbide 36CrNiMo4 alloy steel f, ap, rε Taguchi (9) linear [18] cubic boron nitride AISI H11 steel vc, f, ap, H Box-Behnken (29) quadratic nonlinear model with interactions [19] mixed ceramic AISI 4140 steel vc,, f, ap, rε Box-Behnken (29) quadratic nonlinear ANN [20] polycrystalline cubic boron nitride AISI D2 steel vc, f, ap, H, rε Taguchi (27) quasi-linear power [21] cubic boron nitride 7020 AISI 52100 steel vc, f, ap, H FFD (25) ANN [22] tungsten- coated carbide AISI 4340 alloy steel vc, f, ap, rε FFD (60) GPR SVM ANN [23] multi-layer coated carbide D3 steel vc, f, ap, rε Taguchi (16) linear ANN [24] carbide with PVD micro grain Inconel 718 vc, f, ap, h Taguchi (27) mechanistic [25] TiAlN coated carbide AISI 420 steel vc, f FD with two central points (11) quadratic FFD (Full Factorial Design), FD (Factorial Design), ANN (Artificial Neural Network), GRP (Gaussian Process Regression), SVM (Support Vector Machines), GP (Genetic Programming), vc (cutting speed) m/min, f (feed rate) mm/rev, ap (depth of cut) mm, CT (cutting tool), H (workpiece hardness) HRC, h (chip thickness) mm. Krupa et al. [26] developed a mathematical model and proposed a method for estimating the feed rate impact of stochasticity on the main cutting force in turning. They performed measurements of the main cutting force during turning on universal lathes, with the 45 steel (ISO P) (carbon content 0.45%) workpiece material, a T15K6 cutting tool without coating, and feed rate values of 0.06, 0.1, 0.15 and 0.2 mm/rev. The mathematical model was developed, and dispersion characteristics (mean value, dispersion, and mean square deviation) of the main cutting force were obtained, depending on the corresponding dispersion characteristics of feed rate. Fodor et al. [27] introduced a stochastic extension of the existing cutting force models. They showed that the cutting force is stochastic with a variance of the measured cutting force of around 4-9% of the average value, depending on the selected parameters. Abdellaoui et al. [28] investigated the effects of tool nose radius in the turning process. They developed a thermomechanical model related to the average 4 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ tool/chip contact temperature, which enabled the prediction of all thermomechanical parameters when cutting with a sharp tool. As a function of the value of the minimum uncut chip thickness, two cutting situations were defined (plowing and cutting). Horvath [29] presented a review of the main research results of recent years regarding the cutting force, where the cutting forces in turning depend not only on the material properties and cutting parameters, but also to a large extent on the geometry of the tool edge that determines the shape of the chip (thickness and width). He derived a new cutting force model for fine turning, which can be used to estimate the main cutting force. It can be observed from the literature review that the researchers used different methods to derive cutting force prediction models. Each method makes the prediction fairly accurate, but it comes with limitations such as needing to do more experimental trials, finding the best form of the model, optimizing hyper-parameters, overfitting, and so on, which makes it not so receptive to developing and applying in real production conditions. Therefore, the primary goal of this study is to propose a new cutting force prediction model, developable with a limited number of experimental trials while taking into account different parameters that influence the turning process such as the workpiece and the cutting tool properties as well as main cutting parameters. The approach is based on the principles of dimensional analysis (DA), and in this study these models were developed for the prediction of cutting forces in dry single-pass medium turning of two steels, 20MnCrS5 and S235JRG2. To this aim, Taguchi’s orthogonal experimental design with six trials was used for the arrangement of three dimensionless groups while considering seven input process parameters. Four of them are related to the machining process (depth of cut, feed rate, feed velocity, and cutting speed), two are related to the geometry of the cutting tool (cutting edge angle and rake angle), and one is the workpiece material parameter (tensile strength). The obtained experimental results of the main cutting force were compared with the predicted values to determine the percentage error. The percentage error of additional validation tests was further estimated while using parameter values outside the initial experimental hyperspace. The analysis of the obtained results included which parameter, i.e., dimensionless group, had the greatest influence on the cutting force, as well as determining the correlation coefficients. The percentage error of the experimental and validation tests of the Victor-Kienzle model of the main cutting force was determined with the coefficients (mc, kc1.1) taken from the literature. An extended model of the main cutting force depending on one more cutting parameter, i.e., tool nose radius, was also proposed. Finally, the mutual dependences of all three components of the cutting force were analyzed with respect to the identified dimensionless group with the greatest influence. 