## 4 Discussion and Conclusions  
In this paper, the main deck sections of the GBEB and LDYB are taken as the research objects, and the aerodynamic ROM is established to propose a new analysis method for the aeroelastic effect based on the Volterra series and the Newmark-\(\beta\) method. The flutter critical wind speed and the flutter frequency obtained by the new aeroelastic analysis and other methods are found to be similar, thus verifying the feasibility and accuracy of the method in this paper. Although there is a certain difference between the flutter displacement response calculated by the ROM method proposed in this study and FSI with  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Flutter Critical Wind Speed (m/s)** & **Flutter Frequency (Hz)** \\ \hline Wind tunnel test & & \\ (Wang [32]) & 76.5 & 0.198 \\ FSI (Wang [32]) & 70.0–75.0 & 0.176 \\ ROM (present) & 75.0–80.0 & 0.160 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparisons of flutter results of the girder of LDTB.  
Figure 13: Results of flutter analysis of LDYB based on the ROM method under incoming wind speeds of (**a**) 75 m/s and (**b**) 80 m/s for the prototype main deck section, respectively.  
ANSYS Fluent, the issue can be theoretically alleviated by introducing multi-order Volterra kernels, which can consider more aerodynamic nonlinear effects. Furthermore, the Volterra series-based aerodynamic ROM is suitable to be extended for the three-degrees-of-freedom flutter problem.  
By comparing the flutter critical wind speed and flutter frequency obtained from the aerodynamic ROM based on the Volterra series with those from CFD numerical simulations under both forced and free vibrations, the following conclusions can be drawn:  
1. The aerodynamic ROM of the main girder section is established based on the first-order Volterra series through the CFD numerical simulation. Moreover, the aerodynamic identification accuracy of the main girder is found to be in good agreement with the results of the forced vibration based on CFD numerical simulation. Additionally, the established aerodynamic ROM can consider the nonlinear aerodynamic effect caused by the certain amplitude and frequency of the streamlined main beam section.
2. An aeroelastic analysis method for the main girder section is established based on the Volterra series-based aerodynamic ROM and the Newmark-\(\beta\) method, which enabled efficient FSI calculations and the determination of the flutter critical wind speed and flutter frequency. Moreover, the accuracy and feasibility of the methods in this study are verified by comparing the numerical results of different main deck sections obtained from FSI simulations based on ANSYS Fluent and the existing numerical and experimental results.  
Conceptualization, Z.W., Z.L. and F.H.; methodology, Z.W. and Z.L.; software, Z.W.; validation, Z.W. and Z.L.; formal analysis, F.H.; investigation, Z.W.; resources, Z.L.; data curation, F.H.; writing--original draft preparation, Z.W.; writing--review and editing, Z.L.; visualization, F.H.; supervision, Z.L.; project administration, Z.L.; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 52178475 and 51778225.  
We deeply appreciate the assistance provided by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] in the process of article writing, and the technical support by Hunan Provincial Key Lab of Wind Engineering and Bridge Engineering, Hunan University.  
The authors declare no conflict of interest. The funders of the National Natural Science Foundation of China provided authors with the access to the efficient computing resources.