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Application of the DAE approach to the nonlinear sloshing problem

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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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Acknowledgements

The authors are grateful to an undergraduate student, Shohei Shimizu, for the experimental assistance.

Funding

This study was funded by JSPS KAKENHI (Grant Number 18K04007).

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Authors and Affiliations

  1. Department of Mechanical Engineering, School of Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan

    Kensuke Hara

  2. Department of Mechanical Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa, Japan

    Masahiro Watanabe

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  1. Kensuke Hara
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  2. Masahiro Watanabe
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Correspondence to Kensuke Hara.

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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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