Rocking block stability under periodic and random perturbations

Abstract

We analyze the dynamics of a rigid block subject to ground acceleration, simulating the effects of both an earthquake and the emergence of an eventual earthquake replica. For a single pulse, we find that the escape time τ (elapsed time for the block overturning) shows a periodic dependence with the pulsed duration time. This behavior reveals an asymmetric pattern in the parameter space for the direction of the block overturning. Fixing τ to its maximum value, a second pulsed perturbation (replica) is simulated via two protocols: one describing a kick of fixed amplitude and another described by a perturbation of variable amplitude chosen from a random distribution, which can be considered first with only positive values, and second with positive and negative values. The first protocol does not enhance τ unless the kick is applied immediately after turning off the first pulse. However, using the variable amplitude perturbation, a remarkable enhancement of the duration of the sustained vibrations of the block is observed. More interestingly, we show the emergence of revivals of the rocking motion after the system has initially ceased to vibrate. A frequency–amplitude phase diagram indicates that the system possesses a preferred operational range of parameters from which valuable information could be extracted. This could improve our understanding of the rocking performance of the system.