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Research Papers

Global Analysis of Gravity Gradient Satellite's Pitch Motion in an Elliptic Orbit

[+] Author and Article Information
Dayung Koh

Department of Astronautical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: dayungko@usc.edu

Henryk Flashner

Professor
Department of Aerospace and
Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: hflashne@usc.edu

Manuscript received November 15, 2014; final manuscript received January 14, 2015; published online April 28, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 10(6), 061020 (Nov 01, 2015) (11 pages) Paper No: CND-14-1290; doi: 10.1115/1.4029621 History: Received November 15, 2014; Revised January 14, 2015; Online April 28, 2015

Pitch motion of a gravity gradient satellite in an elliptical orbit is studied. The cell mapping method is employed to find periodic solutions and analyze the global behavior of the system. Stability characteristics of the solutions are established using a point mapping approximation algorithm. The proposed approach does not depend on existence of a small parameter and, therefore, no limitations are imposed on the magnitudes of eccentricity or amplitude of motion. This is in contrast to the perturbation based approaches that require assumptions of small orbital eccentricity and small motion. As a result, stable periodic solutions of twice and three times the orbital period are found for some different eccentricities and inertia parameters. Global behavior and evolution of periodic solutions are also demonstrated using invariant surfaces and bifurcation diagrams.

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References

NASA, 1969, “Spacecraft Gravitational Torques,” NASA Report No. SP-8024.
Hughes, P. C., 1986, Spacecraft Attitude Dynamics, Wiley, New York.
Baker, R. M. L., 1960, “Librations on a Slightly Eccentric Orbit,” ARS J., 30(1), pp. 124–126.
Schechter, H. B., 1963, “Satellite Librations on an Elliptic Orbit,” Rand Corp., Report No. RM-3632-PR.
Schechter, H. B., 1964, “Dumbbell Librations in Elliptic Orbits,” AIAA J., 2(6), pp. 1000–1003. [CrossRef]
Beletskii, V. V., 1965, The Satellite Motion About Center of Mass, Publishing House Science, Moscow, Russia.
Modi, V. J., and Brereton, R. C., 1969, “Periodic Solutions Associated With the Gravity-Gradient-Oriented System: Part 1. Analytical and Numerical Determination,” AIAA J., 7(7), pp. 1217–1225. [CrossRef]
Zlatoustov, V. A., Okhotsimsky, D. E., Sarychev, V. A., and Torzhesky, A. P., 1964, “Investigation of a Satellite Oscillations in the Plane of an Elliptic Orbit,” The Eleventh International Congress of Applied Mechanics, Munich, Germany, pp. 436–439.
Modi, V. J., and Brereton, R. C., 1969, “Periodic Solutions Associated With the Gravity-Gradient-Oriented System: Part 2. Stability Analysis,” AIAA J., 7(8), pp. 1465–1468. [CrossRef]
Sarychev, V. A., Sazonov, V. V., and Zlatoustov, V. A., 1977, “Periodic Oscillations of a Satellite in the Plane of an Elliptic Orbit,” Kosmicheskie Issledovaniya15(6), pp. 809–834 (Russian); Cosmic Res., 15(6), pp. 698–719 (English).
Tong, X., and Rimrott, F., 1991, “Numerical Studies on Chaotic Planar Motion of Satellite in an Elliptic Orbit,” Chaos Solitons Fractals, 1(2), pp. 179–186. [CrossRef]
Karasopoulos, H. A., 1994, “Nonlinear Dynamics of the Planar Pitch Attitude Motion for a Gravity-Gradient Satellite,” Ph.D. thesis, University of Cincinnati, Cincinnati, OH.
Kirchgraber, U., Manz, U., and Stoffer, D., 2000, “Rigorous Proof of Chaotic Behaviour in a Dumbbell Satellite Model,” J. Math. Anal. Appl., 251(2), pp. 897–911. [CrossRef]
Stabb, M. C., and Gray, G. L., 1994, “Chaos in Controlled, Gravity Gradient Satellite Pitch Dynamics Via the Method of Melnikov—Saddle Stabilization,” AIAA Paper No. AIAA-94-1671. [CrossRef]
Fujii, H. A., Ichiki, W., Suda, S., and Watanabe, T. R., 2000, “Chaos Analysis on Librational Control of Gravity-Gradient Satellite in Elliptic Orbit,” J. Guid. Control Dyn., 23(1), pp. 145–146. [CrossRef]
Flashner, H., and Hsu, C. S., 1983, “A Study of Nonlinear Periodic Systems Via the Point Mapping Method,” Int. J. Numer. Methods Eng., 19(2), pp. 185–215. [CrossRef]
Hsu, C. S., 1980, “A Theory of Cell-to-Cell Mapping Dynamical Systems,” ASME J. Appl. Mech., 47(4), pp. 931–939. [CrossRef]
Hsu, C. S., 1980, “An Unravelling Algorithm for Global Analysis of Dynamical Systems: An Application of Cell-to-Cell Mappings,” ASME J. Appl. Mech., 47(4), pp. 940–948. [CrossRef]
Bate, R. R., Mueller, D., and White, J. E., 1971, Fundamentals of Astrodynamics, Dover Publications, New York.
Flashner, H., and Guttalu, R. S., 1989, “Analysis of Non-Linear Non-Autonomous Systems by Truncated Point Mappings,” Int. J. Nonlinear Mech., 24(4), pp. 327–344. [CrossRef]
Guttalu, R. S., and Flashner, H., 1996, “Stability Analysis of Periodic Systems by Truncated Point Mapping,” J. Sound Vib., 189(1), pp. 33–54. [CrossRef]
Hsu, C. S., 1977, “On Non-Linear Parametric Excitation Problems,” Adv. Appl. Mech., 17, pp. 245–301. [CrossRef]
Golat, M., and Flashner, H., 2002, “A New Methodology for the Analysis of Periodic Systems,” Nonlinear Dyn., 28(1), pp. 29–51. [CrossRef]
Zlatoustov, V. A., and Markeev, A. P., 1973, “Stability of Planar Oscillations of a Satellite in an Elliptic Orbit,” Celestial Mech., 7(1), pp. 31–45. [CrossRef]

