Dynamic Analysis of an Elevator Traveling Cable Using a Singularity-Free Beam Formulation

Abstract

An elevator traveling cable is modeled using a singularity-free beam formulation and its static and dynamic behaviors are analyzed. The beam is assumed to be an extensible Euler-Bernoulli beam, and the configuration of the beam is described by Euler parameters, which can resolve the singularity problem of Euler angles, and the normal strain of the centroid line of the beam. The position of the centroid line of the beam is integrated from its slope. Governing equations of the beam and constraint equations are derived using Lagrange’s equations for systems with constraints. The current formulation is used to calculate the equilibrium and dynamic responses of an elevator traveling cable with arbitrarily moving ends. Equilibria of a traveling cable with different elevator car positions are calculated. Natural frequencies and corresponding mode shapes of the traveling cable are calculated and they are in excellent agreement with those calculated by ABAQUS. In-plane natural frequencies of the traveling cable do not change much with the car position compared with its out-of-plane ones. Dynamic responses of the traveling cable are calculated using the current formulation and compared with those from commercial multibody dynamics software RecurDyn, and they are in good agreement with each other. Free responses of the traveling cable due to vertical motion of the car and forced responses with inplane and out-of-plane building sways are simulated, and their effects on dynamic responses of the traveling cable are investigated. While the vertical motion of the car can affect the in-plane lateral response of the traveling cable, it has almost no effect on its out-of-plane response. Building sways can affect both lateral and out-of-plane responses of the traveling cable, but they have little effect on its vertical response.