# Modeling and Numerical Computation of the Longitudinal Non-Linear Dynamics of High-Speed Elevators
## 5 Conclusions  
This paper employs a combined approach of dynamic theoretical modeling and experimental control to develop a high-speed elevator model. The accuracy of this model is verified, serving as the foundation for investigating the longitudinal time-domain response of a high-speed elevator. The following conclusions are drawn from this study:  
1. This paper introduces a novel approach to analyze elevator system dynamics by dividing it into eight subsystems, developing a dynamics model for each and then integrating them based on shared degrees of freedom, providing an efficient and adaptable methodology. The time-domain calculation of the model employs stepwise integration, utilizing the variable-step fourth-order Runge-Kutta method. Comparison of the calculation results with experimental data confirms the feasibility of the model for time-domain analysis. 2. This paper introduces a novel computational approach for handling the non-linear aspects of elevator dynamics. The linear and non-linear systems are separated during modeling, updating only the non-linear part in the numerical solution. This method circumvents the need for re-establishing the dynamics model, streamlining the computational process. 3. With varying car heights, the length of each wire rope section fluctuates, consequently significantly altering the natural frequency of the elevator system. Specifically, when the car is positioned centrally, the first-order natural frequency of the elevator increases, leading to a reduction in its vibration response. 4. Among the four common elevator control strategies, the triangle and trapezoidal control strategies result in significant elevator vibration response due to their abrupt initial and braking acceleration, rendering them unsuitable for high-speed elevators. Conversely, the parabolic and parabolic-linear control strategies exhibit minimal vibration responses, accompanied by gentle acceleration and deceleration. Hence, the parabolic-linear control strategy emerges as the preferable operational law for high-speed elevators. 5. The load variation significantly impacts the elevator's time-domain response, notably evident during the uniform speed phase. The elevator's vibration response increases and the natural frequency diminishes with an increasing load during operation at a uniform speed. Conversely, the vibration response remains similar across different loads during the acceleration phase. In the deceleration phase, no load results in the lowest response fluctuations, while a half load exhibits the most significant fluctuations. 6. The elevator's vibration response is influenced by the changing mechanical properties of the wire rope at different car heights. During acceleration, the highest-level car exhibits the largest vibration response, while those at middle and lowest levels are similar. In the uniform speed phase, the highest-level car experiences larger vibration responses compared to those at the middle and lowest levels, with the middle-level car exhibiting the smallest response. Similarly, during deceleration, the highest-level car displays a greater vibration response than the middle- and lowest-level cars. In conclusion, this paper establishes a theoretical foundation for computing the time-domain response of high-speed elevators and offers an efficient research approach for investigating the parameters that influence the responses in other high-speed elevators. Given the complexity of high-speed elevators, involving numerous parts and parameters, further studies on parameter optimization are warranted. Achieving the optimal elevator configuration and effective vibration damping necessitates continued research utilizing the model presented herein. Conceptualization, H.H. and Z.T. ; methodology, H.H. ; software, H.H. and Z.T. ; validation, H.H. and Y.Z. ; formal analysis, H.H. and Z.T. ; investigation, Y.Z. ; resources, Z.T. ; data curation, H.H. ; writing--original draft preparation, H.H. ; writing--review and editing, H.H. and Z.T. ; supervision, Z.T. ; project administration, Z.T.