## 4 Modal Analysis of the 3# Surface Office Gate
### Modal Analysis Based on Numerical Simulation
It should be pointed out that the frequently used Block Lanczos method is applied to the modal analysis of the structures without considering the fluid element. The asymmetric mass and stiffness matrices will be generated when the fluid elements are involved in the numerical modal analysis, which makes the Block Lanczos method no longer applicable. In this case, the asymmetric method proposed by ANSYS software is used to perform the modal analysis. As shown in Figure 17, the contact area between the 3# surface orifice gate and discharge flow decreases with the increase of its opening from 0% to 50% and the jet velocity also changes during the opening process, which will affect the dynamic characteristic of the hydraulic gate. In the modal analysis considering the interaction between fluid and solid (i.e., cases 4, 5 and 6), the ratio between the length of the water body and the submergence depth of the surface outlet gate equals 4, which is proved to be appropriate according to the related research on both similar flap gates [39] and steel radial gates [40]. As shown in Figure 16d, the contact area between the water body and the gate leaf is considered as the fluid-solid interface [38] and the upper surface of the water body is constraint-free. Meanwhile, the zero normal displacement constraints are applied on the remaining four areas of the water body and zero displacement and rotation constraints are applied to the right and left edges of the surface orifice gate. Moreover, the zero displacement constraints are applied to the supporting articulations at the rotation centers of the supporting arms and, in case 4, the displacement and rotation of the bottom of the gate is also constrained. When the opening is equal to or greater than 75%, only the  
Figure 16: Numerical models for the concerned structures. (**a**) Mid-level orifice gate; (**b**) arch dam; (**c**) surface orifice gate; (**d**) surface orifice gate considering fluid–structure coupling. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multicolumn{3}{c}{**Natural Frequency**} & \multirow{2}{*}{**Order**} & \multicolumn{3}{c}{**Natural Frequency**} \\ \cline{2-2} \cline{4-9}  & **Dry Mode** & & **Wet Mode** & & **Difference** & **Dry Mode** & **Wet Mode** & **Difference** \\ \hline
1 & 1.78 & 1.55 & 12.98\% & 208 & 26.08 & 25.61 & 1.80\% \\
2 & 2.59 & 2.27 & 12.54\% & 222 & 26.93 & 26.28 & 2.41\% \\
5 & 3.07 & 2.97 & 3.37\% & 223 & 27.05 & 26.47 & 2.14\% \\
6 & 3.39 & 3.12 & 7.89\% & 237 & 27.94 & 27.32 & 2.22\% \\
9 & 3.77 & 3.54 & 6.10\% & 238 & 28.04 & 27.49 & 1.96\% \\
10 & 3.8 & 3.70 & 2.60\% & 254 & 28.99 & 28.64 & 1.21\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of the arch dam. dry modes are considered in the numerical simulation and the geometrical constraint condition of the hydraulic gate is the same as the aforementioned constraint conditions in cases 5 and 6. Table 3 gives the natural frequency distribution of the 3# surface orifice gate, and there are always several vibration modes with frequencies which are quite close to the actual vibration frequency in different cases. Two features of the vibration of the 3# surface outlet gate are found according to the analysis results of the prototype test data. Firstly, the vibration of measuring points T7, T20 and T17 is more intense than the vibration of other measuring points on the gate leaf, and the z-directional vibration of the gate leaf is significantly larger than the vibration in the other two directions. Secondly, the x- and z-directional vibrations of the supporting arms are larger than the y-directional vibration. Based on the numerical analysis, the aforementioned features are consistent with the vibration mode of which the frequency ranges from 25 to 28 Hz.