## 4 Modal Analysis of the 3# Surface Office Gate
### Correlation Analysis of the Prototype and Numerical Vibration Modes
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multicolumn{4}{c}{**Natural Frequency**} & \multicolumn{4}{c}{**Natural Frequency**} \\ \cline{2-10}  & **Case 4 (wet)** & **Case 5 (wet)** & **Case 6 (wet)** & **Case 7/8** & **Case 4 (dry)** & **Case 5 (wet)** & **Case 6 (wet)** & **Case 7/8** \\ \hline
4 & 19.70 & 19.69 & 20.31 & 20.36 & 9 & 27.29 & 28.39 & 28.83 & 43.17 \\
5 & 19.79 & 20.62 & 20.69 & 25.77 & 10 & 28.00 & 30.18 & 36.53 & 45.17 \\
6 & 20.65 & 20.67 & 25.15 & 27.81 & 11 & 28.37 & 32.42 & 45.32 & 45.43 \\
7 & 20.71 & 26.11 & 28.08 & 27.92 & 12 & 33.84 & 35.02 & 45.40 & 56.04 \\
8 & 23.12 & 28.30 & 28.30 & 40.98 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Natural frequencies of the 3# surface outlet gate. According to the mode shapes listed in Table 5, the values of modal assurance criterion (MAC) [44] and subspace of order 2 modal assurance criterion (S2 MAC) [45] are calculated and illustrated in Figure 19 to describe the correlation between the prototype and numerical vibration modes. As shown in Figure 19, the MAC values corresponding to the numerical vibration modes of orders 9 to 11 are higher than those corresponding to other numerical vibration modes, which demonstrates the fact that only the vibration modes with frequencies quite close to the actual dominant frequency can significantly contribute to the prototype vibration. According to recent research [46; 47], if a set of closely spaced natural vibration modes is obtained from the numerical simulation, it is more reasonable to estimate the correlation between the prototype and numerical vibration modes by calculating the MAC value between the prototype mode shape and the linear combination of the numerical mode shapes corresponding to closely spaced frequencies. This indicates that the closely spaced natural vibration modes have significant influence on the actual  
Figure 19: MAC and S2 MAC values between the prototype and numerical mode shapes. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Measuring Point**} & \multicolumn{6}{c}{**Mode Shape Obtained by Prototype Data and Numerical Simulation**} \\ \cline{2-11}  & \multicolumn{2}{c}{**Prototype**} & \multicolumn{6}{c}{**Numerical Frequency (Hz)/Order**} \\  & \multicolumn{2}{c}{**(Hz)/Order**} & & & & & & & & \\ \cline{2-11}  & **27.51/2** & **19.70/4** & **19.79/5** & **20.65/6** & **20.71/7** & **23.12/8** &  
vibration in the form of linear combination. Therefore, the S2 MAC values are calculated based on the prototype mode shape and two numerical mode shapes that correspond to relatively high MAC values, and the maximum S2 MAC value [45] is 0.6713. Due to the various error sources in the analysis (such as the mode mixing generated by the very high- and low-frequency vibrations, the signal interferences that frequently occur in complex environments and the calculation errors in the theoretical and numerical analysis), the maximum S2 MAC is considered to be able to reflect the relatively good correlation between the prototype and numerical vibration modes. Therefore, the conclusion can be drawn that the resonances of specific natural vibration modes with frequencies close to the prototype dominant frequency have made a major contribution to the actual vibration of the 3# surface orifice gate. A better simulation can be achieved if more than two numerical vibration modes are used in the calculation of high order MAC. However, the calculation of the coefficients of different numerical modes in the linear combination is complex and attention should be paid to avoid over-fitting the results [47]. As this is not the focus of this paper, the correlation between the prototype and numerical modes is not further analyzed and discussed. In order to save space, the analysis results of the vibration mode correlation under cases 5 to 8 are obtained but not included in the present paper, and the calculated results of the other working conditions are similar to those of working condition 4.