# Modeling and Numerical Computation of the Longitudinal Non-Linear Dynamics of High-Speed Elevators
## 2 Elevator System Dynamics Modeling
### Non-Linear Subsystems  
In traditional modeling, the elevator wire rope is typically considered as a fixed member with constant stiffness and damping arrays. However, in this paper, we treat the wire rope as a non-linear model. Both its stiffness and damping matrices are functions of time, and they are updated dynamically to reflect changes over time. The elevator system has four sections of wire rope, each of which constitutes a non-linear subsystem, namely, the car side traction rope subsystem \((L_{1})\), the counterweight traction rope subsystem \((L_{2})\), the car side tensioning rope subsystem \((L_{3})\), and the counterweight side tensioning rope subsystem \((L_{4})\). As shown in Figure 3, each section of the rope is divided into a number of units, each of which corresponds to a small unit consisting of a mass, a spring, and damping. The vertical displacement of nodes serves as the degrees of freedom in the discretized model of the wire rope, where neighboring units are connected by nodes. Generally, increasing the number of nodes enhances the accuracy of the dynamics model, albeit with a corresponding increase in computational workload. In this paper, we take \(n=5\). The wire rope in the elevator structure is divided into five small units in each of its four sections. The mass and stiffness of these units are non-linear and vary based on the elevator's position during operation. The mass of the small unit of the four sections of wire rope can be expressed as:  
\[ \begin{array}{l}m_{L_{11}}=m_{L_{12}}=\ldots=m_{L_{15}}=N_{1}\rho_{1}l_{11}/ 5\\ m_{L_{21}}=m_{L_{22}}=\ldots=m_{L_{25}}=N_{1}\rho_{1}(l_{1}-l_{1})/5\\ m_{L_{31}}=m_{L_{32}}=\ldots=m_{L_{35}}=N_{2}\rho_{2}l_{2}/5\\ m_{L_{41}}=m_{L_{42}}=\ldots=m_{L_{45}}=N_{2}\rho_{2}(l_{2}-l_{22})/5\end{array}  \tag{2}\]  
where \(m_{L_{11}},m_{L_{12}},m_{L_{13}},m_{L_{44}},m_{L_{15}}\) is the mass of the five small units of \(L_{1}\), and the other three ropes are the same. \(N_{1}\), \(N_{2}\) is the number of traction wire ropes and tensioning wire ropes, \(\rho_{1},\rho_{2}\) is the wire density of the hauling and tensioning ropes, \(l_{1},l_{2}\) is the total length of the traction wire rope and tensioning wire rope, and \(l_{11},l_{2}\) is the real-time length of the car side traction wire rope and car side tensioning wire rope. It can be expressed as:  
\[ \begin{array}{l}l_{t1}=h_{1}-h_{2}-h_{t}+h_{3}\\ l_{t2}=h_{t}+h_{4}\end{array}  \tag{3}\]  
Figure 3: Discrete model of a wire rope. where \(h_{1}\) is the height of the shaft, \(h_{2}\) is the height of the car frame, \(h_{3}\) is the height of the top floor, \(h_{4}\) is the height of the pit, and \(h_{t}\) is the height of the car after the car starts running from the lowest floor and is a function of time \(t\). At this point, the total contributing mass matrix of the four wire rope subsystem models can be expressed as:  
\[[M_{L}]= \begin{pmatrix}m_{L_{11}}&&&\\ &m_{L_{12}}&&\\ &&\ddots&&\\ &&&m_{L_{44}}&\\ &&&m_{L_{45}}\end{pmatrix}  \tag{4}\]  
where \(m_{L_{11}},m_{L_{12}},\cdots,m_{L_{44}},m_{L_{45}}\) are all functions of time \(t\), so \([M_{L}]\) will change with time. In this paper, we use matrix updating to express the time-varying nature of the mass matrix, and its expression is:  
\[= \begin{pmatrix}m_{L_{11}(t)}&&&\begin{bmatrix}M_{L(t+1)}\\ &m_{L_{12}(t)}\\ &&\ddots\\ &&&m_{L_{44}(t)}\end{bmatrix}+\begin{pmatrix}m_{L_{11}(t+1)}&&&\\ &m_{L_{12}(t+1)}&&\\ &&\ddots\\ &&&m_{L_{44}(t+1)}\end{pmatrix}\end{pmatrix}  \tag{5}\]  
Similarly, the stiffness of the four sections of the wire rope is also affected by the length of the wire rope; the effective length of the wire rope changes and the stiffness also changes in real time, as expressed by the formula:  
\[ \begin{array}{l}k_{L_{11}}=k_{L_{12}}=\ldots=k_{L_{15}}=EA_{1}/l_{t1}\\ k_{L_{21}}=k_{L_{22}}=\ldots=k_{L_{25}}=EA_{1}/(l_{1}-l_{t1})\\ k_{L_{31}}=k_{L_{32}}=\ldots=k_{L_{35}}=EA_{2}/l_{t2}\\ k_{L_{41}}=k_{L_{42}}=\ldots=k_{L_{45}}=EA_{2}/(l_{2}-l_{t2})\end{array}