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Research Papers

Autoparametric Resonances of Elastic Structures Coupled With Two Sloshing Modes in a Square Liquid Tank

[+] Author and Article Information
Takashi Ikeda

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

Masaki Takashima

Imabari Shipbuilding Co., Ltd.,
1-4-52 Koura-cho, Imabari,
Ehime 799-2195Japan

Yuji Harata

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

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 30, 2011; final manuscript received March 22, 2012; published online June 14, 2012. Assoc. Editor: Eric A. Butcher.

J. Comput. Nonlinear Dynam 8(1), 011007 (Jun 14, 2012) (13 pages) Paper No: CND-11-1167; doi: 10.1115/1.4006531 History: Received September 30, 2011; Revised March 22, 2012

Nonlinear vibrations of an elastic structure coupled with liquid sloshing in a square tank subjected to vertical sinusoidal excitation are investigated. In the theoretical analysis, the ratios of the natural frequencies of the structure and two sloshing modes satisfy 2:1:1. The equations of motion for the structure and seven sloshing modes are derived using Galerkin’s method while considering the nonlinearity of sloshing. The linear damping terms are then incorporated into the modal equations to consider the damping effect of sloshing. The frequency response curves are determined using van der Pol’s method. The influences of the liquid level, the aspect ratio of the tank cross-section, the deviation of the tuning condition, and the excitation amplitude are investigated. When the liquid level is high, and depending on the excitation frequency, there are three patterns of sloshing: (i) both (1,0) and (0,1) sloshing modes appear at identical amplitudes; (ii) these two modes appear at different amplitudes; and (iii) either (1,0) or (0,1) mode appears. Small deviations of the tuning condition may cause Hopf bifurcations to occur followed by amplitude modulated motion including chaotic vibrations. Bifurcation sets are also calculated to illustrate the influence of the system parameters on the response of the system. It is found that for low liquid levels, square tuned liquid dampers (TLDs) work more effectively than rectangular TLDs. Experiments were also conducted in order to confirm the validity of the theoretical results and were in good agreement with the experimental data.

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References

Ibrahim, R. A., 2005, Liquid Sloshing Dynamics, Cambridge University Press, Cambridge, UK.
Hagiuda, H., 1989, “Oscillation Control System Exploiting Fluid Force Generated by Water Sloshing,” Mitsui Zosen Tech. Rev., 137, pp. 13–20 (in Japanese).
Fujino, Y., Pacheco, M., Sun, L.-M., Chaiseri, P., and Isobe, M., 1989, “Simulation of Nonlinear Wave in Rectangular Tuned Liquid Damper (TLD) and Its Verification,” Trans. Jpn. Soc. Civ. Eng., 35A, pp. 561–574 (in Japanese).
Ikeda, T., and Nakagawa, N., 1997, “Non-Linear Vibrations of a Structure Caused by Water Sloshing in a Rectangular Tank,” J. Sound Vib., 201(1), pp. 23–41. [CrossRef]
Kaneko, S., and Ishikawa, M., 1999, “Modeling of Tuned Liquid Damper With Submerged Nets,” J. Pressure Vessel Technol, 121(3), pp. 334–343. [CrossRef]
Ibrahim, R. A., and Barr, A. D. S., 1975, “Autoparametric Resonance in a Structure Containing Liquid, Part I: Two Mode Interaction,” J. Sound Vib., 42(2), pp. 159–175. [CrossRef]
Ikeda, T., 2003, “Nonlinear Parametric Vibrations of an Elastic Structure With a Rectangular Liquid Tank,” Nonlinear Dyn., 33(1), pp. 43–70. [CrossRef]
Ikeda, T., 2007, “Autoparametric Resonances in Elastic Structures Carrying Two Rectangular Tanks Partially Filled With Liquid,” J. Sound Vib., 302(4,5), pp. 657–682. [CrossRef]
Ikeda, T., 2011, “Nonlinear Dynamic Responses of Elastic Structures With Two Rectangular Liquid Tanks Subjected to Horizontal Excitation,” ASME J. Comput. Nonlinear Dyn., 6(2), p. 021001. [CrossRef]
Ikeda, T., 2010, “Non-Linear Dynamic Responses of Elastic Two-Story Structures With Partially Filled Liquid Tanks,” Int. J. Non-Linear Mech., 45(3), pp. 263–278. [CrossRef]
Ikeda, T., 2010, “Vibration Suppression of Elastic Structures Utilizing Internal Resonance of Liquid Sloshing in a Rectangular Tank,” Proceedings of the ASME 2010 PVP/ K-PVP Conference, Bellevue, WA, July18–22, pp. 1–10.
Stoker, J. J., 1950, Nonlinear Vibrations, John Wily & Sons, New York.
Awrejcewicz, J., and Krysko, V. A., 2006, Introduction to Asymptotic Methods, Chapman and Hall, CRC Press, Boca Raton, NY.
Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B., and Wang, X., 1997, Continuation and Bifurcation Software for Ordinary Differential Equations (With HomCont), AUTO97, Concordia University, Canada.

Figures

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Fig. 1

Theoretical model of an SDOF structure coupled with a TLD

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Fig. 2

Shapes of two predominant sloshing modes. (a) (1,0) mode, (b) (0,1) mode.

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Fig. 3

Frequency response curves of (a) z0, (b) (1,0) mode, and (c) (0,1) mode when ν0 = 0.93,ν1 = 0.07, k = 4.0, c = 0.02, h = 0.6, w = 2.0, ζij = 0.015, and F0 = 0.0013

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Fig. 4

Same as Fig. 3, but w = 1.0

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Fig. 5

Phase angle curves corresponding to Fig. 4

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Fig. 6

Stationary time histories including ηx at (x, y) = (0, w/2) and ηy at (x, y) = ( 1/2, 0) in Fig. 4 (a)ω = 2.035, (b) ω = 2.033, (c) ω = 2.0 (inphase), (d) ω = 2.0 (out-of-phase)

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Fig. 7

Same as Fig. 4, but h = 0.35

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Fig. 8

Same as Fig. 4, but h = 0.28

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Fig. 9

Same as Fig. 8, but w = 1.01

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Fig. 10

Bifurcation sets in the (ω, w) plane, including the cases of Figs. 8 and 9. Solid curve, pitchfork bifurcation set; dashed curve, Hopf bifurcation set; dashed-dotted curve, saddle-node bifurcation set.

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Fig. 11

Same as Fig. 8, but k = 4.13

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Fig. 12

Same as Fig. 8, but k = 4.20

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Fig. 13

Stationary time histories including ηx at (x, y) = (0, w/2) and ηy at (x, y) = ( 1/2, 0) at ω = 2.055 in Fig. 12

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Fig. 14

Same as Fig. 4, but k = 3.8

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Fig. 15

Stationary time histories including ηx at (x, y) = (0, w/2) and ηy at (x, y) = ( 1/2, 0) at ω = 1.935 in Fig. 14

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Fig. 16

Same as Fig. 4, but F0 = 0.0010

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Fig. 17

Same as Fig. 4, but F0 = 0.0017

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Fig. 18

Experimental setup

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Fig. 19

Photo of the experimental apparatus

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Fig. 20

Comparison between the theoretical and experimental results for apparatus A

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Fig. 21

Stationary time histories in Fig. 20(a)f = 5.441 Hz (inphase), (b) f = 5.441 Hz (out-of-phase), (c) f = 5.540 Hz

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Fig. 22

Comparison between the theoretical and experimental results for apparatus B

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Fig. 23

Comparison between the theoretical and experimental results for apparatus C

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Fig. 24

Stationary time histories at f = 5.250 Hz in Fig. 23

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