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Research Papers

Intrinsic Localized Modes of Principal Parametric Resonances in Pendulum Arrays Subjected to Vertical Excitation

[+] Author and Article Information
Takashi Ikeda

Department of Mechanical Systems Engineering,
Institute of Engineering,
Hiroshima University,
1-4-1, Kagamiyama, Higashi-Hiroshima,
Hiroshima 739-8527, Japan
e-mail: tikeda@hiroshima-u.ac.jp

Yuji Harata

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

Chongyue Shi

Ship Machinery Department,
Miura Co., Ltd.,
7, Horie-cho,
Matsuyama,
Ehime 799-2696, Japan

Keisuke Nishimura

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

Manuscript received November 24, 2014; final manuscript received March 24, 2015; published online April 28, 2015. Assoc. Editor: Daniel J. Segalman.

J. Comput. Nonlinear Dynam 10(5), 051017 (Sep 01, 2015) (12 pages) Paper No: CND-14-1298; doi: 10.1115/1.4030215 History: Received November 24, 2014; Revised March 24, 2015; Online April 28, 2015

Intrinsic localized modes (ILMs) are investigated in an N-pendulum array subjected to vertical harmonic excitation. The pendula behave nonlinearly and are coupled with each other because they are connected by torsional, weak, linear springs. In the theoretical analysis, van der Pol's method is employed to determine the expressions for frequency response curves for the principal parametric resonance, considering the nonlinear restoring moment of the pendula. In the numerical results, frequency response curves for N = 2 and 3 are shown to examine the patterns of ILMs, and demonstrate the influences of the connecting spring constants and the imperfections of the pendula. Bifurcation sets are also calculated to show the excitation frequency range and the conditions for the occurrence of ILMs. Increasing the connecting spring constants results in the appearance of Hopf bifurcations. The numerical simulations reveal the occurrence of ILMs with amplitude modulated motions (AMMs), including chaotic motions. ILMs were observed in experiments, and the experimental data were compared with the theoretical results. The validity of the theoretical analysis was confirmed by the experimental data.

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References

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Figures

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Fig. 2

Frequency response curves of amplitudes for N = 2: (a) pendulum 1 and (b) pendulum 2 when μi = 1.0, li = 1.0, ci = 0.02, K1 = 0.03, and a0 = 0.013

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Fig. 3

Frequency response curves of phase angles corresponding to Fig. 2: (a) pendulum 1 and (b) pendulum 2

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Fig. 4

Stationary time histories at ω = 1.90 in Fig. 2: (a) pattern I on branches ai, (b) pattern II-1 on branches bi, and (c) pattern II-2 on branches ci

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Fig. 5

Frequency response curves of amplitudes for N = 2: (a) pendulum 1 and (b) pendulum 2 when μi = 1.0, li = 1.0, ci = 0.02, K1 = 0.015, and a0 = 0.013. (c) Enlarged view of (a), (d) Enlarged view of (b).

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Fig. 6

Frequency response curves of amplitudes for N = 2: (a) pendulum 1, and (b) pendulum 2 when μi = 1.0, li = 1.0, ci = 0.02, K1 = 0.06, and a0 = 0.013

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Fig. 7

Stationary time histories on branches bi in Fig. 6: (a) ω = 1.935, and (b) ω = 1.925

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Fig. 8

Lyapunov exponents along branches di and bi in Fig. 6

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Fig. 9

Bifurcation sets in the (ω, K1) plane including Figs. 2, 5, and 6

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Fig. 10

Frequency response curves of amplitudes for N = 2: (a) pendulum 1 and (b) pendulum 2 when μi = 1.0, l1 = 1.0, l2 = 0.999, ci = 0.02, K1 = 0.03, and a0 = 0.013

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Fig. 11

Frequency response curves of amplitudes for N = 3 (a) pendulum 1, (b) pendulum 2, and (c) pendulum 3 when μi = 1.0, li = 1.0, ci = 0.02, Kj = 0.02, and a0 = 0.013

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Fig. 12

Spatial profiles of oscillations on branches ai–hi at ω = 1.90 in Fig. 11

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Fig. 13

Stationary time histories at ω = 1.90 in Fig. 12: (a) pattern II-1 on branches bi, and (b) pattern IV-1 on branches di

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Fig. 14

Frequency response curves for N = 3 when μi = 1.0, li = 1.0, ci = 0.02, Kj = 0.06, and a0 = 0.013

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Fig. 15

Stationary time histories at ω = 1.935 in Fig. 14

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Fig. 16

Sketch of the experimental setup

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Fig. 17

Photo of the experimental apparatus

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Fig. 18

Comparison between the theoretical and experimental results for apparatus A

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Fig. 19

Comparison between the theoretical and experimental results for apparatus B

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Fig. 20

Stationary time histories at f = 4.50 Hz in Fig. 19: (a) pattern II-1, and (b) pattern II-2

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Fig. 21

Comparison between the theoretical and experimental results for apparatus C

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Fig. 22

Stationary time histories in Fig. 21: (a) on branches ci at f = 4.74 Hz and (b) On branches bi f = 4.93 Hz

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