## 2 Ease-Off Flank Modification Methodology for the Pinion  
### Tooth Representation of Pinion Globally Conjugated with the Gear  
When the pinion tooth surface is fully conjugated with a gear tooth surface, the drive ratio is equal to the nominal drive ratio of the gear pair. The relationship between the meshing angle of gear \(\theta_{2}\) and that of pinion \(\theta_{1}\) is as follows:  
\[\theta_{2}=N_{1}/N_{2}(\theta_{1}-\theta_{10})+\theta_{20} \tag{1}\]  
Here, \(\theta_{10}\) is the meshing rotating angles of pinion and \(\theta_{20}\) is the gear at the design reference points; while \(N_{1}\) and \(N_{2}\) is the teeth number of pinion and gear, respectively. The coordinate system of gear and pinion meshing is shown in Figure 1, and detailed instructions of it are detailed in [20]. \(S_{\text{s}}\), \(S_{\text{r}}\), and \(S_{\text{q}}\) are the reference coordinates, and \(S_{2}\) is the gear coordinate system. Moreover, \(V\) is the offset distance, while \(H_{1}\) and \(H_{2}\) are the axial setting. \(\Sigma\) represents the shaft angle. Based on the space meshing theory and coordinate transformation, the gear tooth surface, which is considered as an imaginary cutter is transformed into the coordinate system \(S_{1}\) of the pinion tooth surface, and then the position vector \(r_{10}\) and normal vector \(n_{10}\) of the pinion are determined by the following formula:  
\[\left\{ \begin{array}{l}r_{10}(u,\beta,\theta_{1},)=M_{1p}(\theta_{1})M_{pq}M _{qr}M_{rs}M_{s2}(\theta_{2})r_{2}(u,\beta)\\ n_{10}(u,\beta,\theta_{1},)=L_{1p}(\theta_{1})L_{4p}L_{4r}L_{rs}L_{s2}(\theta_{ 2})n_{2}(u,\beta)\\ f_{1}(u,\beta,\theta_{1})=n_{10}\cdot(\partial r_{10}/\partial\theta_{1})\\ \end{array} \right. \tag{2}\]  
Here, \(r_{2}\) and \(n_{2}\) are the position vector and normal vector of the gear, respectively;  
\(r_{10}\) and \(n_{10}\) are the position vector and normal vectors of pinion globally conjugated with the gear, respectively; and \(r_{10}\) is taken with respect to \(S_{1}\). \(u\) and \(\beta\) are parameters of the tooth surface; \(M_{\text{lp}}\), \(M_{\text{pq}}\), \(M_{qr}\), \(M_{rs}\), and \(M_{s2}\) are coordinate transformation matrices; and \(L_{1p}\), \(L_{pq}\), \(L_{qr}\), \(L_{rs}\), and \(L_{s2}\) are corresponding 3 \(\times\) 3 submatrices. \(f_{1}\) refers to the equation of meshing \(f_{1}=n_{10}\cdot 9_{10}\), where \(v_{10}\) is the relative velocity between the cutter blade and the work gear in coordinate system \(S_{1}\). Tooth Representation of Pinion with Ease-Off Flank Modification Only Depending on Transmission Error  
Tooth modification changes the initial contact clearance of conjugate tooth surface. The geometric transmission error reflects the initial clearance between contact pairs tooth. It does not change the length of contact line and contact path. Yet, the variation of the load distribution and loaded deformation between different meshing positions could pose a great impact on vibration (such as the profile modification of spur gears). The normal clearance of tooth surface can change the length of contact line and contact path to avoid certain edge stress concentration (for example, the longitudinal modification of spur gears), which are likely to affect the sensitivity of assembly error. The modification design for tooth surface contact clearances follows the principles. 1 In order to reduce the impact of meshing in and out, there should be enough parabolic transmission error near the meshing in and out end. 2 In order to ensure that the meshing transformation point is as smooth as possible, and the loaded deformation at different meshing positions is basically the same, the backlash of several pairs of teeth in contact at the same time should not vary too much.