## 2 Framework of Nonlinear Flutter Analysis
### Flutter Motion Equations for Suspension Bridges  
The current research on flutter utilizes the motion expression proposed by [PERSON] [28]. This model considers bridges as Euler-Bernoulli beams, thereby neglecting shear deformations. Furthermore, it assumes that the plane (cross-section), which is perpendicular to the beam's central axis before deformation, remains planar after deformation, and that regardless of pre- and post-deformation processes, the plane of the cross-section remains perpendicular to the beam's axis. Because the centroid of the bridge's cross-section coincides with its shear center, the vibration equation for the beam, derived based on the Euler-Bernoulli beam theory and the method of separation of variables, can be expressed in terms of its natural modes, mass, and stiffness. The first-mode response of the bridge contributes significantly to its vibration, resulting in a two-degree-of-freedom cross-sectional motion model consisting of a first-order vertical bending and torsional frequencies of the bridge, along with their corresponding masses and stiffness.  
As shown in Figure 1, the cross-section of the bridge deck of the Nansha Bridge over the Nizhou Channel is allowed to undergo motion in the vertical bending and torsional degrees of freedom. It experiences aerodynamic self-excitation forces in a uniform flow field, including aerodynamic lift and aerodynamic torque, acting at the centroid of the cross-section [15; 28]. Therefore, flutter motion equations can be expressed as follows. On the left side of the equation lies the motion equation representing the characteristics of the bridge, while on the right side lies the aeroelastic self-exciting force exerted on the bridge:  
\[m_{h}\ddot{h}+2\xi_{h}m_{h}\omega_{h}\dot{h}+m_{h}\omega_{h}^{2}h=\rho U^{2}B \left(KH_{1}^{*}\frac{\dot{h}}{U}+KH_{2}^{*}\frac{B\dot{\alpha}}{U}+K^{2}H_{3} ^{*}\alpha+K^{2}H_{4}^{*}\frac{h}{B}\right) \tag{1}\]  
\[I\ddot{\alpha}+2\xi_{a}I\omega_{a}\dot{\alpha}+I\omega_{a}^{2}\alpha=\rho U^{2} B^{2}\left(KA_{1}^{*}\frac{\dot{h}}{U}+KA_{2}^{*}\frac{B\dot{\alpha}}{U}+K^{2}A_{3}^ {*}\alpha+K^{2}A_{4}^{*}\frac{h}{B}\right) \tag{2}\]  
where \(m\) and \(I\) are mass and moment of inertia per unit length of the bridge deck, respectively. \(h\), \(\dot{h}\), and \(\ddot{h}\) represent the displacement, velocity, and acceleration of the vertical bending degree of freedom, while \(\alpha\), \(\dot{\alpha}\), and \(\ddot{\alpha}\) represent the angular displacement, angular velocity, and angular acceleration of the torsional degree of freedom. \(\xi_{h}\) and \(\xi_{a}\) are the damping ratios of the suspension bridge in the vertical bending and torsional modes, respectively. \(\omega_{h}\) and \(\omega_{a}\) are the first-order circular frequencies of vertical bending and torsion, respectively. On the right-hand side of Equation (1) is the aerodynamic lift, and on the right-hand side of Equation (2) is the aerodynamic torque. \(\rho\) is air density, \(U\) is fluid flow velocity, \(B\) is the width of the cross-section of the stiffened beam, and \(K\) is reduced frequency, defined as \(K=B\omega/U\). \(H_{i}^{*}\) and \(A_{i}^{*}\) are eight flutter derivatives, which are dimensionless quantities related to the reduced frequency or reduced wind speed and can be expressed as functions.  
\[H_{i}^{*},A_{i}^{*}=f(V_{r})=f(K) \tag{3}\]  
where \(V_{r}\) is the reduced wind speed, defined as \(V_{r}=U/Bf\). This study focuses on the Nansha Bridge over the Nizhou Channel, which is a suspension bridge with a main span of 1688 m and a bridge deck width of 41.7 m. The relevant design parameters of the bridge are listed in Table 1[29].