## 2 Loaded Tooth Contact Analysis of Spiral Bevel Gear Pairs
### Loaded Tooth Contact Analysis Considering Assembly Error
 \tag{3}\]  
where, \(\mathbf{r}_{\text{p}}^{(\text{f})}\) and \(\mathbf{r}_{\text{g}}^{(\text{f})}\) are the coordinates of pinion and gear in the meshing coordinate system, respectively; \(\mathbf{r}_{\text{1}}\) and \(\mathbf{r}_{\text{2}}\) are the coordinates of pinion coordinate system and gear coordinate system, respectively; \(\mathbf{n}_{\text{p}}^{(\text{f})}\) and \(\mathbf{n}_{\text{g}}^{(\text{f})}\) are the normal vector of tooth surface of pinion and gear in the meshing coordinate system, respectively; \(\mathbf{n}_{\text{1}}\) and \(\mathbf{n}_{\text{2}}\) are the normal vectors of pinion coordinate system and gear coordinate system, respectively; \(L_{\text{f1}}\) is the first three order transformation matrix \(M_{\text{f1}}\); and \(L_{\text{f2}}\) is the first three order transformation matrix of matrix \(M_{\text{f2}}\). Given the rotor angle of the pinion (\(\Phi_{\text{p}}\)), solving Equation (4) can obtain the rotor angle of the gear (\(\Phi_{\text{g}}\)) and the coordinate of the initial contact point when unloaded. The contact trajectory is composed of the initial contact points  
\[\left\{ \begin{array}{l}\mathbf{n}_{\text{2}}\mathbf{v}^{(\text{2})}\ =\ 0\\ \mathbf{r}_{\text{p}}^{(\text{f})}-\mathbf{r}_{\text{g}}^{(\text{f})}\ =\ 0\\ \mathbf{n}_{\text{p}}^{(\text{f})}+\mathbf{n}_{\text{g}}^{(\text{f})}\ =\ 0\end{array} \right. \tag{4}\]  
where \(\mathbf{v}^{(\text{2})}\) denotes the relative velocity of the virtual generating gear face and the machined gear face is represented in \(\mathcal{S}_{\text{2}}\). The unloaded transmission error can be calculated as follows:  
\[\text{s}te\ =\ \Phi_{g}-\frac{z_{p}}{z_{g}}\Phi_{p} \tag{5}\]  
where \(z_{\text{p}}\) and \(z_{\text{g}}\) are the tooth numbers of the pinion and gear, respectively. We did not consider lubrication in our analysis because the presented method was a quasi-static approach. According to LTCA, the TVMS can be calculated by iterating the equation of compatibility which takes normal forces of contact points and the angle of pinion as a solution. The equation governing compatibility at an arbitrarily chosen meshing moment is as follows:  
\[ \begin{bmatrix}-(\mathbf{\lambda}_{\text{c}}+\mathbf{\lambda}_{\text{b}})&R_{n\times 1 }\\ R_{1\times n}&0\end{bmatrix}\begin{bmatrix}F_{n\times 1}\\ \theta_{\text{lte}}\end{bmatrix} \ =\ \begin{bmatrix}\mathbf{\varepsilon}_{n\times 1}\\ T\end{bmatrix}  \tag{6}\]  
where \(\mathbf{\lambda}_{\text{b}}\) is the global compliance matrix (see in [31]); \(\mathbf{\lambda}_{\text{c}}\) is the contact compliance matrix (see in [31]); \(R_{n\times 1}\) and \(R_{1\times n}\) are the distance from the initial contact point corresponding to the meshing moment of the rotating axis of pinion or gear; \(\mathbf{\varepsilon}_{n\times 1}\) is the clearance between the corresponding contact points at pinion and gear at the meshing moment; \(T\) is the input torque; \(\theta_{\text{lte}}\) is the rotation angle induced by the load of the pinion at different meshing moments; and \(F_{n\times 1}\) is the distributed normal force of potential contact points. Following this, we can obtain the contact area (see Figure 3), which consists of potential contact points for which the distributed normal force exceeds zero for various meshing moments of SBG. Figure 3a is the rotating projection of a healthy pinion without assembly error and tooth tip chipping, and Figure 3b depicts the contact area on the rotating projection of the pinion considering the gear axial error \(\Delta\text{AG}\) = 0.5 mm. Observations reveal that when the gear axial error \(\Delta\text{AG}\) is 0.5 mm, the contact area shifts towards the tooth tip and heel. As shown in Figure 4, a three-tooth model is established in MATLAB (R2021b) software. The proposed method needs only to get the global compliances obtained by the traversal loading method. Therefore, we constrain all degrees of nodes at the bottom surface of the pinion and gear when simulating the working conditions of SBG pairs. The elements used are 8-node hexahedral elements.