# Modeling and Numerical Computation of the Longitudinal Non-Linear Dynamics of High-Speed Elevators
## 2 Elevator System Dynamics Modeling
### Linear Subsystem  
Apart from the wire rope, all subsystems in the elevator system are linear. [PERSON]'s law and [PERSON]'s equation are applied to model the centralized mass of each subsystem. The traction and tension subsystems in the elevator are complex, requiring the use of Lagrange equations to model their dynamics. Figure 4 illustrates the simplification of the elevator traction subsystem's vibration isolation rubber to a spring damping system. Additionally, the mass of other components is concentrated to create a three-degree-of-freedom dynamics model for the traction subsystem. In Figure 4, \(x_{1},x_{2},x_{3}\) are the three degrees of freedom of the traction subsystem, \(x_{1}\) is the centralized mass displacement of the tractor, \(x_{2}\) is the displacement of the contact point of the traction sheave with the rope on the car side, and \(x_{3}\) is the displacement of the contact point of the traction sheave with the rope on the counterweight side, and is specified in a generalized coordinate system, with the car moving upwards in the positive direction. The kinetic energy of the traction subsystem can be expressed as:  
\[T=\frac{1}{2}m_{1}\dot{x_{1}}^{2}+\frac{1}{2}m_{2}\bigg{(}\frac{\dot{x}_{2}- \dot{x}_{3}}{2}\bigg{)}^{2}+\frac{1}{2}I_{1}\bigg{(}\frac{\dot{x}_{2}+\dot{x}_ {3}}{2r_{1}}\bigg{)}^{2} \tag{9}\]  
The elastic potential energy of the traction subsystem can be expressed as:  
\[V=\frac{1}{2}k_{1}x_{1}^{2}+\frac{1}{2}k_{2}\bigg{(}\frac{x_{2}-x_{3}}{2}-x_{ 1}\bigg{)}^{2} \tag{10}\]  
The dissipated energy of the traction subsystem can be expressed as:  
\[W=\frac{1}{2}c_{1}\dot{x_{1}}^{2}+\frac{1}{2}c_{2}\bigg{(}\frac{\dot{x}_{2}- \dot{x}_{3}}{2}-\dot{x}_{1}\bigg{)}^{2} \tag{11}\]  
Substituting the above energy equation into the Lagrange equation:  
\[\frac{d}{dx}\bigg{(}\frac{\partial T}{\partial\dot{x}}\bigg{)}-\frac{\partial T }{\partial x}+\frac{\partial V}{\partial x}+\frac{\partial W}{\partial\dot{x} }=F \tag{12}\]  
Solving this yields the differential equation of motion for the traction subsystem:  
\[M\ddot{x}+C\dot{x}+Kx=F \tag{13}\]  
Figure 4: Structure of traction and tensioning sheaves. (**a**) Traction subsystem; (**b**) tension subsystem. where the contributing mass matrix of the trailing subsystem is:  
\[[M_{1}]=\frac{1}{4}\left(m_{2}+\frac{I_{1}}{r_{1}^{2}}\right) \begin{pmatrix}1&0&0 \\ 0&1&0\\ 0&0&1\end{pmatrix} +\frac{1}{4}\left(-m_{2}+\frac{I_{1}}{r_{1}^{2}}\right) \begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&0\end{pmatrix} +m_{1} \begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&1\end{pmatrix}  \tag{14}\]  
The contributing stiffness matrix is:  
\[[K_{1}]=\frac{1}{4}k_{2} \begin{pmatrix}1&-1&-2\\ -1&1&2\\ -2&2&4\end{pmatrix} +k_{2} \begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&1\end{pmatrix}  \tag{15}\]  
The contribution damping matrix is:  
\[[C_{1}]=\frac{1}{4}c_{2} \begin{pmatrix}1&-1&-2\\ -1&1&2\\ -2&2&4\end{pmatrix} +c_{1} \begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&1\end{pmatrix}  \tag{16}\]  
In the above formula, \(m_{1}\) is the sum of the mass of the motor and the traction wheel support. \(m_{2}\) is the equivalent mass of the traction wheel and the guide wheel. \(k_{1},k_{2}\) and \(c_{1},c_{2}\) are the stiffness and damping of the vibration isolation rubber on the upper and lower part of the traction wheel support, respectively. \(I_{1}\) is the equivalent rotational moment of the traction wheel and \(r_{1}\) is the radius of the traction wheel. The tension subsystem functions similarly to the traction subsystem. It counterbalances the dynamic imbalance arising from elevator operation, connected with the compensation rope and located beneath the car and counterweight. Unlike the traction sheave, the tension sheave lacks vibration isolation rubber, with damping assumed to be 0. The same Lagrange equation can be used to obtain the contributing mass matrix \([M_{2}]\), the contributing stiffness matrix \([K_{2}]\), and the contributing damping matrix \([C_{2}]\) of the tension subsystem.