Why does narrow band chaos in impact oscillators disappear over a range of frequencies?

Abstract

In impacting mechanical systems, a large-amplitude chaotic oscillation develops close to the grazing condition, and it is explained in terms of “square root singularity” in the discrete-time model. It has been shown that this singularity vanishes when the forcing frequency is an integer multiple of twice the natural frequency of the system. It has also been shown that a dangerous border collision bifurcation occurs at the grazing condition causing divergence of the orbit to the unstable manifold of a preexisting saddle fixed point. In this paper, we show that the onset of large-amplitude chaotic oscillation can be avoided over finite ranges of the forcing frequency, because the unstable periodic orbits that cause the dangerous border collision bifurcation move away and do not coexist with the main orbit over a wide range of the forcing frequency.