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Research Papers

Intrinsic Localized Modes of Harmonic Oscillations in Nonlinear Oscillator Arrays

[+] Author and Article Information
Takashi Ikeda

e-mail: tikeda@hiroshima-u.ac.jp

Keisuke Nishimura

Department of Mechanical Systems Engineering Institute of Engineering,
Hiroshima University
1-4-1, Kagamiyama,
Higashi-Hiroshima, Hiroshima 739-8527 Japan

Contributed by Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received November 27, 2012; final manuscript received February 18, 2013; published online March 26, 2013. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 8(4), 041009 (Mar 26, 2013) (12 pages) Paper No: CND-12-1211; doi: 10.1115/1.4023866 History: Received November 27, 2012; Revised February 18, 2013

Intrinsic localized modes (ILMs) are investigated in an array with N Duffing oscillators that are weakly coupled with each other when each oscillator is subjected to sinusoidal excitation. The purpose of this study is to investigate the behavior of ILMs in nonlinear multi-degree-of-freedom (MDOF) systems. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are shown for N = 2 and 3 and compared with the results of the numerical simulations. Basins of attraction are shown for a two-oscillator array with hard-type nonlinearities to examine the possibility of appearance of ILMs when an oscillator is disturbed. The influences of the connecting springs for both hard- and soft-type nonlinearities on the appearance of the ILMs are examined. Increasing the values of the connecting spring constants may cause Hopf bifurcation followed by amplitude modulated motion (AMM) including chaotic vibrations. The influence of the imperfection of an oscillator is also investigated. Bifurcation sets are calculated to show the influence of the system parameters on the excitation frequency range of ILMs. Furthermore, time histories are shown for the case of N = 10, and many patterns of ILMs may appear depending on the initial conditions.

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References

Figures

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Fig. 4

Stationary time histories at ω = 1.10 in Fig. 2. (a) Pattern I on branches ai; (b) pattern II-1 on branches bi; (c) pattern II-2 on branches ci; and (d) pattern III on branches di.

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Fig. 3

Frequency response curves of the phase angles corresponding to Fig. 2

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Fig. 8

Stationary time histories on branches bi in Fig. 7(a)ω = 1.090; and (b) ω = 1.0871

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Fig. 9

Lyapunov exponents along branches bi in Fig. 7

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Fig. 2

Frequency response curves of the amplitudes for N = 2 when μi = 1.0, ki = 1.0, ci = 0.02, βi = 0.1, K1 = 0.01, and F = 0.05

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Fig. 13

Same as Fig. 11, but K1 = 0.04

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Fig. 16

Same as Fig. 13, but k2 = 0.995

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Fig. 17

Frequency resonance curves for N = 3 when μi = 1.0, ki = 1.0, ci = 0.02, βi = 0.1, Kj = 0.01, and F = 0.05 (i = 1, 2, 3; j = 1, 2)

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Fig. 18

Enlarged view of Fig. 17

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Fig. 5

Basins of attraction for Fig. 2 when oscillator 1 is disturbed on branches ai at (a) ω = 1.064, (b) ω = 1.10, and (c) ω = 1.15, and on branches di at (d) ω = 1.15, (e) ω = 1.10, and (f) ω = 1.064. Basin of attraction for ILMs.

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Fig. 6

Same as Fig. 2, but K1 = 0.02

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Fig. 7

Same as Fig. 2, but K1 = 0.04

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Fig. 10

Bifurcation sets in the (ω, K1) plane, including Figs. 2, 6, and 7. — Saddle-node bifurcation set; – – - pitchfork bifurcation set; – - – Hopf bifurcation set.

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Fig. 11

Frequency response curves of the amplitudes for N = 2 when μi = 1.0, ki = 1.0, ci = 0.02, βi = -0.1, K1 = 0.005, and F = 0.03

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Fig. 12

Same as Fig. 11, but K1 = 0.01

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Fig. 14

Bifurcation sets in the (ω, K1) plane including Figs. 11, 12, and 13. — Saddle-node bifurcation set; – – - pitchfork bifurcation set; – - – Hopf bifurcation set.

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Fig. 15

Same as Fig. 7, but k2 = 0.995

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Fig. 19

Stationary time histories at ω = 1.08 in Fig. 17(a) pattern II-1 on branches bi; and (b) pattern III-1 on branches ei

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Fig. 22

Bifurcation sets in the (ω, Kj) plane including Figs. 17, 20, and 21. — Saddle-node bifurcation set; – – - pitchfork bifurcation set; – - – Hopf bifurcation set

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Fig. 23

Stationary time histories for ILMs at ω = 1.08 when μi = 1.0, ki = 1.0, ci = 0.02, βi = 0.1, Kj = 0.01, and F = 0.05. (a) Six oscillators with high amplitudes; (b) three oscillators with high amplitudes; (c) one oscillator with high amplitudes

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Fig. 20

Same as Fig. 17, but Kj = 0.03

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Fig. 21

Same as Fig. 17, but Kj = 0.066

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