On the resonance conditions of rigid rocking blocks

The dynamic behavior of rigid free-standing blocks subjected to earthquake ground motions is highly non-linear and sensitive to small perturbations of various parameters. Many difficulties arise in defining reliable response spectra for such systems and these are well known in the literature. This paper deals with the resonance conditions in order to highlight to what extent the ground motion details and the system parameters can influence the rocking response. The first step is the construction of an artificial input implying amplitude resonance for the motion, which is analyzed by means of a simplified equation of motion introduced by Housner (1963). The coefficient of restitution is assumed to be a variable of the problem to account also for other damping effects (e.g. local plastic deformations). A stabilized phase of the motion is identified for which an upper-bound of the maximum rotation angle of the block can be defined in closed form. The results are plotted in resonance spectra which point out the influence of the coefficient of restitution and the size and slenderness of the block. An interesting comparison with the response of an elastic damped SDOF oscillator in analogous resonance conditions is also presented.