## 2 Static Equilibrium Equation of the Slewing Ball Bearing
### Geometric Model  
Figure 1 shows the sectional view of a single-row of four-point contact ball bearings. The \(z\)-axis is the axial direction of the bearing and the \(x\)-axis and \(y\)-axis are the radial direction of the bearing. Each ring of a four-point contact ball bearing consists of two arcs, each row of raceways having four arcs. Here, \"inner upper\" raceway is defined as the upper raceway arc of the inner ring; and \"inner lower\" raceway as the lower raceway arc of the outer ring. In like manner, \"outer upper\" raceway is defined as the upper raceway arc of the outer ring; and \"outer lower\" raceway as the lower raceway arc of the outer ring. The contact of the ball with \"inner upper\" and \"outer lower\" raceways forms contact pair 1, and the contact of the ball with \"inner lower\" and \"outer upper\" raceways forms contact pair 2. \(O_{0}\) is the geometric center of the bearing, \(O\) is the center of the ball, \(D_{pw}\) is the pitch diameter of the bearing, \(D_{w}\) is the diameter of the ball, \(\alpha_{0}\) is the original contact angle of the bearing, \(g_{0}\) is the clearance of the bearing, and _id, it_ stand for inner ring upper and lower raceway centers, and _od, ou_ stand for outer ring upper and lower raceway centers, respectively.\(o_{i,r}\), \(o_{i,a}\), \(o_{o,r}\), \(o_{o,a}\) are the nominal radial and axial distances from the center of the rolling element to the center of the inner and outer raceways, respectively. \[o_{i,r} =\left(r_{i}-\frac{D_{w}}{2}\right)\cos\psi \tag{1}\] \[o_{i,a} =\left(r_{i}-\frac{D_{w}}{2}\right)\sin\psi\] (2) \[o_{o,r} =\left(r_{o}-\frac{D_{w}}{2}\right)\cos\psi\] (3) \[o_{o,a} =\left(r_{o}-\frac{D_{w}}{2}\right)\sin\psi \tag{4}\]In Equations (1)-(4), \(\psi\) is the azimuth angle of the bearing rolling element; \(r_{i}\) is the curvature radius of the inner raceway, and \(r_{o}\) is the curvature radius of the outer raceway--the value of which can be calculated according to the curvature coefficient of the inner and outer raceways and the diameter of the ball given by the bearing parameters [24]. Irregular bearing geometry is specified by the center of the irregularly shaped raceway [19]. The position relationship between the center of the bearing inner raceway and the center of the ball is shown in Figure 2. Based on the traditional circular raceway center, additional radial and axial displacements \(\Delta_{i,r}\) and \(\Delta_{i,a}\) are added to represent the arbitrary geometry of the inner raceway, where the additional displacement a is a function of the azimuth b of the rolling element, and the functional relationship between them can be obtained by other methods such as fitting. When a certain functional relationship is given, any reasonable geometry of the inner raceway of the bearing can be specified. The geometric shape of the outer raceway is the same, and the corresponding additional displacements are \(\Delta_{o,r}\) and \(\Delta_{o,a}\). Therefore, the raceway center corresponding to any shape of the inner and outer rings of the bearing can be written as:  
\[\overset{\rightarrow}{r_{tu}}=\overset{\rightarrow}{r_{ou}}= \begin{bmatrix} \left(\frac{D_{pw}}{2}\pm o_{i,r}\right)\cos\psi+\Delta_{i,r}\big{|}_{\psi} \\ \left(\frac{D_{pw}}{2}\pm o_{i,r}\right)\sin\psi+\Delta_{i,r}\big{|}_{\psi}\\ o_{i,a}+\Delta_{i,a}\big{|}_{\psi}\\ 1\end{bmatrix}  \tag{5}\]  
\[\overset{\rightarrow}{r_{id}}=\overset{\rightarrow}{r_{od}}= \begin{bmatrix} \left(\frac{D_{pw}}{2}\pm o_{o,r}\right)\cos\psi+\Delta_{o,r}\big{|}_{\psi}\\ \left(\frac{D_{pw}}{2}\pm o_{o,r}\right)\sin\psi+\Delta_{o,r}\big{|}_{\psi}\\ \left.-o_{o,a}+\Delta_{o,a}\big{|}_{\psi}\\ 1\end{bmatrix}  \tag{6}\]  
where '+' (respectively '\(-\)') means that the calculated distance from the center point of the non-circular raceway to the bearing center \(O_{0}\) is greater (respectively less) than the distance  
Figure 1: Sectional view of single-row four-point contact ball bearing. Figure 2: Irregular deformation of bearing inner raceway [19].