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Research Papers

Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study

[+] Author and Article Information
Francesco Romeo

Associate Professor
Department of Structural
and Geotechnical Engineering,
SAPIENZA University of Rome,
Via Gramsci 53,
Rome 00197, Italy
e-mail: francesco.romeo@uniroma1.it

Grigori Sigalov

College of Engineering,
University of Illinois at Urbana–Champaign
Champaign, IL 61820

Lawrence A. Bergman, Alex F. Vakakis

Professor
Mem. ASME
College of Engineering,
University of Illinois at Urbana–Champaign
Champaign, IL 61820

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received June 10, 2013; final manuscript received March 17, 2014; published online September 12, 2014. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 10(1), 011007 (Sep 12, 2014) (13 pages) Paper No: CND-13-1133; doi: 10.1115/1.4027224 History: Received June 10, 2013; Revised March 17, 2014

The conservative and dissipative dynamics of a 2DOF, system composed of a grounded linear oscillator coupled to a lightweight mass by means of both strongly nonlinear and linear negative stiffnesses is investigated. Numerical studies are presented aiming to assess the influence of this combined coupling on the transient dynamics. In particular, these studies are focused on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to the nonlinear bistable lightweight attachment. It is shown that the main feature of the proposed configuration is the ability of assuring broadband efficient energy transfer over a broad range of input energy. Due to the bistability of the attachment, such favorable behavior is triggered by different nonlinear dynamic mechanisms depending on the energy level. For high energy levels, strongly modulated oscillations occur, and the dynamics is governed by fundamental (1:1) and superharmonic (1:3) resonances; for low energy levels, chaotic cross-well oscillations of the nonlinear attachment as well as subharmonic resonances lead to strong energy exchanges between the two oscillators. The results reported in this work indicate that properly designed attachments of this type can be efficient absorbers and dissipators of impulsively induced vibration energy.

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Figures

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Fig. 1

The model consisting of a linear oscillator of mass m1 coupled to a lightweight attachment m2 through negative linear (k2) and cubic nonlinear (k3) stiffnesses

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Fig. 2

Potential energy surface: (a) without linear stiffness (C0 = 0.0, C = 1.0); (b) with negative linear stiffness (C0 = 0.03, C = 1.0)

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Fig. 3

Approximate frequency-energy plot for different values of negative stiffness (ɛ = 0.05, C = 1.0): (a) S11± in the frequency range (0.0–2.0 rad/s); (b) energy range ΔE between the S11 saddle nodes A and B; (c) S11+ branches (0.0–1.0 rad/s); (d) S11 branches (0.95–2.0 rad/s)

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Fig. 4

Dependence of the optimal TET energy intervals ΔE with respect to ΔW: energy interval ΔE  for C = 1 and 0.1 C0 0.0

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Fig. 5

Comparison between analytical (solid) and numerical (dashed) frequency-energy plots for varying negative stiffness: (a) C0 = 0.0; (b) C0 = 0.03; (c) C0 = 0.12

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Fig. 6

Frequency-energy plot for C0 = 0.03. The points highlighted correspond to nonlinear normal modes for different energy levels. Region III, E = 0.05 (A1,A2); region II, E = 0.006 (B1, B2), and region I, E = 0.0008 (C1).

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Fig. 7

Poincaré section at energy level E = 0.05 (C0 = 0.03) and associated NNMs, v (red/light gray, x (blue/dark gray): (a) NES phase-space v,v·; (b) in-phase NNM (A1), and (c) out-of-phase NNM (A2)

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Fig. 8

Poincaré section at energy level E = 0.006 (C0 = 0.03) and associated NNMs, v (red), x (blue): a) NES phase-space v,v·; (b) in-phase NNM (B1); (c) LPT as predicted in Ref. [1], and (d) out-of-phase NNM (B2)

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Fig. 9

Poincaré section at energy level E = 0.0008 (C0 = 0.03) and associated NNMs, v (red/light gray), x (blue/dark gray): (a) NES phase-space v,v·; (b) in-phase NNM (C1); (c), and (d) nonconventional NNMs

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Fig. 10

Poincaré section at energy level E = 1.0 × 10−6 (C0 = 0.03) and associated NNMs, v (red/light gray), x (blue/dark gray): (a) NES phase-space v,v·; (b) in-well in-phase NNM; (c) in-well low amplitude nonlinear beat, and (d) in-well out-of-phase NNM

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Fig. 11

Time series of the responses of the linear and nonlinear oscillators for different values of input excitation; (a), (c), and (e) linear oscillator; (b), (d), and (f) nonlinear oscillator; (a) and (b) low input energy X = 0.05; (c) and (d) intermediate input energy X = 0.12; (e) and (f) high input energy X = 0.5

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Fig. 12

Main transition stages of the dissipative system displacement and corresponding representation in the configuration space in the background of the potential energy surface: (a) X = 0.05; (b) X = 0.12, and (c) X = 0.5

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Fig. 13

Time-frequency analysis of nonlinear and linear oscillator responses for C0 = 0.03, X = 0.5. (a) and (b) LO and NES displacement time histories; (c) and (d) wavelet spectra of the LO and NES responses; (e) total instantaneous energy of the system, and (f) wavelet spectrum of the NES displacement superimposed to the FEP of the underlying Hamiltonian system.

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Fig. 14

Time-frequency analysis of nonlinear and linear oscillator responses for C0 = 0.05, X = 0.5. (a) and (b) LO and NES displacement time histories; (c) and (d) wavelet spectra of the LO and NES responses; (e) total instantaneous energy of the system, and (f) wavelet spectrum of the NES displacement superimposed to the FEP of the underlying Hamiltonian system.

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Fig. 15

Time-frequency analysis of nonlinear and linear oscillator responses for C0 = 0.07, X = 0.5. (a) and (b) LO and NES displacement time histories; (c) and (d) wavelet spectra of the LO and NES responses; (e) total instantaneous energy of the system, and (f) wavelet spectrum of the NES displacement superimposed to the FEP of the underlying Hamiltonian system.

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Fig. 16

Dissipated energy for varying negative stiffness

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Fig. 17

Time of energy decay to 1% of the initial energy: λ1 = λ2 = 0.015 ((a) and (b)); λ1 = λ2 = 0.0015 ((c) and (d)); mass ratio ɛ = 0.05 ((a) and (c)); mass ratio ɛ = 0.1 ((b) and (d)); v0 represents the initial velocity of the linear oscillator which is identical to the intensity of the applied impulse

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