Controlled onset of low-velocity collisions in a vibro-impacting system with friction

This paper investigates the onset of low-velocity, near-grazing collisions in an example vibro-impacting system with dry friction with particular emphasis on feedback control strategies that regulate the grazing-induced bifurcation behaviour. The example system is characterized by a twofold degeneracy of grazing contact along an extremal stick solution that is shown to result in a locally one-dimensional and piecewise-linear description of the near-grazing dynamics. Explicit control strategies are derived that ensure a persistent, low-impact-velocity, steady-state response across the critical parameter value corresponding to grazing contact even in instances where the dynamics in the absence of control exhibit a sudden transition to a high-impact-velocity response.

Discontinuity-Induced Bifurcations in Systems With Hysteretic Force Interactions

This paper presents the application of the discontinuity-mapping technique to the analysis of discontinuity-induced bifurcations of periodic trajectories in an example hybrid dynamical system in which changes in the vector field associated with the crossing of a discontinuity-surface depend on the direction of crossing. The analysis is motivated by a hysteretic model of the capillary force interactions between an atomic-force-microscope cantilever probe tip and a nanoscale sample surface in the presence of a thin liquid film on the tip and the surface and operating in intermittent-contact mode. The analysis predicts the sudden termination of branches of periodic system responses at parameter values corresponding to grazing contact with the onset of the hysteretic force interactions. It further establishes the increase beyond all bounds of the magnitude of one of the eigenvalues of the linearization of a suitably defined Poincaré mapping, indicating the destabilizing influence of near-grazing contact.

Discontinuity-Induced Bifurcations in Systems With Hysteretic Force Interactions

This paper presents the application of the discontinuity-mapping technique to the analysis of discontinuity-induced bifurcations of periodic trajectories in an example piecewise smooth system in which changes in the vector field associated with the crossing of a discontinuity-surface depend on the direction of crossing. The analysis is motivatived by a hysteretic model of the capillary force interactions between an atomic-force-microscope cantilever probe tip and a nanoscale sample surface in the presence of a thin liquid film on the tip and the surface and operating in intermittent-contact mode. The analysis predicts the sudden termination of branches of periodic system responses at parameter values corresponding to grazing contact with the onset of the hysteretic force interactions. It further establishes the increase beyond all bounds of the magnitude of one of the eigenvalues of the linearization of a suitably defined Poincare´ mapping indicating the destabilizing influence of near-grazing contact.

Periodic Motions and Bifurcations of a Plastic Impact Machine

A mathematical model is developed to describe the characteristic behavior of a small vibro-impact pile driver. Dynamics of the small vibro-impact pile driver is represented by a three-dimensional map. The map is of piecewise property due to synchronous and non-synchronous motion of the driver and pile immediately after the plastic impact, and singularities caused by grazing contact of the driver and pile. The pile driver exhibits two different types of single-impact periodic motions in different regions of the forcing frequency due to the plastic impacts. Transition of two types of single-impact periodic motions is demonstrated, and the influence of the piecewise property, singularities and various parameters on the performance of the pile driver is analyzed.