# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 2 Flutter \"Incentive-Feedback\" Mechanism Theory
### Vertical Modal Damping and Stiffness
\begin{array}{l}F_{L1}\big{(}\dot{\alpha}_{1},\kappa_{1}\big{)}=H_{2}\dot{ \alpha}_{1}(t)+H_{3}\kappa_{1}(t)\\ =\left[\begin{array}{c}\frac{H_{2}A_{4}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^ {2}}}(\cos\theta_{2}+\xi_{sh}\sin\theta_{2})-\frac{H_{3}A_{4}}{\sqrt{\gamma_{ sh}^{2}+\Gamma_{sh}^{2}}}.\\ \frac{1}{\omega_{sh}}\sin\theta_{2}\end{array}\right]\dot{h}+\left[\begin{array} []{c}\frac{H_{2}A_{4}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}\big{(}1+\xi_ {sh}^{2}\big{)}\omega_{sh}\sin\theta_{2}+\frac{H_{3}A_{4}}{\sqrt{\gamma_{sh}^ {2}+\Gamma_{sh}^{2}}}.\\ (\cos\theta_{2}-\xi_{sh}\sin\theta_{2})\end{array}\right]\dot{h}\\ \end{array} \right]\dot{h}+\left[ \begin{array}{c}-\frac{H_{2}A_{4}}{\sqrt{ \gamma_{sh}^{2}+\Gamma_{sh}^{2}}}(\cos\theta_{2}+\xi_{sh}\sin\theta_{2})\\ +\frac{H_{3}A_{1}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}\big{(}1+\xi_{sh}^{ 2}\big{)}\omega_{sh}\sin\theta_{2}\end{array} \right]\dot{h}\\ \end{array} \tag{42}\]  
Substituting the torsional vibration displacement response expression (40) and the corresponding velocity response expression (41) into the right side of Equation (1), the torsional vibration response feedback self-excited lift is  
\[ \begin{array}{l}F_{L2}\big{(}\dot{\alpha}_{2},\alpha_{2}\big{)}=H_{2}\dot{ \alpha}_{2}(t)+H_{3}\alpha_{2}(t)\\ =\left[\begin{array}{c}\frac{H_{2}A_{1}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^ {2}}}\Bigg{(}\begin{array}{c}\sin\theta_{2}-2\xi_{sh}\cos\theta_{2}\\ -\xi_{sh}^{2}\sin\theta_{2}\end{array}\Bigg{)}\omega_{sh}\\ +\frac{H_{3}A_{1}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}(\cos\theta_{2}+\xi_ {sh}\sin\theta_{2})\end{array}\right]\dot{h}+\left[\begin{array}{c}-\frac{H_ {2}A_{1}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}(\cos\theta_{2}+\xi_{sh} \sin\theta_{2})(1+\xi_{sh}^{2})\omega_{sh}^{2}\\ +\frac{H_{3}A_{1}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}\big{(}1+\xi_{sh}^{ 2}\big{)}\omega_{sh}\sin\theta_{2}\end{array}\right]\dot{h}\\ \end{array}  \tag{43}\]  
The damping and stiffness of the vertical vibration system can be obtained by moving expression (42) and expression (43) of the torsional vibration response self-excited lift force on the right side of the vertical vibration equation to the left side of the vertical vibration equation, as shown in the following Equations (44) and (45), respectively:  
\[ \begin{array}{l}C_{h\kappa}=C_{h0}-H_{1}-\frac{H_{2}A_{1}}{\sqrt{\gamma_{sh} ^{2}+\Gamma_{sh}^{2}}}\big{(}\sin\theta_{2}-\xi_{sh}^{2}\sin\theta_{2}-2\xi_{ sh}\cos\theta_{2})\omega_{sh}-\frac{H_{3}A_{1}}{\sqrt{\gamma_{sh}^{2}+ \Gamma_{sh}^{2}}}(\cos\theta_{2}+\xi_{sh}\sin\theta_{2})\\ -\frac{H_{2}A_{4}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}(\cos\theta_{2}+\xi_ {sh}\sin\theta_{2})+\frac{H_{3}A_{4}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}} \frac{1}{\omega_{sh}}\sin\theta_{2}\end{array}  \tag{44}\]  
\[ \begin{array}{l}K_{h\kappa}=K_{h0}-H_{4}+\frac{H_{2}A_{1}}{\sqrt{\gamma_{sh}^ {2}+\Gamma_{sh}^{2}}}\big{(}\cos\theta_{2}+\xi_{sh}^{2}\sin\theta_{2}\big{)} \big{(}1+\xi_{sh}^{2}\big{)}\omega_{sh}^{2}-\frac{H_{3}A_{1}}{\sqrt{\gamma_{sh}^ {2}+\Gamma_{sh}^{2}}}\big{(}1+\xi_{sh}^{2}\big{)}\omega_{sh}\sin\theta_{2}\\ -\frac{H_{2}A_{4}}{\sqrt{\gamma_{sh}^{2}+\Gamma_{sh}^{2}}}\big{(}1+\xi_{sh}^{ 2}\big{)}\omega_{sh}\sin\theta_{2}-\frac{H_{3}A_{4}}{\sqrt{\gamma_{sh}^{2}+ \Gamma_{sh}^{2}}}\big{(}\cos\theta_{2}-\xi_{sh}\sin\theta_{2}\big{)}\end{array}  \tag{45}\]  
The vertical modal damping ratio and modal frequency can be obtained by Equations (46) and (47), respectively:  
\[\zeta_{sh}=\frac{C_{h\kappa}}{2m_{h}\omega_{h\kappa}} \tag{46}\]  
\[\omega_{sh}=\omega_{h\kappa}\sqrt{1-\xi_{sh}^{2}} \tag{47}\]  
where \(\omega_{h\kappa}\) is the frequency of the vertical vibration system, \(\omega_{h\kappa}=\sqrt{\frac{K_{h\kappa}}{m_{h}}}\).