Reduced-Order and Equivalent Mechanical Models for Sloshing in Arbitrarily Shaped Containers

Abstract

We present a systematic approach to derive reduced-order (modal) models that accurately capture the sloshing dynamics in arbitrarily shaped, liquid-filled containers.The reduced models are constructed from finite element representations of an incompressible, inviscid fluid with a free surface. By comparing results in both time and frequency domains, we demonstrate that high accuracy can be achieved using only a small number of modes. We show that, beyond the conventional dynamic sloshing modes, the inclusion of frozen modes -- associated with infinite eigenvalues from the singular mass matrix -- is essential for faithfully reproducing the dynamics. These frozen modes correspond to inertial effects often captured through frozen, convective, or repulsive masses in equivalent mechanical models such as pendulums or mass-spring systems. Building on this insight, we propose a direct method to parametrize equivalent mechanical models from the modal description of the finite element system. We analyze containers of varying geometric complexity, from doubly symmetric to general shapes. While standard mass-spring or pendulum models accurately represent sloshing in flat-bottomed or symmetric containers, they fail for containers with angled or asymmetric bases. To address this gap, we introduce a novel mass-spring-screw model that correctly captures transverse tilting moments induced by longitudinal accelerations -- an effect critical for containers of general shape. This new model extends the applicability of mechanical equivalent models for sloshing dynamics and provides a foundation for improved design and analysis of complex liquid-filled structures.