1 Introduction

With the rapid development of micro-electromechanical systems (MEMS) and micro systems technology (MST), the demand for large amounts of micro components or devices is dramatically increasing [1]. Thereinto, the micro molding technology provides a feasible method to achieve the mass production of micro parts [2, 3]. Whereas, the material properties of the micro molds must be deliberately considered to meet different application requirements. Considering the high energy-efficiency and forming precision especially for micro scale part, the precision cold forming is very suitable for machining micro components [4]. In this case, SKD11 die steel in Japanese specification is one of the most appropriate materials due to its characteristics of high strength and toughness, good wear-resistance and excellent heat balance [5].

However, this kind of material is difficult to be machined by conventional manufacturing processes such as turning, drilling and milling. Thus, non-conventional machining technologies can be employed, including lithography, electroplating, and molding (LIGA), micro electrical discharge machining (micro-EDM), micro electrochemical machining (micro-ECM) and laser processing etc. Compared with the above technologies, micro reciprocated wire electrical discharge machining (wire-EDM) provides a promising alternative to fabricate the micro molds made from SKD11 due to its distinctive superiorities of high machining accuracy, cost-effective and non-contact machining on difficult-to-cut metal materials [6,7,8]. When machining micro molds by micro reciprocated wire-EDM, much attention should be paid to processing accuracy because it plays a critical role in determining forming precision [9]. On the other hand, the machining efficiency is also significant for the rapid production of various molds. However, the two performances are in conflict with each other in micro reciprocated wire-EDM process [10]. The investigation and optimization on machining performance and processing parameters for SKD11 attract many researchers’ attention.

Zhao et al. [11] studied the WEDM process of SiC and SKD11, and found that under the same machining condition, the MRR and tool wear ratio of SiC were superior to that of SKD11. Ming et al. [12] investigated the energy distribution ratio during the EDM of Al 6061, Inconel 718 and SKD11, and found that the discharge energy transferred to the three kind of workpiece were all small. Zheng [13] analyzed the energy characteristics in WEDM, and established a non-pulsed auxiliary energy costing and feeding energy prediction model to save energy. An attempt had been conducted to obtain good surface quality in LS-WEDM process, and it was found that the high rotating speed and peak current generated poor surface quality [14]. Other investigations have also been conducted for high machining accuracy. Based on the analysis between the kerf width and machining parameters, Di et al. [15] established the mathematical model of wire lateral vibration and cut 30.8 μm width slot on stainless steel. The wire movement in fine wire-EDM process was recorded using high-speed camera and the wire vibration was also analyzed under different machining condition, it was found that the machining accuracy could be improved by increasing offset [16], and arranging the workpiece at the position of 3rd order node [17].

In order to find the optimal solutions of the machining performance and parameters, many researches have been carried out. The hybrid approach combining grey relational analysis with principle component analysis was used to identify the optimal parameters for the WEDM process of nanostructured hardfacing material [18]. The production efficiency and part quality of alloy steel was improved by multi-objective optimization of machining parameters based on artificial neural networks [19]. The high material removal rate with better surface finish was obtained by multi-objective genetic algorithm for machining nickel-titanium-based shape memory alloy [20]. According to the above summary, it is found that the systematic investigation on MRR and machining gap for machining SKD11 by micro reciprocated wire-EDM has not been given yet.

In this paper, the effects of open voltage, discharge capacitance, pulse duration and feed rate on MRR and machining gap of SKD11 are studied by the experiments designed based on central composite design (CCD) of response surface method (RSM). The mathematical models for the two responses have been developed, and whilst verified through confirmation experiments. Furthermore, the multi-objective optimization of the machining parameters is conducted using the improved non-dominated sorting genetic algorithm (NSGA-II) for different machining performances. Based on the above systematical investigation, a typical experimental sample, micro gear is successfully machined with high machining efficiency and accuracy using one of the optimal machining conditions.

