Resonance in Cooker's sloshing experiment


 Department of Mathematics
 University of Surrey
 UK

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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  

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