## 2 Static Equilibrium Equation of the Slewing Ball Bearing
### Bearing Load State  
Figure 3 shows the vector description of the bearing after loading. Among them, the bearing outer ring is fixed; and the loads on the inner ring, \(Q_{1}\), \(Q_{2}\) are the contact force between ball and raceway in the contact pairs 1 and 2, respectively. Before the bearing is loaded, the curvature centers of the inner and outer raceways can be expressed as:  
\[\overrightarrow{r_{iu}}=\overrightarrow{r_{ou}}= \begin{bmatrix}\left( \frac{D_{pw}}{2}\pm o_{i,r}\right)\cos\psi\\ \left(\frac{D_{pw}}{2}\pm o_{i,r}\right)\sin\psi\\ o_{i,a}\\ 1\end{bmatrix}  \tag{7}\]  
\[\overrightarrow{r_{id}}=\overrightarrow{r_{od}}= \begin{bmatrix}\left( \frac{D_{pw}}{2}\pm o_{o,r}\right)\cos\psi\\ \left(\frac{D_{pw}}{2}\pm o_{o,r}\right)\sin\psi\\ -o_{o,a}\\ 1\end{bmatrix}  \tag{8}\]  
After the bearing is loaded, under the action of axial force \(F_{a}\), radial force \(F_{r}\) and overturning moment \(M\), the displacements of the inner raceway relative to the outer raceway along the x-, y-, and z-axes are \(u\), \(v\), and \(w\), respectively; and the deflection angles around the x- and y-axes are \(\varphi_{x}\) and \(\varphi_{y}\), respectively. These movements can be given by the coordinate transformation matrix shown in Formula (9). \[T= \begin{bmatrix}\cos\varphi_{y}&\sin\varphi_{x}\sin\varphi_{y}&\cos\varphi_{ x}\sin\varphi_{y}&u\\ 0&\cos\varphi_{x}&-\sin\varphi_{x}&v\\ -\sin\varphi_{y}&\cos\varphi_{y}\sin\varphi_{x}&\cos\varphi_{x}\cos\varphi_{y}& w\\ 0&0&0&1\end{bmatrix}  \tag{9}\]  
Figure 3: Vector description of load acting on bearing raceway. Assuming that the deflection angle of the bearing due to load deformation is very small [17], where \(\sin\varphi=\varphi\) and \(\cos\varphi=1\), the coordinate transformation matrix \(T\) can be simplified as follows:  
\[T\approx \begin{bmatrix}1&\varphi_{x}\varphi_{y}&\varphi_{y}&u\\ 0&1&-\varphi_{x}&v\\ -\varphi_{y}&\varphi_{x}&1&w\\ 0&0&0&1\end{bmatrix}  \tag{10}\]  
The position of the curvature center after the inner raceway movement can be described as:  
\[r_{iu,T}^{\rightarrow}=T\cdot r_{iu}^{\rightarrow}r_{id,T}^{\rightarrow}=T \cdot\overset{\rightarrow}{r_{id}} \tag{11}\]  
After the inner raceway movement, the direction of contact force \(Q_{1}\), \(Q_{2}\) can be expressed by the unit vectors \(e_{Q_{1},T}\) and \(e_{Q_{2},T}\), respectively. \[e_{Q_{1},T}^{\rightarrow}=\frac{r_{iu,T}^{\rightarrow}-r_{od}^{\rightarrow} }{\left\|r_{iu,T}^{\rightarrow}-r_{od}^{\rightarrow}\right\|}e_{Q_{2},T}^{ \rightarrow}=\frac{r_{id,T}^{\rightarrow}-r_{ou}^{\rightarrow}}{\left\|r_ {id,T}^{\rightarrow}-r_{ou}^{\rightarrow}\right\|} \tag{12}\]  
Therefore, after the inner raceway movement, the action point of the contact force on the inner and outer raceways of the bearing can be calculated as:  
\[Q_{1}=\begin{cases}r_{c_{1},1}^{\rightarrow}=r_{od}^{\rightarrow}+e_{Q_{1}, T}^{\rightarrow}\cdot r_{o}&Q_{2}= \begin{cases}r_{c_{2},2}^{\rightarrow}=r_{ou}^{ \rightarrow}+e_{Q_{2},T}^{\rightarrow}\cdot r_{o}\\ r_{c_{2},T}^{\rightarrow}=r_{id}^{\rightarrow}-e_{Q_{2},T}^{\rightarrow} \cdot r_{i}\end{cases}  \tag{13}\]  
The size of the contact force depends on the contact deformation of the raceway and the rolling element. Therefore, the premise of calculating the contact force is to calculate the relative approach \(\delta\) between the inner and outer raceways after bearing loading. Since the influence of bearing clearance is neglected in this paper, the initial relative approach degree is the rolling element diameter before bearing loading; after the bearing is loaded, the relative proximity a can be calculated as:  
\[\delta_{1}=D_{w}-\left\|r_{c_{1},1,T}^{\rightarrow}-r_{c_{1}}^{\rightarrow} \right\|\delta_{2}=D_{w}-\left\|r_{c_{2},T}^{\rightarrow}-r_{c_{2},2}^{ \rightarrow}\right\| \tag{14}\]  
When \(\delta\leq 0\), it means that the distance between the two raceways after loading is greater than the diameter of the rolling element, that is, the ball and the raceway are not in contact. In this case, the contact force is 0.