## 4 Tooth Meshing Characteristics for Compound Transmission
### Tooth Contact Analysis
\right] \tag{34}\]  
\[\overset{\rightarrow}{n}_{2L}^{(f)}\left(\theta_{2}^{\prime},\phi_{s},u_{s} \right)=M_{f2}\left(\theta_{2}^{\prime}\right)M_{2s}(\theta_{s})\overset{ \rightarrow}{n}_{2L}^{(2)}(\phi_{s},u_{s})=\left[ \begin{array}{c}-\cos( \theta_{2}-\theta_{2}^{\prime})\sin(\theta_{s}+\phi_{s}-\alpha_{0s})\\ \cos(\theta_{s}+\phi_{s}-\alpha_{0s})+l(\theta_{2})-ll(\theta_{1}^{\prime})\\ -\sin(\theta_{2}-\theta_{2}^{\prime})\sin(\theta_{s}+\phi_{s}-\alpha_{0s})\\ 1\end{array} \right], \tag{35}\]  
where, \(\theta_{1}^{\prime}\) and \(\theta_{2}^{\prime}\) are correspondingly the angular displacements of the cylindrical gear and the curve-face gear in meshing. When \(\theta_{2}^{\prime}\) is taken as the input parameter, then the vector equations of tangency of contacting surfaces can be described as:  
\[\left\{ \begin{array}{l}r_{1}\left[\cos(\theta_{1}^{\prime}+\phi_{1})+(\phi_ {1}-\phi_{01})\cos\alpha_{01}\sin(\theta_{1}^{\prime}+\phi_{1}-\alpha_{01}) \right]\ldots\\ =r_{s}\cos(\theta_{2}-\theta_{2}^{\prime})[\cos(\theta_{s}+\phi_{s})+(\phi_{ s}-\phi_{0s})\cos\alpha_{0s}\sin(\theta_{s}+\phi_{s}-\alpha_{0s})]+\sin(\theta_{2}- \theta_{2}^{\prime})u_{s}(\theta_{s},\phi_{s})\\ r_{1}\left[\sin(\theta_{1}^{\prime}+\phi_{1})-(\phi_{1}-\phi_{01})\cos\alpha_{01 }\cos(\theta_{1}^{\prime}+\phi_{1}-\alpha_{01})\right]\ldots\\ =r_{s}[\sin(\theta_{s}+\phi_{s})-(\phi_{s}-\phi_{0s})\cos\alpha_{0s}\cos( \theta_{s}+\phi_{s}-\alpha_{0s})]+l(\theta_{2})-ll(\theta_{1}^{\prime})\\ \sin(\theta_{1}^{\prime}+\phi_{1}-\alpha_{01})=\cos(\theta_{2}-\theta_{2}^{ \prime})\sin(\theta_{s}+\phi_{s}-\alpha_{0s})\\ \cos(\theta_{1}^{\prime}+\phi_{1}-\alpha_{01})=\cos(\theta_{s}+\phi_{s}-\alpha_ {0s})+l(\theta_{2})-ll(\theta_{1}^{\prime})\\ 0=-\sin(\theta_{2}-\theta_{2}^{\prime})\sin(\theta_{s}+\phi_{s}-\alpha_{0s}) \end{array} \right. \tag{36}\]  
It is easy to verify that, while the two tooth surfaces are indeed in tangency with each other, the angular displacement \(\theta_{2}\) of curve-face gear being generated is equal to its angular displacement \(\theta_{2}^{\prime}\) for compound transmission. And the angular displacement \(\theta_{1}^{\prime}\) of the cylindrical gear for compound transmission can obtained based on the desired gear ratio as:  
\[\theta_{1}^{\prime}\left(\theta_{2}^{\prime}\right)=\frac{\int_{0}^{\theta_{2}^{ \prime}}\sqrt{R^{2}+ll^{\prime 2}(\theta)}d\theta}{r_{1}} \tag{37}\]  
The angular displacement \(\theta_{s}(\theta_{2}^{\prime})\) of shaper is:  
\[\theta_{s}\left(\theta_{2}^{\prime}\right)=\frac{\int_{0}^{\theta_{2}^{\prime}} \sqrt{R^{2}+l^{\prime 2}(\theta)}d\theta}{r_{s}} \tag{38}\]Thus, the Gaussian coordinates \([\phi_{s}(\theta_{2}^{\prime}),u_{s}(\theta_{2}^{\prime})]\) of the involute shaper surface is represented by implicit function:  
\[\left\{ \begin{array}{l}p(\phi_{s})=(r_{1}-r_{s})\sin(\theta_{s}+\phi_{s})- \big{[}r_{b1}\big{(}\phi_{s}-\phi_{01}+\theta_{s}-\theta_{1}^{\prime})-r_{bs}( \phi_{s}-\phi_{0s})\big{]}\cos(\theta_{s}+\phi_{s}-\alpha_{0s})=0\\ u_{s}(\theta_{2}^{\prime})=\frac{r_{bs}+i_{2}(\theta_{2})l^{\prime}(\theta_{2} )\cos(\theta_{s}+\phi_{s}-\alpha_{0s})}{i_{2s}(\theta_{2})\sin(\theta_{s}+ \phi_{s}-\alpha_{0s})}\\ \end{array} \right. \tag{39}\]  
And the Gaussian coordinates \([\phi_{1}(\theta_{2}^{\prime}),u_{1}(\theta_{2}^{\prime})]\) of the cylindrical gear surface is:  
\[\left\{ \begin{array}{l}q(\phi_{s})=(r_{1}-r_{s})\sin(\theta_{1}^{\prime}+ \phi_{1})-\big{[}r_{b1}\big{(}\phi_{1}-\phi_{01})-r_{bs}\big{(}\phi_{1}-\phi_{ 0s}+\theta_{1}^{\prime}-\theta_{s}\big{)}\big{]}\cos(\theta_{1}^{\prime}+ \phi_{1}-\alpha_{0s})=0\\ u_{s}(\theta_{2}^{\prime})=\frac{r_{bs}+i_{2s}(\theta_{1})l^{\prime}(\theta_{2 })\cos(\theta_{s}+\phi_{s}-\alpha_{0s})}{i_{2s}(\theta_{2})\sin(\theta_{s}+ \phi_{s}-\alpha_{0s})}\end{array} \right.