Figures

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Fig. 1

Geometry of satellite motion

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Fig. 2

Equilibrium point x* and P − 3 solutions X*(i), i = 1, …, K for K = 3

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Fig. 3

Cell state space for N = 2

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Fig. 4

Equilibrium cell z* and P − K cells z *(i), i = 1, …, K for K = 3

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Fig. 5

Small perturbation z of periodic solutions z * at cell j. (a) Perturbation of P − 1 solution and (b) perturbation of P − K solution.

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Fig. 6

Result of the cell mapping algorithm showing global behavior in a cell state space for e = 0.01, k2 = 0.1

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Fig. 7

P − 2 cell groups for e = 0.01, k2 = 0.1 and convergence of solution B1 and B2 using extended cell mapping

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Fig. 8

Three different ways to verify P − 3 solutions in Table 1 (e = 0.01, k2 = 0.1). (a) Phase plane trajectories, (b) fast Fourier analysis, (c) response of a stable solution S1, and (d) response of an unstable solution U1.

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Fig. 9

P − 1 to P − 1 bifurcation for different eccentricities (solid line: stable solution, dashed line: unstable solution)

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Fig. 10

Invariant surfaces in phase plane when bifurcation from a stable P − 1 solution to two stable P − 1 solutions and one unstable P − 1 solution occurs (e = 0.01)

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Fig. 11

Bifurcation sequence P − 1 → P − 2 for k2 = 0.1. (a) Three-dimensional representation, (b) projection on e − x1 plane, and (c) projection on e − x2 plane.

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Fig. 12

Invariant surface in phase plane for P − 1 to P − 2 bifurcation (k2 = 0.1)

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Fig. 13

Floquet multipliers confirming the P − 1 → P − 2 bifurcation condition

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Fig. 14

Bifurcation sequence P − 1 → P − 3 → P − 9 for e = 0.1. (a) Three-dimensional representation, (b) projection on x1 − k2 plane, and (c) projection on x2 − k2 plane.

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Fig. 15

(a) Trajectory of P − 3 solutions in phase plane (gi) and trajectory of a P − 1 solution in phase plane (+), (b) pitch angle response versus true anomaly in number of revolutions for a P − 3 solution, (c) trajectory of P − 9 solutions in phase plane: numbers 0–9 show the points at n·T, and (d) pitch angle response versus true anomaly in number of revolutions for a P − 9 solution

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Fig. 16

Invariant surfaces in phase plane when bifurcation from a stable P − 1 solution to two stable P − 3 solutions and then one unstable P − 9 solution occurs (e = 0.1)

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