# Equivalent mechanical model of liquid linear sloshing under microgravity

DOI: 10.34759/trd-2022-126-05

### Аuthors

Yu Z. *, Temnov A. N.**

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: yuzhaokai933@mail.ru
**e-mail: antt45@mail.ru

### Abstract

A numerical approach to develop the equivalent mechanical models representing liquid sloshing is established and the effects of surface tension are taken into account which are predominant in low-gravity environment. An appropriate model is a pendulum that has a mass which represents the liquid fraction that participates in the fundamental model of the sloshing. Furthermore, the pendulum must be attached to the tank through a torsional spring which represents the stiffening effect of surface tension. A formulation is derived from the linearization of the motion equations of the liquid near its initial equilibrium position considering pressure jump on the free surface and free-end boundary condition at the three-phase contact line. The continuous problem domain is discretized by the finite element method and its discretization gives a classical generalized eigenvalue problem, whose solutions are natural frequencies and mode shapes. Expressions for the parameters of the mechanical model are obtained by the principle of dynamic similarity. Several examples illustrate the influence of Bond number and fill levels on the behavior of liquid in toroidal tanks. Comparing numerical results with the experimental measurements obtained under ground conditions, it is found that the non-dimensional eigenvalue and slosh masses increases as Bond number increases, but the spring moment and length of pendulum decreases. The results obtained in this paper can be used in the coupling dynamic analysis of the spacecraft with propellant tanks.

### Keywords:

microgravity, equivalent mechanical model, toroidal tank, surface tension, finite element method

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