Periodic Motions of Coupled Impact Oscillators

Abstract

We study the existence and stability of time-periodic oscillations in a chain of coupled impact oscillators, for rigid impacts without energy dissipation. We formulate the search of periodic solutions as a boundary value problem incorporating unilateral constraints. This problem is solved analytically in the vicinity of the uncoupled limit and numerically for larger coupling constants. Different solution branches corresponding to nonlinear localized modes (breathers) and normal modes are computed.

D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

“…Non-smooth modal analysis (NSA) is used to obtain resonant vibratory responses and frequencies for structures prone to unilateral contact conditions [6,18]. While linear modes of vibration are obtained by solving eigenvalue problems, non-smooth modes are seen as invariant manifolds supporting periodic autonomous solutions [16].…”

Nonsmooth modal analysis of a varying cross-sectional area bar in unilateral contact

“…For vibrations of structures prone to unilateral contact, families of periodic solutions are indicative of resonance frequencies and behaviours and are, in fact, referred to as nonsmooth modes of vibration (NSMs) [1][2][3][4]. In essence, NSMs are families of periodic motions of the unforced and undamped investigated structure [5][6][7][8]. In the present work, we focus on nonsmooth modal analysis of the bar of varying (cross-section) area in unilateral contact with a rigid structure.…”