2 Experimental Detail

2.1 Experimental Setup

Figure 1 shows the photograph of micro-WEDM setup with reciprocated traveling wire electrode. It mainly consists of reciprocated wire-moving system, RC micro energy pulse generator with precise control, servo control system based on piezoelectric ceramics motors drive, constant tension control unit, vibration inhibition unit, working liquid-circulation system and CCD camera system. The X/Y stage with 0.1 μm in resolution is composed of precise servomotor and driver which is driven by piezoelectric ceramic motors, utilizing a high-precision grating ruler as the position feedback component. Because the micro energy generator is utilized and the ultrafine wire electrode is circularly travelling, the wear of wire electrode is negligible. The working liquid-circulation system is used to cool electrodes and expel the debris particles from the machining gap easily, and the CCD camera system is applied to observe the micro-WEDM process.

Fig. 1
figure 1

Micro reciprocated wire-EDM setup

Full size image

2.2 Experimental Material

The material employed in this study is SKD11 in Japanese specifications considering its properties of high hardness and toughness, good hardenability, as well as excellent wear-resistance and thermostability. Its harness is 56–58 HRC, impact toughness is 9.1 J cm−2 and other mechanical properties are given in Table 1. The as-received workpiece is a plate with 2 mm thickness. To avoid the overcut of the fillet radius, the tool electrode used in experiments is the ultrafine tungstenic wire with 30 μm diameter, which has high melting point of 3410 °C and stable chemical properties. Commercial EDM oil is used as working medium because of the good deionization insulation.

Table 1 Mechanical properties of SKD11
Full size table

2.3 Evaluation for MRR and Machining Gap

The experiments are performed on SKD11 plate with 2 mm thickness. The cutting length is 120 μm in each experiment. The micro wire with 30 μm diameter is selected as tool electrode in all tests. MRR is expressed by the removed volume of SKD11 material in unit time, as shown in Eq. (1).

$$MRR = {{\begin{array}{*{20}c} {{\text{Removed}}\;{\text{volume}}} \\ \end{array} } \mathord{\left/ {\vphantom {{\begin{array}{*{20}c} {{\text{Removed}}\;{\text{volume}}} \\ \end{array} } {{\text{Machining}}\;{\text{time}}}}} \right. \kern-0pt} {{\text{Machining}}\;{\text{time}}}}$$
(1)

The microscope system VHX-6000 made by KEYENCE is applied to measure the kerf width, and seven measurements are conducted for each slot along the cutting direction as shown as the red part in Fig. 2a. The average value of every kerf width is calculated as the final value. Figure 2b indicates the schematic diagram of the cut kerf. When the distance Mb becomes small enough, the dielectric fluid in the clearance will be broke down by the voltage between the wire electrode and workpiece, and simultaneously the discharge happens. The discharge force is randomly exerted on the micro wire, causing the micro wire to vibrate in the discharge gap. The maximum amplitude of the wire lateral vibration is represented by Ap. When the micro wire (diameter d) is selected, the kerf width D is mainly determined by the breakdown gap Mb and the wire lateral vibration Ap, which can be expressed as Eq. (2).

$$D = d + 2(M_{\text{b}} + A_{\text{p}} )$$
(2)
Fig. 2
figure 2

Measurement and analysis for the kerf width: a partly machined kerf and b schematic diagram of the machined kerf

Full size image

The breakdown gap Mb is mainly determined on the electric field intensity and insulation medium between the two electrodes. But, it is difficult to be calculated because the breakdown condition is affected by the purity of the insulation medium. Besides, the magnitude, occurrence time and location of the discharge force acting on the micro wire is very complicated, thus the wire lateral vibration Ap cannot be easily figured out. According to Eq. (2), the algebraic sum of Mb and Ap can be easily calculated when the kerf width D and wire diameter d are obtained. That is defined as the machining gap Mg and expressed in Eq. (3).

$$M_{\text{g}} = {{(D - d)} \mathord{\left/ {\vphantom {{(D - d)} 2}} \right. \kern-0pt} 2}$$
(3)

3 Experimental Design and Optimization

3.1 Experimental Design

During micro reciprocated wire-EDM, MRR is the main machining characteristic reflecting the processing efficiency. The machining accuracy of the machined micro tools has significant influence on the subsequent application, and it can be represented by machining gap. Thus, MRR and machining gap are mainly discussed in this paper. However, a challenge is caused because MRR and machining gap must be considered simultaneously. The mathematical model should be established in order to predict the two responses under any given input factors. To improve the predicting capability of the established models, the reasonable input factors have to be properly selected. Because the RC micro-energy pulse power is used in the study, the open voltage and discharge capacitance affect the single pulse discharge energy. Besides, the pulse duration determines the discharge time and the feed rate influences the discharge frequency. Therefore, the above four parameters are chosen as the input factors. Based on the literature survey [21], machine capability and many preliminary studies, the levels of the four input factors and other constant parameters are shown in Table 2.

