# Impact of Horse Hoof Wall with Different Solid Surfaces
## 4 Discussion  
There are two basic approaches for studying impact. The algebraic approach assumes rigid body collision and divides impact into two intervals, compression and restitution. The impact is considered to occur instantaneously. The other approach considers the impact force as continuous and the colliding bodies are allowed to undergo local and global deformations. The impact motion is solved by direct numerical integration of the equations of motion during collision without using the coefficient of restitution. Different models have been developed to represent the variations of the contact force [30; 45]. For this study, we first experimentally estimated the coefficient of restitution and the effective coefficient of friction of the hoof wall with different solid surfaces. Next, we presented a differential model based on two phases of the impact, compression and restitution. For compression  
Figure 8: Experimental and simulated kinetic energy during an oblique impact on steel. Figure 7: Mean experimental and simulated velocities of the center of mass after oblique impact on steel (red markers), concrete (blue markers) and asphalt (green markers) with impact angles (**a**) \(\phi=10^{\circ}\), (**b**) \(\phi=20^{\circ}\) and (**c**) \(\phi=30^{\circ}\). Error bars indicate the standard deviations. phase, a spring-damper was used to model the contact force. Restitution phase was considered to be fully elastic when no energy was dissipated in the direction normal to the impact surface. Based on results of this study, the coefficient of restitution varied with surface, but it was not significantly affected by the impact angles and the initial impact velocities. The effective coefficient of friction was affected by the impact surface and impact angles, but was not significantly affected by the initial impact velocities. In each experiment, the effective coefficient of friction were computed as the ratio of the tangential and normal impulse over the impact. Our results were consistent with those of a study that investigated the ground reaction force between hoof and different solid substrates and reported the lowest dynamic coefficient of friction on steel and slightly higher dynamic coefficient of friction on concrete than on asphalt [24]. The dampening effect was suggested to be the main cause of the lower loading on the soft surface compared with the hard surface for horses at the walk and trot [46]. In this study, the mean coefficients of restitution obtained from normal impacts of the hoof wall on steel, concrete and asphalt were 0.84 (s.d. 0.041), 0.72 (s.d. 0.039) and 0.51 (s.d. 0.071), respectively. In normal impact, where the sliding friction is negligible, the initial kinetic energy is partially transferred to the rebound kinetic energy and partially dissipated to the ground surface. Smaller coefficients of restitution means less rebound kinetic energy which indicates more dissipated energy. In this study, with the estimated stiffness and damping coefficients, nondimensional damping ratios (\(\zeta=\dfrac{b}{2\sqrt{km}}\)) could be computed as 0.11, 0.26 and 0.61 on steel, concrete and asphalt, respectively. In contrast to the simulated results obtained from steel, the simulated results for concrete and asphalt deviated more from the experimental investigation as shown in Figure 7. Since the impact was modelled as planar motion, disagreement between the simulation and the experimental data could be caused by the out-of-plane rotation of the hoof wall before and after impact which is more likely to happen on a rough surface than on a smooth surface. This also explains the less varied coefficient of restitution on steel than on concrete and asphalt as shown in Figure 4. Regardless, the selected contact forces for compression and restitution are generally valid for the impact of the horse hoof. The contact force model presented in this study can be used to refine the simulation of equine locomotion. More specimens of hoof wall from different breeds are expected to validate the model.