## 3 Model of Tooth Contact Analysis with Shaft Misalignments  
### Determination of Tooth Contact Coordinate System  
Based on the installation position of the gear, the tooth contact coordinate systems with a center distance error and installation deflection are set up in Figure 10.  
Figure 9: The corresponding counterpart rack of a gear pair. (**a**) Involute tooth profile counterpart rack; (**b**) S-shaped tooth profile counterpart rack.  
1. The axial plane determined by the theoretical axis of the gear and pinion is established. The two vertical planes pass the theoretical axis of the two gears and are perpendicular to the axial plane, respectively. 2. The center of one end face of the gear is taken as the coordinate origin The axis of the gear is the \(Z\) axis, and the plane \(XZ\) coincides with the axial plane. The absolute coordinate system \(S(\mathrm{O}-XYZ)\) is established. The initial position of coordinate system \(S_{2}(\mathrm{O}_{2}-X_{2}Y_{2}Z_{2})\) fixed to the gear coincides with the absolute coordinate system, as shown in Figure 5. 3. Firstly, the reference coordinate system \(S_{w}^{1}(\mathrm{O}_{w}^{1}-X_{w}^{1}Y_{w}^{1}Z_{w}^{1})\) of the pinion with only the center distance error is obtained by translating the origin \(\mathrm{O}\) of the absolute coordinate system \(S(\mathrm{O}-XYZ)\) by the distance \(a+\Delta a\) to the shaft end center of the pinion. Secondly, the coordinate system \(S_{w}^{1}\) is deflected around \(Y_{w}^{1}\) axis in the axial plane. The integrated deflection angle is \(\mu=\arcsin(\delta_{x}/b)\) and \(\delta_{x}\) is the axis error deflection of the axial plane. Accordingly, the reference coordinate system \(S_{w}^{2}(\mathrm{O}_{w}^{2}-X_{w}^{2}Y_{w}^{2}Z_{w}^{2})\) with installation error angle in the axial plane is obtained. Thirdly, the coordinate system \(S_{w}^{2}\) is rotated around the \(X_{w}^{2}\) axis in the vertical plane where the integrated deflection angle is \(v=\arcsin(\delta_{y}/b)\). And \(\delta_{y}\) is the axis error deflection of the vertical plane. Thus, the reference coordinate system with an installation error angle in the vertical plane is obtained, which is the actual installation coordinate system \(S_{1}(\mathrm{O}_{1}-X_{1}Y_{1}Z_{1})\) of the pinion at the same time.  
The transformation matrix \(M_{01}^{(\mathrm{P})}\) was used to transform the pinion from the coordinate system \(S_{1}(\mathrm{O}_{1}-X_{1}Y_{1}Z_{1})\) to the absolute coordinate system \(S(\mathrm{O}-XYZ)\). Assuming a pinion rotate angle \(\kappa_{1}\) relative to the initial position during the transmission, the transformation matrix \(M_{K1}^{(\mathrm{P})}\) could transform the pinion from the coordinate system \(S_{1}(\mathrm{O}_{1}-X_{1}Y_{1}Z_{1})\) to the initial position. And the position vector of the pinion in the absolute coordinate system is given by  
\[\mathbf{r}_{\Sigma 1}^{(\mathrm{P})}=\mathbf{M}_{01}^{(\mathrm{P})}.(\bm  
\[M_{\kappa 1}^{(p)}=\begin{bmatrix}\cos\kappa_{1}&-\sin\kappa_{1}&0&0\\ \sin\kappa_{1}&\cos\kappa_{1}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \tag{27}\]  
The gear can always rotate by an angle \(\kappa_{2}\) as the pinion rotation angle passes the angle \(\kappa_{1}\) which can make the gear pair satisfy the condition of conjugate contact again. The transformation matrix \(M_{\kappa 2}^{(g)}\) is to transform the gear from the coordinate system \(S_{2}(\mathrm{O}_{2}-X_{2}Y_{2}Z_{2})\) to the absolute coordinate system \(S(\mathrm{O}-XYZ)\). Thus, the position vector of the gear in the absolute coordinate system is given by  
\[r_{\Sigma 2}^{(g)}=M_{\kappa 2}^{(g)}.r_{2}^{(g)} \tag{28}\]  
\[M_{\kappa 2}^{(g)}=\begin{bmatrix}\cos\kappa_{2}&-\sin\kappa_{2}&0&0\\ \sin\kappa_{2}&\cos\kappa_{2}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \tag{29}\]  
In the above formula, \(\Delta a\) represents the center distance error, \(\kappa_{1}\) represents the rotation angle of the pinion, and \(\kappa_{2}\) represents the rotation angle of the gear.