# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 2 Flutter \"Incentive-Feedback\" Mechanism Theory
### Vertical Modal Damping and Stiffness  
If the vertical displacement is \(h=h_{0}e^{-\varepsilon_{sh}t}\cos\omega_{sh}t\), then the vertical velocity is  
\[\dot{h}=-\varepsilon_{sh}h-h_{0}e^{-\varepsilon_{sh}t}\omega_{sh}\sin\omega_{ sh}t \tag{33}\]  
where \(\varepsilon_{sh}\) is the vertical mode attenuation rate, \(\varepsilon_{sh}=\xi_{sh}\omega_{sh}\); \(\xi_{sh}\) is the vertical modal damping ratio; and \(\omega_{sh}\) is the vertical mode circular frequency. Under the excitation of vertical displacement self-excited lifting moment \(A_{4}h\), the torsional vibration system response is  
\[I\ddot{\alpha}+(C_{a0}-A_{2})\dot{\alpha}+(K_{a0}-A_{3})\alpha=A_{4}h \tag{34}\]  
The vertical displacement \(h=h_{0}e^{-\varepsilon_{sh}t}\cos\omega_{sh}t\) is substituted into the right side of the torsional vibration Equation (34), and the torsional vibration displacement and velocity response are solved, as shown in Equation (35) and Equation (36), respectively:  
\[\alpha_{1}(t)=-\frac{A_{4}}{\sqrt{\gamma_{ah}^{2}+\Gamma_{ah}^{2}}}\frac{1}{ \omega_{sh}}\sin\theta_{2}\dot{h}+\frac{A_{4}}{\sqrt{\gamma_{ah}^{2}+\Gamma_{ ah}^{2}}}(\cos\theta_{2}-\xi_{sh}\sin\theta_{2})h \tag{35}\]  
\[\dot{\alpha}_{1}(t)=\frac{A_{4}}{\sqrt{\gamma_{ah}^{2}+\Gamma_{ah}^{2}}}(\cos \theta_{2}+\xi_{sh}\sin\theta_{2})\dot{h}+\frac{A_{4}}{\sqrt{\gamma_{ah}^{2}+ \Gamma_{ah}^{2}}}\Big{(}1+\xi_{sh}^{2}\Big{)}\omega_{sh}\sin\theta_{2}h \tag{36}\]  
where \(\gamma_{ah}\) and \(\Gamma_{ah}\) are the stiffness parameters of the torsional vibration system, respectively: \(\gamma_{ah}=(K_{a0}-A_{3})+I\big{(}\xi_{sh}^{2}-1\big{)}\omega_{sh}^{2}-(C_{a0 }-A_{2})\xi_{sh}\omega_{sh}\); \(\Gamma_{ah}=-2I\xi_{sh}\omega_{sh}^{2}+(C_{a0}-A_{2})\omega_{sh}\). \(\theta_{2}\) is the phase difference between the vertical displacement self-excited moment and the torsional vibration displacement response, \(\theta_{2}=\arctan(\Gamma_{ah}/\gamma_{ah})\). The sine function value \(\sin\theta_{2}\) and cosine function value \(\cos\theta_{2}\) of the phase difference \(\theta_{2}\) are as follows:  
\[\sin\theta_{2}=\frac{\Gamma_{ah}}{\sqrt{\gamma_{ah}^{2}+\Gamma_{ah}^{2}}} \tag{37}\]  
\[\cos\theta_{2}=\frac{\gamma_{ah}}{\sqrt{\gamma_{ah}^{2}+\Gamma_{ah}^{2}}} \tag{38}\]  
Under the excitation of vertical velocity self-excited moment \(A_{1}\dot{h}\), the response of the torsional vibration system can be expressed by Equation (39):  
\[I\ddot{\alpha}+(C_{a0}-A_{2})\dot{\alpha}+(K_{a0}-A_{3})\alpha=A_{1}\dot{h} \tag{39}\]  
Substituting the vertical velocity \(\dot{h}=-\varepsilon_{sh}h-h_{0}e^{-\varepsilon_{sh}t}\omega_{sh}\sin\omega_ {sh}t\) into the right side of the torsion Equation (39), the torsional vibration displacement and velocity response are solved, as shown in Equation (40) and Equation (41), respectively:  
\[\alpha_{2}(t)=\frac{A_{1}}{\sqrt{\gamma_{ah}^{2}+\Gamma_{ah}^{2}}}(\cos\theta_ {2}+\xi_{sh}\sin\theta_{2})\dot{h}+\frac{A_{1}}{\sqrt{\gamma_{ah}^{2}+\Gamma_ {ah}^{2}}}\Big{(}1+\xi_{sh}^{2}\Big{)}\omega_{sh}\sin\theta_{2}h \tag{40}\]\[\dot{\alpha}_{2}(t)=\frac{A_{1}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}\left( \begin{array}{c}\sin\theta_{2}-2\xi_{sh}\cos\theta_{2}\\ -\xi_{sh}^{2}\sin\theta_{2}\end{array} \right)\omega_{sh}\dot{h}-\frac{A_{1}}{ \sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}\left( \begin{array}{c}(\cos\theta_{2 }+\xi_{sh}\sin\theta_{2})\\ (1+\xi_{sh}^{2})\end{array} \right)\omega_{sh}^{2}h \tag{41}\]  
Substituting the torsional vibration displacement response expression (35) and the corresponding velocity response expression (36) into the right side of Equation (1), the torsional vibration response feedback self-excited lift force is  
\[