Abstract
The nonlinear sloshing problem is an important issue for design of liquid storage tanks, liquid cargo transportations, tuned liquid dampers and so on. This paper is concerned with development of a novel approach for the nonlinear sloshing problem based on the Hamiltonian mechanics. In particular, this study is aimed at developing the method available to analyze the nonlinear liquid surface behavior like a traveling wave observed in the small liquid depth. In the present formulation, the fluid is assumed to be inviscid incompressible and irrotational flow. Then, the liquid surface motion is described by nonlinear multimodal models. However, since the sloshing problem based on such assumptions yields an irregular Lagrangian, it makes formulation difficult in a straightforward way. Therefore, the present approach employs the constrained Hamiltonian mechanics with the Lagrange’s method of undetermined multipliers to derive equations of motion. The resulting system is comprised of differential equations and algebraic equations, referred to as differential algebraic equations (DAEs). In addition, the present method takes full account of the nonlinear mode-to-mode interactions without reduction methods focusing on the predominant sloshing modes. However, the multimodal models without such reduction methods suffer from severe numerical stiff problem. Therefore, the numerical integration techniques based on implicit schemes (DAE solver) are incorporated as remedies for the stiff problem. Specifically, discrete forms of the equations of motions are derived by employing the Galerkin method and a discrete derivative. The proposed approach is validated by comparisons with an existing model and an experiment in time domain analysis.
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References
Kim, Y., Shin, Y.-S., Lee, K.H.: Numerical study on slosh-induced impact pressures on three-dimensional prismatic tanks. Appl. Ocean Res. 26, 213–226 (2004)
Ardakani, H.A., Bridges, T.J.: Dynamic coupling between shallow-water sloshing and horizontal vehicle motion. Eur. J. Appl. Math. 21, 479–517 (2010)
Miles, J.W.: Internally resonant surface waves in a circular cylinder. J. Fluid Mech. 149, 1–14 (1984)
Miles, J.W.: Resonantly surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31 (1984)
Takahara, H., Hara, K., Ishida, T.: Nonlinear liquid oscillation in a cylindrical tank with an eccentric core barrel. J. Fluid Struct. 35, 120–132 (2012)
Ibrahim, R.A.: Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press, Cambridge (2005)
Ibrahim, R.A., Barr, A.D.S.: Autoparametric resonance in a structure containing a liquid, part I: two mode interaction. J. Sound Vib. 42(2), 159–179 (1975)
Ibrahim, R.A., Barr, A.D.S.: Autoparametric resonance in a structure containing a liquid, part II: three mode interaction. J. Sound Vib. 42(2), 181–200 (1975)
Ikeda, T.: Nonlinear parametric vibrations of an elastic structure with a rectangular liquid tank. Nonlinear Dyn. 33, 43–70 (2003)
Ikeda, T.: Autoparametric interaction of a liquid surface in a rectangular tank with an elastic support structure under 1:1 internal resonance. Nonlinear Dyn. 60, 425–441 (2010)
Farid, M., Gendelman, O.V.: Internal resonances and dynamic responses in equivalent mechanical model of partially liquid-filled vessel. J. Sound Vib. 379, 191–212 (2017)
Farid, M., Gendelman, O.V.: Response regimes in equivalent mechanical model of moderately nonlinear liquid sloshing. Nonlinear Dyn. 92, 1517–1538 (2018)
Kaneko, S., Yoshida, O.: Modeling of deepwater-type rectangular tuned liquid damper with submerged nets. Trans. ASME J. Press. Vessel Technol. 121, 413–422 (1999)
Kaneko, S., Ishikawa, M.: Modeling of tuned liquid damper with submerged nets. Trans. ASME J. Press. Vessel Technol. 121, 334–343 (1999)
Tait, M.J., Damatty, A.A.E.I., Isyumov, N., Siddique, M.R.: Numerical flow models to simulate tuned liqud dampers (TLD) with slat screens. J. Fluid Struct. 20, 1007–1023 (2005)
Love, J.S., Tait, M.J.: Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique. J. Fluid Struct. 26, 1058–1077 (2010)
