## 2 Principle and Multi-Field Simulation of VPECM
Figure 4: _Cont._  
Figure 3: Nephograms of the bubble rate and temperature of four parameter combinations. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Simulation Parameters** \\ \hline Vibration amplitude (mm) & 0.3 \\ Electrolyte temperature (\({}^{\circ}\)C) & 30 \\ Electrolyte & 200 g/L NaNO\({}_{3}\) \\ Inlet pressure (MPa) & 0.8 \\ Outlet pressure (MPa) & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Other general parameters of the multi-field simulation. In parameter combination 3, the feed rate is increased to 0.35 mm/min, and the machining gap is reduced accordingly, which helps to improve the ECM replication accuracy. Moreover, the surface quality will be improved due to the increased current density. However, this also results in an increase in product. As shown in Figure 3e,\(f\), the maximum local bubble rate is 17.0%, and the maximum temperature rise is 11.9 K. As shown in Figure 4e,\(f\), the maximum average bubble rate is 2.7%, and the maximum average temperature of the section is 303.3 K. These results indicate a disadvantage for processing stability. In parameter combination 4, the duty cycle and vibration frequency are changed to 1/3 and 40 Hz at the same time. The machining gap is increased due to the increased duty cycle, while the electrolyte flushing is enhanced due to the increased vibration frequency, so the product accumulation is reduced. The maximum local bubble rate is 7.6%, and the maximum temperature rise is 4.9 K. As shown in Figure 4g,h, the maximum average bubble rate is 1.4%, and the maximum average temperature of the section is 296.8 K.  
In general, the above simulation results reveal the following phenomena. By comparing parameter combinations 1 and 2, it can be found that as the voltage increases from 20 V to 30 V, the bubble rate increases to 1.20 times, and the temperature rise increases to 1.63 times. Therefore, adjusting the voltage parameters will have a certain influence on the temperature and the bubble rate, and the influence on the temperature rise will be greater than that of the bubble rate. By comparing parameter combinations 1 and 3, it can be found that as the feed rate increases from 0.25 mm/min to 0.35 mm/min, the bubble rate increases to 1.35 times, and the temperature rise increases to 1.38 times. Therefore,  
Figure 4: Graph of bubble rate and temperature in machining gap versus time for four parameter combinations. adjusting the feed rate will have an approximate effect on the temperature and the bubble rate. Considering that both the bubble rate and temperature rise will have an effect on the machining results, the influence of voltage and feed speed on the machining results is comparable. By comparing parameter combinations 1 and 4, it can be found that by adjusting the duty cycle and frequency, the bubble rate in parameter combination 4 decreases to 60% of that in parameter combination 1, while the temperature rise in parameter combination 4 decreases to 57% of that in parameter combination 1. Therefore, the effect of duty cycle and frequency on the machining process is also critical. If the pulse width is increased at the same frequency, the situation will effectively be the same as increasing the peak voltage. Thus, this condition is not analyzed specifically. It can be seen that the feed rate, pulse voltage, duty cycle, and tool vibration frequency have different effects on the accumulation of products in the machining gap, which will influence the machining accuracy and surface quality [23,24]. Therefore, to further analyze the influence of the four parameters of VPECM, orthogonal experiments with the factors of peak voltage, feed rate, duty cycle, and tool vibration frequency were designed and carried out.