## 3 Theoretical Analyses
### The Guide Wall Vibration under FSI
\begin{array}{cc}\mathbf{K}&-\mathbf{B}^{ \mathrm{T}}\\ 0&\mathbf{H}\end{array} \right]\left[ \begin{array}{c}\mathbf{x}\\ \mathbf{p}\end{array} \right]=\left[ \begin{array}{c}\mathbf{-}\mathbf{f}_{0}\\ -\mathbf{q}_{0}\end{array} \right] \tag{8}\]  
As an arbitrary load can be Fourier transformed into the sum of harmonic excitations with different frequencies, the input excitation \(\mathbf{q}_{0}\) can be considered as a harmonic force,  
\[q_{0}=Q_{0}e^{i\omega t} \tag{9}\]  
Matching the form of the input excitation, the solution of Equation (7) can be expressed as follows:  
\[\left\{ \begin{array}{c}x=x_{0}e^{i\omega t}\\ p=p_{0}e^{i\omega t}\end{array} \right. \tag{10}\]  
As mentioned above, the fluid is considered to be incompressible ( \(C_{w}\rightarrow\infty\)), which means \(\mathbf{A}\to 0\); the effect of the free surface is usually neglected, so \(\mathbf{E}\to 0\). Moreover, the external excitation \(\mathbf{f}_{0}\) is equal to zero in this case as the excitation is flow excitation only. Substituting into Equation (8), the equation can be obtained as follows:  
\[\left[ \begin{array}{cc}-\omega^{2}\mathbf{M}_{s}+i\omega\mathbf{C}+\mathbf{K}&-\mathbf{B}^ {\mathrm{T}}\\ -\rho\omega^{2}\mathbf{B}&\mathbf{H}\end{array} \right]\left[ \begin{array}{c}\mathbf{x}_ {0}\\ \mathbf{p}_{0}\end{array} \right]=\left[ \begin{array}{c}0\\ -\mathbf{Q}_{0}\end{array} \right] \tag{11}\]  
The displacement \(\mathbf{x}_{0}\) can be obtained as follows:  
\[\left[\omega^{2}\left(\mathbf{M}_{s}+\rho\mathbf{B}^{\mathrm{T}}\mathbf{H}^{-1}\mathbf{B} \right)-i\omega\mathbf{C}-\mathbf{K}\right]\mathbf{x}_{0}=\mathbf{B}^{\mathrm{T}}\mathbf{H}^{-1}\bm {Q}_{0} \tag{12}\]  
Equation (12) can be rewritten as follows:  
\[(\mathbf{M}_{s}+\mathbf{M}_{a})\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{P} \tag{13}\]  
where  
\[\mathbf{M}_{a}=\rho\mathbf{B}^{\mathrm{T}}\mathbf{H}^{-1}\mathbf{B}\mathbf{P}=-\mathbf{B}^{\mathrm{T}} \mathbf{H}^{-1}\mathbf{q} \tag{14}\]  
Equation (13) is a common form of EOM. It is noted that \(\mathbf{P}\) is a coupled input excitation. In practical prototype tests, the value of \(\mathbf{P}\) can be measured as the fluctuation pressure acting on the fluid-structure interface, which is an actual value that can take all the coupled characteristics into consideration. Therefore, the effect of FSI can be concluded as \(\mathbf{M}_{a}\), which is defined as the added mass in existing studies. The value of added mass is difficult to calculate. In existing studies, several models have been established to calculate the approximate value of the added mass. For large hydraulic structures such as dams, the most classic method is the Westergaard method [31; 32]. The added mass can be calculated as follows:  
\[M_{a}=\frac{7}{8}\rho\sqrt{\hbar z} \tag{15}\]  
where \(\rho\) denotes the water density; \(h\) denotes the depth of water above the base; \(z\) denotes the distance from the surface. Concluded from practical hydraulic projects [5], the value in Equation (15) is larger than reality. In the engineering field, the added mass is usually considered as follows:  
\[M_{a}=0.5\rho\sqrt{\hbar z} \tag{16}\]  
With an added mass, the effect of FSI can be taken into consideration, and a more simplified theoretical model of the guide wall can be established.