Application of Central Composite and Orthogonal Array Designs for Predicting the Cutting Force

Transcription

1 Application of Central Composite and Orthogonal Array Design... Application of Central Composite and Orthogonal Array Designs for Predicting the Cutting orce Srinivasa Rao G.* 1 and Neelakanteswara Rao A. 2 1 Mechanical Engineering Department, RVR&JC College of Engineering, Guntur , A.P., India, gsrao_rvr@rediffmail.com 2 Mechanical Engineering Department, National Institute of Technology, Warangal , A.P.,India, neelu@nitw.ernet.in Abstract: Development of empirical models for cutting forces in turning operation has been a major activity in metal cutting research. This paper presents an application of central composite face centered (CC) and L 18 ( ) orthogonal array designs for developing cutting force model in case of turning AISI 1015 steel with HSS tool. Three process parameters namely cutting speed, feed, depth of cut and three tool parameters namely side cutting edge angle, inclination angle, normal rake angle were considered in developing the cutting force model. Each parameter was set at three levels. Based on the experimental data, a mathematical model in terms of process and tool parameters was developed for main cutting force using multiple linear regression. Confirmation tests were performed to verify the predictability of the developed model. Keywords: Turning, cutting force, central composite, orthogonal array, multiple regression. 1. INTRODUCTION In a machining process, turning operation plays an important role in reducing a particular work piece from the original stock to the desired shape and size. In order to achieve economic objective of *Corresponding Author: gsrao_rvr@rediffmail.com Journal of Metallurgical Engineering, 1(1-2) January-December

2 Srinivasa Rao G. and Neelakanteswara Rao A. this process, optimal cutting conditions have to be determined. Although one can determine the desirable cutting conditions based on experience or hand book data, it does not ensure that the conditions obtained will be optimal or near optimal for that particular work-tool combination. Thus, in order to determine the optimal cutting conditions, reliable mathematical models need to be established. To ensure the effectiveness of the models, the design of experimental techniques should be used to plan the machining experiments efficiently and multiple regression methods can then be used for the particular work-tool combination based on the machining data collected on a specific machine. orce modeling in turning is important for a multitude of purposes including tool life estimation, chatter prediction, tool wear monitoring, thermal analysis, etc. Empirical approach has been a popular approach in developing cutting force model. Cutting force in turning has been found to be influenced in varying amounts by a number of factors such as speed, feed, depth of cut, work material characteristics, tool geometry, use of cutting fluids, etc [1, 2]. In the early phases of empirical work, researchers used one-factor-at-atime strategy, i.e., varying one factor while keeping all other factors constant. This strategy cannot provide generalized conclusions about factor effects. urther, interactions between factors cannot be studied. To overcome these limitations, researchers shifted their strategy to full factorial designs and orthogonal array designs. ull factorial designs provide more complete information, but require large number of experiments. Orthogonal array designs, which are in some way fractional factorial designs, require less number of experiments, but provide lesser information compared to full factorial designs. Experiments based on 3 3 full factorial designs were conducted [3] for prediction of tool life, force and power in terms of speed, feed and depth of cut. Optimization of the cutting speed, feed rate, and depth of cut with considerations of multiple performance characteristics including tool life, cutting force and surface roughness was done [4] using L 9 orthogonal array (OA) design experiments. 18 Journal of Metallurgical Engineering, 1(1-2) January-December 2011

