# Geometric Design of a Face Gear Drive with Low Sliding Ratio
## 2 Mathematical Models of the Face Gear Pair with Low Sliding Ratio
### Construction of Face Gear Tooth Surface  
The tooth surface of the face gear can be generated by the shaper. For this, first, four coordinate systems are set up, as depicted in Figure 4. \(S_{50}(o_{50},x_{50},y_{50},z_{50})\) and \(S_{70}(o_{70},x_{70},y_{70},z_{70})\) are fixed coordinate systems of shaper and face gear, respectively. Likewise, \(S_{s}(o_{5},x_{5},y_{5},z_{5})\) and \(S_{7}(o_{7},x_{7},y_{7},z_{7})\) are rigidly attached to the shaper and the face gear. Then, the origins of the four coordinate systems are at the same position, and the shaper and face gear perform rotation about the axis \(z_{s}\) and \(z_{7}\), respectively. The kinematic relationship between shaper and face gear can be described as follows: when the shaper rotates from coordinate system \(S_{s0}\) to \(S_{s}\) by angle \(\phi_{s}\), the face gear rotates from \(S_{70}\) to \(S_{s}\) by angle \(\phi_{f}\). Then, the basic transformation matrixes between the coordinate systems are as follows:  
\[\mathbf{M}_{s0,s}=\left( \begin{array}{cccc}\cos\phi_{s}&-\sin\phi_{s}&0&0\\ \sin\phi_{s}&\cos\phi_{s}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right) \tag{14}\]  
\[\mathbf{M}_{f0,s0}=\left( \begin{array}{cccc}0&-1&0&0\\ 0&0&1&0\\ -1&0&0&0\\ 0&0&1\end{array} \right) \tag{15}\]  
\[\mathbf{M}_{f,f0}=\left( \begin{array}{cccc}\cos\phi_{f}&\sin\phi_{f}&0&0\\ -\sin\phi_{f}&\cos\phi_{f}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right) \tag{16}\]  
\[\mathbf{M}_{f,s}=\mathbf{M}_{f,f0}\cdot\mathbf{M}_{f0,s0}\cdot\mathbf{M}_{s0,s} \tag{17}\] \[=\left( \begin{array}{cccc}-\sin\phi_{s}\cos\phi_{f}&-\cos\phi_ {s}\cos\phi_{f}&\sin\phi_{f}&0\\ \sin\phi_{s}\sin\phi_{f}&\cos\phi_{s}\sin\phi_{f}&\cos\phi_{f}&0\\ -\cos\phi_{s}&\sin\phi_{s}&0&0\\ 0&0&0&1\end{array} \right)\]  
where \(M_{s0,s}\) represents the transformation from coordinate system \(S_{s}\) to coordinate system \(S_{s0}\); the others can then be compared. Figure 3: Tooth profile and sliding ratio of gear pair. In view of the production method, the two surfaces of the shaper are often set symmetrically with respect to the plane \(x_{s}o_{s}z_{s}\). Then, the equation of shaper in \(S_{s}(o_{s}x_{s}y_{s})\) is expressed as  
\[\mathbf{r}_{s}(x,h)=\left( \begin{array}{c}-x\sin\varphi_{0}+(a_{1}x+a_{2}x^{3}+r_ {s})\cos\varphi_{0}\\ \pm[x\cos\varphi_{0}+(a_{1}x+a_{2}x^{3}+r_{s})\sin\varphi_{0}]\\ h\\ 1\end{array} \right) \tag{18}\]  
where '\(\pm\)' represents the left and the right side of the alveolar surface, respectively. \(\varphi_{0}\) is defined as  
\[\varphi_{0}=\frac{-\left((a_{1}^{2}+1)x+\frac{4}{3}a_{1}a_{2}x^{3}+\frac{3}{5} a_{2}^{2}x^{5}\right)}{r_{s}}+\frac{\pi}{2N_{s}} \tag{19}\]  
According to the differential geometry theory, the unit normal vector \(\mathbf{n}_{s}\) of shaper tooth profile can be defined as  
\[\mathbf{n}_{s}=\frac{\frac{\partial\mathbf{r}_{s}}{\partial x}\times\frac{\partial\bm {r}_{s}}{\partial h}}{\left|\frac{\partial\mathbf{r}_{s}}{\partial x}\times\frac{ \partial\mathbf{r}_{s}}{\partial h}\right|} \tag{20}\]  
Figure 4: Generation of the face gear. Then, the meshing equation between the shaper and the face gear is expressed as follows:  
\[f(x,h,\phi_{\rm s})=\mathbf{n}_{s}\cdot\mathbf{v}_{s}^{(sf)}=0 \tag{21}\]  
where \(\mathbf{v}^{(sf)}\) is the relative velocity vector of the shaper and the face gear in \(S_{\rm s}(o_{s}\mathbf{x}_{\rm s}\mathbf{y}_{\rm s})\). \[\mathbf{v}_{s}^{(sf)}=\mathbf{v}_{s}^{(s)}-\mathbf{v}_{s}^{(f)} \tag{22}\]  
where \(\mathbf{v}^{(s)}\) and \(\mathbf{v}^{(f)}\) are the velocity vector of the shaper and the face gear.