## Appendix A The Derivation Process of the Frequency-Domain Transfer Matrix in Section 3  
Following the derivation given in Refs. [11; 12], the FSI 14-equation model is rewritten in matrix form as  
\[\mathbf{A}(\partial/\partial t)\mathbf{\Phi}(z,t)+\mathbf{B}(\partial/ \partial z)\mathbf{\Phi}(z,t)+\mathbf{C}\mathbf{\Phi}(z,t)=\mathbf{r}(z,t) \tag{1}\]  
where, matrix \(\mathbf{A}\), \(\mathbf{B}\) are the 14-order constant term matrix of time differential and spacial differential respectively, represents the gradient of each variable to time and space. \(\mathbf{C}\) is the friction and viscous damping coefficients, vector \(\mathbf{r}(z,t)\) is the external excitation of the pipeline, \(\mathbf{\Phi}(z,t)\) is the variable vector of the axial section position of the pipeline, and its expression is as follows  
\[\mathbf{\Phi}(z,t)=\left(V,P,\dot{u}_{z},f_{z},\dot{u}_{y},f_{y},\dot{\partial }_{x},M_{x},\dot{u}_{x},f_{x},\dot{\partial}_{y},M_{y},\dot{\partial}_{z},M_{ z}\right)^{\mathrm{T}} \tag{2}\]  
Equation (1) can be obtained via Laplace transform on the Equation (1)  
\[s\mathbf{A}^{\star}(s)\widetilde{\mathbf{\Phi}}(z,s)+\mathbf{B}(\partial/ \partial z)\widetilde{\mathbf{\Phi}}(z,s)=\widetilde{\mathbf{r}}(z,s)+ \mathbf{A}\mathbf{\Phi}(z,0) \tag{3}\]  
The expression of \(\mathbf{A}^{\star}(s)\) is as follows  
\[\mathbf{A}^{\star}(s)=\mathbf{A}+\mathbf{C}/s \tag{4}\]The expression of \(\widetilde{\mathbf{\Phi}}(z,s)\) is as follows  
\[\widetilde{\mathbf{\Phi}}(z,s)=\mathbf{S}(s)\widetilde{\mathbf{\eta}}(z,s) \tag{10}\]  
Then, Equation (10) can be simplified as  
\[s\widetilde{\mathbf{\eta}}(z,s)+\mathbf{\Lambda}(s)(\partial/\partial z) \widetilde{\mathbf{\eta}}(z,s)=s\widetilde{\mathbf{\eta}}_{r}(z,s) \tag{11}\]  
where \(\mathbf{\Lambda}(s)\) and \(\widetilde{\mathbf{\eta}}_{r}(z,s)\) are as follows  
\[\mathbf{\Lambda}(s)=\mathbf{S}^{-1}(s)\mathbf{\Lambda}^{*-1}(s)\mathbf{B} \mathbf{S}(s) \tag{12}\]  
\[\widetilde{\mathbf{\eta}}_{r}(z,s)=(1/s)\mathbf{S}^{-1}(s)\mathbf{\Lambda}^{* -1}(s)[\mathbf{r}(z,s)+\mathbf{A}\mathbf{\Phi}(z,0)] \tag{13}\]  
where \(\mathbf{\Lambda}(s)\) is a diagonal matrix composed of the eigenvalues of \(\mathbf{\Lambda}^{*-1}(s)\mathbf{B}\), that is  
\[\text{det}(\mathbf{B}-\lambda(s)\mathbf{\Lambda}^{*}(s))=0 \tag{14}\]  
\[\mathbf{\Lambda}(s)=\left( \begin{array}{cccc}\lambda_{1}(s)&&&\\ &\lambda_{2}(s)&&\\ &&\ddots&\\ &&&\lambda_{14}(s)\end{array} \right) \tag{15}\]  
where \(\mathbf{S}(s)\) is the eigenvector matrix of \(\mathbf{\Lambda}^{*-1}(s)\mathbf{B}\)  
\[\mathbf{S}(s)=(\xi_{1}(s),\xi_{2}(s),\cdots,\xi_{14}(s)) \tag{16}\]  
The general solution form of Equation (11) is  
\[\widetilde{\mathbf{\eta}}(z,s)=\mathbf{E}(z,s)\widetilde{\mathbf{\eta}}_{0}(s )+\widetilde{\mathbf{\eta}}_{r}^{*}(z,s) \tag{17}\]  
In Equation (17), \(\widetilde{\mathbf{\eta}}_{0}(s)\) is a constant related to boundary conditions, it could be written as  
\[\mathbf{E}(z,s)=\left( \begin{array}{cccc}e^{-sz/\lambda_{1}(s)}&&&\\ &e^{-sz/\lambda_{2}(s)}&&&\\ &&\ddots&\\ &&&e^{-sz/\lambda_{14}(s)}\end{array} \right) \tag{18}\]  
\[\widetilde{\mathbf{\eta}}_{ri}(z,s)=\frac{se^{-sz/\lambda_{i}(s)}}{\lambda_{i} (s)}\int^{\Xi}\widetilde{\mathbf{\eta}}_{ri}(z^{*},s)e^{-sz^{\prime}/\lambda_{ i}(s)}\text{d}z^{*},i=1,2,\cdots,14 \tag{19}\]  
When \(\widetilde{\mathbf{\eta}}_{ri}(z,s)\) is not related to z, then \(\widetilde{\mathbf{\eta}}_{ri}(z,s)\) is equal to \(\widetilde{\mathbf{\eta}}_{ri}(z,s)\). For axially distributed pipeline systems, \(\widetilde{\mathbf{r}}(z,s)=0\), that is \(\mathbf{\Phi}(z,0)=0\). According to Equation (13), Equation (10) and Equation (19), \(\widetilde{\mathbf{\eta}}_{r}(z,s)=\widetilde{\mathbf{\eta}}_{r}^{*}(z,s)=0\).