3 Application of Central Composite and Orthogonal Array Design... The effects of shape of cutting edge, work piece hardness, feed rate and cutting speed on surface roughness and resultant forces were experimentally investigated [5] by using a four-factor two-level factorial design. An experimental investigation was conducted to determine the effects of cutting speed, feed, effective rake angle and nose radius on the surface roughness in the finish hard turning of the bearing steel based on a 3 4 full factorial deign [6]. An L 9 orthogonal array has been used [7] to determine the optimum levels for the parameters insert radius, feed rate, and depth of cut on surface roughness while turning AISI 1030 steel bars. An abductive network technique was adopted [8] to construct a prediction model for surface roughness and cutting force and once the process parameters (cutting speed, feed rate and depth of cut) are given, the surface roughness and cutting force can be predicted by this network. Artificial neural network approach has been proposed [9] for modeling cutting forces. By seeing the literature, it can be observed that the study of factorial effects and the empirical model building was performed based on OA or factorial designs, particularly in case of force modeling. urther, tool related factors are excluded from most of the studies. Central composite designs, which are a combination of 2-level factorials and one-factor-at-a-time experiments, are known to be best fit for fitting a second-order model [10]. In recent past, researchers have been using central composite designs for developing surface roughness models [11-13]. One of the key issues in empirical based research is selection of experimental designs. It has been observed that researchers adopted different experimental designs for similar works. It might be of curiosity to verify the performance of orthogonal array design and Central composite designs with respect to cutting force. In this work, an attempt has been made to study the effectiveness of the two designs for the same work-tool combination. urther, in the present work, both process parameters, namely, cutting speed, feed and depth of cut, and tool parameters, namely, side cutting edge angle, inclination angle and normal rake Journal of Metallurgical Engineering, 1(1-2) January-December

4 Srinivasa Rao G. and Neelakanteswara Rao A. angle are considered for the development of the cutting force model while turning AISI 1015 steel with HSS tool. Experiments were conducted based on central composite face centered design and L 18 orthogonal array design. Separate models were developed using the data obtained from each design. Confirmation experiments were conducted to verify the adequacy of the models. 2. PROCEDURES AND METHODS 2.1 Cutting orce Model The proposed relationship between the cutting force and machining independent variables can be represented by the following: orce = ka m B n C p D q E r s (1) Where is the main cutting force, k, m, n, p, q, r, s are the constants, A (side cutting edge angle), B (inclination angle), C (normal rake angle) are tool parameters, and D (cutting speed), E (feed rate), (depth of cut) are process parameters. To facilitate the determination of constants and parameters, the mathematical model is linearized by performing a logarithmic transformation. The logarithmic transformed mathematical model is given by: ln (orce) = ln k + m ln A + n ln B + p ln C + q ln D + r ln E + s ln Though a multiplicative model like equation (1) implicitly incorporates interaction effects, upon its logarithmic transformation, equation (2) becomes a simple linear form without any interaction terms. The second order model also is useful when the second order effects and the two way interactions amongst the process parameters and tool parameters were significant. The general second order model for six parameters is of the form shown below: Y = c 0 + c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 + c 6 x 6 + c 11 x c 22 x c 33 x c 44 x c 55 x c 66 x c 12 x 1 x 2 + c 13 x 1 x 3 + c 14 x 1 x 4 + c 15 x 1 x 5 + c 16 x 1 x 6 + c 23 x 2 x 3 + c 24 x 2 x 4 + c 25 x 2 x 5 + c 26 x 2 x 6 + c 34 x 3 x 4 + c 35 x 3 x 5 + c 36 x 3 x 6 +c 45 x 4 x 5 + c 46 x 4 x 6 + c 56 x 5 x 6 (3) 20 Journal of Metallurgical Engineering, 1(1-2) January-December 2011 (2)

5 Application of Central Composite and Orthogonal Array Design... Where x 1, x 2, x 3, x 4, x 5, x 6 are the independent variables, c 0, c 1, c 2,.. c 56 are the constants and y is the response. Equation (3) is useful when the second order effects of variables and the two way interactions among the variables are significant. In the present study, the parameters of equations (2) and (3) have been estimated by the multiple linear regression using a SPSS software package. 2.2 Materials and Processes The experiments were carried out on TMX-2030 engine lathe. The HSS tools (Co- 10%, W-9.3%, Cr-4.0%, Mo-3.6%, C-1.26%) with required cutting angles were ground on a tool and cutter grinding machine using the standard procedure [1]. Tool geometry used was as follows: end cutting edge angle 10, normal side clearance angle 8, and normal end clearance angle 8, were fixed and remaining side cutting edge angle, inclination angle and normal rake angles were changed during experimentation. Table 1 The Chemical Composition of AISI 1015 Steel Element C Si Mn P S Cr Ni Mo e % Table 2 Level of Control actors actor actor Level1(-1) Level2 (0) Level3 (+1) symbol A Side cutting edge angle(degrees) B Inclination angle(degrees) C Normal rake angle (degrees) D Cutting speed(m/min) E eed (mm/rev) Depth of cut(mm) Journal of Metallurgical Engineering, 1(1-2) January-December