Table 2 Processing conditions in micro wire-EDM process21
Full size table

Response surface methodology (RSM) was originally developed by Box and Wilson to explore the potentials of statistical design in industrial experiments [22]. The experimentation along with the regression analysis facilitates the development of the mathematical model between the desired response and the input factors. In RSM, the mathematical relationship of the output responses versus the input factors can be expressed by Eq. (4):

$$y_{\text{i}} = f(x_{1} ,x_{2} ,x_{3} , \ldots ,x_{\text{n}} ) \pm \varepsilon$$
(4)

where \(y_{\text{i}}\) is the desired response, f is the response function and \(x_{1}\), \(x_{2}\), \(x_{3}\), …, \(x_{\text{n}}\) are the input parameters, ε is the fitting error which measures the experimental errors. In generally, the quadratic model is proposed to present the approximation of the response function, which can be expressed as following:

$$y_{\text{i}} = a_{0} + \sum\nolimits_{i = 1}^{n} a_{i} x_{i} + \sum\nolimits_{i = 1}^{n} {a_{ii} x_{i}^{2} } + \sum {\sum\nolimits_{i < j}^{n} {a_{ij} x_{i} x_{j} \pm \varepsilon } }$$
(5)

where \(a_{0}\) is the constant term determined by the method of least squares,\(a_{\text{i}}\),\(a_{\text{ii}}\) and \(a_{\text{ij}}\) represent the coefficients of liner, quadratic and interactive terms, respectively. The response \(y_{\text{i}}\) may include linear, quadratic and interaction terms.

The central composited design (CCD) fits the quadratic responses very accurately. Besides, it is widely used in various RSM designs due to its advantages of high flexibility and efficiency in providing the overall experimental error in minimum tests. Therefore, it is employed to design the experiments in this study. The experimental design matrix and the responses are shown in Table 3.

Table 3 Experimental design matrix and relevant responses
Full size table

3.2 Optimization Technique

NSGA-II is one of the most popular multi-objective genetic algorithms because it has the advantages of high running speed and good convergence of the solution set. Thus, it is used to obtain the objectives of maximum MRR and minimum machining gap in this paper. The two responses are the functions of input factors and will be discussed in Sects. 4.1 and 4.2. NSGA-II is an improved multi-objective genetic algorithm based on NSGA, proposed by Dev et al. [23]. The procedure of multi-objective optimization using NSGA-II is shown in Fig. 3. Specifically, the initially parent population P of size N is randomly generated. Then, offspring Q of size N generated through the genetic operation including selection, crossover and mutation, which is created from parents by simulated binary crossover. The random mutation operator is used to create a solution from the entire search space. When the objective function of all individuals in R has been combined parent P and offspring Q, the solutions for the objective function are sorted into diverse non-dominated fronts. The better solutions are chosen based on the crowded comparison operator for better new population P. The above process is repeated until the required generation is reached. In this paper, the initial population size N with 100 and the maximum generation with 200 is used for better convergence and efficiency.

Fig. 3
figure 3

Flow chart of multi-objective optimization using NSGA-II

Full size image

4 Results and Discussions

4.1 Analysis of MRR for Machining SKD11

To analyze the machining process and optimize the machining parameters, a mathematical model between the output response and the input factors is established using RSM. In this paper, the statistical analysis software of Designer-Expert 8.0 is applied to compute the regression coefficients of the developed models. The analysis of variance (ANOVA) technique is employed to find out the factors and the interaction between various factors that have significant impact on the responses. The “P value” in ANOVA result is used to check the appropriateness and significance of the model by judging if it is less than 0.05 and 0.001, respectively. From the ANOVA result of MRR in Table 4, it can be seen that the “P value” for the model is < 0.0001, which indicates that the model is significant. Besides, the “P value” of the terms including A, B, C, D, A2, C2 are less than 0.05, revealing that they are statistically significant for MRR model. Other terms with the “P value” of larger than 0.05 are not significant and can be removed from the MRR model. The final simplified equation for MRR model is expressed in Eq. (6).

