###### Abstract  
Bearing preload significantly affects the running performance of a shaft-bearing system including the fatigue life, wear, and stiffness. Due to the mounting error, the bearing rings are often angularly misaligned. The effects of the combined bearing preload and angular misalignment on the fatigue life of ball bearings and a shaft-bearing system are analyzed in this paper. The contact force distribution of angular contact ball bearings in the shaft-bearing system is investigated based on the system model. The system model includes the bearing model, and the shaft model is verified by comparing with the manufacturer's manual and the results from other theoretical models, with the difference between the results from the present bearing model and manufacturer manual within 3%. The global optimization method is used to replace the Newton-Raphson algorithm to solve the ball elements' displacements and friction coefficients, which improves the computation efficiency of the system model. The fatigue life of each bearing is evaluated with the consideration of the two preload methods and two angular misalignment cases. The fatigue life results show that the system life at the optimal angular misalignment is more than 1.5 times that without angular misalignment at the low preload value, and this ratio decreases as the preload value increases. The optimal angular misalignment of both the shaft-bearing system and the misaligned bearing is not always consistent, which depends on the preload value and bearing life. Both the constant-displacement preload and constant-force preload do not cause a significant difference in the highest system life. The different misaligned bearings can lead to different highest system lives as the preload value is low. shaft-bearing system; angular contact ball bearing; bearing preload; angular misalignment; fatigue life +
Footnote †: journal: Applied  
Among various causes that affect the fatigue life of the rolling element bearing, both preload and angular misalignment are two common and important factors. Many researchers have conducted numerous studies on the fatigue life of the rolling element bearings considering the preload and angular misalignment. [PERSON] (1985), as one of first researchers, presented the dependence of the fatigue life of a cylindrical roller bearing having crowned rolling elements on the angular misalignment. [PERSON] (1996) investigated the relation between the preload and service life for an angular contact ball bearing by theoretical and experimental methods. [PERSON] and [PERSON] (1997) reviewed the three categories of preload technologies and clarified that the determination of proper preload should take the fatigue life, stiffness, and temperature of rolling element bearings into account. Considering the significance of the pressure distribution of roller elements on the fatigue life estimation, [PERSON] et al. (1998) extended the 3D elastic contact method to simulate the contact pressure and analyzed the fatigue life of a tapered roller bearing with the consideration of the angular misalignment effect. [PERSON] et al. (1999) analyzed the effects of the combined external loads and angular misalignment on the double-row tapered roller bearing, and the results demonstrated that the external load, rotation speed, and angular misalignment had a significant influence on the fatigue life of a double-row tapered roller bearing. [PERSON] et al. (1999, 2000) investigated the effect of the correction parameters of roller generators and angular misalignment on the fatigue life of the radial cylindrical roller bearing, in which the bearing radial clearance and complex loads were both taken into account. The above research works were mainly limited to a single bearing, and many works investigated the relations among the preload, fatigue life, temperature, and stiffness in a shaft-bearing system. [PERSON] et al.