Abstract
A rigid, slender, rectangular block is an apparently simple example of a stability problem. In a static sense, the upright position remains stable providing the restoring moment due to gravity is sufficient to offset any applied overturning moment. Rotated about either corner, the block will overturn if a critical angle is exceeded. However, if the (quiescent) block is subjected to a horizontal (dynamic) base excitation, and assuming that no sliding, bouncing, or out-of-plane motion occurs, a typical block may either: (i) remain stationary relative to the base, (ii) rock, or (iii) overturn.
The conditions under which overturning occurs are of particular interest here, i.e., the boundary between categories (ii) and (iii) above. Combinations of forcing parameters, i.e., magnitude and frequency of the sinusoidal excitation, are categorized according to whether, and how quickly, overturning occurs. The transition when the block changes from rotation about one corner to the other, the energy loss at impact, and large-angle geometric effects are the sources of nonlinearity in this problem.
Based on numerical integration, and taking great care to accurately model the impact condition, it is shown that transient rocking prior to overturning exhibits an extreme sensitivity and dependence on certain system parameters. A number of counter-intuitive features are observed. Some elementary experiments conducted on a relatively slender, rigid block placed on a shake table confirm some of the simulation results, but also point out the modeling difficulties encountered in such inherently sensitive systems.
