# Influences of an Arrow's Aerodynamic Pressure Center on Recurve Bow Shooting
## 3 Results  
### Influences of Changes in \(\gamma\) on \(C_{\rm P}\)  
\(C_{\rm P}\) and \(C_{\rm m}\) positions of an arrow do not generally overlap. The offset between the \(C_{\rm P}\) and \(C_{\rm m}\) positions may affect the stability and flight performance of the arrow. When the \(\gamma_{\rm 0}\) and \(\gamma_{\rm 0}\) change in flight, the aerodynamic moment on arrows also varies, causing differences in the force distribution on the arrow and, therefore, variation in the position of \(C_{\rm P}\). For a recurve bow, the arrow features a non-zero \(\gamma\) on the vertical plane when it leaves the bow, which facilitates the disengagement of the arrow from the bow, guarantees a short shooting time for the arrow in contact with the bow, and therefore maintains the stability of the arrow. Since the arrow moves very fast, the force of air on the arrow is also very strong. It can be seen from the arrow surface pressure diagram in Figure 13 that the arrowhead and fletching are the positions that bear the greatest pressure. The frictional resistance along the arrow shaft and the resistance from the arrow fletching are the main components of the fletching force. Relative to the lift on the arrow under small \(\gamma\), the vertical force on the arrowhead is small. The lift on the fletching cannot be ignored: this dominates the spin of the arrow and helps maintain flight stability, while the drag on the fletching also induces a slight, non-negligible, lateral displacement of the arrow. The specific forces acting on the arrow are shown in Figure 14.  
Figure 14: _Cont._  
Figure 13: Contour lines for surface pressure on the arrow. Contour line: surface pressure (Pa). Figure 15 illustrates changes in \(C_{\rm p}\) with \(\gamma\), and \(\gamma_{\rm\phi}\) and \(\gamma_{\rm\beta}\) are the pitch angle and yaw angle, respectively, providing a more detailed differentiation of the attack angle, \(\gamma\). \(\gamma_{\rm\theta}\) expresses the variation in the attack angle in the simulation with a change only in the arrow's upward and downward angular deflection, while \(\gamma_{\rm\beta}\) expresses the variation in the attack angle in the simulation with a change only in the arrow's horizontal angular deflection on the plane. In the free flight process of the arrow, \(C_{\rm p}\) varies a little (by millimeters). Under the conditions of a zero and small \(\gamma\), the \(C_{\rm p}\) is concentrated in the neck region, ranging from 0.478 to 0.498 m. The moment (drag and lift on the fletching and on the arrow shaft to the rear of the center of mass) on the rear part of the arrow is always greater than that on the front part of the arrow (the point and the arrow shaft in front of the center of mass). When the angle of attack is large, the arrow will become unstable. Since the angle of attack of the arrow in international archery competitions generally does not exceed \(\pm 3^{\circ}\), the change in the pressure center point when the angle of attack is large was not studied in the present research. As shown in Figure 15a, \(C_{\rm p}\) separately reaches its minimum and maximum values when the \(\gamma_{\rm\theta}\) is \(-1^{\circ}\) and \(1^{\circ}\). This is because, when \(\gamma_{\rm\theta}\) is negative, to align the \(C_{\rm m}\) velocity with the arrowhead, the air drag on the neck turns to down-pressure, thus forcing the arrow to lift. The forward shift in the \(C_{\rm p}\) position occurs to reduce the moment and therefore decelerate the lifting speed of the arrow, thus maintaining a more stable flight of the arrow. If \(\gamma_{\rm\theta}\) is positive, because the \(C_{\rm m}\) velocity is below the flight direction, the air drag on the arrow turns to lift that is conducive to the arrow inducing a stable spinning effect, which is enhanced via the backward shift of the \(C_{\rm p}\) position, thus decreasing \(\gamma_{\rm\theta}\). When \(\gamma_{\rm\theta}\) decreases, the \(C_{\rm p}\) position moves forward again, which reduces the moment on the arrow again, thus weakening the above spinning effect.