# Modeling and Numerical Computation of the Longitudinal Non-Linear Dynamics of High-Speed Elevators
## 1 Introduction  
The increase in high-rise buildings and the adoption of high-speed and ultra-high-speed elevators have heightened the awareness of elevator system vibrations. When the running speed reaches or exceeds 6 m/s, it is considered a high-speed elevator. Components of high-speed elevators, such as the mass of the car and counterweight and wire rope length, contribute to the low natural frequency and dense frequency distribution of the elevator system. Meanwhile, as the elevator speed increases, the excitation frequency from components like the traction system, guide rail, and wire rope also rises. These factors exacerbate elevator system vibration issues, making resonance more likely to occur if not addressed properly. This can lead to vibration noise that exceeds standards and increased component loads, impacting the component lifespan as well as elevator comfort and safety [1]. A typical high-speed elevator system comprises numerous components, including the car, car frame, rope head spring, wire rope, traction motor, motor double-layer vibration isolation, counterweight, tension sheave, etc. These components exhibit diverse mechanical properties, and most of the stiffness components can generate and transmit vibrations. During elevator operation, the length of the wire rope undergoes continuous changes, rendering the elevator system a complex non-linear system, thereby complicating dynamics modeling and numerical calculations. Currently, some scholars address the elevator vibration issue through dynamic modeling. [PERSON] enhanced the discrete model of an elevator lift system by centralizing the parameters, creating a 27-degree-of-freedom model, and validated itsalignment with an actual system [2]. [PERSON], considering the discrete properties of the steel wire rope and the time-varying aspects of the elevator system, incorporated elements like vibration-damping rubber sheets and tensioning systems. They developed a 19-degree-of-freedom vertical vibration model with a 1:1 traction ratio and verified its accuracy [3]. [PERSON] et al. discretized the wire rope into multiple spring-damped systems and developed a dynamic model for a three-degree-of-freedom traction system. This model was utilized to investigate the dynamic behavior of the traction system under normal operation [4]. [PERSON] conducted vibration characterization experiments using an elevator with a traction ratio of 1:1, placing sensors on the car frame, car, and traction rope to test the vibration of the elevator during ascent and descent. It was concluded that the longitudinal vibrations of the car frame are low-frequency vibrations [5]. [PERSON] developed a seven-degree-of-freedom dynamics model for an elevator, considering the time-varying stiffness of the wire rope and exploring the vibration characteristics of the elevator car under various starting and braking speeds and different loads [6]. [PERSON] simplified the elevator traction system to a damped vibration system, validated its damping characteristics, and conducted experiments on an operational elevator to verify the method's accuracy [7]. [PERSON] proposed a time-varying wire rope modeling approach with non-complete boundary conditions and demonstrated its viability through testing on a mine hoist in Serbia [8]. [PERSON] utilized the finite volume method and [PERSON]'s theorem to accurately investigate the vibration response of a high-speed elevator and the response differences during different operating processes. The feasibility of the model and modeling method was verified, and the responses during different operation processes were obtained and analyzed [9]. [PERSON] et al. developed a nine-degree-of-freedom model for vertical elevator dynamics. They numerically simulated the elevator dynamics at three positions--low, middle, and top floors--under both no-load and full-load conditions. The findings indicate that the vertical dynamic behavior of high-speed elevators varies over time [10].