Performance analysis of µed-milling process using various statistical techniques

Transcription

1 Int. J. Machining and Machinability of Materials, Vol. 11, No. 2, Performance analysis of µed-milling process using various statistical techniques G. Karthikeyan Department of Mechanical Engineering, Indian Institute of Technology, Kanpur , UP, India J. Ramkumar* Department of Mechanical Engineering, Indian Institute of Technology, Kanpur , UP, India Fax: *Corresponding author Shalabh Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur , UP, India Fax: shalab@iitk.ac.in S. Aravindan Department of Mechanical Engineering, Indian Institute of Technology, Delhi , India Fax: aravindhans@hotmail.com Abstract: µed-milling is a newly evolved process for machining complex 3D profiles in various micro devices. To achieve desired machining performance of the process, it is necessary to understand the influence of parameters on responses such as MRR and TWR. The parameters which are expected to control the process are tool rotational speed, feed rate and aspect ratio along with the fundamental parameter as energy. This work presents the use of various statistical tools on experimental data to study single and multiple objective performance characteristics of the process. Initially, an empirical equation for MRR and TWR are generated by multiple regression analysis with a confidence interval of 95 and 99% for each coefficient to generalise the models. To study the influence of individual parameter on single performance characteristics, tools such as S/N ratio and ANOVA are employed. Further using grey relational algorithm (GRA) technique, the multi performance characteristic of maximising MRR and minimising TWR is solved. The result shows that for single performance study parameters energy and speed are superior whereas for multi performance analysis feed contributes significantly. Copyright 2012 Inderscience Enterprises Ltd.

2 184 G. Karthikeyan et al. Keywords: regression; electric discharge machining EDM; µed-milling, machining materials; wear; material removal rate; MRR; tool wear rate; TWR; parameter; S/N ratio; analysis of variance; ANOVA; grey relational algorithm; GRA. Reference to this paper should be made as follows: Karthikeyan, G., Ramkumar, J., Shalabh and Aravindan, S. (2012) Performance analysis of µed-milling process using various statistical techniques, Int. J. Machining and Machinability of Materials, Vol. 11, No. 2, pp Biographical notes: G. Karthikeyan is pursuing his PhD in Department of Mechanical Engineering, Indian Institute of Technology, Kanpur. He graduated in Mechanical Engineering from Bharathidasan University, India in 1997 and he received his Masters degree in Manufacturing Technology from Regional Engineering College (presently called as National Institute of Technology) Tiruchirappalli, India in J. Ramkumar is currently an Associate Professor in the Department of Mechanical Engineering, Indian Institute of Technology, Kanpur India. He graduated in Mechanical Engineering from Regional Engineering College, (presently called as National Institute of Technology) Tiruchirappalli, India and he received his Masters degree in Mechanical Engineering. He received his PhD from Indian Institute of Technology, Madras, India. He has published over 60 research papers in refereed journals and conference proceedings. His area of interest includes micromachinig, composites, non-traditional machining and finishing and optimisation. Shalabh is currently an Associate Professor in the Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, India. He graduated in Mathematical Statistics from Lucknow University, India in He has published over 60 research papers in refereed journals and co-authored three books. His area of interest includes linear regression models and econometric models. S. Aravindan is currently an Associate Professor in the Department of Mechanical Engineering, Indian Institute of Technology, Delhi India. He graduated in Mechanical Engineering from Bharathidasan University India. He received his Masters degree in Mechanical Engineering from Annamalai University and his PhD from Indian Institute of Technology, Madras, India. He has published over 40 research papers in refereed journals and conference proceedings. His area of interest includes ceramics, composites, welding and nano-manufacturing. 1 Introduction Electric discharge machining (EDM) is a well known process used for fabricating components of both micro and macro dimensions applicable to variety of industrial applications. Recently, a new machining strategy called µed-milling (Saito et al., 1986; Kaneko and Tsuchiya, 1988) has evolved with introduction of computer numerical control (CNC) and four axis servo-controller (Ding et al., 2006) in µedm machine tool. In this process, a desired feature is achieved by moving a rotating cylindrical tool along a programmed path similar to conventional CNC milling which eventually eliminating the

