# Dynamic Characteristics of the Bouc-Wen Nonlinear Isolation System
## 4 Analysis of the Influence Factors of the Nonlinear Isolation System  
### Influence of the Isolation Control Parameters  
For the isolation structure using the Bouc-Wen model, \(\beta\) and \(\gamma\) can effectively reflect the dynamic characteristics of the isolation system. Therefore, the steady-state responses of each particle of the system under different control parameters were analyzed in the two cases, as shown in Table 1.  
Figure 3(a) shows that when \(\gamma\geq 0\), all particles of the isolation system drift to the left, which shows the soft mechanical properties. With the increase in the ratio of the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(\mathbf{m_{1}}\) & \(\mathbf{m_{2}}\) & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \(\mathbf{k_{1}}\) & \(\mathbf{k_{2}}\) \\ \hline
**Values** & **1** & **8** & **0.2** & **0.4** & **4** & **2** \\ \hline \multirow{3}{*}{\(\gamma\geq 0\)} & \(\beta\) & 0.25 & 0.5 & 0.75 & 1.0 \\  & \(\gamma\) & 0.75 & 0.5 & 0.25 & 0 \\ \cline{1-1}  & \(\beta\) & 0.1 & 0.25 & 0.5 & 0.75 \\ \cline{1-1}  & \(\gamma\) & \(-\)0.9 & \(-\)0.75 & \(-\)0.5 & \(-\)0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Value table of the model control parameters.  
control parameters \(\beta/\gamma\), the amplitude response of each particle of the structural system becomes smaller, which proves that the energy dissipation effect of the structure is better and the energy transmission characteristics are more complete under this parameter. By comprehensive comparison with Figure 4, it can be found that in the isolation system of Bouc-Wen model with this parameter, there is no unstable solution in the first- and second-order resonance region of each particle, and there is no amplitude jump phenomenon in the peak curve. The whole system is stable.  
Figure 4: Amplitude–frequency characteristic curve of the isolation structure under different control parameters (\(\gamma\geq 0\)) (**a,b**) Amplitude–frequency characteristic curve of the substructure; (**c,d**) Amplitude–frequency characteristic curve of the superstructure; (**e,f**) Amplitude–frequency characteristic curve of the hysteretic element.  
Figure 5 shows that when \(\gamma<0\), all particles of the isolation system drift to the right, which shows the hardened mechanical properties. At this time, the material characteristics' change will make the system particles appear as having an unstable solution with jumping phenomena. By comparing Figure 5a-c, it can be found that as the ratio of the control parameters \(\beta/|\gamma|\) decreases, the amplitude response of each particle of the structural system gradually increases. In this case, the material in the isolation system appears as aging or hardening characteristics, which is easy to lead to the overall destruction of the isolation structure.  
Figure 5: Amplitude–frequency characteristic curve of the isolation structure under different control parameters (\(\gamma\leq 0\)) (**a,b**) Amplitude–frequency characteristic curve of the substructure; (**c,d**) Amplitude–frequency characteristic curve of the superstructure; (**e,f**) Amplitude–frequency characteristic curve of the hysteretic element.  
By comprehensive comparison of Figures 4 and 5, it can be found that when \(\gamma\leq 0\), the isolation bearing appears as aging, corrosion, low-temperature hardening and other phenomena [36; 37]. The steady-state response of each particle of the system has an unstable solution, and the peak curve of the first-order resonance region shows an amplitude jump phenomenon. In addition, it can be observed that if and only if \(\beta/|\gamma|\)<1, the structural system has an unstable solution and produces a large response. Therefore, the influence of the control parameter should be fully considered in the actual design.