## 1 Introduction
[PERSON] [33] simplified the assumption of straightness of contact curvature of the elliptical raceway ball bearing and analyzed the influence of maximum contact stress of the raceway and elliptical curvature. [PERSON] et al. [34] considered different radial clearance values caused by the elliptical raceway, and analyzed the calculation error of the maximum contact stress between the elliptical raceway and rolling body with different macro geometry through the finite element method. [PERSON] et al. [35] took the load distribution as the research object for the design of non-standard bearings, and determined the maximum contact stress model between the rolling element and the inner and outer rings from the perspective of isostress. [PERSON] et al. [36] considered the contact stress law by studying the contact state of the four contact points and the distribution of the load in the contact area, which combined the change in the contact point between the rolling element and the raceway for the asymmetric four-point contact ball bearing. The research on the contact stress of the variable curvature raceway mostly focuses on the determination and influencing factors of the maximum contact stress, but there is little information on the three-dimensional stress distribution in the micro contact area. Combining the characteristics of raceway structure and contact state changes when the rolling elements pass through the variable diameter raceway, this study proposes a three-dimensional contact stress calculation method between the rolling element and the raceway of a cageless ball bearing based on the rolling contact theory. This method macroscopically analyzes the influence of variable diameter raceway structure and different working conditions on the contact stress of cageless bearing. The narrow-band method was used to calculate stress components, and the three-dimensional contact stress numerical model in the contact area was established. This study determines the analytical solution of stress and reveals the characteristics of stress components in the contact area, and obtains the action relationship between the factors affecting the sudden change in stress. Finally, strain experiments were used to verify the validity of the stress model. This provides feasible theoretical guidance for the design and application of cageless bearing with a local variable diameter raceway, provides ideas for optimizing the structure to improve the maximum stress concentration, and ensures the discrete effect of the variable diameter raceway. Our study mainly provides the following contributions:  
First, the macroscopic contact stress distribution model was established with the raceway structure as a parameter, which provides a wide applicability for the study of raceway contact stress of non-standard bearings. The composition and distribution model of the microscopic three-dimensional stress in the contact area was established, which fills the gap in the research on the contact characteristics of non-standard bearings. Then, the narrow-band method was used to calculate each stress component of the three-dimensional stress distribution in the contact area, and determines the analytical solution of the stress component. Next, the dimensionless numerical solution of the stress component was carried out to provide an accurate law of stress distribution. Finally, through experimental comparative analysis, it was verified that our model and method are instructive in the contact stress distribution between the rolling element and the variable diameter raceway of the cageless bearing.