# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 2 Flutter \"Incentive-Feedback\" Mechanism Theory  
This text discusses the modal decoupling of a bending-torsion-coupled vibration system using an \"incentive-feedback\" mechanism, with the identification of aerodynamic damping for flutter vertical modes and torsional modes. The bending-torsional coupling of flutter [25] is shown in Equations (1) and (2):  
\[m_{h}\ddot{h}+(C_{h0}-H_{1})\dot{h}+(K_{h0}-H_{4})h=H_{2}\dot{\alpha}+H_{3}\alpha \tag{1}\]  
\[I\ddot{\alpha}+(C_{a0}-A_{2})\dot{\alpha}+(K_{a0}-A_{3})\alpha=A_{1}\dot{h}+A_ {4}h \tag{2}\]  
where \(m_{h}\) and \(I\) represent the unit mass per length in the vertical direction and the mass moment of inertia, respectively; \(C_{h0}\) and \(C_{a0}\) represent the inherent damping in the vertical and torsional directions, respectively; \(C_{h0}=2m_{h}\xi_{h0}\omega_{h0}\), \(\omega_{h0}=2\pi f_{h0}\), \(\xi_{h0}\), and \(\omega_{h0}\) are the vertical natural damping ratio and circular frequency, respectively; \(C_{a0}=2l\xi_{a0}\omega_{a0}\), \(\omega_{a0}=2\pi f_{a0}\), \(\xi_{a0}\), and \(\omega_{a0}\) are the torsional natural damping ratio and circular frequency, respectively; \(K_{h0}\) and \(K_{a0}\) are the inherent stiffness in the vertical and torsional directions, respectively, \(K_{h0}=m_{h0}\omega_{h0}^{2}\), \(K_{a0}=I\omega_{h0}^{2}\); \(\ddot{h}\), \(\dot{h}\), and \(h\), respectively, represent vertical acceleration, velocity, and displacement; \(\ddot{\alpha}\), \(\dot{\alpha}\), and \(\alpha\) are torsional acceleration, velocity, and displacement, respectively; \(H_{i}(i=1,2,3,4)\) are the aerodynamic derivatives of the lift; \(A_{i}(i=1,2,3,4)\) are the aerodynamic derivatives of the lifting moment; \(H_{1}\dot{h}+H_{4}h\) and \(H_{2}\dot{\alpha}+H_{3}\alpha\) are the vertical self-excited lift and the coupled torsional self-excited lift, respectively; and \(A_{2}\dot{\alpha}+A_{3}\alpha\) and \(A_{1}\dot{h}+A_{4}h\) are the torsional self-excited lift moment and the coupled vertical self-excited lift moment, respectively. Among them, the torsional self-excited lift force \(H_{2}\dot{\alpha}+H_{3}\alpha\) and the vertical self-excited lift moment \(A_{1}\dot{h}+A_{4}h\) cause the vertical vibration system and the torsional vibration system to couple with each other, forming a bending-torsion-coupled vibration system.  
The relationship between aerodynamic derivatives and dimensionless flutter derivatives of aerodynamic forces is presented in Equations (3) and (4):  
\[\begin{cases}H_{1}=\rho B^{2}\omega H_{1}^{*}&H_{2}=\rho B^{3}\omega H_{2}^{*}\\ H_{3}=\rho B^{3}\omega^{2}H_{3}^{*}&H_{4}=\rho B^{2}\omega^{2}H_{4}^{*}\\ \end{cases} \tag{3}\]  
\[\begin{cases}A_{1}=\rho B^{3}\omega A_{1}^{*}&A_{2}=\rho B^{4}\omega A_{2}^{*} \\ A_{3}=\rho B^{4}\omega^{2}A_{3}^{*}&A_{4}=\rho B^{3}\omega^{2}A_{4}^{*}\\ \end{cases} \tag{4}\]  
where \(\rho\) is air density, \(U\) is incoming average flow velocity, and \(B\) is structural cross-sectional width; \(H_{i}^{*}(i=1,2,3,4)\) are the dimensionless flutter derivatives of lift; and \(A_{i}^{*}(i=1,2,3,4)\) are dimensionless flutter derivatives of lift moments.