## 5 Results
### Vibration-Induced Fatigue Performance  
After the finite element simulations were completed, the von Mises stresses of root points in all bolts within a complete vibration cycle were extracted. The obtained stress time series of different bolted joints are shown in Figure 22.  
It is worth mentioning that the three bolted joints with double nuts had the same preload distribution, that is, the upper and lower nuts were both assigned 15 kN. It can be seen from the figure that the stresses in all bolted joints fluctuate within a vibration cycle. The locations of the peaks and valleys in the curves are almost the same in the six threaded connection structures, i.e., in the regular bolted joint, as well as in the bolted joints installed with a plain washer, variable-diameter nut, wedge-locking nut, regular double nuts, and a double-thread bolt. However, the magnitudes of stresses of the peaks and valleys in the curves are different for the six abovementioned types of bolted joints. We can also see that the locations of the peaks and valleys in the curve for the bolt installed  
Figure 21: Changes in preloads with an increase in vibration cycles for eight different bolted joints.  
with an eccentric double nut are almost the opposite of those of the six abovementioned bolted joints. In addition, the change in the curves of the bolted joint installed with a wedge washer considerably differs from that of other bolted joints, causing the locations and stresses of the peaks and valleys in the curve to significantly differ.  
With the rainflow counting method, the stress time series of threaded connection structures were counted cyclically; the stress range histograms are shown in Figure 23. Under the same preload level, the majority of stress cycles extracted via the rainflow algorithm are small-amplitude stress cycles, while there are few stress cycles with amplitudes higher than 100 MPa.  
According to [PERSON]'s cumulative damage criterion, the larger the fatigue damage index under the same condition, the shorter the fatigue life. Therefore, the fatigue resistance performance of different bolted joints can be estimated according to the fatigue damage index accumulated within one vibration cycle. Using the stress range histograms and the S-N curve used for M10 \(\times\) 1.5 bolts shown in Figure 17, the fatigue damage indexes accumulated within a complete cycle were calculated, as shown in Table 4. The calculation results demonstrate that the fatigue index of regular bolted joint is the highest, representing the worst fatigue resistance performance under the conditions of the same vibration  
Figure 23: Stress range histograms obtained from different bolted joints.  
Figure 22: Stress time series in root points of bolts.  
cycles. Moreover, the fatigue resistance performance of the bolted joints installed with a variable-diameter nut and wedge-locking nut is the best among all the threaded connection structures, i.e., these bolts have the longest fatigue life under cyclic vibration. Additionally, among the three bolted joints with double nuts, the double-thread bolt connection shows the best fatigue resistance performance, whereas that of the eccentric double nut is the worst.