Love, J.S., Tait, M.J.: Non-linear multimodal model for tuned liquid dampers of arbitrary tank geometry. Int. J. Non Linear Mech. 46, 1065–1075 (2011)
Love, J.S., Tait, M.J.: Nonlinear multimodal model for TLD of irregular tank geometry and small fluid depth. J. Fluids Struct. 43, 83–99 (2013)
Faltinsen, O.M., Rognebakke, O.F., Lukovsky, I.A., Timokha, A.N.: Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J. Fluid Mech. 407, 201–234 (2000)
Faltinsen, O.M., Timokha, A.N.: An adaptive multimodal approach to nonlinear sloshing in a rectangular tank. J. Fluid Mech. 432, 167–200 (2001)
Faltinsen, O.M., Timokha, A.N.: Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth. J. Fluid Mech. 470, 319–357 (2002)
Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N.: Resonant three-dimensional nonlinear sloshing in a square base basin. J. Fluid Mech. 487, 1–42 (2003)
Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N.: Classification of three-dimensional nonlinear sloshing in a square-base tank with finite depth. J. Fluids Struct. 20, 81–103 (2005)
Faltinsen, O.M., Timokha, A.N.: A multimodal method for liquid sloshing in a two-dimensional circular tank. J. Fluid Mech. 665, 457–479 (2010)
Faltinsen, O.M., Timokha, A.N.: Multimodal analysis of weakly nonlinear sloshing in a spherical tank. J. Fluid Mech. 719, 129–164 (2013)
Craig, W., Sulem, C.: Numerical simulation of gravity waves. J. Comput. Phys. 108, 78–83 (1993)
Craig, W., Groves, M.D.: Hamiltonian long-wave approximation to the water-wave problem. Wave Motion 19, 367–389 (1994)
Craig, W., Guyenne, P., Kalisch, H.: Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. LVIII, 1587–1641 (2005)
Hara, K., Takahara, H.: Hamiltonian formulation of surface and interface sloshing in a tank containing two fluids. J. Syst. Des. Dyn. 2(1), 299–310 (2008)
Hara, K., Takahara, H.: Hamiltonian formulation for nonlinear sloshing in layered two immiscible fluids. J. Syst. Des. Dyn. 2(5), 1183–1193 (2008)
Hara, K., Takahara, H.: Hamiltonian formulation of free surface and interface motions in a tank (n-wave resonant interaction caused by nonlinearity of system). J. Syst. Des. Dyn. 2(6), 1218–1229 (2008)
Hara, K., Watanabe, M.: Formulation of the nonlinear sloshing-structure coupled problem based on the Hamiltonian mechanics for constraint systems. J. Fluids Struct. 62, 104–124 (2016)
Bauchau, O.A.: Computational schemes for flexible, nonlinear multi-body systems. Multibody Syst. Dyn. 2, 169–225 (1998)
Betsch, P., Steinmann, P.: Conservation properties of a time FE method—part III: mechanical systems with holonomic constraints. Int. J. Numer. Method Eng. 53, 2271–2304 (2002)
Sugiyama, H., Escalona, J.L., Shabana, A.A.: Formulation of three-dimensional joint constraints using the absolute nodal coordinates. Nonlinear Dyn. 31, 167–195 (2003)
Dufva, K., Kerkkänen, K., Maqueda, L.G., Shabana, A.A.: Nonlinear dynamics of three-dimensional belt drives using the finite-element method. Nonlinear Dyn. 48, 449–466 (2007)
Seliger, R.L., Whitham, G.B.: Variational principles in continuum mechanics. Proc. R. Soc. Lond. A 305, 1–25 (1968)
Borri, M., Trainelli, L., Croce, A.: The embedded projection method: a general index reduction procedure fo constrained system dynamics. Comput. Methods Appl. Mech. Eng. 195, 6974–6992 (2006)
Dirac, P.A.M.: Lectures on Quantum Mechanics. Yeshiva University Press, New York (1964)
Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996)
Miles, J.W.: Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475 (1967)
Acknowledgements
The authors are grateful to an undergraduate student, Shohei Shimizu, for the experimental assistance.
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This study was funded by JSPS KAKENHI (Grant Number 18K04007).
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Hara, K., Watanabe, M. Application of the DAE approach to the nonlinear sloshing problem. Nonlinear Dyn 99, 2065–2081 (2020). https://doi.org/10.1007/s11071-019-05399-3
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DOI: https://doi.org/10.1007/s11071-019-05399-3