## 2 High-Speed Elevator Car Vibration Model and Sky-Hook Damping Control Strategy  
The guiding system of a high-speed elevator is composed of guide shoes and guide rails, and the guide shoe structure has a great impact on the lateral vibration of the elevator [18]. Compared with traditional guide shoes, electromagnetic active sliding guide shoes, and electromagnetic active rolling guide shoes have better vibration reduction performance. The electromagnetic active sliding guide shoes form a gap between the guide shoes and the guide rails through the electromagnetic force. Thus, the car is suspended on the guide rails so as to effectively reduce the lateral vibration of the high-speed elevator. However, it requires high control technology and reliability [19]. Therefore, it is not widely used in practice. Electromagnetic active rolling guide shoes use an electromagnet to exert active control force on the basis of rolling guide shoes. Its structural diagram is shown in Figure 1. Since this structure has little change to the original high-speed elevator, it owns a wider range of applications. The introduction of electromagnetic active rolling guide shoes in the vibration control of high-speed elevators aims to convert the damping related to car vibration into an adjustable one. Figure 1: Structure diagram of electromagnetic active rolling guide shoes. In order to effectively reduce the vibration of the elevator car, the electromagnetic active rolling guide shoe structure is introduced. Inspired by the sky-hook damping control strategy in high-speed train cabin vibration control, the strategy is also introduced into the vibration control of high-speed elevators. The sky-hook damping control method compensates for the shortcomings of traditional dampers in that the damping force generated by the damper increases the vibration of the car when the car's motion speed and relative speed (between the car and the frame) are opposite. The strategy of sky-hook damping control is to make the damping force of the damper zero or reduce it to a minimum when the car's motion speed and the relative speed between the car and the frame are different, thereby reducing the vibration of the car in the above-mentioned situation. The calculation for the sky-hook damping control is shown as Formula (1). \[C=\left\{ \begin{aligned} & C_{S}\left|\frac{\dot{x}_{c}}{\left( \frac{\dot{x}_{c}}{x_{c}-\dot{x}_{w}}\right)}\right|,\quad\dot{x}_{c}\left( \dot{x}_{c}-\dot{x}_{w}\right)\geq 0\\ & 0,\qquad\qquad\qquad\qquad\qquad\text{else}\end{aligned} \right. \tag{1}\]  
where \(\dot{x}_{c}\) and \(\dot{x}_{w}\) respectively denote the horizontal speed and frame of the elevator car, and \(C_{s}\) represents the damping coefficient of transverse shock absorber. In practice, the relative speed between the car and the car frame may be zero or a minimum. According to Formula (1), in such cases, the value of \(C\) will tends to infinity, exceeding the maximum value of the actual adjustable damper and damping coefficient. Since the damping force that can be generated by the actual shock absorber has a range, it can only approximately achieve some characteristics of the ideal sky-hook damper. Therefore, the values of \(C_{\text{min}}\) and \(C_{\text{max}}\) are set in this case. The actual application of sky-hook damping control in lateral vibration control of high-speed elevators is represented by Formula (2). \[C=\left\{ \begin{aligned} & C_{\text{max}}& C_{S}\frac{\dot{x}_{c}}{ \left(\dot{x}_{c}-\dot{x}_{w}\right)}\geq C_{\text{max}}\\ & C_{S}\frac{\dot{x}_{c}}{\left(x_{c}-\dot{x}_{w}\right)}& C_{\text{min}}<\frac{\dot{x}_{c}}{\left(x_{c}-\dot{x}_{w} \right)}<C_{\text{max}}\\ & C_{\text{min}}&\frac{\dot{x}_{c}}{\left(x_{c}-x_{w} \right)}\leq C_{\text{min}}\end{aligned} \right. \tag{2}\]  
The high-speed elevator car system consists of three parts, namely, the car frame, the car, and the vibration-damping rubber block at the bottom.