Fractal Behavior of an Asymmetric Rigid Block Overturning Due to Harmonic Motion of a Tilted Foundation

The motion of a slender rigid block with a flat or concave base resting on a rigid and flat foundation is analyzed. The block may be symmetric or asymmetric, and the foundation may be horizontal or tilted. The foundation oscillates harmonically for a finite period of time, and the block exhibits planar motion: it may rotate about either of its bottom corners, it may rock from one corner to the other, and it may overturn. Sliding and bouncing are not considered. Energy is lost during the impact when the point of rotation switches from one corner to the other. The number of impacts prior to overturning is computed, and results for horizontal foundation acceleration are plotted in the plane of excitation amplitude versus excitation frequency. The boundaries separating regions associated with different numbers of impacts, and in particular the boundary between overturning and nonoverturning regions, are fractal.