# Steady-Suction-Based Flow Control of Flutter of Long-Span Bridge
## 3 Results and Discussion  
### Experiment Validation  
To verify the positive effect of the steady-suction control on the bridge flutter performance, two types of information should be collected: (1) the geometric information of the model and (2) important aerodynamic parameters, including the FDs and critical flutter speed obtained through the FDs. Table 2 lists the structural information of the actual bridge and bridge section model, including the geometric dimensions, mass characteristics, and vertical and torsional frequencies. Flutter derivatives are important aerodynamic parameters for the evaluation of the vibration tendency of a bluff body under flow action, and they provide a better insight into the flutter sensitivity of bridge decks. Therefore, FD identification is an indispensable step. [PERSON] and [PERSON] (2017) proposed the application of FDs to define a linear aeroelastic system. The FDs are identified in experiments and used to estimate the self-excited forces acting on the vibrating body, which can be expressed as linear combinations of the body motions:  
\[F_{se}=\left[ \begin{array}{c}\frac{1}{2}\rho U^{2}B\left(KH_{1}^{*}\frac{ \dot{h}}{U}+KH_{2}^{*}\frac{\dot{B}\dot{u}}{U}+K^{2}H_{3}^{*}\alpha+K^{2}H_{4}^ {*}\frac{h}{B}\right)\\ \frac{1}{2}\rho U^{2}B\left(KA_{1}^{*}\frac{\dot{h}}{U}+KA_{2}^{*}\frac{\dot{B} \dot{u}}{U}+K^{2}A_{3}^{*}\alpha+K^{2}A_{4}^{*}\frac{h}{B}\right)\end{array} \right] \tag{1}\]  
where \(F_{se}\) represents the motion-related self-excited forces, including the aeroelastic lift and pitching moment forces; \(B\) denotes the section width, \(\rho\) is the air density, \(U\) is the far-field wind speed, \(K\) is the reduced frequency defined as \(K=2\pi fB/U\) with \(f\) as vibration frequency of the model; \(H_{i}^{*}\) and \(A_{i}^{*}\) (\(i=1-4\)) are the FDs; and \(h\) and \(\alpha\) are the vertical and torsional displacements, respectively. Numerous theories have been developed to determine the FDs in wind tunnel tests ([PERSON] et al., 2016; [PERSON] et al., 2017; [PERSON] et al., 2018). In this study, the iterative least-squares (ILS) theory proposed by [PERSON] and [PERSON] (2017) was adopted to determine the FDs. Specifically, the experiments were conducted with the free-vibration technique, and the initial forced displacements were applied to the bridge model in the vertical and rotational directions. Then, the free-decay oscillations under different wind speeds were measured (inlet wind \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Wind Velocity (m/s)** & **Position** & **Interval** & **Flow Rate** \\ \hline \(s\)00 & 2, 4, 6, 8, 10, 12, 14 & - & - & - \\ _l-2d_-10 & 2, 4, 6, 8, 10, 12, 14 & windward & 2\(H\) & 10 L/min \\ _m-2d_-10 & 2, 4, 6, 8, 10, 12, 14 & middle & 2\(H\) & 10 L/min \\ _b-2d_-10 & 2, 4, 6, 8, 10, 12, 14 & leeward & 2\(H\) & 10 L/min \\ _l-1d_-5 & 2, 4, 6, 8, 10, 12, 14 & windward & 1\(H\) & 5 L/min \\ _l-2d_-5 & 2, 4, 6, 8, 10, 12, 14 & windward & 2\(H\) & 5 L/min \\ _l-3d_-5 & 2, 4, 6, 8, 10, 12, 14 & windward & 3\(H\) & 5 L/min \\ _l-4d_-5 & 2, 4, 6, 8, 10, 12, 14 & windward & 4\(H\) & 5 L/min \\ _l-2d_-7 & 2, 4, 6, 8, 10, 12, 14 & windward & 2\(H\) & 7 L/min \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test cases for flutter performance of section model with various control parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Unit** & **Real Bridge** & **Similarity Ratio** & **Section Model** \\ \hline Length/L & m & 1624 & 0.7 \\ Breadh/B & m & 31 & 1/80 & 0.3875 \\ Height/H & m & 4 & 1/80 & 0.05 \\ Equivalent mass & kg/m & 23,687 & 1/802 & 3.701 \\ Equivalent mass moment of inertia & kg\(\cdot\)m\({}^{2}\)/m & \(2.501\times 105\) & 1/804 & 0.061 \\ Frequency of heaving mode & Hz & 0.097 & 26.66 & 2.586 \\ Frequency of pitching mode & Hz & 0.27 & 25.19 & 6.8 \\ Frequency ratio & & 2.78 & 0.95/1 & 2.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of bridge section model. speeds of 2, 4, 6, 8, 10, 12, and 14 m/s were considered).