The nature of the normal form map for soft impacting systems

Soft impacting mechanical systems—where the impacting surface is cushioned with a spring–damper support—are common in engineering. Mathematically such systems come under the description of switching dynamical systems, where the dynamics toggle between two (or more) sets of differential equations, determined by switching conditions. It has been shown that the Poincaré map of such a system would have a power of 1/2 (the so-called square-root singularity) if the vector fields at the two sides of the switching manifold differ, and a power of 3/2 if they are the same. These results were obtained by concentrating on the leading order terms in a Taylor expansion of the zero-time discontinuity map, and are true in the immediate neighbourhood of a grazing orbit. In this paper we investigate how the character of the two-dimensional map changes over a large parameter range as the system is driven from a non-impacting orbit to an impacting orbit. This study leads to vital conclusions regarding the character of the normal form of the map not only in the immediate vicinity of the grazing orbit, but also away from it, as dependent on the system parameters. We obtain these characteristics by experiment and by simulation.