3 Performance analysis of µed-milling process 185 need for manufacturing complex tools. µed-milling process is primarily used to generate complex 3D profile on thin and difficult to machine materials such as hard die steel, inconel, titanium, etc. The unique ability of this process is that, it is possible to generate high aspect ratio (AR) profile with reasonably good surface integrity. Due to increasing demand for micro channels in various micro devices, new techniques of machining such as µed milling finds significant importance in micro manufacturing processes. Micro channels of dimension less than 1 mm and greater than 1 µm has wide applications in micro heat exchangers and MEMS devices (Mohamed, 2002). To make heat exchanger compact and efficient, micro channels are used because they have large surface to volume ratio which leads to high rate of heat and mass transfer. For MEMS devices, micro channels are used to transport biological materials such as proteins, DNA, cells and embryos. In order to apply µed-milling process for such applications, it is first necessary to understand its behaviour at various conditions. In µed-milling, it is expected that along with the fundamental parameters such as energy (E), others like rotational speed (S), feed rate (F) and AR can play a significant role on the process responses such as material removal rate (MRR) and tool wear rate (TWR). The E factor by itself consist of various other factors such as voltage, current, Pulse on time, duty cycle, etc., and these are further reduced to voltage and capacitance in case of resistance capacitance (RC) circuit which is commonly used in micro domain (Jahan et al., 2008). Since the process is new and the parameters considered are not much dealt in literatures, it has become necessary to conduct experiments and further analyse the influence of parameters on the required output. Various statistical tools such as regression analysis, S/N ratio, analysis of variance (ANOVA), etc., are employed rigorously to solve single objective performance characteristics of parameters (Montgomery, 2001). These techniques are well accepted and often used for various type of data analysis (Tseng, 1998; Davim et al., 2004; Mohammadi et al., 2008). But for most of the machining processes, the objectives are several like max. MRR, min. TWR, min. surface roughness (Ra), min. overcut, etc., and it is essential to study the influence of process parameters on all these objectives together. Such an analysis is called multi objective performance optimisation where the optimal value of the parameters is obtained for more than one objective. Unlike in single objective study which gives a unique solution, there will be many optimal solutions for a multi objective problem and the exception is when the objectives are not conflicting in which case only one unique solution is expected. Hence, multiple objective performance study is not straightforward and requires some special techniques. The various techniques for multi objective optimisations include genetic algorithm (GA), simulated annealing (SA), ant colony algorithm (ACO), particle swap algorithm (PSO), grey relational algorithm (GRA), etc. However, as stated by Chan and Wu (1998) it is not easy to say which method is more reliable and reasonable for any multi objective problem. Among the above techniques GRA is rigorously used for solving wide variety of engineering problems because the analysis is based upon the original data and, the calculations are simple and easy to understand (Kuo et al., 2008). This theory was proposed by Deng in 1982 and proved to be a powerful tool to analyse the process with multiple performance characteristics. The work of various author have revealed the effectiveness of grey algorithm in optimisation. Authors (Tosun, 2006; Yang et al., 2006; Palanikumar et al., 2006) used GRA for drilling, end milling and turning operations to optimise the cutting parameters on the multiple performance characteristics including MRR, tool wear and surface roughness. GRA has also been successfully applied for

