# Modeling and Numerical Computation of the Longitudinal Non-Linear Dynamics of High-Speed Elevators
## 1 Introduction
[PERSON] established four kinds of external excitation models, i.e., sinusoidal excitation, triangular excitation, step excitation, and impulse excitation. The factors affecting the vibration response were analyzed by solving the vibration acceleration of the coupled system [11]. [PERSON] investigated the application of dynamic iterations to non-linear differential equations using implicit time integration. It was shown that dynamic iteration affects the rate of convergence of the solution of a system of non-linear differential equations [12]. [PERSON] considered the time-varying parameters of an elevator system and established a model for the KLK2 high-speed elevator using Gaussian exact integration. Vibration acceleration was obtained in three directions through simulations. The proposed EVM method was compared with traditional methods [13]. [PERSON] established a four-degree-of-freedom model to study the vibration response under guide rail deformation excitation, and introduced a non-linear energy absorber to effectively reduce elevator car vibrations [14]. [PERSON] applied the fourth-order Runge-Kutta method to handle higher-order non-linear differential equations. Additionally, genetic algorithms were introduced to investigate the dynamic response of the formation relaxation sieve across multiple degrees of freedom [15]. [PERSON] proposed an improved elevator fault improvement method, which can be processed via multidomain extraction of features, and time-domain feature extraction of elevator vibration signals [16]. [PERSON] employed [PERSON]'s principle to process elevator vibration signals, expressing the dynamic response of multiple ropes under boundary perturbation and multiple weight constraints with hybrid partial differential equations and ordinary differential equations. The equations were solved using the finite difference method, and the validity of the hybrid model was verified through simulation in elevator response analysis [17]. [PERSON] investigated how different speed profiles affect the elevator operation response and concluded that determining the elevator operation speed profile is crucial when studying elevator dynamic responses [18]. [PERSON] took a single winding elevator as the object, established an Adams elevator operation model, and designed the possible traction machine micro rotation imbalance disturbance and guide rail disturbance during the operation of the elevator system according to the actual working conditions [19]. In the investigation of elevator emergency braking, [PERSON] examined how the load affects the effective braking distance and descending speed during emergency braking and provided calculation results [20]. [PERSON] applied time-varying vibration analysis theory to investigate an elevator hoisting system, determining that downward emergency braking is essential for studying an elevator's dynamic response. Furthermore, [PERSON] concluded that the load does not impact the peak deceleration of the car frame and wire rope during downward emergency braking [21]. While these studies offer valuable insights into elevator system vibrations, they often concentrate on analyzing the wire rope and car vibrations, overlooking the impact of the wire rope's unique time-varying characteristics on the overall elevator system vibrations. This paper employs finite element theory to develop a multi-degree-of-freedom dynamics model considering the time-varying properties of the wire rope. The model is solved using a time-domain numerical method, aiming to accurately predict high-speed elevator system vibrations and offering support for elevator design, installation, and commissioning.