Skip to main content Accessibility help
×
Home

Nonlinear energy transfer between fluid sloshing and vessel motion

  • M. R. Turner (a1) and T. J. Bridges (a1)

Abstract

This paper examines the dynamic coupling between a sloshing fluid and the motion of the vessel containing the fluid. A mechanism is identified that leads to an energy exchange between the vessel dynamics and fluid motion. It is based on a 1:1 resonance in the linearized equations, but nonlinearity is essential for the energy transfer. For definiteness, the theory is developed for Cooker’s pendulous sloshing experiment. The vessel has a rectangular cross-section, is partially filled with a fluid and is suspended by two cables. A nonlinear normal form is derived close to an internal 1:1 resonance, with the energy transfer manifested by a heteroclinic connection, which connects the purely symmetric sloshing modes to the purely antisymmetric sloshing modes. Parameter values where this pure energy transfer occurs are identified. In practice, this energy transfer can lead to sloshing-induced destabilization of fluid-carrying vessels.

Copyright

Corresponding author

Email address for correspondence: M.Turner@surrey.ac.uk

References

Hide All
Adee, B. H. & Caglayan, I. 1982 The effects of free water on deck on the motions and stability of vessels. In Proceedings of the Second International Conference on the Stability of Ships and Ocean Vehicles, Tokyo, Japan. Springer.
Alemi Ardakani, H. & Bridges, T. J. 2010 Dynamic coupling between shallow-water sloshing and horizontal vehicle motion. Eur. J. Appl. Maths 21, 479517.
Alemi Ardakani, H., Bridges, T. J. & Turner, M. R. 2012a Resonance in a model for Cooker’s sloshing experiment. Eur. J. Mech. (B/Fluids) 36, 2538.
Alemi Ardakani, H., Bridges, T. J. & Turner, M. R. 2012b Resonance in a model for Cooker’s sloshing experiment – extended version. Technical Report, University of Surrey.http://personal.maths.surrey.ac.uk/st/T.Bridges/SLOSH/RESONANCE.
Caglayan, i. & Storch, r. l. 1982 Stability of fishing vessels with water on deck: a review. J. Ship Res. 26, 106116.
Cooker, M. J. 1994 Water waves in a suspended container. Wave Motion 20, 385395.
Cotter, C. S. 1986 The 1:1 resonance and the Hénon–Heiles family of Hamiltonians. PhD thesis, University of California, Santa Cruz, USA.
Cushman, R. & Rod, D. L. 1982 Reduction of the semisimple 1:1 resonance. Physica D 6, 105112.
Dillingham, J. 1981 Motion studies of a vessel with water on deck. Wave Motion 18, 3850.
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2003 Resonant three-dimensional nonlinear sloshing in a square-base basin. J. Fluid Mech. 487, 142.
Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.
Feng, Z. C. 1998 Coupling between neighbouring two-dimensional modes of water waves. Phys. Fluids 10 (9), 24052411.
Feng, Z. C. & Sethna, P. R. 1989 Symmetry breaking bifurcations in resonant surface waves. J. Fluid Mech. 199, 495518.
Frandsen, J. B. 2005 Numerical predictions of tuned liquid tank structural systems. J. Fluids Struct. 20, 309329.
Graham, E. W. & Rodriguez, A. M. 1952 The characteristics of fuel motion which affect airplane dynamics. J. Appl. Mech. 19, 381388.
Herczyński, A. & Weidman, P. D. 2012 Experiments on the periodic oscillation of free containers driven by liquid sloshing. J. Fluid Mech. 693, 216242.
Holmes, P. J. & Marsden, J. E. 1983 Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math. J. 32, 273309.
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics. Cambridge University Press.
Ikeda, T., Ibrahim, R. A., Harata, Y. & Kuriyama, T. 2012 Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation. J. Fluid Mech. 700, 304328.
Ikeda, T. & Nakagawa, N. 1997 Non-linear vibrations of a structure caused by water sloshing in a rectangular tank. J. Sound Vib. 201, 2341.
Linton, C. M. & McIver, P. 2001 Handbook of Mathematical Techniques for Wave–Structure Interaction. Chapman & Hall/CRC.
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395397.
Mayer, H. C. & Krechetnikov, R. 2012 Walking with coffee: why does it spill? Phys. Rev. E 85, 046117.
Moiseyev, N. N. & Rumyantsev, V. V. 1968 Dynamic Stability of Bodies Containing Fluid. Springer.
Struble, R. A. & Heinbockel, J. H. 1963 Resonant oscillations of a beam–pendulum system. J. Appl. Maths 30, 181188.
Tadjbakhsh, I. & Keller, J. B. 1960 Standing surface waves of finite amplitude. J. Fluid Mech. 8, 442451.
Taylor, G. I. 1974 The interaction between experiment and theory in fluid mechanics. Annu. Rev. Fluid Mech. 6, 116.
Yu, J. 2010 Effects of finite water depth on natural frequencies of suspended water tanks. Stud. Appl. Maths 125, 337391.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Type Description Title
PDF
Supplementary materials

Turner and T. J. Bridges
Supplementary material

 PDF (449 KB)
449 KB

Nonlinear energy transfer between fluid sloshing and vessel motion

  • M. R. Turner (a1) and T. J. Bridges (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed