Formulation of the nonlinear sloshing-structure coupled problem based on the Hamiltonian mechanics for constraint systems

• A new coupling strategy for a sloshing-structure problem was developed.
• Hamiltonian mechanics for constraint systems was introduced for the formulation.
• A nonlinear model with the fourth-order polynomials was derived.
• A validity of the proposed model was evaluated by an experiment.
• Nonlinear hydrodynamic sloshing forces in the shallow water depth was discussed.

This paper describes a formulation of a nonlinear sloshing problem based on the Hamiltonian mechanics. In particular, we focus on behavior of a liquid surface and a hydrodynamic force arising from the nonlinear sloshing in shallow water depth. It is well known that the water wave in shallow water depth shows the characteristic behaviors such as the solitary wave by inherent nonlinearities. Therefore, the effect of nonlinearity is significantly crucial for accurate predictions of the wave height and the hydrodynamic force. Although many researches have been studied for the feature of the nonlinear sloshing in shallow water depth, the theoretical analysis is essentially difficult because a lot of higher order nonlinear terms and sloshing modes have to be taken into account for accurate numerical predictions. Consequently, it yields complicated algebraic procedures. This study presents a formulation of nonlinear sloshing based on the canonical theory for constrained systems. In addition, the Dirichlet–Neumann operators developed by Craig and Sulem (1993) is introduced to obtain an asymptotic description for the kinematic boundary condition of the liquid surface. The proposed approach facilitates the consideration of the nonlinearity in the formulation. This study demonstrates analytical predictions considering up to the fourth-order nonlinear terms and higher-order sloshing modes and discusses adequate truncation orders for them. Moreover, experiments are conducted to measure time histories of the wave height and the nonlinear hydrodynamic force due to the sloshing in a rectangular tank subjected to a horizontal excitation. As the results of frequency analyses for the time histories, many frequency spectra with the integral multiples of the dominant frequency were observed. In particular, the only odd multiples of the dominant frequency were involved in the results of hydrodynamic force. These features were also obtained by the analytical predictions by the proposed method.