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Research Papers

Nonlinear Dynamic Responses of Elastic Structures With Two Rectangular Liquid Tanks Subjected to Horizontal Excitation

[+] 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(2), 021001 (Oct 15, 2010) (15 pages) doi:10.1115/1.4002382 History: Received July 22, 2009; Revised May 28, 2010; Published October 15, 2010; Online October 15, 2010

Nonlinear vibrations of an elastic structure with two partially filled liquid tanks subjected to horizontal harmonic excitation are investigated. The natural frequencies of the structure and sloshing satisfy the tuning condition 1:1:1 when tuned liquid dampers are used. The equations of motion for the structure and the modal equations of motion for the first, second, and third sloshing modes are derived by using Galerkin’s method, taking into account the nonlinearity of the sloshing. Then, van der Pol’s method is employed to determine the frequency response curves. It is found in calculating the frequency response curves that pitchfork bifurcation can occur followed by “localization phenomenon” for a specific excitation frequency range. During this range, sloshing occurs at different amplitudes in the two tanks, even if the dimensions of both tanks are identical. Furthermore, Hopf bifurcation may occur followed by amplitude- and phase-modulated motions including chaotic vibrations. In addition, Lyapunov exponents are calculated to prove the occurrence of both amplitude-modulated motions and chaotic vibrations. Bifurcation sets are also calculated to show the influence of the system parameters on the frequency response. Experiments were conducted to confirm the validity of the theoretical results. It was found that the theoretical results were in good agreement with the experimental data.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Theoretical model

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Figure 2

Frequency response curves for a single-tank system when ν=0.94, k=1.0, c=0.013, γ=0.15, h1=0.6, d1=0.667, ζn=0.015, and F0=0.0015: (a) structure, and (b) tank 1

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Figure 3

Frequency response curves of the displacements for a dual-tank system when ν=0.94, k=1.0, c=0.013, γ=0.15, h1=h2=0.6, d1=d2=0.333, ζn=ζ¯n=0.015, and F0=0.0015: (a) structure, (b) tank 1, and (c) tank 2

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Figure 4

As Fig. 3 when h1=h2=0.8 and d1=d2=0.25: (a) structure, (b) tank 1,and (c) tank 2

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Figure 5

As Fig. 3 when h1=h2=1.0 and d1=d2=0.20: (a) structure, (b) tank 1, and (c) tank 2

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Figure 6

Enlarged curves of Fig. 5: (a) structure, (b) tank 1, and (c) tank 2

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Figure 7

Phase angle curves corresponding to Fig. 6: (a) structure, (b) tank 1, and (c) tank 2

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Figure 8

Stationary time histories of the displacements for Fig. 6: (a) ω=0.950 (constant amplitudes), (b) ω=0.930 (period-one modulation), (c) ω=0.923 (period-two modulation), and (d) ω=0.913 (chaotic vibration)

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Figure 9

Poincaré maps for Fig. 6: (a) ω=0.930 (period-one modulation), (b) ω=0.923 (period-two modulation), and (c) ω=0.913 (chaotic vibrations)

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Figure 10

Lyapunov exponents corresponding to Fig. 6

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Figure 11

Bifurcation sets in the (ω,h1=h2) plane for F0=0.0015, including the cases of Figs.  34612.

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Figure 12

As Fig. 3 when h1=h2=1.7 and d1=d2=0.1176: (a) structure, (b) tank 1, and (c) tank 2

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Figure 13

Frequency response curves of the displacements for a dual-tank system when ν=0.97, k=1.0, c=0.013, γ=0.15, h1=h2=0.28, d1=d2=0.357, ζn=ζ¯n=0.015, and F0=0.0020: (a) structure, (b) tank 1, and (c) tank 2

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Figure 14

Bifurcation sets in the (ω,F0) plane for h1=h2=1.0, including the case of Fig. 6

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Figure 15

As Fig. 6 when h1=1.0, h2=1.01, d1=0.200, and d2=0.198: (a) structure, (b) tank 1, and (c) tank 2

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Figure 16

Experimental setup

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Figure 17

A picture of the experimental apparatus

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Figure 18

The comparison between the experimental and theoretical results for a single-tank system: (a) structure and (b) tank 1

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Figure 19

The comparison between the experimental and theoretical results for a dual-tank system: (a) structure, (b) tank 1, and (c) tank 2

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Figure 20

Time histories at f=2.561 Hz of Fig. 1: (a) the case during decreasing the excitation frequency and (b) the case after adding the disturbance

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Figure 21

The influence of the imperfection of the tank dimensions on the response curves: (a) structure, (b) tank 1, and (c) tank 2

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