## 4 Conclusions  
In this paper, the first five natural frequencies and modal shapes of the finite element model of the premium connection are obtained by using the time domain modal parameter identification method, and the results are verified by experiments. The tubing with modal vibration mode is analyzed, and the energy dissipation and sealing performance of the premium connection are studied. The main conclusions are as follows:  
1. The finite element analysis shows that the vibration modes of the first five natural frequencies mainly occur in the x-axis direction of the tubing, and the amplitude of the vibration mode in the y-axis direction is small. The test verifies the accuracy of the finite element analysis results.
2. In the displacement cycle, the gross slip stage occupies a large proportion, resulting in the hysteresis curve being a \"parallelogram\". Under the second-order mode shape, the gross slip ratio is 94.1%, which is slightly higher than the 93.9% of the first-order mode shape. As the displacement amplitude increases, the macroscopic slip region and energy dissipation in the hysteresis curve increase.
3. After applying the displacement amplitude, the effective sealing length of the sealing surface of the premium connection is reduced by about 21%, and the normal contact pressure is reduced by about 30%. As the displacement amplitude increases, the sealing performance tends to be stable, indicating that the vibration load will lead to a decrease in air tightness, but the sealing performance tends to be stable under a larger displacement amplitude.  
Figure 20: Normal contact pressure of the sealing surface of the second mode shape.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Internal** & **Sealing Index Before** & **Sealing Index After** & \\
**Pressure/MPa** & **Loading/mm\(\cdot\)MPa\({}^{1.4}\)** & **Loading/mm\(\cdot\)MPa\({}^{1.4}\)** & **Decrease Ratio/\%** \\ \hline
80 & 4670 & 1704 & 63.51 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Second-order mode shape sealing surface sealing index.  
**Author Contributions:** Conceptualization, Y.Y. and Y.L.; methodology, Y.Y. and Y.L.; software, Y.Y. and Y.C.; validation, Y.C. and Y.D.; formal analysis, Y.Y.; investigation, Y.L.; resources, Y.Y.; data curation, Y.L.; writing--original draft preparation, Y.Y.; writing--review and editing, Y.Y., Y.L., Y.C. and J.T.; visualization, Y.Y.; supervision, Y.D. and Q.Y.; project administration, Y.L.; funding acquisition, J.T and Y.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China (NSFC) (Nos. 52274006 and 52374039), and Natural Science Basic Research Program of Shaanxi (No. 2023-JC-YB-337).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request.  
**Conflicts of Interest:** Authors [PERSON] and [PERSON] were employed by the company CNPC Baoji Petroleum Steel Pipe Co., Ltd. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.