JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICSVol. 16, No. 4, July-August 1993

Dynamic Characteristics of Liquid Motion in Partially FilledTanks of a Spinning Spacecraft

Brij N. Agrawal*Naval Postgraduate School, Monterey, California 93943

This paper presents a boundary-layer model to predict dynamic characteristics of liquid motion in partiallyfilled tanks of a spinning spacecraft. The solution is obtained by solving three boundary-value problems: aninviscid fluid problem, a boundary-layer problem, and a viscous correction problem. The boundary-layersolution is obtained analytically, and the solutions to inviscid and viscous correction problems are obtained byusing 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 ingeneral will contain significant circulatory motion due to Coriolis forces except in the first azimuth and firstelevation 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 tankfill fractions. This phenomenon was also present during on-orbit liquid slosh tests and ground air-bearing tests.

I. Introduction

A RECENT trend in geosynchronous spacecraft design isto use liquid apogee motors, which results in liquid con-

stituting almost half of the spacecraft mass during transferorbit. In these spacecraft, liquid motion significantly influ-ences the spacecraft attitude stability and control. LEAS AT,a geosynchronous spacecraft with liquid apogee motor,launched in September 1984, experienced attitude control mo-tion instability1 during the pre-apogee injection phase, imme-diately following the activation of despin control. The instabil-ity was found to be the result of interaction between liquidlateral sloshing modes and the attitude control. This experi-ence demonstrated that the analysis of dynamic interactionbetween liquid slosh motion and attitude control is critical inthe attitude control design of these spacecraft. To perform thisanalysis, accurate determination of liquid dynamic character-istics, such as natural frequencies, mode shapes, damping, andmodal masses becomes important. Accurate prediction ofliquid dynamic characteristics is, however, a difficult problembecause of the complexity of the hydrodynamical equationsof motion.

Several investigators have analyzed the fluid motion in ro-tating containers. Greenspan2 analyzed the transient motionduring spin up of an arbitrarily shaped container filled withviscous imcompressible fluid. Stewartson3 developed a stabil-ity criterion for a spinning top containing fluid. This stabilitycriterion was corrected by Wedemeyer4 by considering fluidviscosity. Nayfeh and Meirovitch5 analyzed a spinning rigidbody with a spherical cavity partially filled with liquid. Viscouseffects are considered only for a boundary layer near thewetted surface. Hendricks and Morton6 analyzed the stabilityof a rotor partially filled with a viscous incompressible fluid.Stergiopoulous and Aldridge7 studied inertial waves in a par-tially filled cylindrical cavity during spin up. Pfeiffer8 intro-duced the concept of homogeneous vorticity to the problem ofpartially filled containers. El-Raheb and Wagner9 developed afinite element model based on a homogeneous vorticity as-

Presented as Paper 90-0997 at the AIAA/ASME/ASCE/AHS/ASC31st Structures, Structural Dynamics, and Materials Conference,Long Beach, CA, April 2-4, 1990; received Oct. 29, 1990; revisionreceived Dec. 11, 1992; accepted for publication Dec. 11, 1992. Copy-right © 1993 by the American Institute of Aeronautics and Astronau-tics, Inc. All rights reserved.

*Professor, Department of Aeronautics and Astronautics. Associ-ate Fellow AIAA.

sumption. These investigations, however, have not resulted inthe development of accurate models to predict liquid dynamiccharacteristics in a spinning spacecraft.

Because of lack of availability of accurate models to predictliquid dynamic characteristics in a spinning spacecraft, simpli-fied models, such as rigid pendulum models, have been used bythe space industry. These models have been found to be signif-icantly inaccurate, resulting in some cases in unstable attitudecontrol designs.1 This has provided impetus to the space indus-try to develop improved analytical models to predict liquiddynamic characteristics in a spinning spacecraft.