6 Srinivasa Rao G. and Neelakanteswara Rao A. Table 3 The Experimental Layout: L 18 Orthogonal Array Exp.No. A B C D E orce(kgf) The material used in the tests for controlled machining was AISI 1015 steel. The chemical composition of the AISI 1015 steel with a 137 HB hardness, is given in Table 1. The steel bar stock was 65 mm diameter, 300 mm length and these bars were trued, centered and cleaned by removing a 2 mm depth of cut from the outside surface, prior to the actual machining tests. The main cutting force was measured with a multi component digital force indicator (IEICOS made and model 652). The force signals were amplified by a 3-channel charge amplifier. The range of force measurement is kg. 2.3 Experimental Design Design of experimental techniques was used for execution of the plan of experiments, for six variables at three levels, whereby the levels are the values taken by the factors. The factors to be studied 22 Journal of Metallurgical Engineering, 1(1-2) January-December 2011

7 Application of Central Composite and Orthogonal Array Design... and the level of each factor are given in Table 2. The levels of each factor were selected based on machining data hand book [14]. Experiments were conducted as per central composite face centered (CC) design [10] and orthogonal array based designs [15]. This research assumes that the three-four- and five-factor interactions are negligible, because high order interactions are normally assumed highly impossible in practice. or six factors, the CC design consists of 45 runs, which includes a 26-1 fractional factorial portion (32 experiments), 12 axial points and a central point. or six 3-level factors, the smallest (in terms of number of experiments) orthogonal array design is L 18 ( ), which consists of eight columns, and one 2-level factor and seven 3-level factors can be accommodated. or the present work, the factors side cutting edge angle ( A), inclination angle (B), normal rake angle (C), cutting speed (D), feed rate (E), and depth of cut () were assigned to columns 2, 3, 4, 5, 7 and 8, and columns 1 and 6 were kept empty. The empty columns provide the necessary degrees of freedom for error estimation. The experimental layouts along with the response (main cutting force) obtained are shown in Tables 3 and 4. In the tables, 1 indicates level 1 of the factor, 0 indicates level 2 of the factor and +1 indicates level 3 of the factor. Experiments were conducted in random order to avoid any bias. Table 4 The Experimental Layout : CC (6) Design Exp.No. A B C D E orce(kgf) Table Cont d Journal of Metallurgical Engineering, 1(1-2) January-December

8 Srinivasa Rao G. and Neelakanteswara Rao A. Table 4 Cont d DATA ANALYSIS AND DISCUSSION O RESULTS The plan of tests was developed aiming at determining the relation between the process parameters and tool parameters with the cutting force. The analysis of experiments was made into two phases. The 24 Journal of Metallurgical Engineering, 1(1-2) January-December 2011

9 Application of Central Composite and Orthogonal Array Design... first one concerned the analysis of the effects of factors and of the Interactions. Models for cutting force in terms of process parameters and tool parameters were developed in second phase. inally, the comparison between the models has been made. 3.1 Analysis of the actors and Interactions rom L 18 orthogonal array design, factor affects at three levels can be obtained and interaction affects cannot be studied. Since the experimental design is orthogonal, it is then possible to separate out the effect of each parameter at different levels. The influence of each control factor on the response considered i.e. cutting force, has been performed with level mean analysis. A level mean of a factor is the average of the response value of experiments in which the factor is at the particular level. or example, the mean value of the response for the side cutting edge angle at level 1, 2 and 3 can be calculated by averaging the response for experiments 1-3 & 10-12, 4-6 & and 7-9 & respectively. The mean of the response for each level of the other cutting parameters can be computed in a similar manner. The control factor with the strongest influence is determined by the difference between mean values of the factor at high and low levels. rom the factorial portion of CC design, both factor and interaction effects (at two levels) can be obtained. It can be observed from axial and central portion of CC design, considering experiments from 33 to 45, factor effects (at three levels) of each factor can be obtained when all other factors are at 0 levels. Using the experimental data, level means have been calculated. The level means obtained from L 18 design, factorial portion of CC design, and axial portion of CC design are given in Table 5 to 7. The influence of each control factor can be more clearly presented with response graphs. A response graph shows the change of the response when the settings of the control factor are changed from one level to the other. The slope of the line determines the power of influence of a control factor. Corresponding response plots are presented in igs. 1 to 3. Journal of Metallurgical Engineering, 1(1-2) January-December

