When the carrier-based aircraft is subjected to impact excitation during landing, the vibration of the engine rotor system is intensified, and the conventional squeeze film damper may fail to reduce vibration. To improve the impact resistance of the damper, the staggered double elastic ring squeeze film damper (SDERSFD) is proposed in this paper. The design of the double elastic ring structure enables the damper to maintain good stability when it is impacted. Based on the finite element method, the dynamic model of the rotor-support system considering the impact load excitation is established, and the influence of the impact speed on the transient response of the rotor system is analyzed. The results show that the load impact at different speeds will arouse the critical speed of the rotor system, and lead to the change of the resonance point of the rotor system and the sudden increase of the rotor vibration. The SDERSFD can reduce the vibration amplitude of the rotor system, and the maximum vibration reduction ratio can reach 67.7%, which can realize the effective load reduction of the impact load and improve the stability of the rotor system under the impact condition.