# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 2 Flutter \"Incentive-Feedback\" Mechanism Theory
### Torsional Modal Damping and Stiffness
\begin{array}{l}M_{1}\Big{(}\dot{h}_{1},h_{1}\Big{)}=A_{1}\dot{h}_{1}(t)+A_{ 4}h_{1}(t)\\ =\left[\begin{array}{c}\frac{A_{1}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2 }}}(\cos\theta_{1}+\xi_{sa}\sin\theta_{1})\\ -\frac{A_{4}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}\frac{1}{\omega_{sa} }\sin\theta_{1}\end{array}\right]\dot{\alpha}+\left[\begin{array}{c}\frac{A _{1}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}(1+\xi_{sa}^{2})\omega_{sa}^ {2}\sin\theta_{1}\\ +\frac{A_{4}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}(\cos\theta_{1}-\xi_ {sa}\sin\theta_{1})\end{array}\right]\alpha\end{array}  \tag{14}\]  
Substituting the vertical vibration displacement response expression (12) and the corresponding velocity response expression (13) into the right side of the torsional vibration Equation (2), the vertical vibration response feedback self-lifting moment is:  
\[ \begin{array}{l}M_{2}\Big{(}\dot{h}_{2},h_{2}\Big{)}=A_{1}\dot{h}_{2}(t)+A _{4}h_{2}(t)\\ =\left[\begin{array}{c}\frac{A_{1}H_{2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{ 2}}}\left(\begin{array}{c}\omega_{sa}\sin\theta_{1}-2\xi_{sa}\omega_{sa}\cos \theta_{1}\\ -\xi_{sa}^{2}\omega_{sa}\sin\theta_{1}\end{array}\right)\\ +\frac{A_{4}H_{2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}(\cos\theta_{1}+\xi_ {sa}\sin\theta_{1})\end{array}\right]\dot{\alpha}+\left[\begin{array}{c}- \frac{A_{1}H_{2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}(\cos\theta_{1}+\xi _{sa}\sin\theta_{1})\big{(}1+\xi_{sa}^{2}\big{)}\omega_{sa}^{2}\\ +\frac{A_{4}H_{2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}\big{(}1+\xi_{sa}^{2 }\big{)}\omega_{sa}\sin\theta_{1}\end{array}\right]\alpha\end{array}  \tag{15}\]  
Moving the vertical vibration response feedback expression (14) and expression (15) from the right-hand side of the torsional vibration equation to the left-hand side yields the damping and stiffness of the torsional vibration system, respectively, as shown in Equations (16) and (17):  
\[ \begin{array}{l}C_{ah}=C_{a0}-A_{2}-\frac{A_{1}H_{2}}{\sqrt{\gamma_{ha}^{2}+ \Gamma_{ha}^{2}}}\big{(}\sin\theta_{1}-2\xi_{sa}\cos\theta_{1}-\xi_{sa}^{2} \sin\theta_{1}\big{)}\omega_{sa}-\frac{A_{4}H_{2}}{\sqrt{\gamma_{ha}^{2}+ \Gamma_{ha}^{2}}}(\cos\theta_{1}+\xi_{sa}\sin\theta_{1})\\ -\frac{A_{1}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}(\cos\theta_{1}+\xi_ {sa}\sin\theta_{1})+\frac{A_{4}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}} \frac{1}{\omega_{sa}}\sin\theta_{1}\\ K_{ah}=K_{a0}-A_{3}+\frac{A_{1}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}( \cos\theta_{1}+\xi_{sa}\sin\theta_{1})\big{(}\xi_{sa}^{2}+1\big{)}\omega_{sa}^ {2}-\frac{A_{4}H_{2}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}\big{(}\xi_{sa}^{2 }+1\big{)}\omega_{sa}\sin\theta_{1}\\ -\frac{A_{1}H_{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}\big{(}\xi_{sa}^{2}+1 \big{)}\omega_{sa}\sin\theta_{1}-\frac{A_{4}H_{3}}{\sqrt{\gamma_{ha}^{2}+ \Gamma_{ha}^{2}}}(\cos\theta_{1}-\xi_{sa}\sin\theta_{1})\end{array}  \tag{17}\]  
The torsional modal damping ratio and the modal frequency can be obtained by Equation (18) and Equation (19), respectively:  
\[\xi_{sa}=\frac{C_{ah}}{2I\omega_{ah}} \tag{18}\]  
\[\omega_{sa}=\omega_{ah}\sqrt{1-\xi_{sa}^{2}} \tag{19}\]  
where \(\omega_{ah}\) is the frequency of the torsional vibration system, \(\omega_{ah}=\sqrt{\frac{K_{sa}}{I}}\). The aerodynamic derivatives \(H_{i}\) and \(A_{i}\) in the damping of the torsional vibration system are expressed by the dimensionless flutter derivatives \(H_{i}^{*}\) and \(A_{i}^{*}\), respectively, and the damping of the torsional vibration system is as follows:  
\[C_{ah,1}=C_{a0} \tag{20}\]  
\[C_{ah,2}=-\rho B^{4}\omega_{sa}A_{2}^{*}=\rho^{2}B^{6}\frac{\omega_{sa}^{3}}{ \sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}A_{2}^{*}\Bigg{(}-\frac{1}{\rho B^{2}} \frac{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^{2}}}{\omega_{sa}^{2}}\Bigg{)} \tag{21}\]  
\[C_{ah,3}=\rho^{2}B^{6}\frac{\omega_{sa}^{3}}{\sqrt{\gamma_{ha}^{2}+\Gamma_{ha}^ {2}}}A_{1}^{*}H_{2}^{*}\Bigg{(} \begin{array}{c}-\sin\theta_{1}+\zeta_{sa}^{2 }\sin\theta_{1}\\ +2\xi_{sa}\cos\theta_{1}\end{array}