## 3 Verification with Numerical Examples
### Frame with Concentrated Damper
The relative errors of the first 3 natural frequencies and damping ratios obtained by MPM are within 0.146% when \(n=11\), which are obviously less than the PDM and FPM. Therefore, the additional number of modes \(\Delta n=8\) is enough to obtain sufficiently accurate results. The errors of damping ratios of the other three methods may exceed 25%. Therefore, the proposed method can deal with the highly non-proportionally damped system for the equivalent proportionally damped system reduces the effect of the deviation damping matrix. Appendix A lists the proportionally damping matrix \(\left[c_{p}\right]\) and deviation damping matrix \(\left[\Delta c\right]\) after static condensation for the understanding the effect of Equation (8), although \(\left[c_{p}\right]\) and \(\left[\Delta c\right]\) need not to be explicitly determined in the proposed method. The main calculation time for the proposed method contains two parts, one is for solving the characteristic equation of undamped system and the other one is for solving the nonlinear algebraic equations. For the frame with concentrated damper, the first part consumed 0.057 ms, and the second part consumed 0.309 ms, the total time is 0.366 ms. The PDM only needed to solve the characteristic equation of undamped system which consumed 0.057 ms. To solve the characteristic equation of state space, the exact method consumed 0.470 ms. The FPM [23] needs to solve the characteristic equation of undamped system and to calculate the perturbation of undamped system which totally consumed 0.132 ms. The FPM [27] needs to optimize proportional damping coefficient and to solve the characteristic equation of undamped system. It totally consumed 0.123 ms. It can be seen that the time consumed by proposed method is shorter than state space method and longer than the other three existing methods. The advantages of the proposed method will become more prominent as the degrees of the system increase because it only needs to solve the characteristic equation of undamped system. \begin{table}
\begin{tabular}{c} **FPM** \\ **(\(\mathbf{n}\) = 11)** \\ \end{tabular} \\ \hline
1 & 0 & 0.001 & 0.470 & 0.458 & 0.66 & 0 & 0.132 & 27.161 & 27.149 & 25.734 \\
2 & 0 & 0.042 & 3.395 & 2.41 & 3.395 & 0 & 0.146 & 2.028 & 0.999 & 1.964 \\
3 & 0 & 0.000 & 0.000 & 0 & 0.478 & 0 & 0.001 & 8.599 & 8.696 & 8.696 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The errors of natural frequencies and damping ratios of the frame system. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{4}{c}{**Frequencies (rad/s)**} \\ \cline{2-7}  & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**MPM (\(\mathbf{n}\) = 16)**} & \multicolumn{1}{c}{**MPM (\(\mathbf{n}\) = 11)**} & \multicolumn{1}{c}{**PDM**} & \multicolumn{1}{c}{**FPM [23]**} & \multicolumn{1}{c}{**FPM [27]**} \\ \hline
1 & 7.645171 & 7.645171 & 7.645066 & 7.60922 & 7.610119 & 7.695591 \\
2 & 9.049425 & 9.049425 & 9.04580 & 8.74224 & 8.831296 & 8.74224 \\
3 & 12.725162 & 12.725162 & 12.725139 & 12.72513 & 12.72513 & 12.785979 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The natural frequencies (rad/s) of the frame system. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{4}{c}{**Damping Ratios**} \\ \cline{2-7}  & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**MPM (\(\mathbf{n}\) = 16)**} & \multicolumn{1}{c}{**MPM (\(\mathbf{n}\) = 11)**} & \multicolumn{1}{c}{**PDM**} & \multicolumn{1}{c}{**FPM [23]**} & \multicolumn{1}{c}{**FPM [27]**} \\ \hline
1 & 0.012089 & 0.012089 & 0.012105 & 0.015373 & 0.015371 & 0.0152 \\
2 & 0.140255 & 0.140255 & 0.140460 & 0.1431 & 0.141656 & 0.14301 \\
3 & 0.000506 & 0.000506 & 0.000507 & 0.00055 & 0.00055 & 0.00055 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The damping ratios of the frame system. Although the other three methods consumed less time, the accuracy of the proposed method is much higher, especially for high non-proportional damping systems.