Near-resonant dynamics, period doubling and chaos of a 3-DOF vibro-impact system

Abstract

AbstractA mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. A conception of the stability analysis of the periodic motions suited for this theoretical approach is presented. The frequency–response curves and force–response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response are verified via a series of numerical experiments and computation of Lyapunov exponents. Finally, the limitations and extensibility of the approach are discussed.