# Coupled Vibration Analysis of Submerged Floating Tunnel System in Wave and Current
## 3 Case Study
### Analysis Results
#### 3.2.2 Auto-Parametric Resonance of the System without Damping  
In order to more directly reflect the coupling effect between the cables and the tube, Figure 4 gives the displacement time history curves of these structural components, in the case of undamped parametric vibration without hydrodynamic effects. The kinetic energy of the system is transferred among the cables and the tube, an obvious \"beating\" phenomenon of the system during parametric resonance is observed.  
The initial displacements of the cable and the tube are set as 0.010 m and 0.050 m, respectively, but the maximum amplitude of the cable reaches 0.949 m. This is significantly greater than the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Components** & **Parameter** & **Symbol** & **Value** \\ \hline \multirow{8}{*}{The \(\hat{\pi}\) pair of cables} & Length & \(l_{i}\) & 161.11 m \\  & Diameter & \(\hat{d}_{i}\) & 0.424 m \\  & Mass per unit length & \(m_{ci}\) & 1108.39 kg/m \\  & Elastic modulus & \(E_{i}\) & \(2.1\times 10^{11}\) Pa \\  & Initial tension & \(T_{0}\) & \(4.04\times 10^{7}\) N \\  & Inclined morning angle & \(\xi_{i1}\) & 90\({}^{\circ}\) \\  & Damping ratio & \(\xi_{i1}\) & 0.0018 \\  & Fundamental frequency & \(\omega_{i1}\) & 3.493 rad/s \\ \hline \multirow{8}{*}{Tunnel tube} & Length & \(L\) & 150 m \\  & Outside diameter & \(D\) & 20 m \\  & Mass per unit length & \(m_{t}\) & \(2.04\times 10^{5}\) kg/m \\  & Flexual rigidity & \(l_{t}\) & 2701 m\({}^{4}\) \\  & Elastic modulus & \(E_{t}\) & \(3.45\times 10^{10}\) Pa \\  & Gravity-buoyancy ratio & \(\eta\) & 0.65 \\  & Damping ratio & \(\xi_{t1}\) & 0.01 \\  & Fundamental frequency & \(\omega_{t1}\) & 6.986 rad/s \\ \hline \multirow{8}{*}{Hydrodynamic environment} & Density & \(\rho_{m}\) & 1000 kg/m\({}^{3}\) \\  & Flow velocity & \(v_{0}\) & 1.18 m/s \\ \cline{1-1}  & Installation depth & \(z_{0}\) & \(-\)20 m \\ \cline{1-1}  & Water depth & \(d\) & 190 m \\ \cline{1-1}  & Wave Height & \(H\) & 8.2 m \\ \cline{1-1}  & Wave Period & \(T\) & 10.8 s \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic parameter settings of the submerged floating tunnel (SFT) model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **With Wave Force** & **Without Wave Force** & **Relative Difference (Based on** **Results without Wave Force)** \\ \hline \multirow{2}{*}{Cable amplitude/m} & 0.402 & 0.402 & \(\rightarrow\)0 \\  & Tube amplitude/m & 0.0185–0.0206 & 0.0194 & \(-\)4.64\(\omega\)\(\rightarrow\)6.19\% \\ \cline{1-1}  & Cable stress variation/MPa & 20.675–23.076 & 22.061 & \(-\)6.28\%\(\rightarrow\)+4.60\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculation results of the steady resonance state due to vortex-induced vibration (VIV) of the cables.  
initial state. As the mass of the cable is much smaller than the tube, the cable is more prone to severely oscillating.  
It should be noted that in this working condition, Equation (9) is simplified into the same form reported in the existing references [16; 35], where the tube was modeled as a spring-mass lump with one degree of freedom, rather than a slender beam. This means that the previous model is a special case of this tube-cable system. More importantly, it also proves the reliability and the superiority of the proposed model in this paper.