SUMMARYThe space structures such as satellites, probes or space stations generally contain large amounts of liquids, which can be propellants, cooling liquids, etc. The motion of these liquids can influence the vibrational behavior of the main structure and can potentially disturb the trajectory controller or the stabilization procedures. To avoid hazardous coupling between the inner liquid sloshing and the main structure movements, engineers need to know precisely the evolution of the inner liquid eigenmodes and eigenfrequencies during the mission. In this aim, we propose here a numerical approach to solve the three-dimensional linear sloshing problem of an incompressible inviscid liquid taking into account the effects of surface tension which are predominant in low-gravity environment. A variational formulation is derived from the linearization of the motion equations of the liquid near its initial equilibrium state (meniscus) and its discretization by the finite element method gives a classical spectral problem, whose solutions are the sloshing eigenmodes. While the majority of authors consider simple geometries of tanks, the advantage of the formulation presented here is to be easily applicable to real three-dimensional cases as it will be demonstrated in some application examples.

To study the vibrations of a tank partially filled with a liquid in low-gravity environment, we first have to find the static position of the liquid. In this paper, we present a three-dimensional finite element approach to find this equilibrium configuration for any tank geometry. Both gravity and capillary effects are taken into account. The nonlinear equations of this problem are derived from the differentiation of the total potential energy of the system, then the problem is transformed into a liquid free surface form-finding. The well-known singularity of this kind of problems is regularized using the updated reference strategy. The equations of the regularized problem are discretized using the finite element method and solved by the Newton-Raphson algorithm. Several examples illustrate the effectiveness of this method, even for complex cases, and two validation tests are presented. The linear sloshing vibrations of the liquid are finally studied near this equilibrium position and two validation cases are proposed for the eigenvalue dynamic problem.