Resonance in Cooker's sloshing experiment

 Department of Mathematics
 University of Surrey


Resonance in Cooker's sloshing experiment
  Cooker's sloshing experiment is a prototype for studying the dynamic coupling between fluid sloshing and vessel motion. It involves a container, partially filled with fluid, suspended by two cables and constrained to remain horizontal while undergoing a pendulum-like motion. In this paper the fully-nonlinear equations are taken as a starting point, including a new derivation of the coupled equation for vessel motion, which is a forced nonlinear pendulum equation. The equations are then linearized and the natural frequencies studied. The coupling leads to a highly nonlinear transcendental characteristic equation for the frequencies. Two derivations of the characteristic equation are given, one based on a cosine expansion and the other based on a class of vertical eigenfunctions. These two characteristic equations are compared with previous results in the literature. Although the two derivations lead to dramatically different forms for the characteristic equation, we prove that they are equivalent. The most important observation is the discovery of an internal 1:1 resonance in the fully two-dimensional finite depth model, where symmetric fluid modes are coupled to the vessel motion. Numerical evaluation of the resonant and nonresonant modes are presented. The implications of the resonance for the fluid dynamics, and for the nonlinear coupled dynamics near the resonance are also briefly discussed.
  H. Alemi Ardakani, T.J. Bridges & M.R. Turner. Resonance in a model for Cooker's sloshing experiment. Euro. J. Mech. B/Fluids 36 25-32 (2012)   Finalform Preprint    
  H. Alemi Ardakani, T.J. Bridges & M.R. Turner. Resonance in a model for Cooker's sloshing experiment - the extended version -, Technical Report (2012).    

Proof of equivalence
  The boundary-value problem for the linear horizontally-forced sloshing problem can be solved using two different classes of eigenfunction expansions. The first will be referred to as the "cosine" expansion since the organizing centre is a cosine series in the x-direction (the horizontal direction), and the second is called the ``vertical eigenfunction expansion'' since the organizing centre is a class of y-direction (the vertical direction) eigenfunctions. These two methods lead to very different forms of solution, and their equivalence is not obvious. This technical report gives the details of a proof of equivalence. The strategy of the proof was suggested to the authors by Phil McIver (Loughborough).
  H. Alemi Ardakani, T.J. Bridges & M.R. Turner. Details of the proof of equivalence: the ``cosine'' versus ``vertical eigenfunction'' representations. Technical Report (2012)    

1:1 resonance in shallow water
  In this report an explicit proof is given showing how the 1:1 resonance arises in the shallow water model for dynamic coupling in Cooker's sloshing experiment.
  H. Alemi Ardakani & T.J. Bridges. 1:1 resonance in the shallow-water model for Cooker's sloshing experiment. Technical Report (2011)    

Baffling and a multifold 1:---:1 resonance
  Strong internal resonance, or 1:1 resonance, in a linearized model for a physical system is a precursor to dramatic dynamics in the nonlinear problem. Such resonances have been found in fluid models, mechanical systems and in celestial mechanics. It is generally believed that higher order, 1:1:1 or 1:1:1:1 internal resonances are rare. However, we have discovered a physical system that has an (n+1)-fold 1: --- :1 resonance for any natural number n. These resonances occur at natural parameter values and an experimental configuration is constructible. Moreover, the system is of great practical importance as it is a model for dynamic coupling between vessel motion and fluid sloshing for a container with baffles.
  M.R. Turner, H. Alemi Ardakani, & T.J. Bridges. Dynamic coupling in Cooker's sloshing experiment with baffles. Physics of Fluids 25 112102 (2013)   PoF Website  

Compendium of references

  1. H. Alemi Ardakani. Rigid-body motion with interior shallow-water sloshing, PhD Thesis, University of Surrey (2010).

  2. H. Alemi Ardakani & T.J. Bridges. Dynamic coupling between shallow water sloshing and horizontal vehicle motion, Europ. J. Appl. Math. 21 479-517 (2010).

  3. H. Alemi Ardakani \& T.J. Bridges. Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions, J. Fluid Mech. 667 474-519 (2011).

  4. H. Alemi Ardakani \& T.J. Bridges. Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in two dimensions, Euro. J. Mech. B/Fluids 31 30-43 (2012).

  5. H. Alemi Ardakani & T.J. Bridges. The Euler equations in fluid mechanics relative to a rotating-translating reference frame Technical Report, Department of Mathematics, University of Surrey (2010).

  6. J.P. Boyd. Chebyshev and Fourier Spectral Methods, Second Edition, Dover Publications: New York (2001).

  7. P.G. Chamberlain & D. Porter. On the solution of the dispersion relation for water waves, Appl. Ocean. Res. 21 161-166 (1999).

  8. M.J. Cooker. Water waves in a suspended container Wave Motion 20 385-395 (1994).

  9. O.M. Faltinsen & A.N. Timokha. Sloshing, Cambridge University Press (2009).

  10. J.B. Frandsen. Numerical predictions of tuned liquid tank structural systems, J. Fluids & Structures 20 309-329 (2005).

  11. Z.C. Feng. Coupling between neighboring two-dimensional modes of water waves, Phys. Fluids 10 2405-2411 (1998).

  12. Z.C. Feng & P.R. Sethna. Symmetry breaking bifurcations in resonant surface waves, J. Fluid Mech. 199 495-518 (1989).

  13. Z.C. Feng & P.R. Sethna. Global bifurcation and chaos in parametrically forced systems with 1:1 resonance, Dynamical Systems: an Inter. Journal 5 201-225 (1990).

  14. E.W. Graham & A.M. Rodriguez. The characteristics of fuel motion which affect airplane dynamics, J. Appl. Mech. 19 381-388 (1952).

  15. R.A. Ibrahim. Liquid Sloshing Dynamics Cambridge University Press (2005).

  16. T. Ikeda & N. Nakagawa. Non-linear vibrations of a structure caused by water sloshing in a rectangular tank, J. Sound & Vibration 201 23-41 (1997).

  17. C.M. Linton & P. McIver. Handbook of Mathematical Techniques for Wave-Structure Interaction, Chapman & Hall/CRC: Boca Raton (2001).

  18. H. Ockendon, J.R. Ockendon & D.D. Waterhouse. Multi-mode resonances in fluids, J. Fluid Mech. 315 317-344 (1996).

  19. H. Ockendon & J.R. Ockendon. Nonlinearity in fluid resonances, Meccanica 36 297-321 (2001).

  20. R.A. Struble & J.H. Heinbockel. Resonant oscillations of a beam-pendulum system, J. Appl. Mech. 30 181-188 (1963).

  21. G.I. Taylor. The interaction between experiment and theory in fluid mechanics, Ann. Rev. Fluid Mech. 6 1-17 (1974).

  22. J.-M. Vanden-Broeck. Nonlinear gravity-capillary standing waves in water of arbitrary uniform depth, J. Fluid Mech. 139 97-104 (1984).

  23. J. Yu. Effects of finite water depth on natural frequencies of suspended water tanks, Stud. Appl. Math. 125 373-391 (2010).

--- Sloshing mainpage ---