## 5 Results and Discussion
### Force Coefficient  
According to the quasi-steady theory by [PERSON] [59; 60], the drag force \(F_{d}\) and lift force \(F_{l}\) at each attack angle \(\alpha\) in the process of oscillation are the same as the values measured at the corresponding steady attack angle \(\alpha\). The relative attack angle \(\alpha=\arctan(\dot{y}/U)\) and forces were defined, as shown in Figure 9. The transverse force is the resultant of both drag force \(F_{d}\) and lift force \(F_{l}\)  
\[F_{y}=F_{L}\cos\alpha-F_{D}\sin\alpha \tag{19}\]  
Transverse force \(F_{y}\) was directly measured in the wind tunnel using the force balance. The transverse force coefficient \(C_{Fy}\) is expressed as  
\[C_{Fy}=\frac{F_{y}}{\frac{1}{2}\rho_{a}U_{eff}^{2}A} \tag{20}\]  
where \(\rho_{a}\) is the air density, \(A\) is the projected area of the bluff body, and \(U_{eff}\) is effective wind velocity. The differential equation governing the galloping oscillation is  
\[m\ddot{y}+C\dot{y}+Ky=F_{y}=\frac{1}{2}C_{Fy}\rho_{a}U_{eff}^{2}A \tag{21}\]  
where \(m\), \(C\), and \(K\) are the mass, damping, and stiffness modified by an electromechanical decoupled model [61]. The net damping ratio of the system is  
\[d=C\dot{y}-\frac{1}{2}C_{Fy}\rho_{a}U_{eff}^{2}A \tag{22}\]  
If \(d>0\), the system tends toward stability. When \(\dot{y}\) and \(C_{Fy}\) are in the same direction, as shown in Figure 10, the aerodynamic force is negative. Negative aerodynamic damping is an essential prerequisite for the galloping instability. With the increase of the wind velocity, \(d\) decreases gradually and may be negative, and then the system is prone to instability. In terms of the well-known [PERSON] criterion [62; 63], the necessary condition for the occurrence of the galloping instability is \(\left.\frac{\partial C_{Fy}}{\partial\alpha}\right|_{\alpha=0^{\circ}}>0\). In other words, a positive slope of the \(C_{Fy}\) versus \(\alpha\) curve at \(\alpha=0^{\circ}\) is an essential prerequisite for galloping instability. Figure 9: Variation of output power with wind velocity for WEHs with prism and arc-plate-shaped bluff bodies. The variation of transverse force coefficient \(C_{Fy}\) with wind attack angle \(\alpha\) for bluff bodies with different arc angles is shown in Figure 11. The slopes of the transverse force coefficient of all the three bluff bodies are positive when wind attack angle \(\alpha\) = 0\({}^{\circ}\). As a result, the corresponding WEHs work at high wind velocity, as shown in Figure 7. With the increase of the wind attack angle, the transverse force coefficient of shape 18 is much more significant than those of shapes 30 and 42. This is because the transverse force provides the negative aerodynamic force, and then the output power of the WEH with the shape 18 is much larger than those with the shapes 30 and 42. The variation of transverse force coefficient \(C_{Fy}\) with wind attack angle \(\alpha\) for bluff bodies with different tail lengths is shown in Figure 12. The slope of the transverse force coefficient of shape 17 is positive at \(\alpha\) = 0\({}^{\circ}\). In other words, shape 17 satisfies the essential prerequisite of galloping. The transverse force coefficient remains positive, but the value of the transverse force coefficient is relatively small, less than 0.1. The WEH with the shape 17 hardly works, because the bluff body cannot produce the aerodynamic damping force. The slope of the transverse force coefficient of shape 19 is positive at \(\alpha\) = 0\({}^{\circ}\). The bluff body also satisfies the essential prerequisite of galloping. The transverse force coefficient of shape 19 increased at the initial wind attack angle and decreased with the wind attack angle \(\alpha\) > 6\({}^{\circ}\). Although the WEH with the shape 19 can work, the performance was barely satisfactory. Figure 11: Variation of transverse force coefficient \(C_{Fy}\) with wind attack angle \(\alpha\) for bluff bodies with different arc angles.