$$\begin{aligned} MRR & = 1 2. 3 8 0 2+ 0. 2 4 5 6\times U + 0.000 3\times C + 0.16278 \times T_{\text{on}} \\ & \quad + 0. 0 5 8 5 8 7\times F - 0.000902 \times U^{ 2} - 0.006571 \times T_{\text{on}}^{ 2} \\ \end{aligned}$$
(6)
Table 4 ANOVA results of MRR
Full size table

The lack of fit is a measure to evaluate whether the model is adequate or not. The model is inadequate if there is a significant lack of fit [24]. The “P value” of lack of fit is 0.3175, implying that the lack of fit is insignificant and the MRR model is advisable. The “Predicted R-square” of 0.9043 is in reasonable agreement with the “Adjust R-square” of 0.9603. All the data show that the MRR model provides good accuracy. The normal probability plot of the residuals for MRR is shown in Fig. 4. It can be observed that the residuals generally fall on a straight line, which implies that the errors are normally distributed.

Fig. 4
figure 4

Normal plot of residuals for MRR

Full size image

The perturbation plot in Fig. 5 shows the comparative effects of open voltage (A), discharge capacitance (B), pulse duration (C) and feed rate (D) on MRR. By default, Design Expert 8.0 software sets the reference point at the midpoint (coded value 0) of all factors. It can be seen from Fig. 5 that MRR increases rapidly with open voltage increase in the initial stage, but it tends to be stable when open voltage reaches a certain level. The reason is that MRR depends on the discharge energy which is mainly determined by open voltage. The high value of open voltage will lead to much material to be removed. However, when the open voltage increases to a certain value, the single discharge energy is so large that much material are removed and the generated debris particles are accumulated in the discharge gap, thus deteriorating the discharge state and decreasing MRR. The rise in discharge capacitance causes the increase of MRR. The reason for the phenomenon is in the same way as that of open voltage, but MRR is less sensitive to discharge capacitance than open voltage. As for the plot of pulse duration and MRR, the large value of pulse duration results in high MRR. It can be explained that the pulse duration increases the discharge time, which causes a large amount of discharge energy. The high feed rate generates large MRR. The reason is that the high feed rate increases the discharge frequency, thus increasing the MRR.

Fig. 5
figure 5

Perturbation plot of MRR

Full size image

4.2 Analysis of Machining Gap for Machining SKD11

ANOVA is performed using Designer-Expert 8.0 software to statistically analyze the results of machining gap model, individual model terms and lack of fit. The ANOVA result of the machining gap for machining SKD11 is shown in Table 5. It can be observed that the “F-value” of machining gap model is 96.94, which indicates that there is only a 0.01% chance that a “Model F-value” this large could occur due to noise. The model term is significant if it’s “P value” is less than 0.05. In this case, the terms A, B, C, D, BC and A2 are significant for machining gap model. The “P value” of “Lack of fit” is 0.1670, implying that the lack of fit is insignificant. Non-significant lack of fit is good because it enables the model to match the experimental data. The determination coefficient “R-squared” is 0.9620, which indicates that the quadratic model can explain 96.20% variation in the machining gap. The “Pred R-Squared” of 0.9444 is in reasonable agreement with the “Adj R-Squared” of 0.9520. The “Adeq Precision” measures the signal to noise. The value of 42.111 for “Adeq Precision” indicates an adequate signal because a ratio larger than 4 is desirable. Therefore, the quadratic model of machining gap can be used to navigate the design space. The developed quadratic model for the machining gap is expressed in Eq. (7).

$$\begin{aligned} M_{\text{g}} & = 1.9456{ + 0} . 0 3 3 2\times U + 0.000 2 3\times C{ + 0} . 0 4 9 7 3\times T_{\text{on}} \\ & \quad - 0. 0 1 6 6\times F - 0.0000558 \times C \times T_{on} - 0.0000838 \times U^{2} \\ \end{aligned}$$
(7)
Table 5 ANOVA of machining gap for machining SKD11
Full size table

From Eq. (7), it is found that there is interaction between capacitance and pulse width on machining gap. The variation of machining gap under the interaction of the two factors can be obtained by Design Expert software, and the result is shown in Fig. 6. It presents that when other conditions remain unchanged, the machining gap increases with the increase of pulse width and capacitance. The reason is that the discharge energy is proportional to capacitance, and the pulse width determines the discharge time. Thus, the discharge energy will be increased when using large capacitance and pulse width, leading to the severe vibration of micro wire.

