Dynamics of a small vibro-impact pile driver

A mathematical model is developed to study periodic-impact motions and bifurcations in dynamics of a small vibro-impact pile driver. Dynamics of the small vibro-impact pile driver can be analyzed by means of a three-dimensional map, which describes free flight and sticking solutions of the vibro-impact system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the Poincaré map. The piecewise property is caused by the transitions of free flight and sticking motions of the driver and the pile immediately after the impact, and the singularity of map is generated via the grazing contact of the driver and the pile immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularities and parameter variation on the performance of the vibro-impact pile driver is analyzed. The global bifurcation diagrams for the impact velocity of the driver versus the forcing frequency are plotted to predict much of the qualitative behavior of the actual physical system, which enable the practicing engineer to select excitation frequency ranges in which stable period one single-impact response can be expected to occur, and to predict the larger impact velocity and shorter impact period of such response.