# Dynamic Characteristics of the Bouc-Wen Nonlinear Isolation System
## 3 Theoretical Analysis of the Nonlinear Isolation System
 \tag{8}\]  
where \(K_{nd}\) and \(C_{nd}\) are the partial derivatives of the nonlinear term \(K_{n}\) with respect to \(x\) and \(\dot{x}\), which can be written in the Jacobian matrix form as follows:  
\[K_{nd}=\frac{\partial K_{n(i)}}{\partial X_{(i)}}=\left[ \begin{array}{cc} \frac{\partial K_{n(1)}}{\partial x_{1}}&\frac{\partial K_{n(1)}}{\partial x_ {2}}&\frac{\partial K_{n(1)}}{\partial\dot{x}}\\ \frac{\partial K_{n(2)}}{\partial\dot{x}_{1}}&\frac{\partial K_{n(2)}}{ \partial\dot{x}_{2}}&\frac{\partial K_{n(2)}}{\partial\dot{x}_{2}}\\ \frac{\partial K_{n(3)}}{\partial\dot{x}_{1}}&\frac{\partial K_{n(3)}}{ \partial\dot{x}_{1}}&\frac{\partial K_{n(3)}}{\partial\dot{x}_{2}}&\frac{ \partial K_{n(3)}}{\partial\dot{x}}\end{array} \right]=\left[ \begin{array}{ ccc}0&0&0\\ 0&0\\ 0&0&\beta\big{|}\dot{x}_{2}-\dot{x}_{1}\big{|}+\frac{\gamma\left(\dot{x}_{2}- \dot{x}_{1}\right)}{|z|}\end{array} \right] \tag{9}\]  
\[C_{nd}=\frac{\partial K_{n(i)}}{\partial\dot{X}_{(i)}}=\left[ \begin{array}{ ccc}\frac{\partial K_{n(1)}}{\partial\dot{x}_{1}}&\frac{\partial K_{n(1)}}{ \partial\dot{x}_{2}}&\frac{\partial K_{n(1)}}{\partial\dot{x}_{2}}\\ \frac{\partial K_{n(2)}}{\partial\dot{x}_{1}}&\frac{\partial K_{n(2)}}{ \partial\dot{x}_{2}}&\frac{\partial K_{n(2)}}{\partial\dot{x}_{2}}\\ \frac{\partial K_{n(3)}}{\partial\dot{x}_{1}}&\frac{\partial K_{n(3)}}{ \partial\dot{x}_{2}}&\frac{\partial K_{n(3)}}{\partial\dot{x}}\end{array}  \right]=\left[ \begin{array}{cccc}0&0&0\\ 0&0&0\\ -\frac{\beta\left(\dot{x}_{2}-\dot{x}_{1}\right)}{|\dot{x}_{2}-\dot{x}_{1}|z }-\gamma|z|&\frac{\beta\left(\dot{x}_{2}-\dot{x}_{1}\right)}{|\dot{x}_{2}- \dot{x}_{1}|z}+\gamma|z|&0\end{array} \right] \tag{10}\]  
Suppose the steady-state solution of the system is  
\[ \begin{array}{c}x_{i0}=\sum\limits_{k=1}^{n}a_{ik}\cos(2k-1)\tau_{1}+\sum \limits_{k=1}^{n}b_{ik}\sin(2k-1)\tau_{1}=DA_{i}\\ \Delta x_{i0}=\sum\limits_{k=1}^{n}\Delta a_{ik}\cos(2k-1)\tau_{1}+\sum \limits_{k=1}^{n}\Delta b_{ik}\sin(2k-1)\tau_{1}=D\Delta A_{i}\hskip 28.452756 pt(i=1,2,3) \end{array}  \tag{11}\]  
among which  
\[ \begin{array}{c}D=(\cos\tau_{1},\cos 3\tau_{1},\ldots,\cos(2n-1)\tau_{1},\sin \tau_{1},\sin 3\tau_{1},\ldots,\sin(2n-1)\tau_{1})\\ A_{i}=(a_{i1},a_{i2},\ldots a_{in},b_{i1},b_{i2},\ldots b_{in})^{T}\\ \Delta A_{i}=(\Delta a_{i1},\Delta a_{i2},\ldots\Delta a_{in},\Delta b_{i1}, \Delta b_{i2},\ldots\Delta b_{in})^{T}\end{array}  \tag{12}\]  
The steady-state solution of the system is equivalent to the following Equation (13):  
\[x=SA,\delta x=S\delta A \tag{13}\]  
among which  
\[ \begin{array}{c}A=\left[\begin{array}{cc}A_{1}^{T}&A_{2}^{T}\end{array} \right]^{T},\Delta A=\left[\begin{array}{cc}\Delta A_{1}^{T}&\Delta A_{2}^{T} \end{array}\right]^{T}\\ S=\left[\begin{array}{cc}D\\ &D\end{array}\right]\end{array}  \tag{14}\]Substitute Equation (13) into Equation (8), and apply the [PERSON] algorithm on both sides of the equation to obtain  
\[ \begin{array}{l}\int_{0}^{2\pi}\delta x^{T}\big{[}\omega_{0}^{2}M\Delta\ddot{x }+\omega_{0}C\Delta\dot{x}+K\Delta x+K_{nd}\Delta x+C_{nd}\Delta\dot{x}\big{]}d \tau=\\ \int_{0}^{2\pi}\delta x^{T}\big{\{}\big{[}\Phi-\big{(}2\omega_{0}M\ddot{x}_{0}+ C\dot{x}_{0}+K_{n}(\dot{x}_{0})\big{)}\big{]}\big{\}}\Delta\omega d\tau\end{array}  \tag{15}\]  
Substitute Equation (7) into the formula:  
\[ \begin{array}{l}\int_{0}^{2\pi}S^{T}\Big{[}\omega_{0}^{2}M\ddot{S}+\omega_{0 }C\dot{S}+KS+K_{nd}S+\omega_{0}C_{nd}\dot{S}\Big{]}d\tau\Delta A=\int_{0}^{2 \pi}\Phi d\tau-\\ \int_{0}^{2\pi}S^{T}\Big{[}2\omega_{0}M\ddot{S}+C\dot{S}+K_{n}\Big{]}d\tau A \Delta\omega\end{array}  \tag{16}\]  
Order:  
\[ \begin{array}{l}K_{mc}=\int_{0}^{2\pi}S^{T}\Big{[}\omega_{0}^{2}M\ddot{S}+ \omega_{0}C\dot{S}+KS+C_{nd}\dot{S}+\omega_{0}K_{nd}S\Big{]}d\tau\\ R_{mc}=\int_{0}^{2\pi}S^{T}\Big{[}2\omega_{0}M\ddot{S}+C\dot{S}+K_{n}\Big{]}d \tau\\ R=\int_{0}^{2\pi}\Phi d\tau\end{array}