4 186 G. Karthikeyan et al. non-conventional machining techniques (Chiang and Chang, 2006; Huang and Liao, 2003; Caydas and Hascalık, 2008) with different parameters and data sets, and the authors have shown that the method is highly feasible for performance analysis. In EDM process, Lin and Lin (2002) and Singh et al. (2004) used GRA to optimise parameter for multi objective characteristics and the results are proved to be effective. The detailed literature survey on GRA has revealed that this technique can be an effective tool. In the present work, a regression model is generated for MRR and TWR by considering four important parameters namely E, F, S and AR. Confidence intervals for regression coefficients are constructed to generalise the model. Using S/N ratio and ANOVA, an analysis is conducted on the data to determine the influence of individual parameter levels and their contribution to achieve the desired machining performance such as maximising MRR or minimising TWR. Further to conduct multi objective performance study, i.e., maximum MRR with minimum TWR, a grey regression analysis is performed. These analyses will present the importance and the effect of individual parameters on single and multi objective performance characteristics. 2 Experimental details A multipurpose precision micro EDM machine with a CNC control is used to conduct the experiments. The equipment works on RC-based power supply with three axes servo control. A cylindrical tungsten electrode is taken as a tool and it is rigidly fixed in the spindle ensuring that it is straight and rotates without wobbling. The rotation aids not only uniform wear of tool but also enhances dielectric flushing. To ensure tool base flatness, the tool tip is spark machined with a sacrificial block at the start of each experiment. A die material EN24 is chosen as workpiece and is grinded for flatness on all surfaces before clamping it in a vice. Both tool and workpiece is immersed in dielectric during machining. Conventional EDM oil is used as dielectric and it is supplied through nozzle at certain pressure for continuous recirculation of fluid. Figure 1(a) shows the experimental setup and Table 1 shows the choice of workpiece, tool and dielectric considered for experiment. For µed-milling process of channel machining, the electrode is rotated and fed in a predefined path along y-axis. The schematic diagram in Figure 1(b) shows the µed-milling of channel and the chosen parameters. In the figure, L, w, d and R represents length of cut, width of channel, depth of the channel and radius of the tool respectively, whose values are given in Table 2. The SEM picture in Figure 2 shows the top view of the channels machined by µed-milling process. The channel dimension is 500 µm width and 1.5 mm length. The taper in the channel is because of tool wear and this varies for different machining conditions. Experiments are conducted with general factorial design considering four parameters with different levels. The parameters chosen are E, S, F and AR and they are varied in different levels as shown in Table 3. The details of the choice of parameters levels and their individual behaviour are discussed elsewhere (Karthikeyan, 2010). As discussed, E has been studied exhaustively for EDM process in literatures and our objective is to study the combination effect of E with other parameters. Hence, two values of energies were chosen, i.e., 500 µj and 2,000 µj. It is found that machining time increases enormously below 500 µj and for the present machine the maximum E value falls closer to 2,000 µj. So it has been decided to set these values as the E levels. Also the intermittent values are not possible as the machine has discrete values of capacitance. By using OFAT, S values

5 Performance analysis of µed-milling process 187 were changed from 0 rpm to 1,200 rpm. At 0 rpm, continuous short circuiting was observed without any machining. When the rpm is increase from 100 rpm, the machining time decreases and saturates after 800 rpm. Hence, the levels of S s were taken as 100, 500 and 800 rpm. In case of F, the values are changed from 10 µm/s to 80 µm/s. The observation was similar to as that of S and the limiting value here is 60 µm/s. Hence, the levels are 10, 25, 45 and 60 µm/s. Four levels are considered since the F-parameter is very new to µedm process. Since for a constant diameter of tool electrode, the AR can only be changed to maximum of 2 for micro process. Hence, the levels are 0.5, 1.0, 1.5 and 2.0. Figure 1 Schematic diagram shows (a) experimental setup (tool Ø500 µm) and (b) the parameters used (see online version for colours) Spindle Tool Workpiece Fixture (a) (b) Source: Karthikeyan et al. (2010)

6 188 G. Karthikeyan et al. Table 1 Factors Workpiece Tool Dielectric Table 2 Fixed parameters Slot dimensions Specification EN 24 (1.5 mm thick) (250 HB hardness and E = 25 GPa) Tungsten (Ø500 µm) EDM oil Dimension Value Length of cut (L) 1,500 µm Width of slot (w) 500 µm ± 15 µm Depth of slot (d) 250, 500, 750 and 1,000 µm Radius of tool (R) 250 µm Figure 2 SEM picture showing the channels machined by µed-milling process in EN24 die steel material (see online version for colours) Table 3 Parameters and levels Symbols Factors Units Levels E Discharge energy µj 500 2,000 F Feed rate µm/s S Speed rpm A Aspect ratio

7 Performance analysis of µed-milling process 189 The two important responses which are considered in a spark discharge processes are MRR and TWR. They are measured as TWE = V T and MRR V W t = t (1) where V T volume of material removed from tool, V W volume of material removed from workpiece and t machining time. Figure 3 Surface plots showing the relationship between the parameters and MRR (see online version for colours) Figure 4 Surface plots showing the relationship between the parameters and TWR (see online version for colours)

