## 2 High-Speed Elevator Car Vibration Model and Sky-Hook Damping Control Strategy
In order to establish a mathematical model for the lateral vibration of the elevator car in a high-speed elevator, the vibration-damping rubber blocks in the car system and the rolling guide shoes in the guiding system are simplified as a spring-damping system, while the other parts can be considered as rigid bodies. By simplification, a schematic diagram of the simplified structure of the elevator car and guiding system in a high-speed elevator can be obtained, as shown in Figure 2.  
According to the theory of dynamics, a mathematical model for lateral vibration of a high-speed elevator with four degrees of freedom, considering the separation of the car frame and the car, and the upper and lower guide shoes, is established. Formulas (3)-(5) correspond to the translational, pitch, and roll motion modes shown in Figure 2.  
\[x_{1}=x_{2}=x_{3}=x_{4} \tag{3}\]  
\[\left(x_{1}=x_{2}\right)\
eq\left(x_{3}=x_{4}\right) \tag{4}\]  
\[\left(x_{1}=x_{3}\right)\
eq\left(x_{2}=x_{4}\right) \tag{5}\]  
where \(x_{1},x_{2},x_{3}\) and \(x_{4}\) represent the horizontal displacements of the four guide shoe wheels. The lateral vibration simulation model of a high-speed elevator car can be obtained by synthetically establishing the equations of the three motion modes. Different motion modes involve different motion equations as follows [20]:  
1. Translational motion In the case of translational motion, the motion equation of the high-speed elevator car frame is shown in Formula (6): \[m_{w}\cdot\ddot{x}_{w}=-c\sum\limits_{n=1}^{4}\left(\dot{x}_{w}-\dot{x}_{n} \right)-k\sum\limits_{n=1}^{4}\left(x_{w}-x_{n}\right)-c_{c}\left(\dot{x}_{w}- \dot{x}_{c}\right)-k_{c}\left(x_{w}-x_{c}\right)\] (6) The car motion equation of a high-speed elevator is shown in Formula (7): \[m_{c}\cdot\ddot{x}_{c}=-c_{c}\left(\dot{x}_{c}-\dot{x}_{w}\right)-k_{c}\left(x_ {c}-x_{w}\right)\] (7) where \(m_{w}\) is the mass of the car frame, \(x_{w}\) is the displacement of the mass center of the car frame, \(m_{c}\) is the mass of the car, \(x_{c}\) is the displacement of the car, \(k_{c}\) is the lateral stiffness coefficient of the vibration-damping rubber block at the car bottom, \(c_{c}\) is the damping coefficient of the vibration damping rubber block, \(k\) is the stiffness coefficient of the rolling guide shoes, and \(c\) is the damping coefficient of the rolling guide shoes. 2. Pitching motion When pitching, the motion equation of the high-speed elevator car frame is drawn in Formula (8). \[ \begin{array}{ll}I_{w}\cdot\ddot{\varphi}_{w}&=-c\sum\limits_{n=1}^{2}L_{1} \cdot\left(\dot{\varphi}_{w}\cdot L_{1}-\dot{x}_{n}\right)-k\sum\limits_{n=1}^ {2}L_{1}\cdot\left(\varphi_{w}\cdot L_{1}-x_{n}\right)\\ &-c\sum\limits_{n=1}^{2}L_{2}\cdot\left(\dot{\varphi}_{w}\cdot L_{2}-\dot{x}_{ n+2}\right)-k\sum\limits_{n=1}^{2}L_{2}\cdot\left(\varphi_{w}\cdot L_{2}-x_{n+2} \right)\\ &-c_{c}L_{5}\left(\dot{\varphi}_{w}\cdot L_{5}-\dot{\varphi}_{c}\cdot L_{5} \right)-k_{c}L_{5}\left(\varphi_{w}\cdot L_{5}-\varphi_{c}\cdot L_{5}\right) \end{array} \] (8) The motion equation of a high-speed elevator car is shown in Formulas (9)-(11): \[I_{c}\cdot\ddot{\varphi}_{c}=-c_{c}\cdot L_{5}\left(\dot{\varphi}_{c}\cdot L_ {5}-\dot{\varphi}_{w}\cdot L_{5}\right)-k_{c}\cdot L_{5}\left(\varphi_{c} \cdot L_{5}-\varphi_{w}\cdot L_{5}\right)\] (9)  
Figure 2: Structure diagram of the high-speed elevator car and guide system (**a**) Translational motion (**b**) Pitching motion (**c**) Roll motion. \[I_{w}=\frac{2}{3}m_{w}\cdot L_{1}^{2} \tag{10}\]  
\[I_{c}=\frac{2}{3}m_{c}\cdot L_{5}^{2} \tag{11}\]  
where \(L_{1}\) to \(L_{5}\) are the dimensions of the high-speed elevator car, \(I_{c}\) is the car rotational inertia under the mode, \(I_{w}\) is the car frame rotational inertia under the mode, \(\varphi_{c}\) is the car angle, and \(\varphi_{w}\) is the car frame angle. (c) Roll motion  
When doing Roll motion, the motion equation of the high-speed elevator car frame is drawn in Formula (12):  
\[