Mechanical Models of Low Gravity Sloshing Taking Into Account Viscous Damping

Abstract

Mechanical models of damped lowgravity sloshing are developed using a proposed analytical method for arbitrary axisymmetric tanks. It is shown that (a) the complex amplitudes of the force and moment caused by the conventional mechanical model do not coincide with the complex amplitudes of the force and moment calculated from the modal equation of sloshing and (b) these differences arise not only from the damping ratio but also from the surface tension although the surface tension does not cause energy dissipation. A mechanical model for correcting these differences is developed. The mass of this correction model is found to be equal to the mass of the liquid that fills the domain bounded by the meniscus and the plane that includes the contact line of the meniscus with the tank wall. With decreasing Bond number, the correction model mass as well as the damping ratio increase and, therefore, the correction becomes important. The force and moment caused by the conventional uncorrected mechanical model have phase lag with respect to the force and moment calculated from the modal equation of sloshing near the resonant frequency. Therefore, the correction is important for the dynamics and control analysis of a space vehicle.