## 2 Optimization Formulation of Flutter Derivatives Identification  
According to [PERSON]'s model for flutter analysis, a 2 DOF linear oscillator representing the bridge deck section vibrating in vertical displacement \(h(t)\) and torsional angle \(\alpha(t)\) is shown in Figure 1. The equations of motion can be written as follows:  
\[m(\ddot{h}+2\zeta_{h}\omega_{h}\dot{h}+\omega_{h}^{2}h)=L_{h} \tag{1}\]  
\[I\left(\ddot{\alpha}+2\zeta_{\alpha}\omega_{\alpha}\dot{\alpha}+\omega_{\alpha }^{2}\alpha\right)=M_{\alpha} \tag{2}\]  
where \(m\) and \(I\) are the mass and mass moment of inertia per unit length; \(\omega_{h}\) and \(\omega_{\alpha}\) are the circular frequencies of the vertical bending and torsional modes; \(\zeta_{h}\) and \(\zeta_{\alpha}\) are the damping ratios of the vertical bending and torsional modes; \(L_{h}\) and \(M_{\alpha}\) are the lift and pitching moment per unit length. Generally, the terms of the lift and pitching moment on the right-hand side involve three components of mean, buffeting, and self-excited forces; nevertheless, for free vibration tests, only the self-excited forces are retained since the motion is dominated by free decaying vibration around a mean position and thus the load due to signature turbulence can be neglected. The self-excited forces \(L_{se}\) and \(M_{se}\) can be expressed in [PERSON]'s form as follows:  
\[L_{se}=\frac{1}{2}\rho U^{2}B[KH_{1}^{*}(K)\frac{\dot{h}}{U}+KH_{2}^{*}(K) \frac{B\dot{\alpha}}{U}+K^{2}H_{3}^{*}(K)\alpha+K^{2}H_{4}^{*}(K)\frac{h}{B}] \tag{3}\]  
\[M_{se}=\frac{1}{2}\rho U^{2}B^{2}[KA_{1}^{*}(K)\frac{\dot{h}}{U}+KA_{2}^{*}(K) \frac{B\dot{\alpha}}{U}+K^{2}A_{3}^{*}(K)\alpha+K^{2}A_{4}^{*}(K)\frac{h}{B}] \tag{4}\]  
where \(\rho\) is the air density, \(H_{i}^{*}\) and \(A_{i}^{*}\) (\(i=1,\,2,\,3,\,4\)) are the flutter derivatives, \(B\) is the deck width, \(U\) is the mean wind velocity, and \(K=B\omega/U\) is the reduced frequency of oscillation. Substituting Equations (3) and (4) into Equations (1) and (2), respectively, yield:  
\[\ddot{h}+2\zeta_{h}\omega_{h}\dot{h}+\omega_{h}^{2}\dot{h}=H_{1}\dot{h}+H_{2}\dot{ \alpha}+H_{3}\alpha+H_{4}h \tag{5}\]  
\[\ddot{\alpha}+2\zeta_{a}\omega_{a}\dot{\alpha}+\omega_{a}^{2}\dot{\alpha}=A_{1} \dot{h}+A_{2}\dot{\alpha}+A_{3}\alpha+A_{4}h \tag{6}\]  
where \(H_{1}=\frac{1}{2}\rho B^{2}\omega H_{1}^{*}(K)/m\), \(H_{2}=\frac{1}{2}\rho B^{3}\omega H_{2}^{*}(K)/m\), \(H_{3}=\frac{1}{2}\rho B^{3}\omega^{2}H_{3}^{*}(K)/m\), \(H_{4}=\frac{1}{2}\rho B^{2}\omega^{2}H_{4}^{*}(K)/m\), \(A_{1}=\frac{1}{2}\rho B^{3}\omega A_{1}^{*}(K)/I\), \(A_{2}=\frac{1}{2}\rho B^{4}\omega A_{2}^{*}(K)/I\), \(A_{3}=\frac{1}{2}\rho B^{4}\omega^{2}A_{3}^{*}(K)/I\), \(A_{4}=\frac{1}{2}\rho B^{3}\omega^{2}A_{4}^{*}(K)/I\). Rewriting Equations (5) and (6) in matrix form, yields:  
\[\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=0 \tag{7}\]  
where \(\mathbf{x}=[h,\,\alpha]^{\mathrm{T}}\), \(\mathbf{C}=\left[ \begin{array}{cc}2\zeta_{h}\omega_{h}-H_{1}&-H_{2}\\ -A_{1}&2\zeta_{a}\omega_{a}-A_{2}\end{array} \right]\), \(\mathbf{K}=\left[ \begin{array}{cc}\omega_{h}^{2}-H_{4}&-H_{3}\\ -A_{4}&\omega_{\alpha}^{2}-A_{3}\end{array} \right]\). Using a state vector \(\mathbf{y}=[\mathbf{x},\mathbf{\dot{x}}]^{\mathrm{T}}\), Equation (7) can be rewritten in state-space form as follows:  
\[\dot{\mathbf{y}}=\mathbf{A}\mathbf{y} \tag{8}\]  
where \(\mathbf{A}=[\mathbf{O},\mathbf{E};-\mathbf{K},-\mathbf{C}]\), \(\mathbf{O}\) is a \(2\times 2\) zero matrix and \(\mathbf{E}\) is a \(2\times 2\) identity matrix. The flutter derivatives are involved in matrix \(\mathbf{A}\), and the free vibration response can be obtained by solving Equation (8). Neglecting the aerodynamic damping and stiffness in still air, the aerodynamic matrices are obtained by comparing the measured matrices in wind flow and in still air.