# Research on an Improved Method for Galloping Stability Analysis Considering Large Angles of Attack
## 2 Theoretical Galloping Force Coefficient Equation
### The Improved Method
 \tag{14}\]  
when the section is axially symmetrical along the direction of vibration, the range of change in the wind attack angle can be regarded as \(-90^{\circ}<\alpha<90^{\circ}\), and the galloping force coefficient can only be calculated in this range. In Equation (13), when \(-90^{\circ}<\alpha<90^{\circ}\), \(\cos\alpha>0\); when \(Q\geq 0\), \(\Delta F_{y}\) is opposite to \(\dot{y}\), and the self-excited aerodynamic force is negative. So, in the state of micro-vibration, the aerodynamic force causes the dissipation of energy in the structural system, and the vibration is limited. When \(Q<0\), \(\Delta F_{y}\) and \(\dot{y}\) have the same sign, and the self-excited aerodynamic force performs positive work and continuously inputs energy into the structural system. Then, the amplitude of the structure gradually increases, which causes the phenomenon of vibration divergence. From another point of view, when \(Q\geq 0\), positive aerodynamic damping is generated, which means that the total damping of the system is positive, and galloping instability does not occur in the structure. When \(Q<0\), negative aerodynamic damping occurs, but the negative damping needs to be greater than the structure's own damping to cause galloping instability. Therefore, \(Q<0\) is a necessary condition for judging the galloping instability. Therefore, \(\cos\alpha C_{D}+\sin\alpha\frac{dC_{D}}{d\alpha}-\sin\alpha C_{L}+\cos\alpha \frac{dC_{L}}{d\alpha}\) is the criterion for quasi-steady galloping stability. Accordingly, the differential equation of motion can be expressed as follows:  
\[m\Big{(}\ddot{y}+2\zeta\omega\dot{y}+\omega^{2}y\Big{)}=-\frac{1}{2}\rho U^{2 }D\Bigg{(} \begin{array}{c}\cos\alpha C_{D}+\sin\alpha\frac{dC_{D}}{d\alpha} \\ -\sin\alpha C_{L}+\cos\alpha\frac{dC_{L}}{d\alpha}\end{array} \Bigg{)}\frac{ \dot{y}\cos\alpha}{\sqrt{\big{(}\dot{y}\big{)}^{2}+U^{2}+2U\dot{y}\sin\alpha}} \tag{15}\]  
where \(\zeta\), \(\omega\), and \(m\) denote the damping ratio, circular  
Then, the damping of the system can be expressed as  
\[d=2m\zeta\omega+\frac{1}{2}\rho UD\frac{dC_{V}}{d\alpha} \tag{20}\]  
when \(d=0\), the critical wind speed for galloping at a wind attack angle of \(\alpha\) can be expressed as  
\[U_{cg}=\frac{-4m\zeta\omega}{\rho D\frac{dC_{V}}{d\alpha}} \tag{21}\]  
where \(C_{V}\) is equal to the statistical average value of the lift coefficient on the body axis in a steady flow field, which can be calculated through wind tunnel tests or computational fluid dynamics (CFD) simulations. The meaning of replacing \(\cos\alpha C_{D}+\sin\alpha\frac{dC_{D}}{d\alpha}-\sin\alpha C_{L}+\cos\alpha \frac{dC_{L}}{d\alpha}\) with \(\frac{dC_{V}}{d\alpha}\) is that the possibility of quasi-steady galloping can be directly judged by observing the trend of change in the parameter \(C_{V}\) at every wind attack angle \(\alpha\). When \(dC_{V}/d\alpha<0\), the structure may gallop, and the galloping stability should be analyzed on the bridge. When \(dC_{V}/d\alpha>0\), the structure does not gallop. In theory, the calculation result of Equation (17) is consistent with that of \(dC_{V}/d\alpha\), but inevitable errors often occur during data processing, as there are too many parameters in Equation (17), leading to a relatively large error between the two equations. Therefore, the calculation process for \(dC_{V}/d\alpha\) is concise enough to improve the calculation accuracy.