2. DIMENSIONAL ANALYSIS OF THE CUTTING FORCE In physics and engineering, many parameters and measurements take the form of numerical quantities with corresponding dimensional units [30]. DA is a powerful tool that aids in the design, ordering, performance, analysis, and synthesis of the resulting data [31]. In general, it is a mathematical tool used to simplify a problem by reducing the number of variables to the smallest number of essential parameters and a technique that enables the conversion of physical quantities from one unit of measurement to another, and the conversion of the basic physical quantities of the SI system of measurements with their dimensions (length-L, time-T, mass-M, thermodynamic temperature-θ, electric current-I, A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 5 luminous intensity-J, amount of substance-N) [32]. Parameters are described by dimensionless groups (π groups) that connect independent variables with dependent variables. The primary benefit of this method is its significant reduction in the number of experimental trials needed to determine the relationship between variables, thereby reducing both costs and development time. This study uses the Buckingham π-theorem as the main tool for the reduction of variables. The Buckingham π-theorem, as a key concept of DA, helps simplify complex physical problems by reducing the number of variables involved. It is necessary to select the repeating and non-repeating variables for model development [33]. According to the Buckingham π-theorem, “the number of independent dimensionless groups that may be employed to describe a phenomenon known to involve n variables is equal to the number n–m, where m is usually the number of basic dimensions needed to express the variables dimensionally” [33]. It is necessary to fulfill two basic criteria before using DA [34]:  principle of dimensional homogeneity, and  all relevant variables are included in the proposed relationship. The principle of dimensional homogeneity is based on the fact that in order to establish equality, the dimensions and units in each term of an equation must be the same. The equation is dimensionally homogeneous if this condition is satisfied DA [33]. In the present study, DA was used to derive a model for predicting the cutting force in turning, initially for predicting the main cutting force (Fc), Figure 1. Fig. 1 Cutting force components in turning The dimensions of the cutting force and the parameters that are defined based on the basic dimensions, i.e. length (L), time (T), mass (M), which affect the main cutting force, are as follows:  Main cutting force (Fc): MLT-2  Tensile strength (Rm): ML-1T-2 6 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ  Feed rate (f): L  Cutting speed (v): LT-1  Feed velocity (vf): LT-1  Depth of cut (ap): L  Rake angle (γ0): –  Cutting edge angle (κ): – Rake angle and cutting edge angle have no dimensions. The dependent parameter is the main cutting force, and the independent parameters are tensile strength, depth of cut, cutting speed, feed rate, feed velocity, rake angle and cutting edge angle. Feed velocity, feed rate, and rake angle were selected as repeating parameters. By applying Buckingham π–theorem, the π1 group can be represented by the following form: dcb f a mc fvRF 01   (1) By changing the dimensions, equation 1 has the following form:        c ba LLTTMLMLT   1212 1 (2) Only those values with the same dimension are added. Addition by MLT is shown in the following forms: 01:  aM (3) 01:  cbaL (4) 022:  baT (5) By solving the system of three equations with three variables, the values of the unknowns are obtained: a=–1, b=0 and c=–2. Based on the obtained values for each dimension in the initial π1 group, equation (1), the is replaced with the obtained values and the value of the π1 group can be represented by the following form: 2 21 1 fR F fRF m c mc    (6) In this case, there are