# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 2 Flutter \"Incentive-Feedback\" Mechanism Theory
### Torsional Modal Damping and Stiffness  
Set the torsional displacement to \(\alpha=\alpha_{0}e^{-\varepsilon_{sa}t}\cos\omega_{sa}t\); then, the torsional velocity is:  
\[\dot{\alpha}=-\varepsilon_{sa}\alpha-\alpha_{0}e^{-\varepsilon_{sa}t}\omega_{ sa}\sin\omega_{sa}t \tag{5}\]  
where \(\varepsilon_{sa}\) is the torsional mode attenuation rate, \(\varepsilon_{sa}=\xi_{sa}\omega_{sa}\); \(\xi_{sa}\) is the torsional mode damping ratio; and \(\omega_{sa}\) is the torsional mode circle frequency. The vertical vibration system response under the torsional displacement self-excited lift \(H_{3}\alpha\) is:  
\[m_{h}\ddot{h}+(C_{h0}-H_{1})\dot{h}+(K_{h0}-H_{4})h=H_{3}\alpha \tag{6}\]  
Substituting the torsional displacement \(\alpha=\alpha_{0}e^{-\varepsilon_{sa}t}\cos\omega_{sa}t\) into the right side of the vertical vibration Equation (6), the vertical vibration displacement and velocity response are solved, as shown in Equation (7) and Equation (8), respectively:  
\[h_{1}(t)=\frac{H_{3}}{\sqrt{\gamma_{h\alpha}^{2}+\Gamma_{h\alpha}^{2}}}(\cos \theta_{1}-\xi_{sa}\sin\theta_{1})\alpha-\frac{H_{3}}{\sqrt{\gamma_{h\alpha}^{ 2}+\Gamma_{h\alpha}^{2}}}\frac{1}{\omega_{sa}}\sin\theta_{1}\dot{\alpha} \tag{7}\]  
\[\dot{h}_{1}(t)=\frac{H_{3}}{\sqrt{\gamma_{h\alpha}^{2}+\Gamma_{h\alpha}^{2}}} (\cos\theta_{1}+\xi_{sa}\sin\theta_{1})\dot{\alpha}+\frac{H_{3}}{\sqrt{ \gamma_{h\alpha}^{2}+\Gamma_{h\alpha}^{2}}}\Big{(}1+\xi_{sa}^{2}\Big{)} \omega_{sa}^{2}\sin\theta_{1}\alpha \tag{8}\]  
where \(\gamma_{h\alpha}\) and \(\Gamma_{h\alpha}\) are stiffness parameters of the vertical vibration system, which are as follows: \(\gamma_{h\alpha}=(K_{h0}-H_{4})+m_{h}\big{(}\xi_{sa}^{2}-1\big{)}\omega_{sa}^{2 }-(C_{h0}-H_{1})\xi_{sa}\omega_{sa}\), \(\Gamma_{h\alpha}=-2m_{h}\xi_{sa}\omega_{sa}^{2}+(C_{h0}-H_{1})\omega_{sa}\). \(\theta_{1}\) is the phase difference between the torsional displacement self-excited lift force and the vertical vibration displacement response, \(\theta_{1}=\arctan(\Gamma_{h\alpha}/\gamma_{h\alpha})\). The sine function value and cosine function value of phase difference \(\theta_{1}\) are:  
\[\sin\theta_{1}=\frac{\Gamma_{h\alpha}}{\sqrt{\gamma_{h\alpha}^{2}+\Gamma_{h \alpha}^{2}}} \tag{9}\]  
\[\cos\theta_{1}=\frac{\gamma_{h\alpha}}{\sqrt{\gamma_{h\alpha}^{2}+\Gamma_{h \alpha}^{2}}} \tag{10}\]  
The vertical vibration system response under the excitation of torsional speed-induced lift \(H_{2}\dot{\alpha}\) is:  
\[m_{h}\ddot{h}+(C_{h0}-H_{1})\dot{h}+(K_{h0}-H_{4})h=H_{2}\dot{\alpha} \tag{11}\]Substituting the torsional velocity \(\dot{\alpha}=-\varepsilon_{sa}\alpha-\alpha_{0}e^{-\varepsilon_{sa}t}\omega_{sa}\sin \omega_{sa}t\) into the right side of the vertical vibration in Equation (11), the vertical vibration displacement and velocity response are solved, as shown in Equation (12) and Equation (13), respectively:  
\[h_{2}(t)=\frac{H_{2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}\Big{(}1+\xi_{sa}^{ 2}\Big{)}\omega_{sa}\sin\theta_{1}\alpha+\frac{H_{2}}{\sqrt{\gamma_{ha}^{2}+ \Gamma_{ha}^{2}}}(\cos\theta_{1}+\xi_{sa}\sin\theta_{1})\dot{\alpha} \tag{12}\]  
\[\dot{h}_{2}(t)=\frac{H_{2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}\Bigg{(} \begin{array}{c}\sin\theta_{1}-2\xi_{sa}\cos\theta_{1}\\ -\xi_{sa}^{2}\sin\theta_{1}\end{array} \Bigg{)}\omega_{sa}\dot{\alpha}-\frac{H_ {2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}\Bigg{(} \begin{array}{c}(\cos \theta_{1}+\xi_{sa}\sin\theta_{1})\\ (1+\xi_{sa}^{2})\omega_{sa}^{2}\alpha\end{array} \Bigg{)} \tag{13}\]  
Substituting the vertical vibration displacement response expression (7) and the corresponding velocity response expression (8) into the vertical self-excited lifting moment on the right side of the torsional vibration Equation (2), the vertical vibration response feedback self-lifting moment is:  
\[