8 190 G. Karthikeyan et al. The behaviour of process parameters on the responses MRR and TWR are depicted in Figure 3 and Figure 4 respectively. Each 3D plot shows the effect of two parameters on response and further the parameters combination is varied in each plot as shown in Figure 3(a) to Figure 3(d). Figure 3 shows that the variation of parameter on response is non-linear and the nature of curve in all plots remains almost same. A detailed discussion of the parameter behaviour is presented in Karthikeyan et al. (2010). These plots are helpful to decide which type of regression equation can be applied to the experimental data in order to generate a MRR and TWR model which will work satisfactorily for wide values of selected parameters. Figure 4 shows the plots for TWR with all the process parameters and the discussion is same. 3 Regression modelling 3.1 Multi linear regression analysis To establish relationship for MRR and TWR with respect to E, F, S and AR(A), the technique of multiple linear regression analysis is considered. The idea is to present a model which can work satisfactory over a large range of values of parameter levels. In a multiple linear regression model, it is assumed that the relationship between the observation on response variable y and p explanatory parameters X 1, X 2,, X p is linear and it is denoted in matrix notations as y = Xβ + u where y = (y 1, y 2,,y n ) is a n 1 vector of n observations on response variable, X = ((x ij )); i = 1, 2,,n; j = 1, 2,, p; is a n p matrix of n observations on each of the p explanatory variables, β = (β 1, β 2,,β p ) is a p 1 vector of the regression coefficients associated with the explanatory variables and u = (u1, u2,,u n ) is a n 1 vector of random error component. Further, it is assumed that X is a non-stochastic matrix, β is fixed, rank of X is p (n > p), and u follows a multivariate normal distribution with n 1 mean vector 0 and n n positive definite covariance matrix σ 2 I where σ 2 is unknown and I is an n n identity matrix. When applying the principle of least squares, the least square estimator of β turns out to be ˆ 1 β = ( XX ) Xy This ˆβ is the point estimate of β which means that ˆβ is estimating β at a point (Rao et al., 2008). Alternatively, the interval estimate of β j, j = 1, 2,, p in terms of confidence intervals can be obtained under the normality of u as ˆ β σ β ˆ β σ 2 2 j t, n p ˆ Cjj j j + t, n p ˆ Cjj α α 2 2 where C jj is the (j, j) th element of (X X) 1,

9 Performance analysis of µed-milling process ymy SS ˆ σ = = Res, n p n p 1 M = I X( XX ) X, SS Res is the sum of squares due to residuals and t α, n p 2 is the upper α 2 % points on a t-distribution with (n p) degrees of freedom. By fitting a multiple linear regression equation, the model turned out to be unsatisfactory because the number of mismatched observations between the observed and fitted values of y. Hence, several fits were attempted and finally the fit of log MRR versus log E, log F, log S and log A turned out to be the best fitting. 3.2 Fitting of MRR versus process parameters Considering fitting of the model, log MRR = β + β log E + β log F + β log S + β log A + u the following output of regression analysis is produced. log MRR = log E log F log S log A R-sq = 85.9% R- sq( adj) = 85.3% The value of coefficient of determination denoted by R-sq and its adjusted value denoted by R-sq (adj) infer about the quality of model. Both these values are nearly 85% which means that the model is 85% capable of explaining the variability in the values of response variable due to the explanatory variables. Table 4 Regression analysis of MRR Predictor Coefficient Standard error of coefficient t-value p-value Constant log E log F log S log A Table 5 ANOVA for regression analysis of MRR Source Degree of freedom Sum of squares Mean square F-value p-value Regression Residual error Total In Table 4, the t-values are corresponding to the test of hypothesis in order to test the null hypothesis about the significance of individual regression coefficient. The corresponding p-values indicate that all the regression coefficients are significant. Thus, the variables log E, log S, log F and log A are important and contributes significantly to MRR. The

10 192 G. Karthikeyan et al. standard errors of coefficients are not very high, which explains that the parameters are linearly independent. Table 5 shows the output of ANOVA which also indicates that the sum of squares due to residual error is which is small in comparison to other sum of squares and thus validates a good fitting of the model. The F-value in the ANOVA corresponds to the testing of hypothesis about the equality of regression coefficients. The corresponding p-value indicates that the regression coefficients are not equal. The fitted model is represented as, ( ) MRR = E F S A (2) Different types of residual plots for log MRR shown in Figure 5 are analysed. Note that residual is defined as the difference between the observed and fitted value of y obtained through the fitted model. This is also indicated in the scatter diagram of residuals versus fitted values that the relationship between response variable and explanatory variables is non-linear. Here, the non-linear relationship is approximated by converting it into a linear function by using the log transformation and then reverting it back. The normal probability plot for the residual confirms that most of the points are lying in a straight line which indicates that the random error components are following the normal distribution and this is very well confirmed by the histogram of the residuals. The plot of residuals versus the order of the data indicates the absence of autocorrelation in the data. Figure 5 Residual plots for MRR model (see online version for colours) The model presented above does not take into account of the variability in the process. Any two experiments in real environment will not provide a single response value. In order to accommodate this variability into the model, a confidence interval estimation of

