## 4 Dynamic Model Updating Examples
### Proof Examples: Dynamic Model Updating of a Thin Plate
#### 4.1.1 Updating Based on the Simulated \"Experimental\" Data  
The \"experimental\" model, generating the simulated \"experimental\" modal data, was obtained by introducing certainly known discrepancies in the non-dimensional ratios of material parameters based on the initial FE model, as used in Reference [23], shown in Table 1.  
The first ten (ignore the rigid modes) natural frequencies and mode shapes corresponding to the fifteen \"response\" points (see Figure 3b) were determined. Table 2 shows that all the MAC values approximated 1. However, the differences regarding natural frequencies between the initial analytical and simulated \"experimental\" results were remarkable, and the FRE values were about 40%. Furthermore, the \(RMSE_{FRE}\), \(RMSE_{MAC}\), and \(CV\) values were 41.23%, 1.26%, and 1.07%, respectively. To update the initial FE model, the lower and upper boundaries of the updating parameters are given in Table 3.  
The pre-analysis for the MUUM method was performed based on the updating parameters generated by central composite design [36]. As illustrated in Figure 4a, the swapping between the third and the fourth mode occurs frequently. The same goes for the sixth and the seventh modes, as shown in Figure 4b. Thus, four uncorrelated mode pairs are specified for the MUUM method: (A4, E3), (A3, E4), (A7, E6), and (A6, E7). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model Type**} & \multirow{2}{*}{**Area Type**} & \multirow{2}{*}{
\begin{tabular}{c} **Non-Dimensional Ratio** \\ **of Density** \\ \end{tabular} } & \multicolumn{2}{c}{**Non-Dimensional Ratio**} & \multicolumn{2}{c}{**Non-Dimensional Ratio**} \\  & & **of Density** & **of Modulus of Elasticity** & **of Poisson’s Ratio** \\ \hline \multirow{2}{*}{Initial FE} & Hole & 0.5 & 1.2 & 0.75 \\  & Main & 1.0 & 1.0 & 1.0 \\ \hline \multicolumn{1}{l}{’Experimental’} & Hole & 0.89 & 1.05 & 0.97 \\ \multicolumn{1}{l}{’Experimental’} & Main & 0.64 & 0.88 & 1.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters for the initial FE and simulated “experimental” model of the thin plate. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Mode** \\ **No.** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **”Experimental”Frequency (Hz)** \\ \end{tabular} } & \multicolumn{4}{c}{**Initial Analytical Modal Analysis**} \\ \cline{3-6}  & & & **Frequency (Hz)** & **FRE (\%)** & **MAC** & **1-MAC** \\ \hline
1 & 161.13 & 227.18 & 41.00 & 1.00 & 0.00 \\
2 & 209.56 & 303.10 & 44.64 & 1.00 & 0.00 \\
3 & 445.72 & 624.76 & 40.17 & 1.00 & 0.00 \\
4 & 4551.18 & 658.21 & 44.61 & 1.00 & 0.00 \\
5 & 732.77 & 1016.49 & 38.72 & 0.98 & 0.02 \\
6 & 780.83 & 1124.19 & 43.97 & 1.00 & 0.00 \\
7 & 855.55 & 1175.48 & 37.40 & 0.99 & 0.01 \\
8 & 892.67 & 1222.67 & 36.97 & 0.97 & 0.03 \\
9 & 1154.68 & 1629.08 & 41.08 & 0.99 & 0.01 \\
10 & 1219.74 & 1742.04 & 42.82 & 1.00 & 0.00 \\ \multicolumn{1}{l}{_RMSE_} & & 41.23 & & 0.0126 \\ \multicolumn{1}{l}{_CV_} & & & 0.0107 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparisons between the initial analytical and simulated “experimental” modal results. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Area**} & \multirow{2}{*}{**Boundary**} & \multicolumn{2}{c}{**Non-Dimensional Ratio**} & \multicolumn{2}{c}{**Non-Dimensional Ratio**} & \multicolumn{2}{c}{**Non-Dimensional Ratio**} \\  & & **of Density** & **of Modulus of Elasticity** & **of Poisson’s Ratio** \\ \hline \multirow{3}{*}{Main} & Lower & 0.1 & 0.1 & 0.1 \\  & Upper & 1.5 & 1.5 & 1.2 \\ \cline{2-5}  & Lower & 0.1 & 0.1 & 0.1 \\ \cline{1-1}  & Upper & 1.5 & 1.5 & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The lower and upper boundaries of updating parameters for the thin plate.