# Fatigue Life Analysis of Cylindrical Roller Bearings Considering Elastohydrodynamic Lubrications
## 1 Introduction  
As critical components in rotating machinery, the accurate prediction of fatigue life for cylindrical roller bearings is essential for ensuring equipment reliability. Thus, predicting and extending bearing service life remains a central challenge in bearing design and application [1, 2, 3]. The classical Lundberg-Palmgren (L-P) theory [4, 5] establishes a statistical framework for rolling bearing life calculation. However, to achieve higher accuracies, more precise determination of internal contact load distributions within bearings is required. In recent years, numerous scholars have proposed various related models. [PERSON] [6] developed a quasi-static analysis model for rolling bearings by accounting for the elastic deformation of shafs and supporting structures, as well as centrifugal and gyroscopic loads on rollers under high-speed operation. [PERSON] [7] revealed the roles of roller geometry, operating clearance, misalignment angle, and working loads in shaping contact pressure distributions between rollers and their corresponding inner/outer ring. [PERSON] et al. [8] established a dynamic model for high-speed-train bearing-rotor systems under variable-speed conditions, incorporating unbalanced excitation caused by rotor mass eccentricity. Experimental validation on bearings with outer and inner race faults under diverse operating conditions confirmed the model's effectiveness. [PERSON] et al. [9] proposed a novel damage evolution equation based on octahedral shear stress amplitude, enabling the numerical simulation of crack initiation, propagation, and spalling. They systematically explored the correlation between contact stress and fatigue damage, along with its impact on material fatigue life. [PERSON] et al. [10], guided by race control theory [11], constructed an improved quasi-static model by analyzing the contact characteristics between balls and races, thereby elucidating the relationship between preload and bearing fatigue life. However, the presence of lubricant films significantly alters contact zone stresses and fatigue failure mechanisms. Existing quasi-static models often neglect lubrication effects, leading to deviations in life predictions. For line contact elastohydrodynamic lubrication problems, solutions typically involve coupled Reynolds, film thickness, Hertzian contact pressure, squeeze, and dynamic load equations. [PERSON] [12] and [PERSON] [13] laid the foundation for numerical solutions, with subsequent scholars [14; 15; 16] refining elastohydrodynamic lubrication models to address non-Newtonian fluids, surface roughness, and thermal effects. Recent studies [17; 18; 19] predominantly employ the multigrid method for solving elastohydrodynamic lubrication problems. Since its foundational work by [PERSON] [20], the multigrid method has evolved into a core technique for solving large-scale partial differential equations. The finite difference method, capitalizing on its regular grid structure, intuitive discretization approach, and inherent compatibility with the multigrid transfer process, functions as the principal platform for the initial application and advancement of multigrid techniques. Within the framework of the finite difference method, the multigrid method matures rapidly, extending its scope from linear elliptic partial differential equations to encompass strongly nonlinear systems, parabolic problems, and adaptive mesh refinement. It now finds extensive application across computational physics and engineering disciplines [21; 22; 23]. Compared with alternative iterative computational approaches, the multigrid method demonstrates significant advantages in computational efficiency, solution accuracy, and problem applicability, particularly in the analysis of lubrication problems involving complex contact conditions. This study proposes a fatigue life prediction method for cylindrical roller bearings that combines elastohydrodynamic lubrication and quasi-static analysis.