Equivalent Mechanical Models for Sloshing
Francesco Capolupo
TL;DR
This work addresses propellant sloshing in spacecraft by formulating rigorous equivalent mechanical models, using pendulum-based dynamics and a mass-spring-damper (MSD) representation. It derives both nonlinear and linearized eight-DOF multi-body dynamics for single and multiple pendulums, incorporating a nominal longitudinal force along the vehicle axis, and shows exact equivalence between the linearized pendulum and the MSD model through modal analysis. The authors validate the approach with MATLAB Simscape Multibody simulations, confirming time-domain accuracy for the nonlinear model and near-perfect frequency-domain matches for the linearized models, thereby enabling reliable, control-oriented sloshing design. The methods offer a practical framework for GNC subsystem design under high-g slosh and microgravity conditions, with extensions to zero-g regimes and multiple sloshing modes.
Abstract
Propellant sloshing is a well-known, but not completely mastered phenomenon in space vehicles. It is particularly critical in both microgravity environments - such as interplanetary spacecraft requiring high pointing stability - and high-g conditions, as encountered during launch, re-entry, and landing. In both cases, sloshing can significantly affect vehicle performance and stability, and must often be explicitly considered in the design of the guidance, navigation, and control (GNC) subsystem. For stability analysis and control design, the most common approach to modeling sloshing is through an equivalent mechanical representation, where the moving propellant is treated as a mechanical system interacting with the rigid (or flexible) spacecraft. Pendulum-based models and mass-spring-damper systems are widely used by control analysts to assess sloshing-induced perturbations on vehicles subjected to persistent non-gravitational acceleration along one of their body axes. In this work, we present a rigorous mathematical formulation of pendulum dynamics, starting from a single spherical pendulum attached to a rigid spacecraft. We derive the nonlinear equations of motion for this 8-degree-of-freedom multi-body system, and then extend the formulation to include multiple pendulums, representing multiple sloshing modes within a tank and/or multiple tanks on the same vehicle. Furthermore, we derive the corresponding linearized equations of motion, explicitly accounting for a nominal longitudinal force acting on the vehicle - consistent with the high-g sloshing regime - expressed in either the inertial or body frame. Finally, we demonstrate the mathematical equivalence between the pendulum and mass-spring-damper models and validate the proposed models through time-domain simulation and frequency-domain analysis.