Recently, a finite element model has been developed, underINTELSAT sponsorship by MBB, to predict liquid naturalfrequencies, mode shapes, damping ratios, nutational fre-quency, and nutation time constant for spinning spacecraftwith partially filled tanks. This model has been extensivelyused to study the effects of liquid motion on the attitude dy-namics and control of INTELSAT VI, a dual-spin spacecraftwith a liquid apogee motor. This paper presents the boundary-layer model, an analytical prediction of liquid dynamic charac-teristics and nutation time constants using this model, and acomparison of theoretical and experimental results.

II. Equations of MotionAn analytical model is developed for the spacecraft configu-

ration shown in Fig. 1. The tank is offset from the spin axisand is partially filled with liquid. The tank can be of rotationalsymmetry but its contour may be of arbitrary shape. The sym-metry axis may be inclined with respect to the spin axis. Thecoordinate system XYZ is fixed in the spinning spacecraft withorigin at the center of mass of the spacecraft, including liquid,and the Z axis along the spin axis in steady state. The coordi-nate system x"y"z" is fixed in the tank with origin at thecenter of the tank and the z" axis along the tank symmetryaxis.

General equations of motion for the liquid are representedby Navier-Stokes equations as follows:

—dt

+ 2 Q x u

+ / + (1)

where u is the relative velocity of the liquid with respect to thetank, R the position vector of liquid particle, 12 the angularvelocity, p the liquid density, p the liquid pressure, / the bodyforce per unit mass, and v the kinematic viscosity.

636

AGRAWAL: LIQUID MOTION IN SPINNING SPACECRAFT 637

Assuming the liquid to be incompressible, we may write thecontinuity equation as

V - w = 0 (2)

There are two sets of boundary conditions. The first condi-tion which enforces no flow through the tank wall is

u >n = 0 (3)

where n is a unit vector normal to the tank wall. The second setof boundary conditions enforces constant pressure and surfaceequation on the free surface as

p = const

dFdt = 0

(4)

(5)

where F is the free surface equation. The steady-state equi-librium surface is a paraboloid of revolution. It should benoted that for a spinning spacecraft, centrifugal force domi-nates and the surface tension effects are small and are ne-glected in this formulation.

The equation of motion of a spinning spacecraft can bewritten as

= T (6)

where / is the inertia dyadic of the spacecraft with empty tanksand T is the torque exerted by the liquid on the spacecraft andis given by

{ [»R x np ds + pv\ R x f(/i • v)[/i x ( n xu)]\ ds (7)

Sw «J Sw

The integrals are taken over the wetted surface Sw, where thefirst integral is the torque due to the liquid pressure and thesecond integral is the torque due to shear stress on the wall ofthe tank.

III. Analytical ModelThe solution of general equations of motion of liquid, Eqs.

(1) and (2), with associated boundary conditions, Eqs. (3-5), isvery complex. Therefore, several simplified analytical modelshave been used to study the dynamic behavior of liquids in aspinning spacecraft. A simple model, commonly used by thespace industry, is the rigid pendulum model, where the liquidpropellant is treated as a distributed mass pivoting about thecenter of the tank with the total liquid mass located at the masscenter of the liquid. This model has been found to be veryinaccurate in the prediction of liquid natural frequencies, re-sulting in some cases in unstable attitude control design.1Abramson's model10 is based on an ideal fluid executing anirrotational motion. The centrifugal force is represented by anequivalent constant gravitational force and the Coriolis effectsare neglected.

In the homogeneous vortex model,9 the simplifying assump-tions are that the liquid vorticity is independent of the spatialcoordinates and only time dependent, and the Coriolis acceler-ation is retained inside the liquid but is neglected in the freesurface boundary condition to obtain an integral. In theboundary-layer model developed by Pohl,11 three boundary-value problems are solved: an inviscid fluid problem, a viscousboundary-layer problem, and a viscous correction problem.For large Reynolds numbers, Re>0(lQ5), the boundary-layerapproximation is appropriate. For INTELSAT VI parameters,the Reynolds number is greater than 106. During the INTEL-SAT VI study,12 all of the analytical models discussed herewere compared and the conclusion reached was that theboundary-layer model gives the most accurate prediction ofliquid dynamic characteristics. The boundary-layer model isbriefly discussed in the following section.

