# Flutter Mechanism of a Thin Plate Considering Attack Angles
## 6 Conclusions  
This study employs the coupled flutter closed-form solution theory to conduct a detailed analysis of the flutter mechanisms and the main factors influencing the flutter performance of the thin plate at attack angles of 0\({}^{\circ}\), 3\({}^{\circ}\), 5\({}^{\circ}\), and 7\({}^{\circ}\). The main conclusions are as follows:  
1. At attack angles ranging from 0\({}^{\circ}\) to 7\({}^{\circ}\), the thin plate exhibits flutter characterized by bending-torsion coupling, with torsion mode branching dominating the flutter behavior.
2. At 0\({}^{\circ}\) and 3\({}^{\circ}\), the increase in coupled aerodynamic negative damping is the primary factor driving the occurrence of flutter. At 5\({}^{\circ}\), the significant reduction in flutter speed results from the combined effects of decreased non-coupled aerodynamic positive  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**AAOA**} & \multicolumn{2}{c}{**Coupled Flutter**} & \multicolumn{2}{c}{**Single-Degree-of-Freedom Flutter**} \\ \cline{2-5}  & \(\mathbf{V=Ull/B}\) & \(\mathbf{U}\) & \(\mathbf{V=Ull/B}\) & \(\mathbf{U}\) \\ \hline
0\({}^{\circ}\) & 10.3 & 11.87 & \(\infty\) & \(\infty\) \\
7\({}^{\circ}\) & 6.3 & 7.76 & 7.3 & 8.94 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Critical flutter with reduced velocity.  
damping and increased coupled aerodynamic negative damping. At 7\({}^{\circ}\), the drastic reduction in non-coupled aerodynamic positive damping is the main reason for the substantial decline in flutter performance.
3. At attack angles of 0\({}^{\circ}\), 3\({}^{\circ}\), and 5\({}^{\circ}\), the phase value of the non-coupled damping term at the critical flutter point is close to 1, indicating that a dramatic bending-torsion coupling effect is necessary to trigger flutter. However, at 7\({}^{\circ}\), the phase value decreases to 0.7, suggesting that a weaker coupling effect can significantly diminish flutter performance, resulting in a notable decrease in the critical flutter wind speed.  
In addition, in the absence of experimental or numerical identification of actual flutter derivatives, the flutter derivatives of the flat plate model under various wind conditions can provide an alternative option for estimating the flutter initiation velocity of high aspect ratio (>10) streamlined box girders in the initial design stage.  
Conceptualization, Q.W.; methodology, L.H.; software, C.Y. and B.W.; validation, L.H.; formal analysis, C.Y. and B.W.; investigation, L.H.; resources, Q.W.; data curation, L.H.; writing--original draft preparation, C.Y.; writing--review and editing, L.H; visualization, B.W.; supervision, Q.W.; project administration, Q.W.; funding acquisition, C.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key Research and Development Program of China (Grant No. 2022 YFC3005304), the Science and Technology Plan Projects of the Zhejiang Provincial Department of Transportation (Grant No. 2021069), and the National Natural Science Foundation of China (Grant No. 52378537).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
Author [PERSON] was employed by the Sichuan Road and Bridge Group Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.