The performance of rotary control head or rotary blowout preventer will directly affect the success or failure of the underbalanced construction of oil and gas well. Once the failure occurs in the construction process, there may be blowout and other big drilling accidents. Based on the Yeoh constitutive equation, the dynamic sealing model of rubber core is established in this paper, and the sealing process of rubber core during tripping and tripping is calculated and analyzed. Then, the effect of the bubble defect in the core on the sealing performance of the core under different drilling fluid pressure was analyzed. Finally, the effect of metal degumming on the sealing performance of the core was analyzed. The results showed that the larger the initial sealing pressure and contact stress value of the Mises core was, the better the sealing effect of the Mises core was, while the larger the Mises stress value was, the more likely the Mises core was to suffer fatigue failure. The drilling fluid pressure significantly affects the maximum equivalent stress of the core, and the maximum equivalent stress appears in the bubble position inside the core at different times and under different drilling fluid pressure. The degumming position is placed in the contact position between metal and rubber core. When the tensile stress exceeds the limit of rubber core material, radial and circumferential cracks will occur on the upper surface of rubber core. With the increase of drilling times, rubber is prone to fatigue cracks, which eventually lead to mutual tearing.