A simplified equation of motion for free rocking rigid blocks

In this paper a novel formulation for the free rocking motion of slender rigid blocks is proposed and compared with the classical theory. In particular, it is assumed that the free motion between two subsequent impacts can be represented as a uniformly accelerated/decelerated motion and an ad hoc approximation is introduced to evaluate the constant acceleration at each half cycle. The dissipation of energy at each impact is represented by the coefficient of restitution, herein considered as a variable parameter in order to take into account different damping effects. The limits of validity of the proposed formulation are tested through an extensive comparison with the results obtained by Housner model in terms of moment-rotation relationship, critical angular velocity, normalized amplitude and duration of the half-cycles. The comparison is also developed in terms of time histories of non-dimensional angular acceleration, velocity and displacement confirming a good agreement with Housner model.