## 3 Theoretical Analyses  
### The Guide Wall Vibration under FSI  
It is noted that the guide wall is a hydraulic structure that is partially submerged in water, as shown in Figure 5. For such structures, the vibration is nonlinear and under the effect of the fluid-structure interaction (FSI) [27; 28], which means the structural loads and responses are mutually influential. To investigate the effect of FSI, the coupled equations of motion (EOM) of FSI are discussed below. Suppose that the fluid around the guide wall is homogenous, incompressible, non-viscous and irrotational. Therefore, the hydrodynamic pressure and velocity potential of the fluid satisfy the following relationships:  
\[\
abla^{2}\mathbf{p}=0 \tag{1}\]  
\[\
abla^{2}\phi=0 \tag{2}\]  
where \(\mathbf{p}\) denotes the hydrodynamic pressure field; \(\phi\) denotes the velocity potential; \(\
abla\) is the del operator. Employing the [PERSON] method to discretize Equation (1), the discrete fluid motion equation can be deduced as follows [29; 30]:  
\[\mathbf{H}\mathbf{p}+\mathbf{A}\dot{\mathbf{p}}+\vec{E}\mathbf{\dot{p}}+\rho\mathbf{B}\vec{\mathbf{r}}+\mathbf{ \eta}_{0}=0 \tag{3}\]  
where \(\rho\) denotes the fluid density; \(\mathbf{r}\) denotes the displacement vector; \(\mathbf{q}_{0}\) denotes the input excitation vector; and  
\[ \begin{array}{l}\mathbf{H}=\iint\limits_{\Omega}\
abla\mathbf{N}\mathbf{\ast}\
abla\mathbf{N }^{\text{T}}\text{d}\Omega\\ \mathbf{A}=\frac{1}{C_{\text{av}}}\iint\limits_{S_{r}}\mathbf{N}\mathbf{N}^{\text{T}}\text {d}S_{r}\\ \mathbf{E}=\frac{1}{C_{\text{av}}^{2}}\iint\limits_{\Omega}\mathbf{N}\mathbf{N}^{\text{T}} \text{d}\Omega+\frac{1}{S_{f}}\iint\limits_{S_{f}}\mathbf{N}\mathbf{N}^{\text{T}} \text{d}S_{f}\\ \mathbf{B}=\left(\iint\limits_{S_{l}}\mathbf{N}\mathbf{N}_{s}^{\text{T}}\text{d}S_{l} \right)\mathbf{\Lambda}\end{array}  \tag{4}\]  
where \(\mathbf{N}\) and \(\mathbf{N}_{s}\) denote the shape functions of the fluid and fluid-structure interface, respectively; \(S_{r}\), \(S_{f}\), and \(S_{l}\) denote the areas of infinite boundary, free surface, and the fluid-structure interface, respectively; \(\Omega\) denotes the volume of the fluid; \(C_{\text{av}}\) denotes the compression wave velocity; \(g\) denotes the gravity acceleration; \(\mathbf{\Lambda}\) denotes the coordinate transformation matrix; the superscript T denotes the transpose of the corresponding matrix. It is noted that the matrix \(\mathbf{H}\) represents the \"stiffness\" of the fluid, and the matrix \(\mathbf{B}\) represents the coupled characteristics on the fluid-structure interface. Figure 5: The left guide wall of TGHS in still water. The EOM of a structure in the fluid is given as the following:  
\[\mathbf{M}_{s}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}+\mathbf{f}_{p}+\mathbf{f}_{0}=0 \tag{5}\]  
where \(\mathbf{x}\) is the displacement vector; \(\mathbf{M}_{s}\), \(\mathbf{C}\), and \(\mathbf{K}\) denote the mass, stiffness and damping matrices of the structure, respectively; \(\mathbf{f}_{p}\) denotes the hydrodynamic nodal forces on the fluid-structure interface; \(\mathbf{f}_{0}\) denotes the external excitation apart from \(\mathbf{f}_{p}\). Gathering the contributions of each fluid element, \(\mathbf{f}_{p}\) can be obtained as the following:  
\[\mathbf{f}_{p}=-\mathbf{B}^{\mathrm{T}}\mathbf{p} \tag{6}\]  
and Equation (5) can be rewritten as follows:  
\[\mathbf{M}_{s}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}-\mathbf{B}^{\mathrm{T}}\bm {p}+\mathbf{f}_{0}=0 \tag{7}\]  
It is noted that the dynamic behavior of a fluid-structure system considering FSI can be obtained as the combination of Equations (3) and (7). The combined EOM is given as follows:  
\[\left[ \begin{array}{cc}\mathbf{M}_{s}&0\\ \rho\mathbf{B}&\mathbf{E}\end{array} \right]\left[ \begin{array}{c}\ddot{\mathbf{x}}\\ \ddot{\mathbf{p}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{C}&0\\ 0&\mathbf{A}\end{array} \right]\left[ \begin{array}{c}\dot{\mathbf{x}}\\ \ddot{\mathbf{p}}\end{array} \right]+\left[