Development of RSM Model in Surface Modification of EN-31 Die Steel Material Using Copper-Tungsten Powder Metallurgy Semi-Sintered Electrodes by EDM Process

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Table 2. Machining conditions used during experimentation work

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The process parameters, output measures and machining conditions used during experimentation are listed in Table 2. These were chosen through literature survey, reviews of experience, and some preliminary investigations. Moreover, some pilot experiments were conducted to fix the levels among the available options.

The work-piece was cleaned with acetone before and after each experiment and then dried with a dryer and precisely weighed. A digital balance (make: Essae Teraoka) with 0.1 mg resolution was used to calculate the SDR. The centreline average value (R_a) for SR was measured by portable roughness tester ‘Surftest SJ-301’ (make: Mitutoyo) . The cut-off length was set at 2.5mm with an evaluation length of 10 mm .The average value of six SR measurements was recorded in different directions for each specimen and these values has been used for development of SR model. The surface roughness (Ra) value of the work surface has been obtained in the wide range of 5.079 to 7.804 µm.

2.1. Design of Experiments

A central composite design with five levels for each independent variable was used for planning the experimentation work. The design initially chosen was factorial design 2⁴ with 6 central point’s which provides the information about curvature existence, consequently carrying out a total number of 22 experiments. The addition of six central points was allowed to carry out lack-of-fit test for the proposed first order model. In case first order model fails to reveal the behaviour of response that was studied, then it was further widened by addition of eight axial runs located on each axis at two alpha levels. So in case of second order CCD model a total of 30 experiments were carried out. Based on CCD, experiments were conducted to develop mathematical model for SDR and SR in terms of four input parameters; peak current, pulse-on time, pulse-off time and powder compaction pressure. Each input parameter had five levels of variation (±2, ±1, 0).

The input process parameters along with their levels, used in experimentation are listed in Table 3. The range of input parameters, i.e. peak current (I_p), pulse-on time (T_on), pulse-off time (T_off) and powder compaction pressure (P_c) was selected after performing a number of pilot experiments.

Table 3. Experimental input variables along with selected levels

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Table 4 contains the design of experiment (DOE) matrix having actual and coded values of the input variables along with two output responses.

Table 4. Design of Experiment matrix and machining characteristics

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2.2. Mathematical Modelling with RSM

Response Surface methodology is an interaction of mathematical and statistical techniques that are useful for developing, analysing, improving and optimizing the processes ^[12]. In the present work, by the use of DOE and applying regression analysis, surface deposition rate and surface roughness prediction model with independent input variable has been obtained. If all variables are assumed to be measurable, the first order model can be written as;

(1)

Ŷ is the response function (or response surface) and Y is the measured value of corresponding response functions SDR and SR. The model coefficients, b0, b1, b2, b3, b4 are to be estimated and where ε is the fitting error of the experimental observations. In this study, four variables (i.e., I_p, T_on, T_off, P_c) are under consideration. A second order polynomial regression model has been proposed as follows;

(2)

The adequacy for the model was checked by using the analysis of variance (ANOVA) technique.

2.3. Modelling of SDR by RSM

A second order model was created for mapping four input parameters, such as I_p, T_on, T_off and P_c with single output parameter, SDR. The ANOVA and Fisher’s statistical test was used to check the adequacy of the model as well as significance of each independent input variable. Table 5 shows ANOVA results of the proposed model for SDR. The ANOVA table contains sum of squares (SS), degree of freedom (df), mean squares (MS), F-value and P-value, where MS was obtained by dividing the SS of each of the sources of variation by the respective degree of freedom df. The F-value is the ratio of MS of the model terms to the MS of the residual, whereas the residual is the difference between actual and predicted value. The P-value is the smallest level of significance at which the data is significance.

Table 5. ANOVA for SDR

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The model F-value of 219.16 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise. Values of "Prob > F" less than 0.0500, indicates, that model terms are significant. In this case I_p, T_on, T_off, P_c, I_pT_of_f, I_p², T_on², P_c² are significant model terms. Values greater than 0.1000 indicate that the model terms are not significant. The "Lack of Fit F-value" of 57.02 implies the Lack of Fit is significant. There is only a 0.02% chance, that a "Lack of Fit F-value" this large could occur due to noise. The "Pred R-Squared" of 0.9722 is in reasonable agreement with the "Adj R-Squared" of 0.9906. "Adeq Precision" measures the signal to noise ratio. A ratio greater than 4 is desirable. The ratio of 53.342 indicates an adequate signal. From the above analysis and after eliminating the non-significance terms, the final equation for SDR is:

(3)

Experimental values and predicted values by above equation was plotted in Figure 1. Plot and ANOVA reveals that model and Eq. (3) were highly significant and adequate to represent the actual relationship between input variables and output response. Furthermore the residuals V/s normal plots for SDR obtained shown in Figure 2. Normal distribution of residuals depicts data spread, that was approximately along a straight line, which show a good correlation between experimental and predicted values. As a whole analysis of a normal plot of residuals for SDR confirms the adequacy and prediction capability of the model.

Figure 1. Variation of actual V/s predicted response for SDR

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Figure 2. Normal plot of residuals for SDR

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2.4. Mathematical Modelling with SR

A second order model for SR was developed to correlate input variables, I_p, T_on, T_offand P_c with one output parameter i.e., surface roughness (SR). Table 6 shows the ANOVA results for proposed model and significant model terms.

F-value of 640.09 implies that the model is significant. Values of "Prob > F" less than 0.0500 indicate that model terms I_p, T_on, T_off, P_c, I_pT_on, I_pT_off, I_pP_c, T_onP_c, T_offP_c, I_p², T_on², T_off², P_c² are significant. The "Pred R-Squared" of 0.9936 is in reasonable agreement with the "Adj R-Squared" of 0.9968. "Adeq Precision" of 109.946 indicate an adequate signal. ANOVA analysis reveals that Eq. (4) was highly significant that can be used to navigate the design space. Final equation in terms of coded factors is as follows:

(4)

Table 6. ANOVA for Response SR

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Variation of actual and predicted values using Eq. (4) are plotted in Figure 3 and residuals V/s normal plots for SF are shown in Figure 4. Comparison of predicted value of SR with actual values SR indicates very good agreement, ANOVA and analysis of a normal plot of residuals for SF also confirms the adequacy and prediction capability of the model.

Figure 3. Variation actual and predicted response for SR

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Figure 4. Plot of residuals for SR

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