## 4 Modal Analysis of the 3# Surface Office Gate
### Correlation Analysis of the Prototype and Numerical Vibration Modes  
The combinational analysis method of using the random decrement technique (RDT) and eigensystem realization algorithm (ERA) is a frequently used method for estimating the modal parameters of structures subjected to environmental vibration excitation. This algorithm has the advantages that the analysis is efficient, accurate results can be obtained and little data on the structural free vibration (or impulse response) in a short time are required for the calculation [41]. Therefore, structural modal identification is performed using ERA in the present paper. Before the application of ERA, the structural free vibration data is obtained by RDT based on the testing data of the actual structural dynamic responses. The detailed theoretical deduction process of RDT and ERA methods is described in references [41; 42; 43]. In the following analysis, the vibration mode of the 3# surface orifice gate under working condition 4 is calculated by both the theoretical and numerical methods. In order to calculate the gate modal parameters based on the prototype dynamic response data under unknown environmental excitations, the RDT is first employed to obtain the structural free vibration responses, then the free-decay vibration histories are used as input for the ERA to calculate the minimal realization and the vibration mode of the hydraulic gate system. The first seven modal frequencies calculated by the RDT and ERA joint method in different cases are listed in Table 4. Due to the reduction in the degrees of freedom, it is easy to understand that a large number of vibration modes cannot be identified and the intervals between adjacent modal frequencies are significantly increased. However, there always exists a vibration mode of which the frequency is very close to the dominant frequency of actual structure's dynamic response in every operating condition. Thus, the resonances of these specific vibration modes are naturally considered to have made a major contribution to the actual vibration. In order to further investigate the contributions of specific natural vibration modes to actual structural dynamic responses, the correlation analysis between the mode shapes obtained by the prototype data and numerical simulation is carried out. It is noted that the dominant frequency obtained by the prototype vibration data distributed in a narrow range from 27.5 to 28.5 Hz and the power spectral density of the dominant frequency is much larger than that of other frequency components. Thus, only the vibration modes with similar frequencies to the dominant frequency may significantly contribute to the actual vibration, and the frequencies and shapes for only one vibration mode in the prototype (order 2) and eight vibration modes in the numerical simulation (orders 4 to 12) are included in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode No.**} & \multicolumn{4}{c}{**Modal Frequencies in Different Cases**} & \multirow{2}{*}{**Mode No.**} & \multicolumn{4}{c}{**Modal Frequencies in Different Cases**} \\ \cline{3-10}  & **Case 4** & **Case 5** & **Case 6** & **Case 7** & **Case 8** & & **Case 4** & **Case 5** & **Case 6** & **Case 7** & **Case 8** \\ \hline
1 & 0.65 & 0.67 & 0.86 & 0.59 & 0.60 & 5 & 127.40 & 119.73 & 159.84 & 157.90 & 157.49 \\
2 & **27.51** & **27.50** & **27.53** & **28.18** & **28.07** & 6 & 128.74 & 173.88 & 170.40 & 182.55 & 180.19 \\
3 & 44.22 & 63.21 & 52.48 & 91.61 & 92.86 & 7 & 193.29 & 201.29 & 200.07 & 200.03 & 199.73 \\
4 & 67.49 & 95.06 & 111.54 & 110.09 & 110.25 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Modal frequencies calculated by the random decrement technique (RDT) and eigensystem realization algorithm (ERA) joint method in different cases.