## 3 Theoretical Analyses
### Parameter Sensitivity Analyses  
To investigate the influence of every parameter on the amplitude coefficient given in Equation (25), sensitivity analyses for each parameter are performed, and reasonable values are set for all the parameters except the target one beforehand. Referring to the minimum value specified in the code for seismic design of hydraulic structures [37], the damping ratio of the structure made of concrete is considered to be 0.05, i.e., \(\xi_{1}=\xi_{2}=\xi_{3}=0.05\). The excitation amplitude ratio \(\gamma\) is considered to be 1.5, which is concluded from previous flow excitation studies [13; 38; 39]. According to the practical structures of TGHS [40] and the added mass of each part of structures, the coupled mass ratio \(\mu\) is calculated to be 31. The parameter \(\lambda_{13}\) is considered to be 0.18, which is deduced from the coupled mass ratio. It is noted that the values of these parameters are only referential values for qualitative studies of each parameter, and the values can be changed to apply this analysis to other practical projects. Based on these predefined values, the dynamic response (amplitude coefficient) with the variation of \(\lambda_{12}\) is shown in Figures 9-11. As shown in Figure 9, the influence of the parameter \(\lambda_{12}\) on the dynamic response is investigated for the case of \(\lambda_{12}\leq\lambda_{13}=0.2\). Two peaks appear when the external excitation frequency is roughly equal to the natural frequency of the primary structure and the accessory structure in the AV system (i.e., \(\omega_{0}=\omega_{1}\) and \(\omega_{0}=\omega_{2}\)). With the increase of \(\lambda_{12}\), the peak value of \(\omega_{0}=\omega_{1}\) decreases, while the peak value of \(\omega_{0}=\omega_{2}\) shows no apparent change. The peak of \(\omega_{0}=\omega_{1}\) demonstrates that the vibration of the accessory structure synchronizes with the primary structure, i.e., AV is generated; and the peak value of \(\omega_{0}=\omega_{1}\) represents the intensity of AV. The significant decrease in the amplitude coefficient when \(\omega_{0}\
eq\omega_{1}\) can be explained as the reduction of vibrational synchronization between the vibrations of the primary structure and the accessory structure. It is noticeable that the excitation frequency of the first peak is slightly greater than \(\omega_{1}\), which is considered to be induced by the constraint between the accessory structure and the foundation. As shown in Figure 10, while \(0.2<\lambda_{12}\leq 5\), there are still two peaks on the curve, but the peak corresponding to \(\omega_{0}=\omega_{1}\) tends to disappear with the increase of \(\lambda_{12}\). Different from Figure 9, the second peak no longer occurs when \(\omega_{0}=\omega_{2}\), but is generated when \(\omega_{0}=\omega_{3}\), and its value increaseswith an increasing \(\lambda_{12}\). This phenomenon can be illustrated as the recession of AV: as \(\lambda_{12}\) increases, which means the frequency \(\omega_{2}\) gradually approaches and exceeds \(\omega_{1}\), the vibration synchronization is weakened, causing a further decrease in the vibration induced by the primary structural vibration. Simultaneously, with a larger \(\omega_{2}\) and a smaller \(\omega_{3}\), the FV of the accessory structure subjected to external excitation becomes more significant so that the peak value of \(\omega_{0}=\omega_{3}\) increases with the increase in \(\lambda_{12}\). Figure 11: The dynamic response of the accessory structure with varying parameter \(\lambda_{12}(\lambda_{12}>5)\). (**a**) The entire graph; (**b**) partial enlarged view of the peak. Figure 9: The dynamic response of the accessory structure with a varying parameter \(\lambda_{12}(\lambda_{12}\leq\lambda_{13}=0.2)\). Another situation where \(\lambda_{12}>5\) is shown in Figure 11. In this situation, the curve becomes single-peaked.