## 3 Result and Discussion
### Sailing Performance Analysis of the Premium Connection Under Modal Shapes  
Figure 18 illustrates the von Mises stress distribution for the premium connection. Considering the high-cycle displacement loading experienced by the tubing during extended service in the well, such displacement amplitudes can accelerate the formation of small cracks between the sealing surfaces. The prolonged reduction in the combined stiffness of the sealing surfaces inevitably leads to significant stress concentrations at the root of the bearing surfaces of the first and second threads of the threaded connection, resulting in substantial plastic deformation. As shown in Figure 18, the maximum von Mises stress at the thread of the premium connection is 850 MPa, while the yield strength of the premium connection in this paper is 758 MPa. The accumulation of plastic deformation damages the connection's auxiliary sealing structure, thereby impairing the sealing performance of the premium connection. Figure 19 shows the normal contact pressure distribution of the sealing surface of the premium connection before and after the first-order vibration mode loading. From the analysis of the figure, it is observed that the normal contact pressure on the sealing surface before the application of the loading displacement amplitude is higher than that after the application. The maximum normal contact pressure is reduced by approximately 30%, while the effective sealing length decreases from 1.4 mm to 1.1 mm, representing a reduction of about 21%. The parameters from Figure 17 are substituted into Equations (14) and (16) to calculate the variation in the sealing index values of the sealing surface before and after the application of the loading displacement amplitude, as shown in Table 4. The  
Figure 17: Second-order vibration mode hysteresis curve. sealing index value is reduced by 2948 (mm\(\cdot\)MPa1.4) from before to after loading, which constitutes a reduction of 63.12% relative to the sealing index before loading. Figure 20 illustrates the distribution of normal contact pressure on the sealing surface of the premium connection both before and after the application of the second-order vibration mode loading. As shown in the figure, the maximum normal contact pressure decreases by approximately 31% after loading with the second-order vibration mode, and the effective sealing length is reduced by about 22%. According to the data presented in Table 5, the sealing index of the sealing surface before and after loading decreased by 2966 (mm\(\cdot\)MPa1.4), accounting for 63.51% of the sealing index before loading. A comparison with the analysis in Figure 19 reveals that, as the applied displacement amplitude increases, the maximum normal contact force on the sealing surface remains almost unchanged, with only a slight reduction. Additionally, there is no significant change in the effective sealing length, and the effective sealing length under the first-order mode shape is essentially the same. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Internal** & **Sealing Index Before** & **Sealing Index After** & **Decrease Ratio/\%** \\
**Pressure/MPa** & **Loading/mm\(\cdot\)MPa1.4** & **Loading/mm\(\cdot\)MPa1.4** & **Decrease Ratio/\%** \\ \hline
80 & 4670 & 1722 & 63.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: First-order mode shape sealing surface sealing index. Figure 19: Normal contact pressure of the sealing surface of the first mode shape. Figure 18: von Mises stress contour diagram of the premium connection under cyclic load. Based on the above analysis, it is concluded that after the application of the displacement amplitude, a significant reduction in the normal contact pressure on the sealing surface occurs, which leads to a decrease in the gas-tight sealing performance of the premium connection. The effective sealing length of the sealing surface initially increases with time and the displacement load, but subsequently remains unchanged.