# Flutter Mechanism of a Thin Plate Considering Attack Angles
## 2 Coupled Flutter Calculation Theory  
For a given dynamic system, the vibration equations of an N-degree-of-freedom system can be solved using the following vibration equation:  
\[Mq+Cq+Kq=F_{se} \tag{1}\]  
Here, \(M\) represents the mass matrix, \(C\) represents the damping matrix, \(K\) represents the stiffness matrix, \(q\) represents the displacement vector, and \(F_{se}\) represents the aerodynamic self-excitation force matrix. Figure 1: The research flowchart. For a segmental model considering only the vertical and torsional degrees of freedom, the equation of motion can be simplified as follows:  
\[\ddot{h}+2\xi_{h0}\omega_{h0}\dot{h}+\omega_{h0}^{2}h=F_{se} \tag{2}\]  
\[\ddot{\alpha}+2\xi_{a0}\omega_{a0}\dot{h}+\omega_{a0}^{2}h=\frac{1}{2}\rho U^{2 }(2b^{2})/I\times kA_{1}^{*}\frac{\dot{h}}{U}+kA_{2}^{*}\frac{b\dot{\alpha}}{U} +k^{2}A_{3}^{*}\alpha+k^{2}A_{4}^{*}\frac{h}{b} \tag{3}\]  
In this equation, \(h\) and \(\alpha\) represent the vertical displacement and torsion at the section's center of rotation, respectively; \(\ddot{h}\), \(\dot{h}\), \(\ddot{\alpha}\), and \(\ddot{\alpha}\) denote the vertical acceleration, vertical velocity, torsional acceleration, and torsional velocity at the center of rotation, respectively; \(\omega_{h0}\) and \(\omega_{a0}\) are the natural circular frequencies of the system in the vertical and torsional directions; \(\xi_{h0}\) and \(\xi_{a0}\) are the structural damping ratios in the vertical and torsional directions; \(U\) is the incoming wind speed; \(\rho\) is the air density; \(b\) is the half-width of the section; \(m\) and \(I=mr^{2}\) are the mass per unit length and the mass moment of inertia per unit length of the system, respectively, with \(r\) as the radius of gyration; \(A_{i}^{*}\) and \(H_{i}^{*}\) are the dimensionless aerodynamic derivatives related to the reduced wind speed; \(k=\omega b/U\) is the reduced frequency; and \(\omega\) is the system's circular frequency. To accurately describe the damping characteristics and periodic properties of the system in each degree of freedom during vibration, it is assumed that both the vertical motion and the torsional motion follow an exponential form as given by \(q=q_{0}e^{st}\). Substituting this into Equations (1) and (2) yields the following vibration equations in the Laplace domain:  
\[ \begin{array}{l}\big{(}s^{2}+2\xi_{h0}^{2}\omega_{h0}s+\omega_{h0}^{2}\big{)} h=\frac{1}{2}\rho U^{2}(2b)/m\\ \times\left(kH_{1}^{*}\frac{\dot{h}}{U}+kH_{2}^{*}\frac{\dot{h}\dot{h}\dot{h} \dot{h}}{U}+k^{2}H_{3}^{*}\alpha+k^{2}H_{4}^{*}\frac{h}{b}\right)\end{array}  \tag{4}\]  
\[ \begin{array}{l}\big{(}s^{2}+2\xi_{a0}\omega_{a0}s+\omega_{a0}^{2}\big{)} \alpha=\frac{1}{2}\rho U^{2}(2b^{2})/I\\ \times\left(kA_{1}^{*}\frac{\dot{h}}{U}+kA_{2}^{*}\frac{\dot{h}\dot{h}}{U}+k^ {2}A_{3}^{*}\alpha+k^{2}A_{4}^{*}\frac{h}{b}\right)\end{array}  \tag{5}\]  
Here, \(s=-\xi\omega+i\omega\) denotes the complex frequency in the Laplace domain represents the system's vibration damping ratio; \(\omega\) is the system's natural circular frequency; \(i\) is the imaginary unit \(i^{2}=-1\). In general, the structural damping ratio is small; therefore, based on this fact, the following simplification can be made:  
\[s^{2}\approx-\omega^{2}-2\xi\omega^{2}i \tag{6}\]  
\[2\xi_{h0}\omega_{h0}s\approx 2\xi_{h0}^{2}\omega_{h0}\omega i \tag{7}\]  
\[sh\approx i\omega h \tag{8}\]  
\[sa\approx\omega si \tag{9}\]  
Refer to the coupling flutter closed-form solution theory, which aims to provide readers with a clearer understanding of the flutter mechanism and process. Assume that during the system's vibration process, the vertical degrees of freedom of the vibration system (vertical mode branch) are subjected to aerodynamic lift due to vertical motion (uncoupled terms) and aerodynamic lift due to torsional motion (coupled terms).