A cascade of autoresonances in an accelerating elevator cable system

Abstract

In this project the transversal vibrations of an
accelerating elevator cable system are studied, with the aim to find the
resonance times, the resonance duration and the resonance amplitude. The
elevator cable is modelled as an axially moving string, with length given by
l(t) = l0 + 1/2 at2, with a the acceleration and t the time. The cable is
sinusoidally excited at the top and fixed at the bottom. It is assumed that the
axial acceleration is small compared to the transversal acceleration, that the
cable mass is small compared to the car mass, and that the excitation amplitude
is small compared to the length of the cable. Using these estimations, the
solution for the transversal displacement u is approximated up to O(ε) with ε a
small parameter. The elevator cable goes through a cascade of autoresonances:
the eigenfrequencies of the cable are varying because the cable length is
varying, and at several times an eigenfrequency matches the excitation
frequency. These are the resonance times, and they have been found as t+ =
(2/εa1l0)1/2arccos((Ωl0/χk)1/2), with t+ a measure of oscillation of t, Ω the
angular excitation frequency, l0 the initial length, χk the eigenfrequency of
mode k and εa1 = a. The duration of the resonances (the timescale) is shown to
be O(ε-1/4) if χk≠Ωl0 and O(ε-1/6) if χk=Ωl0 (a bifurcation of the problem).
The amplitude scale is thus O(ε3/4) or O(ε5/6), respectively, and solutions for
the amplitude are calculated both outside and inside the resonance zone.