An asymptotic approximation of the chattering time for the rocking problem

Abstract

Abstract The dynamic behaviour of a freestanding block rocking on a rigid base when subjected to a strong ground excitation is a classical problem of nonlinear dynamics that has recently gained considerable momentum. However, when a rocking block is subjected to a low amplitude (i.e. weak) ground motion has drawn little attention due to the belief that it has marginal effect on the response trajectory. Despite its apparent structural simplicity, a rocking block undergoes complex nonlinear and nonsmooth dynamics phenomena under both low and high amplitude ground motions. This study focuses on one of the least investigated nonlinear phenomena of rocking dynamics that might appear during the low amplitude forced oscillations of a block, called chattering. Chattering can be complete or incomplete. Complete chattering occurs when a block undergoes a theoretically infinite sequence of impacts in finite time, that eventually bring the block to the state of persistent (continuous) contact even under a nonzero excitation. On the contrary, incomplete chattering does not bring the block to rest after a theoretically infinite number of impacts. A challenging problem that arises during complete chattering is the accurate estimation of the time needed for the block to reach the state of persistent (continuous) contact, i.e. chattering time. Thus, this paper presents an iterative algorithm that approximates chattering time using asymptotic analysis and perturbation methods.