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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on June 19, 2006
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbl009
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© The author 2006. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org
Received October 14, 2005
Revised April 10, 2006
Accepted May 9, 2006

Article

ASYMPTOTIC ANALYSIS OF CHATTERING OSCILLATIONS FOR AN IMPACTING INVERTED PENDULUM

Lucio Demeio 1 * and Stefano Lenci 2

( 1 Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 1, I-60131 Ancona, Italy
2 Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica delle Marche, Via Brecce Bianche 1, I-60131 Ancona, Italy
)

* To whom correspondence should be addressed.
Lucio Demeio, E-mail: demeio{at}dipmat.univpm.it


  Abstract

The chattering oscillations for an inverted pendulum impacting between lateral walls, a prototype of a class of impact dampers, are analysed. Attention is focused on the periodic chattering appearing when the rest positions cease to be attracting, and the aim is that of computing the time {tau} required by the micro-oscillations to come to rest. An algorithm is proposed to compute {tau}, and it is shown how this time depends on the excitation amplitude. When {tau} becomes equal to the excitation period, the considered chattering is observed to lose attractivity. This occurs for a certain excitation amplitude threshold, which is computed and whose (very weak) dependence on the excitation frequency is illustrated.

An asymptotic estimate of the chattering time with respect to a small parameter proportional to the excitation amplitude is given. This is obtained by an appropriate expansion in Taylor series of the relevant quantities involved in the analysis. It is shown that {tau} is asymptotically proportional to the square root of the excitation amplitude.


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