## 3 FEM Case Analysis
As shown in Figure 9, in the dynamic calculation results, there is a certain deviation when the bolt mesh size is four, and the results at other sizes are very close. As with the calculation results of the static, under the dynamic calculation, when the flange mesh size is 10, the maximum acceleration response amplitude deviates greatly, while the results under other mesh sizes have little difference. In summary, considering the computational efficiency and the highest possible accuracy of the results, the mesh size of the bolts used in this paper was 0.5, and the mesh size of the flanges was 5. Figure 8: Force-displacement curve (varies with mesh size). Figure 7: Load schematic. The connection structure was assembled with six evenly distributed bolts, and static loads (20 kN top tensile and compressive loads) and continuous impulse loads were applied to the top of the connection to calculate the static and dynamic properties of the structure, respectively. The static load calculation shows that when the bolt is severely loosened, the stiffness at the measuring point significantly decreases under a small load. As the load increases (Figure 10), it tends to be consistent, with no loosening. When the compressive load is applied, whether the bolt is loose or not has little effect on the force-displacement curve of the overall connection structure. The following two equations can be obtained by polynomial fitting of the force-displacement curve data in the tensile load interval:  
\[y_{molosen}=-0.019x^{3}+0.1x^{2}+0.015x+0.0025 \tag{10}\]  
\[y_{loosen}=-0.042x^{3}+0.1x^{2}+0.083x+0.0010 \tag{11}\]  
In Equations (10) and (11), \(y\) is the displacement and \(x\) is the tensile force. The bolts in Equation (10) are not loosened, and the bolts in Equation (11) are severely loosened. It is evident that in the polynomial fitting results, when the tensile force exceeds 18 kN, the slopes of the force-displacement curves of the two are almost the same, and the stiffness at this time is almost the same. However, when the tensile force is small, the stiffness of the two is very different, which indicates that the bolt preload mainly affects the stiffness  
Figure 10: Force-displacement curve (static loads). Figure 9: Acceleration response (varies with mesh size). in the initial stage of loading. Under the top tensile load, the local warpage of the shell is also significantly different due to the difference in the bolt preload (Figure 11). This initial stiffness damage is very detrimental to the structure's safety and reduces the load-bearing performance of the structure. The acceleration response results obtained in the dynamic calculation of the connection structure without loosening are shown in Figure 12. Since reductions in bolt preload in certain locations will cause local stiffness loss, this will inevitably affect the overall continuous impulse response of the connection structure. Therefore, this section adopts the implicit dynamic calculation process to analyze the artificial loosening setting, that is, after the preload is greatly reduced. The response spectrum is obtained after performing a short-time Fourier transform (SFFT) on the acceleration response, as shown in Figure 13, with the color representing the amplitude level. At different times, when a single bolt is loose, the first-order frequency does not significantly change, but it significantly changes with a high frequency. There will be different peaks in the frequency response, which indicate the presence of a loose bolt. This kind of response change in the frequency domain is not intuitive, and although there is an obvious degree of distinction, it is still impossible to directly judge. Figure 5:0 Comparison of stress clouds near fastening bolt and loosening bolts under static load; (**a**) No loosening; (**b**) One bolt loosened to 0.3 kN. Figure 5:1 Constant-amplitude continuous impulse excitation acceleration response.