11 Performance analysis of µed-milling process 193 the parameters β 0, β 1, β 2, β 3, β 4 is carried out. The values of lower and upper limits of the confidence interval are obtained with 95% and 99% confidence coefficient is show in Table 6. Based on the nature of application of the model, the user can choose any one of the confidence coefficient. Table 6 Confidence interval of regression coefficient for MRR model 95% confidence coefficient 99% confidence coefficient Lower limit Upper limit Lower limit Upper limit β β β β β The interpretations of values in Table 6 are as follows. For example, the confidence interval for β 1 at 95% confidence coefficient is P ( β ) = This means that the value of β 1 will lie in the interval ( ) with 95% chances. Similarly, β 1 at 99% confidence interval is ( ). Similar interpretation of other values of β 1 s also hold. Thus the fitted model turns out to be, β β β β MRR CE F S A = (3) where the values C, β 1, β 2, β 3, β 4 lie between the lower and upper limits as stated in Table 6 at a desired level of confidence coefficient. So the only difference between the earlier model based on point estimate and this model based on interval estimates is that the values of the parameters β 0, β 1,, β 4 are estimated at an interval. 3.3 Fitting of TWR versus process parameters Similarly for TWR model, * * * * * * logtwr = β + β log E + β log F + β log S + β log A + u and the output of regression analysis is produced as follows, logtwr = log E log F log S log A R-sq = 82.4% R- sq( adj) = 81.6% Table 7 Regression analysis of TWR Predictor Coefficient Standard error of coefficient t-value p-value Constant log E log F log S log A

12 194 G. Karthikeyan et al. Table 8 ANOVA for regression analysis of TWR Source Degree of freedom Sum of squares Mean square F-value p-value Regression Residual error Total The t- and p-values from Table 7 indicate that all variables are significant in the model. The ANOVA also rejects the null hypothesis of the equality of regression coefficient. The value of coefficient of determination and its adjusted values are nearly 82%. The sum of squares due to residuals shown in Table 8 is also small. This indicates that the model is good fit. The analysis of residual plot in Figure 6 is conducted on the same lines as fitting of MRR and the conclusion are same. Figure 6 Residual plots for TWR model (see online version for colours) The model based on point estimation of regression coefficient is, ( ) TWR = E F S A (4) The model based on the confidence interval estimation of regression coefficient is * * * * * 1 2 β3 4 β β β TWR = C E F S A (5) * * * * * where the values of C, β1, β2, β3, β 4 lie in intervals at 95% and 99% confidence coefficients as given in Table 9. Thus, the equation (3) and (5) represents the generated empirical models for MRR and TWR respectively and this can be applied for varying conditions and parameter values.

13 Performance analysis of µed-milling process 195 Table 9 Confidence interval of regression coefficient for TWR model 95% confidence coefficient 99% confidence coefficient Lower limit Upper limit Lower limit Upper limit * β * β * β * β * β Statistical analysis 4.1 ANOVA and F-test As discussed in the last section, the regression analysis only establishes a relationship for MRR and TWR as a function of four parameters. From the coefficient values of model, it is not possible to directly say that a particular parameter is contributing more than the other on the responses. In order to determine the significant process parameter, it is necessary to apply a statistical technique like ANOVA (Montgomery, 2001; Toutenburg and Shalabh, 2009). The method of ANOVA enables to determine the amount of variance attributable to various causes and decide whether or not the parameters have produced any significant effects on response. Using F-test the significance parameter is determined by testing F A > F 0.05, n1, n2 at 5% level of significance with n 1 and n 2 degree of freedom (Huang and Liao, 2003). In ANOVA the measurements are taken on each experimental parameter pertaining to character of interest and its variance is calculated. But in order to analyse the influence of parameters on single performance characteristics of responses such as maximising MRR or minimising TWR, two characteristics namely, HB (higher is better) and LB (lower is better) are considered. The formulae for calculating HB and LB are same as that used in S/N ratio calculation and they are given as (Chattopadhyay et al., 2009), n 1 1 HB : η = 10 log 1 2 N i= 1 yi n 1 2 LB : η = 10 log 1 yi N i= 1 In these relations, i = 1, 2,,n represent the number of repeated experiment and N is total number of tests. Since in the present study i = 1, S/N ratio for HB and LB can be represented as { } 2 1 HB : η = 10 log y and LB : η = 10 log 2 y (6)

