## 3 Experiments
### Experimental Results and Grey Relational Analysis
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Grey Relational Coefficients**} & \multicolumn{3}{c}{**Grey Relational Grades**} \\ \cline{2-6}  & **SR** & **RA** & **ACD** & **Average Value** & **Order** \\ \hline
1 & 0.7456 & 0.9630 & 0.6245 & 0.7777 & 4 \\
2 & 0.4980 & 0.9109 & 0.4220 & 0.6103 & 11 \\
3 & 0.9196 & 0.9055 & 0.3826 & 0.7359 & 6 \\
4 & 0.7777 & 0.8460 & 0.3661 & 0.6633 & 9 \\
5 & 0.7500 & 0.7198 & 0.8386 & 0.6178 & 10 \\
6 & 0.9473 & 1.0000 & 0.7770 & 0.9081 & 1 \\
7 & 0.6738 & 0.6884 & 0.3593 & 0.5738 & 12 \\
8 & 0.9619 & 0.9096 & 0.4157 & 0.7624 & 5 \\
9 & 0.7000 & 0.4383 & 0.3488 & 0.4957 & 13 \\
10 & 0.6428 & 0.3333 & 0.3415 & 0.4392 & 15 \\
11 & 0.7283 & 0.9827 & 1.0000 & 0.9037 & 2 \\
12 & 1.0000 & 0.9684 & 0.5715 & 0.8466 & 3 \\
13 & 0.6428 & 0.4140 & 0.3333 & 0.4634 & 14 \\
14 & 0.3333 & 0.4183 & 0.3738 & 0.3751 & 16 \\
15 & 0.7000 & 0.8483 & 0.4873 & 0.6785 & 8 \\
16 & 0.7683 & 0.8643 & 0.4533 & 0.6953 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of GRC and GRG. From Figure 10, it was found that parameter combination A\({}_{2}\)B\({}_{4}\)C\({}_{1}\)D\({}_{4}\) was the best based on the analysis of the mean GRG. Compared with the other three parameters, the duty cycle had the largest range of mean GRG, which showed that the pulse width was the key factor and had the highest relative importance to SR, RA, and ACD in the VPECM of TiAl TNM. By using the optimized parameter combination, namely an applied voltage of 25 V, an electrode feed rate of 0.35 mm/min, a duty cycle of 1/12, and a tool vibration frequency 40 Hz, the SR, RA, and ACD of the machined specimen were Ra 0.35 \(\mu\)m, 0.0175 mm, and 173 A/cm\({}^{2}\), respectively. To validate the results of the grey relational analysis, single-factor experiments were performed. The grey relational analysis above shows that the duty cycle has a relatively large influence on the VPECM results and the vibration frequency has a small influence on the machining results. Therefore, single-factor experiments with these two parameters as variables were carried out to verify the above-mentioned grey correlation analysis results. The basic parameters of the single-factor test are shown in Table 9. The parameters of the single-factor experiment and their values are shown in Table 10.  
Consistent with the evaluation indicators of the orthogonal experiments, the surface roughness (SR), replication accuracy (RA), and average current density in pulse width (ACD) were selected to analyze the processing results of the single-factor experiments. The results are shown in Figure 11. It can be seen from Figure 11 that with the change in duty cycle, SR, RA, and ACD will also change significantly, while vibration frequency has little influence on SR, RA, and ACD. With the increase in the duty cycle, the average current density in a single pulse width (ACD) increases, the surface roughness (SR) decreases, the replication accuracy (RA) improves, and the surface machining quality of the workpiece is improved. However, as the vibration frequency increases, the average current density in a \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{4}{c}{**Parameter Values**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline Duty cycle & 1/12 & 1/6 & 1/4 & 1/3 \\ Tool vibration frequency (Hz) & 10 & 20 & 30 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Parameter values of single-factor experiments. Figure 10: Mean GRGs of parameters. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Parameter Type**} & \multicolumn{1}{c}{**Processing Parameters**} \\ \hline Applied voltage (V) & 20 \\ Feed rate (mm/min) & 0.2 \\ Duty cycle & 1/4 \\ Tool vibration frequency (Hz) & 10 \\ Electrolyte temperature & 30 \({}^{\circ}\)C \\ Inlet pressure & 0.8 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Basic parameters of the single-factor experiments.