## 3 Numerical Simulations
### Influence of Excitation Amplitude on Vibration Characteristics  
In order to study the influence of excitation amplitude on piezoelectric energy harvesters, under the condition of \(f=11\) Hz, the influence of different excitation amplitudes on the response produced by IBPEH and BPEH is compared and analyzed. Figure 11 is a bifurcation diagram with excitation amplitude as a parameter. Figure 12 is the RMS voltage diagram. It can be seen from Figure 11 that BPEH and IBPEH can always achieve a large-amplitude interwell motion state with the increase in excitation amplitude. When the excitation amplitude is in the range of 4-9.7 m/s\({}^{2}\), it is difficult for the BPEH system to obtain enough energy to eliminate the limitation of the potential well and to oscillate within the well. However, IBPEH presents a state of chaotic motion. Chaotic motion is an irregular motion that is composed of small-amplitude intrawell and large-amplitude interwell. Although it is difficult for IBPEH to maintain high-energy yield, the RMS voltage produced by IBPEH is greater than that of BPEH. Increasing the excitation amplitude to more than 9.7 m/s\({}^{2}\), the BPEH maintains a small-amplitude intrawell motion, and the IBPEH system has enough energy to cross the bistable potential barrier to achieve a large-amplitude periodic oscillation; therefore, the piezoelectric energy harvester can continuously produce high energy. Continuing to increase the excitation amplitude, when \(A=12.2\) m/s\({}^{2}\), BPEH can end the small-amplitude intrawell motion state, and the transition realizes the large-amplitude interwell periodic motion state. After that, with the increase in excitation amplitude, the piezoelectric energy harvester will continuously output high energy. Within the analyzed excitation amplitude range, the excitation level threshold for the BPEH system to enter a large-amplitude interwell periodical motion is \(A=12.2\) m/s\({}^{2}\), and when \(A=9.7\) m/s\({}^{2}\), IBPEH can continuously output high energy. Compared with BPEH, the excitation amplitude threshold of IBPEH to achieve a large-amplitude periodic motion state is reduced by 2.5 m/s\({}^{2}\). Figure 10: Simulation results of time-domain velocity versus the displacement and voltage–time diagram when \(A=14\) m/s\({}^{2}\) and \(f=12\) Hz. (**a**) Simulation results of time-domain velocity versus the displacement; (**b**) voltage–time diagram. To further explore the influence of excitation amplitude on the response of BPEH and IBPEH, and keep \(f~{}=~{}11\) Hz unchanged, based on the above research and analysis, within the range of excitation amplitudes studied, we take excitation amplitudes \(A~{}=~{}\frac{8\text{m}}{\text{s}^{2}}\), \(A~{}=~{}\frac{11\text{m}}{\text{s}^{2}}\), and \(A~{}=~{}14\) m/s\({}^{2}\). When \(f~{}=~{}11\) Hz and \(A~{}=~{}8\) m/s\({}^{2}\), Figure 13a,b shows the simulation results of time-domain velocity versus the displacement and the voltage-time diagram, respectively. It can be seen from Figure 13a and the bifurcation diagram in Figure 11 that IBPEH is a chaotic motion state under the action of this low excitation level, while BPEH is a small-amplitude intrawell oscillation state, and the response displacement and vibration velocity are much smaller than IBPEH. It can be seen from Figure 13b that the RMS voltage produced by IBPEH is 13.18 V, while that of BPEH is only 2.14 V.  
When \(f=11\) Hz, increase the excitation amplitude to \(A=11\) m/s\({}^{2}\). Figure 14a,b shows the simulation results of time-domain velocity versus the displacement and the voltage-time diagram, respectively. It can be seen from Figure 14a that the BPEH is still in a small-amplitude intrawell motion state, while IBPEH has achieved a large-amplitude interwell periodical motion state, and the response displacement and velocity have been greatly improved. Combined with Figure 14b, IBPEH can continuously produce voltage with high energy, and the RMS voltage produced is 8.07 times that of BPEH.