## 2 Loaded Tooth Contact Analysis of Spiral Bevel Gear Pairs
### Loaded Tooth Contact Analysis Considering Assembly Error  
The method proposed in this paper is based on loaded tooth contact analysis of spiral bevel gear pairs. The detailed processes of TCA and LTCA are stated in the paper published by [PERSON] et al. [31]. Here is a brief introduction to the key parts of TCA and LTCA:  
The TCA is divided into two parts: (1) calculating discrete points of pinion and gear by accounting for both the engagement relationship between the tooth and cutter, which are in conjugate contact, and the projection relationship between three-dimensional space and the rotating projection plane; and (2) calculating the corresponding rotor angle and initial contact points when pinion and gear are meshing according to the conditions of continuous tangency between pinion and gear in the assembly coordinate system. Figure 1: Tooth tip chipping of spiral bevel gear. The LTCA can be generalized into four parts: (1) establishing a model and meshing grid according to the discrete points of pinion and gear; (2) determining the points along the long axis of the contact ellipse, which represents the intersection between the normal vector and the tooth surface, at all moments by translating the normal vector; (3) obtaining the global compliance matrix by traversal loading method and interpolation; and (4) calculating the distributed force and the rotation angle induced by the torque of pinion by solving the equation governing compatibility. Assembly errors do not directly impact the tooth, therefore, here we will only introduce the method for bringing four assembly errors into TCA. As shown in Figure 2, \(\mathbf{S}_{1}(x_{1},y_{1},z_{1})\) and \(\mathbf{S}_{2}(x_{2},y_{2},z_{2})\) are coordinate systems of pinion and gear, respectively; \(\mathbf{S}_{\rm b1}(x_{\rm b1},y_{\rm b1},z_{\rm b1})\) and \(\mathbf{S}_{\rm b2}(x_{\rm b2},y_{\rm b2},z_{\rm b2})\) are the auxiliary coordinate systems; and \(\mathbf{S}_{\rm f}(x_{\rm f},y_{\rm f},z_{\rm f})\) is the meshing coordinate system. When transforming coordinate systems of pinion and gear into the meshing coordinate system, it can introduce assembly errors through the transfer matrix. This can be described as:  
\[M_{f1}\;=\; \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&\Delta AP\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}\cos\Phi_{\rm p}&-\sin\Phi_{\rm p}&0&0\\ \sin\Phi_{\rm p}&\cos\Phi_{\rm p}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{1}\]  
\[M_{f2}\;=\; \begin{bmatrix}1&0&0&0\\ 0&1&0&\Delta\text{E}\\ 0&0&1&\Delta\text{AG}\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}\cos(\Sigma+\Delta\Sigma)&0&-\sin(\Sigma+ \Delta\Sigma)&0\\ 0&0&0&0\\ \sin(\Sigma+\Delta\Sigma)&0&\cos(\Sigma+\Delta\Sigma)&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}\cos\Phi_{\rm g}&-\sin\Phi_{\rm g}&0&0\\ \sin\Phi_{\rm g}&\cos\Phi_{\rm g}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{2}\]  
where \(\Delta\)AP is the axial assembly error of pinion; \(\Delta\)AG is the axial assembly error of gear; \(\Delta\)E is the offset error; \(\Delta\)E is the shaft angle error; and \(\Phi_{\rm p}\) and \(\Phi_{\rm g}\) are the rotation angle of pinion and gear. Figure 2: Coordinate systems of SBG with assembly errors. Then, the tooth surface equation and the normal vector of pinion and gear in the meshing coordinate system considering assembly errors can be calculated as:  
\[ \begin{cases}\mathbf{r}_{\text{p}}^{(f)}&=\ M_{\text{f1}}\mathbf{r}_{\text{1}},\ \ \ \mathbf{n}_{\text{p}}^{(f)}&=\ L_{\text{f1}}\mathbf{n}_{\text{1}}\\ \mathbf{r}_{\text{g}}^{(\text{f})}&=\ M_{\text{f2}}\mathbf{r}_{\text{2}},\ \ \ \mathbf{n}_{\text{g}}^{(\text{f})}&=\ L_{\text{f2}}\mathbf{n}_{ \text{2}}\end{cases}