## 3 Verification with Numerical Examples
### Frame with Concentrated Damper  
Although the modal perturbation solution would converge to the exact solution based on the complete eigenvector set of the undamped system, usually only the first lower order eigenvectors can be obtained, especially for large-scale system with hundreds and thousands of DOF. Relatively few modes can provide sufficiently accurate modal superposition results in engineering applications, therefore the number \(n\) of undamped eigenvalues in Equation (23) is only a little larger than the truncated mode number. Similar to the subspace iteration method, to calculate the \(j\)th eigenvalue of the non-proportionally damped system, the number of modes \(n\) must be:  
\[n=j+\Delta n, \tag{38}\]  
As the additional number of modes, \(\Delta n\), increases, the result gradually converges to the exact solution. The second example, a simple frame structure [12; 35], is used to investigate the effect of \(\Delta n\) on the accuracy of eigenvalues. As schematically shown in Figure 2, the frame structure is idealized as 9 beam elements with 24 DOFs. The Young's modulus and density of material are 500 N/m\({}^{2}\), and 1 kg/m\({}^{3}\). The cross-section inertia and area are 1m\({}^{4}\) and 1m\({}^{2}\). The length of each beam element is 1 m. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**\(\alpha\)**} & \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**MPM**} & \multicolumn{2}{c}{**FDM**} & \multicolumn{2}{c}{**FPM [23]**} & \multicolumn{2}{c}{**FPM [27]**} \\ \cline{3-10}  & & & \(e_{\omega}\) **(\%)** & \(e_{\zeta}\) **(\%)** & \(e_{\omega}\) **(\%)** & \(e_{\zeta}\) **(\%)** & \(e_{\omega}\) **(\%)** & \(e_{\zeta}\) **(\%)** & \(e_{\omega}\) **(\%)** & \(e_{\zeta}\) **(\%)** \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{0.237} & 1 & 0 & 0 & 1.728 & 0.603 & 0.273 & 0.865 & 0.957 & 0.181 \\  & & 2 & 0 & 0 & 1.758 & 1.226 & 4.359 & 3.688 & 1.910 & 1.374 \\ \hline \multirow{2}{*}{2} & \multirow{2}{*}{0.708} & 1 & 0 & 0 & 5.360 & 0.230 & 3.722 & 1.927 & 3.063 & 2.595 \\  & & 2 & 0 & 0 & 5.664 & 2.271 & 7.825 & 4.230 & 6.123 & 2.694 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The coupling index \(\alpha\) and relative error \(e\) of the 2-DOFs system. Using the lumped-mass, the frame has 16 possible real mode shapes. The non-proportional damping arises from a damper attached to node 3 and the other attached to node 8. The damping coefficients of the two dampers are \(c_{1}=10\) N.s/m, \(c_{2}=25\) N.s/m. The coupling index \(\alpha\) of the frame structure is equal to 1, which indicates that system is highly non-proportionally damped. The relative errors of the first three natural frequencies and damping ratios by MPM for various number of modes \(\Delta n\) are presented in Figure 3. It can be seen that the MPM leads to monotonically convergent solutions and gradually converges to the exact solution with the increase of \(\Delta n\) even if it is a highly non-proportionally damped system. It represents that the equivalent proportionally damped system reduces the effect of the deviation damping matrix so that the method can deal with the highly non-proportionally damped system. The curves of error are almost merging into one when \(\Delta n\) is greater than 8. Figure 3: Relative errors of the first three natural frequencies and damping ratios. Figure 2: A frame with concentrated damper. Then, the first three natural frequencies and damping ratios of the frame system by the proposed perturbation method with \(n=11\) and \(n=16\) are shown in Tables 4-6. As a comparison, those results obtained by exact method, PDM, FPM [23], and FPM [27] are also listed. Obviously, the first three natural frequencies and damping ratios of perturbation method are same as the exact solution when \(n=16\). This is because all of the 16 modes of the frame system are the complete eigenvector set.