## 1 Introduction
[PERSON] et al. [5] optimized the parameters of a guide shoe to suppress car vibration by studying the relationship between the deformation amount of the guide shoe and the restoring force. [PERSON] et al. [6] studied the mechanical model of the elevator guiding system and obtained the relationship between the load and deformation of the guide shoe. In elevator hoisting systems, the investigation into vibration-damping mechanisms primarily centers on hydraulic floating sky wheels, electromagnetic active guide shoes, hydraulic active guide shoes, servomotor active guide shoes, and magnetic levitation guide shoes [7]. The roller cage shoe includes an active guide shoe and a passive guide shoe. Passive roller cage shoes primarily rely on a spring damping system to gradually diminish car vibrations. While effective to an extent, they have limitations in swiftly suppressing vibrations and enhancing the comfort of the lift ride [8]. On the other hand, active roller cage shoes designed for vibration control are intricate, expensive, and demand a continuous supply of external energy. In comparison, semi-active control systems offer damping effects akin to active control but can modify their damping characteristics with minimal energy input, resulting in lower costs and enhanced stability. Among semi-active control devices, MRDs stand out as being highly promising due to their outstanding attributes. The magnetorheological damper (MRD) utilizes magnetorheological fluid (MRF) to absorb shocks. By adjusting the current in the excitation coil, different magnetic field forces are achieved. This causes the fluid's rheological properties to change, generating resistance to movement and effectively damping vibrations [9]. As a highly effective semi-active damper, it can be extensively used in transportation, structural seismic applications, and other fields. [PERSON] et al. [10] explored the feasibility of employing magnetorheological group dampers in an airplane landing-gear system to diminish unintended shocks and vibrations during the landing and taxiing phases. [PERSON] et al. [11] conducted an experimental validation of the seismic performance of an elevator equipped with MRF dampers during an earthquake. This system's effectiveness was verified through numerical simulations in elevator applications. The MRD principle is also widely utilized for vibration damping in elevator hoisting systems. [PERSON] et al. [12] developed an elevator MRD based on a permanent magnetic field, investigating factors like the magnetic field strength, distribution, descent speed, deceleration time, and distance. The results demonstrated that the eddy currents induced by the permanent magnetic field effectively achieved elevator deceleration. [PERSON] et al. [13] introduced a variable-domain fuzzy control method for semi-active roller cage shoes equipped with magnetorheological dampers; the elevator experiments conclusively demonstrated that this controller proficiently mitigates horizontal vibrations. In summary, the controller proves highly effective in suppressing horizontal vibrations, markedly enhancing the ride experience in high-speed elevators. [PERSON] et al. [14] and colleagues investigated the horizontal response of vertical transportation systems with nonlinearities under guideway deformation excitation. They put forward a strategy for controlling the dynamic horizontal behavior of a three-degree-of-freedom model using a controllable MRD, which demonstrated an effective reduction in vertical transport vibrations. In the mine hoisting system, [PERSON] et al. [15] introduced a novel type of MRD vibrating screen damper; experimental evaluations demonstrated a superior vibration-dampingperformance. Additionally, [PERSON] et al. [16] proposed a tension equalizer based on MRDs, and its potential for vibration reduction was confirmed through numerical simulations. The literature above highlights a predominant focus on analyzing the vibration characteristics and control strategies in mine hoisting.