# Surface Patterns for Drag Modification in Volleyballs
## 2 Experiments  
As for surface patterns on volleyballs, we selected conventional and surface-modified volleyballs. Conventional volleyballs (Adidas AV514 RB and Mizuno 9 OV80027) and surface-modified volleyballs (Mikasa MVA200 CEV and Molten V5M5000) were purchased commercially. Conventional and surface-modified balls had same diameters of 21 cm, same circumferences of 65\(\sim\)67 cm and same weights of 260\(\sim\)280 g, except for different surface panel numbers and patterns as: six-panel and smooth surface for conventional balls (Adidas AV514 RB and Mizuno 9 OV80027), while six-panel and hexagon-patterned surface (Molten V5M5000) and eight-panel and dimpled surface (Mikasa MVA200 CEV) for surface-modified balls. For the wind tunnel experiments, a circuit flow-type wind tunnel (San Technologies Co., Ltd., Tochigi, Japan) was used at the University of Tsukuba. The maximum wind speed was 55 m/s, the blow out size was 1.5 \(\times\) 1.5 m\({}^{2}\) and the wind speed and the turbulence distributions were controlled within \(\pm\)0.5% and less than 0.1%, respectively. For the volleyball installed in the wind tunnel, the aerodynamic characteristics were measured with respect to the panel direction. The ball was supported from the back support that was made of a rod with 0.8 m length and 0.02 m width. The aerodynamic force of each ball was measured in the range of wind speeds (\(U\)) from 7 to 35 m/s using a Sting-type 6-division detector (LMC-61256, Nissho Electric Works). The aerodynamic coefficients of the drag coefficient (\(C_{d}\)), the lift force coefficient (\(C_{l}\)) and the side force coefficient (\(C_{s}\)) were calculated as \(C_{d}=\frac{2D}{\rho U^{2}A}\), \(C_{l}=\frac{2L}{\rho U^{2}A}\) and \(C_{s}=\frac{2S}{\rho U^{2}A}\), respectively. Here \(\rho\) is the density of air (\(\rho=1.2\) kg/m\({}^{3}\)), \(A\) is the cross-sectional area of the volleyball (\(A=\pi d^{2}/4\approx 0.0346\) m\({}^{2}\) for the diameter of the volleyball \(d=0.21\) m [15]), and the \(D\), \(L\), and \(S\) quantities are the longitudinal, transverse and side components of the aerodynamic forces [6].  
Two intrinsic forces act on a ball moving through the air by competition of gravity and aerodynamics. The first force acts down on the ball, namely, the ball weight (\(mg\)), where \(m\) is the ball mass and \(g=9.80\) m/s\({}^{2}\) is the constant magnitude of gravitational acceleration near the Earth surface. The second force is the drag force, whose direction is opposite the ball velocity. By considering that the \(x\) axis point along the horizontal and the \(y\) axis point vertically upward, the [PERSON]'s second law is reduced to \(\dot{x}=-\beta vC_{d}\dot{x}\) and \(\ddot{y}=-\beta vC_{d}\dot{y}-g\) where \(\beta=\frac{\rho A}{2m}\), \(v=\sqrt{\dot{x}^{2}+\dot{y}^{2}}\) and the dot signifies the time derivative. These relations can be numerically solved under appropriate initial conditions using a fourth-order Runge-Kutta algorithm. For the speed-dependent C\({}_{d}\), a linear interpolation between experimental wind-tunnel data points was used. The drag coefficients of each ball taken from the wind tunnel experiments and the corresponding Reynolds numbers were obtained at the same initial speed of 15 m/s and the takeoff angle of 25\({}^{\circ}\).  
The flight characteristics were investigated by measuring the landing point after being hit. The hitting robot aimed at the center of a non-rotating ball (e.g., float serve) at the typical ball speed of 15 m/s in real games [25; 26] with a target (1 \(\times\) 1 m\({}^{2}\)) located on the floor 15 m away. The position at which the volleyball hits the floor was denoted as the landing point with its flight distance along the horizontal and lateral distances. Data analysis was performed from the landing point measurements from the repeated 30 throws for four balls and two panel directions for a total dataset of 240 throws.