10 Srinivasa Rao G. and Neelakanteswara Rao A. Table 5 Average Response for L 18 Design A B C D E Level Level Level Table 6 Average Response for actorial Portion of CC Design A B C D E Level Level Table 7 Average Response for One actor at a Time Analysis A B C D E Level Level Level ig. 1: Response Plot for L 18 Design ig. 2: Response Plot for actorial of CC Design 26 Journal of Metallurgical Engineering, 1(1-2) January-December 2011

11 Application of Central Composite and Orthogonal Array Design... ig. 3: Response Plot for One actor at a Time Analysis rom the tables of level mean analysis, it can be observed that the feed rate and depth of cut have been showing consistent behavior. As the feed and depth of cut increases from 1 level to +1 level, the force increases, whereas other factors do not show the same consistency. or example, as the side cutting edge angle changes from 1 level to +1 level in the L 18 data, the cutting force increases from 1 level to 0 level then decreases from 0 level to +1 level. The same is not the case with one-factor-at-a-time analysis portion of CC design (Table 7). Perhaps, the reason might have been the presence of interaction effects. The analysis of interactions gives additional information about the process. Interaction effects can be obtained by calculating all combinations of two control factors. or example, the interaction Ax D has four possible combinations of control factor settings: A1D1, A1D2, A2D1 and A2D2. The interaction matrix enables the construction of interaction graphs, which indicate the existence or non-existence of interaction between two control factors. If the lines in the interaction graph are parallel, it indicates non-existence of interaction. The interaction matrix and interaction graphs are shown in Table 8 and ig. 4. rom the interaction graphs, it can be observed that the interactions between the feed and depth of cut, feed and normal rake angle, depth of cut and normal rake angle are quite prominent. The interaction graph of control factors speed and feed shows a lesser influence, and remaining interactions are not prominent. In essence, it can be concluded that feed and depth of cut are having strong impact on cutting force followed by rake angle and speed. Journal of Metallurgical Engineering, 1(1-2) January-December

12 Srinivasa Rao G. and Neelakanteswara Rao A. Table 8 Interaction Matrices for the Cutting orce AXB B1 B2 AXC C1 C2 AXD D1 D2 AXE E1 E2 A A A A A A A A AX 1 2 BXC C1 C2 BXD D1 D2 BXE E1 E2 A B B B A B B B BX 1 2 CXD D1 D2 CXE E1 E2 CX 1 2 B C C C B C C C DXE E1 E2 DX 1 2 EX 1 2 D D E D D E ig. 4: Interaction Graphs for Parameters 28 Journal of Metallurgical Engineering, 1(1-2) January-December 2011

13 Application of Central Composite and Orthogonal Array Design... ig. 4(a): Interaction Graphs (Continued) 3.2 Model for L 18 Design As stated earlier L 18 design does permit estimation of interaction effects, and accordingly L 18 design data has been used to fit the first order model. The input data to SPSS software is provided in the Journal of Metallurgical Engineering, 1(1-2) January-December

14 Srinivasa Rao G. and Neelakanteswara Rao A. logarithmic transformation of actual values of factors. Backwards linear regression, which eliminates the insignificant factors one at a time, option of SPSS is used to estimate the parameters and the final results are shown in Table 9. Table 9 Parameter Estimates Variable Parameter estimate Standard error t Sig. Intercept ln(c) ln(e) ln() Analysis of variance for the model Source df Sum of squares Mean Square -value Sig. Model Error E E-02 Total R-square = Based on the above results, the cutting force model developed is given by: ln (orce) = ln (C) ln (E) ln () orce = C E The R-square value of indicates that 98.60% of the variability in cutting force was explained by the model. It can be observed that normal rake angle, feed and depth of cut are only coming into model. Based on the mathematical model, it can be observed that feed, depth of cut and normal rake angle are showing a prominent effect on the cutting force. The other factors have no significant effect on the cutting force. Perhaps it is consistent with the level means analysis of L 18 data. or 30 Journal of Metallurgical Engineering, 1(1-2) January-December 2011