IV. Boundary-Layer ModelThe equation of motion of the liquid, Eq. (1), is first nondi-

mensionalized. For nondimensionalization, a reference veloc-ity U = a -&„ is defined, where a denotes a reference length ofthe tank and cow is the nutation frequency of the rigid space-craft. The following dimensionless quantities are introduced.

= R/a, (8)

Re = ula2/i>

Here Re denotes Reynolds number and A is the eigenvalueto be determined. The angular velocity is written as

Q = &k + co (9)

where 0 is steady-state angular velocity. Introducing the non-dimensional parameters into Eq. (1) and neglecting body force/, it can be written as

Aw + 2Qk x u + co x rX + Q [d> x (k x r) + k x (co x ?)]

= -VP + Re~lV2u

where

P = p +

S2- — (kxR)(kxR)

2*

The spacecraft equation (6) becomes

\I •w + £lkxl-u + axl-k = f

(10)

(U)

(12)

Fig. 1 Coordinate system.

638 AGRAWAL: LIQUID MOTION IN SPINNING SPACECRAFT

where

7 = I /pa5

i rt

R x np ds + — \ R x ((n • V)[« x (n x u)]\ ds

For a multitank configuration, the contributions from thetanks are summed up.

The effects of viscosity are included in the liquid motionthrough a boundary-layer analysis. Using the procedure pro-posed by Hendricks and Morton,6 the unknown parametersare expanded in the power of Re~l/2 as follows:

U = UQ + UQ + Re ~ Y2(Ui + HI(13)

where quantities with a tilde refer to boundary-layer varia-bles. Substituting these perturbation expressions into Eq. (10)and equating powers of Re~l/z yields three boundary-valueproblems corresponding to an inviscid problem, a boundary-layer problem, and the correction to the inviscid solution. Bysolving these boundary-layer problems, the complete solutionis obtained.

The unknown variables for the inviscid solution are thepressure function P0, the velocity w 0 > angular velocity co0, andthe eigenvalue X0. Neglecting the viscous terms in Eq. (10), wemay solve for the velocity components in terms of pressure.Substituting the velocity component, the continuity equationbecomes

a2P0 a2p0a*2 dy2+\

a2p0x2 dz2 (14)

The boundary conditions are also expressed in terms of thepressure. From Eq. (14) and the boundary conditions, thepressure and the eigenvalues of the liquid motion are deter-

Transverse Axis

Spin Axis

mined by using a finite element method. The boundary-layerequations are written with respect to the wetted surface polarnormal coordinate. The unknown variables P0» "o» P\ > and HIare determined analytically as described in Ref. 11.

The equations for the viscous correction are obtained bycomparing all terms of order Re~l/2. The equations for theunknown viscous correction parameters PI , u\, and \\ are

a2p0a*2 dz2 (15)

Fig. 2 Finite element model.

The finite element method is used to solve Eq. (15) withappropriate boundary conditions to determine viscous correc-tion to the liquid natural frequencies and mode shapes.

V. Numerical SolutionThe finite element computer program based on the bound-

ary-layer model calculates liquid natural frequencies, modeshapes, damping ratios, torque exerted by the liquid on thespacecraft, energy dissipation rates, spacecraft nutation fre-quency, and nutation time constant. This model has been usedextensively on the INTELSAT VI program12'13 to calculateliquid natural frequencies, mode shapes, and spacecraft nuta-tion time constants.

