## 4 Tooth Meshing Characteristics for Compound Transmission  
### Tooth Contact Analysis  
The tooth surface contact between the cylindrical gear and the curve-face gear in meshing without installation error will be explored. Similar to the generation of face gears, the number of shaper is more than that of cylindrical gear to maintain point contact between the surfaces of cylindrical gear and curve-face gear. The contact coordinate systems of tooth surfaces are established as shown in Figure 11, which is the same as the motion coordinate system. As shown in Figure 11, while cylindrical gear fixed inputs a rotation counterclockwise about \(Z\)-axis, the curve-face gear performs a screw motion clockwise with respect to the coordinate system \(S\), and \(Y\)-axis is the axis of the screw motion. In this case, the left surfaces of cylindrical gear and curve-face gear are in meshing. As a result, the left tooth surface of the two input and output gears are taken as the explanation objects in this section, and the calculation method of the right tooth surfaces are similar. By analogy to the equation of shaper tooth surface, the equation of the left involute cylindrical gear tooth surface in its rigidly connected coordinate system \(S_{1}\) can be inferred, which the Gaussian coordinates \((\phi_{1},u_{1})\) are used to replace \((\phi_{S},u_{S})\). When the two gears in mesh, two tooth surfaces must be indeed in tangency with each other, so the position vector and normal vector of cylindrical gear coincide with those of the curve-face gear at any instant. The continuous tangency of contacting surfaces at the contact point \(M^{\prime}\) are expressed by using the frame coordinate system \(S\), and the relevant coordinate transforma  
Figure 11: The contact coordinate systems of the curve-face gear compound transmission. tion matrixes has been shown in Section 3.3. Consequently, using similar considerations, the position vector \(\overset{\rightarrow}{r}_{1L}^{(f)}\) and normal vector \(\overset{\rightarrow}{n}_{1L}^{(f)}\) of the cylindrical gear are derived in the coordinate system \(S\) as:  
\[\overset{\rightarrow}{r}_{1L}^{(f)}\left(\theta_{1}^{\prime},\phi_{1},u_{1} \right)=M_{f1}\left(\theta_{1}^{\prime}\right)\overset{\rightarrow}{r}_{1L}^ {(1)}(\phi_{1},u_{1})=\left[ \begin{array}{c}r_{1}\left[\cos(\theta_{1}^{ \prime}+\phi_{1})+(\phi_{1}-\phi_{01})\cos\alpha_{01}\sin(\theta_{1}^{\prime}+ \phi_{1}-\alpha_{01})\right]\\ r_{1}\left[\sin(\theta_{1}^{\prime}+\phi_{1})-(\phi_{1}-\phi_{01})\cos\alpha_{01 }\cos(\theta_{1}^{\prime}+\phi_{1}-\alpha_{01})\right]\\ u_{1}\\ 1\end{array} \right] \tag{32}\]  
and  
\[\overset{\rightarrow}{n}_{1L}^{(f)}\left(\theta_{1}^{\prime},\phi_{1},u_{1} \right)=M_{f1}\left(\theta_{1}^{\prime}\right)\overset{\rightarrow}{n}_{1L}^ {(1)}(\phi_{1},u_{1})=\left[ \begin{array}{c}-\sin(\theta_{1}^{\prime}+\phi_ {1}-\alpha_{01})\\ \cos(\theta_{1}^{\prime}+\phi_{1}-\alpha_{01})\\ 0\\ 1\end{array} \right]. \tag{33}\]  
And the position vector \(\overset{\rightarrow}{r}_{2L}^{(f)}\) and normal vector \(\overset{\rightarrow}{n}_{2L}^{(f)}\) of the curve-face gear can be represented in the coordinate system \(S\) as follows:  
\[=\left[ \begin{array}{c}\overset{\rightarrow}{r}_{2L}^{(f)}\left(\theta_{2} ^{\prime},\phi_{s},u_{s}\right)=M_{f2}(\theta_{2}^{\prime})\overset{ \rightarrow}{r}_{2L}^{(2)}(\phi_{s},u_{s})\\ \\ \overset{\rightarrow}{n}_{2L}^{(f)}\left(\theta_{2}^{\prime},\phi_{s},u_{s} \right)=M_{f2}\left(\theta_{2}^{\prime}\right)M_{2s}(\theta_{s})\overset{ \rightarrow}{n}_{2L}^{(2)}(\phi_{s},u_{s})=\left[\begin{array}{c}-\cos( \theta_{2}-\theta_{2}^{\prime})\sin(\theta_{s}+\phi_{s}-\alpha_{0s})\\ \cos(\theta_{s}+\phi_{s}-\alpha_{0s})+l(\theta_{2})-ll(\theta_{1}^{\prime})\\ -\sin(\theta_{2}-\theta_{2}^{\prime})\sin(\theta_{s}+\phi_{s}-\alpha_{0s})\\ 1\end{array}\right],\end{array}