# Pattern Classification and Gear Design of Spatial Noncircular
## 4 Tooth Profile Classification Design of Spatial Noncircular Gear
### Transmission Principle and Pitch Curve  
According to the gear meshing principle, the pitch curve of the curve face gear pair is independent of the tooth profile. Figure 10 is the meshing coordinate system of the pitch curve of the curved face gear pair. \(\sigma_{1}\) and \(\sigma_{2}\) are the corresponding pitch curves, respectively. \(\omega_{1}\) and \(\omega_{2}\) are the corresponding angular velocities, respectively. The coordinate system \(S_{f}\) (\(O_{f}\)-\(X_{f}Y_{f}Z_{f}\)), abbreviated as \(S_{f}\), is the fixed coordinate system of the noncircular gear frame. The coordinate system \(S_{1}\) (\(O_{1}\)-\(X_{1}Y_{1}Z_{1}\)), abbreviated as \(S_{1}\), is the follow-up coordinate system of the noncircular gear. The coordinate system \(S_{m}\) (\(O_{m}\)-\(X_{m}Y_{m}Z_{m}\)), abbreviated as \(S_{m}\), is the fixed coordinate system of the curve face gear frame. The coordinate system \(S_{2}\) (\(O_{2}\)-\(X_{2}Y_{2}Z_{2}\)), abbreviated as \(S_{2}\), is the follow-up coordinate system of the curve face gear. Figure 9: Tooth profile classification of curve face gear pair. When the gear pair is in the initial meshing situation A, \(S_{1}\) and \(S_{f}\) coincide and \(S_{2}\) and \(S_{m}\) coincide. When the gear pair changes from situation A to situation B, the rotation axes \(Z_{1}\) and \(Z_{2}\) of the two gears are fixed, and the meshing point of the pitch curve changes from P\({}_{1}\) to P\({}_{2}\). \(\theta_{1}\) and \(\theta_{2}\) are the rotation angles of the noncircular gear and curve face gear respectively. \(r_{1}(\theta_{1})\) and \(r_{2}(\theta_{2})\) are the pitch curve parameter equation of the noncircular gear and curve face gear, respectively. \(R\) is the distance from the projection of point P\({}_{1}\) on the pitch curve of curve face gear in plane \(X_{m}O_{m}Y_{m}\) to \(O_{2}\). \(V_{1}\) and \(V_{2}\) are the speeds at the meshing points of the two curves, respectively. According to the principle of spatial coordinate transformation, the homogeneous transformation matrix from the coordinate system \(S_{1}\) to \(S_{f}\) is  
\[M_{f1}=\left[ \begin{array}{cccc}\cos\theta_{1}&\sin\theta_{1}&0&0\\ -\sin\theta_{1}&\cos\theta_{1}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]. \tag{4}\]  
The homogeneous transformation matrix from the coordinate system \(S_{f}\) to \(S_{m}\) is  
\[M_{mf}=\left[ \begin{array}{cccc}0&0&1&-R\\ 0&1&0&0\\ -1&0&0&E\\ 0&0&0&1\end{array} \right]. \tag{5}\]  
The homogeneous transformation matrix from the coordinate system \(S_{m}\) to \(S_{2}\) is  
\[M_{2m}=\left[ \begin{array}{cccc}\cos\theta_{2}&\sin\theta_{2}&0&0\\ -\sin\theta_{2}&\cos\theta_{2}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]. \tag{6}\]  
Figure 10: Pitch curve coordinate system of the curve face gear pair. Then, the homogeneous transformation matrix from the coordinate system \(S_{1}\) to \(S_{2}\) can be expressed as  
\[M_{21}=M_{2m}M_{mf}M_{f1}=\left[ \begin{array}{cccc}-\sin\theta_{1}\sin\theta_{2 }&\cos\theta_{1}\sin\theta_{2}&\cos\theta_{2}&-R\cos\theta_{2}\\ -\cos\theta_{2}\sin\theta_{1}&\cos\theta_{1}\cos\theta_{2}&-\sin\theta_{2}&R \sin\theta_{2}\\ -\cos\theta_{1}&-\sin\theta_{1}&0&E\\ 0&0&0&1\end{array} \right]. \tag{7}\]  
The coordinates of point \(P_{2}\) in \(S_{1}\) are \([r_{1}(\theta_{1})\text{cos}(\theta_{1})\), \(r_{1}(\theta_{1})\text{sin}(\theta_{1})\),0,1]T, then the coordinates of point \(P_{2}\) in \(S_{2}\) can be expressed as  
\[\left[ \begin{array}{c}x_{2}\\ y_{2}\\ z_{2}\\ 1\end{array} \right]=M_{21}\left[ \begin{array}{c}r_{1}(\theta_{1})\cos \theta_{1}\\ r_{1}(\theta_{1})\sin\theta_{1}\\ 0\\ 1\end{array} \right]=\left[ \begin{array}{c}-R\cos\theta_{2}\\ R\sin\theta_{2}\\ E-r_{1}(\theta_{1})\\ 1\end{array} \right]. \tag{8}\]  
Therefore, the pitch curve equation of the curve face gear can be obtained as  
\[\left\{ \begin{array}{c}x_{2}=-R\cos\theta_{2}\\ y_{2}=R\sin\theta_{2}\\ z_{2}=E-r_{1}(\theta_{1})\end{array} \right.. \tag{9}\]  
It can be seen from Equation (9) that the pitch curve of the curve face gear is a cylindrical curve with \(R\) as the radius.