## 2 Mathematical Models of an S-Shaped Tooth Profile Gear  
### Generation of an S-Shaped Tooth Profile  
According to the [PERSON] theorem and the principle of conjugate motion, a pair of gear pairs processed by a pair of mutual rack cutters which achieve gomphosis on tooth shape can realize conjugate contact. The involute gear pair can be processed by a pair of linear mutual gomphosis rack cutters. While the line of the blade is replaced by a sine curve, an S-shaped tooth profile gear pair is obtained. A rack cutter with sine a curve tooth profile is shown in Figure 1. It can be seen that the two rack cutters are in mutual gomphosis when their tooth profiles are mutually fitted with the same sine curve, then the geometry of the cutting parts of the two rack cutters is exactly the same; therefore, one tool can process a pair of mutually conjugate meshing gear pairs exclusively. The gear tooth profile shown in Figure 2 can be obtained by the above-mentioned rack cutter using an envelope processing method. A pair of mutually conjugate meshing gear pairs as shown in Figure 3 can also be processed by the same cutter. It can be seen that the special meshing of convex and concave tooth profiles lengthens the relative radius of curvature of the contacting point, which reduces the contact stress on the teeth and improves the contact ratio effectively. Figure 1: A pair of mutual gomphosis rack cutters. Figure 2: S-shaped tooth profile envelope processed by sine shape cutter. (**a**) The pinion; (**b**) the gear. As shown in Figure 4, there is a coordinate system \(O_{1}x_{1}y_{1}\) fixed to the rack cutter. In order to make the equations of the rack cutter intuitive and simple, a new coordinate system \(O_{0}x_{0}y_{0}\) is established based on the basic coordinate system, which has the same origin with \(O_{1}x_{1}y_{1}\), and the angle between them is \(\alpha\); the tooth profile equations of the rack cutter from point \"a\" to point \"b\" are established on the coordinate system \(O_{0}x_{0}y_{0}\). \[\left\{ \begin{array}{ll}x_{0}=u\\ y_{0}=\lambda\text{sin}\;\omega u\end{array} \right.-\pi/\omega\leq u\leq\pi/\omega \tag{1}\]  
\[\boldsymbol{\lambda}=\boldsymbol{\lambda}^{\prime}\boldsymbol{m} \tag{2}\]  
where \(\omega=\pi\cos\alpha/(h_{a}^{*}+c^{*})\); \(h_{a}^{*}\) is the addendum coefficient; \(c^{*}\) is the tip clearance coefficient; \(m\) represents the modules; and \(\boldsymbol{\lambda}^{\prime}\) is the derivative coefficient, which shows the amplitude degree of the curve. As shown in Figure 5, the rack cutter equation in the coordinate system \(O_{0}x_{0}y_{0}\) is converted to the coordinate system \(O_{2}x_{2}y_{2}\) fixed with gears. With the help of the coordinate system \(\boldsymbol{P}_{xy}\), the tooth profile equation can be obtained, and the transformation matrices are given by  
\[\boldsymbol{M}_{10}= \begin{pmatrix}sin\;\alpha&-cos\;\alpha&0\\ cos\;\alpha&sin\;\alpha&0\\ 0&0&1\end{pmatrix}  \tag{3}\]  
\[\boldsymbol{M}_{P1}= \begin{pmatrix}1&0&-r_{2}\varphi_{2}\\ 0&1&0\\ 0&0&1\end{pmatrix}  \tag{4}\]  
\[\boldsymbol{M}_{2P}= \begin{pmatrix}cos\;\varphi_{2}&-sin\;\varphi_{2}&r_{2}sin \;\varphi_{2}\\ sin\;\varphi_{2}&cos\;\varphi_{2}&r_{2}cos\;\varphi_{2}\\ 0&0&1\end{pmatrix}  \tag{5}\]  
Figure 4: Coordinate system of the rack cutter. Figure 3: A pair of mutually meshing S-shaped tooth profile gears. \[ \begin{bmatrix}x_{2}\\ y_{2}\\ t_{2}\end{bmatrix} =M_{2P}M_{P1}M_{10} \begin{bmatrix}x_{0}\\ y_{0}\\ t_{0}\end{bmatrix} =M_{20} \begin{bmatrix}x_{0}\\ y_{0}\\ t_{0}\end{bmatrix}  \tag{6}\]  
Substituting Equation (1) into Equation (6) and simplifying can obtain the tooth profile equations of the gear. According to the forming theory of an S-shaped tooth profile, a pair of S-shaped gears that can be meshed with each other have the same tooth profile equation. As shown in Figure 6, in the coordinate system \(O_{1}x_{1}y_{1}\), \(O_{2}x_{2}y_{2}\) fixed with gears and based on the origin of the gear center, the parameter equations of the mutually conjugate meshing gear tooth profile can be unified as  
\[