## 2 Volterra Model and FSI Method
### Aeroelastic Analysis Based on the Volterra Model and Newmark-\(\beta\) Method  
The vibration differential equations of a two-degree-of-freedom system shown in Figure 5 can be expressed as  
\[\ddot{y}(t)+2\xi_{h}\omega_{h}\dot{y}(t)+\omega_{h}^{2}y(t)=F_{L}(t)/m \tag{7a}\] \[\ddot{\alpha}(t)+2\xi_{a}\omega_{a}\dot{\alpha}(t)+\omega_{a}^{2}\alpha(t)=M_ {\alpha}(t)/I_{m} \tag{7b}\]  
where \(m\) and \(I_{m}\) represent the mass and torsion mass moment of inertia of the structure, respectively; \(\zeta_{h}\) and \(\zeta_{a}\) represent the vertical bending and torsional damping radio, respectively. \(\omega_{h}\) and \(\omega_{a}\) represent the circular frequency of vertical and torsional vibration, respectively. \(F\) and \(M\) represent the vertical and torsional load, respectively; \(y\) and \(\alpha\) represent the vertical displacement and the torsional displacement, respectively.  
Figure 4: Mesh near the GBEB girder section model.  
Figure 3: Computational domain, boundary conditions, and blocks.  
The Newmark-\(\beta\) method assumes that the structural displacement, velocity, and acceleration of the discrete-time system can be expressed by introducing coefficients \(\beta\) and \(\gamma\) as follows  
\[\overline{k}y_{t+\Delta t}=\overline{F}_{t+\Delta t} \tag{8a}\] \[\dot{y}_{t+\Delta t}=\frac{\gamma}{\beta\Delta t}(y_{t+\Delta t}-y_{t})+(1- \frac{\gamma}{\beta})\dot{y}_{t}-(\frac{\gamma}{2\beta}-1)\ddot{y}_{t}\Delta t\] (8b) \[\ddot{y}_{t+\Delta t}=\frac{1}{\beta\Delta t^{2}}(y_{t+\Delta t}-y_{t})-\frac {1}{\beta\Delta t}\dot{y}_{t}-(\frac{1}{2\beta}-1)\ddot{y}_{t} \tag{9}\]  
where the expressions of \(\overline{k}\) and \(\overline{F}_{t+\Delta t}\) are shown in Equation (9)  
\[\overline{k}=k+\frac{m}{\beta\Delta t^{2}}+\frac{\gamma c}{\beta\Delta t} \tag{9a}\] \[\overline{F}_{t+\Delta t}=F_{t+\Delta t}+m\left\{\frac{y_{t}}{\beta\Delta t^{ 2}}+\frac{\dot{y}_{t}}{\beta\Delta t}+(\frac{1}{2\beta}-1)\ddot{y}_{t}\right\} +c\left\{\frac{\gamma y_{t}}{\beta\Delta t}+(\frac{\gamma}{\beta}-1)\dot{y}_ {t}+(\frac{\gamma}{2\beta}-1)\Delta t\ddot{y}_{t}\right\} \tag{9b}\]  
In the aerodynamic model based on the Volterra series, the aerodynamic lift force coefficient and torsion moment coefficient are related to the velocity of heaving and pitch motion. Therefore, the aerodynamic self-excited force can be expressed as follows  
\[L(t)=\frac{1}{2}\rho U^{2}BC_{L}(t) \tag{10a}\] \[M(t)=\frac{1}{2}\rho U^{2}B^{2}C_{M}(t)\] (10b) \[C_{L}(t)=\int\limits_{0}^{t}h_{1 uL}(\tau_{1})\dot{u}(t-\tau_{1})dt+\int \limits_{0}^{t}h_{1\theta L}(\tau_{1})\dot{\theta}(t-\tau_{1})dt+C_{L0}\] (11a) \[C_{M}(t)=\int\limits_{0}^{t}h_{1 uM}(\tau_{1})\dot{u}(t-\tau_{1})dt+\int \limits_{0}^{t}h_{1\theta M}(\tau_{1})\dot{\theta}(t-\tau_{1})+C_{M0} \tag{12}\]  
In this case, the Volterra series-based ROMs are models of four first-order Volterra kernels to be identified. Accordingly, due to the advantage of fewer parameters to be solved, the computational cost to establish the ROMs can be greatly reduced compared with flutter derivative theory and some other flutter analysis methods, e.g., aerodynamic describing-function-based models ([PERSON] et al. [27]). Moreover, the vertical-torsional-coupling effect has been considered, significantly improved over single-degree-of-freedom (SDOF) torsional flutter simplified models (e.g., [PERSON] et al. [28]).  
Figure 6 shows the working process of the aeroelastic numerical simulation for flutter analysis established in this study based on the proposed aerodynamic model via the Volterra  
Figure 5: Aerodynamic forces and displacements of the girder section model.  
series and Newmark-\(\beta\) method. Moreover, the aerodynamic force at time \(t\) is calculated from the structural displacement at time \(t-\Delta t\), and then, the structural displacement at time \(t\) is calculated from the aerodynamic force at time \(t\) by the Newmark-\(\beta\) method.