A Case Study on the Use of Box-Behnken Design to Improve an Electrical Discharge Machining

2.1 One-electrode SDS Approach

Conventional EDM research has focused on using a single electrode and a single discharge step (SDS), as shown in Figure 2(a). The use of one electrode for both roughing and finishing is more economical than the two-electrode approach. One-electrode SDS approach is appropriate if the removal volume is very small. However, this approach is inappropriate when machining large volumes; the use of a single electrode can increase the machining errors caused by severe electrode wear.

Figure 2.

Electrodes and Discharge StepsSDS: single discharge step, MDS: multiple discharge steps, RCD: roughing current difference, FCD: finishing current difference

2.2 Two-electrode SDS Approach

The sharp edge of the electrode becomes worn after extensive machining, causing errors in the concave edges of the workpiece. The industry has adopted two-electrode roughing and finishing stages to enhance both productivity and surface quality. In the roughing stage, a high discharge current is adopted to remove material rapidly with one electrode, and then a low discharge current is applied in the finishing stage with the other electrode. The approach is shown in <Figure 2(b)>.

2.3 Two-electrode MDS Approach

We try a new two-electrode approach based on multiple discharge steps (MDS) to improve productivity and surface quality. During both roughing and finishing, discharges are applied over three steps, to increase die-sinking EDM efficiency. The three steps are shown in <Figure 2(c)>. High discharge energy during roughing shortens the machining time but increases surface roughness. The discharge current is increased in the first step by an amount equivalent to the roughing current difference (RCD), fixed during the second step, and decreased during the third step by an amount equivalent to the RCD. The high discharge energy of the first step increases the material removal rate, and the low energy of the third step reduces surface roughness. The low discharge guarantees a good surface finish but increases the finishing time. The finishing stage is also divided into three steps. At each step, the discharge current decreases by the amount of the finishing current difference (FCD). The high discharge energy of the first step reduces the finishing time, and the low energy of the third step improves the surface finish.

3.1 Material and Equipment

The composition of the workpieces to be machined is as follows: 0.50~0.55 C, 0.15~0.35 Si, 0.75~0.90 Mn, and a maximum of 0.50 Ni (all wt%). The workpieces are milled to create 20 × 20 × 50 mm “boxes” before EDM. The graphite electrodes are shaped into a convex wedge with a transverse area of 10mm × 10mm. The die-sinking EDM machine is used to fabricate concave wedge shapes on the mold steel using graphite electrodes, as shown in <Figure 3(a)>. The workpiece is placed on a magnetic table, and the graphite electrode is bonded to the head of the machine. The dielectric fluid jet is flushed in six directions to remove debris in the gap between the electrode and the workpiece. Electrode edge wear is measured using a digital microscope, and the roughness value of the inclined wedge surface is measured by a stylus roughness tester in <Figure 3(b)>.

Figure 3.

Experimental Equipment and Measurement Instruments

3.2 Experimental Conditions

The discharge current in the roughing stage is set to 14 A based on the results of Kiyak and Cakır (2017), while the discharge current at the finishing stage is set to 5 A, according to Jeong et al. (2017). The other experimental parameters are determined based on the equipment maker’s recommendation.

The experimental conditions and results of the three approaches are shown in <Table 1>. With the one-electrode SDS approach, roughing and finishing are performed by the same electrode. The two-electrode SDS approach uses one electrode for roughing and the other for finishing. The normal discharge current in the roughing stage is 14 A, to remove material quickly, and it changes to 5 A at the finishing stage for good surface quality. With the two-electrode MDS approach, the discharge currents in the roughing stage are 16 A for the first step, 14 A for the second step, and 12 A for the third step. The discharge currents in the finishing stage are 6 A for the first step, 5 A for the second step, and 4 A for the third step. All three approaches are repeated three times and the average values of the surface roughness, edge wear, and machining time are shown in <Table 1>.

Table 1.

Comparative Experimental Conditions and Results of the Three EDM Approaches

3.3 Edge wear

<Figure 4(a)> presents the edge wear of the one-electrode SDS approach. The substantial edge wear of 0.216 mm results from the fact that one electrode is used to remove the entire 4-mm-deep metal. <Figure 4(b)> shows the edge wear of the two-electrode SDS approach. The edge wear of the finishing electrode is reduced to 0.065 mm. <Figure 4(c)> shows the edge wear of the two-electrode MDS approach. The edge wear of the finishing electrode of the two-electrode MDS approach is 0.086 mm, which is a little larger than that of the two-electrode SDS approach.

Figure 4.

