## 4 Modeling Ball/Pad Impact
### Experiments/Simulations Comparisons  
While the ball's simulated incident trajectory was an input, its reflected trajectory needed to be determined in order to compute the aforementioned metrics. Thus, the position of the center of the ball led to the determination of deviation angle and the linear coefficient of restitution and was calculated from the coordinates at any time increment of two nodes, denoted M and N in Figure 7a, located away from the contact region. The center was half way between M and N, assuming that these two nodes corresponded at any time to the ball's diameter. With this assumption, the rotation angle \(\beta\) of the ball with respect to the out of plane _x-axis_ was determined as follows:  
\[\beta=\mathrm{atan}\bigg{(}\frac{\mathrm{z_{N}}-\mathrm{z_{M}}}{\mathrm{y_{N} }-\mathrm{y_{M}}}\bigg{)}, \tag{4}\]  
with \(\{\mathrm{y_{N}}\), \(\mathrm{z_{N}}\}\) and \(\{\mathrm{y_{M}}\), \(\mathrm{z_{M}}\}\) the in-plane y- and z-coordinates of the N and M points respectively. The associated angular velocity (spin) was then determined from two consecutive time increments:  
\[\omega=\frac{\Delta\beta}{\Delta\mathrm{t}}, \tag{5}\]  
with \(\Delta\beta\) and \(\Delta\mathrm{t}\) the angle and time increments, respectively. Figure 9 presents the simulated vertical displacement mapping for a ball impacting the _foam_ + _compact_ target at a \(60^{\circ}\) angle, \(14\) m/s and \(-66\) rev/s. Four screenshots are displayed corresponding to the initial time (\(\mathrm{t_{0}}\)), first contact (\(\mathrm{t_{1}}\)), contact at maximum depth (\(\mathrm{t_{2}}\)) and end time (\(\mathrm{t_{end}}\)). A close up view of the through-thickness deformation in the contact region at time \(\mathrm{t_{2}}\) is added to evidence the complex deformation pattern of the target layers. It is worth noting that no buckling of the ball occurred in this case and that the pimples were significantly sheared. The positions of the N and M equatorial points, and a fictitious line connecting them are also displayed. As discussed above, their time positions were necessary for the metrics calculations. Also, the trajectories of the M, N and calculated C points are plotted in Figure 10. The positions for the times selected in Figure 9 are presented and the spin inversion is clearly evidenced. In the previous study focusing on the normal impact with no spin, the linear coefficient of restitution was the sole relevant metric monitored. Also, finite element simulations with rough contact conditions resulted in fair predictions of the experimental data compared to simulations with frictionless ones. A similar study had been conducted in this work for the _foam_ sample impacted under various incidences and spins. The results are presented in Figure 11 where the experimental and simulated linear coefficient of restitutions are compared. Figure 11a focuses on impacts with fixed backspin (\(-66\) rev/s) and varying incidence and Figure 11b presents the results for impacts at constant incidence (\(30^{\circ}\)) and varying backspin. Fair predictions using rough contact conditions were obtained whereas frictionless simulations clearly over predicted the experimental results. With this broad set of impact conditions, the numerical predictions capability using rough contact conditions could be further challenged when tracking the two additional metrics that had been defined. The experiment/simulation comparisons are presented in Figure 12 for impact tests at constant velocities (V\({}_{\rm i}\) = 14 m/s and \(\omega_{\rm i}\) = \(-\)66 rev/s) and varying incidence, performed on the _foam_ (Figure 12a) and _foam_ + _compact_ (Figure 12b) targets. For both the _foam_ and the _foam_ + _compact_ targets, the magnitude of the linear coefficient of restitution and its decrease with increasing the incident angle were fairly predicted (top figures in Figure 12). Figure 10: In-plane trajectories of the M, N and calculated C points of the ball. (\(\alpha_{\rm i}\) = 60\({}^{\circ}\) angle, V\({}_{\rm i}\) = 14 m/s and \(\omega_{\rm i}\) = \(-\)66 rev/s).