# Flutter Mechanism of a Thin Plate Considering Attack Angles
## 2 Coupled Flutter Calculation Theory
Assume that the vertical system is initially subjected to the aerodynamic force of the uncoupled term and undergoes motion with a frequency of \(\overline{\omega}_{1}\) and a damping ratio of \(\overline{\xi}_{1}\); this can be regarded as a new system with \(\overline{\omega}_{1}\) and \(\overline{\xi}_{1}\) as its inherent dynamic parameters. Derive the vibration equation of this system under the aerodynamic force of the coupled term from Equation (4):  
\[ \begin{array}{l}\big{(}s^{2}+2\overline{\xi}_{1}\overline{\omega}_{1}s+ \overline{\omega}_{1}^{2}\big{)}h\\ =\mu\omega^{2}\times\big{(}H_{3}^{*}+iH_{2}^{*}\big{)}b\alpha\end{array}  \tag{10}\]For the torsional motion system (torsional mode branch), a similar approach can be used to form a new vibration system with a frequency of \(\overline{\omega}_{2}\) and a damping ratio of \(\overline{\xi}_{2}\). The motion equation of this system under the aerodynamic torque produced by vertical motion can be obtained from Equation (5):  
\[ \begin{array}{c}\big{(}s^{2}+2\overline{\xi}_{2}^{2}\overline{\omega}_{2}s+ \overline{\omega}_{2}^{2}\big{)}b\alpha\\ =\omega\omega^{2}\ \times\big{(}A_{4}^{*}+iA_{1}^{*}\big{)}h\end{array}  \tag{11}\]  
Among these, the four dynamic parameters of the new motion system determined by the uncoupled aerodynamic forces are given by the following four equations:  
\[\overline{\omega}_{1}=\omega_{h0}\Big{[}1-\mu(\omega/\omega_{h0})^{2}H_{4}^{ *}\Big{]}^{1/2} \tag{12}\]  
\[ \begin{array}{c}\overline{\xi}_{1}=\xi_{h0}(\omega_{h0}/\overline{\omega}_{ 1})-\\ 0.5\mu(\omega/\overline{\omega}_{1})H_{1}^{*}-\xi(\omega/\overline{\omega}_{1}) \end{array}  \tag{13}\]  
\[ \begin{array}{c}\overline{\omega}_{2}=\omega_{a0}\Big{[}1-v(\omega/\omega_{ a0})^{2}A_{3}^{*}\Big{]}^{1/2}\end{array}  \tag{14}\]  
\[ \begin{array}{c}\overline{\xi}_{2}=\xi_{a0}(\omega_{a0}/\overline{\omega}_{ 2})-\\ 0.5v(\omega/\overline{\omega}_{2})A_{2}^{*}-\xi(\omega/\overline{\omega}_{2}) \end{array}  \tag{15}\]  
In the equations, \(\mu=\rho b^{2}/m\) and \(v=\rho b^{4}/I\) represent the dimensionless mass and dimensionless moment of inertia, respectively. For the vertical mode branch, the aerodynamic torque caused by vertical motion induces torsional displacement in the torsional system, with a displacement amplitude ratio of \(b\alpha/h=\Phi e^{i\phi}\). Here, \(\Phi\) represents the amplitude ratio of torsional motion to vertical motion, \(\phi\) is the phase difference between the two motions, and \(\phi\) indicates that the vertical motion lags behind the torsional motion. Let the ratio of the vibration frequency to the new system frequency be \(\beta_{h}=\omega_{1}/\overline{\omega}_{2}\); from Equation (11), we have  
\[\Phi=vR_{d1}\sqrt{\big{(}A_{4}^{*}\big{)}^{2}+\big{(}A_{1}^{*}\big{)}^{2}} \tag{16}\]  
\[\phi=\theta_{1}+\theta_{2} \tag{17}\]  
\[\theta_{1}=\tan^{-1}(A_{1}^{*}/A_{4}^{*}) \tag{18}\]  
\[\theta_{2}=-\tan^{-1}\Big{\{}2\overline{\xi}_{2}^{2}\beta_{h}/\Big{(}1-\beta_ {h}^{2}\Big{)}\Big{\}} \tag{19}\]  
Here, \(R_{d1}\) represents the dynamic amplification factor of the torsional system under the influence of the coupled terms, with its value determined by Equation (20); \(\theta_{1}\) denotes the phase difference by which the coupling torque generated by vertical motion lags behind the vertical motion; \(\theta_{2}\) represents the phase difference between the coupling torque and the torsional motion when the coupling torque is in phase with the vertical motion.