## 3 Model of Tooth Contact Analysis with Shaft Misalignments
### Analysis of Contact Pattern without Loading  
In the process of the transmission of the gear, the path of the contact area on the tooth flank is called the tooth contact pattern. The distribution of the tooth contact pattern is an important basis for judging the load characteristics of the tooth flank and the dynamic characteristics of the gear [25]. To determine the tooth contact pattern, an arbitrary cross section of the tooth was taken as an example. It is assumed that the tooth flank of the gear is painted with a layer of marking compound with a thickness of \(D_{\rm e}\). According to the contact state of Figure 12a, the surface of the marking compound contacts the surface of the pinion at the point \(M\). As shown in Figure 12b, the gear installation position is fixed, and the pinion is rotated around the shaft until the point \(M\) on the tooth flank of the pinion in contact with the point \(M^{\prime}\) on the gear tooth flank which is the theoretical meshing point. In this case, \(\widehat{AB}\) on the pinion tooth flank is the interference area produced by the tooth surface of the gear and the marking compound, which is the tooth contact pattern, and the \(\widehat{A^{\prime}B^{\prime}}\) is the tooth contact pattern of the gear. According to the above theories, the tooth profile equation based on any section plane of the gear which was parallel to plane OXY could be obtained. Figure 13a shows the dynamic coordinate system where the section plane is located, the \(O_{\rm g}Z_{\rm g}\) axis coincides with the \(OZ\) axis, and the \(O_{\rm g}X_{\rm g}\), \(O_{\rm g}Y_{\rm g}\) axis in the section plane is respectively parallel to the \(OX\) axis, \(OY\) axis. The transformation matrix from the absolute coordinate system \(S({\rm O}-XYZ)\) to the dynamic coordinate system \(S_{\rm g}({\rm O}_{\rm g}-X_{\rm g}Y_{\rm g}Z_{\rm g})\) was \(M_{\rm g}^{(\rm pg)}\). \[M_{\rm g}^{(\rm pg)}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&-p_{1}v_{1}&0\\ 0&0&0&1\end{bmatrix}  \tag{35}\]In the coordinate system \(S_{\xi}(O_{\xi}-X_{\xi}Y_{\xi}Z_{\xi})\), \(\mathbf{r}_{\xi 1}^{(\rm p)}\) is the tooth profile vector of the pinion, \(\mathbf{r}_{\xi 2}^{(\rm g)}\) is the tooth profile vector of the gear with a layer of marking compound, and \(\mathbf{n}_{\xi 2}^{(\rm g)}\) is the tooth profile normal vector of the gear. \[\mathbf{r}_{\xi 1}^{(\rm p)}=\mathbf{M}_{\xi}^{(\rm pg)}\cdot\mathbf{r}_{\Sigma 1}^{(\rm p)} \tag{36}\]  
\[\mathbf{n}_{\xi 2}^{(\rm g)}=\mathbf{M}_{\xi}^{(\rm pg)}\cdot\mathbf{n}_{\Sigma 2}^{(\rm g)} \tag{37}\]  
\[\mathbf{r}_{\xi 2}^{(\rm g)}=\mathbf{M}_{\xi}^{(\rm pg)}\cdot\mathbf{r}_{\Sigma 2}^{(\rm g)}+\mathbf{n}_{ \xi 2}^{(\rm g)}\cdot D_{e} \tag{38}\]  
Figure 12: Analysis method of a contact pattern. (**a**) Initial stage of tooth contact; (**b**) analysis of contact. Figure 13: Contact area conditions. (**a**) Section plane of a gear; (**b**) contact area. According to the geometric relation in Figure 8b, for a tooth profile vector \(r_{\xi 1}^{(\mathrm{p})}\) corresponding to an arbitrary global variable \(u_{1}\), and if there is a unique global variable \(u_{2}\) such that \(r_{\xi 2}^{(\mathrm{g})}\) is collinear with \(r_{\xi 1}^{(\mathrm{p})}\), under this condition, if and only if the modules length of \(r_{\xi 2}^{(\mathrm{g})}\) is greater than or equal to the module length of \(r_{\xi 1}^{(\mathrm{p})}\), the tooth contact area appears on the tooth surface. Therefore, the condition equations of generating the contact area are given by  
\[\left\{ \begin{array}{l}p_{1}v_{1}=p_{2}v_{2}\\ \dfrac{r_{\xi 2}^{(\mathrm{g})}}{|r_{\xi 2}^{(\mathrm{g})}|}=\dfrac{r_{\xi 1 }^{(\mathrm{p})}}{|r_{\xi 1}^{(\mathrm{p})}|}\\ \left|r_{\xi 2}^{(\mathrm{g})}\right|\geq\left|r_{\xi 1}^{(\mathrm{p})} \right|\end{array} \right. \tag{39}\]  
The actual calculation process is as follows.