## 3 Validations and Applications
### Numerical Results of Wind-Induced Vibration Responses  
To further verify the accuracy of the ROM established in this study, the wind-induced vibration responses of the main deck section of the GBEB for different wind speeds near the flutter critical wind speed are calculated by ROM and FSI, respectively. The main parameters of the main deck section of GBEB are given in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(y_{h}\)/**m**} & \multirow{2}{*}{\(f_{h}\)/**Hz**} & **Root Variance of** & **First-Order** & **Numerical** & \multirow{2}{*}{**Relative Error, \(\delta\)/\%**} \\  & & & **Force Coefficient** & **Approximation** & **Simulation** & \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{0.004} & \multirow{2}{*}{1} & \(C_{L}\)\({}^{\prime}\) & 0.0145 & 0.0140 & 3.57 \\  & & & \(C_{M}\) & 0.0038 & 0.0036 & 5.55 \\  & & & \(C_{L}\)\({}^{\prime}\) & 0.1447 & 0.1396 & 3.65 \\  & & & \(C_{M}\)\({}^{\prime}\) & 0.0378 & 0.0359 & 5.29 \\
3 & 0.04 & 4 & \(C_{L}\)\({}^{\prime}\) & 0.4990 & 0.4796 & 4.05 \\ \hline \multirow{2}{*}{3} & \multirow{2}{*}{0.04} & \multirow{2}{*}{4} & \(\text{Note: }y_{h}=\text{oscillation amplitude; }f_{h}=\text{oscillation frequency; relative error }\delta=\left|Y_{1}-Y_{2}\right|/Y_{1}\right|\times 100\%; Y_{1}=\text{the results of first-order approximation; }Y_{2}=\text{the results of numerical simulation.}\) \\ \cline{1-1} \cline{2-5}  & & & & & \\ \end{tabular}
\end{table} 
Table 1: Parameters of heaving motion and comparisons of results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(\theta_{a}\)/\({}^{\circ}\)} & \multirow{2}{*}{\(f_{a}\)/**Hz**} & **Root Variance of** & **First-Order** & **Numerical** & \multirow{2}{*}{**Relative Error, \(\delta\)/\%**} \\  & & & **Force Coefficient** & **Approximation** & **Simulation** & \\ \hline \multirow{2}{*}{4} & \multirow{2}{*}{3} & \multirow{2}{*}{1} & \(C_{L}\)\({}^{\prime}\) & 0.1460 & 0.1474 & 0.95 \\  & & & \(C_{M}\) & 0.0371 & 0.0392 & 5.36 \\ \multirow{2}{*}{5} & \multirow{2}{*}{6} & \multirow{2}{*}{1} & \(C_{L}\)\({}^{\prime}\) & 0.2919 & 0.2963 & 1.48 \\  & & & \(C_{M}\)\({}^{\prime}\) & 0.0743 & 0.0786 & 5.47 \\ \multirow{2}{*}{6} & \multirow{2}{*}{6} & \multirow{2}{*}{4} & \(C_{L}\)\({}^{\prime}\) & 0.2761 & 0.2910 & 5.12 \\  & & & \(C_{M}\)\({}^{\prime}\) & 0.0919 & 0.0871 & 5.51 \\ \hline \hline \end{tabular} Note: \(\theta_{a}=\text{oscillation amplitude; }f_{a}=\text{oscillation frequency; relative error }\delta=\left|Y_{1}-Y_{2}\right|/Y_{1}\right|\times 100\%; Y_{1}=\text{the results of first-order approximation; }Y_{2}=\text{the results of numerical simulation.}\) \\ \end{tabular}
\end{table} 
Table 2: Parameters of pitch motion and comparisons of results. In the FSI numerical simulation, the turbulence intensity is set to 0.5%, the turbulent viscosity ratio is set to 2, and the initial uniform flow field turbulence characteristic is set for the whole calculation domain. The incoming wind speed is set to 4.735 m/s, 5 m/s, and 5.625 m/s, respectively, corresponding to the wind speeds of 35 m/s, 40 m/s, and 45 m/s for the prototype main deck section, respectively. Figure 9 shows the numerical results of wind-induced vibration responses of the main deck section by FIS numerical simulation for different incoming wind speeds, respectively. As shown in Figure 9, when the incoming wind speed is 35 m/s for the prototype main deck section, the torsional displacement and vertical displacement of the prototype girder have no obvious flutter divergence phenomenon. However, when the incoming wind speed is 40 m/s for the prototype main deck section, the torsional displacement and vertical displacement of the main deck section are stable, and the displacement attenuation rate is 0.05%, which is close to 0, which can be considered close to the flutter critical wind speed.