The parametrically excited upside-down rod: an elastic jointed pendulum model

J Galan, WB Fraser, DJ Acheson, AR Champneys

Research output: Contribution to journalArticle (Academic Journal)peer-review

13 Citations (Scopus)
245 Downloads (Pure)

Abstract

A model is studied which consists of a chain of N identical pendulums coupled by damped elastic joints subject to vertical sinusoidal forcing of its base. Particular attention is paid to the stability of the upright equilibrium configuration with a view to understanding recent experimental results on the stabilization of an unstable stiff column under parametric excitation. It is shown via an appropriate scaling argument how the continuum rod model arises by taking the limit N→∞.

The effect of the inclusion of bending stiffness is first studied via asymptotics and numerics for the case N=1, showing how the static bifurcation of the pendulum varies with the four dimensionless parameters of the system; damping, bending stiffness and amplitude and frequency of excitation. For the multiple pendulum system, the bifurcation behaviour of the upright position as a function of the same four parameters is studied via numerical methods applied to the linearized equations. The damping term is found to be crucial in destroying many of the resonant instabilities that occur in the limit as N→∞. At realistic damping levels only a few instabilities remain, which are shown to be largely independent of N. These instabilities agree qualitatively with the experiments.
Translated title of the contributionThe parametrically excited upside-down rod: an elastic jointed pendulum model
Original languageEnglish
Pages (from-to)359-377
Number of pages19
JournalJournal of Sound and Vibration
Volume280
Issue number1-2
Early online date2 Mar 2004
DOIs
Publication statusPublished - Feb 2005

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