[13] analyzed the sealing pressure of the premium con  
Figure 1: Schematic diagram of the structure of the API and premium connection. (**a**) API threaded connection; (**b**) Premium connection. Figure 2: Wear of the premium connection. (**a**) Scratches and wear on the outer wall of the connection; (**b**) Sealing surface wear. nection filled with threaded grease. Based on the gas state equation, the time at which the leakage gas broke through the sealing pressure of the thread grease and escaped from the coupling was analyzed. However, the above research is mainly based on a static analysis of the sealing performance of the premium connection, which can not effectively solve the problem of the decrease in the sealing performance of the premium connection caused by the vibration of the tubing. Under vibration conditions, the sealing surface will experience gross slip and generate energy dissipation. With the passage of time, the energy dissipation of the connection will accumulate, resulting in fretting wear or the fatigue failure of the sealing surface [14], thus affecting its sealing performance. In the micro slip state, the energy dissipation is usually represented by the load-displacement hysteresis curve [15]. As shown in Figure 3, the load-displacement hysteresis curve is categorized into three types: linear, parallelogram, and elliptical [16]. The linear curve typically occurs under very small displacement amplitudes or large normal loads, where the contact surfaces experience only adhesion or microscopic slip. The parallelogram curve occurs under larger displacement amplitudes or smaller normal loads, where the macro-slip of the contact surfaces is observed. The elliptical curve mainly occurs in the later stages of the micro-slip cycle, where significant plastic deformation is usually present on the friction surfaces [17]. [PERSON] et al. [18; 19] study the hysteresis characteristics of connection surfaces using analytical and experimental methods, defining the stiffness and slip characteristics of connection surfaces under preload, and investigate the horizontal displacement of connection surfaces under different loading and unloading conditions, plotting load-displacement curves. [PERSON] [20] develops a contact stress model for the sealing surface and analyzes the impact of different sealing cone angles on energy dissipation in the premium connection under oscillating loads. YU [PERSON] et al. [21] establish a micro-slip model for the shear layer of the sealing surface and analyze the impact of the shear layer coefficient on the slip state and energy dissipation of the sealing surface under dynamic loads. [PERSON] et al. [22] study how the coupling of the neglected normal and in-plane elastic responses affects the tribological properties when the Hertzian or random rough indenter slides over the elastomer. This coupling will affect the gap morphology and thus affect the leakage. These studies provide new insights into the analysis of sealing performance for the premium connection; however, the models established are based on certain simplifications and assumptions, and do not include vibration modal analysis or its impact on the sealing performance. During the service of the premium connection in high-temperature and high-pressure gas wells, vibration induced by changes in the internal flow rate and pressure can further exacerbate the risk of leakage at the natural frequency [23; 24; 25]. Figure 3: Typical micro-slip load–displacement curves. (**a**) Linear; (**b**) Parallelogram-shaped curve; (**c**) Elliptical-shaped curve. According to the above content, this paper considers the energy dissipation and sealing performance of the premium connection under vibration conditions, establishes a finite element model of the premium connection with a cone-to-cone surface, and carries out a modal analysis experiment on the premium connection to obtain the modal vibration mode and natural frequency. The accuracy of the finite element analysis results is verified by experiments.