Fig. 6
figure 6

Interaction of pulse width and capacitance on machining gap under 100 V open voltage and 32.9 μm/min feed rate

Full size image

Figure 7 presents the normal probability plot of the residuals for machining gap, and it is observed that the residuals are lying on a straight line, which means that the errors are normally distributed. Figure 7 shows the perturbation plot of machining gap versus open voltage (A), discharge capacitance (B), pulse duration (C) and feed rate (D). The large slope for open voltage and feed rate indicate that the machining gap is highly sensitive to the two factors while the relative flat lines for discharge capacitance and pulse duration means that the machining gap is less sensitive to these factors. It can be observed from Fig. 8 that the machining gap increases with the rise in open voltage. On one hand, the single discharge energy increases when using high open voltage, and that causes large discharge force and electromagnetic force, thus resulting in fierce wire vibration. On the other hand, the breakdown gap between micro wire and workpiece is proportional to open voltage [25, 26], thus generating larger machining gap. The above two factors result in this phenomenon. The effects of discharge capacitance and pulse duration on machining gap are in the same way as open voltage. The reason for the phenomenon is the same as the first one of open voltage. But the machining gap is less sensitive to discharge capacitance and pulse duration compared with open voltage. As for the effect of feed rate on machining gap, the increase in feed rate leads to the decrease of machining gap. The reason is that when using low feed rate, the discharge frequency is so low that the single pulse discharge force is large and exerts much influence on the micro wire vibration. With the increase of feed rate, the discharge frequency increases and the single pulse discharge force decreases, thus reducing the vibration amplitude of the micro wire.

Fig. 7
figure 7

Normal plot of residuals for machining gap

Full size image
Fig. 8
figure 8

Perturbation plot of machining gap

Full size image

4.3 Comparison of Experimental and Predicted Responses

Confirmation experiments are performed for 30 experiments to validate the quadratic models of MRR and machining gap expressed in Eqs. (6) and (7), respectively. The experimental conditions are the same as that in Table 3. Every experiment is performed three times and the average value is calculated as the final result of each run. The comparison between the experimental and predicted results for MRR and machining gap is shown in Fig. 9a, b. It is noted that the experimental MRR and machining gap are following the predicted values calculated from the developed models. The relative deviation between the experimental and predicted results for MRR and machining gap are respectively figured out. It is found that the maximum deviation rate of MRR is 8.38%, and that of machining gap is 6.70%. According to the above results, it can be concluded that the developed models of MRR and machining gap are reliable, and they can be used for the advance prediction for the purpose of high efficiency and accuracy.

Fig. 9
figure 9

a Experimental and predicted results for MRR and b Experimental and predicted results for machining gap

Full size image

4.4 Multi-objective Optimization Based on NSGA-II

In order to obtain the best fitness values of the objective function, NSGA-II is implemented by the codes written in MATLAB R2015b. The codes have been implemented using eight different functions to conduct the steps in NSGA-II. For better convergence, the population size N with 100 and the generation number with 200 are entered as the input augments to the functions. A set of 100 non-dominated solutions are obtained finally. Figure 10 shows the pareto-optimal front i.e. the final set of solutions. Because the performance studied in this paper are conflicting in nature, the machining accuracy decreases with the increase of MRR. The same performance characteristic can be clearly observed in the obtained solutions presented in Fig. 10. The partial objective fitness values and their decision variables of non-dominated solution are enlisted in Table 6. Since none of the solutions in the pareto-optimal set is completely superior to others, any one of them can be acceptable. The selection of one solution depends on the working requirement. The solutions in Region-A will be selected if the machining accuracy is more desirable than MRR. Accordingly, MRR is compromised due to the conflicting nature of the two responses. The solutions in Region-C are suitable if MRR is more desirable than machining gap. But the machining accuracy will be compromised at the same time. The solutions in Region-B should be considered if the machining accuracy and MRR are treated equally.

