## 4 Simulations of Tightening, Vibration-Induced Loosening, and Fatigue
### Vibration-Induced Fatigue  
In engineering applications, another failure form of bolted joints is the fatigue fracture of bolts under long-term cyclic vibration [45; 46]. Fatigue life is denoted by the number of loading cycles before specimen failure. The presence of some anti-loosening bolted connections, such as washers, can increase the fatigue lives of bolted joints [47]. In order to study the fatigue characteristics of different anti-loosening bolted connections, the cyclic transversal vibration applied to movable plates was simulated using the same method deployed for the simulation of vibration-induced loosening. Due to the large calculations and small changes in stress amplitude under the same simulation condition required for FE models, only one vibration period of simulation was performed, and the vibration amplitude was set to 0.5 mm. It has been validated that the bolt thread root located within one pitch of the bearing surface has the maximum stress level, which is where fatigue cracks tend to occur most easily [20]. Therefore, the root point in the bolt was extracted as shown in Figure 15 in order to analyze the fatigue resistance performance of different bolted joints. Von Mises stress, which takes the first, second, and third principal stresses into account, is often used to evaluate fatigue resistance performance. After the simulations are completed, the von Mises stresses of root points in all bolts within a complete vibration cycle are extracted. The stress time series obtained via FEA is a continuous random process. The issue of how to account for complex loading history is a basic problem in fatigue life estimation [48]. Cycle counting is an effective way to simplify complicated loading history into simple load sequences with a cyclic concept for the purpose of estimating fatigue life using existing knowledge [49]. The rainflow counting method is one of the typical methods developed to calculate the number of cycles  
Figure 14: Changes in preloads under the vibration cycle for FE models with different mesh densities. Figure 15: Root point of the first engaged thread in a bolt. in complex loading history, and this technique has been successfully applied in fatigue life prediction for the automotive, railway, and aircraft industries [50]. This counting method uses all the information that impacts the fatigue damage of materials [51]. In order to carry out statistical processing for subsequent fatigue analysis, a MATLAB program based on the rainflow counting method was developed. Figure 16 shows the flow chart of the program to explain it in detail. For the purpose of cyclic counting, the irregular stress fluctuation is converted into a series of peaks and valleys. Then, a stress range histogram can be obtained via the rainflow counting method. There are several fatigue damage cumulative theories. Among them, linear damage cumulative theory is often used to estimate fatigue life due to the simplification of the fatigue damage mechanism and the convenience of its calculation. This theory assumes that the fatigue damage generated under various levels of stress is independent of other forms, meaning that the cumulative total damage can be obtained using linear superposition  
Figure 16: Flow chart of rainflow cycle counting. theory. [PERSON]'s linear damage cumulative rule [52; 53] given in Equation (2) is used in this study to evaluate fatigue damage of a threaded fastener. \[D=\sum_{i}\frac{n_{i}}{N_{i}} \tag{2}\]  
where \(D\) is the fatigue damage index, \(n_{i}\) is the number of the existing stress cycles in each stress range, and \(N_{i}\) is the number of stress cycles in each range that could lead to fatigue failure. The data needed in the rule can be extracted from the histograms by referring to S-N diagrams, which indicate the relationship between the stress ranges and fatigue life within set stress ranges.