Liquid Natural Frequencies and Mode ShapesThe finite element model used in the analysis is shown in

Fig. 2. It consists of 81 node points. The inviscid solution giveseigenvalues representing liquid natural frequencies and com-plex eigenvectors, representing modes shapes in terms of threevelocity components at each node point. The complex velocitycomponents are represented in terms of amplitudes and phaseangles. The liquid motion can be considered as a combinationof two types of natural oscillations: sloshing waves and inertialwaves. The sloshing waves are characterized by free surfaceliquid oscillation. Mathematically, slosh motion can be ana-lyzed on the basis of an ideal fluid executing irrotational mo-tion. These modes are characterized by velocity componentseither in phase or out of phase. Inertial waves represent adynamic interaction between Coriolis forces and pressureforces. Inertial waves are circulatory in the liquid interior andthey may or may not have any apparent motion at the freesurface. Inertial wave frequencies are less than twice the spinrate.3 These waves are characterized by velocity components ata node having 90-deg phase difference, creating a circulatorymotion. In general, a mode will have a combination of sloshand inertial wave motion. The viscous correction solution cal-culates damping for the modes.

The calculated natural frequency ratios, ratios of naturalfrequencies to the spin rate, as a function of fraction fill forINTELSAT VI parameters are given in Table 1. It should benoted that only the modes near the frequency ratio = 2 arecalculated. There will be additional modes present below andabove the calculated modes. The parameters are tank ra-dius =0.42 m, radial distance of the tank center from the spinaxis= 1.31 m, and spin rate = 30 rpm. The modes are dividedinto three groups: inertial, slosh, and higher modes containingboth inertial and slosh components. In the slosh modes, twomodes are present: azimuth and elevation. In the azimuthmode, the liquid motion is in a plane normal to the spin axiswith a major velocity component in the transverse direction.The azimuth mode produces a reaction torque along the spinaxis and therefore can cause interaction with the despin con-trol. In the elevation mode, the major velocity component is inthe axial direction. The elevation mode produces a reactiontorque normal to the spin axis and can interact with the nuta-tion control.

Figures 3 and 4 show typical mode shapes of first azimuthand first elevation modes, respectively. To simplify the plot,the velocity is plotted only for the node points along the x axisand on the free surface parallel to the z axis. In the azimuthmode, the liquid motion is in the xy plane with the major

AGRAWAL: LIQUID MOTION IN SPINNING SPACECRAFT 639

Table 1 Liquid natural frequency ratios

Slosh modesFractionfill, %

95908070605040302010

Inertial modes1.996 2.01.9921.9921.9911.9921.9921.989

.997

.996

.996

.996

.996

.9961.989 1.9951.979 1.9931.993

2.0092.0022.02.02.02.02.02.01.997a

1.997a

Firstazimuth2.8762.572.3142.1182.1092.0562.0232.0072.003a

2.005a

Firstelevation

32222222

04784556434351287233

.1872.1422.09

4333333222

Higher modes114778366287167070089590849

4333333332

42998694552396278170901.92

4.814.2553.7923.5803.533.5073.5123.5363.573.6

aDifficult to distinguish between inertial and first azimuth mode.

Table 2

Fill fraction,102030508090

Analytical results

% Frequency2.082.122.162.262.522.74

for elevation mode

ratio Damping ratio0.00880.00640.00250.00400.00230.0019

Fig. 3 First azimuth slosh mode.

Fig. 4 First elevation slosh mode.

velocity along the y axis. In the elevation mode, the liquidmotion is in the xz plane, with the major component in theaxial z direction. The inertial and higher modes are difficult toplot due to their circulatory motion characteristic.

Scaled Model TestsUnder the INTELSAT VI program, Hughes Aircraft Com-

pany performed scaled model tests to determine liquid dy-namic characteristics. The boundary-layer model was alsoused to determine liquid dynamic characteristics. The bound-ary-layer model was also used for the scaled test parametersto analytically predict liquid dynamic characteristics. Ana-lytical results are given in Table 2. The parameters used weretank diameter = 0.146m, radial distance of the tank centerfrom spin axis =0.218 m, liquid density = 1 g/ml, and viscos-ity =0.009 cmVs. The damping ratio is defined as a ratio ofdamping to critical damping. Some of the experimental dataare presented in Ref. 1, and regarding the INTELSAT bound-ary-layer model it is stated in the paper that "Unpublishedfinite element eigenmode modeling for the off-axis case atINTELSAT is beginning to compare well with measured data,promising that spinning liquid modal tests may soon followrather than lead analysis—as is the practice in the more estab-lished area of structural mechanics."

