On the reliability of the linear oscillator and systems of coupled oscillators

L. A. Bergman; J. C. Heinrich

doi:10.1002/nme.1620180902

Sustained High-Frequency Dynamic Instability of a Nonlinear System of Coupled Oscillators Forced by Single or Repeated Impulses: Theoretical and Experimental Results

Journal of Vibration and Acoustics ◽

10.1115/1.4025605 ◽

2013 ◽

Vol 136 (1) ◽

Cited By ~ 8

Author(s):

Kevin Remick ◽

Alexander Vakakis ◽

Lawrence Bergman ◽

D. Michael McFarland ◽

D. Dane Quinn ◽

...

Keyword(s):

Periodic Orbits ◽

High Frequency ◽

Coupled Oscillators ◽

Dynamic Instability ◽

Specific Impulse ◽

Resonance Scattering ◽

Experimental Results ◽

Linear Oscillator ◽

Finite Frequency ◽

Energy Plane

This report describes the impulsive dynamics of a system of two coupled oscillators with essential (nonlinearizable) stiffness nonlinearity. The system considered consists of a grounded weakly damped linear oscillator coupled to a lightweight weakly damped oscillating attachment with essential cubic stiffness nonlinearity arising purely from geometry and kinematics. It has been found that under specific impulse excitations the transient damped dynamics of this system tracks a high-frequency impulsive orbit manifold (IOM) in the frequency-energy plane. The IOM extends over finite frequency and energy ranges, consisting of a countable infinity of periodic orbits and an uncountable infinity of quasi-periodic orbits of the underlying Hamiltonian system and being initially at rest and subjected to an impulsive force on the linear oscillator. The damped nonresonant dynamics tracking the IOM then resembles continuous resonance scattering; in effect, quickly transitioning between multiple resonance captures over finite frequency and energy ranges. Dynamic instability arises at bifurcation points along this damped transition, causing bursts in the response of the nonlinear light oscillator, which resemble self-excited resonances. It is shown that for an appropriate parameter design the system remains in a state of sustained high-frequency dynamic instability under the action of repeated impulses. In turn, this sustained instability results in strong energy transfers from the directly excited oscillator to the lightweight nonlinear attachment; a feature that can be employed in energy harvesting applications. The theoretical predictions are confirmed by experimental results.

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Irreversible Passive Energy Transfer in the Damped Response of Coupled Oscillators With Essential Nonlinearity

Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2005-84498 ◽

2005 ◽

Author(s):

G. Kerschen ◽

Y. S. Lee ◽

A. F. Vakakis ◽

D. M. McFarland ◽

L. A. Bergman

Keyword(s):

Energy Transfer ◽

Periodic Solutions ◽

Coupled Oscillators ◽

Topological Structure ◽

Seismic Isolation ◽

Degrees Of Freedom ◽

Linear Oscillator ◽

Resonance Capture ◽

Low Frequencies ◽

Flutter Suppression

We study, numerically and analytically the dynamics of passive energy transfer from a damped linear oscillator to an essentially nonlinear end attachment. This transfer is caused either by fundamental or subharmonic resonance capture, and in some cases is initiated by nonlinear beat phenomena. It is shown that, due to the essential nonlinearity, the end attachment is capable of passively absorbing broadband energy both at high and low frequencies, acting, in essence, as a passive broadband boundary controller. Complicated transitions in the damped dynamics can be interpreted based on the topological structure and bifurcations of the periodic solutions of the underlying undamped system. Moreover, complex resonance capture cascades are numerically encountered when we increase the number of degrees of freedom of the system. The ungrounded, essentially nonlinear end attachment discussed in this work can find application in numerous practical settings, including vibration and shock isolation of structures, seismic isolation, flutter suppression and packaging.

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