# Dynamic Characteristics of the Bouc-Wen Nonlinear Isolation System
## 3 Theoretical Analysis of the Nonlinear Isolation System  
The structure presents strong nonlinear characteristics under the action of an actual strong earthquake. For the isolation structure, the material properties of the isolation layer differ greatly from the structure, so the isolation structure can be divided into the superstructure and the substructure. The characteristics of the isolation layer are coupled and simplified by the Bouc-Wen nonlinear model, in which the horizontal stiffness of the superstructure is large and usually presents a translational state. In order to study the nonlinear dynamic characteristics of the isolation system, a strong nonlinear solution method--Incremental Harmonic Balance Method (IHB) [35] as adopted in this paper, to analyze the Bouc-Wen nonlinear model and to study the influence law of the different control parameters under the steady-state response of the system. The Bouc-Wen nonlinear isolation model with two degrees of freedom under harmonic excitation is shown in Figure 3a,b. The dynamic equations of a two-degree-of-freedom Bouc-Wen system are as follows:  
\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+c_{1}\dot{x}_{1}+k_{1}x_{1}+c_{2}( \dot{x}_{1}-\dot{x}_{2})-F_{BW}=f\cos\omega t\\ m_{2}\ddot{x}_{2}+c_{2}(\dot{x}_{2}-\dot{x}_{1})+F_{BW}=0\\ \dot{z}=\dot{x}_{2}-\dot{x}_{1}-\beta|\dot{x}_{2}-\dot{x}_{1}|z-\gamma\big{(} \dot{x}_{2}-\dot{x}_{1})|z|\\ F_{BW}=\text{a}k_{2}(x_{2}-x_{1})+(1-\alpha)k_{2}z\end{array} \right. \tag{4}\]  
Written in matrix form:  
\[\left[ \begin{array}{ccc}m_{1}&0&0\\ 0&m_{2}&0\\ 0&0&0\end{array} \right]\left[ \begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\\ \dot{z}\end{array} \right]+\left[ \begin{array}{ccc}c_{1}+c_{2}&-c_{2}&0\\ -c_{2}&c_{2}&0\\ 1&-1&1\end{array} \right]\left[ \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{z}\end{array} \right]+\left[ \begin{array}{ccc}\alpha k_{2}+k_{1}&-ak_{2 }&-(1-\alpha)k_{2}\\ -\alpha k_{2}&\alpha k_{2}&(1-\alpha)k_{2}\\ 0&0&0\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\\ z\end{array} \right]+ \tag{5}\] \[\left[ \begin{array}{c}0\\ 0\\ \beta|\dot{x}_{2}-\dot{x}_{1}|z+\gamma\big{(}\dot{x}_{2}-\dot{x}_{1})|z|\\ 0\end{array} \right]=\left[ \begin{array}{c}f\\ 0\\ 0\end{array} \right]\cos\omega t\]  
among which  
\[M=\left[ \begin{array}{ccc}m_{1}&0&0\\ 0&m_{2}&0\\ 0&0&0\end{array} \right]C=\left[ \begin{array}{ccc}c_{1}+c_{2}&-c_{2}&0\\ -c_{2}&c_{2}&0\\ 1&-1&1\end{array} \right]K=\left[ \begin{array}{ccc}\alpha k_{2}+k_{1}&- \alpha k_{2}&-(1-\alpha)k_{2}\\ -\alpha k_{2}&\alpha k_{2}&(1-\alpha)k_{2}\\ 0&0&0\end{array} \right]\]  
\[K_{n}=\left[ \begin{array}{c}K_{n(1)}\\ K_{n(2)}\\ K_{n(3)}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\\ \beta|\dot{x}_{2}-\dot{x}_{1}|z+\gamma\big{(}\dot{x}_{2}-\dot{x}_{1})|z|\\ \end{array} \right]\]  
Figure 3: Simplified analysis model of the isolation system. (**a**) Simplified model; (**b**) Analysis model. where \(M\), \(C\), \(K\) and \(K_{n}\), respectively, represent the structural mass, damping, linear stiffness and nonlinear hysteretic matrix, and introduce the time scale. Then, Equation (5) can be written as  
\[\omega^{2}M\ddot{x}+\omega C\dot{x}+Kx+\omega K_{n}=F\cos\tau \tag{6}\]  
among which  
\[F=\left[ \begin{array}{cccc}f&0&0\end{array} \right]^{T}x=\left[ \begin{array}[ ]{cccc}x_{1}&x_{2}&z\end{array} \right]^{T}\]  
Applying incremental processes:  
\[X_{(i)}=X_{(i0)}+\Delta X_{(i)},\ X_{(1)}=x_{1},\ X_{(2)}=x_{2},\ X_{(3)}=z, \ \omega=\omega_{0}+\Delta\omega \tag{7}\]  
where \(\Delta X_{(i)}\) and \(\Delta\omega\) are weak increments. Substitute Equation (7) into Equation (6), and then ignore the high-order trace, and only retain the items below the first order, as shown in Equation (8). \[ \begin{array}{c}\omega_{0}^{2}M\Delta\ddot{x}+\omega_{0}C\Delta\dot{x}+K \Delta x+K_{nd}\Delta x+C_{nd}\Delta\dot{x}=\Phi-\left[2\omega_{0}M\ddot{x}_{0 }+C\dot{x}_{0}+K_{n}(\dot{x}_{0})\right]\Delta\omega\\ \Phi=F\cos(\tau)-\omega_{0}K_{n}(\dot{x}_{0})-\left(\omega_{0}{}^{2}M\ddot{x}_ {0}+\omega_{0}C\dot{x}_{0}+Kx_{0}\right)\end{array}