# Coupled Vibration Analysis of Submerged Floating Tunnel System in Wave and Current
## 2 Modeling of the SFT System
### Solution of Equations  
In line with the references [18; 31], we assume that the parameter excitations on anchor-cables mainly come from the tension variations in cables, rather than the displacements of the upper end connected to the tube. This is reasonable when the displacement of the tube is much smaller than the length of cables, because of the strong stiffness of the mooring system and the length of the tethers, which reach up to 200 m or even longer.  
In terms of the mode superposition method, the displacements of the structure can be approximately expressed in standard forms as follows [18; 31]:  
\[\left\{\begin{array}{l}v_{i}(x_{i},\ t)=\sum\limits_{m_{i}}^{\infty}v_{i}(t) \sin\left(\frac{m_{i}\tau x_{i}}{l_{i}}\right)m_{i}=1,\ 2,\ 3\ldots\\ z(x,\ t)=\sum\limits_{m}^{\infty}z(t)\sin\left(\frac{m\tau x}{L}\right)m=1,\ 2,\ 3 \ldots\end{array}\right. \tag{8}\]  
where \(v_{i}(t)\) and \(z(t)\) are the generalized coordinates, \(\sin(m_{i}\tau x_{i}/l_{i})\) and \(\sin(m\tau x/L)\) are the shape functions.  
[PERSON] et al. [32] pointed out that the first-order vibration of the SFT tube contributes the most to the displacement response. [PERSON] [33] concluded that the first-order mode dominates when external excitation is applied at the end of the cable. Hence, in this article, we deal with the fundamental mode vibration, and then take \(m=m_{i}=1\).  
Substituting Equation (8) into Equation (1), we get:  
\[\begin{array}{l}\tilde{v}_{i}+\left[\omega_{11}^{2}+\frac{E_{i}A_{ci}\pi^{ 2}}{m_{i}l_{i}^{3}}z\cdot\sin\left(\frac{\pi\tau x_{i}}{L}\right)\cdot\sin \alpha_{i}\right]\cdot v_{i}+2\xi_{i1}\omega_{i1}\hat{v}_{i}+\frac{E_{ci}A_{ci} \pi^{4}}{4m_{i}l_{i}^{4}}\
u_{i}^{3}\\ =\frac{2}{m_{ci}\pi}\rho_{w}d_{i}C_{Li}v_{0}^{2}\cos(2\pi f_{i}t)-\frac{4}{3m_ {ci}\pi}\rho_{w}d_{i}C_{Di}\hat{v}_{i}|\hat{v}_{i}|\qquad i=1,\ 2,\ \ldots,\ n\\ \qquad\tilde{z}+\left\{\overline{\omega}_{t1}{}^{2}+2\sum\limits_{i}^{n} \Bigl{[}\frac{2}{m_{i}L}\frac{E_{ci}A_{ci}}{l_{i}}\cdot\sin\alpha_{i}\cdot\sin ^{2}\bigl{(}\frac{\pi\hat{v}_{i}\pi}{L}\bigr{)}\Bigr{]}\Bigr{\}}z+2\xi_{i1} \omega_{t1}\hat{z}\\ =-2\sum\limits_{i}^{n}\frac{2}{m_{i}L}\left[\frac{E_{ci}A_{ci}\pi^{2}}{4l_{i}^ {2}}v_{i}^{2}\right]\cdot\sin\bigl{(}\frac{\pi\hat{v}_{i}\pi}{L}\bigr{)}\cdot \sin\alpha_{i}+\frac{1}{m_{i}L}\pi D^{2}\rho_{w}(1+C_{m})\hat{w}_{z}-\frac{D_{w }(t)}{m_{i}L}\end{array} \tag{9}\]  
where \(\omega_{i1}\) is the first-order natural frequency of the \(i^{\#}\) pair of cables, \(\omega_{i1}=\Bigl{[}(\pi/l_{i})^{2}T_{0}/\overline{m}_{ci}\Bigr{]}^{1/2}\), \(\overline{m}_{ci}=m_{ci}+(\pi d_{i}^{2}/4)\rho_{w}C_{mi}\); \(\xi_{i1}\) is the first-order damping ratio of the \(i^{\#}\) cables; \(\overline{\omega}_{t1}\) is the fundamental frequency of the tube without the cables, \(\overline{\omega}_{t1}=\Bigl{[}E_{i}l_{i}(\pi/L)^{4}/\overline{m}_{t}\Bigr{]} ^{1/2},\overline{m}_{t}=m_{t}+(\pi D^{2}/4)\rho_{w}C_{m}\); \(\xi_{t1}\) is the first-order damping ratio of the tube; \(D_{w}(t)\) is the item associated with the hydraulic resistance on the tube, and can be computed as: \(D_{w}(t)=\frac{1}{2}\rho_{w}DC_{D}\int_{0}^{L}\left(w_{z}-\dot{z}\sin\frac{ \pi\tau x}{L}\right)\cdot\bigl{|}w_{z}-\dot{z}\sin\frac{\tau x}{L}\bigr{|}\cdot \sin\bigl{(}\frac{\pi\tau x}{L}\bigr{)}\mathrm{d}x\).  
It can be found in Equation (9) that the motions of the cables and tube are coupled to each other, forming an autoparametric vibration system. Additionally, the tube-cable system is also subjected to the external hydrodynamic forcing excitations (i.e., vortex-induced force on the cables, and wave force on the tube). This better reflects the nonlinear vibration characteristics of the SFT system in a marine environment. The fourth-order Runge-Kutta method is adopted to solve Equation (9) within MATLAB software with a calculation step length of 0.01 s to guarantee the convergence of the integration method [24].