## 2 Experimental and Finite Element Modal Analysis
### Finite Element Modal Analysis  
First, a three-dimensional model of the plate was built (Figure 5). The reference material parameters and design dimensions of the plate are listed in Table 3. To automate the model-updating process, the FE model was parameterized, i.e., the material parameters and plate dimensions (including the length and thickness) were selected as the design variables. In addition, the constraint condition of the structure is another important factor affecting the FE modal-analysis results; thus, it needs to be taken into account. In this study, the constraint state of the bolted joint was simplified into two cases: perfect rigid constraint and elastic support constraint. The FE modal analysis with the perfect rigid constraint was performed first and the first 40 natural frequencies and displacement mode shapes of the plate are depicted in Figure 6. The red triangles in Figure 6 indicate the FE modes corresponding to the experimental modes, while the natural frequency errors between the experimental and FE modal-analysis results are listed in Table 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode** & **Frequency (Hz)** & **Mode Shape** \\ \hline
1 & 221.44 \\
2 & 429.77 \\
3 & 684.89 \\
4 & 1117.92 \\ \end{tabular}
\end{table}
Table 2: Experimental natural frequencies and mode shapes.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Symbol** & **Item** & **Value** \\ \hline \multirow{3}{*}{Material parameters} & E & Young’s modulus (GPa) & 70.0 \\  & \(\mu\) & [PERSON]’s ratio & 0.33 \\  & \(\rho\) & Density (kg/m\({}^{3}\)) & 2720 \\ \hline \multirow{3}{*}{Design dimensions} & \(L\) & Length (mm) & 300 \\  & \(T\) & Thickness (mm) & 2 \\ \cline{1-1}  & \(D\) & Center hole diameter (mm) & 9 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Reference material parameters and design dimensions.  
According to the results, there was a significant difference in the dynamic characteristics between the initial FE model and the actual structure; therefore, the FE model needs to be updated. In addition, the modes obtained by FE modal analysis were more than those measured in the experimental modal analysis. This difference is related to the direction and position of the excitation force. In the experimental modal analysis, the direction of the excitation force was unidirectional; thus, only the mode in that direction could be excited. On the other hand, in the FE modal analysis, modes in all directions can be easily obtained. Moreover, when the position of the excitation force is just near the node of the mode shape, where the vibration amplitude is close to zero, the mode will not be excited successfully. For instance, the FE mode shapes not measured in the experimental modal analysis have one thing in common; that is, the excitation position was in the low amplitude region (blue area in Figure 6) over the entire surface of the plate. Mode identification or mode-tracking techniques can effectively solve this problem of mode mismatch between experimental and FE modal-analysis results. Consequently, the research on a model-updating method with mode identification is worthy of attention.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Experimental Mode** & **Frequency (Hz)** & **FE Mode** & **Frequency (Hz)** & **Relative Error (\%)** \\ \hline
1 & 221.4 & 9 & 254.3 & +14.82 \\
2 & 429.8 & 14 & 489.9 & +14.00 \\
3 & 684.9 & 20 & 786.0 & +14.76 \\
4 & 1117.9 & 29 & 1270.4 & +13.64 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Natural frequency errors between the experimental and FE modal-analysis results.  
Figure 5: Three-dimensional model of the plate with dimensions.