On the motions of the offset impact oscillator

Abstract

The motions of a periodically forced, damped, harmonic oscillator with one degree of freedom, impacting with no energy loss against a wall, are considered. In the 'offset' case considered here, the wall is not at the rest point of the oscillator. The regions of the stability of single impact orbits are calculated as functions of the forcing amplitude and frequency. A digital simulation is carried out, producing multi-impact orbits, flip and tangent bifurcations and chaotic attractors. Hysteresis between attractors is observed, and crises of the attractor are found which are not immediately identifiable with the collisions of unstable periodic and chaotic attractors. Some return maps for chaotic motions are plotted, providing evidence that the attractors do not have integer dimension; the families of curves do, however, show local structures which may be identified with nearby stable motions.