2. Understanding Sloshing as a Complex Asymptotically Reduced Dynamical System

The lecture notes present a ‘guided tour’ over the so-called nonlinear multimodal method in the liquid sloshing dynamics. The method is, in fact, a version of the Reduced Order Modelling based on combining variational and asymptotic approaches to the original free-surface (sloshing) problem. I start with mathematical fundamentals of how to derive a discrete approximate model of sloshing from the original fluid dynamics formulation by using the Bateman-Luke variational principle. The derived Euler-Lagrange equations (here, the Miles-Lukovsky-type modal system) couple the generalised hydrodynamic coordinates and velocities associated with time-dependent coefficients in functional series representing the free surface and velocity potential, respectively. Linearising the modal system yields an infinite set of uncoupled linear oscillators which, naturally, have an analytical solution for prescribed tank motions. The linear modal theory becomes physically irrelevant for resonant excitations. Analytical approaches to nonlinear resonant sloshing can be based on asymptotic methods reducing the infinite-dimensional Miles-Lukovsky-type system to finite dimensions. The subsequent text centres around how to construct and analyse those finite-dimensional asymptotic modal equations for rectangular and square base tanks. A particular focus is on the so-called Moiseev’s (Duffing-like) third-order asymptotic ordering and resonant steady-state wave regimes which are associated with asymptotic periodic solutions of modal equations when the forcing frequency is close to the lowest natural sloshing frequency. These solutions are (semi-) analytically constructed, their stability is examined, the results are validated by experiments.