four dimensionless groups, π1, π2, π3, π4, other π groups are obtained in the same way and presented in Table 2. Table 2 Four dimensionless (π) groups π1 π2 π3 π4 2fR F m c  fv v f a p 0  If one sets: π1=f(π2, π3, π4), the main cutting force as the function of dimensionless groups can be represented by the following model: A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 7 432 0 2 1 xx p x f mc f a v v fRxF                             (7) 3. EXPERIMENTAL SETUP The experiments of dry longitudinal single-pass turning were performed on a “POTISJE” PA–C 30 universal lathe, with the power of 11 kW, spindle speed range of 20÷2000 rpm, and longitudinal feed rate range of 0.04÷9.136 mm/rev, in laboratory conditions. The cutting tool was Sandvik Coromant PCLNR 3225P 12 tool holder (cutting edge angle of κ=95°, rake angle of γoh=–6°) with cutting inserts CNMG 120408E-SF (γoi=14.5°, tool nose radius of rε=0.8mm) and CNMG 120408E-NF (γoi=25°, tool nose radius of rε=0.8mm), manufactured by Dormer Pramet, of T8430 grade (PVD coated fine grained cemented carbide, suitable for higher feed rates and medium to lower cutting speeds). The workpiece materials used in this experiment were 20MnCrS5 and S235JRG2 steel. 20MnCrS5 is an alloy steel with excellent mechanical properties, which makes it suitable for a wide range of applications in various industries. Its strength, toughness, and wear resistance ensure the durability and reliability of components made of this material. It is most widely used in the automotive and mechanical industries (for the production of machine parts that suffer heavy loads and wear, including spindles, gears, and bearings). According to Dormer Pramet, it belongs to the workpiece material group P3.2 (alloy steels (carbon steels with an alloying content ≤ 10%) with a hardness of 180-260 HB). S235JRG2 is a carbon construction steel that has a specific application thanks to its mechanical properties and chemical composition. It is most widely used in the construction industry, infrastructure, mechanical engineering, etc. According to Dormer Pramet, it belongs to the workpiece material group P2.1 (plain carbon steels (steels comprised of mainly iron and carbon) containing < 0.25% C with a hardness of < 180 HB). Chemical composition of the workpiece materials 20MnCrS5 and S235JRG2 are shown in Table 3. The sample of cylindrical rods used in these experiments had different diameters (55 and 59 mm) and the same length of 250 mm. Table 3 Chemical composition of the workpiece materials Chemical composition [%] 20MnCrS5 C Mn Si P S Cr Ni Mo Al 0.018 1.11 0.24 0.016 0.026 1.07 0.08 0.018 0.026 Cu As V Ti B N Cev. Ca Sn 0.19 0.007 0.007 0.024 0.0003 0.007 0.602 - - S235JRG2 C Mn Si P S Cr Ni Mo Al 0.16 0.46 0.176 0.013 0.005 0.08 0.04 0.01 0.012 Cu As V Ti B N Cev. Ca Sn 0.08 0.03 <0.001 <0.001 0.0005 0.008 0.25 0.002 0.05 The HRB method was used to determine the hardness of workpiece materials. The hardness was measured on a Rockwell device scale B with standard indenter and test force. Tensile strength (Rm) was indirectly determined by using Conversion table A.1– 8 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ Standard ISO 18265:2003 Metallic materials–conversion of hardness values. For 20MnCrS5 hardness is 156 N/mm2 and tensile strength is 525 N/mm2, while for S235JRG2 hardness is 163 N/mm2 and tensile strength is 540 N/mm2. Components of cutting force were measured using a three-component piezoelectrical force transducer, basic unit KISTLER Type 9265A and a tool holder KISTLER Type 9441, which was mounted on the lathe via a custom tool holder adapter creating a very rigid tooling fixture. The load signal is generated by the dynamometer and a Kistler type 5007 charge amplifier. Through the HBM QuantumX MX840 universal measuring amplifier- measuring data were acquired, visualized, and analyzed using the “Catman” data acquisition software. The experimental setup included the machine tool, tool holder, cutting inserts, three- component dynamometer, charge amplifier, data acquisition unit, computer, software, input parameters, workpieces, and hardness measurement device. The experimental setup is shown in Fig. 2. Fig. 2 Schematic representation of the experimental setup The experiment was planned according to the form of the proposed main cutting force prediction model. In the study, seven parameters were considered, four related to the machining process (depth of cut, feed rate, feed velocity, and cutting speed), two related to the geometry of the cutting tool (cutting edge angle and rake angle), and one material parameter (tensile strength). Moreover, tool nose radius was also considered in additional experimental trials so as to determine the correction coefficient for the proposed cutting force prediction model. Parameter ranges and levels were selected based on the tool manufacturer's recommended cutting conditions for the inserts. The experimental design used was the Taguchi orthogonal design (L6) 33 ×21 [35]. Parameters with their names, symbols, units, A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 9 and values are shown in Table 4. The parameters were varied in accordance with the proposed main cutting force prediction model, i.e., defined by dimensionless π groups. Cutting speed and cutting edge angle for both experimental designs were taken as constant values because their influence on the main force is very small, almost negligible, as discussed by Stephenson and Agapiou [36], for most materials, except soft metals and temperature sensitive materials. Table 4 Parameters with their names, symbols, units and values Parameter Symbol Unit 20MnCrS5 S235JRG2 Values Cutting speed v m/min 157 245 Feed rate f mm/rev 0.196 0.249 0.321 Depth of cut ap mm 1.2 1.8 1.9 2.0 2.9 Rake angle γo ° 8.5 19 Cutting edge angle κ ° 95 Tool nose radius rε mm 0.8 1.2 4. RESULTS AND DISCUSSION 4.1 DA Model of the Main Cutting Force The main cutting force values obtained experimentally in laboratory conditions are shown in Tables 5 and 6 for 20MnCrS5 steel and S235JRG2 steel, respectively. Based on the experimental data of the main cutting force and by minimizing the sum of squared errors using Matlab, the unknown coefficients of the proposed main cutting force prediction models were determined. Main cutting force prediction models in dry longitudinal single–pass turning of two workpiece materials, 20MnCrS5 and S235JRG2 steels with a tool nose radius of 0.8 mm, were obtained in the following form: (20MnCrS5) 01669.0 0 89362.009816.0 25254865.2                             f a v v fF p f c (8) (S235JRG2) 05254.0 0 04605.110746.0 25404551.1                              f a v v fF p f c (9) The predicted main cutting force values were compared with the main cutting force values obtained experimentally in order to determine the mean absolute percentage error (MAPE). The MAPE value of the DA model for steel 20MnCrS5 is about 0.38%, while for steel S235JRG2 is about 0.65%. Based on these results, it can be said that the developed main cutting force prediction models obtained by applying DA for both workpiece materials give quite good results from an engineering point of view. 10 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ Table 5 Experimental design 33×21 of the main cutting force of 20MnCrS5 steel No. A B C v/vf π2 ap/f π3 κ/γo π4 Fc [N] experimentally Fc [N] DA model 1 0 0 -1 693.93 7.63 5.00 969 971.2 2 0 0 1 693.93 7.63 11.18 977 984.7 3 1 -1 -1 881.57 6.12 5.00 508 506.1 4 1 1 1 881.57 9.18 11.18 742 736.8 5 -1 1 -1 538.28 9.03 5.00 1831 1830.4 6 -1 -1 1 538.28 6.23 11.18 1334 1331.4 Table 6 Experimental design 33×21 of the main cutting force of S235JRG2 steel No. A B C v/vf π2 ap/f π3 κ/γo π4 Fc [N] experimentally Fc [N] DA model 1 0 0 -1 744.39 7.63 5.00 895 904.1 2 0 0 1 744.39 7.63 11.18 939 943.1 3 1 -1 -1 945.68 6.12 5.00 465 456.5 4 1 1 1 945.68 9.18 11.18 731 727.8 5 -1 1 -1 577.43 9.03 5.00 1747 1744.5 6 -1 -1 1 577.43 6.23 11.18 1235 1233.7 With both steel materials, the influence of parameters A, B and C, i.e., the influence of dimensionless groups, π2, π3 and π4, is quite the same for the main cutting force. Therefore, with an increase in the value of dimensionless groups, π2, π3 and π4, the main cutting force increases, where the influence of ap/f is dominant and the most pronounced, followed by groups π4 and π2, for 20MnCrS5 steel, that is, groups π2 and π4, for S235JRG2 steel. Since the exponents of dimensionless groups are different, it is possible to see an increase in the main cutting force with an increase in the value of dimensionless group π4, i.e., a decrease in rake angle, more pronounced when turning S235JRG2 steel compared to 20MnCrS5 steel. The same applies to dimensionless group π3. Namely, the increase in the ap/f ratio (chip slenderness ratio) rapidly increases the main cutting force when turning S235JRG2 steel compared to 20MnCrS5 steel. Both investigated materials have approximately the same tensile strength (Rm), but the calculated coefficients of the DA model (C, x1, x2, x3) are different, which indicates different machinability and the fact that one DA model cannot be used for accurate prediction of cutting force. If only