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Nonlinear dynamics of coupled oscillators in 1:2 internal resonance: effects of the non-resonant quadratic terms and recovery of the saturation effect

Article dans une revue avec comité de lecture
Author
SHAMI, Zein Alabidin
543315 Laboratoire d’Ingénierie des Systèmes Physiques et Numériques [LISPEN]
ccSHEN, Yichang
421305 Institut des Sciences de la mécanique et Applications industrielles [IMSIA - UMR 9219]
GIRAUD-AUDINE, Christophe
13338 Laboratoire d’Électrotechnique et d’Électronique de Puissance - ULR 2697 [L2EP]
ccTOUZÉ, Cyril
421305 Institut des Sciences de la mécanique et Applications industrielles [IMSIA - UMR 9219]
THOMAS, Olivier
543315 Laboratoire d’Ingénierie des Systèmes Physiques et Numériques [LISPEN]

URI
http://hdl.handle.net/10985/22697
DOI
10.1007/s11012-022-01566-w
Date
2022-08
Journal
Meccanica

Abstract

This article considers the nonlinear dynamics of coupled oscillators featuring strong coupling in 1:2 internal resonance. In forced oscillations, this particular interaction is the source of energy exchange, leading to a particular shape of the response curves, as well as quasi-periodic responses and a saturation phenomenon. These main features are embedded in the simplest system which considers only the two resonant quadratic monomials conveying the 1:2 internal resonance, since they are the proeminent source allowing one to explain these phenomena. However, it has been shown recently that those features can be substantially modified by the presence of non-resonant quadratic terms. The aim of the present study is thus to explain the effect of the non-resonant quadratic terms on the dynamics. To that purpose, the normal form up to the third order is used, since the effect of the non-resonant quadratic terms will be transferred into the resonant cubic terms. Analytical solutions are detailed using a second-order mutliple scale expansion. A thorough investigation of the backbone curves, their stability and bifurcation, and the link to the forced–damped solutions, is detailed, showing in particular interesting features that had not been addressed in earlier studies. Finally, the saturation effect is investigated, and it is shown how to correct the detuning effect of the cubic terms thanks to a specific tuning of non-resonant quadratic terms and resonant cubic terms. This choice, derived analytically, is shown to extend the validity of the saturation effect to larger amplitudes, which can thus be used in all applications where this effect is needed e.g. for control.

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