Interaction Between Coexisting Orbits in Impact Oscillators

Abstract

Impact oscillators exhibit an abrupt onset of chaos close to grazing due to the square-root singularity in their discrete time maps. In practical applications, this large-amplitude chaotic vibration needs to be avoided. It has been shown that this can be achieved if the ratio of the natural frequency of the oscillator ω0 and the forcing frequency is an even integer. But, in practice, it is difficult to set a parameter at such a precise value. We show that in systems with square-root singularity (prestressed impacting surface), there exists a range of ω0 around the theoretical value over which the chaotic orbit does not occur, and that this is due to an interplay between the main attractor and coexisting orbits. We show that this range of forcing frequency has exponential dependence on the amount of prestress as well as on the stiffness ratio of the springs.