14 196 G. Karthikeyan et al. Table 10 shows the sum of square, mean square and F-values for various parameters. The significance of the parameters on responses is measured by comparing the calculated F-value with the table value obtained for 95% confidence level. Table 11 shows that the calculated F-values are greater than the table values for all the parameters indicating that each parameter is contributing to responses. Table 10 ANOVA results Parameters DF MRR (HB values) TWR (LB values) SS MS F SS MS F Energy Feed Speed 2 1, , Aspect ratio Error Total 95 2, , Table 11 F-test values Parameters F-test % Contribution F- (MRR) F- (TWR) Table F (0.05,dof) MRR TWR Energy Feed Speed Aspect ratio Figure 7 Percentage contributions of process parameters on MRR and TWR (see online version for colours)

15 Performance analysis of µed-milling process 197 The % contribution for each parameter is then calculated and the same is plotted in Figure 7. The histogram shows that the contribution of speed and energy is significantly high for maximising MRR. The contribution of speed is about 15% more than the energy indicating its importance in µed-milling process in achieving the desired MRR. While for the case of minimising TWR, the influence of energy is higher and it contributes 10% more than the speed. Thus the energy and speed are two parameters which are important for achieving desired response when compared with feed rate and AR. During machining only energy and speed contribute to the responses directly. Feed rate comes into play only when the process encounters short circuiting and for a given cut the AR is kept constant. Hence, for the case of single objective, the parameters energy and speed are important in achieving the desired performance. To substantiate the results of ANOVA, certain conditions are considered from the experiments as shown in Table 12. In each condition, only one parameter is varied from lower to higher level keeping other parameters constant at their optimum level. Table 12 shows that, the maximum percentage increase in MRR and TWR is due to change is speed and energy respectively. Thus, the significant µed-milling process parameters are energy and rotation speed. Table 12 Percentage improvement in responses for various experimental conditions Objective S. no. Conditions MRR TWR Parameter varied R 1 * R 2 * Response % improvement in response 1 E & F high; A=1.5 S L1 to L E & S high; A=1.5 F L1 to L S & F high; A=1.5 E L1 to L E, S & F high A L1 to L E, F & A low S L1 to L E, S & A low F L1 to L S, F & A low E L1 to L E, S & F low A L1 to L Note: *R 1 and R 2 are responses at lower and higher levels (L) respectively 4.2 Signal to noise ratio Another tool of statistic which is widely used for data evaluation is signal to noise (S/N) ratio. This tool determines the level of individual parameter which affects the desired response significantly. Since here again the objective is single performance characteristic of responses (i.e., maximising MRR or minimising TWR), the HB and LB value [from equation (6)] is used to calculate the S/N ratio value. According to Taguchi methodology of analysis, the HB and LB are primarily used for S/N ratio calculation. The S/N ratio value for each parameter is the summation of HB or LB values corresponding to the level of parameters considered. The optimal level of the process parameters for both the objectives is the level with the highest value of S/N ratio. Table 13 shows the values of HB and LB determined for each level of parameter. It is evident from the table that higher level of energy, speed and feed rate, and desire level of AR (A = 1.5) can be used for maximum MRR whereas the lower level of all parameters

16 198 G. Karthikeyan et al. gives minimum TWR. Since energy determines the strength of the spark and tool rotation speed decides the flushing efficiency, it is understandable that higher values of these parameters realise maximum MRR. Correspondingly there lower values reduces the removal rate thereby realises minimum TWR. The performance of feed rate is similar to energy and speed even though from experiments it is found that both MRR and TWR stabilises after the feed rate of 45 µm/s (see Figure 3 and Figure 4). The results of feed rate are justifiable because the increase in feed rate reduces the machining time which increases the rate of machining. The behaviour of AR from experiment revealed that as AR increases to 1.5, both MRR and TWR increases and further increase in AR causes drop in responses. This is seen in Figure 3(b) and Figure 4(b). Such a behaviour is attributed to the change in sparking surface area which causes increase in debris content resulting in poor flushing even at higher speed, Thus, AR of 1.5 gives maximum MRR and as the case of other parameters lower value of AR gives minimum TWR. Table 13 Signal to noise ratio results Machining performance Levels Energy Feed Speed AR MRR , * * * TWR Note: *Optimum values * * * * * The ANOVA and S/N ratio is used to analyse single performance characteristic with one objective of either maximising MRR or minimising TWR. However, optimisation of multiple performance characteristics cannot be straightforward as in the optimisation of a single performance characteristic. The higher S/N ratio for one performance characteristic may correspond to the lower S/N ratio for another performance characteristic. As the result, the overall evaluation of the S/N ratio is required for the optimisation of a multiple performance characteristics which will be discussed by using grey analysis. 5 Grey relational analysis Grey theory is a simple and accurate method for multiple attributes decision problems. Grey, when thought as a colour is a blend of black and white. In grey analysis, the black is represented as lack of information whereas the white is full of information. Thus, the information that is either incomplete or undetermined is called as grey. A system having incomplete information is called grey system (Tsai et al., 2003). The grey relational analysis uses information from the grey system to dynamically compare each factor

