## 1 Introduction  
With the vigorous development of China's economy and the dynamic implementation of urbanization, high-rise, and super-high-rise buildings are growing up. According to the statistics of the Council on Tall Buildings and Urban Habitat (CTBUH), by the end of 2022, there are 11 of the world's top 20 skyscarepers in China, and Shanghai Tower, with a total height of 632 m, ranks the second in the list [1]. In high-rise and ultra-high-rise buildings, conventional elevators often operate at slow speeds, resulting in low transportation efficiency. This has prompted elevators to evolve for higher speeds and greater comfort. At present, the high-speed elevators used in Shanghai Tower have reached 18 m/s. The increase in speed of these elevators meets the vertical transportation needs of high-rise buildings and improves traffic efficiency. However, along with the speed upgrade, there are some negative effects. For example, high-speed elevators experience greater wear and tear on their components during operation, which requires higher wear resistance and strength of structural materials. The higher running speed of the elevators results in more noticeable vibrations and makes them more susceptible to external stimuli, leading to abnormal vibrations that affect ride comfort [2,3]. Previous research has shown that slight vibration generally will not affect the safety and health of passengers. However, when the amplitude reaches a certain value, it will trigger the resonance phenomenon of the organs of passengers, which results in irreversible damage to their health [4]. Additionally, the human body has a sensitive range for cabin vibrations, which is 4-8 Hz vertically and 1-2 Hz horizontally with a higher sensitivity [5]. Vibration control, as one of the core technologies of high-speed elevators, directly influences the comfort of elevator rides and is also an important criterion for evaluating the quality of high-speed elevators. With the large scale of existing high-speed elevators and demanding requirements for safety and comfort, effective control of lateral vibrations in elevator cars holds great engineering significance for improving the quality of elevator rides. The lateral vibration control of the car can be divided into three main methods: passive control, active control, and hybrid control. Among them, passive control consumes the least energy but can only target vibrations within a narrow frequency band, making it less suitable for dynamic high-speed elevator parameters [6]. On the contrary, active control provides the best results but consumes the most energy. Hybrid control considers the coupling relationships between high-speed elevator structural parameters and active control, achieving a balance between power consumption and vibration reduction effectiveness. Presently, there are few studies on the lateral vibration of high-speed elevator cars. [PERSON] [7] et al. designed a new type of damping element consisting of two rotating motors and an eccentric disc. Experimental results show that the dampers can reduce lateral vibrations, but their actual effect depends on the accuracy of the manufacture and installation of the components. [PERSON] [8] et al. established a model for the horizontal vibration dynamics of high-speed elevator cars by considering the relationship between lateral forces and overturning moments and horizontal displacement, deflection angle displacement, and rated speed. [PERSON] [9] et al. analyzed the vibration spectrum of the guide rail, which indicated that the disturbance of the rail mainly concentrated in the low-frequency areas, and there was a main disturbance frequency which was directly proportional to the running speed of the elevator and inversely proportional to the guide-rail length. [PERSON] [10] et al.