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Research Papers

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

[+] Author and Article Information
Kevin Remick

e-mail: remick2@illinois.edu

Alexander Vakakis

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
1206 W. Green Street,
Urbana, IL 61801

D. Michael McFarland

Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign,
104 S. Wright Street,
Urbana, IL 61801

D. Dane Quinn

Department of Mechanical Engineering,
University of Akron,
Akron, OH 44325

Themistoklis P. Sapsis

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 1, 2013; final manuscript received September 18, 2013; published online November 13, 2013. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 136(1), 011013 (Nov 13, 2013) (15 pages) Paper No: VIB-13-1064; doi: 10.1115/1.4025605 History: Received March 01, 2013; Revised September 18, 2013

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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References

Figures

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Fig. 2

Frequency-energy plot of the underlying Hamiltonian system (1)

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Fig. 1

Configuration of the two-degree-of-freedom coupled oscillator with essential geometric nonlinearity

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Fig. 8

Second cycle of the damped response of Fig. 6, following the second impulse: (a) relative displacement between the nonlinear attachment and the linear oscillator, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) on the Hamiltonian FEP

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Fig. 3

Damped response of the theoretical model for single impulse excitation of normalized intensity I0 = 0.007 m/s: (a) displacement of the nonlinear attachment, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) superimposed on the Hamiltonian FEP

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Fig. 4

Damped response of the theoretical model for single impulse excitation of normalized intensity I0 = 0.010 m/s: (a) displacement of the nonlinear attachment, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) superimposed on the Hamiltonian FEP

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Fig. 5

Damped response of the theoretical model for single impulse excitation of normalized intensity I0 = 0.070 m/s: (a) displacement of the nonlinear attachment, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) superimposed on the Hamiltonian FEP

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Fig. 6

Damped response of the theoretical model for periodic impulse excitation of normalized intensity I0 = 0.010 m/s: (a) relative displacement between the nonlinear attachment and the linear oscillator, and (b) displacement of the linear oscillator

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Fig. 7

First cycle of the damped response of Fig. 6, following the first impulse: (a) relative displacement between the nonlinear attachment and the linear oscillator, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) on the Hamiltonian FEP

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Fig. 9

Third cycle of the damped response of Fig. 6, following the third impulse: (a) relative displacement between the nonlinear attachment and the linear oscillator, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) on the Hamiltonian FEP

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Fig. 10

Initial conditions of the linear oscillator and the nonlinear attachment at the beginning of each of the ten impulsive cycles of the damped response depicted in Fig. 6

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Fig. 11

Experimental realization of the system of Fig. 1: configuration for (a) single impulse excitation, and (b) periodic impulse excitation

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Fig. 12

Damped experimental response for the single impulse excitation of normalized intensity I0 = 0.0198 m/s: (a) displacement of the nonlinear attachment, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) superimposed on the Hamiltonian FEP

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Fig. 13

Damped experimental response for single impulse excitation of normalized intensity I0 = 0.1454 m/s: (a) displacement of the nonlinear attachment, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) superimposed on the Hamiltonian FEP

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Fig. 18

Third cycle of the damped response of Fig. 14, following the third impulse: (a) relative displacement between the nonlinear attachment and the linear oscillator, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) on the Hamiltonian FEP

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Fig. 14

Experimental damped response for periodic impulse excitation of normalized intensities I0 = 0.2262 m/s (first cycle), I0 = 0.3806 m/s (second cycle), and I0 = 0.2829 m/s (third cycle): (a) relative displacement between the nonlinear attachment and the linear oscillator, and (b) displacement of the linear oscillator

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Fig. 15

Experimentally realized impulsive forces for the experimental test

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Fig. 16

First cycle of the damped response of Fig. 14, following the first impulse: (a) relative displacement between the nonlinear attachment and the linear oscillator, (b) displacement of the linear oscillator, (c) wavelet spectrum of relative the displacement, and (d) wavelet spectrum of (c) on the Hamiltonian FEP

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Fig. 17

Second cycle of the damped response of Fig. 14, following the second impulse: (a) relative displacement between the nonlinear attachment and the linear oscillator, (b) displacement of the linear oscillator, (c) wavelet spectrum of the relative displacement, and (d) wavelet spectrum of (c) on the Hamiltonian FEP

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