# 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 Shaper  
According to the gear meshing theory, the shaper or pinion tooth surface can be achieved based on the meshing of the shaper and rack cutter. Additionally, the conjugate tooth profiles can be obtained from a given path of contact. Consequently, to obtain a low sliding ratio, the contact path \(\Sigma\) is represented by a given function, as illustrated in Figure 2. The coordinate systems \(S_{1}(o_{1}x_{1},y_{1})\) and \(S_{2}(o_{2},x_{2},y_{2})\) are rigidly attached to the shaper and rack cutter, respectively. \(S(o,x,y)\) is a fixed coordinate system whose origin \(o\) coincides with the pitch point \(P\). Figure 1: Approximate tooth profile of face gear. The equation of contact path \(\Sigma\) in \(S(o,x,y)\) can be defined as  
\[y=f(x) \tag{1}\]  
It is assumed that the equation of rack profile \(\Sigma_{r}\) in \(S_{2}(o_{2},x_{2},y_{2})\) is expressed as  
\[Y=g(X) \tag{2}\]  
Based on the theory of gearing, the profile \(\Sigma_{r}\) can be defined as  
\[\left\{ \begin{array}{l}X=\int\left(-\frac{y}{x}\right)\mathrm{d}y+C\\ Y=y\end{array} \right. \tag{3}\]  
where \(C\) is an integration constant; let the initial meshing position of conjugated tooth profiles be at the pitch point \(P\), then \(C=0\). For this, Equation (3) can be further expressed as  
\[\left\{ \begin{array}{l}X=\int\left(-\frac{y}{x}\cdot\frac{\mathrm{d}y}{ \mathrm{d}x}\right)\mathrm{d}x\\ Y=y\end{array} \right. \tag{4}\]  
From Figure 2b, the kinematic relationship between gear and rack can be described: when the rack \(\Sigma_{r}\) translates leftwards to the position II from the start position I by the distance \(L\), the shaper \(\Sigma_{s}\) rotates counterclockwise around the point \(o_{1}\) by the angle \(\phi_{1}\) and meshes with \(\Sigma_{r}\) at the point \(B(x,y)\) of contact path \(\Sigma\). Moreover, the point \(B\) corresponds to point \(B_{2}(X,Y)\) of tooth profile \(\Sigma_{r}\) at the position I. It is easy to know that \(BB_{2}=L=X-x\), and \(\phi_{1}\) can be expressed as  
\[\phi_{1}=\frac{L}{r_{s}}=\frac{-x+\int\left(-\frac{y}{x}\cdot\frac{\mathrm{d}y }{\mathrm{d}x}\right)\mathrm{d}x}{r_{s}} \tag{5}\]  
where \(r_{s}\) is the radius of the pitch circle of the shaper. By coordinate transforming and integrating Equation (5), the coordinate of \(B\) can be derived in dashed coordinate system \(S_{1}(o_{1},x_{1},y_{1})\) as  
\[\left\{ \begin{array}{l}x_{1}=x\cos\phi_{1}+y\sin\phi_{1}+r_{s}\sin\phi_{1}\\ y_{1}=-x\sin\phi_{1}+y\cos\phi_{1}+r_{s}\cos\phi_{1}\end{array} \right. \tag{6}\]  
Figure 2: Generation of the shaper. Thus, the work profiles of shaper can be presented as follows:  
\[\left\{ \begin{array}{l}x_{1}^{(w)}=x\cos\phi_{1}+(y+r_{s})\sin\phi_{1}\\ y_{1}^{(w)}=-x\sin\phi_{1}+(y+r_{s})\cos\phi_{1}\\ \phi_{1}=\frac{-x+\int\left(-\frac{y}{\tau}\frac{\mathrm{d}y}{\mathrm{d}x} \right)\mathrm{d}x}{r_{s}}\\ x\in[x_{a},x_{b}]\end{array} \right. \tag{7}\]  
where \(x_{a}\) and \(x_{b}\) are the addendum and dedendum parameters of the shaper, which can be solved by the following Equations (8) and (9), respectively. \[x_{1}^{2}+y_{1}^{2}=(r_{s}+h_{a})^{2} \tag{8}\]  
\[a_{1}x+a_{2}x^{3}=-h_{f} \tag{9}\]  
where \(h_{a}\) and \(h_{f}\) are the addendum height and the dedendum height of shaper, respectively. The fillet profiles of the gear can be obtained by the envelope movement of the generating rack's tip. Therefore, the fillet profile of the shaper is expressed as  
\[\left\{ \begin{array}{l}x_{1}^{(f)}=x\cos\phi_{1}+\left(r_{s}-h_{f}\right) \sin\phi_{1}\\ y_{1}^{(f)}=-x\sin\phi_{1}+\left(r_{s}-h_{f}\right)\cos\phi_{1}\\ \phi_{1}=\frac{-x+\int\left(-\frac{y}{\tau}\frac{\mathrm{d}y}{\mathrm{d}x} \right)\mathrm{d}x}{r_{s}}\\ x\in[0,x_{b}]\end{array} \right. \tag{10}\]  
In order to obtain the rack and gear pair with a low sliding ratio, the relationship between the sliding ratio and meshing curve can be established from the solution method of sliding ratio.