## 2 Principle and Multi-Field Simulation of VPECM  
The principle of VPECM is shown in Figure 1, comprising a cathode tool connected to the negative pole of a power supply and an anode workpiece connected to the positive pole. The cathode tool feeds toward the workpiece and creates a reciprocating small-amplitude vibration. When the cathode tool moves close to the workpiece, the power is turned on for processing. When the cathode tool moves away from the workpiece, the power is turned off for electrolyte scouring. During the pulse interval, the increased machining gap can increase the electrolyte flow rate and improve its ability to carry away electrolytic products, i.e., bubbles, metal ions, and insoluble products, etc. With the full removal of electrolytic products within the pulse interval, VPECM can process stably at a smaller machining gap and achieve higher machining accuracy than the DC and pulsed ECM methods. In order to reveal the influence of machining parameters on the distribution of machined products in the machining gap, a multi-field simulation of VPECM was carried out using COMSOL Multiphysics. The two-dimensional multi-field simulation model is shown in Figure 2. In the simulation analysis, the electric field distribution in the machining gap satisfied the Laplace equation:  
\[\
abla^{2}\varphi=0 \tag{1}\]Taking into account the influence of hydrogen bubbles and heat, the conductivity of the electrolyte (\(\kappa\)) is defined as follows:  
\[\kappa=\kappa_{0}(1+\varepsilon(T-T_{0}))(1-\beta)^{n} \tag{2}\]  
where \(\kappa_{0}\) is the initial conductivity of the electrolyte, \(\beta\) is the bubble rate contained in the electrolyte, \(T\) is the temperature of the electrolyte, \(T_{0}\) is the initial temperature of the electrolyte, \(n\) is the influence coefficient of the bubble rate on the conductivity, and \(\varepsilon\) is the temperature-dependent gradient [21; 22]. Considering the processing characteristics of VPECM, the parameters of peak voltage, feed rate, vibration frequency, and duty cycle were selected as the analysis objects, and multi-field simulations with different parameter combinations were carried out. The selected parameter combinations of the multi-field simulations are shown in Table 1, and other general parameters are shown in Table 2. Nephograms of the bubble rate and temperature distribution when the cathode tool moves to the minimum inter-electrode gap are shown in Figure 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **1** & **2** & **3** & **4** \\ \hline Applied peak voltage (V) & 20 & 30 & 20 & 20 \\ Feed rate (mm/min) & 0.25 & 0.25 & 0.35 & 0.25 \\ Duty cycle & \(1/6\) & \(1/6\) & \(1/6\) & \(1/3\) \\ Vibration frequency (Hz) & 20 & 20 & 20 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected parameter combinations of the multi-field simulation. Figure 1: Principle of pulse vibration ECM. Figure 2: Two-dimensional multi-field simulation model. In parameter combination 1, the peak voltage is 20 V, the feed rate is 0.25 mm/min, the duty cycle is 1/6, and the vibration frequency is 20 Hz. As shown in Figure 2(a), when the machining gap is the smallest, the maximum local bubble rate is 12.6%, and the maximum temperature rise is 8.6 K. The average bubble rate and temperature of the section perpendicular to the flow in the machining gap changing with time are shown in Figure 3(a), respectively. The maximum average bubble rate is 2.3%, and the maximum average temperature of the section is 300.4 K.  
In parameter combination 2, as the voltage is set to 30 V, the product is increased slightly compared to that at 20 V. The maximum local bubble rate is 15.1%, and the maximum temperature rise is 14.0 K. As shown in Figure 3(c), the maximum average bubble rate is 2.7%, and the maximum average temperature of the section is 304.3 K. The increase in product accumulation can lead to a decrease in machining accuracy. In addition, due to the increase in voltage, the machining gap has increased, which is not conducive to accuracy control.