A dynamical systems approach to the overturning of rocking blocks

A rocking block resting on a horizontal rigid foundation and excited by a periodic excitation can topple if the excitation amplitude is sufficiently high. This question is addressed in this work by the combined use of dynamical systems arguments and numerical tools. The problem is first addressed from a theoretical point of view, with the objective of analytically detecting the most relevant critical thresholds. We succeed in obtaining closed form and manageable criteria for overturning. Then, numerical computations are performed, aimed at verifying the analytical thresholds, and understanding the overall overturning behavior. Furthermore, attention is paid to studying how toppling is modified by the excitation phase, whose role is very important and was not underlined in previous works, and by other problem parameters. The analytical criteria are shown to be bounds for the first overturning, which corresponds to engineering failure of the structure: a relevant improved upper bound is also obtained, still by means of invariant manifolds arguments.