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Research Papers

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum Arrays Subjected to Horizontal 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, Keisuke Nishimura

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

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 2, 2013; final manuscript received September 1, 2014; published online January 12, 2015. Assoc. Editor: Carmen M. Lilley.

J. Comput. Nonlinear Dynam 10(2), 021007 (Mar 01, 2015) (11 pages) Paper No: CND-13-1239; doi: 10.1115/1.4028474 History: Received October 02, 2013; Revised September 01, 2014; Online January 12, 2015

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. 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 presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant above which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

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References

Figures

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Fig. 2

Frequency response curves of amplitudes and NNMs for N = 2. (a) Pendulum 1 and (b) pendulum 2 when μ1 = μ2 = 1, l1 = l2 = 1, c1 = c2 = 0.02, K1 = 0.01, and a0 = 0.05.

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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 = 0.92 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. 5

Same as Fig. 2, but K1 = 0.03

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Fig. 6

Same as Fig. 2, but K1 = 0.08

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Fig. 7

Stationary time histories on branches GiHi in Fig. 6. (a) ω = 0.985, (b) ω = 0.968, and (c) ω = 0.945.

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Fig. 8

Lyapunov exponents for Fig. 6

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Fig. 9

Bifurcation sets in the (ω, K1) plane including Figs. 2, 5, and 6. ---- Saddle-node bifurcation set; -- -- - pitchfork bifurcation set; and --- - -- Hopf bifurcation set.

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Fig. 10

Same as Fig. 6, but l2 = 0.999

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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 = 0.01, and a0 = 0.05 (i = 1, 2, 3; j = 1, 2).

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Fig. 12

Stationary time histories at ω = 0.92 in Fig. 11. (a) Pattern II-1 on branches “bi” and (b) pattern III-1 on branches “ei.

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Fig. 13

Same as Fig. 11, but Kj = 0.03

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Fig. 14

Experimental setup

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Fig. 15

Photo of the experimental apparatus

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Fig. 16

Comparison between the theoretical and experimental results for apparatus A

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Fig. 17

Comparison between the theoretical and experimental results for apparatus B

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Fig. 18

Stationary time histories at f = 2.50 Hz in Fig. 17. (a) Pattern II-1 and (b) pattern II-2.

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Fig. 19

Comparison between the theoretical and experimental results for apparatus C

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Fig. 20

Stationary time histories at f = 2.51 Hz in Fig. 19. (a) Pattern II-1 and (b) pattern II-2.

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