# Influences of an Arrow's Aerodynamic Pressure Center on Recurve Bow Shooting
## 1 Introduction  
The use of bows and arrows dates back nearly 20,000 years. Archery has been listed as an official event since the first modern Olympic Games in 1896. Athletes compete with recurve bows in Olympic sports. Accuracy is key to winning any competition, and it is related to factors including the initial velocity, boundary-layer state, center of aerodynamic pressure (\(C_{\mathrm{p}}\)), center of mass (\(C_{\mathrm{m}}\)) position, and angle of attack (\(\gamma\)) of arrows [1, 2, 3, 4]. Therefore, aerodynamics research on arrows is not only critically significant for the design and manufacturing of bows and arrows but can also help excellent archers improve their archery. Scholars have conducted preliminary theoretical research on the mechanics of bows and arrows since the early 20 th century [3, 4, 5, 6]. However, the mathematical model is too simple to apply to recurve bows. It is 1987, when [PERSON] [6] divided the ballistic trajectory of arrows into two stages, that marks the research on archery entering a more systematic and scientific stage. [PERSON] defined the first stage of archery as the inner ballistic trajectory, in which an arrow interacts with the archer and bow until it disengages from the bowstring,as shown in Figure 1; the second stage is the outer ballistic trajectory, in which the arrow keeps flight after being detached from the bow until hitting the target, as shown in Figure 2. The theory of the outer ballistic trajectory of arrows is more complex than that governing the inner trajectory, so early research focuses on the theory of the inner ballistic trajectory [7; 8; 9; 10]. Researchers [11; 12; 13; 14; 15; 16; 17; 18; 19] have begun to study the outer ballistic trajectory of arrows only in recent years. [PERSON] [14] estimated the lift and drag coefficients of arrows according to the geometry (size and structure) of arrows. On this basis, [PERSON] [15] further established a complex model considering the angle between the longitudinal axis of the arrow and its flight direction, that is, the angle of attack (\(\gamma\)). This represents the deviation of the arrowhead from its flight path, as shown in Figure 2. [PERSON] [16] and [PERSON] [17] modeled an arrow by dividing the arrow into four modules, namely the arrowhead, shaft, neck, and fletching, thus investigating the influences of boundary-layer transition on the shooting accuracy of the arrow. Afterwards, [PERSON] et al. [18] conducted water-channel experiments to further verify that the boundary-layer transition greatly influences the shooting accuracy of arrows. [PERSON] [19] measured the drag coefficients, the lift coefficients, and their relationships with the Reynolds numbers of two arrows under different initial velocities and \(\gamma\) by performing wind tunnel experiments and flight experiments. In this way, they verified that the shape and \(\gamma\) of arrowheads affect the laminar-turbulent transition in the boundary layer. Figure 2 illustrates that the arrow's angle of attack, \(\gamma\), changes dynamically during its flight, and this variability results in a fluctuating aerodynamic center of pressure, \(C_{\rm p}\). Aerodynamics significantly affects the trajectories of sports projectiles [20; 21; 22; 23; 24; 25]. For example, [PERSON] et al. [25] studied javelins of a larger mass and found that changes in the \(C_{\rm p}\) position of javelins influence the pitch moment of javelins, thus altering their trajectory. Figure 1: Diagram of the inner ballistic trajectory of an arrow [9]. \(\alpha_{\rm y}\): the angles of a bow’s torsion around the y-axis; \(\alpha_{\rm z}\): the angles of a bow’s torsion around the z-axis; \(\rm{I_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{ \rm{ \rm{    }}}}}}}}}}}}}}}}}\) the length of an initial draw. Figure 2: Diagram of the outer ballistic trajectory of the arrow [9]. However, arrows have a lower mass and a higher Mach number than juveniles. In addition, the presence of the fletching also complicates the computation of \(C_{\rm p}\).