# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 4 Ideal Flat Plate Flutter Analysis
### Ideal Flat Plate Aerodynamic Damping
It can be seen from this that the trend of \(A_{2}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) depends on the derivative of vibration \(A_{2}^{*}\). Although function \(\phi_{ah}(\theta_{1},\xi_{sa})\) has a small impact on the value of \(A_{2}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\), its value is negative, determining that the aerodynamic damping of term \(A_{2}^{*}\) is positive damping. 2. Function \(\psi_{ah}\Big{(}A_{j}^{*},H_{k}^{*}\Big{)}\) is the aerodynamic damping function of the product \(A_{1}^{*}H_{2}^{*}\), which is always negative in value. Its absolute value gradually increases with increasing wind speed and then rapidly decreases. The value of \(\phi_{ah}(\theta_{1},\xi_{sa})\) is always negative, and its absolute value gradually increases with increasing wind speed. Therefore, the aerodynamic damping of \(A_{1}^{*}H_{2}^{*}\) is positive damping, and when the wind speed exceeds 12 m/s, the absolute value of function \(\psi_{ah}\Big{(}A_{j}^{*},H_{k}^{*}\Big{)}\) decreases, resulting in a significant decrease in \(A_{1}^{*}H_{2}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) with increasing wind speed. 3. Function \(\psi_{ah}\Big{(}A_{j}^{*},H_{k}^{*}\Big{)}\) is the aerodynamic damping function of the product \(A_{1}^{*}H_{2}^{*}\), which is always negative in value. Its absolute value increases initially with wind speed and then decreases. As the wind speed increases, \(\phi_{ah}(\theta_{1},\xi_{sa})\) decreases. Its value lies between 0.9 and 1.0. Its impact on \(A_{4}^{*}H_{2}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) is minimal, while its value aligns closely with the product of \(A_{4}^{*}H_{2}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) and the flutter derivative \(A_{4}^{*}H_{2}^{*}\). 4. Function \(\psi_{ah}\Big{(}A_{j}^{*},H_{k}^{*}\Big{)}\) is the aerodynamic damping function of the product \(A_{1}^{*}H_{3}^{*}\), which is always negative in value and gradually decreases with increasing wind speed. \(\phi_{ah}(\theta_{1},\xi_{sa})\) decreases with the increase of wind speed, and its value is between 0.9 and 1.0, which has little effect on \(A_{1}^{*}H_{3}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\), and the value of \(A_{1}^{*}H_{3}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) is basically consistent with the value of flutter derivative \(A_{1}^{*}H_{3}^{*}\). The reason why the aerodynamic damping of item \(A_{1}^{*}H_{3}^{*}\) appears as negative damping is because the vibration derivatives \(A_{1}^{*}\) and \(H_{3}^{*}\) have opposite signs, and function \(\phi_{ah}(\theta_{1},\xi_{sa})\) is positive. 5. Function \(\psi_{ah}\Big{(}A_{j}^{*},H_{k}^{*}\Big{)}\) is the aerodynamic damping function of the product \(A_{4}^{*}H_{3}^{*}\), which always maintains a negative value and gradually decreases with increasing wind speed. The value of \(\phi_{ah}(\theta_{1},\xi_{sa})\) always remains positive, ranging between 0.0 and 0.44, and gradually increases with wind speed. Therefore, the aerodynamic damping of \(A_{4}^{*}H_{3}^{*}\) is negative, and function \(\phi_{ah}(\theta_{1},\xi_{sa})\) leads to a significant decrease in the value of \(A_{4}^{*}H_{3}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) relative to the flutter derivative \(A_{4}^{*}H_{3}^{*}\). 6. It can be seen from (f) in Figure 5 that among the aerodynamic damping of torsional modes, \(A_{1}^{*}H_{3}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) in the aerodynamic damping of \(A_{2}^{*}\phi_{ah}(\theta_{1},\xi_{sa})\) and \(A_{1}^{*}H_{3}^{*}\) in the aerodynamic damping of term \(A_{2}^{*}\) are significantly greater than that of aerodynamic damping \(\psi_{ah}\Big{(}A_{j}^{*},H_{k}^{*}\Big{)}\phi(\theta_{1},\xi_{sa})\), which shows that the aerodynamic damping of flutter torsional mode mainly comes from two aspects: one is the self-excited aerodynamic damping of torsional velocity in the torsional vibration system; the second is the vertical vibration velocity response induced by the torsional displacement lift in the vertical vibration system, and the pneumatic damping in the torsional vibration system is fed back to the torsional vibration system through the vertical velocity self-excited torque.