## 2 Solution Scheme of Perturbation Method
### Modal Perturbation Method  
In the modal perturbation method (MPM), the \(j\)th eigenvalue and its associated mode of vibration with non-proportionally damped system is related to those of the corresponding proportionally damped system by:  
\[\gamma_{j}=s_{j}+\Delta s_{j}, \tag{22}\]  
\[\left\{\Psi_{j}\right\}=\left\{\eta_{j}\right\}+\sum_{k=1,k\
eq j}^{2n}\{ \eta_{k}\}q_{kj}=[\eta]\big{[}q_{j}\big{]}, \tag{23}\]  
For the \(j\)th mode (\(j\leq n\)), substituting Equations (22) and (23) into Equation (4) gives  
\[\left(s_{j}+\Delta s_{j}\right)[A]\left(\{\eta_{j}\}+\sum_{k=1,k\
eq j}^{2n} \{\eta_{k}\}q_{kj}\right)=-[B]\left(\{\eta_{j}\}+\sum_{k=1,k\
eq j}^{2n}\{\eta _{k}\}q_{kj}\right), \tag{24}\]Premultiplying Equation (24) by \(\{\eta_{l}\}^{T}\) gives:  
\[l\
eq j,l\leq n\] \[\left(s_{j}+\Delta s_{j}\right)\!\!\left(C_{lj}+\sum\limits_{k=1,k \
eq j}^{n}\left[C_{lk}+2s_{i}M_{l}\delta_{lk}\right]\!q_{kj}+\sum\limits_{k=n+ 1}^{2n}\left[C_{lj,k-n}+(s_{l}+\bar{s}_{k-n})M_{l}\delta_{lj,k-n}\right]\!q_{kj }\right)\,\] (25) \[=-(\omega_{l}^{2}-s_{l}^{2})M_{l}q_{lj}\] \[l=j\] \[\left(s_{j}+\Delta s_{j}\right)\!\!\left(C_{jj}+2s_{j}M_{j}+\sum \limits_{k=1,k\
eq j}^{n}C_{jk}q_{kj}+\sum\limits_{k=n+1}^{2n}\left[C_{j,k-n}+( s_{j}+\bar{s}_{k-n})M_{j}\delta_{j,k-n}\right]\!q_{kj}\right)\,\] (26) \[=-(\omega_{j}^{2}-s_{j}^{2})M_{j}\] \[\left(s_{j}+\Delta s_{j}\right)\!\!\left(C_{l-n,j}+(\bar{s}_{l-n} +s_{j})M_{l-n}\delta_{l-n,j}+\sum\limits_{k=1,k\
eq j}^{n}\left[C_{l-n,k}+( \bar{s}_{l-n}+s_{k})M_{l-n}\delta_{l-n,k}\right]\!q_{kj}+\sum\limits_{k=n+1}^{2 n}\left[C_{l-n,k-n}+2\bar{s}_{l-n}M_{l-n}\delta_{lk}\right]\!q_{kj}\right)\,\] (27) \[\left\{\Psi_{j}\right\}=\left\{\eta_{j}\right\}+\sum_{k=1,k\
eq j}^{2n}\{ \eta_{k}\}q_{kj}=[\eta]\big{[}q_{j}\big{]}, \tag{23}\]0 After \(x_{k}= \begin{cases}\Delta s_{j}/s_{j}&k=j\\ q_{kj}&k\
eq j\end{cases} \) has been introduced and \(l\) is taken from \(1\) to \(2n\), Equations (25)-(27) can be written in the matrix form:  
\[\left( \begin{bmatrix}[D_{11}]&[D_{12}]\\ &[D_{21}]&[D_{22}]\end{bmatrix} +x_{j}\left[ \begin{array}{cc}[E_{11}]&[E_{12} ]\\ &[E_{21}]&[E_{22}]\end{array} \right]\!\!\right)\!\!\left|x\right\rangle=\left\{ \begin{array}{c}\{R_{1}\}\\ \{R_{2}\}\end{array} \right\}\!\!, \tag{28}\]  
where \([D_{11}]\), \([D_{12}]\), \([D_{21}]\), \([D_{22}]\), \([E_{11}]\), \([E_{12}]\), \([E_{21}]\), and \([E_{22}]\) are all \(n\times n\) square matrices, \(\{R_{1}\}\) and \(\{R_{2}\}\) are \(n\times 1\) vectors, \(\{x\}\) are \(2n\times 1\) vectors. The matrices and vectors are given below:  
\[[D_{11}]=[C]+diag[2s_{l}M_{l}]+diag[\frac{\omega_{l}^{2}-\bar{s}_{l}^{2}}{s_{j }}M_{l}]\]  
\[[D_{12}]=[C]+diag[(s_{l}+\bar{s}_{l})M_{l}]\]  
\[[D_{21}]=[C]+diag[(s_{l}+\bar{s}_{l})M_{l}]\]  
\[[D_{22}]=[C]+diag[2\bar{s}_{l}M_{l}]+diag[\frac{\omega_{l}^{2}-\bar{s}_{l}^{2} }{s_{j}}M_{l}]\]  
\[[E_{11}]=\left[ \begin{array}{cccccc}C_{11}&C_{12}&\cdots&0&\cdots&C_{1n}\\ C_{21}&C_{22}&\cdots&0&\cdots&C_{2n}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ C_{j1}&C_{22}&\cdots&0&\cdots&C_{jn}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ C_{n1}&C_{n2}&\cdots&0&\cdots&C_{nn}\end{array} \right]+diag[2s_{l}M_{l}]\]  
\[[E_{12}]=[C]+diag[(s_{l}+\bar{s}_{l})M_{l}]\]  
\[[E_{21}]=\left[ \begin{array}{cccccc}C_{11}&C_{12}&\cdots&0&\cdots&C_{1n}\\ C_{21}&C_{22}&\cdots&0&\cdots&C_{2n}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ C_{j1}&C_{22}&\cdots&0&\cdots&C_{jn}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ C_{n1}&C_{n2}&\cdots&0&\cdots&C_{nn}\end{array} \right]\!+diag[(s_{l}+\bar{s}_{l})M_{ l}]\]  
\[[E_{22}]=[C]+diag[2\bar{s}_{l}M_{l}]\]\[\{R_{1}\}=-\bigg{\{} \begin{array}{ccccc}C_{1j}&C_{2j}&\cdots&C_{(j-1)j}&C_{jj}+2s _{j}M_{j}+\frac{\omega_{j}^{2}-s_{j}^{2}}{s_{j}}M_{j}&C_{(j+1)j}&\cdots&C_{nj} \end{array} \bigg{\}}^{T}\]  
\[\{R_{2}\}=-\bigg{\{} \begin{array}{ccccc}C_{1j}&C_{2j}&\cdots&C_{(j-1)j}&C_{ jj}+(\widetilde{s}_{j}+s_{j})M_{j}&C_{(j+1)j}&\cdots&C_{nj}\end{array}