## 2 Basic Theories
### Efficiency Calculation of SBG  
#### 2.1.1 Tooth Flank and Contact Model  
Once the original pinion machine settings are defined, position vectors of the cutting cone \(\mathbf{r}_{pt}\) and \(\mathbf{r}_{gt}\) is calculated by cutter radius, cutter head parameters, and profile angle. The meshing coordinate system could be established based on geometry and movement relations [24]. Figure 1a,b show the machining coordinate systems of the pinion and gear. Coordinate systems \(S_{m1}\)-\(O_{m1}\), \(x_{m1}\), \(y_{m1}\), \(z_{m1}\) and \(S_{m2}\)-\(O_{m2}\), \(x_{m2}\), \(y_{m2}\), \(z_{m2}\) are the cradle coordinate systems for the pinion and gear, respectively. Coordinate systems \(S_{1}\)-\(O_{1}\), \(x_{1}\), \(y_{1}\), \(z_{1}\) and \(S_{2}\)-\(O_{2}\), \(x_{2}\), \(y_{2}\), \(z_{2}\) are the coordinate systems of the pinion and gear. Coordinate systems \(S_{c1}\)-\(O_{c1}\), \(x_{c1}\), \(y_{c1}\), \(z_{c1}\), \(S_{d1}\)-\(O_{d1}\), \(x_{d1}\), \(y_{d1}\), \(z_{d1}\), \(S_{c2}\)-\(O_{c2}\), \(x_{c2}\), \(y_{c2}\), \(z_{c2}\), and \(S_{d2}\)-\(O_{d2}\), \(x_{d2}\), \(y_{d2}\), \(z_{d2}\) are auxiliary coordinate systems. The meaning of the symbols are listed as follows: \(\varphi_{p}\), \(\varphi_{g}\) are the cradle rotation angles of the pinion and gear; \(\varphi_{1}\) and \(\varphi_{2}\) are the rotation angle of the pinion and gear; \(q_{1}\) and \(q_{2}\) are the initial cradle angle settings of the pinion and gear; \(S_{r1}\) and \(S_{r2}\) are the cutter radial settings of the pinion and gear; \(X_{b1}\) and \(X_{b2}\) are the sliding base feed setting; \(E_{m1}\) and \(E_{m2}\) are the vertical offset; \(X_{g1}\) and \(X_{g2}\) are the increment of machine center to the back of the pinion and gear. All these machining parameters determine the shape of the tooth surface that means these parameters also directly affect meshing performance. According to the transformation of coordinates, the surface position vector \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) of the pinion and gear could be written as:  
\[\mathbf{r}_{1}=M_{1d1}M_{d1m1}M_{m1c1}M_{c1p}\mathbf{r}_{gt} \tag{1}\]  
\[\mathbf{r}_{2}=M_{2d2}M_{d2m2}M_{m2c2}M_{c2p}\mathbf{r}_{gt} \tag{2}\]Similarly, the normal vector of tooth surface \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\) could be obtained based on the position vectors of the cutting cone \(\mathbf{n}_{pt}\) and \(\mathbf{n}_{gt}\):  
\[\mathbf{n}_{1}=L_{1d1}L_{d1m1}L_{m1c1}\mathbf{L}_{c1p}\mathbf{n}_{pt} \tag{3}\]  
\[\mathbf{n}_{2}=L_{2d2}L_{d2m2}L_{m2c2}\mathbf{L}_{c2g}\mathbf{n}_{gt} \tag{4}\]  
Figure 1: Machining coordinate system of SBG. (**a**) The machining coordinate system of the pinion and (**b**) the machining coordinate system of the gear.  
where \(M_{1d1}\), \(M_{d1m1}\), \(M_{m1c1}\), \(M_{c1p}\), \(M_{2d2}\), \(M_{d2m2}\), \(M_{d2m2}\), \(M_{m2c2}\), \(L_{d1d}\), \(L_{d1m1}\), \(L_{m1c1}\), \(L_{c1p}\), \(L_{2d2}\), \(L_{d2m2}\), \(L_{m2c2}\), \(L_{c2g}\) are coordinate transformation matrixes.  
Tooth surface contact of SBG is a typical point contact form. The contact ellipse parameters of spiral bevel gears are illustrated in Figure 2, which are obtained by tooth contact analysis (TCA). The contact ellipse parameters and curvature parameters are closely related to the machining parameters and tool parameters. As Figure 2 shows, \(a\) and \(b\) represent the major axis and minor axis of the contact ellipse, respectively. \(V\) and \(U\) are relative sliding velocity and entrainment velocity. \(\theta_{v}\) and \(\theta_{u}\) are angles between minor axis of the ellipse and \(V\) and \(U\). Moreover, the radius of curvature in contact with minor and major axis direction \(r_{x}\) and \(r_{y}\) and coordinates of contact point \(M\) under the coordinate system of the pinion and gear \(R_{mp}\) and \(R_{mg}\) are obtained for the calculation of lubrication and efficiency, which has not been detailed here.