## 3 Verification with Numerical Examples  
### Two Degrees-Of-Freedom System  
The first example is a 2 degrees-of-freedom system as shown in Figure 1, which had been extensively studied by many researchers [10]. The mechanical parameters of the system are chosen as follows:  
\[m_{1}=1\text{kg},\ k_{1}=100\text{N}/\text{m},\ m_{2}/m_{1}=0.3,\ (\zeta_{a}+ \zeta_{b})/2=0.2,\ \omega_{a}=\omega_{b}\]  
where \(c_{1}=2\zeta_{a}\omega_{a}m_{1},c_{2}=2\zeta_{b}\omega_{b}m_{2},\ \omega_{a}=\sqrt{k_{1}/m_{1}},\ \omega_{b}=\sqrt{k_{2}/m_{2}}\).  
Obviously, when \(\zeta_{a}=\zeta_{b}\), the 2 DOFs system is proportionally damped. Here, two cases with different damping values for \(\zeta_{a}\) and \(\zeta_{b}\) are analyzed. In case 1, \(\zeta_{a}=0.3\) and \(\zeta_{b}=0.1\); in case 2, \(\zeta_{a}=0.37\) and \(\zeta_{b}=0.03\).  
Tables 1 and 2 compare the proposed perturbation method with other existing methods for the calculation of the \(\widetilde{\omega}_{j}\) natural frequencies and damping ratios \(\widetilde{\zeta}_{j}\) of the 2 DOFs system. The exact solution is the solution of Equation (4). Proportional damping method (PDM) obtained the natural frequencies \(\omega_{j}\) from Equation (5) and the damping ratios \(\zeta_{j}\) from Equation (9) based on the equivalent proportional damping system. [PERSON] and [PERSON] [23] used the first-order perturbation method (FPM) to determine the complex eigenvalues. [PERSON] [27] constructed a proportional damping matrix by least squares approach, then used the first-order perturbation method to calculate the complex eigenvalues. Those results are also showed in Tables 1 and 2.  
For non-proportionally damped systems, the matrix [C] is not diagonal. The relative maximum magnitude of the off-diagonal elements of [C] to the diagonal element can be expressed as [34]:  
\[\alpha{=}\max(\frac{C_{\textit{R}}^{2}}{C_{\textit{I}}C_{\textit{R}}})\ (l\
eq k), \tag{35}\]  
The coupling index \(\alpha\) indicates how much the damping is non-proportional is shown in Table 3. For case 1 and case 2, coupling index \(\alpha\) is equal to 0.237 and 0.708, respectively. Therefore, the damping of case 2 is highly non-proportional. The relative error \(e\) of the \(j\)th modal eigenvalues can be expressed into:  
\[e_{\omega_{j}} =\left|\frac{\widetilde{\omega}_{j}-\widetilde{\omega_{j}}}{ \widetilde{\omega_{j}}}\right|\times 100\%, \tag{36}\] \[e_{\zeta_{j}} =\left|\frac{\widetilde{\zeta}_{j}-\widetilde{\zeta_{j}}}{ \widetilde{\zeta_{j}}}\right|\times 100\%, \tag{37}\]  
in which \(\widetilde{\omega}_{j}\) and \(\widetilde{\omega_{j}}^{\ast}\) are the approximate and exact frequencies, \(\widetilde{\zeta}_{j}\) and \(\widetilde{\zeta_{j}}^{\ast}\) are the approximate and exact damping ratios. The relative errors of the natural frequencies and damping ratios are shown in Table 3. Obviously, the results of modal perturbation method are all converged to the exact solution. This is because all the 4 modes \(\left\{\eta_{j}\right\}\) of the 2 DOFs system are the complete eigenvector set to work out the exact complex eigenvector \(\left\{\psi_{j}\right\}\) by the perturbation method. The errors of the other three methods are generally larger with the increase of the coupling index \(\alpha\). The errors of natural frequencies solved by the other three methods of case 2 exceed 5% which can characterize the non-proportionality of non-proportionally damped systems. Therefore, the MPM is more accurate than both FPM and PDM.