# Research on an Improved Method for Galloping Stability Analysis Considering Large Angles of Attack
## 2 Theoretical Galloping Force Coefficient Equation
### Theoretical Solution  
[PERSON] introduced the equivalent wind attack angle and equivalent wind speed into the structural vibration equation according to their geometric relationship and expanded the equivalent wind attack angle into a linear expression of the structure's transverse wind direction vibration velocity \(\dot{y}\). The critical wind speed of galloping in the wind axis coordinate system can be expressed as  
\[U_{cg-x}=\frac{-4m\zeta\omega}{\rho D\Big{[}\Big{(}\frac{dC_{D}}{da}+C_{L} \Big{)}\sin\alpha\cos\alpha+\frac{dC_{L}}{da}\cos^{2}\alpha+C_{D}\sin^{2} \alpha+C_{D}\Big{]}} \tag{22}\]  
At the same time, [PERSON] et al. derived the formula for calculating the galloping amplitude based on the galloping force coefficient with the wind's angle of attack. The calculation results were in good agreement with the test results, which proved the accuracy of the calculation.  
The classic [PERSON] criterion, the d\(\mathrm{C_{V}}\)/d\(\alpha\) criterion, and the Xie criterion all contain the derivative of the tri-component force coefficient; so, it was necessary to fit the tri-component force coefficient curve shown in Figure 3. However, if the value of the tri-component force coefficient was small, the reliability of the fitted curve would have a great impact on its derivative. To reduce the error, the thir-order, fourth-order, fifth-order, and ninth-order nonlinear polynomial fitting curves of each tri-component force coefficient curve were plotted. Then, the original broken line and each fitting curve were derived. For example, the derivative curve of the lift coefficient \(C_{L}\) in the wind axis coordinate system is shown in Figure 4, and the correlation coefficients are shown in Table 1. The fitted derivative curve with the highest correlation coefficient was selected for the subsequent calculations with the aforementioned formula.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Modality** & **Pearson** & **Spearman** & **Kendall** \\ \hline Third order & 0.92737 & 0.97464 & 0.90389 \\ Fourth order & 0.95118 & 0.97464 & 0.90389 \\ Fifth order & 0.96341 & 0.97464 & 0.90389 \\ Ninth order & 0.94683 & 0.98699 & 0.93978 \\ \hline \end{tabular}
\end{table}
Table 1: Correlation coefficients.  
Figure 3: Tri-component force coefficients of the square section with the wind axis and body axis coordinates.  
By substituting the tri-component force coefficients that were processed above into the Den Hartog criterion, the \(\mathrm{dC_{V}/d\alpha}\) criterion, and the criterion proposed by [PERSON], the galloping force coefficients were calculated and are shown in Figure 5. It could be deduced that the galloping force coefficients of the three criteria were almost the same in the range of smaller wind attack angles. As the absolute value of the wind attack angle increased, the error between the calculation results of the \(\mathrm{dC_{V}/d\alpha}\) criterion and the Den Hartog criterion also gradually increased, indicating that the \(\mathrm{dC_{V}/d\alpha}\) criterion had enough reliability in the range of small wind attack angles. At a larger absolute value of the wind attack angle, the [PERSON] criterion was no longer applicable, thus showing the necessity of considering the influence of the wind attack angle on the galloping force coefficient.