one model was used, for example, the DA model of S235JRG2 steel for 20MnCrS5 steel, the MAPE for all experimental tests would be around 10%, and in the opposite case even 20%. The intercept values of derived models are significantly different (2.48 and 1.45) indicating that 20MnCrS5 steel is more difficult to machine compared to S235JRG2 steel. Parameter A (dimensionless group π2), i.e., the ratio of cutting speed and feed velocity, depends only on feed velocity because cutting speed was kept constant for given workpiece material in the experimentation. Feed velocity depends on feed rate and this ratio represents only the change in the feed rate value. This parameter has a positive correlation with the main cutting force, i.e., increasing parameter A increases the main cutting force. Too much increase can lead to machine tool overload and increased tool wear. In other words, a A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 11 decrease in feed rate results in a decrease in the main cutting force, since the cutting tool removes smaller layer of material. This implies that a smaller volume of material being removed requires less cutting force. Furthermore, removal of smaller volume of material results in reduction of the heat generated during cutting. The reduced temperature helps to maintain the mechanical properties of the cutting tool and material, which further reduces the main cutting force [37]. Parameter B (dimensionless group π3), i.e., the ratio of depth of cut to feed rate, chip slenderness ratio, has a positive correlation with the main cutting force - by increasing the chip slenderness ratio the main cutting force increases sharply [37-39]. Increasing the cross-sectional area of the uncut chip and the volume of the deformed material, either by increasing depth of cut or feed rate, results in an increase in the main cutting force. Denkena et al. [40] have concluded the same about the influence of the undeformed cross-section of the chip and the volume of the deformed material on the main cutting force for alloy and carbon steel. Parameter C (dimensionless group π4), i.e., the ratio of the cutting edge angle and rake angle, also has a positive correlation with the main cutting force, but the effect is less pronounced. Thus, an increase in this parameter, which is accomplished by reducing rake angle, slightly increases the main cutting force. Increasing rake angle increases shear angle and results in a decrease in chip thickness, as well as a decrease in the main cutting force and therefore a decrease in cutting power [41]. Although large rake angles decrease the main cutting force, they simultaneously reduce the strength of the tool tip which ultimately may lead to its fracture. The proposed main cutting force prediction models are visualized in Fig. 3. This figure shows 3D surface plots of the main cutting force (Fc) for both workpiece materials, 20MnCrS5 steel and S235JRG2 steel, depending on different dimensionless groups. The diagrams clearly indicate that there are no interactions between dimensionless groups, i.e., there is no qualitative change in the influence of a certain dimensionless group in relation to another dimensionless groups. The absence of significant interactions of processing parameters (depth of cut, feed rate, cutting speed, feed velocity, rake angle, cutting edge angle) on the main cutting force was also previously reported [39]. The absence of significant interactions indicates the possibility of modeling input-output dependences with simpler models, which was done in this study by creating degree models using DA. Adequate experimental validation of the proposed main cutting force prediction models is a necessary step to ensure that the models generalize well with new data, which is key to their application. Therefore, based on the perceived significance of dimensionless groups the additional cutting regimes were tested, Tables 6 and 7. 