## 4 Verifications
### Numerical Simulation  
Based on the above analysis, it can be concluded that AV can be detected in the guide wall structure, and it indeed amplifies the vibrations of certain blocks on the guide wall during the discharge process. To explore the detailed mechanism of guide wall vibration, numerical simulations for the vibration mode of the dam and the guide wall are proposed using the commercial software ANSYS. Compared with the operational modal parameter identification by eigensystem realization algorithm (ERA) on the same project [42], it is shown that the vibration appears as the vibration of an integral structure, rather than the independent vibration of each block, due to the transmission of shear forces through joints and thermal expansion (the tests were implemented in the summer). Therefore, the cementation between adjacent blocks is considered, with the integral numerical models built, and the frequencies of the dam and the guide wall are calculated for both dry and wet modes. The numerical models and the corresponding material parameters (density \(\rho\), elastic modulus \(E\), and Poisson ratio \(\
u\)) are shown in Figure 19. For the guide wall, the connection to the foundation is simplified as the all-degrees of freedom (all-DOF) constraint applied to the bottom, which corresponds to the constraints in the independent FV system; the connection to the dam is simplified as the all-DOF without the longitudinal displacement constraint for the diving joint between the guide wall and the dam, corresponding to the constraints in the AV system. For the AFCV system, both kinds of constraints are applied. For the dam, the all-DOF constraint is applied to the bottom, and the diving joint between the guide wall and dam is considered as the all-DOF without the longitudinal displacement constraint. For the wet mode, the range of water is considered to be twice as much as the height of the guide wall or the dam. Figure 19: Numerical model of concerned structures. (**a**) Guide wall; (**b**) spillway dam. Table 2 shows several natural frequencies of the dam for both dry and wet modes. There are several modes with frequencies that match actual dominant frequencies. It is noted that the dam is a structure with closely spaced natural frequencies, whether in dry or wet mode. That is, under the flow fluctuation with a broad frequency band, there is always a mode with a natural frequency which is close to the excitation frequency. As mentioned above, by changing constraints, the vibration modes for the AV system, independent FV system and AFCV system can be respectively analyzed. Table 3 illustrates some typical natural frequencies of the guide wall. There are also several modes with frequencies that match the actual dominant frequencies. According to prototype tests, the root mean square (RMS) of the horizontal displacement reaches a maximum in the 6# guide wall block. Through numerical simulation, it can be found that for all three vibration systems, there are always several natural frequencies in the range of 2-7 Hz that are in accordance with actual vibrational characteristics. The corresponding vibration patterns, their orders, and natural frequencies are illustrated in Figure 20. Therefore, these vibration modes can be inferred to participate in the actual vibration. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Order** & **Dry Mode (Hz)** & **Wet Mode (Hz)** & **Order** & **Dry Mode (Hz)** & **Wet Mode (Hz)** \\ \hline
1 & 2.70 & 1.88 & 23 & 8.76 & 6.03 \\
2 & 2.98 & 2.21 & 25 & 9.10 & 6.27 \\
5 & 3.95 & 3.24 & 30 & 9.60 & 6.67 \\
9 & 6.32 & 4.20 & 35 & 9.79 & 6.99 \\
10 & 6.40 & 4.39 & 37 & 9.83 & 7.18 \\ \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of the dam.