Edge Wears of Electrodes and the Machined Workpiece

3.4 Comparison of the Results among Different Approaches

<Figure 5> compares the edge wear, surface roughness, and machining time among the three approaches. The edge wear of the two-electrode SDS approach is about 70% less than that of the one-electrode SDS approach. The edge wear result of the two-electrode MDS approach is inferior to that of the two-electrode SDS approach. The average surface roughness (Ra) values are similar among all three approaches. The two-electrode SDS approach is somewhat better than the other approaches in terms of machining time. A limitation of this comparison is that the current difference of the two-electrode MDS approach is fixed at 2 A in the roughing stage, and at 1 A in the finishing stages, while the gap distance is also fixed at 0.07 mm. To improve the performance of the MDS approach, we use RSM to determine conditions minimizing edge wear, surface roughness, and machining time.

Figure 5.

Comparison Graphs of Three Approaches

4.1 Experimental Plan

The process variables and their respective levels are shown in <Table 2>. We consider three variables: the discharge current differences in the roughing and finishing stages, and the gap distance. In the MDS approach, a high discharge energy during the first step saves machining time, and a low discharge energy during the third step ensures a good surface finish. Therefore, in the roughing stage, the discharge current is increased in the first step by an amount equivalent to the RCD, fixed to 14 A during the second step, and decreased during the third step by an amount equivalent to the RCD. The RCD is set to 0, 2, or 4 A. The minimum RCD is 0 A (equivalent to that of the SCD approach), the intermediate RCD is 2 A (equivalent to the MDS approach before RSM optimization), and the maximum RCD is double the intermediate value. During finishing, the discharge current is increased, fixed, and then decreased in the first, second, and third steps, respectively. The FCD is set to 0, 1, or 2 A, using a method similar to that applied to derive the RCDs. The gap distance, which will be referred to as Gap, W is the clearance between the workpiece and the electrode. The Gap is set to 0.04, 0.07, or 0.10 mm. The intermediate level is the same as that of the SDS and MDS before RSM.

Table 2.

Parameters and Their Levels for the MDS Experiment

<Table 3> shows the BBD experimental matrix and measurement data, where the edge wear is the wear of the electrode after the finishing stage, and the machining time is the sum of the roughing and finishing times.

Table 3.

Box-Behnken Design Matrix and Data for MDS

4.2 Analysis of Surface Roughness

Analysis of variance (ANOVA) is performed to draw a model with significant terms. Criteria for model fitting are the adjusted R² and the smallest S(root mean square error). A model with the largest adjusted R² and smallest S(root mean square error) is persued. Equation (1) is the regression model for the average surface roughness (Ra) with respect to the FCD ΔI_f and Gap W in the uncoded units estimated from the experimental data:

As the roughing stage is followed by the finishing stage, the Ra should be unaffected by the RCD. ANOVA shows that the coefficient of determination R² of the fitted model is 62.51%, the adjusted R² is 52.28%, and S is 0.7413 . Thus, 62.51% of the surface roughness variation is attributable to the linear effects of the FCD, the Gap, and their interaction. Model adequacy checking with studentized residuals reveals that the roughness value of 7.64 at run number 14 is an outlier, which is outside of 6 standard deviation of the studentized residuals. The existence of the outlier leads to the small R2 value, which reflects the possibility of tough surface roughness resulting from imperfect discharging and errors when measuring inclined surfaces. <Figure 6> is the contour plot of the estimated Ra, with respect to FCD ΔI_f and Gap W when RCD is fixed at 2 A. The FCD has a greater influence on Ra. When the FCD is increased, Ra tends to decrease to a minimum. <Figure 6> also shows that the Gap has less effect on Ra when the FCD is increased by up to 2 A. <Figure 7> is the photographed surfaces when FCD is 0 A and 2 A, respectively, when the Gap is fixed to 0.07mm.

Figure 6.

Estimated Response Surface of Roughness vs. FCD and Gap

Figure 7.

Surface Photographed by Optical Microscope, RCD 2A, Gap 0.07 mm

4.3 Analysis of Edge Wear

The edge wear EW of the finishing electrode is influenced by three variables. Equation (2) shows their effects, where ΔI_r is RCD. The adjusted R² and S are and 93.95% and 0.00535, respectively. The R² is 96.97%, which means that 97% of the experimental data variation is explained by Equation (2). Model adequacy checking with studentized residuals shows no severe violations with respect to normality, run order, and fitted values.

<Figure 8(a)> shows the estimated response surface of edge wear with respect to the RCD and FCD when the Gap remains constant at the intermediate value of 0.07 mm. The edge wear tended to be smallest when the RCD is set to 1.5 A and the FCD to either 0 or 2 A. <Figure 8(b)> shows the estimated response surface in terms of the RCD and Gap when the FCD is set to 1 A. The edge wear of the finishing electrode tends to be smallest when the RCD is 1 A and largest when the RCD is 4 A, showing that the roughing variable is linked to edge wear in the finishing stage. <Figure 8(c)> shows the estimated response surface of edge wear in view of the FCD and the Gap. Edge wear tended to decrease as the Gap increases. The FCD exerts a quadratic effect on edge wear.

Figure 8.

