# Impact of Horse Hoof Wall with Different Solid Surfaces
## 2 Materials and Methods
### Mathematical Modeling  
Figure 2 shows the schematic of the hoof wall during the impact with a solid surface. The angle of the impact surface with the horizontal axis is \(\phi\). The contact point on the hoof wall is \(T\). The distance between the center of mass \(C\) and center of the circular end \(E\) of the hoof wall is \(d\). The radius of the circular end is \(R\). The unit vectors \(t_{0}\), \(J_{0}\) and \(\ k_{0}\) with the origin at the contact point \(T\) form a global reference frame. Let \(\iota\), \(\jmath\) and \(\ k\) be the unit vectors that form a local reference frame at the same origin with \(x\)-axis along the impact surface. To characterize the configuration of the hoof wall, the generalized coordinates \(q_{1},q_{2},q_{3}\) were selected. Coordinates \(q_{1},q_{2},q_{3}\) denote the displacement of the contact point  
Figure 1: Sketch of the experiment setup (**a**) front view and (**b**) right view.  
\(T\) along \(x,y,z\) directions, respectively. Forces acting on the hoof wall during impact are gravity \(\mathbf{G}\), normal contact force \(\mathbf{P}\) and friction force \(\mathbf{F}_{f}\).  
The position of the contact point \(T\) was \(\mathbf{r}_{T}=q_{1}\boldsymbol{t}+q_{2}\boldsymbol{J}\). The position of the center of the circular end \(E\) was \(\mathbf{r}_{E}=q_{1}\boldsymbol{t}+(q_{2}+R)\boldsymbol{J}\). The position of the center of mass \(C\) was calculated by \(\mathbf{r}_{C}=(q_{1}+d\,\cos q_{3})\,\boldsymbol{t}+(q_{2}+R+d\,\sin q_{3}) \boldsymbol{J}\). The angular velocity and acceleration of the hoof wall were \(\boldsymbol{\omega}=q_{3}\,\boldsymbol{k}\) and \(\boldsymbol{\alpha}=\dot{\mathbf{q}}_{3}\,\boldsymbol{k}\). The velocity and acceleration of the center of the circular end \(E\) were \(\mathbf{v}_{E}=\frac{d\mathbf{r}_{E}}{dt}\) and \(\mathbf{a}_{E}=\frac{d^{2}\mathbf{r}_{E}}{dt^{2}}\). The velocity and acceleration of the contact point \(T\) were \(\mathbf{v}_{T}=\mathbf{v}_{E}+\boldsymbol{\omega}\times(\mathbf{r}_{T}- \mathbf{r}_{E})\) and \(\mathbf{a}_{T}=\mathbf{a}_{E}+\boldsymbol{\alpha}\times(\mathbf{r}_{T}- \mathbf{r}_{E})+\boldsymbol{\omega}\times[\boldsymbol{\omega}\times(\mathbf{r }_{T}-\mathbf{r}_{E})]\). The velocity and acceleration of the center of mass \(C\) were \(\mathbf{v}_{C}=\mathbf{v}_{E}+\boldsymbol{\omega}\times(\mathbf{r}_{C}- \mathbf{r}_{E})\) and \(\mathbf{a}_{C}=\mathbf{a}_{E}+\boldsymbol{\omega}\times(\mathbf{r}_{C}- \mathbf{r}_{E})+\boldsymbol{\omega}\times[\boldsymbol{\omega}\times(\mathbf{r }_{C}-\mathbf{r}_{E})]\).  
Gravitational force was calculated as \(\mathbf{G}=m\,g\,\sin\phi\,\boldsymbol{t}-m\,g\,\cos\phi\,\boldsymbol{J}\), where \(m\) is the mass of the hoof wall and \(g=9.81\,\mathrm{m/s^{2}}\) is the gravitational acceleration. The normal contact force acting on the contact point \(T\) was along \(y\)-axis with an expression \(\mathbf{P}=P\,\boldsymbol{J}\). The friction force was calculated by \(\mathbf{F}_{f}=-\boldsymbol{\mu}|P|\frac{\mathbf{v}_{T}\boldsymbol{t}}{| \mathbf{v}_{T}\boldsymbol{t}}\boldsymbol{t}\), where \(\boldsymbol{\mu}\) is the kinetic coefficient of friction. Newton-Euler equations of motion were developed for the impact analysis: \(m\,\mathbf{a}_{C}=\mathbf{G}+\mathbf{P}+\mathbf{F}_{f}\) and \(\mathrm{I}_{C}\,\boldsymbol{\alpha}=(\mathbf{r}_{T}-\mathbf{r}_{C})\times( \mathbf{P}+\mathbf{F}_{f})\), where \(I_{C}\) is the mass moment of inertia about the center of mass C.  
The impact was divided into two phases: compression and restitution. The compression phase started when the hoof wall contacted the impact surface with the initial impact velocity \(v_{i}\) and continued until the maximum compression, \(\delta_{m}\), was reached. At the maximum compression, the normal velocity of the contact point \(T\) became zero, i.e., \(\mathbf{v}_{T}\cdot\boldsymbol{J}=0\). Next, the restitution phase began and ended when there was no contact between the hoof wall and the impact surface.