## 4 Verifications
### Vibration Mechanism Analyses for the TGHS Left Guide Wall  
Based on the spectrum analyses, XWT correlation analyses, and numerical simulation, the detailed mechanism of the TGHS left guide wall vibration can be clarified with the proposed AFCV theory. The guide wall vibration consists of the AV induced by the dam vibration and the independent FV caused by flow fluctuation. The contribution of AV depends on the location of the guide wall block. Using the theoretical model detailed in Section 3, the mechanism can be verified as below.  
According to the modal analyses, the dam is a structure with closely spaced modes, therefore, there is always a natural frequency of the dam that matches the excitation frequency. The concerned frequencies for all of the corresponding vibration systems have participated in the vibration, which means they also match the excitation frequencies. Referring to the corresponding natural frequencies in Figure 20, the parameter \(\lambda_{12}\) can be considered as 0.8, and \(\lambda_{13}\) can be considered as 1. Consequently, the dynamic response of the guide wall is shown in Figure 21.  
Figure 21: The dynamic response of the TGHS left guide wall.  
Figure 20: Vibrational modes which are in accordance with actual vibrational characteristics. (**a**) The associated vibration (AV) system; (**b**) the independent forced vibration (FV) system; (**c**) the associated-forced coupled vibration (AFCV) system.  
For this case, parameters \(\xi_{2}\) and \(\gamma\) are considered to be corresponding parameter values of the integral guide wall, as well as the parameter \(\lambda_{12}\), as mentioned above. For upstream guide wall blocks, which are further from the dissipation zone and closer to the dam, the flow fluctuations are weak, and the frequency bands are narrow. As mentioned above, the external excitation matches the natural frequency of the dam; therefore, \(\lambda_{01}\) is located around 1. As shown in Figure 21, there is a peak value at \(\lambda_{01}=1\), indicating that the guide wall vibration is significantly affected by the dam, i.e., AV is significant, corresponding to Figure 18a-c. While, for downstream guide wall blocks, which are closer to the dissipation zone, the flow here demonstrates strong fluctuation and a broad frequency bandwidth. The ratio between the frequencies of external excitation and the dam (i.e., \(\lambda_{01}\)) range from 0.2 to 5 (analogous to Figure 10). It is noted that the two peaks are located in this range, which indicates that both AV and independent FV contribute to the guide wall vibration. Due to the greater peak value of \(\omega_{0}=\omega_{2}\), the contribution from independent FV is more significant than that from AV. Because there are also several natural frequencies of the dam which are in this range, it illustrates a broader correlation between the dam block and the guide wall blocks, as seen in Figure 18d-h.  
Therefore, by the verification of prototype tests, it can be demonstrated that the vibration mechanism of the TGHS left guide wall can be distinctly clarified by the theoretical model, in other words, the proposed AFCV theory is valid and correct.