Abstract
A 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.
Highlights
Vibro-impact systems can be found in many areas of science and engineering
There is a solution strategy that fits in with the asymptotic approach to nonlinear problems and near-resonant dynamics. It consists in combining the multiple time scales method (MTSM) with a use of a sawtooth function which describes the non-smooth nature of the given system
The analytical studies have been restricted to a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance
Summary
Vibro-impact systems can be found in many areas of science and engineering. Even with a simple structure, they are strongly nonlinear. There is a solution strategy that fits in with the asymptotic approach to nonlinear problems and near-resonant dynamics It consists in combining the multiple time scales method (MTSM) with a use of a sawtooth function which describes the non-smooth nature of the given system. For example, by Popplewell et al [29] In all these works, motion of the primary system between impacts is described by the general solution for a piecewise linear oscillator with viscous damping and periodic excitation. Such a model is usually inconsistent with the approximate solution formulated with the method of multiple scales under the assumption of small forcing and/or damping.
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