17 Performance analysis of µed-milling process 199 quantitatively. This approach is based on the level of similarity and variability among all factors to establish their relation. This kind of interaction is mainly through the connection among machining parameters and some conditions that are already known. In grey relational analysis, the optimisation is carried out in different steps. The first step is data pre-processing where the original sequence is transferred to comparable sequence. This is required because the range, unit and objective of data sequence 199 (for a response) vary between each others. Normalisation of data sequence in the range between zero and one is performed using linear data pre-processing methodology (Singh et al., 2004). This is carried out in three different types based on the response characteristics as, Higher is better ( HB) (0) (0) * xi ( k) min xi ( k) xi ( k) = (0) (0) max xi ( k) min xi ( k) Lower is better ( LB) (0) (0) * max xi ( k) xi ( k) xi ( k) = (0) (0) max xi ( k) min xi ( k) Desire value (0) (0) x ( ) * i k x xi ( k) = 1 max (0) ( ) (0) xi k x (7) where * (0) xi ( k ) is the generating value of grey relational analysis, min xi ( k ) is the minimum value of (0) (0) xi ( k ), max xi ( k ) is the maximum value of x (0) i ( k ) and x (0) is the desired value (Huang and Liao, 2003). Next step in grey analysis is to calculate the grey relational coefficient from the normalised experimental data to express the relationship between the ideal (best) and the actual experimental data. Grey relational grade is then calculated by averaging the grey relational coefficients corresponding to each performance characteristic. The term grey relational grade, Γ is used to show the connection among original and comparative series. The procedure for calculating grey relational grade is followed as given by Huang and Liao (2003). Let (X, Γ) be a grey relational space, where X is the collection of grey relational factors, and let x i (k) be the comparative series, x 0 (k) the reference series: x ( k) = x (1), x (2),, x ( n) x ( k) = x (1), x (2),, x ( n) X i i i i where i = 1,,m. Then, the grey relational grade can be calculated by equation (7). Δ min and Δ max is the minimum and maximum value among all the Δ oi values, respectively. Δ oi (k) is the absolute value of difference between x 0 and x i at the k th point. Δ min +Δ Γ oi = Δ+Δ max max (8)

18 200 G. Karthikeyan et al. where 1 i = 1,,m, k = 1,,n, j i 2 x 0 (k): reference sequence, x i (k): comparative sequences 3 Δ oi (k) = x 0 (k) x i (k) 4 min 0 j i k Δ = minmin x ( k) x ( k) j 5 max 0 j i k Δ = max max x ( k) x ( k) j n 2 Δoi ( k) Δ = n k = 1 This grey relational grade is used for the evaluation of the experimental data of the multi response characteristics. Highest grey relational grade corresponds to the optimal level of experiments. Optimisation of the complicated multiple performance characteristics (i.e., maximising MRR with minimum TWR) can be converted into a single grey relational grade. This can be done by calculating the grade values for both the responses separately and then averaging them to generate single grey relational grade. Basically, larger the grey relational grade, better are the multiple performance characteristics. Figure 8 shows the plot for grey relational grade for 96 experiments. Figure 8 Grey relational grades for different experiments (see online version for colours)

