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Research Papers

Energy Transfer and Dissipation in a Duffing Oscillator Coupled to a Nonlinear Attachment

[+] Author and Article Information
R. Viguié1

Department Aerospace and Mechanical Engineering, Structural Dynamics Research Group, University of Liège, Liège 4000, Belgiumr.viguie@ulg.ac.be

M. Peeters

Department Aerospace and Mechanical Engineering, Structural Dynamics Research Group, University of Liège, Liège 4000, Belgiumm.peeters@ulg.ac.be

G. Kerschen

Department Aerospace and Mechanical Engineering, Structural Dynamics Research Group, University of Liège, Liège 4000, Belgiumg.kerschen@ulg.ac.be

J.-C. Golinval

Department Aerospace and Mechanical Engineering, Structural Dynamics Research Group, University of Liège, Liège 4000, Belgiumjc.golinval@ulg.ac.be

1

Corresponding author.

J. Comput. Nonlinear Dynam 4(4), 041012 (Sep 02, 2009) (13 pages) doi:10.1115/1.3192130 History: Received June 14, 2008; Revised November 19, 2008; Published September 02, 2009

The dynamics of a two-degree-of-freedom nonlinear system consisting of a grounded Duffing oscillator coupled to an essentially nonlinear attachment is examined in the present study. The underlying Hamiltonian system is first considered, and its nonlinear normal modes are computed using numerical continuation and gathered in a frequency-energy plot. Based on these results, the damped system is then considered, and the basic mechanisms for energy transfer and dissipation are analyzed.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Linear oscillator coupled to a light-weight NES

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Figure 2

NES performance when coupled to a LO. (a) Energy dissipated in the NES against the linear stiffness of the primary system and the impulse magnitude; (b) contour plot; and (c) two-dimensional section for k1=1 N/m.

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Figure 3

FEP of a LO coupled to an NES (m1=k1=knl2=1, m2=0.05)

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Figure 4

Close-up of the S11+ branch. The NNMs in the configuration space are inset. The horizontal and vertical axes in these plots depict the displacement of the NES and primary system, respectively. Furthermore, the aspect ratio is set so that increments on the horizontal and vertical axes are equal in size, enabling one to directly deduce whether the motion is localized in the linear or the nonlinear oscillator, respectively.

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Figure 5

Pseudo-arclength continuation method

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Figure 6

Duffing oscillator coupled to a light-weight NES

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Figure 7

NES performance when coupled to a Duffing oscillator. (a) Energy dissipated in the NES against the nonlinear stiffness of the Duffing oscillator and the impulse magnitude; (b) contour plot; and (c) two-dimensional section for knl1=1.

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Figure 8

Dynamics in region No. 1 in Fig. 7: knl1=9 N/m3 and ẋ1(0)=0.13 m/s. (a) Duffing oscillator response; (b) NES response; (c) instantaneous total energy in the NES; (d) close-up of the Duffing oscillator and NES responses; (e) energy dissipated in the NES.

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Figure 9

Dynamics in region No. 2 in Fig. 7: knl1=1.3 N/m3 and ẋ1(0)=7 m/s. (a) Duffing oscillator response; (b) NES response; (c) instantaneous total energy in the NES; (d) close-up of the Duffing oscillator and NES responses (early-time responses); (e) energy dissipated in the NES; (f) close-up of the Duffing oscillator and NES responses (late-time responses).

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Figure 10

Dynamics in region No. 3 of Fig. 7: knl1=2.9 N/m3 and ẋ1(0)=1.9 m/s. (a) Duffing oscillator response; (b) NES response; (c) instantaneous total energy in the NES; (d) close-up of the Duffing oscillator and NES responses (early-time responses); (e) energy dissipated in the NES; (f) close-up of the Duffing oscillator and NES responses (late-time responses).

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Figure 11

Frequency-energy plots. (a) Duffing oscillator coupled to an NES; (b) linear oscillator coupled to an NES.

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Figure 12

Close-up of several branches of the FEP of a Duffing oscillator coupled to an NES. (a) S11−, (b) S11+, (c) S13, (d) S31, (e) U54, and (f) U32.

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Figure 13

Locus of impulsive orbits: computed orbits and their estimated loci are represented by circles and a dashed line, respectively

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Figure 14

Interpretation of the dynamical mechanisms related to high energy dissipation. Left column: WT; right column: WT superposed to the FEP. (a) and (b): knl1=9 N/m3 and ẋ1(0)=0.13 m/s (region No. 1); (c) and (d): knl1=1.3 N/m3 and ẋ1(0)=7 m/s (region No. 2); (e) and (f): knl1=2.9 N/m3; and ẋ1(0)=1.9 m/s (region No. 3).

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Figure 15

NES performance. (a) c2=0.002 N s/m and (b) c2=0.01 N s/m.

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