TY - JOUR
T1 - Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities
AU - Vakakis, Alexander F.
AU - Rand, Richard H.
N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.
PY - 2004/9
Y1 - 2004/9
N2 - We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes - NNMs), as well as, asynchronous periodic motions (elliptic orbits - EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets 'captured' in the neighborhood of a damped NNM before 'escaping' and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.
AB - We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes - NNMs), as well as, asynchronous periodic motions (elliptic orbits - EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets 'captured' in the neighborhood of a damped NNM before 'escaping' and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.
KW - Capture
KW - Coupled oscillators
KW - Non-linear
KW - Resonance
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U2 - 10.1016/S0020-7462(03)00098-2
DO - 10.1016/S0020-7462(03)00098-2
M3 - Article
AN - SCOPUS:0344308470
SN - 0020-7462
VL - 39
SP - 1079
EP - 1091
JO - International Journal of Non-Linear Mechanics
JF - International Journal of Non-Linear Mechanics
IS - 7
ER -