On a cascade of autoresonances in an elevator cable system

Abstract

The aim of this paper is to study autoresonance phenomena in a space–time-varying mechanical system. The maximal amplitude of the autoresonant solution and the time of autoresonant growth of the amplitude of the modes of fast oscillations are determined. A vertically translating string with a time-varying length and a space–time-varying tension are considered. The problem can be used as a simple model to describe transversal vibrations of an elevator cable for which the length changes linearly in time. The slowly time-varying length is given by $$l(t)=l_{0}+\varepsilon {t}$$ , where $$l_{0}$$ is a constant and $$\varepsilon $$ is a dimensionless small parameter. It is assumed that the axial velocity of the cable is small compared to nominal wave velocity and that the cable mass is small compared to car mass. The elevator cable is excited sinusoidally at the upper end by the displacement of the building in the horizontal direction from its equilibrium position caused by wind forces. This external excitation has a constant amplitude of order $$\varepsilon $$ . It is shown that order $$\varepsilon $$ amplitude excitations at the upper end result in order $$\sqrt{\varepsilon }$$ solution responses. Interior layer analysis has been provided systematically to show that there exists an unexpected timescale of order $$\frac{1}{\sqrt{\varepsilon }}$$ . For this reason, a three-timescale perturbation method is used to construct asymptotic approximations of the solutions of the initial-boundary value problem.