## 5 Results and Discussion
### The Constant-Displacement Preload Condition  
Figure 11 shows the nonlinear dependence of the basic reference rating life of all three bearing on the angular misalignment in the constant-displacement preload condition. As can be seen, the fatigue life of each bearing varied significantly with the angular misalignment, and most of them were not the highest when the angular misalignment was zero, except for that of Rear Bearing 2 in the Case II condition. For each misaligned bearing, an optimal angular misalignment existed regardless of the preload value at which the misaligned bearing had the highest fatigue life. The optimal angular misalignment values were different for the two misaligned bearings, \(-0.8\) mrad for Rear Bearing 1 and \(0\) mrad for Rear Bearing 2, which mainly depended on the contact force distribution of the misaligned bearings [27]. According to SKF [28], the permissible misalignment angle was \(1.2\) mrad. It could be seen that the fatigue life of all bearings was very low at either \(1.2\) mrad or \(-1.2\) mrad compared with that within the permissible misalignment angle, as shown in Figure 11; the angular misalignment approaching the permissible misalignment angle could lead to a significant reduction of bearing life. For a shaft-bearing system, the failure of any bearing will cause the system to fail. Due to the dispersion of material properties, the bearing with a high rated life may fail earlier than that with a low rated life, but the probability of this phenomenon decreases significantly as the gap between the high rated life and low rated life increases. In general, the bearing with a low rated life is more likely to fail first compared with the bearing with a high rated life. Without losing generality, it is supposed that the system life is mainly related to the bearing with the lowest rated life. The shaft fatigue life is not taken into account here. In the Case I condition, Rear Bearing 1 was misaligned. When the front bearing was preloaded by an axial displacement of \(0.025\) mm, the fatigue life of Rear Bearing 1 was always the lowest among the three bearings as the angular misalignment varied from \(-1.2\) mrad to \(1.2\) mrad, therefore governing the system life. The shaft system had the same optimal angular misalignment, \(-0.8\) mrad, and the same highest fatigue life with Rear Bearing 1. The highest system life was significantly greater than the system life at the angular misalignment of \(0\) mrad, reaching \(1.55\) times. As the preload displacement increased to \(0.050\) mm, the system life depended on the fatigue life of both Rear Bearing 1 and the preloaded front bearing, and the optimal angular misalignment turned to \(-0.21\) mrad. The system optimal angular misalignment was no longer the same as the bearing optimal angular misalignment and approached \(0\) mrad, which was caused by the significant reduction of the fatigue life of the preloaded front bearing with increasing preload value. The system life at the optimal angular misalignment was only \(1.12\) times that at the angular misalignment of \(0\) mrad. The difference between the system life at the optimal angular misalignment and the system life at the angular misalignment of \(0\) mrad decreased with the increasing constant-displacement preload. In the Case II condition, the system life was mainly related to both the front bearing and Rear Bearing 1 due to the high fatigue life of the misaligned Rear Bearing 2. The optimal angular misalignments of the shaft system were \(0.8\) mrad, \(0.8\) mrad, and \(0.21\) mrad, respectively, as the preload displacement varied from \(0.025\) mm to \(0.050\) mm. The optimal angular misalignments of the shaft system for Case I and Case II were nearly opposite each other at the same preload displacement.