## 3 Simulation Analysis of CTEHS  
### Analysis of Bifurcation Characteristics of System Static Solution  
1.1 Influence of End Magnet Mass \(M_{\mathrm{e}}\) on the (\(d_{\mathrm{h}}\), \(u\)) Bifurcation Diagram  
Table 1 lists the specific parameters of the system components used in the numerical simulation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Parameter** & **Value** & **Unit** \\ \hline \multirow{8}{*}{Combined Beam part} & Length & 40 & mm \\  & Width & 8 & mm \\  & Thickness & 0.2 & mm \\  & Density & 8300 & kg/m\({}^{3}\) \\  & Young’s Modulus & 128 & GPa \\  & Length & 40 & mm \\  & Width & 8 & mm \\  & Thickness & 0.11 & mm \\  & Density & 1780 & kg/m\({}^{3}\) \\  & Young’s Modulus & 3 & GPa \\  & Piezoelectric Stress Constant & \(-\)11.5 & C/m\({}^{2}\) \\  & Length & 10 & mm \\  & Width & 10 & mm \\ Magnet part & Thickness & 5 & mm \\  & Mass & 3.75 & g \\  & Magnetization Intensity & \(4.5\times 10^{5}\) & A/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of the system. Let \(\tilde{\gamma}(t)=\dot{\gamma}(t)=\dot{V}(t)=V(t)=\ddot{y}(t)=0\) in the system's motion equation, then the system's equation in a static state is  
\[F_{r}+F_{m}+F_{g}=0 \tag{19}\]  
It can be found from Equation (19) that the static solution of the system depends not only on the nonlinear magnetism and resilience of the system but also on the gravity of the end magnets. The system's static bifurcation diagram, under the consideration of the end magnet's gravity, can be obtained by solving Equation (19). First, the influence of end magnet mass \(M_{\rm e}\) on the system's bifurcation behavior is studied in (\(d_{\rm h}\), \(u\)) plane. Taking \(d_{\rm v}\) = 20 mm, \(M_{\rm e}\) = 0,3,6,9,12,15 g, the bifurcation diagram of the system in the (\(d_{\rm hv}\), \(u\)) plane is shown in Figure 4, where the real lines represent the stable solution and the dotted lines represent the unstable solution. Figure 4a is the bifurcation diagram when \(M_{\rm e}\)= 0 g, that is, the gravity of the end magnet is not considered, and two saddle points S symmetrical about \(u\) = 0 appear in the system. When \(d_{\rm h}>d_{\rm hS}\), there is just one zero-point solution to the static equation. When \(d_{\rm h}<d_{\rm hS}\), there exist three stable solutions and two unstable solutions of the static equation, the system shows as a tristable system at this time. When \(M_{\rm e}\) = 3 g, as shown in Figure 4b, under the influence of the gravity of the magnet, the bifurcation diagram presents asymmetry up and down. The upper saddle point shifts to the left to point S\({}_{1}\), and the lower saddle point shifts to the right to point S\({}_{2}\), and the zero stable position of the system is shifted downward, and the offset is \(\Delta u\). When \(d_{\rm h}>d_{\rm hS1}\), there is just one solution to the static equation, and the gravity effect of the magnet makes this stable solution negative, at which time the system is monostable. When \(d_{\rm hS1}<d_{\rm hS2}\), there exist two stable solutions and one unstable solution of the static equation, and the system shows as a bistable state. When \(d_{\rm h}<d_{\rm hS1}\), there exist five solutions to the static equation, and the system exhibits a tristable characteristic in this range. As shown in Figure 4c,d, when \(M_{\rm e}\) is increased to 6 g and 9 g, the saddle point S\({}_{1}\) continues to move to the left gradually, S\({}_{2}\) continues to move to the right gradually, the middle stable solution continues to shift downward, and the offset \(\Delta u\) becomes larger and larger. In addition, the movement of the saddle points S\({}_{1}\) and S\({}_{2}\) makes the range of the bistable system gradually increase. When \(M_{\rm e}\) = 12 g, as shown in Figure 4e, the middle stable solution continues to move down and coincides with the lower saddle point S\({}_{2}\), forming a fork point P. At this time, the range of the system in a bistable state reaches the maximum.