Nutation Time ConstantDuring several mission phases of INTELSAT VI, active nu-

tation control is used. To design the active nutation controlproperly, the nutation dedamping due to liquid slosh needs tobe determined accurately. Air-bearing tests have been per-formed on the INTELSAT VI program to determine its nuta-tion time constant. The critical parameters, inertia ratio andfill fraction, are kept the same and other parameters are scaleddue to limitations of test vehicles and chamber. In-orbit nuta-tion time constants are extrapolated from the measured nuta-tion time constant by using dimensional analysis.

Anomalous ResonancesOn-orbit liquid slosh tests on INTELSAT IV revealed sharp

reduction of nutation time constants for certain tank fill frac-tions and inertia ratios. Air-bearing tests were later performedand these validated the in-orbit results. These conditions dueto resonances in liquid could not be explained by linear modelsbecause the slosh frequencies were significantly higher than thenutation driving frequency. These resonances were termed asanomalous resonances.

INTELSAT VI air-bearing liquid slosh test results have alsoexhibited sharp reductions in nutation time constants forcertain fill fractions and inertial ratios, defined as ratio ofmoment of inertia about spin axis to that about transverseaxis. To validate the boundary-layer finite element model, itwas used to analytically predict nutation time constants forthese parameters. Nutation time constants are determined bysolving Eq. (6) for eigenvalues,12 nutation frequencies and nu-

640 AGRAWAL: LIQUID MOTION IN SPINNING SPACECRAFT

IDz

I815UJ

O

A = ANALYTICAL• = EXPERIMENTALFF = TANK FRACTION FILL

FF = 20%FF = 80%

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

INERTIA RATIO

Fig. 5 Nutation time constant.

tation time constants, of the spacecraft liquid system. Figure 5shows the plots of nutation time constants determined analyt-ically and experimentally for tank fill fractions of 20 and 80%.Both experimental and analytical results predict a sharp dropin nutation time constants for the same inertia ratio (0.45) andfill fraction (20%). The analytical values of nutation timeconstant are, however, higher than experimental results. Thisresonance condition could be due to excitation of inertialmodes close to the nutation driving frequency.

VI. SummaryThe boundary-layer model to predict dynamic characteris-

tics of liquid motion in a spinning spacecraft with partiallyfilled tanks is presented. The solution is obtained by solvingthree boundary-value problems: an inviscid fluid problem, aboundary-layer problem, and a viscous correction problem.The boundary-layer equations are solved analytically and theinviscid and viscous correction solutions are obtained by usingfinite element methods. The finite element computer programhas been developed by MBB, Germany, under an INTELSATcontract. This program calculated the liquid natural frequen-cies and mode shapes, torque exerted by the liquid on thespacecraft, energy dissipation rates, spacecraft nutation fre-quencies, and time constants. The computer program has beenused extensively on the INTELSAT VI program.

The results indicate that the liquid motion in rotating bodiescan be considered as a combination of two types of naturaloscillations: sloshing modes and inertial modes. The numericalresults show two slosh modes, first azimuth and elevationmodes, and lower modes to be inertial modes and highermodes to be a combination of inertial and slosh modes. There-fore, only these two slosh modes can be accurately representedby pendulum models.

