A semi-analytical approach to approximate chattering time of rocking structures

Abstract

This paper investigates the seismic behaviour of a freestanding rigid rocking block when subjected to low amplitude sinusoidal ground excitations. An important scenario in such cases is the complete chattering the block might exhibit. Complete chattering occurs when the block undergoes a theoretically infinite sequence of decaying impacts that converge to the state of persistent (continuous) contact in finite time, even under a nonzero ground excitation. This study proposes a semi-analytical approach that approximates the (finite) time required for this to happen, i.e. chattering time. Specifically, this paper provides a detailed description of the semi-analytical scheme and shows the influence of the amplitude of the ground acceleration on the approximation of the chattering time. Importantly, the proposed scheme is based on the realisation that, during chattering, and after a sufficiently large number of impacts, the ratio of the time-intervals of every two consecutive impacts becomes constant and equal to the square of the coefficient of restitution. The proposed semi-analytical approach efficiently approximates chattering time providing a state-of-the-art mathematical formulation of the chattering phenomenon for the rocking problem.