# The Aerodynamics of New Design Soccer Balls Using a Three-Dimensional Printer
## 2 Materials and Methods
### Wind Tunnel Experiment  
Wind tunnel experiments were conducted at the University of Tsukuba using a circulatory, low-speed, low-turbulence wind tunnel (San Technologies Co., Ltd., San Diego, CA, USA). Figure 4 shows one of the manufactured balls mounted on a sting in the wind tunnel. The sting was inserted where the hole that began the printing process was located, which meant that the support location for 3D printing was covered by the sting and thus did not adversely affect wind-tunnel measurements. The maximum wind speed is approximately 55 m/s (198 kph or 123 mph) with a jet size of 1.5 m \(\times\) 1.5 m, a wind speed distribution within \(\pm\)0.5%, and a turbulence of less \begin{table}
\begin{tabular}{l l l l l l l l l l l l} \hline \hline
**Ball** & **Seam** & **DS** & \(n\) & \(b\) & \(c\) & \(k\) & \(r_{s}\) & \(c_{s}\) & \(k_{s}\) & \(A_{s}\) & \(m\) \\ \hline A & & & & & & & & & 0.000 & 465 \\ B & shallow & & & & & & 1.0 & 2.063 & 1.0 & 102.6 & 472 \\ C & shallow & conical & 9344 & 1.415 & 3.0 & 1.0 & 1.0 & 2.063 & 1.0 & 102.6 & 466 \\ D & shallow & spherical & 9344 & 1.415 & 3.0 & 1.0 & 1.0 & 2.063 & 1.0 & 102.6 & 470 \\ E & shallow & cylindrical & 9344 & 1.415 & 3.0 & 1.0 & 1.0 & 2.063 & 1.0 & 102.6 & 460 \\ F & shallow & conical (1/2) & 4760 & 3.168 & 3.0 & 1.0 & 1.0 & 2.063 & 1.0 & 102.6 & 469 \\ G & shallow & conical (1/4) & 2504 & 5.637 & 3.0 & 1.0 & 1.0 & 2.063 & 1.0 & 102.6 & 468 \\ H & deep & conical & 9344 & 1.415 & 3.0 & 1.0 & 1.0 & 5.367 & 5.0 & 431.6 & 473 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of fabricated soccer balls: ball label from Figure 1, seam type, dimple shape (DS), number of dimples (\(n\)), distance between dimples (\(b\), in mm), dimple width (\(c\), in mm), dimple depth (\(k\), in mm), seam radius of curvature (\(r_{s}\), in mm), seam width (\(c_{s}\), in mm), seam depth (\(k_{s}\), in mm), total seam area (\(A_{s}\), in cm\({}^{2}\)), and mass (\(m\), in g). Figure 3: Schematic illustration (not to scale) of the radius of curvature, \(r_{s}\), seam width, \(c_{s}\), and seam depth, \(k_{s}\), for seven of the manufactured balls. than 0.1%. The wind speed, \(v\), was varied from 7 m/s to 30 m/s, taking two sets of force measurements at 1-m/s intervals. Each force measurement lasted for 5 s. Because data were taken at the rate of 100 Hz, there were 500 force measurements taken in the 5-s interval. Given the two sets of force measurements, there were a total of 1000 force measurements. An average and standard deviation could be obtained from those 1000 force measurements. All measurements were performed in the temperature range 22 \({}^{\circ}\)C-30 \({}^{\circ}\)C. Temperature variation occurred because of time of day when experiments were performed. The aerodynamic forces acting on the manufactured balls were measured using a sting-type, six-component force detector (LMC-61256, Nissho Electric Works, Osaka, Japan). The force on the ball in the direction of the air flow is the drag force, \(\vec{F}_{D}\); the force in the horizontal direction, perpendicular to the air flow is the side force, \(\vec{F}_{S}\); and the vertical force is the lift force, \(\vec{F}_{L}\). The dimensionless aerodynamic coefficients extracted from the force data were the drag coefficient, \(C_{D}\), the side coefficient, \(C_{S}\), and the lift coefficient, \(C_{L}\). Those coefficients are contained in expressions for the magnitudes of the drag force, side force, and lift force, given by [32]:  
\[F_{D}=\frac{1}{2}\,C_{D}\,\rho\,A\,v^{2}\,, \tag{1}\]  
\[F_{S}=\frac{1}{2}\,C_{S}\,\rho\,A\,v^{2}\,, \tag{2}\]  
and  
\[F_{L}=\frac{1}{2}\,C_{L}\,\rho\,A\,v^{2}\,, \tag{3}\]  
respectively, where \(\rho\simeq 1.2\,\mathrm{kg/m^{3}}\) is the air density, which can vary slightly with temperature [32], and \(A\simeq 0.038\,\mathrm{m^{2}}\) is the cross-sectional area of a ball. Taking data for longer time intervals did not significantly alter the average force. Two measurements provided the opportunity to see how much fluctuation there was in measurements.