## 5 Results and Discussion
### The Constant-Force Preload Condition  
Figure 13 shows the variation of bearing life with the angular misalignment in the constant-force preload condition. In order to analyze the effects of the preload method on the bearing life, the preload force was taken as 500 N, 1000 N, and 1500 N. When the shaft was running without external loads, the preload force of 500 N could cause 0.0355 mm axial displacement of the outer ring of the front bearing, which was very close to 0.035 mm, and the preload force of 1000 N could cause 0.050 mm axial displacement. Therefore, the preload force of 500 N and 1000 N at the constant-force preload could be assumed to be equivalent to the preload displacement of 0.035 mm and 0.050 mm in the constant-displacement preload, respectively. The life comparisons can be made between Figures 11c-f and 13a-d. As can be seen, the two preload methods had a certain effect on the variation of the fatigue life of the preloaded front bearing with the angular misalignment. For the constant-force preload condition, the fatigue life of front bearings was basically linear with angular misalignment, and for the constant-displacement condition, a nonlinear relation could be found. The above difference was mainly due to the outer ring of the front bearing moving slightly with the angular misalignment for the constant-force preload condition, while the outer ring of the front bearing was immovable for the constant-displacement preload condition, which affected the contact force distribution in the front bearing, therefore resulting in different trends of the front bearing life with angular misalignment. With the increasing of the constant-displacement preload, the nonlinear relation between the fatigue life of the front bearing and the angular misalignment became more obvious, as shown in Figure 11. For the constant-force preload condition, the increasing preload force reduced the dependence of the  
Figure 12: The contact force distribution in Rear Bearing 1 at the optimal angular misalignment of the shaft-bearing system. The preload displacements are 0.025 mm ((**a**) and (**b**)) and 0.035 mm ((**c**) and (**d**)), respectively.  
front bearing life on the angular misalignment, and the relation between the front bearing life and the angular misalignment approached a horizontal line at the preload force of 1500 N, as shown in Figure 13. Among the three bearings in the shaft system, the effect of the preload method on the fatigue life of Rear Bearing 1 was the weakest, and the two preload methods only caused a small change in the fatigue life of Rear Bearing 1. From Figures 11c-f and 13a-d, it can be seen that the fatigue life of Rear Bearing 1 was lowest over a wide angular misalignment range, therefore playing a significant role in the system life. Due to the effect of the preload method on the fatigue life of Rear Bearing 1 being very limited, the optimal angular misalignment and highest fatigue life for the shaft system were the same for the two preload methods.  
Figure 13: The variation of the bearing life with the angular misalignment in the Case I condition ((**a**), (**c**) and (**e**)) and in the Case II condition ((**b**), (**d**) and (**f**)) for constant-force preload.  
As the preload force reached 1500 N, the system life depended on the fatigue life of both the front bearing and Rear Bearing 1. The optimal angular misalignment for the shaft system was close to 0.4 mrad in the Case I condition and \(-\)0.4 mrad in the Case II condition. As mentioned above, the fatigue life of the front bearing varied slowly with angular misalignment, which resulted in very small variation of the system life near the optimal angular misalignment. Therefore, a large range, for example from \(-\)0.4 mrad to 0.4 mrad for the Case I condition and from \(-\)0.8 mrad to 0.4 mrad for the Case II condition, could be regarded as the optimal angular misalignment range of the shaft system. Such a phenomenon implied that the increasing constant-force preload could weaken the effects of angular misalignment on the system life.