## 3 Flutter Critical State and Hopf Bifurcation
### Flutter Critical Wind Speed Solution  
Flutter Equations (4) and (5) can be written in the form of state-space equations, i.e., \(\dot{\mathbf{x}}=f(\mathbf{x})\). Let \(\mathbf{x}=\left[h,\,\dot{\mathbf{x}},\dot{h},\dot{\mathbf{x}}\right]^{T}\); thus, the state-space equations of this nonlinear flutter system can be written in the following form:  
\[\dot{\mathbf{x}}=\mathbf{F}(U,\omega,x)=\mathbf{F}_{Linear}\cdot\mathbf{x}+\mathbf{F}_{Nonlinear} (\mathbf{x}) \tag{14}\]  
where \(\mathbf{F}_{Linear}\) is the linear coefficient matrix of the flutter equations, and \(\mathbf{F}_{Nonlinear}\) is the nonlinear term of the flutter equations. Based on the analysis and calculations of the nonlinear system equations in the previous sections, the critical flutter state can be computed using the linear part of Equation (14), and then the complete state-space equations can be used to analyze the nonlinear flutter response. The coefficients of the linear and nonlinear parts of Equation (14) are given as follows:  
\[\mathbf{F}_{Linear}=\rho U^{2}B \begin{bmatrix}0&0&1&0\\ 0&0&0&1\\ \frac{K^{2}H_{4}^{*}}{Bm_{h}}-\frac{k_{h}}{m_{h}}&\frac{K^{2}H_{3}^{*}}{m_{h }}&\frac{KH_{1}^{*}}{Bm_{h}}-\frac{c_{h}}{m_{h}}&\frac{KH_{2}^{*}B}{Um_{h}}\\ \frac{K^{2}A_{4}^{*}}{I}&\frac{BK^{2}A_{4}^{*}}{I}-\frac{k_{h}}{I}&\frac{BKA_{ 4}^{*}}{UI}&\frac{B^{2}KA_{4}^{*}}{UI}-\frac{c_{h}}{T}\end{bmatrix}  \tag{15}\]  
\[\mathbf{F}_{Nonlinear}(\mathbf{x})= \begin{bmatrix}0\\ 0\\ \frac{\rho U^{2}B}{m_{h}}\left(KH_{2,02}^{*}\frac{B^{2}\left|\dot{\mathbf{x}} \right|\dot{\mathbf{x}}}{U^{2}}+K^{2}H_{3,02}^{*}|\alpha|\alpha+K^{2}H_{4,01}^{*}| \alpha|+K^{2}H_{4,02}^{*}\alpha^{2}\right)\\ \frac{\rho U^{2}B^{2}}{I}\left(KA_{2,02}^{*}\frac{B\left|\dot{\mathbf{x}} \right|}{U}+KA_{2,03}^{*}\frac{B^{2}\dot{\mathbf{x}}^{2}}{UI}\right)\frac{B\dot{ \mathbf{x}}}{UI}\end{bmatrix}  \tag{16}\]  
The critical point of a nonlinear motion system can be obtained by solving its linear expression. According to the [PERSON] method, as the wind speed \(V_{r}\) of the system parameters changes, when the eigenvalues of the matrix (15) have positive real parts, the corresponding wind speed \(V_{Cr}\) is the critical value of the system [31]. Therefore, the critical flutter state can be obtained through the following steps:  
1. Assume a small value of frequency \(\omega_{0}\) and substitute it into the matrix (15). Gradually increase the reduced wind speed \(V_{r}\) until the first pair of complex conjugate eigenvalues of the matrix have zero real parts. Record the imaginary part \(Imag(\lambda)\) of this eigenvalue as frequency \(\omega_{i}=Imag(\lambda)\);
2. Take \(\omega_{i}\) as the new frequency value and substitute it into the matrix (15). Again, gradually increase the reduced wind speed \(V_{r}\) until the first complex eigenvalue of \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(H_{\text{H}}^{*}\backslash\mathbf{\wedge}_{\text{H}}^{*}\) & \(H_{\text{2}}^{*}\backslash\mathbf{\wedge}_{\text{H}}^{*}\) & \(H_{\text{3}}^{*}\backslash\mathbf{\wedge}_{\text{H}}^{*}\) & \(H_{\text{4}}^{*}\backslash\mathbf{\wedge}_{\text{H}}^{*}\) \\ \hline entry 1 & 0.29832 & \(-\)0.17132 & 0.03441 & \(-\)0.00224 \\ entry 2 & 0.00214 & \(-\)0.00314 & 3.55170 \(\times\) 10\({}^{-4}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The coefficients in the curve of the fitting quartic function of the flutter derivative. the matrix has zero real parts. Record the imaginary part \(Imag_{i}(\lambda)\) of this eigenvalue as the new flutter frequency \(\omega_{i+1}=Imag_{i}(\lambda)\). ;
3. Compare \(\omega_{i}\) and \(\omega_{i+1}\). Repeat step 2 until \(\left|\omega_{i+1}-\omega_{i}\right|\) approaches zero. At this point, the frequency value is the critical flutter frequency \(\omega_{C}\) of the system, the reduced wind speed value is the critical reduced wind speed \(V_{Cr}\) of the flutter, and the flutter critical wind speed \(U_{C}\) can be obtained using the relationship \(V_{r}=2U\tau/B\omega\).