## 4 Tooth Meshing Characteristics for Compound Transmission
### Relative Displacements Sectioetween Conjugate Gear Surfaces  
As mentioned earlier, there is no pure roll in any spatial position of the curve-face gear compound transmission. That is, relative motion exists between the contact points on the conjugate surfaces, and the velocities of two contact points differ in both magnitude and direction. Therefore, it is necessary to discuss the sliding velocity and rolling velocity of the curve-face gear compound transmission pair at the contact point. Additionally, the relative displacement at the contact point is independent of the input and output displacements. The sliding velocity \(\stackrel{{\rightarrow}}{{v}}_{s}\) is the relative motion velocity between two conjugate surfaces and affects the loss of gear meshing directly. The rolling velocity \(\stackrel{{\rightarrow}}{{v}}_{r}\) is used to measure the speed at which the lubricating oil enters the mesh. In the fixed coordinate system \(S\), along the line of action \(\underline{\mathfrak{S}}_{l}\), the velocity of the cylindrical gear \(\overrightarrow{v}_{m1}^{(f)}\) and the curve-face gear \(\overrightarrow{v}_{m2}^{(f)}\) at the conjugate contact point \(M^{\prime}\) can be resolved into \(\overrightarrow{v}_{\parallel m1}^{(f)}\), \(\overrightarrow{v}_{\perp m1}^{(f)}\) and \(\overrightarrow{v}_{\parallel m2}^{(f)}\), \(\overrightarrow{v}_{\perp m2}^{(f)}\), respectively. Due to the conjugate action of the curve-face gear compound transmission, the parallel components \(\overrightarrow{v}_{\parallel m1}^{(f)}\) and \(\overrightarrow{v}_{\parallel m2}^{(f)}\) of contact points on the two tooth surfaces are equal with each other. Then, the sliding velocity \(\overrightarrow{v}_{s}\) of the curve-face gear compound transmission pair is:  
\[\overrightarrow{v}_{s}=\overrightarrow{v}_{\perp m1}^{(f)}-\overrightarrow{v }_{\perp m2}^{(f)} \tag{44}\]  
And the rolling velocity \(\overrightarrow{v}_{r}\) is expressed by:  
\[\overrightarrow{v}_{r}=\frac{1}{2}\bigg{(}\overrightarrow{v}_{\perp m1}^{( f)}+\overrightarrow{v}_{\perp m2}^{(f)}\bigg{)} \tag{45}\]  
The velocity of the cylindrical gear \(\overrightarrow{v}_{m1}^{(f)}\) and the curve-face gear \(\overrightarrow{v}_{m2}^{(f)}\) at the conjugate contact point \(M^{\prime}\) in the fixed coordinate system \(S\) may be represented using Equations (46) and (47). \[\overrightarrow{v}_{m1}^{(f)}=\overrightarrow{\omega}_{1}\times \overrightarrow{r}_{1L}^{(f)}=r_{1}\omega_{1}\Bigg{[} \begin{array}{c}-\sin (\theta_{1}^{\prime}+\phi_{1})+(\phi_{1}-\phi_{01})\cos\alpha_{0}\cos(\theta_ {1}^{\prime}+\phi_{1}-\alpha_{0})\\ \cos(\theta_{1}^{\prime}+\phi_{1})+(\phi_{1}-\phi_{01})\cos\alpha_{0}\sin( \theta_{1}^{\prime}+\phi_{1}-\alpha_{0})\end{array} \Bigg{]} \tag{46}\]  
\[ \begin{array}{c}\overrightarrow{v}_{m2}^{(f)}=\overrightarrow{\omega}_{2} \times\overrightarrow{r}_{2L}^{(f)}+\overrightarrow{v}_{0}^{(f)}\\ =\omega_{2}\Bigg{[}\begin{array}{c}u_{s}(\theta_{s},\phi_{s})\\ -ll(\theta_{1}^{\prime})\end{array}\\ r_{s}[\cos(\theta_{s}+\phi_{s})+(\phi_{s}-\phi_{0s})\cos\alpha_{0}\sin(\theta_ {s}+\phi_{s}-\alpha_{0})]\end{array} \Bigg{]} \tag{47}\]  
Therefore, the perpendicular components \(\overrightarrow{v}_{\perp m1}^{(f)}\) and \(\overrightarrow{v}_{\perp m2}^{(f)}\) of the cylindrical gear and the curve-face gear (perpendicular to the line of action \(\underline{\mathfrak{S}}_{l}\)) are:  
\[\overrightarrow{v}_{\perp m1}^{(f)}=\overrightarrow{C_{l}}\times \bigg{(}\overrightarrow{v}_{m1}^{(f)}\times\overrightarrow{C_{l}}\bigg{)}=r_ {1}\omega_{1}[-\sin\alpha_{0}+(\phi_{1}-\phi_{01})\cos\alpha_{0}]\Bigg{[} \begin{array}{c}\cos(\theta_{1}^{\prime}+\phi_{1}-\alpha_{0})\\ \sin(\theta_{1}^{\prime}+\phi_{1}-\alpha_{0})\\ 0\end{array} \Bigg{]} \tag{48}\]  
and  
\[\overrightarrow{v}_{\perp m2}^{(f)}=\overrightarrow{C_{l}}\times \bigg{(}\overrightarrow{v}_{m2}^{(f)}\times\overrightarrow{C_{l}}\bigg{)}= \omega_{2}\Bigg{[}