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.

References

References
1.
Ibrahim
,
R. A.
,
2005
,
Liquid Sloshing Dynamics
,
Cambridge University Press
,
Cambridge, UK
.
2.
Hagiuda
,
H.
,
1989
, “
Oscillation Control System Exploiting Fluid Force Generated by Water Sloshing
,”
Mitsui Zosen Tech. Rev.
,
137
, pp.
13
20
(in Japanese).
3.
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).
4.
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
.10.1006/jsvi.1996.0722
5.
Kaneko
,
S.
, and
Ishikawa
,
M.
,
1999
, “
Modeling of Tuned Liquid Damper With Submerged Nets
,”
J. Pressure Vessel Technol
,
121
(
3
), pp.
334
343
.10.1115/1.2883712
6.
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
.10.1016/0022-460X(75)90213-8
7.
Ikeda
,
T.
,
2003
, “
Nonlinear Parametric Vibrations of an Elastic Structure With a Rectangular Liquid Tank
,”
Nonlinear Dyn.
,
33
(
1
), pp.
43
70
.10.1023/A:1025569028213
8.
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
.10.1016/j.jsv.2006.11.037
9.
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
.10.1115/1.4002382
10.
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
.10.1016/j.ijnonlinmec.2009.11.012
11.
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
,
July
18–22
, pp.
1
10
.
12.
Stoker
,
J. J.
,
1950
,
Nonlinear Vibrations
,
John Wily & Sons
,
New York
.
13.
Awrejcewicz
,
J.
, and
Krysko
,
V. A.
,
2006
,
Introduction to Asymptotic Methods
,
Chapman and Hall, CRC Press
,
Boca Raton, NY
.
14.
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
.
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