Summary: Control of a double impacting mechanical oscillator
D. K. Arrowsmith
June 4, 2003
Keywords: impacting systems, resonance, control, anti-control, feedback, chaotic regime.
A model which allows a double impacting regime for a particle undergoing simple harmonic motion is considered
in some detail. The behaviour of the particle in the weak spring limit is considered. Symmetries of the motion
are found and the extent of the resonant dynamical behaviour is considered. Control equations are developed and
strategies are described for both the preservation and the annihilation of resonant periodic orbits.
1 Motivation and Background
The model we consider in this paper is a representation of a doubly constrained impacting system. It is motivated
by a generic problem in engineering associated with the servo-assisted control of mechanical systems with slack.
The original motivation for this study was to understand the nature of non-linear resonances associated with the
interaction of servo-hydraulic pistons with multi-storey buildings during full-scale laboratory simulations used to
analyse their seismic behaviour [8, 15, 16]. During tests, the structural interaction of the building with the pistons
can cause spurious resonant vibrations which, ideally, should not be present during the experiments. In this paper we
conjecture that the underlying causes that generate such behaviour are the mechanical discontinuities that are latent