# Modeling and Numerical Computation of the Longitudinal Non-Linear Dynamics of High-Speed Elevators
## 3 Time Domain Calculation Analysis  
General elevator dynamics calculations often overlook the non-linear effects induced by the rigid dead rope components or linearize the non-linear aspects. Employing instantaneous structural assumptions, the entire system movement is segmented into n time intervals. In each interval, the dynamic system characteristics remain invariant, and the variable coefficient differential equations are transformed into constant coefficient ones for elevator calculation. Although this approach allows for an analysis of elevator dynamic characteristics, it neglects the influence of vibrations on the overall elevator motion, as well as the impact of non-linear components such as the wire rope. This study utilizes the fourth-order Runge-Kutta method with a variable step size for time-domain step-by-step integration of the elevator systems' differential equations with variable coefficients. Utilizing a Fortran algorithm, numerical calculations are conducted. Considering the elevator's overall operational pattern and corresponding vibration effects, time-domain step-by-step integration offers a superior stability compared to division into multiple time intervals, concurrently enhancing the computational accuracy. The system \(x\) is the displacement of each degree of freedom, \(\dot{x}\) is the velocity of each degree of freedom, and \(\ddot{x}\) is the acceleration of each degree of freedom. The elevator dynamics equation is:  
\[M\ddot{x}(t)+C\dot{x}(t)+Kx(t)=F \tag{21}\]  
where \(M\) is the equivalent mass matrix of the elevator system model, \(C\) is the equivalent damping matrix of the elevator system model, \(K\) is the equivalent stiffness matrix of the elevator system model, and \(F\) is the external excitation of the elevator system related to the control strategy. The generalized acceleration vector of the elevator system can be expressed as follows:  
\[\ddot{x}(t)=M^{-1}\big{(}F-C\dot{x}(t)-Kx(t)\big{)} \tag{22}\]  
Considering the elevator degree-of-freedom displacement \(x\) and the elevator degree-of-freedom velocity \(\dot{x}\) as unknown, define a new vector \(X(t)= \begin{bmatrix}x(t)\\ \dot{x}(t)\end{bmatrix} \). Then,  
\[\dot{X}(t)= \begin{bmatrix}\dot{x}(t)\\ \dot{x}(t)\end{bmatrix} = \begin{bmatrix}\dot{x}\\ M^{-1}\big{(}F-C\dot{x}-Kx\big{)}\end{bmatrix}  \tag{23}\]  
Then,  
\[\dot{X}(t)= \begin{bmatrix}0&I\\ -M^{-1}K&-M^{-1}C\end{bmatrix}\begin{bmatrix}x(t)\\ \dot{x}(t)\end{bmatrix} + \begin{bmatrix}0\\ M^{-1}F\end{bmatrix} =AX(t)+F(t) \tag{24}\]  
where  
\[A= \begin{bmatrix}0&I\\ -M^{-1}K&-M^{-1}C\end{bmatrix} ,F(t)= \begin{bmatrix}0\\ M^{-1}F\end{bmatrix}  \tag{25}\]  
Based on the variable-step-length fourth-order Runge-Kutta method, this can be derived as:  
\[X_{i+1}=X_{i}+\frac{1}{6}(K_{1}+2K_{2}+2K_{3}+K_{4}) \tag{26}\]  
Included among these are:  
\[ \begin{split} K_{1}&=f(X_{i},t_{i})\\ K_{2}&=f\Big{(}X_{i}+\frac{h}{2}K_{1},t_{i}+\frac{h}{2}\Big{)}\\ K_{3}&=f\Big{(}X_{i}+\frac{h}{2}K_{2},t_{i}+\frac{h}{2}\Big{)}\\ K_{4}&=f(X_{i}+2 hK_{3},t_{i+1}+h)\end{split}  \tag{27}\]  
The computational flow is shown in Figure 6:  
Figure 6: Flowchart of time domain computation. The elevator control strategy was established with a calculation step of 0.001 s, inputting the elevator's basic parameters and utilizing the dynamics model and numerical algorithm provided in this paper to compute the elevator's displacement and acceleration. The computed values were compared with experimental data obtained from a company. The basic parameters of the high-speed elevator are detailed in Table 1, and the outcomes are depicted in Figures 7 and 8. In this study, an elevator-specific vibration analyzer was employed to gather vibration signals from the experimental tower of a company's elevator. Data were collected using a three-axis acceleration sensor, transmitted to the terminal for processing, and output to obtain the experimental results. Figure 7 demonstrates that the calculated results closely align with the displacement of the experimental test results.