# Flutter Mechanism of a Thin Plate Considering Attack Angles
## 4 Flutter Mechanisms of Thin Plates at Different Angles of Attack
### Variation in Aerodynamic Damping  
Based on the derivation of aerodynamic damping, \(-0.5\
u A_{2}^{*}\), in Section 1, aerodynamic damping is composed of the non-coupling term, which is the aerodynamic damping directly produced by torsional motion, \(-0.5\mu\
u\Psi^{\prime}\sin\psi^{\prime}\), and the coupling terms, which are the aerodynamic damping due to lift generated by torsional motion and the damping due to the torque generated by vertical motion. Figure 7 illustrates the contribution of each aerodynamic damping component to the total damping at different angles of attack. Figure 7 shows that at angles of attack of \(0^{\circ}\), \(3^{\circ}\), and \(5^{\circ}\), the non-coupling term consistently provides aerodynamic positive damping, while the coupling term always provides aerodynamic negative damping. The total system damping reaches zero at a specific reduced wind speed, indicating the critical flutter point. Since aerodynamic negative damping is caused by coupling motions, the flutter mode manifests as a bending-torsion coupling motion. At an angle of attack of \(7^{\circ}\), the flutter derivative \(A_{2}^{*}\) changes to a positive value at a reduced wind speed of 7.5. Consequently, the non-coupling term provides aerodynamic positive damping when the reduced wind speed is less than 7.5, and aerodynamic negative damping when the reduced wind speed exceeds 7.5. At this point, since the coupling aerodynamic forces at a \(7^{\circ}\) angle of attack provide aerodynamic negative damping throughout the entire wind speed range and \(A_{2}^{*}\) provides minimal aerodynamic positive damping, the total damping becomes zero at a reduced wind speed below 7.5, leading to critical flutter. Therefore, even at a \(7^{\circ}\) angle of attack, the thin plate still exhibits flutter characteristics driven by coupling aerodynamic negative damping with bending-torsion coupling flutter. However, its flutter mode differs from those at other angles of attack, which will be discussed in detail later. Table 3 presents the variation of the aerodynamic damping components with the angle of attack at \(V=7\). Figure 6 shows that at angles of attack of \(0^{\circ}\) and \(3^{\circ}\), the distribution patterns of aerodynamic damping and its components are similar. This similarity is attributed to the close values of the critical flutter derivatives \(A_{2}^{*}\), \(A_{2}^{*}\), and \(H_{3}^{*}\) that significantly influence aerodynamic damping at these two angles of attack. Additionally, due to the relatively large value of \(A_{2}^{*}\), the non-coupling term \(-0.5\mu\
u\Psi^{\prime}\sin\psi^{\prime}\) provides substantial positive aerodynamic damping, while the negative aerodynamic damping from the coupling term begins to increase significantly only when the reduced wind speed exceeds 7, resulting in a higher critical flutter speed. At a \(5^{\circ}\) angle of attack, the negative aerodynamic damping from the coupling term increases slightly compared to that at \(0^{\circ}\) and \(3^{\circ}\), while the significant decrease in the non-coupling flutter derivative \(A_{2}^{*}\) leads to a notable reduction in the proportion of positive aerodynamic damping in the total damping. This reduction results in a lower and reduced wind speed at which the total damping becomes zero, significantly weakening the flutter performance. At a \(7^{\circ}\) angle of attack, significant changes in \(A_{1}^{*}\), \(A_{4}^{*}\), and \(H_{3}^{*}\) slightly enhance the negative aerodynamic damping from the coupling term. However, the substantial decrease in \(A_{2}^{*}\) significantly weakens the non-coupling positive aerodynamic damping compared to other angles of attack (with positive damping from coupling decreasing from 2.7% at \(0^{\circ}\) to 0.4% at \(V=7\)). This reduction is the primary reason for the further deterioration in flutter performance at this angle of attack.