Parametric excitation of an internally resonant double pendulum. II
type: Item
instance of: scholarly article
title: Parametric excitation of an internally resonant double pendulum. II
zbMATH Open document ID: 0619.70023
DOI: 10.1007/BF00947912
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1986-01-01T00:00:00Z
review text: The second author [ibid. 36, 337-345 (1985; Zbl 0576.70019)] has examined the weakly nonlinear response of a damped, internally resonant double pendulum to parametric excitation at twice the frequency of the dominant (slower) pendulum. He found that, for all values of frequency offset and dissipation, regular motion is obtained in the form of either parametrically excited oscillations or vertical translation of the system as a rigid body. We now suppose that the driving frequency approximates twice the frequency of the satellite pendulum (four times the frequency of the dominant pendulum) and find that, in contrast to I, chaotic motion may occur. We obtain the evolution equations that govern the envelope of the oscillations by postulating the angular displacements, \(\theta_ n\), \(n=1,2\), of the pendulums in the form of slowly modulated sinusoids. We examine the fixed points and bifurcation structure of these evolution equations through the construction of the resonance curves (energy vs frequency). Finally, in Sect. 5, we present numerical results that support the conclusion that chaotic motion occurs in certain parametric neighbourhoods. The corresponding case in which the internally resonant double pendulum is designed so that the satellite has the lower natural frequency is discussed in an appendix.
Mathematics Subject Classification ID: 70K30
Mathematics Subject Classification ID: 70J30
zbMATH DE Number: 4005671
zbMATH Keywords: weakly nonlinear response
zbMATH Keywords: damped, internally resonant double pendulum
zbMATH Keywords: parametric excitation
zbMATH Keywords: dissipation
zbMATH Keywords: regular motion
zbMATH Keywords: parametrically excited oscillations
zbMATH Keywords: satellite pendulum
zbMATH Keywords: evolution equations
zbMATH Keywords: fixed points
zbMATH Keywords: bifurcation structure
zbMATH Keywords: resonance curves
zbMATH Keywords: chaotic motion
author: Janet M. Becker
author: John Miles
MaRDI profile type: Publication
cites work: Parametric excitation of an internally resonant double pendulum
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description: scientific article; zbMATH DE number 4005671