# Geometric Design of a Face Gear Drive with Low Sliding Ratio
## 4 Sliding Ratio Characteristics  
### Explicit Solution for Meshing Point  
Generally, the face gear drive engages in the form of point contact, which can avoid eccentric load. In this paper, a pinion was chosen with 1-3 teeth less than the shaper as the driving wheel of the face gear pair, which is the same as that of the general face gear drive [28]. At present, the common idea for solving the contact points of the tooth surface is to establish the assembly relationship between the shaper, the pinion, and the face gear. The movement relationship between the pinion and the shaper is regarded as an imaginary internal meshing. Then, in the process of producing, the tooth surfaces of the shaper and face gear are in line contact at any time; the contact lines of shaper and face gear  
Figure 5: Effects of \(a_{1}\) and \(a_{2}\) on tooth width characteristics. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Tooth Width Parameters** & **The Proposed Face Gear (D-3)** & **The General face Gear (D-6)** \\ \hline Inner radius (mm) & 170.490 & 173.059 \\ Outer radius (mm) & 203.135 & 203.231 \\ Tooth width (mm) & 32.645 & 30.172 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of two face gears tooth width. are denoted as \(L_{SF}\) and \(L_{FS}\), as displayed in Figure 6. Similarly, the instantaneous contact lines for the meshing of the shaper and the pinion are denoted as \(L_{SP}\) and \(L_{PS}\), respectively. Therefore, when the pinion meshes with the face gear, the contact that forms between the tooth surfaces is point contact. Additionally, the meshing points of face gear can be obtained by calculating the intersection point of the contact lines \(L_{SF}\) and \(L_{SP}\), supplemented by coordinate transformation. This involves the solution of a system of equations, but the solution requires harsh initial values, which brings inconvenience to the solution [29]. Therefore, an explicit solution method is described below. As for the above-mentioned meshing process, which is expressed in Equation (21), the parameter \(h\) can be obtained as follows:  
\[h=\frac{C}{A\sin(\varphi_{0}\pm\phi_{s})+B\cos(\varphi_{0}\pm\phi_{s})} \tag{43}\]  
with  
\[A=1-(a_{1}x+a_{2}x^{3}+r_{s})(a_{1}^{2}+1+4a_{1}a_{3}x^{2}+3a_{3}^{2}x^{4})/r_ {s} \tag{44}\]  
\[B=-(a_{1}+3a_{2}x^{2})-(a_{1}^{2}x+x+4a_{1}a_{3}x^{3}+3a_{3}^{2}x^{5})/r_{s} \tag{45}\]  
\[C=[-x-(a_{1}x+a_{2}x^{3}+r_{s})(a_{1}+3a_{2}x^{2})]/i_{fs} \tag{46}\]  
\[\varphi_{0}=\frac{-\Big{(}(a_{1}^{2}+1)x+\frac{4}{3}a_{1}a_{2}x^{3}+\frac{3}{ 5}a_{2}^{2}x^{5}\Big{)}}{r_{s}}+\frac{\pi}{2N_{s}}\]  
here, it contains two unknown quantities \(x\) and \(\phi_{s}\). According to the meshing relationship between the shaper and the pinion, the relationship between the parameter \(x\) and rotation angle \(\phi_{s}\) can be determined with Equation (47). Therefore, Equation (43) is further defined as Equation (48), and therefore, the rotation angle \(\phi_{s}\) of the pinion can be expressed, which is the key point. \[\phi_{s}=\frac{(a_{1}^{2}+1)x+\frac{4}{3}a_{1}a_{3}x^{3}+\frac{3}{5}a_{3}^{2} x^{5}}{r_{s}}-\frac{N_{s}}{2\pi}=-\varphi_{0} \tag{47}\]  
\[h=\frac{x+(a_{1}x+a_{2}x^{3}+r_{s})(a_{1}+3a_{2}x^{2})}{i_{fs}[(a_{1}+3a_{2}x ^{2})+(a_{1}^{2}x+x+4a_{1}a_{2}x^{3}+3a_{2}^{2}x^{5})/r_{s}]} \tag{48}\]  
Due to \(\phi_{s}/\phi_{p}=r_{p}/r_{s}\), the rotation angle \(\phi_{p}\) of pinion can be determined. Therefore, it is necessary to solve the value range of \(x\). Combining Equations (7)-(9), it can be known that the parameters \(x\) of meshing point at the tooth tip (\(x=x_{c}\)) and root (\(x=x_{d}\)) of the pinion can be obtained as follows:  
\[x_{1}^{2}+y_{1}^{2}=(r_{p}+h_{a})^{2} \tag{49}\]  
Figure 6.0  
where \(x_{1}\) and \(y_{1}\) are defined by replacing \(r_{s}\) with \(r_{p}\) in Equation (7). Figure 6.1 Instantaneous meshing lines on the shaper. Consequently, according to \(x\in[x_{c},x_{d}]\), the range of angle \(\phi_{p}\) can be solved by Equation (47) and gear ratio, and \(h\) also can be obtained by Equation (48).