# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 4 Ideal Flat Plate Flutter Analysis
### Ideal Flat Plate Aerodynamic Damping  
Based on the formulation of the \"incentive feedback\" mechanism derived in Section 3, the damping ratios of the vertical and torsional vibrations of an ideal flat plate are identified, as shown in Figure 4. The following can be seen from the figure:  
1. The torsional modal damping is mainly derived from terms \(A_{2}^{*}\) and \(A_{1}^{*}H_{3}^{*}\), that is, the feedback aerodynamic damping of torsional velocity self-excited aerodynamic damping in the torsional vibration system and the vertical vibration induced by the torsional displacement self-excited lift force in the torsional vibration system, in which the aerodynamic damping of item \(A_{2}^{*}\) is positive damping and the aerodynamic damping of item \(A_{1}^{*}H_{3}^{*}\) is negative damping, while the intrinsic damping of the torsional vibration system and the aerodynamic damping of other items are significantly smaller than the aerodynamic damping of items \(A_{2}^{*}\) and \(A_{1}^{*}H_{3}^{*}\), and their contribution to the total damping of the torsional vibration system is very small. 2. The vertical modal damping is mainly derived from terms \(H_{1}^{*}\) and \(H_{3}^{*}A_{1}^{*}\), that is, the vertical velocity self-excited aerodynamic damping in the vertical vibration system and the pneumatic damping fed back by the torsional vibration induced by the vertical velocity self-excited moment in the vertical vibration system, among which the \(H_{1}^{*}\) and \(H_{3}^{*}A_{1}^{*}\) terms are both positive damping. However, the intrinsic damping and other aerodynamic damping of the vertical vibration system are significantly smaller than those of \(H_{1}^{*}\) and \(H_{3}^{*}A_{1}^{*}\) aerodynamic damping, and their contribution to the total damping of the vertical vibration system is small. 3. When the wind speed exceeds the flutter critical wind speed, the negative aerodynamic damping of the \"incentive feedback\" torsional displacement of the ideal flat plate leads to the dispersion of its torsional vibration, and its vertical vibration system is forced to disperse due the torsional self-excited lifting force, which leads to the occurrence of the bending-torsional-coupled flutter. Function \(\psi_{ah}\left(A_{j}^{*},H_{k}^{*}\right)\) and Function \(\phi_{ah}(\theta_{1},\zeta_{sa})\)  
The expression for general aerodynamic damping in the torsional vibration system reveals that its positive or negative value is not related to \(\rho^{2}B^{6}\frac{\omega_{ah}^{3}}{\sqrt{2_{ha}^{2}+\Gamma_{ha}^{2}}}\) but entirely depends on whether the values of functions \(\psi_{ah}\left(A_{j}^{*},H_{k}^{*}\right)\) and \(\phi_{ah}(\theta_{1},\zeta_{sa})\) have the same sign. In other words, the sign of each aerodynamic damping term in the torsional vibration system is determined solely by the sign of the product of the aforementioned two functions, \(\psi_{ah}\left(A_{j}^{*},H_{k}^{*}\right)\phi(\theta_{1},\zeta_{sa})\). Figure 5 shows the curves of functions \(\psi_{ah}\left(A_{j}^{*},H_{k}^{*}\right)\) and \(\phi_{ah}(\theta_{1},\zeta_{sa})\) and the product function \(\psi_{ah}\left(A_{j}^{*},H_{k}^{*}\right)\phi(\theta_{1},\zeta_{sa})\) of aerodynamic damping in an ideal flat torsional vibration system as wind speed varies. From the graph, the following can be observed:  
1. The function \(\psi_{ah}\left(A_{j}^{*},H_{k}^{*}\right)\) is the aerodynamic damping function of the flutter derivative \(A_{2}^{*}\), and the flutter derivative \(A_{2}^{*}\) is positive at the wind speed of 1 m/s, while the wind speed at all other levels is negative, and its value gradually decreases with the increase of wind speed. The function \(\phi_{ah}(\theta_{1},\zeta_{sa})\) is always negative, and its value gradually increases with increasing wind speed. Therefore, at a wind speed of 1 m/s,\(A_{2}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) is negative, while at all other wind speeds, it is positive, and its value gradually increases with increasing wind speed.