Modeling the Planar Rocking of a Rigid Body with Irregular Geometry

The nonlinear dynamics of rocking rigid bodies with simple geometries, such as rectangular blocks and cylinders, have been the focus of the rocking community over the last six decades. However, many objects that are prone to rocking or overturning do not conform to such geometries. These objects include museum artifacts and precariously balanced rocks in the natural world. Even in cases where the response of the rocking body is planar, the geometry of the body is much more complicated than the commonly studied geometry of a rocking block or a body with only two rocking corners. This paper introduces a complete model that can examine the planar motion of a body with an irregular in-plane polygonal geometry when subjected to a vibrational excitation, utilizing the geometry of the body as an input-for example, in the form of a stereolithography (STL) file. The model is used for studying the rocking response of an object while taking into account sliding and free flight. The problem is formulated and solved using Newtonian equations of motion, and impacts are treated as hard. A robust framework for integrating the occurring discontinuous equations of motion and for detecting transitions between patterns of motion and impacts, using MATLAB, is presented. Suitably chosen examples demonstrate the importance of accounting for the actual geometry of the studied rocking body, whose dynamic response is substantially richer than an object with simplified geometry.