12 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ Fig. 3 Surface plot (3D plot) of the main cutting force (Fc) dependance on different π groups for 20MnCrS5 steel and S235JRG2 steel A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 13 The validation results for both materials show that predicted values are close to the experimentally obtained values. As can be seen from Tables 7 and 8, trials 3 and 4 are within the included experimental hyperspace, where the cutting force values obtained by the DA models are the closest to the values obtained experimentally. Table 7 Main cutting force values for different validation cutting regimes (20MnCrS5) No. v/vf π2 ap/f π3 κ/γo π4 Fc [N] experimentally Fc [N] DA model 1 538.28 3.74 5.00 739 832.4 2 484.00 3.36 11.18 810 939.3 3 807.42 7.48 5.00 715 714.9 4 606.27 7.72 11.18 1248 1285.8 5 970.72 14.61 5.00 973 915.9 6 807.42 11.21 11.18 1069 1040.9 Table 8 Main cutting force values for different validation cutting regimes (235JRG2) No. v/vf π2 ap/f π3 κ/γo π4 Fc [N] experimentally Fc [N] DA model 1 577.43 3.74 5.00 735 693.1 2 519.20 3.36 11.18 794 791.1 3 866.14 7.48 5.00 658 664.4 4 650.36 7.72 11.18 1184 1232.5 5 1041.31 14.61 5.00 890 944.7 6 866.14 11.21 11.18 1005 1059.3 The MAPE for 20MnCrS5 steel is about 1.5% and for S235JRG2 steel is about 2.5%. The other cutting regimes are outside of the experimental hyperspace, in terms of the chosen parameter values. Taking into account the obtained results, the DA models give fairly satisfactory extrapolation results. For all validation tests, the MAPE for 20MnCrS5 steel is about 6.6%, which is borderline satisfactory from an engineering point of view. Results for S235JRG2 steel are much more favorable and amount to a MAPE value of about 3.78%. Based on the obtained results, it can be concluded that the DA models provide the means to develop fairly accurate models for predicting the main cutting force during the turning process, using a small number of experimental trials, which result in the reduction of the cost of the experiment, savings in materials and energy, as well as reduced time Prediction performances of the proposed DA models in the form of regression are given in Fig. 4. Values from the main experiment and additional validation trials were taken into consideration. The correlation coefficient for both workpiece materials is around 0.99, confirming the considerable accuracy of the developed DA models. 14 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ Fig. 4 Correlations between the predicted (DA model) and the experimentally measured values of the main cutting force using the entire data set The proposed DA models were further verified using residual analysis. Figure 5 shows the residuals on the vertical axis and the values of the main cutting force for both steel materials (20MnCrS5 and S235JRG2) on the horizontal axis by considering experimental data from the main experiment and additional validation trials. The residuals do not follow any recognizable pattern and are scattered above and below zero. Fig. 5 Residual plot of the main cutting force The models of the main cutting force for both steel materials yield good predictions in a wide range of chip slenderness ratio values, where significant changes in chip morphology occur. A higher chip slenderness ratio results in longer and thinner chips [42- 44]. Figure 6 shows that the 20MnCrS5 steel chips are somewhat different, and more unfavorable as their form also depends on the mechanical properties such as hardness and microstructure of the material [45]. Materials with higher hardness have a lower ability to deform, so the chip breaks and is easily removed from the cutting zone, as is the case with 235JRG2 steel. A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 15 Fig. 6 Chip morphology as a function of chip slenderness ratio Based on the initial experimental and validation tests, the main cutting force of 235JRG2 steel remains about 6% higher compared to 20MnCrS5 steel. However, when the values of ap/f are less than 4, the main cutting forces are almost equal, which is not the case with the Victor-Kienzle model [46]. 4.2 Victor-Kienzle Model of the Main Cutting Force The well-known Victor-Kienzle model [15] is still widely used for the calculation of the main cutting force, which has the following form [46-48]: cm cc hkAF   1.11 (10) where A1 (mm2) is the cross-sectional area of the uncut chip, kc1.1 (N/mm2) is the specific cutting force for 1 mm2 cross-sectional area of the cut, h (mm) is the uncut chip thickness, and mc is the exponent of the specific cutting force for the given workpiece material. Values of mc and kc1.1 can be found in the catalogs of cutting tool manufacturers and books. For 20MnCrS5 steel values are mc=0.25 and kc1.1=2140 N/mm2 [49]. The MAPE value of the main cutting force is about 60%, while for the experimental trials and for the validation test, it is about 65%. Steel S235JRG2 belongs to the universal workpiece material group, unalloyed and low-alloyed steels, C > 0.55%, not tempered, according to cutting tool manufacturer Walter (mc=0.25, kc1.1=1700 N/mm2). The same values can be found in the Sandvik Coromant catalog [10]. For S235JRG2 steel values are mc=0.17 and kc1.1=1780 N/mm2 [49]. The MAPE value of the main cutting force for S235JRG2 steel is about 35%, while for the experimental trials and for the validation test, it is about 38%. The exact values of coefficients mc and kc1.1 cannot be easily found because some materials may be thermally treated, while some specific materials have properties that are between the different workpiece materials subgroups offered in catalogs of cutting tool manufacturers or available databases. 