## 2 Motion and Contact State of the Rolling Element  
The rolling elements of cageless bearing are in contact with each other during operation. The literature [37] proposed to design a variable diameter raceway structure in the outer ring raceway of the bearing. As shown in Figure 1, when rolling elements pass through the variable diameter raceway, the discrete effect is achieved. Assuming that the bearing outer ring is fixed, the inner ring rotates at an angular speed \(\omega_{i}\), the radial load is applied, and the axial preload is zero. The inertial coordinate system (\(O_{i}X_{i}Y_{i}Z_{i}\)) is fixed at the rotating axis center of the outer ring. A variable diameter raceway is located in the load area of the bearing outer ring in a symmetrical structure. The design of the variable diameter raceway is related to the number of rolling elements and the diameter of the inner and outer ring raceways of the bearing. The variable diameter raceway structure parameters are expressed as \(\theta_{1}\) and \(\theta_{2}\). The axial span angle \(\theta_{2}\) increases first and then decreases with the increase in the circumferential span angle \(\theta_{1}\). The starting position of the variable diameter raceway forms an included angle \(\theta_{1}/2\) with the negative direction of the \(Z_{i}\)-axis. The circumferential span angle \(\theta_{1}\) and axial span angle \(\theta_{2}\) correspond to the range of \(6\arcsin\left(\frac{r_{i}}{d_{i}}\sin\frac{\theta_{2}}{2}\right)\leq\theta_{1 }\leq 2\arcsin\left(\frac{R_{1}}{d_{i}+d_{o}}\right)\) and \(\theta_{2}<2\arcsin\left(\frac{d_{o}\theta_{1}\tau}{2\times 360\tau_{o}}\right)\), respectively. The radius of curvature of the variable diameter raceway in the \(Y_{i}O_{i}Z_{i}\) plane is \(R_{i}\) in the plane parallel to the \(X_{i}O_{i}Z_{i}\) plane, the radius of curvature is \(r_{i}\). The motion coordinate system of the rolling element mass center \(O_{b}X_{b}Y_{b}Z_{b}\) is established. The \(X_{b}\)-axis coincides with the rolling element rotation axis. The \(Z_{b}\)-axis points to the rotation center of bearing. Under the action of radial load \(F_{r}\) and centrifugal force, the rolling elements rotate around the \(X_{b}\)-axis while revolving circumferentially along the variable diameter raceway, the rolling elements maintain contact with the variable diameter. The normal force between the rolling element and the variable diameter raceway is \(Q_{i}\), the direction of \(Q_{i}\) passes through the rolling element spherical center and is perpendicular to the contact area. The contact surface is generated between the rolling elements and the variable diameter raceway under the radial load and is considered to be an ellipse based on the Hertz theory. The contact area is exaggerated in Figure 1, and its actual dimensions in engineering practice are quite small, based on which the contact coordinate system \(o^{\prime}x^{\prime}y^{\prime}z^{\prime}\) is established. The equivalent contact radius of curvature \(R_{i}\) and \(r_{i}\) are related to the structure of the variable diameter raceway. As the position angle \(\varphi_{i}\) is different, the curvature radius \(R_{i}\) changes from the outer ring groove bottom radius to the radius of the variable diameter raceway transition region, and then to the inner gyration radius of the contact section (\(Y_{i}O_{i}Z_{i}\) plane). \(r_{i}\) changes from the curvature radius of the raceway groove to the radius of the variable diameter raceway transition region. When the rolling elements pass through the variable diameter raceway, combined with the discrete principle, the normal variable load \(Q_{i}\) on the contact area varies with the angle of the rolling element at different positions of the variable diameter raceway. The variable diameter raceway is prone to generate alternating stress under the effect of the alternating load. The alternating stress leads to the wear of the variable diameter raceway, which changes the shape and size of the variable diameter raceway.