19 Performance analysis of µed-milling process 201 The experiments with maximum grade values are extracted and tabulated in Table 14. Experiment numbers 2 and 95 records the maximum grade values of 0.99 indicating that these two experiments yield the best multi performance characteristics. The values of parameters in these two experiments indicate that two very different combinations have given the best results. In experiment 2, the parameters values are set to their lower levels (except for the AR whose value is 1) and this gives a very small TWR with reasonable MRR. Conversely in experiment 95, the values of parameters are set to their maximum levels and the response values shows a very high MRR with reasonable TWR. Since both the experiments proved to have better multi performance characteristics, it is suggested here to use the test 95 for roughing operations as this gives highest MRR while experiment 2 for finishing operation where the MRR and TWR is small. Table 14 List of experiments which gives better performance S. no. Exp. no. E F S A MRR TWR Grade The effect of each process parameter on the grey relational grade can be separated out by calculating the mean of grey relational grade for each parameter. By using the grey coefficient of different parameters, a grey relational grade is calculated and tabulated in Table 15. Table 15 Grey relational grades for process parameters with multiple performance characteristics Parameters MRR TWR Total Energy Feed Speed Aspect ratio The results in above table indicate that the contribution of speed, feed rate and energy are significant in achieving combined performance of maximising MRR and minimising TWR. When comparing the grade values of energy, feed rate and speed, the feed rate has the highest value indicating its superiority over the others. Even though the feed rate does not influence much the single performance characteristics but contributes significantly when combined objective optimisation is performed. As discussed earlier increase in energy and speed increases MRR and to achieve smaller TWR their values has to be reduced. Hence, an intermediate value of energy and speed has to be determined to achieve multi objective which is determined with respect to the set feed rate. Hence, parameter feed rate is found to influence multi objective problem. 6 Conclusions For a newly developed process, it is necessary to study the performance of parameters on responses using different statistical techniques so that the behaviour of process is comprehended. In µed-milling process, parameters such as energy, speed, feed rate and

20 202 G. Karthikeyan et al. AR is considered to be important on studying the behaviour of responses such as MRR and TWR. Through non-linear regression analysis, models for MRR and TWR are obtained to represent experimental data in the form of empirical equations. The result shows that the models are good fitted with very small residual error. Also, to account for variability due to condition of machining, a confidence interval for each coefficient is incorporated which makes the models more generalised. Both 95% and 99% confidence values are calculated and it is left to the user to decide the suitable value based on experimental condition. Using S/N ratio, the level of parameter which influences the desire objective is determined. For maximising MRR, higher level of energy, speed, feed rate and desired value for AR is optimal. For the present case, energy of 2,000 µj, speed of 800 rpm, feed rate of 60 µm/s and AR of 1.5 is optimal. For minimising TWR all values of parameters should be in their lower level that is energy is 500 µj, speed is 100 rpm, feed rate is 10 µm/s and AR is 0.5. The ANOVA shows that, among the chosen parameters, speed is superior in maximising MRR and energy is influential in minimising TWR. It is desired to use other parameters in their optimal value. On applying grey relational analysis, feed rate is found to have maximum grade value (slightly higher than the speed) when the combined objective is considered. Thus, based on these studies it is clear that speed and energy are most influencing parameter for individual objective performance study and feed rate have a certain role to play in combined objective of maximising MRR and minimising TWR. Hence, the parameters which are selected for µed-milling process are proved to be important. Acknowledgements The authors acknowledge the financial support provided by Department of Science and Technology (DST), India, to acquire equipment under FIST program (Project no. DST-ME ). The authors also acknowledge the Indian Space Research Organisation (ISRO), IIT Kanpur cell for Project No. ISRO/ME/ References Caydas, U. and Hascalık, A. (2008) Use of the grey relational analysis to determine optimum laser cutting parameters with multi-performance characteristics, Optics & Laser Technology, Vol. 40, pp Chan, L.K. and Wu, M.L. (1998) Prioritizing the technical measures in quality function deployment, Qual. Eng., Vol. 10, No. 3, pp Chattopadhyay, K.D., Verma, S., Satsangi, P.S. and Sharma, P.C. (2009) Development of empirical model for different process parameters during rotary electrical discharge machining of copper-steel (EN-8) system, Journal of Materials Processing Technology, Vol. 209, pp Chiang, K.T. and Chang, F.P. (2006) Optimization of WEDM process of particle reinforced material with multiply performance characteristics using grey relational analysis, J. Mater Process Technol., Vol. 180, pp Davim, P.J., Reis, P. and Antonio, C.C. (2004) A study on milling of glass fiber reinforced plastics manufactured by hand-lay up using statistical analysis (ANOVA), Composite Structures, Vol. 64, pp Ding, P., Yuan, R., Li, Z. and Wang, K. (2006) CNC electrical discharge rough machining of turbine blades, Journal of Engineering Manufacturing, Vol. 220, pp

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