16 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ However, the aforementioned Victor-Kienzle model gives approximate values of the main cutting force, which may be in poor agreement with the cutting force values obtained experimentally, while the main advantage is fast calculation. 4.3 Extended DA Model of the Main Cutting Force and Correction Coefficient for Tool Nose Radius The cutting forces in the turning process are dependent not only on the properties of the workpiece material and cutting parameters, but also on the geometry of the cutting tool edge, which affects the chip shape (thickness and width). Tool nose radius is an important parameter that affects the cutting force. Parida et al. [50] analyzed the influence of tool nose radius on the main cutting force. Their study revealed that as tool nose radius increases, the main cutting force also increases due to the increased contact area. Increasing tool nose radius does not only increase the main cutting force, but also affects the components of the cutting force, i.e., causing an increase in the radial and a decrease in the axial force [51-53]. The extended main cutting force prediction models with a correction coefficient are given by Eqs. (11) and (12): (20MnCrS5) r p f c C f a v v fF                            01669.0 0 89362.009816.0 25254865.2   (11) (S235JRG2) r p f c C f a v v fF                            05254.0 0 04605.110746.0 25404551.1   (12) where the correction coefficient was Cr = 1 in the case of using the tool nose radius of 0.8 mm or Cr = 1.08 in the case of using the tool nose radius of 1.2 mm. In order to determine the correction coefficient (Cr), additional experimental trials were conducted for both steel materials using a cutting insert with a larger tool nose radius of 1.2 mm (Tables 9 and 10). Table 9 Experimental trials for extended DA model of 20MnCrS5 steel No. v/vf π2 ap/f π3 κ/γo π4 Fc [N] experimentally Fc [N] extended DA model 1 881.57 6.12 5.00 555 546.6 2 538.28 9.03 5.00 1972 1977.6 3 538.28 6.23 11.18 1300 1331.5 Table 10 Experimental trials for extended DA model of S235JRG2 steel No. v/vf π2 ap/f π3 κ/γo π4 Fc [N] experimentally Fc [N] extended DA model 1 945.68 6.12 5.00 505 493 2 577.43 9.03 5.00 1882 1884.1 3 577.43 6.23 11.18 1226 1332.4 A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 17 Increasing tool nose radius increased the main cutting force and its effect was consistent, given the same Cr value was obtained for both materials. 4.4 Analysis of Cutting Force Components Based on the analysis of experimental results it was observed that for all three cutting force components, main cutting force (Fc), passive force, i.e., radial force (Fp), and feed force, i.e., axial force (Ff), the most important parameter was dimensionless group π3, i.e., chip slenderness ratio [43]. The cutting force components values obtained experimentally in laboratory conditions are shown in Tables 11 and 12 for 20MnCrS5 steel and S235JRG2 steel, respectively. The effect of the chip slenderness ratio on cutting force components is given in Fig. 7. For both workpiece materials, by increasing chip slenderness ratio, the main cutting force and feed force increase drastically, while the radial force decreases. Table 11 Experimental design 33×21 of the cutting force components of 20MnCrS5 steel No. v/vf π2 ap/f π3 κ/γo π4 Fc [N] Fp [N] Ff [N] experimentally 1 693.93 7.63 5.00 969 172 478 2 693.93 7.63 11.18 977 197 497 3 881.57 6.12 5.00 508 144 247 4 881.57 9.18 11.18 742 155 404 5 6 538.28 538.28 9.03 6.23 5.00 11.18 1831 1334 178 282 903 634 Table 12 Experimental design 33×21 of the cutting force components of S235JRG2 steel No. v/vf π2 ap/f π3 κ/γo π4 Fc [N] Fp [N] Ff [N] experimentally 1 744.39 7.63 5.00 895 139 330 2 744.39 7.63 11.18 939 163 377 3 945.68 6.12 5.00 465 118 182 4 945.68 9.18 11.18 731 139 326 5 6 577.43 577.43 9.03 6.23 5.00 11.18 1747 1235 181 198 712 451 Analysis of the ratio of cutting force components provides useful insights into the efficiency and stability of the machining process. The ratio of the cutting force components Fc/Ff indicates the efficiency of material removal in relation