Estimated Response Surface of Edge Wear

4.4 Analysis of Machining Time

Machining time (MT) is also influenced by three variables, as shown in Equation (3), with the adjusted R² = 89.23%, S = 0.564, and R² = 93.85%. Model adequacy checking with studentized residuals shows no severe violations.

<Figure 9(a)> shows the estimated response surface of the machining time with respect to the RCD and FCD when the Gap is fixed at its intermediate value of 0.07 mm. The machining time is shortest when the RCD is 2 A and the FCD is 0 or 2 A. The effects of the RCD and Gap are shown in <Figure 9(b)>. When the FCD is set to the intermediate value of 1, the machining time is minimized if the RCD is 2 A and the Gap is 1 mm. The estimated response surface of the machining time with respect to the FCD and Gap is shown in <Figure 9(c)>. The machining time decreases as the Gap increases to 0.1 mm and the FCD is either 0 or 2 A. The response surface of the machining time exhibits a trend similar to that of the edge wear of the finishing electrode, indicating that a longer machining time increases edge wear.

Figure 9.

Estimated Response Surface of Machining Time

4.5 Optimal Conditions

The desirability function approach is implemented to optimize the three response variables affected by the three process variables (Derringer and Suich, 1980). The desirability function approach is most often employed to optimize multiple responses simultaneously (Myers et al., 2016). This approach searches for variable settings that jointly optimize multiple responses by satisfying the requirements for each response under consideration. In this approach, the estimated response values of each response are transformed to scale-free desirability between 0 and 1. The individual desirability (d) for each response to be minimized is obtained by specifying the target value and upper bound required for the response. If the response is larger than the upper bound, d is set at 0. If the response is smaller than the upper bound, d increases from 0 to 1 as the response variable comes closer to the target value. If the response is smaller than the target value, d is determined to be 1. A weight factor, which determines the desirability function shape for each response, is then assigned to each response. Weight can be given as a value between 0.1 and 10. When the weight is 1, the desirability function is linear. When the response needs to be smaller than the upper bound, a weight less than 1 is determined. If the response should be close to the target value, the weight is set at a value greater than 1. In general, if the weight factor is not mentioned, it is set at 1 (Myers et al., 2016).

The individual d is combined into an overall desirability D, which is the geometric mean of the individual d. When the response variables vary in terms of importance, D is the weighted geometric mean of the individual d. The relative importance of response variables is reflected by the ‘importance values’. The optimal compromise among multiple responses is achieved by maximizing D (Derringer and Suich, 1980). The desirability function approach is employed to simultaneously minimize the average surface roughness Ra, the edge wear of the finishing electrode, and the machining time simultaneously. The target values and upper bounds are 3 and 6 μm for Ra, 0.05 and 0.1 mm for edge wear, and 10 and 15 min for the machining time, respectively. As it is more important that the surface roughness and machining time should be lower than their upper bounds than that they reach the important target values, their weights are set to 0.5. Moreover, edge wear is considered to be twice as important as surface roughness and machining time, and so it is assigned an importance value of 2.

Using the response optimizer in Minitab, the optimal conditions are shown to be (RCD, FCD, Gap) = (0.580, 2.0, 0.04) (<Figure 10>). These conditions optimize the three responses simultaneously. The estimated Ra, edge wear of the finished electrode, and the machining time are 3.50 μm, 0.052 mm, and 10.78 min, respectively. As the RCD is controlled in integral increments, we perform additional experiments at RCDs of 0 A and 1 A.

Figure 10.

Optimal Conditions of the Desirability Function Approach

4.6 Follow-up Experiment

As the current can be controlled only in integral units, we test two conditions (RCD, FCD, Gap) = (0 A, 2 A, 0.04 mm) and (1 A, 2 A, 0.04 mm) three times, and compared the data (<Table 4>). The three responses are optimized when the RCD was 1 A.

Table 4.

Comparing two Conditions for Optimal Parameter Settings

4.7 RSM-optimized MDS Results

<Figure 11> shows the average values of the three responses for the two-electrode SDS and two-electrode MDS approaches, before and after RSM. RSM for the two-electrode MDS approach contributes to the improvement of edge wear and surface roughness. Through RSM optimization for the MDS approach, the edge wear of the finishing electrode is improved from 0.072 mm to 0.052 mm, and the average surface roughness is reduced from 4.01 to 3.27. The RSM-optimized MDS approach has reduced edge wear by 20% and average surface roughness by 24% compared to the SDS approach. However, the machining time of the MDS approach has increased by 15% compared to the SDS approach. The machining time of the MDS is longer than that of the SDS because the time saved from the high discharge current in the first step is smaller than the time increased from the low discharge current in the third step. Overall, the two-electrode MDS approach with RSM shows better performance than the two-electrode SDS approach.

Figure 11.

Comparison of SDS and MDS before and after RSM Optimization