## 6 Discussion
### Effect of Preload Distribution on Anti-Loosening Performance  
In order to discuss the influence of preload distributions on the anti-loosening abilities of three bolted joints with double nuts, the preload distributions of the upper nuts were also changed (i.e., 5, 10, 15, 20, 25, and 30 kN) under the same total preload (30 kN), and transversal vibrations of movable plates were simulated via FEA. As shown in Figure 27, the preload decays of the bolted joints installed with regular double nuts vary under different preload distributions. When the preloads assigned to the upper nut are 5 and 10 kN, the total preloads decrease gradually under vibration conditions, indicating that the anti-loosening performance is lost. For the other preloads assigned to the upper nut, the total preloads drop slightly in the initial cycle subsequently remain unchanged with continued vibration. This demonstrates that these regular double nuts can achieve anti-loosening performance. This may be because the thread surface on the upper nut retains a stuck state and that interface slippage is prevented when the proportion of preload assigned to the upper nut is increased. Furthermore, it can be deduced that there is a critical preload assigned to the upper nut below which the anti-loosening ability is lost. The critical value ranges from 10 to 15 kN for a bolted joint installed with regular double nuts. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Preload Assigned to the Upper Nut** & **5 kN** & **10 kN** & **15 kN** & **20 kN** & **25 kN** & **30 kN** \\ \hline First stage (Nmm) & 60,065 & 56,125 & 55,362 & 53,734 & 51,222 & 46,819 \\ Second stage (Nmm) & 11,801 & 21,708 & 31,560 & 40,139 & 49,516 & 58,543 \\ Max torque (Nmm) & 60,065 & 56,125 & 55,362 & 53,734 & 51,222 & 58,543 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Torques of the double-thread bolt connection required under different preload distributions. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Preload Assigned to the Upper Nut** & **5 kN** & **10 kN** & **15 kN** & **20 kN** & **25 kN** & **30 kN** \\ \hline First stage (Nmm) & 89,090 & 88,270 & 81,928 & 80,340 & 75,498 & 74,497 \\ Second stage (Nmm) & 47,766 & 81,099 & 96,540 & 122,718 & 160,249 & 182,401 \\ Max torque (Nmm) & 89,090 & 88,270 & 96,540 & 122,718 & 160,249 & 182,401 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Final torques of the eccentric double nut under different preload distributions. Figure 26: Discrete data points and curves of tightening torques and preloads in the bolted joint installed with an eccentric double nut under different preload distributions. Figures 28 and 29 show the changes in preload as vibration cycles are increased for a bolted joint with a double-thread bolt and an eccentric double nut, respectively. From Figure 28, we can observe that the decline trend of preload is evident when the preloads assigned to the upper nut are 5 kN and 10 kN, while the preloads are almost constant under vibration for the other preloads assigned to the upper nut. This finding is the same for the regular double nuts shown in Figure 27. This indicates that the large preload distribution of the upper nut can help to improve anti-loosening ability. Therefore, the lower nut should be tightened using a small torque, while the upper nut should be tightened using a large torque in order to achieve better anti-loosening performance. Similarly, it can be said that the critical preload assigned to the upper nut is between 10 kN and 15 kN for double-thread bolt connection, below which the anti-loosening ability is lost. Figure 29 shows that only when the upper nut is assigned a value of 5 kN is the preload decreased continuously. Otherwise, the preloads are unchanged with an increase in vibration cycles. Therefore, when the preload assigned to the upper nut is equal to or less than 5 kN, the anti-loosening ability of the eccentric double is lost.