15 Application of Central Composite and Orthogonal Array Design Model for CC Design As stated earlier, central composite designs are best for fitting a second order model, and accordingly CC data is used to fit a second order model. Procedure stated above is applied for developing cutting force model using CC design. The input data to SPSS software is provided in coded form of factors i.e. 1 to +1. To be precise, value of the factor in coded scale is = (actual value of the factor central value in the range)/ (difference between maximum value and central value in the range). or example the cutting speed value of 40 m/min is coded as 40-60/20 = 1. The parameter estimates obtained from SPSS software are shown in Table 11. Table 10 Parameter Estimates Variable Parameter Estimate Standard Error t Sig. Constant C D E CxE Cx Dx E Ex Analysis of variance for the model Source df Sum of Squares Mean Square -value Sig. Model Error Total R-square=0.990 Table 11 Cutting conditions used in confirmation tests Test A B C D E Journal of Metallurgical Engineering, 1(1-2) January-December

16 Srinivasa Rao G. and Neelakanteswara Rao A. The cutting force model developed using CC design was given by: orce = C 0.265D E CxE 0.5Cx 0.219DxE Ex as: Cutting force model in terms of actual factors can be expressed orce = normal rake angle cutting speed feed depth of cut normal rake angle x feed 0.50 normal rake angle x depth of cut cutting speed x feed feed x depth of cut. The R-square value of indicated that 99.00% of the variability in cutting force was explained by the model. In comparison to the previous model, it can be observed that the cutting speed apart from normal rake angle, feed and depth of cut is also coming into the model. urther, four interaction terms are also into the model. This is consistent with the level means and interaction effects analysis of CC design. 3.4 Confirmation Tests Conducting confirmation experiments has been the final step of the design of Experimental (DOE) process. The confirmation is performed by conducting tests using combinations of the factors and levels that are not previously evaluated. Table 12 shows the conditions used in the confirmation tests. Table 13 shows the results obtained where a comparison was done between the foreseen values from the models developed in the present work with the values obtained experimentally. Table 12 Confirmation Tests and their Comparison with the Results Test CC design L 18 design Experiment Model Error (%) Experiment Model Error (%) Journal of Metallurgical Engineering, 1(1-2) January-December 2011

17 Application of Central Composite and Orthogonal Array Design... It can be observed that the error percentages associated with both the models have been within the limits. Therefore, we can consider the empirical models, which correlate the cutting force with the process and tool parameters, with a reasonable degree of approximation within the given working conditions. 4. CONCLUSIONS In this work, the cutting force models have been developed by considering both process and tool parameters. Central composite and orthogonal array designs are used to develop the models. Based on the work, the following conclusions may be drawn: 1. It can be observed from L 18 design that normal rake angle, feed and depth of cut are only coming into model whereas in CC design the cutting speed apart from normal rake angle, feed and depth of cut is also coming into the model. urther, four interaction terms are also into the model. Interactions between feed and depth of cut, feed and normal rake angle, depth of cut and normal rake angle, and cutting speed and feed are having significant impact on the cutting force. 2. Both L 18 and CC (6) designs have been showing same prediction accuracy for cutting force. rom cutting force equation, it can be observed the absence of quadratic effects of factors, and the linear effect of factors has been only identified. In such cases, the L 18 design is sufficient to reveal the information regarding the process. 3. Significance of interaction terms of parameters has been clearly predicted in CC design, whereas none of them are considered in L 18 design. This is owing to the fact that in Taguchi s design interactions between control factors are aliased with their main effects. 4. By explicit incorporation of the interaction terms, the CC design model gives better insight into the process. So, CC design model developed for cutting force serves as a good alternative to the popular multiplicative model. 5. The significance of normal rake angle stresses the need for inclusion of tool parameters in empirical model building studies. Journal of Metallurgical Engineering, 1(1-2) January-December

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