Narrow Results

A model which allows a double impacting regime for a particle undergoing simple harmonic motion is considered in some detail. The behavior of the particle in the weak spring limit is considered. Symmetries of the motion are found and the extent of the resonant dynamical behavior is considered. Control equations are developed and strategies are described for both the preservation and the annihilation of experimental and analytical resonant periodic orbits.

We analyze an important class of engineering systems characterized by the discontinuous motion of a spring-mass constrained by the motion of a feedback-assisted actuator. We show that the combined effects of mechanical restitution coefficient and displacement feedback can be exactly represented by a single equivalent dissipation coefficient. We also show that the topological properties of the surfaces of section of orbits generated by impact oscillators which possess differing proportions of restitution and feedback levels, but whose equivalent dissipation coefficients are equal, are equivalent and universally scalable. The scaling law allows us to interchange the effects of restitution and feedback coefficients and so, effectively, eliminate one of these parameters from the equations of motion. Thus, the topological properties of dissipative feedback-assisted systems can be seen as scaled versions of either purely dissipative, or purely feedback-assisted, oscillators.