Dynamic Characteristics of Liquid Motion in Partially Filled Tanks of a Spinning Spacecraft

This paper presents a boundary-layer model to predict dynamic characteristics of liquid motion in partially filled tanks of a spinning spacecraft. The solution is obtained by solving three boundary-value problems: an inviscid fluid problem, a boundary-layer problem, and a viscous correction problem. The boundary-layer solution is obtained analytically, and the solutions to inviscid and viscous correction problems are obtained by using finite element methods. The model has been used to predict liquid natural frequencies, mode shapes, damping ratios, and nutation time constants for a spinning spacecraft. The results show that liquid motion in general will contain significant circulatory motion due to Coriolis forces except in the first azimuth and first elevation modes. Therefore, only these two modes can be represented accurately by equivalent pendulum models. The analytical results predict a sharp drop in nutation time constants for certain spacecraft inertia ratios and tank fill fractions. This phenomenon was also present during on-orbit liquid slosh tests and ground air-bearing tests. I. Introduction A RECENT trend in geosynchronou s spacecraft design is to use liquid apogee motors, which results in liquid constituting almost half of the spacecraft mass during transfer orbit. In these spacecraft, liquid motion significantly influences the spacecraft attitude stability and control. LEAS AT, a geosynchronous spacecraft with liquid apogee motor, launched in September 1984, experienced attitude control motion instability1 during the pre-apogee injection phase, immediately following the activation of despin control. The instability was found to be the result of interaction between liquid lateral sloshing modes and the attitude control. This experience demonstrated that the analysis of dynamic interaction between liquid slosh motion and attitude control is critical in the attitude control design of these spacecraft. To perform this analysis, accurate determination of liquid dynamic characteristics, such as natural frequencies, mode shapes, damping, and modal masses becomes important. Accurate prediction of liquid dynamic characteristics is, however, a difficult problem because of the complexity of the hydrodynamical equations of motion. Several investigators have analyzed the fluid motion in rotating containers. Greenspan2 analyzed the transient motion during spin up of an arbitrarily shaped container filled with viscous imcompressible fluid. Stewartson3 developed a stability criterion for a spinning top containing fluid. This stability criterion was corrected by Wedemeyer 4 by considering fluid viscosity. Nayfeh and Meirovitch5 analyzed a spinning rigid body with a spherical cavity partially filled with liquid. Viscous effects are considered only for a boundary layer near the wetted surface. Hendricks and Morton6 analyzed the stability of a rotor partially filled with a viscous incompressible fluid. Stergiopoulous and Aldridge7 studied inertial waves in a partially filled cylindrical cavity during spin up. Pfeiffer8 introduced the concept of homogeneous vorticity to the problem of partially filled containers. El-Raheb and Wagner9 developed a finite element model based on a homogeneous vorticity as