## 3 Discrete Time Transfer Matrix Method  
Referencing the deduction process in [9; 18; 44], the time domain solution (DTTMM) of the FSI fourteen-equation model is studied in this section. The fourteen equations in Section 2.1 can be expressed with matrixes as:  
\[\mathbf{A}\frac{\partial\mathbf{\Phi}(z,t)}{\partial t}+\mathbf{B}\frac{ \partial\mathbf{\Phi}(z,t)}{\partial z}+\mathbf{C}\mathbf{\Phi}(z,t)+\mathbf{ D}=\mathbf{y}(z,t) \tag{27}\]  
where \(\mathbf{A}\) and \(\mathbf{B}\) are matrices of constant coefficients about time and space differentiation, respectively, \(\mathbf{C}\) is the vector containing friction and viscous damping, \(\mathbf{D}\) is the coefficient vector including gravity, and the vector \(\mathbf{y}\) describes the external excitation. The state vector \(\mathbf{\Phi}\), containing fourteen variables of system, can be written as:  
\[\mathbf{\Phi}(z,t)=\left[V\;\dot{w}_{z}\;f_{z}\;M_{y}\;\dot{\theta}_{y}\;\dot {w}_{x}\;f_{x}\;M_{x}\;\dot{\theta}_{x}\;\dot{w}_{y}\;f_{y}\;\dot{\theta}_{z} \;M_{z}\right]^{\text{T}} \tag{28}\]  
Then, Equation (27) is discretize in equal time interval, and a new set of the variables vector \(\mathbf{\varphi}(z,t_{i})=\mathbf{\Phi}(z,t)\) is substituted into Equation (27), giving:  
\[\mathbf{A}\frac{\mathbf{\varphi}(z,t_{i})-\mathbf{\varphi}(z,t_{i-1})}{ \Delta t}+\mathbf{B}\frac{\partial\mathbf{\varphi}(z,t_{i})}{\partial z}+ \mathbf{C}\mathbf{\varphi}(z,t_{i})=\mathbf{Y} \tag{29}\]  
Let \(\mathbf{A}^{*}=-\mathbf{B}^{-1}(\frac{1}{\Delta t}\mathbf{A}+\mathbf{C})\), \(\mathbf{y}^{*}=\mathbf{B}^{-1}\Big{(}\mathbf{Y}+\frac{1}{\Delta t}\mathbf{A} \mathbf{\varphi}(z,t_{i-1})\Big{)}\); the Equation (29) can be simplified as:  
\[\frac{\partial\mathbf{\varphi}(z,t_{i})}{\partial z}=\mathbf{A}^{*}\mathbf{ \varphi}(z,t_{i})+\mathbf{y}^{*} \tag{30}\]Let \(\widetilde{\mathfrak{n}}(z,t_{i})=\mathbf{V}^{-1}\boldsymbol{\varphi}(z,t_{i}), \widetilde{\mathfrak{n}}_{r}(z,t_{i})=\mathbf{V}^{-1}\mathbf{y}^{*}\); Equation (30) can be written as:  
\[\frac{\partial\widetilde{\mathfrak{n}}(z,t_{i})}{\partial z}=\mathbf{T} \widetilde{\mathfrak{n}}(z,t_{i})+\widetilde{\mathfrak{n}}_{r}(z,t_{i}) \tag{31}\]  
where \(\mathbf{T}=\text{diag}\{\,\lambda_{1}(t_{i}),\lambda_{2}(t_{i}),\cdots\lambda _{14}(t_{i})\}\) is the eigenvalues of \(\mathbf{A}^{*}\), and  
\(\mathbf{V}=\left[ \begin{array}{cc}\xi_{1}(t)&\xi_{2}(t)&\cdots&\xi_{14}(t) \end{array} \right]^{\mathsf{T}}\) is the eigenvector of \(\mathbf{A}^{*}\). According to the general form of the solution of the first order nonhomogeneous linear differential equations, the general solution form of Equation (31) is:  
\[ \begin{array}{rl}\widetilde{\mathfrak{n}}(z,t_{i})&=\exp[(z-0)\mathbf{T}] \widetilde{\mathfrak{n}}(0,t_{i})+\int_{0}^{z}[(z-0)\mathbf{T}]\widetilde{ \mathfrak{n}}_{r}(z,t_{i})ds\\ &\quad\quad\quad\quad=e^{(z-0)\mathbf{T}}\widetilde{\mathfrak{n}}(0,t_{i})+ \int_{0}^{z}\boldsymbol{\varepsilon}^{(z-0)\mathbf{T}}\widetilde{\mathfrak{n}} _{r}(z,t_{i})ds\\ &\quad\quad\quad\quad\quad=\mathbf{E}(z,t_{i})\widetilde{\mathfrak{n}}(0,t_{i} )+\widetilde{\mathfrak{n}}_{r}^{*}(z,t_{i})\end{array}  \tag{32}\]  
where  
\[ \begin{array}{rl}\mathbf{E}(z,t_{i})=\text{diag}\{\,e\lambda_{1}(t_{i})z,e \lambda_{2}(t_{i})z,\cdots,e\lambda_{14}(t_{i})z\}\\ \widetilde{\mathfrak{n}}_{r}^{*}(z,t_{i})=\int_{0}^{z}\exp[(z-s)\mathbf{T}] \widetilde{\mathfrak{n}}_{r}(z,t_{i})ds=\int_{0}^{z}\exp(\mathbf{T}s) \widetilde{\mathfrak{n}}_{r}(z-s,t_{i})ds\end{array}  \tag{33}\]  
Substituting \(\widetilde{\mathfrak{n}}(z,t_{i})=\mathbf{V}^{-1}\boldsymbol{\varphi}(z,t_{i})\) into Equation (32) gives:  
\[\boldsymbol{\varphi}(z,t_{i})=\mathbf{V}\mathbf{E}(z,t_{i})\widetilde{ \mathfrak{n}}(0,t_{i})+\mathbf{V}\widetilde{\mathfrak{n}}_{r}^{*}(z,t_{i}) \tag{34}\]  
For the pipe system currently studied, there is no spatial distributive excitation.