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Research Papers

Nonlinear Responses of Dual-Pendulum Dynamic Absorbers

[+] Author and Article Information
Takashi Ikeda

Mechanical Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8527, Japantikeda@hiroshima-u.ac.jp

J. Comput. Nonlinear Dynam 6(1), 011012 (Oct 05, 2010) (11 pages) doi:10.1115/1.4002385 History: Received September 29, 2009; Revised June 10, 2010; Published October 05, 2010; Online October 05, 2010

The nonlinear responses of a single-degree-of-freedom system with two pendulum tuned mass dampers under horizontal sinusoidal excitation are investigated. In the theoretical analysis, van der Pol’s method is applied to determine the expressions for the frequency response curves. In the numerical results, the differences between the responses in single- and dual-pendulum systems are shown. A pitchfork bifurcation occurs followed by mode localization where both identical pendula vibrate at constant but different amplitudes. Hopf bifurcations occur, and then amplitude- and phase-modulated motions including chaotic vibrations appear in the identical dual-pendulum system. The Lyapunov exponents are calculated to prove the occurrence of chaotic vibrations. In a nonidentical dual-pendulum system, a perturbed pitchfork bifurcation occurs and saddle-node bifurcation points appear instead of pitchfork bifurcation points. Hopf bifurcations and amplitude- and phase-modulated motions also appear. The deviation of the tuning condition is also investigated by showing the frequency response curves and bifurcation sets. The numerical simulations are shown to be in good agreement with the theoretical results. In experiments, the imperfections of the two pendula were taken into consideration, and the validity of the theoretical analysis was confirmed.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

The model for theoretical analysis

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Figure 2

Frequency response curves for a single-pendulum system when μ0=0.9, k0=1.0, c0=0.01, μ1=0.1, l1=1.0, c1=0.008, and F0=0.045

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Figure 3

Frequency response curves of the amplitudes for a dual-pendulum system when μ0=0.9, k0=1.0, c0=0.01, μi=0.05, li=1.0, ci=0.004, and F0=0.045

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Figure 4

Frequency response curves of the phase angles corresponding to Fig. 3

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Figure 5

Enlarged plots for Fig. 3

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Figure 6

Enlarged plots for Fig. 4

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Figure 8

Stationary time histories for the phase angles of Fig. 5: (a) ω=0.885, (b) ω=0.880, and (c) ω=0.860

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Figure 9

Frequency response curves same as Fig. 3 except k0=0.9

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Figure 10

Enlarged plots for Fig. 9

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Figure 12

Lyapunov exponents of Fig. 1

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Figure 14

Enlarged plots same as Fig. 5 except μ2=0.045

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Figure 15

Bifurcation sets in the (ω,μ2) plane

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Figure 17

Plots same as Fig. 3 except l2=0.70

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Figure 18

Bifurcation sets in the (ω,l2) plane

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Figure 19

Experimental setup

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Figure 20

A picture of the experimental apparatus

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Figure 21

Experimental results for the single-pendulum system

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Figure 22

Experimental results for the dual-pendulum system

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Figure 7

Stationary time histories for the displacements of Fig. 5: (a) ω=0.885, (b) ω=0.880, and (c) ω=0.860

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Figure 11

Stationary time histories of Fig. 1. (a) ω=0.8725, (b) ω=0.870, (c) ω=0.860, and (d) ω=0.850.

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Figure 13

Bifurcation sets in the (ω,k0) plane

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Figure 16

Enlarged plots same as Fig. 5 except l2=0.98

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Figure 23

Time histories of Fig. 2: (a) f=2.210 Hz, (b) f=2.100 Hz, (c) f=2.050 Hz, and (d) f=2.030 Hz

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