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Planar Librational Motion of a Gravity-Gradient Satellite During Deployment

  • Conference paper
Astronautical Research 1971

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Abstract

Two existing analytical methods of predicting the planar librational motion during deployment of a gravity-gradient satellite with extendible booms are studied. The first method by Watson is based on the assumption that during the relatively short time of deployment the integrated effects of gravity-gradient torques can be neglected. The second method by Liu and Mitchell assumes that the librational motion is limited to small amplitudes. The series solution approach of Liu and Mitchell has been adapted here to include a corrected higher order term. Both analytical solutions are applied to predict librational deployment motion of (1) the DODGE satellite during a particular ‘dead-beat’ capture maneuver, and (2) a Dumbbell satellite initially stabilized with respect to the local vertical. The solutions obtained by both analytical methods are compared with the results of numerical integration of the exact deployment equation.

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Abbreviations

A 0, B 0, A 1, B 1 :

Constants appearing in the zeroth order and the first order series solutions respectively and determined from the initial conditions.

C 0, C 1, …, C s :

Coefficients appearing in the series solutions for r = 0

D 0, D 1, …, D s :

Coefficients appearing in the series solutions for r= − 1

F g :

Force due to gravity on satellite end masses acting toward the center of the Earth

I:

Moment of inertia of the central satellite body

I:

Moment of inertia tensor

I i, I f :

Initial and final moments of inertia of the satellite about an axis perpendicular to the orbital plane

I x, I y, I z :

Principal moments of inertia about spacecraft body axes

K:

Arbitrary constant

Kʹ:

Constant determined from initial conditions of the first (energy) integral; \(K' = \omega _0^2({I_x} - {I_z})/{I_y}\)

L, L(t):

Length of the satellite boom at time t

L 0 :

Length of the satellite boom before initiation of extension

Lc :

Angular momentum vector with respect to the center of mass of the system

R c :

Distance from center of the Earth to the satellite center of mass

T:

Dimensionless time; \(T = {\omega _0}t + \beta \)

Tg :

A vector quantity representing the gravitational torque

U 0(t), V 0(T), U 1(T), V 1(T):

Functions to be determined in the variation of parameter solution of the non-homogeneous parts of the zeroth order and the first order deployment equations

V:

Uniform extension rate of the booms; \(V=\dot{L}\)

cʹ:

Constant; \(c' = {({I_i}/2m{V^2})^{1/2}}\)

k:

Radius of gyration of the central satellite body

k 1, k 2 :

Constants to be determined in the variation of parameter technique

m:

Boom end mass

n:

Any integral number; n = 1, 2, 3…

r:

Root of indicial equation

s:

Any integral numbers; s= 1, 2, 3…

t:

Time

t f :

Final time

y 1 , y 2, y 3, y 4 :

Periodic functions of T appearing in the variation of parameter technique

α:

Dimensionless variable; \({a^2} = I{\beta ^2}/(2mL_0^2)\)

β:

Dimensionless variable; \(\beta {\text{ = }}{L_0}{\omega _0}/V\)

φ:

Libration angle in the orbit plane (Pitch angle)

φ 0, φ 1, φ 2 :

Contributions to the libration angle by the zeroth, first and second order approximations respectively

ω:

Angular velocity vector

ω 0 :

Magnitude of orbital angular velocity vector

ω i, ωf :

Magnitude of initial and final angular velocities respectively

ω y :

Component of the inertial angular velocity normal to the orbital plane

References

  1. Klemperer, W. B. and Baker, Jr., R. M., Proceedings of the VIIth International Astronautical Congress, Rome, September, 1956.

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  2. Beletskii, V. V., ‘The Libration of a Satellite’, NASA TTF-10, May, 1960 (Translation).

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  3. Bainum, P. M., ‘On the Motion and Stability of a Multiple Connected Gravity-Gradient Satellite with Passive Damping’, Ph.D. dissertation, Catholic University, Washington, D.C., 1966; also the Johns Hopkins University-Applied Physics Laboratory Technical Report TG-872, January, 1967.

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  4. Fischell, Robert E., ‘A Graviety-Gradient Satellite at Synchronous Altitude’, Second IFAC Symposium on Automatic Control in Space, Vienna, Austria, Sept. 1967.

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  5. Dow, P. C., Scammell, F. H., Murray, F. T., Carlson, N. A., and Buck, J. H., in Proceedings of the AIAA/JACC Guidance and Control Conference, Seattle, August 15–17, 1966, published by AIAA, New York, N.Y., p. 285.

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  6. Watson, D. M., Proceedings of the Symposium on Passive Gravity-Gradient Stabilization, NASA-Ames Research Center, NASA-SP-107, 1966, p. 227.

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  7. Liu, Han-Shou and Mitchell, Thomas P., ‘The Structural and Librational Dynamics of a Satellite Deploying Flexible Booms or Antennas’, AIAA 5th Aerospace Sciences Meeting, New York, N.Y., Jan. 23–26, 1967, Paper No. 67-43.

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  8. Smola, J. F., ‘Momentum Transfer as a Means of Despinning a Rotating Spacecraft’, The Johns Hopkins University, Applied Physics Laboratory, Technical Report TG-885, January 1967.

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  9. Goldstein, Herbert, Classical Mechanics, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1965.

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  10. Private Conversation with Dr Han-Shou Liu, NASA Goodard Space Flight Center, Green-belt, Maryland, October 1969.

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  11. Puri, Virender, ‘Planar Librational Motion of a Gravity-Gradient Satellite During Deployment’, Master’s dissertation, Howard University, Washington, D.C. Jan. 1971.

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  12. Rainville, Earl D., Elementary Differential Equations, The Macmillan Co., New York, 1965.

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Author information

Authors and Affiliations

  1. Dept. of Mechanical Engineering, Howard University, Washington, D.C., 20001, USA

    Virender Puri (Graduate student) & Peter M. Bainum (Associate Professor of Aerospace Engineering)

Authors
  1. Virender Puri
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  2. Peter M. Bainum
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Editor information

L. G. Napolitano P. Contensou W. F. Hilton

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© 1973 D. Reidel Publishing Company, Dordrecht-Holland

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Puri, V., Bainum, P.M. (1973). Planar Librational Motion of a Gravity-Gradient Satellite During Deployment. In: Napolitano, L.G., Contensou, P., Hilton, W.F. (eds) Astronautical Research 1971. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2559-1_6

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  • DOI: https://doi.org/10.1007/978-94-010-2559-1_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-2561-4

  • Online ISBN: 978-94-010-2559-1

  • eBook Packages: Springer Book Archive

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