## 1 Introduction  
With the increasing application of high-speed equipment such as control systems for rocket engines and fans, the cageless bearing is used as the protective bearing of the electromagnetic bearing rotor system, the discrete performance of rolling elements is the premise to ensure the smooth operation of rotor [1, 2]. The design of the local variable diameter raceway in the outer ring of bearing is a method to realize the discretization of rolling elements [3]. Considering the particularity of the local variable diameter raceway structure, when the rolling elements pass through the variable diameter raceway, the contact characteristics between the rolling elements and the bearing raceway change. To determine the influence of the contact characteristics between the rolling elements and the raceway on the discrete effect of the variable diameter raceway, it is necessary to analyze the contact state and contact stress between the rolling elements and the raceway. It is urgent to determine the three-dimensional contact stress distribution and the influencing factors of the contact characteristics of the discrete motion of the rolling elements, and optimize the design of the cageless bearing structure and provide suitable application conditions. For research on the contact stress characteristics between the bearing rolling element and the raceway, [PERSON] et al. [4] combined with the classical Hertz theory to study the bearing contact characteristics, which ignored the friction in the contact area in the research process. [PERSON] et al. [5] simplified the contact problem of the ball bearing and provided an ideal numerical model for the stress analysis in the static stage. [PERSON] [6] considered the dynamic contact stress calculation model of the load, local deformation and rolling element motion state. In recent years, in order to improve the research on the contact stress of rolling bearing, researchers [7; 8; 9] have studied the existence of radial clearance and the influence of the shape of rolling elements on the contact stress, and suggest that having radial negative clearance minimizes the mutation of maximum contact stress, which provides the guidance for the design of a bearing raceway. [PERSON] et al. [10] suggested the calculation model of the maximum stress of rolling element and raceway under different operating speeds, and provided a basis for the setting of the bearing limit speed according to the influence of centrifugal force on the maximum stress of rolling elements and raceways at high speeds. [PERSON] et al. [11] proposed the contact stress distribution model of bearing parts under variable working conditions, and determined the influence of contact load and rotating speed on the variation law of maximum stress. [PERSON] [12] developed a numerical model to calculate the maximum contact stress according to the geometric parameters of the bearing. The model was taken to provide the raceway contact radius with the static bearing capacity based on the maximum contact stress as the objective function. However, the study of bearing contact stress not only considered the distribution of macro maximum contact stress by influencing factors such as speed, load and bearing structure, etc., but also the microscopic stress distribution in the contact area also affects the contact characteristics of the rolling elements and raceway. The stress distribution and composition in the contact area not only inform the sliding between the rolling element and the raceway, but also have important significance for the prediction of bearing wear and life. [PERSON] [13] proposed a three-dimensional stress distribution theory in the elastic half-space of rolling contact based on the Hertzian theory of two-dimensional contact stress analysis. [PERSON] et al. [14] investigated a method for determining the stick-slip boundary in the contact area based on the geometry of linear bearings. The method can determine the influencing factors of the contact friction between the rolling element and the raceway.