## 3 The Spatial Finite Helical Motion for Compound Transmission
### Compound Transmission Axodes
#### 3.3.1 Instantaneous Screw Axis  
The instantaneous screw axis (ISA) can be determined according two mutually perpendicular and intersecting input and output shafts arranged in space illustrated in Figure 6, which is the common generator for the axodes. Note that, unlike ISA, the instantaneous  
Figure 8: The conjugate pitch surfaces of curve-face gear compound transmission. screw (IS) is an instantaneous generator vector with pitch component according to the magnitude of the I/O relationship. In reference to Ref. [9], the main vector of the twist is the rotational direction of the rigid body, and the dual vector essentially represents the instantaneous linear velocity of the point that coincides with the origin \(O\) on the rigid body. Unless otherwise stated hereinafter, the coordinate system using for screw and vector is fixed coordinate system \(S\). According to the definition above, the input twist \(S_{1}\) of cylindrical gear is:  
\[S_{1}=\left( \begin{array}{c}\overset{\rightarrow}{C}_{1}\\ \overset{\rightarrow}{\xi}_{1}\end{array} \right)=(0,0,\omega_{1};0,0,0)^{T} \tag{16}\]  
where, \(\overset{\rightarrow}{C}_{1}\) and \(\overset{\rightarrow}{\xi}_{1}\) are correspondingly the primary part and secondary part of the input twist \(S_{1}\). And the output twist \(S_{2}\) of curve-face gear is:  
\[S_{2}(\theta_{1})=\left( \begin{array}{c}\overset{\rightarrow}{C}_{2}\\ \overset{\rightarrow}{\xi}_{2}\end{array} \right)=(0,-\omega_{2}(\theta_{1}), 0;0,-v_{0}(\theta_{1}),0)^{T} \tag{17}\]  
Similarly, \(\overset{\rightarrow}{C}_{2}\) and \(\overset{\rightarrow}{\xi}_{2}\) are the primary and secondary part of the output twist \(S_{2}\), respectively. According to the theorem of three axes, the homogeneous screw coordinate of the instantaneous screw \(S_{is}\) of the curve-face gear with respect to the cylindrical gear is:  
\[S_{is}(\theta_{1})=S_{2}(\theta_{1})-S_{1}=\left( \begin{array}{c}\overset{ \rightarrow}{C}_{is}\\ \overset{\rightarrow}{\xi}_{is}\end{array} \right)=(0,-\omega_{2}(\theta_{1}),-\omega_{1};0,-v_{0}(\theta_{1}),0)^{T} \tag{18}\]  
Then, the direction vector \(\overset{\rightarrow}{s}_{is}\) of the instantaneous screw axis is \(-(0,i(\theta_{1}),1)/\sqrt{1+i^{2}(\theta_{1})}\); and the relative angular velocity \(\omega_{is}\) between the input and output gear elements is:  
\[\omega_{is}(\theta_{1})=\omega_{1}\sqrt{1+i^{2}(\theta_{1})} \tag{19}\]  
And the pitch \(h_{is}\) of the instantaneous screw \(S_{is}\) is:  
\[h_{is}(\theta_{1})=\frac{\overset{\rightarrow}{C}_{is}\overset{\rightarrow} {\xi}_{is}}{\overset{\rightarrow}{C}_{is}\overset{\rightarrow}{C}_{is}}= \frac{i(\theta_{1})v_{0}(\theta_{1})}{\sqrt{1+i^{2}(\theta_{1})}} \tag{20}\]  
The translation component \(h_{is}\overset{\rightarrow}{\omega}_{is}\) of the instantaneous screw \(S_{is}\) is equal to the axial component of the relative velocity between two gears in the direction of the screw axis. According to the Equation (20), the pitch \(h_{is}\) of the instantaneous screw can never be zero during the whole motion for compound transmission. That is, there is relative motion between two conjugate tooth surfaces along the instantaneous axis direction. Furthermore, the relative motion between the two gears is a spatial finite helical motion along the instantaneous axis. It is obvious that there is no pure rolling contact point whose relative velocity \(\overset{\rightarrow}{v}_{21}\) is zero in the whole space. In the fixed coordinate system \(S\), the instantaneous screw axis equation \(I_{isa}\) of the curve-face gear compound transmission pair can be expressed as:  
\[I_{isa}(\theta_{1},m)=\left\{ \begin{array}{l}x_{isa}=-l^{\prime}(\theta_{1}) /\left[1+i^{2}(\theta_{1})\right]\\ y_{isa}=-i(\theta_{1})m\\ z_{isa}=-m\end{array} \right. \tag{21}\]Here, \(t\) is the distance from any point on the instantaneous screw axis to the position vector \(\overset{\rightarrow}{r}_{isq}\).