# Research on an Improved Method for Galloping Stability Analysis Considering Large Angles of Attack
## 2 Theoretical Galloping Force Coefficient Equation  
### The Improved Method  
In this section, a square section is taken as the research object. The structure has two main vibration axes: X and Y. Galloping generally occurs in conditions with a wind attack angle near 0\({}^{\circ}\) or 90\({}^{\circ}\). When the wind attack angle is near 0\({}^{\circ}\), the structure gallops in the Y direction; when the wind attack angle is around 90\({}^{\circ}\), the galloping vibration of the structure is in the X direction, and the vibration direction is perpendicular to the main axis of the model. The research methods for these two cases are the same [21]. Therefore, this study chose the situation in which the wind attack angle is around 0\({}^{\circ}\), and the vibration is in the Y direction. The parameter \(\dot{y}\) is the speed in the vibration direction. When the wind attack angle with respect to the direction of vibration is \(\alpha\), due to the vibration of the structure, an additional relative wind attack angle \(\Delta\alpha\) is generated between the incoming flow and the structure, as shown in Figure 1.  
Figure 1: Diagram of the additional wind attack angle. According to the sine theorem, the following is obtained:  
\[\frac{U_{r}}{\sin(90+\alpha)}=\frac{\dot{y}}{\sin\Delta\alpha} \tag{1}\]  
\[\sin\bigtriangleup\alpha=\frac{\dot{y}}{U_{r}}\cos\alpha \tag{2}\]  
According to the cosine theorem, the following is obtained:  
\[U_{r}=\sqrt{\left(\dot{y}\right)^{2}+U^{2}+2U\dot{y}\sin\alpha} \tag{3}\]  
Equation (2) is substituted into Equation (3), which yields  
\[\sin\bigtriangleup\alpha=\frac{\dot{y}\cos\alpha}{\sqrt{\left(\dot{y}\right)^{ 2}+U^{2}+2U\dot{y}\sin\alpha}} \tag{4}\]  
\[\bigtriangleup\alpha=\arcsin\frac{\dot{y}\cos\alpha}{\sqrt{\left(\dot{y} \right)^{2}+U^{2}+2U\dot{y}\sin\alpha}} \tag{5}\]  
Considering that the vibrations are micro-vibrations, Equation (5) can be approximately expressed as  
\[\bigtriangleup\alpha\approx\frac{\dot{y}\cos\alpha}{\sqrt{\left(\dot{y} \right)^{2}+U^{2}+2U\dot{y}\sin\alpha}} \tag{6}\]  
Under a uniform flow at an angle \(\alpha\), the force acting on the structure can be expressed as  
\[F_{y}(\alpha)=-F_{D}(\alpha)\sin\alpha-F_{L}(\alpha)\cos\alpha \tag{7}\]  
where \(F_{D}\) (\(\alpha\)) and \(F_{L}\) (\(\alpha\)) denote the drag force and the lift force under the wind axis coordinate system, respectively, which can be expressed as  
\[F_{D}(\alpha)=\frac{1}{2}\rho U^{2}DC_{D}(\alpha) \tag{8}\]  
Figure 1:0  
where \(\rho\) and \(D\) represent the air density, which is equal to 1.225 kg/m\({}^{3}\), and the characteristic length of a column with a square cross section, respectively. Equations (8) and (9) are substituted into Equation (7), which yields  
Figure 1:1  
The derivative of \(F\) with respect to X can be expressed as  
\[\frac{dF_{y}}{d\alpha}=-\frac{1}{2}\rho U^{2}D\left( \begin{array}{c}\cos \alpha C_{D}+\sin\alpha\frac{dC_{D}}{d\alpha}\\ -\sin\alpha C_{L}+\cos\alpha\frac{dC_{L}}{d\alpha}\end{array} \right) \tag{11}\]  
The self-excited aerodynamic force generated by the addition of a relative wind angle \(\Delta\alpha\) can be expressed as  
\[ \begin{array}{rl}\bigtriangleup F_{y}&=\frac{dF_{y}}{d\alpha}\bigtriangleup \alpha\\ &=-\frac{1}{2}\rho U^{2}D\left(\begin{array}{c}\cos\alpha C_{D}+\sin\alpha \frac{dC_{D}}{d\alpha}\\ -\sin\alpha C_{L}+\cos\alpha\frac{dC_{L}}{d\alpha}\end{array}\right)\bigtriangleup \alpha\end{array}  \tag{12}\]Equation (5) is substituted into Equation (12), which yields  
\[\triangle F_{y}=\frac{dF_{y}}{d\alpha}\triangle\alpha=-\frac{1}{2}\rho U^{2}D \Bigg{(} \begin{array}{c}\cos\alpha C_{D}+\sin\alpha\frac{dC_{D}}{d\alpha}\\ -\sin\alpha C_{L}+\cos\alpha\frac{dC_{L}}{d\alpha}\end{array} \Bigg{)}\frac{ \dot{y}\cos\alpha}{\sqrt{\big{(}\dot{y}\big{)}^{2}+U^{2}+2U\dot{y}\sin\alpha}} \tag{13}\]  
The galloping force coefficient can be expressed as \(Q\):  
\[ \begin{array}{ll}Q&=\cos\alpha C_{D}+\sin\alpha\frac{dC_{D}}{d\alpha}\\ &-\sin\alpha C_{L}+\cos\alpha\frac{dC_{L}}{d\alpha}\end{array}