## 4 Dynamic Model Updating Examples
### Proof Examples: Dynamic Model Updating of a Thin Plate
#### 4.1.2 Updating Based on the Impact Modal Test Data
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{**Experimental**} & \multirow{2}{*}{**Modal Damping Ratios (\%)**} & \multicolumn{4}{c}{**Initial Analytical Modal Analysis**} \\ \cline{3-8}  & & & **Frequency (Hz)** & **FRE (\%)** & **MAC** & **1-MAC** \\ \hline
1 & 145.56 & 0.34 & 227.18 & 56.08 & 0.9549 & 0.0451 \\
2 & 193.65 & 0.24 & 303.10 & 56.52 & 0.9728 & 0.0272 \\
3 & 397.43 & 0.30 & 624.76 & 57.20 & 0.7745 & 0.2255 \\
4 & 403.57 & 0.40 & 658.21 & 63.10 & 0.9286 & 0.0714 \\
5 & 627.03 & 0.32 & 1016.49 & 62.11 & 0.9918 & 0.0082 \\
6 & 695.33 & 0.50 & 1124.19 & 61.68 & 0.7069 & 0.2931 \\
7 & 718.01 & 0.55 & 1175.48 & 63.71 & 0.9631 & 0.0369 \\
8 & 773.24 & 0.41 & 1222.67 & 58.12 & 0.6200 & 0.3800 \\
9 & 976.84 & 0.62 & 1629.08 & 66.77 & 0.9492 & 0.0508 \\
10 & 1073.56 & 0.54 & 1742.04 & 62.27 & 0.6504 & 0.3496 \\ _RMSE_ & & & & & 0.2038 \\ _CV_ & & & & & 0.1724 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison between the initial analytical and impact modal results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{4}{c}{**MUUM**} & \multicolumn{4}{c}{**AMPES**} \\ \cline{2-9}  & **Frequency (Hz)** & **FRE (\%)** & **MAC** & **1-MAC** & **Frequency (Hz)** & **FRE (\%)** & **MAC** & **1-MAC** \\ \hline
1 & 141.47 & \(-\)2.81 & 0.9489 & 0.0511 & 145.72 & 0.11 & 0.9537 & 0.0463 \\
2 & 195.02 & 0.70 & 0.9725 & 0.0275 & 183.65 & \(-\)5.16 & 0.9716 & 0.0284 \\
3 & 383.20 & \(-\)3.58 & 0.7507 & 0.2493 & 393.05 & \(-\)1.10 & 0.7951 & 0.2049 \\
4 & 413.39 & 2.43 & 0.9301 & 0.0699 & 396.65 & \(-\)1.71 & 0.9276 & 0.0724 \\
5 & 614.27 & \(-\)2.04 & 0.9917 & 0.0083 & 632.10 & 0.81 & 0.9842 & 0.0158 \\
6 & 702.44 & 1.02 & 0.7191 & 0.2809 & 692.68 & \(-\)0.38 & 0.7244 & 0.2756 \\
7 & 722.52 & 0.63 & 0.9723 & 0.0277 & 719.50 & 0.21 & 0.9562 & 0.0438 \\
8 & 746.54 & \(-\)3.45 & 0.6106 & 0.3894 & 766.81 & \(-\)0.83 & 0.6559 & 0.3441 \\
9 & 1019.28 & 4.34 & 0.9588 & 0.0412 & 1029.38 & 5.38 & 0.9437 & 0.0563 \\
10 & 1087.17 & 1.27 & 0.6394 & 0.3606 & 1094.62 & 1.96 & 0.6657 & 0.3343 \\ _RMSE_ & & 2.55 & 0.2082 & & 2.55 & & 0.1902 \\ _CV_ & & & & & & 0.1552 & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of the updated results by the MUUM and CMPES methods based on the impact modal test data. results, as listed in Table 8. The initial FE model was updated by the MUUM method and the CMPES method with three random initial populations, respectively. The \(RMSE_{FRE}\), \(RMSE_{MAC}\), and \(CV\) of the updated model obtained by the two methods are listed in Table 9. All the values of the updated model obtained by the CMPES method were better than or equal to those obtained by the MUUM method. The \(RMSE_{FRE}\) obtained by the MUUM method fluctuated wildly from 2.55% to 7.46%, compared with almost constant \(RMSE_{FRE}\) obtained by the CMPES method. Similarly, the fluctuations of the \(RMSE_{MAC}\) and \(CV\) obtained by the MUUM method were greater than the CMPES method. Therefore, the robustness of the CMPES method is much better than the MUUM method. In this example, the FE model updating of the thin plate was carried out by the MUUM and CMPES methods. The comparisons demonstrate that both the updating methods can work well. The accuracy, effectiveness, and robustness of the proposed CMPES method are better than the MUUM method.