## 3 Simulation Analysis of CTEHS
### Analysis of Potential Energy  
From Equation (5), the system's potential energy includes the magnetism potential energy and gravitational potential energy generated by magnets, as well as the resilient potential energy of the combined beam, the expression of which is  
\[U=U_{r}+U_{m}+U_{g} \tag{20}\]  
To study the effect of the end magnet's gravity on the total potential energy of the tristable system, take \(d_{\rm h}=11\) mm, \(d_{\rm v}=16\) mm, \(M_{\rm e}=0\)-\(20\) g, according to Equation (20), the 3D potential energy surface of the system with the change of end magnet mass \(M_{\rm e}\) can be drawn as shown in Figure 6. It can be found from Figure 6 that the change law of system potential energy on both sides of \(u=0\) is opposite. When \(u<0\), as \(M_{\rm e}\) increases, the system's potential energy decreases gradually. When \(u>0\), the system's potential energy increases gradually with the increase of \(M_{\rm e}\).  
To deeply analyze the variation features of the potential energy of the tri-stable system, \(M_{\rm e}\) is taken as 0, 4, 8, 12, and 16 g, respectively. The system's potential energy curves under different \(M_{\rm e}\) are shown in Figure 7. When \(M_{\rm e}=0\) g, that is, when the gravity of the magnet is not considered, the system's left and right potential wells have identical shapes, and the potential energy curve is symmetrical about \(u=0\). When \(M_{\rm e}=4\) g, the left potential well of the system moves downward and becomes deeper, the right potential well moves upward and becomes shallower, and the potential energy curve is no longer symmetric. As \(M_{\rm e}\) continues to increase, the system's left potential well continues to move down, the right potential well continues to move up, and the asymmetry of the potential well becomes more and more obvious. In addition, it can be found that under the action of magnet gravity, with the increase of \(M_{\rm e}\), the lowest points of the system's three potential wells, that is, the system's three stable positions, all gradually move towards the negative half-axis of \(u\). This is accordant with the study results of the system's bifurcation behavior.  
Figure 6: 3D potential energy surface of the system.  
Figure 7: System’s potential energy curve under different \(M_{\rm e}\).  
Keeping other parameters constant, change the excitation acceleration \(A\) to 14 m/s\({}^{2}\). As shown in Figure 9c, when \(M_{\rm e}\) = 4 g, the system derives sufficient kinetic energy to pass the potential barrier and achieve a large oscillation of the tri-stable state due to the increase of the vibration acceleration. Compared with Figure 9a, the vibration velocity and displacement are greatly improved at this time. As shown in Figure 9d, when \(M_{\rm e}\) = 8 g, the system can more readily traverse the potential barrier to realize large-amplitude oscillations between wells. Comparing Figure 9b,d, it can be found that the system's vibration velocity and displacement increase as the excitation acceleration increases. Figure 9d, compared with 9c, shows that the larger the end magnet mass, the larger the vibration velocity and displacement of the system for the same initial excitation conditions. However, the large magnet mass will make the vibration displacement of the system too large, which may cause the combined beam to be damaged during the movement. Therefore, it is essential to select the appropriate end magnet mass for actual use.