Abstract
The coupled motion between shallow water sloshing in a moving vessel with baffles and the vessel dynamics is considered. Here the vessel dynamics is restricted to horizontal motion such as in Tuned Liquid Dampers. It was shown by Turner {____it et al.} ____cite{tbaa} (Phys. Fluids (2013) {____bf 25}(10) 112102) that partitioning a moving vessel into $n$ separate compartments leads to an interesting dynamical behaviour of the system. Also, under particular input parameter values an internal $(n+1)$-fold $1:____cdots:1$ resonance can be generated, where the frequency of the sloshing fluid in each compartment is equal, and equal to the frequency of the vessel itself. Here the form of the sloshing eigenmodes at this resonance are derived in the shallow-water limit. Using the Lagrangian formulation of the problem, an efficient numerical algorithm is implemented to solve the fully nonlinear system of equation based on the implicit midpoint rule. This algorithm is simple, fast and maintains the energy partition between the vessel and the fluid over long times. In this work numerical results are presented for dynamical vessel/sloshing motion attached to a nonlinear spring.