## 2 Solution Scheme of Perturbation Method
### Corresponding Proportionally Damped System  
For a corresponding undamped system, the real eigenvalue problem can be expressed as:  
\[\Big{(}[k]-\omega_{j}^{2}[m]\Big{)}\{\phi_{j}\}=\{0\}, \tag{5}\]  
in which \(\omega_{j}\) and \(\left\{\phi_{j}\right\}\) are the \(j\)th undamped circular natural frequency and its associated real mode shape. Assume the undamped circular natural frequency is distinct, then the real modes satisfy the following orthogonality relation:  
\[\Big{\{}\phi_{l}\Big{\}}^{T}[m]\Big{\{}\phi_{k}\Big{\}}=M_{l}\delta_{lk}, \tag{6}\]  
\[\Big{\{}\phi_{l}\Big{\}}^{T}[k]\Big{\{}\phi_{k}\Big{\}}=M_{l}\omega_{l}^{2} \delta_{lk}, \tag{7}\]  
in which \(\delta_{lk}=\begin{cases}1&l=k\\ 0&l\
eq k\end{cases}\) is Kronecker's Delta function, \(M_{l}\) is the modal mass.  
The non-proportional damping matrix can be split into the sum of two matrices:  
\[[c]=\Big{[}c_{p}\Big{]}+[\Delta c], \tag{8}\]  
in which \(\Big{[}c_{p}\Big{]}\) is a proportional damping matrix, \([\Delta c]\) is the deviation from \(\Big{[}c_{p}\Big{]}\). The proportional damping matrix \(\Big{[}c_{p}\Big{]}\) can be diagonalized by the undamped mode shape, and satisfies the following orthogonality relation:  
\[\zeta_{l}=\frac{\Big{\{}\phi_{l}\Big{\}}^{T}[c]\Big{\{}\phi_{l}\Big{\}}}{2M_ {l}\omega_{l}}, \tag{9}\]  
\[\Big{\{}\phi_{l}\Big{\}}^{T}\Big{[}c_{p}\Big{]}\Big{\{}\phi_{k}\Big{\}}=\left\{ \begin{matrix}2\zeta_{l}M_{l}\omega_{l}&l=k\\ 0&l\
eq k\end{matrix},\right. \tag{10}\]  
in which \(\zeta_{l}\) is the modal damping ratio. In fact, \(\Big{[}c_{p}\Big{]}\) is the superposition of modal damping matrix, which can be solved but need not to be explicitly determined in the proposed method. Therefore, the system composed of \([m]\), \([c_{p}]\), and \([k]\) can be called as equivalent proportionally damped system, which is close to the non-proportionally damped system to reduce the effect of the deviation damping matrix. By defining:  
\[[A_{0}]=\left[\begin{array}{cc}\zeta_{p}&[m]\\ [m]&[0]\end{array}\right], \tag{11}\]  
the eigenvalue problem in state vector form for the corresponding proportionally damped system with \([m]\), \([c_{p}]\), and \([k]\) can be expressed as:  
\[\left(s_{j}[A_{0}]+[B]\right)\{\eta_{j}\}=\{0\}, \tag{12}\]  
in which \(s_{j}=-\zeta_{j}\omega_{j}+i\omega_{j}\sqrt{1-\zeta_{j}^{2}}\), \(\{\eta_{j}\}=\left\{\begin{array}{c}\left\{\phi_{j}\right\}\\ s_{j}\left\{\phi_{j}\right\}\end{array}\right\}\)\((j=1,\,2,\,\ldots,\,n)\). For \(j>n,s_{j}=\overline{s}_{j-n}\), \(\{\eta_{j}\}=\left\{\overline{\eta}_{j-n}\right\}\). \(n\leq N\) is the first \(n\) modes determined from Equation (5). \"\(\cdot\)\" denotes complex conjugation, \(i=\sqrt{-1}\) is the unit complex.  
On introducing the symbol \(C_{lk}\), defined by  
\[C_{lk}=\left\{\phi_{l}\right\}^{T}[c]\left\{\phi_{k}\right\}, \tag{13}\]  
and making use of the orthogonality conditions of real mode \(\left\{\phi\right\}\), the complex mode \(\{\eta\}\) will satisfy the following relationships:  
\[\text{for}\;l\leq n,\;k\leq n,\;\{\eta_{l}\}^{T}[A]\{\eta_{k}\}=C_{lk}+2s_{l}M _{l}\delta_{lk}, \tag{14}\]  
\[\text{for}\;l\leq n,\;k>n,\;\{\eta_{l}\}^{T}[A]\{\eta_{k}\}=C_{l,k-n}+(s_{l}+ \overline{s}_{k-n})M_{l}\delta_{l,k-n}, \tag{15}\]  
\[\text{for}\;l>n,\;k\leq n,\;\{\eta_{l}\}^{T}[A]\{\eta_{k}\}=C_{l-n,k}+( \overline{s}_{l-n}+s_{k})M_{l-n}\delta_{l-n,k}, \tag{16}\]  
\[\text{for}\;l>n,\;k>n,\;\{\eta_{l}\}^{T}[A]\{\eta_{k}\}=C_{l-n,k-n}+2\overline {s}_{l-n}M_{l-n}\delta_{lk}, \tag{17}\]  
\[\text{for}\;l\leq n,\;k\leq n,\;\{\eta_{l}\}^{T}[B]\{\eta_{k}\}=\left(\omega_ {l}^{2}-s_{l}^{2}\right)M_{l}\delta_{l,k}, \tag{18}\]  
\[\text{for}\;l\leq n,\;k>n,\;\{\eta_{l}\}^{T}[B]\{\eta_{k}\}=0, \tag{19}\]  
\[\text{for}\;l>n,\;k\leq n,\;\{\eta_{l}\}^{T}[B]\{\eta_{k}\}=0, \tag{20}\]  
\[\text{for}\;l>n,\;k>n,\;\{\eta_{l}\}^{T}[B]\{\eta_{k}\}=\left(\omega_{l-n}^{2} -\overline{s}_{l-n}^{2}\right)M_{l-n}\delta_{l,k}, \tag{21}\]