Journal article
Non-resonant damped transitions resembling continuous resonance scattering in coupled oscillators with essential nonlinearities
Physica. D, Vol.241(10), pp.964-975
05/15/2012
DOI: 10.1016/j.physd.2012.02.009
Abstract
We study a peculiar damped nonlinear transition of a system of two coupled oscillators into a state of sustained nonlinear resonance scattering. This system consists of a grounded, weakly damped linear oscillator attached to a light, weakly damped oscillator with essential (nonlinearizable) stiffness nonlinearity of the third degree, and linear or nonlinear damping. We find that under specific forcing conditions the damped response of this system locks into a damped, non-resonant transition resembling continuous resonance scattering, whereby the transient damped dynamics closely follows an impulsive orbit manifold of the dynamics in the frequency–energy plane. This manifold is formed by a countable infinity of periodic orbits and an uncountable infinity of quasi-periodic orbits of the underlying Hamiltonian system, with each of these orbits representing the response of the Hamiltonian system being initially at rest and forced by an impulse applied to the linear oscillator. Hence, the damped transitions reported here appear to lock in sustained resonance scattering from a countable infinity of periodic orbits along the impulsive orbit manifold. Such transitions represent an anti-resonance state, where the dynamics is farthest away from resonance. We conjecture that such transitions are only made possible by the essential (nonlinearizable) stiffness nonlinearity of the nonlinear attachment and cannot be realized in linearizable nonlinear dynamics where resonance captures prevent sustained resonance scattering. Our findings are supported by numerical, analytical and experimental results.
► A phenomenon resembling sustained resonance scattering is examined. ► The system dynamics closely follow an impulsive orbit manifold. ► Analytical work predicts the response during sustained resonance scattering. ► Experimental tests confirm the existence of sustained resonance scattering.
Details
- Title: Subtitle
- Non-resonant damped transitions resembling continuous resonance scattering in coupled oscillators with essential nonlinearities
- Creators
- David Andersen - University of Illinois Urbana-ChampaignYuli Starosvetsky - University of Illinois at Urbana–ChampaignMercedes Mane - University of Illinois at Urbana–ChampaignSean Hubbard - University of Illinois at Urbana–ChampaignKevin Remick - University of Illinois at Urbana–ChampaignXingyuan Wang - University of Illinois at Urbana–ChampaignAlexander Vakakis - University of Illinois at Urbana–ChampaignLawrence Bergman - University of Illinois at Urbana–Champaign
- Resource Type
- Journal article
- Publication Details
- Physica. D, Vol.241(10), pp.964-975
- Publisher
- Elsevier B.V
- DOI
- 10.1016/j.physd.2012.02.009
- ISSN
- 0167-2789
- eISSN
- 1872-8022
- Language
- English
- Date published
- 05/15/2012
- Academic Unit
- Electrical and Computer Engineering; Physics and Astronomy
- Record Identifier
- 9984197319302771
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