to the side load of the cutting tool. A high value of this ratio is desirable because it indicates an efficient process with reduced side load, which can improve the quality of the machined surface and extend tool life. For both materials, it ranges from 1.8 to 2.7, which indicates more efficient cutting with less feed force. The ratio of the cutting force components Fc/Fp varies between 3.1 and 10.2, indicating that most of the energy is directed to the removal of material, potentially reducing wear and vibration. The ratio of the cutting force components Ff/Fp varies around 2, which indicates a stable process with reduced lateral load and vibrations, which can improve surface quality and extend tool life. 18 J. STANOJKOVIĆ, M. MADIĆ, M. TRIFUNOVIĆ, P. JANKOVIĆ, D. PETKOVIĆ Fig. 7 The influence of the chip slenderness ratio (dimensionless group π3) on the cutting force components The ratios of the cutting force components for both steel materials (20MnCrS5 and S235JRG2) are analogous. The ratio of the cutting force components Fc/Ff decreases, while the ratios Fc/Fp and Ff/Fp increase with an increase in the chip slenderness ratio. It is not possible to predict all cutting force components using only one extended DA model, considering that the ratios of the cutting force components change. 5. CONCLUSION This study proposed a novel approach to predicting the cutting force in turning using dimensional analysis. Experimental investigation in dry longitudinal single-pass turning of two steel materials, 20MnCrS5 steel and S235JRG2 steel, was considered using Taguchi's 33×21 design with additional validation trials. Based on the conducted analyses, experimental and modeling results, the following conclusions may be drawn:  The proposed models derived using DA and limited experimental data proved to be able to accurately predict the main cutting force.  Additional validation tests with parameter values outside the initial experimental hyper-space showed very good extrapolation capabilities of the proposed models for predicting the main cutting force.  Chip slenderness ratio was the most important parameter affecting the main cutting force, followed by the ratio of cutting speed and feed velocity, while the effect of the ratio of the cutting edge angle and rake angle was the least pronounced.  The well-known Victor-Kienzle model can be used to predict the cutting force if the exact values of mc and kc1.1 coefficients are known for the particular material used in the cutting process.  The derived extended main cutting force prediction models showed a consistent positive correlation between tool nose radius and the main cutting force.  The proposed models proved to be valid for predicting all three cutting force components with high accuracy. It was found that by increasing the chip slenderness ratio, the main cutting force and feed force increased drastically, while the radial force decreased. In this regard, the ratios Fc/Fp and Ff/Fp are subject to change, A Novel Aproach for Predicting the Cutting Force in Turning Using Dimensional Analysis 19 considering the selected values of ap and f (or chip slenderness ratio), while the ratio Fc/Ff is approximately constant. It can be concluded that DA-based models have a very good potential for reliable and accurate estimation of the cutting force in turning. The discussed approach can take into account a large number of cutting parameters using a smaller number of experimental trials, which plays a significant role not only regarding prediction performance but also in great economy, reduced experimentation time, and saving materials and energy for conducting experiments. The limitation of the proposed DA-based models is that it would be necessary to extend these models in case of significant parameter interactions. In addition, the existence of multiple dimensionless groups with a number of parameters makes it difficult to define levels of variation and selection of appropriate experimental designs. In order to derive the greatest benefit from the calculations and experimental methods used in the present work, it is also necessary to open the door to new possibilities of DA modeling, i.e., the application of the DA model for determining the model of surface roughness, cutting temperature, etc. Future research will probably be in the direction of determining the correction coefficients of the cutting parameters and predicting other performances based on the cutting force assessment (power consumption, cutting temperature, tool wear, friction, etc.). 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