## 2 Overview of the TMM
### The Theory of the MVTMM
#### 2.3.2 FSI Frequency Domain Solution of Hydraulic Pipeline Based on MVTMM  
The boundary conditions play a crucial role in determining the characteristics of a pipeline. Figure 2 illustrates the schematic representation of the boundaries and constraints within the hydraulic pipeline system. In such systems, fluid boundary conditions are typically induced by pulsating pressure at one end, while they are constrained by a throttle valve at the opposite end. The pipeline's boundary condition must be established based on the actual installation state of the pipeline, which can be modeled as a spring-damping system with six degrees of freedom. This paper focuses on the hydraulic pipeline associated with ship steering gear, where boundary conditions at both ends of the pipeline can be considered as fixed constraints. The boundary conditions and excitations can be expressed in the form of a boundary matrix and an excitation vector. For the system illustrated in Figure 2:  
The boundary conditions and excitation vector at \(l=0\) are as follows:  
\[\mathbf{D_{s}}(s)= \begin{bmatrix}1&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0&-Z_{\text{zs}}&0&0&0&0&0\\ 0&0&-Z_{\text{ys}}&0&0&0&0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0&0&0&-Y_{\text{xs}}&0&0&0\\ 0&0&0&0&1&0&0&0&0&0&-Z_{\text{zs}}&0&0\\ 0&0&0&0&0&-Y_{\text{ys}}&0&0&0&0&0&1&0\\ 0&0&0&0&0&1&0&0&0&0&0&0&-Y_{\text{zs}}\end{bmatrix} _{\mathcal{J}\times 14} \tag{27}\]  
\[\mathbf{Q_{s}}(s)= \begin{bmatrix}P&0&0&0&0&0&0\end{bmatrix}  \tag{28}\]  
Figure 2: A schematic diagram of hydraulic pipeline boundaries and constraints. The boundary conditions and excitation vector at \(l=L\) are as follows:  
\[\mathbf{D_{e}}(s)= \begin{bmatrix}1&0&0&0&0&0&0&-Z_{v}&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0&Z_{ze}&0&0&0&0&0\\ 0&0&Z_{ye}&0&0&0&0&0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0&0&0&0&Y_{xe}&0&0&0\\ 0&0&0&0&1&0&0&0&0&0&0&Z_{xe}&0&0\\ 0&0&0&0&0&Y_{y}&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1&0&0&0&0&0&0&Y_{ze}\end{bmatrix}  \tag{29}\]  
\[Q_{e}(s)=[0&0&0&0&0&0&0] \tag{30}\]  
\[Z_{i}=\frac{k_{i}}{s}+c_{i}+ms \tag{31}\]  
\[Y_{i}=\frac{k_{ti}}{s}+c_{ti}+J_{i}s \tag{32}\]  
In Equations (31) and (32), \(Z_{i}\) and \(Y_{i}\) denote the velocity impedance coefficients and angular velocity impedance coefficients, respectively, \(i=x,y,z\). Additionally, \(k,c,k_{t},c_{t},m,J\) represent linear stiffness, linear damping, rotational stiffness, rotational damping, mass, and the moment of inertia. For the boundary conditions of fixed constraint, it is sufficient to assign maximum values to \(k\) and \(k_{t}\), with a value of \(1\times 10^{12}\) utilized in this study. Here, \(P\) signifies the Laplace transform of the fluid excitation signal and \(Z_{v}\) represents the fluid impedance associated with the throttle valve, which can be expressed as follows:  
\[Z_{v}=2A_{f}\frac{\triangle P}{Q} \tag{33}\]  
where \(\triangle P\) represents the pressure differential across the throttle valve, and \(Q\) denotes the flow rate within the piping system. The recursive algorithm of MVTMM  
For the chain system illustrated in Figure 11, it is necessary to derive a recursive algorithm to obtain the global transfer matrix. The mixed variable transfer matrix for the pipeline segment \(i\) can be expressed as follows:  
\[ \begin{bmatrix}\mathbf{Var}(l_{i},s)\\ \mathbf{Var_{c}}(l_{i-1},s)\end{bmatrix} = \begin{bmatrix}\mathbf{X}_{i,i-1}&- \mathbf{Z}_{i,j-1}\\ \mathbf{Y}_{i,i-1}&\mathbf{W}_{i,i-1}\end{bmatrix}\begin{bmatrix}\mathbf{ Var}(l_{i-1},s)\\ \mathbf{Var_{c}}(l_{i},s)\end{bmatrix}  \tag{34}\]  
The mixed variable transfer matrix for the pipeline segment \(i+1\) can be expressed as follows:  
\[ \begin{bmatrix}\mathbf{Var}(l_{i+1},s)\\ \mathbf{Var_{c}}(l_{i},s)\end{bmatrix} = \begin{bmatrix}\mathbf{X}_{i+1,i}&- \mathbf{Z}_{i+1,i}\\ \mathbf{Y}_{i+1,i}&\mathbf{W}_{i+1,i}\end{bmatrix}\begin{bmatrix}\mathbf{ Var}(l_{i},s)\\ \mathbf{Var_{c}}(l_{i+1},s)\end{bmatrix}  \tag{35}\]  
Combining Equations (34) and (35) yields the following:  
\[ \begin{bmatrix}\mathbf{Var}(l_{i+1},s)\\ \mathbf{Var_{c}}(l_{i-1},s)\end{bmatrix} =