The analytical results predict a sharp drop in nutation timeconstant for certain inertia ratios and tank fill fractions. Thisphenomenon, known as anomalous resonance, was also pres-ent during INTELSAT IV in-orbit liquid slosh tests andground air-bearing tests for INTELSAT IV and VI. The nuta-tion time constant depends on two liquid characteristics, en-ergy dissipation, and resonance of normal modes due to nu-tation driving frequency. Since the slosh frequencies aresignificantly higher than the nutation driving frequencies, it

appears that the resonance condition is due to excitation ofinertial modes. Further work is required to identify the inertialmodes which contribute to this resonance.

VII. ConclusionsThe boundary-layer model accurately predicts liquid natural

frequencies and mode shapes. The analytical results show thatin general a liquid mode will be a combination of slosh andinertial modes. Only very few modes can be considered as pureslosh modes, such as first azimuth and first elevation modes inthe example presented in this paper. Therefore, only theseslosh modes can be represented accurately by equivalent pen-dulum models. The boundary-layer model also predicted asharp reduction of nutation time constant for certain fill frac-tions and inertia ratios as revealed in ground and on-orbitliquid slosh data. It appears to be a liquid resonance due toexcitation of certain inertia mode(s). Further work is necessaryto explain this phenomenon.

AcknowledgmentsThis author wishes to express his deepest appreciation to

A. Pohl for development of boundary-layer model software.The support of D. Sakoda in the preparation of this paper isalso sincerely appreciated.

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the Payload Control System of Dual-Spin Spacecraft," Journal ofGuidance, Control, and Dynamics, Vol. 11, No. 4, 1988, pp. 343-351.

2Greenspan, H. P., "On Transient Motion of a Contained RotatingFluid," Journal of Fluid Mechanics, Vol. 20, Pt. 4, 1964, pp. 673-696.

3Stewartson, K., "On the Stability of a Spinning Top ContainingLiquid," Journal of Fluid Mechanics, Vol. 5, Pt. 4, 1959, p. 577.

4Wedemeyer, E. H., Viscous Correction to Stewartson's StabilityCriterion, AGARD Conf. Proceeding, No. 10, 1966, pp. 99-116.

5Nayfeh, A. H., and Meirovitch, L., "The Stability of Motion of aSatellite with Cavities Partially Filled with Liquid," AIAA 12th Aero-space Sciences Meeting, Washington, DC, AIAA Paper 74-168, Jan.-Feb. 1974.

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7Stergiopoulos, S., and Aldridge, K. D., "Inertial Waves in a FluidPartially Filling a Cylindrical Cavity During Spin-Up from Rest,"Geophysics Astrophysics Fluid Dynamics, Vol. 21, 1982, pp. 89-112.

8Pfeiffer, F., "Bin Naherungsverfahren fur FliissigkeitsgefullteKreisel," Ingeniur-Archiv, Vol. 43, 1974, pp. 306-316.

9El-Raheb, M., and Wagner, P., "Vibration of a Liquid with a FreeSurface in a Spinning Spherical Tank," Journal of Sound and Vibra-tion, Vol. 76, No. 1, 1981, pp. 83-93.

10Abramson, H., "The Dynamic Behavior of Liquids in MovingContainers," NASA SP-106, 1966.

11 Pohl, A., "Dynamic Effects of Liquid on Spinning Spacecraft,"Proceedings of the First INTEL S A T/ESA Symposium on DynamicEffects of Liquids on Spacecraft Attitude Control, INTELSAT,Washington, DC, April 1984, pp. 121-147.

12Agrawal, B. N., "Interaction Between Propellant Dynamics andAttitude Control in Spinning Spacecraft," Proceedings of AIAA Dy-namics Specialist Conference (Monterey, CA), AIAA, New York,1987, pp. 774-790 (AIAA Paper 87-0928).

13Agrawal, B. N., "Analytical Prediction of Nutation Time Con-stant for Spinning Spacecraft with Partially Filled Tanks," Proceed-ings of Seventh VPI & SU/AIAA Symposium on Dynamics and Con-trol of Flexible Large Structures (Blacksburg, VA), May 1989, pp.439-452.