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Research Papers

Bifurcation Phenomena Caused by Multiple Nonlinear Vibration Absorbers

[+] Author and Article Information
Takashi Ikeda

Department of Mechanical System Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8527, Japantikeda@hiroshima-u.ac.jp

J. Comput. Nonlinear Dynam 5(2), 021012 (Mar 02, 2010) (15 pages) doi:10.1115/1.4000790 History: Received January 05, 2009; Revised April 27, 2009; Published March 02, 2010

The characteristics of two, three, and four nonlinear vibration absorbers or nonlinear tuned mass dampers (NTMDs) attached to a structure under harmonic excitation are investigated. The frequency response curves are theoretically determined using van der Pol’s method. When the parameters of the absorbers are equal, it is found from the theoretical analysis that pitchfork bifurcations may occur on the part of the response curves, which are unstable in the multi-absorber systems, but are stable in a system with one NTMD. Multivalued steady-state solutions, such as three steady-state solutions for a dual-absorber system with different amplitudes, five steady-state solutions for a triple-absorber system, and seven steady-state solutions for a quadruple-absorber system, appear near bifurcation points. The NTMDs behave in that one of them vibrates at high amplitudes while the others vibrate at low amplitudes, even if the dimensions of the NTMDs are identical. Namely, “localization phenomenon” or “mode localization” occurs. After the pitchfork bifurcation, Hopf bifurcations may occur depending on the values of the system parameters, and amplitude- and phase-modulated motions, including chaotic vibrations, appear after the Hopf bifurcation when the excitation frequency decreases. Lyapunov exponents are numerically calculated to prove the occurrence of chaotic vibrations. Bifurcation sets are also calculated to investigate the influence of the system parameters on the response of the systems.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Theoretical model

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Figure 2

Frequency response curves for a single absorber system when μ0=0.88, k0=0.88, c0=0.008, μ1=0.12, k1=0.12, c1=0.015, β1=−0.2, and F0=0.02; (a) structure and (b) absorber 1

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Figure 3

Frequency response curves of the displacements for a dual-absorber system when μ0=0.88, k0=0.88, c0=0.008, μ1=μ2=0.06, k1=k2=0.06, c1=c2=0.0075, β1=β2=−0.1, and F0=0.02; (a) structure, (b) absorber 1, and (c) absorber 2

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Figure 4

Frequency response curves of the phase angles for the dual-absorber system corresponding to Fig. 3: (a) structure, (b) absorber 1, and (c) absorber 2

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Figure 5

Enlarged curves of Fig. 3: (a) structure, (b) absorber 1, and (c) absorber 2

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Figure 6

Enlarged curves of Fig. 4: (a) structure, (b) absorber 1, and (c) absorber 2

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Figure 7

Stationary time histories of the displacements for Fig. 5: (a) ω=0.840, (b) ω=0.820, (c) ω=0.804, and (d) ω=0.7998

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Figure 8

Stationary time histories of the phase angles for Fig. 6: (a) ω=0.840, (b) ω=0.820, (c) ω=0.804, and (d) ω=0.7998

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Figure 9

Lyapunov exponents corresponding to Fig. 5

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Figure 10

As Fig. 2 when c1=0.006 and β1=0.2

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Figure 11

As Fig. 3 when c1=c2=0.003 and β1=β2=0.1: (a) structure, (b) absorber 1, and (c) absorber 2

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Figure 12

As Fig. 3 when β1=−0.1 and β2=−0.105

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Figure 13

As Fig. 3 when β1=−0.1 and β2=0: (a) structure, (b) absorber 1, and (c) absorber 2

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Figure 14

Bifurcation sets for the dual-absorber system. (a) Loci of the bifurcation points in the (ω,β1=β2) plane, including the case of Fig. 5. (b) Loci of the bifurcation points in the (ω,β2) plane when β1=−0.1, including the case of Figs.  51213.

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Figure 15

As Fig. 3 when k0=0.82(σ1=σ2=−0.035)

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Figure 16

As Fig. 3 when k0=0.96(σ1=σ2=0.044)

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Figure 17

Frequency response curves for a triple-absorber system when μ0=0.88, k0=0.88, c0=0.008, μi=0.04, ki=0.04, ci=0.005, βi=−0.0667, and F0=0.02

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Figure 18

Stationary time histories for Fig. 1: (a) ω=0.820 (b) ω=0.7812

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Figure 19

Lyapunov exponents corresponding to Fig. 1

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Figure 20

Bifurcation sets in the (ω,β3) plane for the triple-absorber system, including the cases of Figs.  1721

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Figure 21

As Fig. 1 when β3=−0.02: (a) structure, (b) absorber 1, (c) absorber 2, and (d) absorber 3

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Figure 22

Frequency response curves for a quadruple-absorber system when μ0=0.88, k0=0.88, c0=0.008, μi=0.03, ki=0.03, ci=0.00375, βi=−0.05, and F0=0.02

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Figure 23

Stationary time histories for Fig. 2: (a) ω=0.800 (b) ω=0.7818

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Figure 24

Lyapunov exponents corresponding to Fig. 2

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Figure 25

Bifurcation sets in the (ω,β4) plane for the quadruple-absorber system, including the cases of Figs.  2226

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Figure 26

As Fig. 2 when β4=−0.02: (a) structure, (b) absorber 1, (c) absorber 2, (d) absorber 3, and (e) absorber 4

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