# Flutter Mechanism of a Thin Plate Considering Attack Angles
## 4 Flutter Mechanisms of Thin Plates at Different Angles of Attack
### Variation of the Phase Difference between Motions  
Phase difference intuitively reflects the coupling motion characteristics of vertical and torsional movements during flutter and determines the coupling aerodynamic negative damping value. Figure 8 shows the phase differences between the coupling moment and vertical motion \(\theta_{1}\), coupling lift and torsional motion \(\theta_{3}\), coupling lift and vertical motion \(\theta_{4}\), and torsional motion and vertical motion, as well as the total phase difference in the coupling aerodynamic damping under attack angles of \(0^{\circ}\), \(3^{\circ}\), \(5^{\circ}\), and \(7^{\circ}\) as a function of the reduced wind speed. It can be observed that within the attack angles of \(0^{\circ}\) to \(5^{\circ}\), the variation patterns of the phase differences are quite similar, with \(\theta_{1}\) and \(\theta_{3}\) decreasing with the reduced wind speed, while \(\theta_{4}\) increases with the reduced wind speed. \(\theta_{1}\) has a large value and is the most significant component of \(\psi^{\prime}\). \(\theta_{4}\) has a minimal value in the low reduced wind speed range below 7, contributing weakly to \(\psi^{\prime}\) and \(\psi\). However, at and above the critical flutter wind speed, it occupies a significant portion of the phase difference, \(\psi\), and it becomes the second most significant component of the aerodynamic damping phase difference, \(\psi^{\prime}\). This phenomenon is pronounced at attack angles of \(0^{\circ}\) and \(3^{\circ}\). \(\theta_{3}\) has a small value in the high wind speed range and contributes the least to \(\varphi\) and \(8\), particularly at attack angles of \(0^{\circ}\) and \(3^{\circ}\). However, its contribution increases with the attack angle. At an attack angle of \(7^{\circ}\), the variation patterns of phase differences are entirely different from those at other attack angles. \(\theta_{1}\), \(\theta_{3}\), and \(\theta_{4}\) all increase with the reduced wind speed. \(\theta_{1}\), \(\theta_{3}\), and \(\theta_{4}\) all contribute significantly to \(\psi^{\prime}\), with \(\theta_{1}\) and \(\theta_{4}\) having a more pronounced effect on increasing \(\psi\) compared to \(\theta_{3}\). More importantly, as the attack angle increases, the phase difference between the motions decreases from positive values at small angles to zero and then becomes negative. The results indicate that with increasing attack angle, the flutter mode transitions from torsional motion lagging behind vertical motion to vertical motion lagging behind torsional motion, representing a significant shift in flutter characteristics at different attack angles. Considering the composition of \(-0.5\mu\psi\Psi^{\prime}\)/\(sin\psi^{\prime}\), the closer the total phase difference, \(\psi^{\prime}\), of the coupling aerodynamic damping is to \(90^{\circ}\), the greater the aerodynamic negative damping it provides. Figure 9 directly illustrates the variation of \(sin\psi^{\prime}\) under different attack angles and conditions. It can be seen from Figures 9 and 10 that at attack angles of \(0^{\circ}\), \(3^{\circ}\), and \(5^{\circ}\), the value of \(sin\psi^{\prime}\) is above 0.95 and can be approximated as 1. The reason for this is that at these angles, aerodynamic positive damping plays a significant role in suppressing flutter. Therefore, coupling must achieve maximum effectiveness to counteract the aerodynamic positive damping as much as possible, thus inducing flutter. Since \(\theta_{1}\) contributes the most to the total phase difference, flutter energy is primarily provided by the coupling moment term generated by vertical motion. At an attack angle of \(7^{\circ}\), the value of \(sin\psi^{\prime}\) is more minor, dropping to 0.71 at the critical wind speed, indicating a reduction in the coupling effect between the motions. The reason for this is that, at this angle, the aerodynamic positive damping is minimal, and the aerodynamic negative damping produced by coupling dominates.