## 2 Framework of Nonlinear Flutter Analysis
### Expression of Nonlinear Aerodynamic Self-Excited Force  
Based on the aerodynamic self-excited force model proposed by [PERSON] (Equations (1) and (2)), [PERSON] et al. proposed a nonlinear aerodynamic self-excited force model [18]. The expressions for the nonlinear self-excited forces \(F(L)\) and \(F(M)\) in this model are as follows:  
\[F(L)=\rho U^{2}B\left[KH_{1}^{\frac{h}{2}}\frac{\dot{h}}{U}+\left(KH_{2}^{2}+KH _{2,02}^{\frac{B|\dot{a}|}{U}}\right)\frac{B\dot{a}}{U}+(K^{2}H_{3}^{2}+K^{2}H _{3|\alpha|}|\alpha|)\alpha+K^{2}H_{4|\alpha|}^{\frac{h}{2}}+K^{2}H_{4|\alpha |}\alpha^{2}\right] \tag{10}\]  
\[F(M)=\rho U^{2}B^{2}\left[KA_{1}^{\frac{s}{2}}\frac{\dot{h}}{U}+\left(KA_{2}^ {s}+KA_{2,02}^{\frac{B|\dot{a}|}{U}}\right)+KA_{2,03}^{\frac{B^{2}\dot{a}^{2}} {U^{2}}}\right)\frac{B\dot{a}}{U}+K^{2}A_{3}^{s}\alpha+K^{2}A_{4}^{\frac{s}{2 }}\frac{h}{B}\right] \tag{11}\]  
where \(F_{L}\) represents aerodynamic lift force, \(F_{M}\) represents aerodynamic torque, \(\rho\) is air density, \(U\) is fluid velocity, \(B\) is the cross-sectional width of the girder, \(K\) is the reduced frequency, \(K=B\omega/U\). \(H_{i}^{s}\) and \(A_{i}^{s}\) are flutter derivatives, and \(H_{2,02}^{s}\), \(H_{3,02}^{s}\), \(H_{4,01}^{s}\), \(H_{4,02}^{s}\), \(A_{2,02}^{s}\), \(A_{2,03}^{s}\) are small perturbation flutter derivatives. These flutter derivatives are dimensionless quantities that depend on the reduced frequency or reduced wind speed. Similarly, these flutter derivatives can be expressed as functions of the reduced wind speed or reduced frequency. For computational convenience, in this study, these flutter derivatives are expressed as functions of the reduced wind speed.  
\[H_{i}^{s},A_{i}^{s}=f(V_{r})=f(K)=H,A_{i1}^{s}\cdot V_{r}+H,A_{i2}^{s}\cdot V_ {r}^{2} \tag{12}\]  
\[H_{2}^{s}\lor A_{2}^{s}=f(V_{r})=f(K)=H_{i1}^{s}\lor A_{i1}^{s}\cdot V_{r}+H_ {i2}^{s}\lor A_{i2}^{s}\cdot V_{r}^{2}+H_{3}^{s}\lor A_{i3}^{s}\cdot V_{r}^{2} +H_{4}^{s}\lor A_{i4}^{s}\cdot V_{r}^{2} \tag{13}\]  
Nonlinear flutter self-excited forces (10) and (11) not only include linear first-order components but also nonlinear second-order components. In the nonlinear self-excited force model proposed by [PERSON] et al., the discrete results of flutter derivatives were obtained through wind tunnel tests on a bridge deck section model [18]. In the flutter motion equation proposed by [PERSON], the flutter derivatives in the flutter self-excited forces are only related to the cross-sectional shape of the bridge deck panel. Therefore, it can be assumed that the numerically obtained results of the box girder section model by [PERSON] et al. can be used to analyze the nonlinear self-excited forces of the Nansha Bridge over the Nizhou Channel in this study. To facilitate a continuous solution of the flutter motion  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cases Name** & \(c_{h2}\) & \(c_{\alpha 2}\) & \(k_{\alpha 2}\) \\ \hline Reference case & 0 & 0 & 0 \\ Case A & 0 & \(10c_{\alpha 1}\) & 0 \\ Case B & 0 & \(10c_{\alpha 1}\) & \(1000m_{h}\omega_{\alpha}^{2}\) \\ Case C & \(10c_{h1}\) & \(10c_{\alpha 1}\) & \(1000m_{h}\omega_{\alpha}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of cubic damping and stiffness terms for each case.  
equation, the flutter derivatives are fitted as a function of the reduced wind speed using the least squares method, as shown in the form of (12). The discrete values of the first-order flutter derivatives and the corresponding function graph are shown in Figure 2, where the black dots represent the discrete flutter derivative values obtained from wind tunnel tests, and the black solid line represents the fitted flutter derivative function curve obtained through the least squares method.  
Figure 2: Discrete points and fitted curves of the first-order flutter derivatives of aero self-excited force.