On the Lateral Vibrations of a Vertically Moving String With a Harmonically Varying Length

In this paper, the free lateral vibrations of a vertically translating continuum modeled as a taut string with variable length are studied. The time-varying length of the cable is described by a harmonically varying function about a constant mean length. This model can be used to describe the lateral vibrations of an elevator cable with changing length. A Fourier series approach is used to predict resonant frequencies arising because of the length fluctuation. The amplitudes of vibrations are represented by the infinitely dimensional system of coupled ordinary differential equations. This system is studied by Galerkin’s truncation method for the lowest resonant frequency. The obtained results are in agreement with the energy analysis, but the truncation technique leads to inaccurate approximations on long timescales. Hence, an alternative analytical method is proposed promising an accurate approximation of the response on long timescales for the general resonant frequency.

Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System

This paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive. The governing equation of motion of the objective system is developed by virtue of Lagrangian’s method. A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system. The obtained equation is an example of a well-known category of nonlinear oscillators, namely, non-natural systems. Due to the complex terms in the governing equation, perturbation methods cannot directly extract any closed form expressions for the natural frequency. Unavoidably, different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency. Energy balance method, modified energy balance method and variational approach are utilized for frequency analyzing of the system. Frequency-amplitude relationships are analytically obtained for nonlinear vibration of the elevator’s drum. In order to examine accuracy of the obtained results, exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases. In a parametric study for different nonlinear parameters, variation of the natural frequencies against the initial amplitude is investigated. Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.

An Accurate Spatial Discretization and Substructure Method With Application to Moving Elevator Cable-Car Systems—Part II: Application

This paper uses the methodology developed in Part I of this work to study the longitudinal, transverse, and their coupled vibrations of moving elevator cable-car systems. A suspension cable is a one-dimensional length-variant distributed-parameter component. When there is only one suspension cable connected to the car, the car is modeled as a point mass. When there are multiple suspension cables, the car is modeled as a rigid body, and the rotation of the car is considered. There are complicated matching conditions between the cable and car, which cannot be satisfied in the classical assumed modes method but can be satisfied in the current method. Hence, not only the longitudinal and transverse displacements but also the internal forces/moment, such as the axial force, the bending moment, and the shear force, which are related to the spatial derivatives of the longitudinal and transverse displacements, are accurately calculated. The results from different choices of boundary motions and trial functions are essentially the same, and the convergence is much faster than that of the assumed modes method. The longitudinal-transverse coupled vibrations of a moving cable-car system are also studied using the current method, and the results are compared with those from the linear models. While the result from the linear model for the transverse vibration agrees well with that from the nonlinear coupled model, the axial force from the linear model can significantly differ from that from the nonlinear model when the car approaches the top of the hoistway.