Fig. 10
figure 10

Pareto-optimal front between MRR and machining gap

Full size image
Table 6 Pareto-optimal solutions (only 30 solutions are given)
Full size table

4.5 Application of Optimal Solutions

A micro gear with the tip diameter of 1200 μm and the module of 10 is chosen as the machining sample. To keep the balance between high MRR and good machining accuracy, the solution labelled No. 14 is selected to fabricate the experimental sample. The actual machining gap is 3.83 μm obtained under No. 14 processing condition, and whilst it is compensated in the tool path planned by UG NX 8.0 software. The tool paths without and with compensating machining gap are presented in Fig. 11a, b, and the relevant machined samples are shown in Fig. 12a, b, respectively. The tip diameters of the two samples are measured by VHX-6000 on five pair of teeth, and the average values are 1186.64 μm and 1194.38 μm, respectively. It can be noted that the deviation of the two samples is 7.74 μm, which is nearly the same as the double machining gap under No. 14 solution.

Fig. 11
figure 11

Tool path of the micro gear: a without compensating machining gap and b with compensating machining gap

Full size image
Fig. 12
figure 12

Machined micro gear: a without compensating machining gap and b with compensating machining gap

Full size image

To understand the wear of the repeatedly used micro wire, the new micro wire before machining and the used one continuously working about 17 h to fabricate micro gear are inspected by scanning electron microscopy and microscope system VHX-6000, respectively. Figure 13a shows the smooth surface of new micro wire. There are some micro stripes on the surface along the axial direction of new micro wire as shown in Fig. 13b, which are formed in the drawing process of wire electrode. From Fig. 14a, it can be seen that there is no large wear on the used micro wire due to using micro-energy power supply in micro-WEDM. However, there are many eroded pits on the surface of the micro wire after machining micro gear as shown in Fig. 14b. Nevertheless, the diameters of the micro pits are within 2 μm, and some are even in submicron. The diameters of the micro wire before and after machining are measured by microscope system VHX-6000 with 29.6 μm and 29.1 μm, respectively. The above reveal that the micro wire wear is negligible and it has little effect on machining accuracy.

Fig. 13
figure 13

SEM images of micro wire before machining: a overall view and b enlarged view

Full size image
Fig. 14
figure 14

SEM images of micro wire after machining: a overall view and b enlarged view

Full size image

5 Conclusions

In this study, the MRR and machining gap during machining SKD11 by micro reciprocated wire-EDM are investigated under different machining parameters including open voltage, discharge capacitance, pulse duration and feed rate. What’s more, the optimal solutions are obtained based on NSGA-II, and finally a typical experimental sample is successfully machined with high machining accuracy. Detailed conclusions can be drawn below:

  1. 1.

    MRR initially increases with the increase of open voltage, and then tends to be constant when open voltage reaches a certain level. The reason is that high open voltage generates large discharge energy. But, overlarge one will lead to much material to be removed and debris particles accumulate in the machining gap, deteriorating the discharge state. The rise in discharge capacitance, pulse duration and feed rate cause high MRR. For machining gap, it increases with the increase of open voltage, discharge capacitance and pulse duration. That is because large discharge force and electromagnetic force are caused by high levels of the factors, resulting in fierce wire vibration. As for feed rate, the large one leads to the decrease of machining gap.

  2. 2.

    The mathematical models of MRR and machining gap are established by ANOVA, which are used to predict and optimize the output responses. Meanwhile, the verification experiments are conducted in the all 30 experiments and the deviation rate of MRR and machining gap are within 8.38% and 6.70%, respectively. Thus, it is concluded that the developed models are reliable. Subsequently, the multi-objective optimization based on NSGA-II is carried out using the established functions and the pareto-optimal front is obtained.

  3. 3.

    To meet different machining performance, the optimal solutions are divided into three regions i.e. Region-A, B and C for small machining gap, compromised machining gap and MRR, and high MRR, respectively. One of the optimal machining conditions labelled No. 14 is used to machine the experimental sample i.e. micro gear. The unilateral machining gap with 3.83 μm is compensated in tool path and the micro gear with high machining accuracy of 1194.38 μm tip diameter is obtained.