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Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation

Published online by Cambridge University Press:  01 May 2012

Takashi Ikeda ,
Raouf A. Ibrahim ,
Yuji Harata  and
Tasuku Kuriyama
Takashi Ikeda*
Affiliation:
Department of Mechanical Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8527, Japan
Raouf A. Ibrahim
Affiliation:
Department of Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA
Yuji Harata
Affiliation:
Department of Mechanical Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8527, Japan
Tasuku Kuriyama
Affiliation:
Terumo Corporation, 2-44-1, Hatagaya, Shibuya-ku, Tokyo 151-0072, Japan
*
†Email address for correspondence: [email protected]
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Abstract

Nonlinear responses of surface waves in rigid square and nearly square tanks partially filled with liquid subjected to obliquely horizontal, sinusoidal excitation are investigated theoretically and experimentally. Two predominant modes of sloshing are significantly coupled nonlinearly because their natural frequencies are nearly identical resulting in 1:1 internal resonance. Therefore, if only one of these modes is directly excited, the other mode is indirectly excited due to the nonlinear coupling. In the nonlinear theoretical analysis, the modal equations of motion are derived for the two predominant sloshing modes as well as five higher sloshing modes. The linear viscous terms are incorporated in order to consider the damping effect of sloshing. The expressions for the frequency response curves are determined using van der Pol’s method. The influences of the excitation direction and the aspect ratio of the tank cross-section on the frequency response curves are numerically examined. Planar and swirl motions of sloshing, and Hopf bifurcations followed by amplitude modulated motions including chaotic motions, are predicted when the excitation frequency is close to one of the natural frequencies of the two predominant sloshing modes. Lyapunov exponents are calculated and reveal the excitation frequency range over which liquid chaotic motions occur. In addition, bifurcation sets are shown to clarify the influences of the parameters on the change in the structural stability. The theoretically predicted results are in good agreement with the measured data, thus the theoretical analysis was experimentally validated.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 700 , 10 June 2012 , pp. 304 - 328
Copyright
Copyright © Cambridge University Press 2012

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References

1. Abramson, H. N. 1966 The dynamic behavior of liquids in moving contains. NASA SP-106.Google Scholar
2. Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B. & Wang, X. 1997 Continuation and bifurcation software for ordinary differential equations (with HomCont), AUTO97. Concordia University.Google Scholar
3. Faltinsen, O. M., Rognebakke, O. F., Lukovsky, I. A. & Timokha, A. N. 2000 Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J. Fluid Mech. 407, 201–234.Google Scholar
4. Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2003 Resonant three-dimensional nonlinear sloshing in a square-base basin. J. Fluid Mech. 487, 1–42.CrossRefGoogle Scholar
5. Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2005 Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 2. Effect of higher modes. J. Fluid Mech. 523, 199–218.CrossRefGoogle Scholar
6. Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2006a Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 3. Base ratio perturbations. J. Fluid Mech. 551, 93–116.CrossRefGoogle Scholar
7. Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2006b Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square base tank with a finite fluid depth. Phys. Fluids 18 (1), 012103.CrossRefGoogle Scholar
8. Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.Google Scholar
9. Feng, Z. C. 1997 Transition to travelling waves from standing waves in a rectangular container subjected to horizontal excitations. Phys. Rev. Lett. 73 (3), 415–418.CrossRefGoogle Scholar
10. Feng, Z. C. 1998 Coupling between neighbouring two-dimensional modes of water waves. Phys. Fluids 10 (9), 2405–2411.CrossRefGoogle Scholar
11. Funakoshi, M. & Inoue, S. 1988 Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192, 219–247.CrossRefGoogle Scholar
12. Hayama, S., Aruga, K. & Watanabe, T. 1983 Nonlinear response of sloshing in rectangular tanks (1st report, nonlinear response of surface elevation). Bull. JSME 26 (219), 1641–1648.Google Scholar
13. Hill, D. F. 2003 Transient and steady-state amplitudes of forced waves in rectangular basins. Phys. Fluids 15 (6), 1576–1587.CrossRefGoogle Scholar
14. Hutton, R. E. 1963 An investigation of resonant, nonlinear, nonplaner free surface oscillations of a fluid. NASA Tech. Note D-1870, 1–64.Google Scholar
15. Ibrahim, R. A. 2005 Liquid Sloshing Dynamics. Cambridge University Press.CrossRefGoogle Scholar
16. Ikeda, T. 2003 Nonlinear parametric vibrations of an elastic structure with a rectangular liquid tank. Nonlinear Dyn. 33 (1), 43–70.CrossRefGoogle Scholar
17. Ikeda, T. & Ibrahim, R. A. 2008 Nonlinear surface waves in a square liquid tank under obliquely horizontal excitation. In Proceedings of the 22nd International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, August 24–29. ISBN 978-0-9805142.Google Scholar
18. Ikeda, T. & Nakagawa, N. 1997 Non-linear vibrations of a structure caused by water sloshing in a rectangular tank. J. Sound Vib. 201 (1), 23–41.CrossRefGoogle Scholar
19. Kimura, K., Takahara, H. & Ogura, H. 1996 Three-dimensional sloshing analysis in a rectangular tank subjected to pitching excitation. Trans. Japan Soc. Mech. Engng C 62 (596), 1285–1294 (in Japanese).CrossRefGoogle Scholar
20. Miles, J. W. 1984 Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.Google Scholar
21. Moiseev, N. N. 1958 To the theory of nonlinear oscillations of a limited liquid volume. Appl. Math. Mech. (PMM) 22, 612–621 (in Russian).CrossRefGoogle Scholar
22. Royon-Lebeaud, A., Hopfinger, E. J. & Cartellier, A. 2007 Liquid sloshing and wave breaking in circular and square-base cylindrical containers. J. Fluid Mech. 577, 467–494.CrossRefGoogle Scholar
23. Stoker, J. J. 1950 Nonlinear Vibrations. John Wiley & Sons.Google Scholar
24. Takahara, H. & Kimura, K. 2002 Frequency response of sloshing in a rectangular tank subjected to pitching excitation. Trans. Japan Soc. Mech. Engng C 68 (667), 738–746 (in Japanese).CrossRefGoogle Scholar
25. Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. 1985 Determining Lyapunov exponents from a time series. Physica D 16, 285–317.CrossRefGoogle Scholar
26. Yoshimatsu, K. & Funakoshi, M. 2001 Surface waves in a square container due to resonant horizontal oscillations. J. Phys. Soc. Japan 70 (2), 394–406.Google Scholar