## 3 Numerical Simulations
### Influence of Excitation Frequency on Vibration Characteristics  
In order to study the output performance of IBPEH and BPEH, we set \(d=18\) mm, unchanged, and take \(s=2\) mm and \(dg=8.5\) mm. Under different excitation frequencies and excitation amplitudes, the Runge-Kutta algorithm is used to numerically solve the dynamic equations, and the vibration characteristics of IBPEH and BPEH are analyzed and compared. In order to compare and analyze the influence of different excitation frequencies on the response characteristics of BPEH and IBPEH, we set \(A=14\) m/s\({}^{2}\), unchanged, and study the system performance through numerical simulation. Figure 6 shows bifurcation diagrams of BPEH and IBPEH systems with excitation frequency as the parameter. It can be seen from Figure 6 that the change in excitation frequency causes the piezoelectric energy harvester to exhibit diverse motion states. It can be seen from Figure 6a that BPEH experiences small-amplitude intrawell, large-amplitude interwell, chaos, and small-amplitude intrawell various motion states. It can be seen from Figure 6b that IBPEH undergoes complex motion states such as chaos, large-amplitude interwell, chaos, and small-amplitude intrawell oscillation in sequence. To further explore the response output of the piezoelectric energy harvester, the root mean square (RMS) voltage is used for comparison, and the calculation formula is as follows:  
\[U_{RMS}=\sqrt{\frac{\sum_{i=1}^{n}U_{i}^{2}}{n}} \tag{17}\]  
Figure 5: Potential energy curve of the system. Figure 6: Bifurcation diagram with excitation frequency as parameter: (**a**) BPEH; (**b**) IBPEH. In the formula, \(n\) is the total number of sampling points, and \(U_{i}\) represents the real-time voltage of the \(i\)-th sampling point. Sampling in the range of excitation frequency under study, the RMS voltage diagram with excitation frequency as a parameter is obtained in Figure 7.  
It can be seen from Figure 6 that when the excitation frequency is in the range of 2-6.3 Hz, the BPEH system performs small-amplitude intrawell oscillation, and the IBPEH system performs chaotic motion. Combined with Figure 7, it can be seen that the RMS voltage produced by IBPEH is larger than that of BPEH, which is caused by the existence of some large-amplitude interwell and small-amplitude intrawell motions in chaotic motion. When the excitation frequency is greater than 6.3 Hz, IBPEH can adapt to this excitation frequency, and the system changes from chaotic motion to large-amplitude interwell periodic motion. Combined with Figure 7b, it can be seen that the RMS voltage produced by IBPEH is improved, while BPEH still maintains small-amplitude intrawell oscillation. When the excitation frequency continues to increase to \(f=10.2\) Hz, BPEH transitions from small-amplitude intrawell motion to large interwell periodic motion state and the RMS voltage increases significantly, while the IBPEH system continues to maintain large-amplitude interwell periodic oscillation. The operating frequency bandwidth of BPEH is 3.2 Hz in the state of large-amplitude interwell periodic motion, and the operating frequency bandwidth of IBPEH is 6.1 Hz. Continuing to increase the excitation frequency, after the large-amplitude interwell periodic motion of the system ends, the BPEH and IBPEH will decay from the large-amplitude interwell periodic motion state to the chaotic motion state. Because the system is difficult to adapt to the higher excitation frequency, small-amplitude intrawell motion is finally performed. The above analysis shows that before the system realizes large-amplitude motion, the IBPEH behaves in a state of chaotic motion. In this state, the response output of the energy harvester is between the motion state of small-amplitude intrawell and large-amplitude interwell. Compared with BPEH, IBPEH can achieve high-energy yield at a lower excitation frequency, and the operating frequency bandwidth is broadened by 93.55%.