Unfolding Degenerate Grazing Dynamics in Impact Actuators

In this paper, a selected analysis of the dynamics in an example impact microactuator is performed through a combination of numerical simulations and local analysis. Here, emphasis is placed on investigating the system response in the vicinity of the so-called grazing trajectories, i.e. motions that include zero-relative-velocity contact of the actuator parts, using the concept of discontinuity mappings that account for the effects of low-relative-velocity impacts and brief episodes of stick–slip motion. The analysis highlights the existence of isolated co-dimension-two grazing bifurcation points and the way in which these organize the behaviour of the impacting dynamics. In particular, it is shown how higher-order truncations of local maps of the near-grazing dynamics predict and enable the computation of global bifurcation curves emanating from such degenerate bifurcation points, thereby unfolding the near-grazing dynamics. Although the numerical results presented here are specific for the chosen model of an electrically driven and previously experimentally realized impact microactuator, the methodology generalizes naturally to arbitrary systems with impacts. Moreover, the qualitative nature of the near-grazing dynamics is expected to generalize to systems with similar nonlinearities.