## 2 Overview of the TMM
### The Theory of the MVTMM
#### 2.3.2 FSI Frequency Domain Solution of Hydraulic Pipeline Based on MVTMM
\begin{bmatrix}\mathbf{X}_{i+1,j-1}& -\mathbf{Z}_{i+1,j-1}\\ \mathbf{Y}_{i+1,j-1}&\mathbf{W}_{i+1,j-1}\end{bmatrix}\begin{bmatrix}\mathbf{ Var}(l_{i-1},s)\\ \mathbf{Var_{c}}(l_{i+1},s)\end{bmatrix}  \tag{36}\]  
where\[ \begin{split}\mathbf{X}_{i+1,i-1}&=\mathbf{X}_{i+1,i}( \mathbf{I}+\mathbf{Z}_{i,i-1}\mathbf{Y}_{i+1,i})^{-1}\mathbf{X}_{i,i-1}\\ \mathbf{Z}_{i+1,i-1}&=\mathbf{Z}_{i+1,i}+\mathbf{X}_ {i+1,i}(\mathbf{Z}_{i-1,i-1}^{-1}+\mathbf{Y}_{i+1,i})^{-1}\mathbf{X}_{i+1,i} ^{H}\\ \mathbf{Y}_{i+1,i-1}&=\mathbf{Y}_{i,i-1}+\mathbf{X}_ {i,i-1}^{H}(\mathbf{Y}_{i-1,i}^{-1}+\mathbf{Z}_{i,i-1})^{-1}\mathbf{X}_{i,i-1} \\ \mathbf{W}_{i+1,i-1}&=\mathbf{W}_{i,i-1}(\mathbf{I}+ \mathbf{Y}_{i+1,i}\mathbf{Z}_{i,i-1})^{-1}\mathbf{W}_{i+1,i}\end{split}  \tag{37}\]  
Therefore, the global transfer matrix for a hydraulic pipeline comprising \(N\) pipe segments is as follows:  
\[ \begin{split}\begin{bmatrix}\mathbf{Var}(l_{N},s)\\ \mathbf{Var}_{\mathbf{c}}(l_{0},s)\end{bmatrix}=\mathbf{MVT}_{global} \begin{bmatrix}\mathbf{Var}(l_{0},s)\\ \mathbf{Var}_{\mathbf{c}}(l_{N},s)\end{bmatrix}=\begin{bmatrix}\mathbf{X}_{N \_0}&-\mathbf{Z}_{N\_0}\\ \mathbf{Y}_{N\_0}&\mathbf{W}_{N\_0}\end{bmatrix}\begin{bmatrix}\mathbf{Var}(l_ {0},s)\\ \mathbf{Var}_{\mathbf{c}}(l_{N},s)\end{bmatrix}\end{split}  \tag{38}\]  
\[ \begin{split}\mathbf{X}_{N\_0}&=\mathbf{X}^{N}( \mathbf{I}+\mathbf{Z}_{N-1,0}\mathbf{Y}^{N})^{-1}\mathbf{X}_{N-1,0}\\ \mathbf{Z}_{N\_0}&=\mathbf{Z}^{N}+\mathbf{X}^{N}(\mathbf{Z}_{N-1,0}^{-1}+ \mathbf{Y}^{N})^{-1}\mathbf{W}^{N}\\ \mathbf{Y}_{N\_0}&=\mathbf{Y}_{N-1,0}+\mathbf{W}_{N-1,0}( \left(\mathbf{Y}^{N}\right)^{-1}+\mathbf{Z}_{N-1,0})^{-1}\mathbf{X}_{N-1,0}\\ \mathbf{W}_{N\_0}&=\mathbf{W}_{N-1,0}(\mathbf{I}+\mathbf{Y}^{N} \mathbf{Z}_{N-1,0})^{-1}\mathbf{W}^{N}\end{split} \end{split} \tag{39}\]  
In Equation (39), \(\mathbf{X}^{n}\) denotes the transfer matrix of the n-th pipe segment, and \(\mathbf{X}_{N-1,0}\) signifies the global transfer matrix of the preceding \(N-1\) pipe segments. Consequently, the global transfer matrix for the entire pipeline system can be derived by iteratively applying Equations (34)-(37). Furthermore, Equations (27)-(30) are modified in accordance with the principles governing conjugate variables, as illustrated in Equation (40). Equation (41) can then be resolved through integration with Equation (39). Notably, the dimension of the global coefficient matrix in (41) remains constant at \(28\times 28\), irrespective of the number of pipe segments involved. This characteristic contributes to a higher computational efficiency for the MVTMM compared to the PTMM. \[ \begin{split}\begin{bmatrix}0&\mathbf{D}_{s}^{V}(s) \mathbf{Var}(l_{0},s)+\mathbf{D}_{s}^{V_{c}}(s)\mathbf{Var}_{\mathbf{c}}(l_{0},s)=\mathbf{Q}_{s}(s)\\ \mathbf{D}_{\mathbf{c}}^{V}(s)\mathbf{Var}(l_{N},s)+\mathbf{D}_{\mathbf{c}}^ {V_{c}}(s)\mathbf{Var}_{\mathbf{c}}(l_{N},s)=\mathbf{Q}_{\mathbf{c}}(s)\end{split} \end{split} \tag{40}\]  
\[ \begin{bmatrix}0&\mathbf{D}_{s}^{V_{c}}&\mathbf{D}_{s}^{V}&0\\ \mathbf{I}_{Y\times 7}&0&\mathbf{X}_{N\_0}&-\mathbf{Z}_{N\_0}\\ 0&\mathbf{I}_{Y\times 7}&\mathbf{Y}_{N\_0}&\mathbf{W}_{N\_0}\\ \mathbf{D}_{\mathbf{c}}^{V}&0&0&\mathbf{D}_{\mathbf{c}}^{V_{c}}\\ \end{bmatrix} _{28\times 28} \begin{bmatrix}\mathbf{Var}(l_{N},s)\\ \mathbf{Var}_{\mathbf{c}}(l_{0},s)\\ \mathbf{Var}(l_{0},s)\end{bmatrix} _{28\times 1}= \begin{bmatrix}\mathbf{Q}_{s}(s)\\ 0\\ \mathbf{Q}_{\mathbf{c}}(s)\end{bmatrix} _{28\times 1}\end{bmatrix} \tag{41}\]  
Based on the above theoretical derivation, the process of the MVTMM-based research method for FSI frequency-domain vibration characteristics of hydraulic pipelines proposed in this paper can be intuitively expressed using Figure 3: