A Periodically Forced Impact Oscillator With Large Dissipation

Abstract

We consider the simple harmonic oscillator with harmonic excitation and a constraint that restricts motions to one side of the equilibrium position. Thus, on the achievement of a specified displacement, the direction of motion is reversed using the simple impact rule. The coefficient of restitution for this impact, r, is taken to be small. For r = 0 the motions of the system can be studied using a one-dimensional mapping. Analysis of this map shows that stable periodic orbits exist at almost all forcing frequencies but that transient nonperiodic or chaotic motions can also occur. Moreover, over certain (narrow) frequency windows arbitrarily long stable periodic motions exist. These results are then extended to the case r ≠ 0, small.