## 3 Numerical Simulations  
### Potential Energy Analysis of the System  
In order to analyze the influence of system parameters on the vibration characteristics of BPEH and IBPEH systems, the potential energy characteristics of the system are first analyzed, and the structural parameters of the BPEH and IBPEH are shown in Table 1.  
The depth of the system potential well has a certain influence on the output of the piezoelectric energy harvester. The potential well of the potential energy of the BPEH system is deep at this time, and a large excitation is required to realize the motion between the wells. In this paper, based on the control variable method, by adjusting the positions of magnets C and D, the distance between the BPEH and IBPEH wells are kept the same, and the height of the potential barrier is reduced.  
Figure 4: Lumped parameter model of IBPEH.  
The total potential energy of the BPEH system is \(U_{1}=U_{m1}+U_{r}\). We selected \(d=18\) mm and set the highest point of the barrier \((y_{1},U_{11})\) and the lowest point of the potential well \((y_{2},U_{12})\). From the potential energy curve of the BPEH system, we can obtain  
\[\left\{\begin{array}{l}{U_{11}}^{\prime}(y_{1})=F_{m1}(y_{1})+F_{r}=0\\ {U_{12}}^{\prime}(y_{2})=F_{m1}(y_{2})+F_{r}=0\end{array}\right. \tag{14}\]  
Since the potential energy curve of the system is symmetric about the \(U\) axis, the values of \(y_{1}\) and the lowest point \(y_{2}\) of the potential well on either side are substituted into the following equation:  
\[\left\{\begin{array}{l}{U_{21}}^{\prime}(y_{1})=F_{m}(y_{1},d_{g},s)+F_{r}=0 \\ {U_{22}}^{\prime}(y_{2})=F_{m}(y_{2},d_{g},s)+F_{r}=0\end{array}\right. \tag{15}\]  
When the position of magnet C and magnet D is greater than a certain distance from magnet A, magnet A is not affected by the magnetic force of magnet C and magnet D. At this time, the potential energy curves of IBPEH and BPEH systems are consistent, and the equations have countless solutions. Therefore, within a certain range, the initial value is based on the fact that the distances between magnets C and D and magnet B are zero, respectively. We take the values near \(s=0\) mm and \(d_{g}=4\) mm as the initial values, use the Newton-Raphson iteration method to solve the nonlinear equation system through MATLAB, and obtain the distance value when the system potential well is the shallowest through verification:  
\[\left\{\begin{array}{l}s=2\;\text{mm}\\ d_{g}=8.5\;\text{mm}\end{array}\right. \tag{16}\]  
Figure 5 shows the potential energy curve of the system when \(d=18\) mm, \(s=2\) mm, and \(d_{g}=8.5\) mm. It can be seen from Figure 5 that the potential well depth of the BPEH system is 0.23 mJ, while that of IBPEH system is 0.12 mJ.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Substrate layer & & & \\ Density & \(\rho_{s}\) & 8300 & kg/m\({}^{3}\) \\ Arc-shaped radius & \(R_{s}\) & \(10\times 10^{-3}\) & m \\ Length & \(L\) & \(40\times 10^{-3}\) & m \\ Height & \(h_{s}\) & \(2\times 10^{-4}\) & m \\ Width & \(b_{s}\) & \(8\times 10^{-3}\) & m \\ PVDF & & & \\ Density & \(\rho_{p}\) & \(1780\) & kg/m\({}^{3}\) \\ Length & \(L_{p}\) & \(40\times 10^{-3}\) & m \\ Height & \(h_{p}\) & \(1.1\times 10^{-4}\) & m \\ Width & \(b_{p}\) & \(8\times 10^{-3}\) & m \\ Magnet & & & \\ Mass & \(m_{0}\) & \(3.8\times 10^{-3}\) & kg \\ Length & \(l_{a},l_{b},l_{c},l_{d}\) & \(4\times 10^{-3}\) & m \\ Height & \(h_{a},h_{b},h_{c},h_{d}\) & \(10\times 10^{-3}\) & m \\ Width & \(w_{a},w_{b},w_{c},w_{d}\) & \(10\times 10^{-3}\) & m \\ Magnetization strength & \(M_{a},M_{b},M_{c},M_{d}\) & \(5.5\times 10^{6}\) & A/m \\ Vacuum permeability & \(\mu_{0}\) & \(4\pi\times 10^{-7}\) & H/m \\ \hline \end{tabular}
\end{table}
Table 1: Structural parameters of BPEH and IBPEH.