A universal nonlinear equivalent model and its stability analysis for large amplitude liquid sloshing in rotationally symmetric tanks

Modern spacecraft usually carry rotationally symmetric liquid propellant storage tanks. The study of equivalent dynamic models is significant for obtaining the dynamic laws and designing controllers for large amplitude liquid sloshing. In this article, the moving pulsating ball model (MPBM), which was originally only applicable to liquid sloshing in spherical tanks in weightless environments, is extended to arbitrary rotational symmetric tanks and arbitrary gravity environments. Firstly, by combining the Newton–Euler equation of the pulsating ball with the energy relationship of ‘breathing’, the expression for the force exerted by the liquid on the wall in an arbitrary rotationally symmetric storage tank is derived. Then, taking the ellipsoidal tank as an example, the validity of the model is verified by the computational fluid dynamics (CFD) software Flow3D through the calculation examples of liquid sloshing in both weightlessness and constant gravity environments respectively. The comparison results show that the improved MPBM (IMPBM) can save up to 98.2% of computation time, greatly improving simulation efficiency while ensuring accuracy. Finally, the stability of the pulsating ball model in a microgravity environment was studied. The dynamic equation of the pulsating ball when subjected to small disturbances is derived and reduced to the standard Mathieu equation. Based on the Mathieu equation and Linz Tad-Poincare perturbation method, the stability of the pulsating ball is studied and the mathematical expressions for the boundary of the stable domain are calculated, which can provide some inspiration for the study of liquid distribution in spacecraft storage tanks in a microgravity environment.