## 3 Simulation Analysis of CTEHS
### Analysis of Potential Energy
#### 3.3.2 Influence of Initial Vibration Point \(u_{0}\) on System Response Characteristics  
As can be seen from Figure 7, when the gravity of the end magnet is considered, the left potential well of the tri-stable system incrementally falls and the right potential well incrementally rises, and the potential energy curve exhibits asymmetric characteristics.  
Next, under the consideration of the gravity of the magnet, the influence of initial vibration points on the system's output characteristics is studied. Take the vibration frequency \(f=10\) Hz, \(d_{\text{h}}=12\) mm, \(d_{\text{v}}=16\) mm, \(M_{\text{e}}=10\) g, the initial velocity of the system is 0, and the initial vibration point \(u_{0}\) is \(-14\), \(-1\), and \(12\) mm, respectively (that is, the lowest point of the three potential wells). Figure 10 shows the change law of the system's RMS voltage with the vibration acceleration \(A\) under different \(u_{0}\). From Figure 10, it can be found that the excitation acceleration threshold for the system to produce high-energy output is different when the initial vibration point \(u_{0}\) of the system is different. When \(u_{0}\) is \(-14\) mm, the acceleration threshold for the system to produce high-energy output is \(12.8\) m/s\({}^{2}\). When \(u_{0}\) is \(-1\) and \(12\) mm, the acceleration threshold of the system is reduced to \(8.2\) m/s\({}^{2}\) and \(8\) m/s\({}^{2}\), respectively. In addition, it can be found that the output voltage when the system realizes a large response is consistent regardless of the stable balance point from which the oscillation starts.  
To further study the effect of the initial vibration point \(u_{0}\) on the response characteristics of the system, take \(A=9\) m/s\({}^{2}\), \(f=10\) Hz, \(M_{\text{e}}=10\) g, and the initial vibration point \(u_{0}\) is \(-14\) mm and \(12\) mm, respectively. The 3D time domain simulation results of the system are shown in Figure 11a,b. When \(u_{0}\) is \(-14\) mm, as shown in Figure 11a, the system is situated in a large potential trough and cannot escape the bondage of the potential trough. Therefore, the system can only perform intra-well oscillation, and the response velocity and displacement at this time are relatively small. When \(u_{0}\) is \(12\) mm, as shown in Figure 11b, the system is situated in a small potential trough, which can readily escape the confinement of the potential trough, thereby realizing a large oscillation between the wells. At the moment, the system's vibration velocity and displacement are relatively large.  
Keeping other initial parameters unchanged, improve the excitation acceleration \(A\) to \(13\) m/s\({}^{2}\). As shown in Figure 11c, when \(u_{0}\) is \(-14\) mm, the excitation acceleration at the moment reaches the acceleration threshold of the system's large response. Therefore, the system realizes the large periodic oscillation of the tri-stable state, and the response velocity and displacement both show large amplitude. As shown in Figure 11d, when \(u_{0}\) is \(12\) mm, the system also performs periodic oscillation between wells under the excitation acceleration at this time. In addition, the comparison between Figure 11b,d can again show that increasing the initial vibration acceleration can improve the system's vibration velocity and displacement with the same other initial parameters.  
Figure 10: Excitation acceleration threshold of the system under different \(u_{0}\).