## 5 The Processing Method of Conjugate Pinions
### Machining Method for Conjugate Tooth Surface  
In order to machine the ideal conjugate tooth surface, in addition to the basic movements mentioned above, the radial movement \(\Delta L\) of the hob and the additional rotation \(\Delta\varphi\) of the workpiece should be considered simultaneously, as shown in Figure 13. So as  
Figure 12: Coordinate systems for pinion generation. to obtain additional motion laws, \(\Delta\varphi\) and \(\Delta L\) are represented by a Taylor polynomial at \(\varphi_{\rm h}=0\), and the equations are as follows:  
\[\left\{ \begin{array}{l}\Delta\varphi=a_{0}\varphi_{\rm h}^{0}+a_{1}\varphi_{ \rm h}^{1}+a_{2}\varphi_{\rm h}^{2}+a_{3}\varphi_{\rm h}^{3}+a_{4}\varphi_{\rm h }^{4}+a_{5}\varphi_{\rm h}^{5}+a_{6}\varphi_{\rm h}^{6}\\ \Delta L=b_{0}\varphi_{\rm h}^{0}+b_{1}\varphi_{\rm h}^{1}+b_{2}\varphi_{\rm h }^{2}+b_{3}\varphi_{\rm h}^{3}+b_{4}\varphi_{\rm h}^{3}+b_{5}\varphi_{\rm h}^{ 3}+b_{6}\varphi_{\rm h}^{6}\end{array} \right. \tag{37}\]  
where \(a_{\rm n}\) and \(b_{\rm n}\) are the nth (n = 0, 1, 2,\(\ldots\), and 6) order coefficients for the additional rotation angle \(\Delta\varphi\) and the additional movement \(\Delta L\), respectively. Therefore, the position vector and normal vector of the conjugate pin are represented by the following equation:  
\[\left\{ \begin{array}{l}R_{1}^{\rm c}(\varphi_{\rm h},L_{\rm s},\theta_{\rm h },u_{\rm h})=M_{\rm 1h}(a_{\rm n},b_{\rm n},\varphi_{\rm h},L_{\rm s})R_{\rm h}( \theta_{\rm h},u_{\rm h})\\ n_{1}^{\rm c}(\varphi_{\rm h},L_{\rm s},\theta_{\rm h},u_{\rm h})=M_{\rm 1h}(a_{ \rm n},b_{\rm n},\varphi_{\rm h},L_{\rm s})n_{\rm h}(\theta_{\rm h},u_{\rm h}) \end{array} \right. \tag{38}\]  
where \(M_{\rm 1h}\) is the coordinate transformation matrix after additional motions. In order to solve the additional motion coefficients, a sensitivity matrix can be listed. The theoretical conjugate tooth surface is dispersed into a grid, and its value for each point is calculated, which can be written as a column vector \(\delta_{1}^{\rm c}=\delta_{i}\) (\(i\) = 1, 2,\(\ldots\), \(q\)); the coefficients \(a_{\rm n}\) and \(b_{\rm n}\) are merged into a column vector \(\mathbf{c}=c_{\rm j}\) (\(j\) = 0, 1, 2,\(\ldots\), \(p\), \(p\) = 7 \(\times\) 2 = 14), and \(\Delta\mathbf{c}\) is set as an increment that causes the change in the tooth surface position vector. Therefore, the sensitivity matrix equation can be expressed as:  
\[\delta_{1(q\times 1)}^{\rm c}=\mathbf{J}_{(q\times p)}\cdot\mathbf{\Delta c}_{(p\times 1)} \tag{39}\]  
where \(\mathbf{J}\) is the sensitivity matrix, which is composed by partial derivatives, and it represents the degree of response of the target modification value when the increment changes, which can be expressed as follows:  
\[\mathbf{J}=\frac{n_{\rm c}\cdot[R_{1}^{\rm c}(\mathbf{c}_{\rm j}+\Delta\mathbf{c}_{\rm j })-R_{\rm c}]}{\Delta\mathbf{c}_{\rm j}} \tag{40}\]  
where \(\Delta\mathbf{c}_{\rm j}\) is the unit increment of coefficient \(\mathbf{c}_{\rm j}\). Due to the deviation values being too small, Equation (39) is usually ill conditioned, and the singular value decomposition method is used to iteratively solve the equation, where the singular value decomposition formula of the sensitivity matrix is:  
\[\mathbf{\Gamma}\mathbf{\Omega}\mathbf{\Lambda}=svd(\mathbf{J}) \tag{41}\]  
when the following conditions are met, the iteration stops and the parameters are obtained. \[\left\{ \begin{array}{l}\mathbf{H}=\delta_{1}^{\rm c}-n_{\rm c}\cdot[R_{1}^{\rm c }(\mathbf{c}_{\rm j}+\Delta\mathbf{c}_{\rm j})-R_{\rm c}]\\ \sqrt{\mathbf{H}^{T}\mathbf{H}}\leq 10^{-3}\end{array} \right. \tag{42}\]  
Figure 13: The machining motions of conjugate surface and overcutting phenomenon.