Abstract
This paper studies the attitude dynamics of a rigid body in a Keplerian orbit. We show that the use of Classical Rodrigues Parameters for the attitude motion of the rigid body subject to gravity-gradient torques enables us to characterize the equilibria associated with the rotational motion about its mass center. A parametric study of the stability of equilibria is conducted to show that large oscillations are induced due to the energy exchange between the pitch and roll–yaw motions, specifically near the 2:1 resonant commensurability regions. A visualization tool is developed to study these pitch oscillations and gain insight into the rigid body motion near internal resonance conditions. A measure of coupling between the pitching and roll–yaw motions is developed to quantify the energy exchange utilizing information from the state transition matrix.
References
Battin, R.H.: An introduction to the mathematics and methods of astrodynamics. AIAA (1999)
Beletsky, V.V.: The libration of a satellite on an elliptic orbit. In: Roy, M. (ed.) Dynamics of Satellites/Dynamique des Satellites, pp. 219–230. Springer, Berlin (1963)
Beletskii, V.V.: Motion of Artificial Satellite About its Center of Mass. NASA TT F-429 (1966)
Breakwell, J.V., Pringle Jr, R.: Nonlinear resonance affecting gravity-gradient stability (1966)
Brereton, R.C., Modi, V.J.: Periodic solutions associated with the gravity-gradient-oriented system. I-analytical and numerical determination. AIAA J. 7(7), 1217–1225 (1969)
Brereton, R.C., Modi, V.J.: Periodic solutions associated with the gravity-gradient-oriented system. II-stability analysis. AIAA J. 7(8), 1465–1468 (1969)
da Silva, M.R.C.: Attitude stability of a gravity-stabilized gyrostat satellite. Celest. Mech. 2(2), 147–165 (1970)
DeBra, D.B., Delp, R.H.: Rigid body attitude stability and natural frequencies in a circular orbit. J. Astron. Sci. 8(1), 14–17 (1961)
Eapen, R.T., Frueh, C.: Averaged solar radiation pressure modeling for high area-to-mass ratio objects in geosynchronous orbits. Adv. Space Res. 62(1), 127–141 (2018)
Eapen, R., Majji, M., Alfriend, K.T.: Equilibria and stability of the attitude motions of a gravity gradient satellite. In: Presented at the AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, August 19–23 (2018)
Floquet, G.: Sur les equations differentielles lineaires. Ann. ENS 12, 47–88 (1883)
Früh, C., Schildknecht, T.: Attitude motion of space debris objects under influence of solar radiation pressure and gravitiy. In: 63rd International Astronautical Congress (2012)
Früh, C., Kelecy, T.M., Jah, M.K.: Coupled orbit-attitude dynamics of high area-to-mass ratio (HAMR) objects: influence of solar radiation pressure, Earth’s shadow and the visibility in light curves. Celest. Mech. Dyn. Astron. 117(4), 385–404 (2013)
Gerlach, O.H.: Attitude stabilization and control of earth satellites. Report VTH 122. Technische Hogeschool Delft Vliegtuigbouwkunde, Delft The Netherlands (1965)
Goldstein, H., Poole, C.P., Safko, J.L.: Classical Mechanics, 3rd edn. Addison-Wesley, Boston (2001). ISBN: 978-0-201-65702-9
Gouliaev, V.I., Zubritskaya, A.L., Koshkin, V.L.: Universal sequence of bifurcation of doubling of the oscillation period for a satellite in an elliptical orbit. Mech. Solids 24, 1–6 (1989)
Hitzl, D.L.: Nonlinear attitude motion near resonance. AIAA J. 7(6), 1039–1047 (1969)
Hughes, P.C.: Spacecraft Attitude Dynamics. Dover Publications, INC, New York (2004)
Junkins, J.L., Singla, P.: How nonlinear is it? A tutorial on nonlinearity of orbit and attitude dynamics. Adv. Astron. Sci. 115(SUPPL.), 1–45 (2003)
Kane, T.R.: Attitude stability of Earth-pointing satellites. AIAA J. 3(4), 726–731 (1965)
Karasopoulos, H.A., Richardson, D.L.: Numerical investigation of chaos in the attitude motion of a gravity-gradient satellite. Astrodynamics 1993, 1851–1870 (1994)
Karasopoulos, H.A.: Nonlinear dynamics of the planar pitch attitude motion for a gravity-gradient satellite (No. WL-TR-94-3123). Wright Lab Wright-Patterson AFB OH (1994)
Koch, B.P., Bruhn, B.: Chaotic and periodic motions of satellites in elliptic orbits. Z. Naturforschung A 44(12), 1155–1162 (1989)
Kuryakov, V.A.: A method of constructing polhodes of an intermediate motion in the dynamics of a rigid body. J. Appl. Math. Mech. 50(5), 666–669 (1986)
Landau, L.D., Lifshitz, E.M.: Mechanics. Course of Theoretical Physics, vol. 1, 3rd edn. Reed Educational and Professional Publishing Ltd, Oxford (1976)
Malkin, I., Gilévich.: Theory of Stability of Motion. United States Atomic Energy Commission, Office of Technical Information, Oak Ridge (1958)
Meirovitch, L., Wallace, F.: Attitude instability regions of a spinning unsymmetrical satellite in a circular orbit. J. Astron. Sci. 14, 123 (1967)
Schaub, H., Junkins, J.L.: Stereographic orientation parameters for attitude dynamics: a generalization of the Rodrigues parameters. J. Astron. Sci. 44(1), 1–19 (1996)
Tong, X., Rimrott, F.P.: Numerical studies on chaotic planar motion of satellites in an elliptic orbit. Chaos Solitons Fractals 1(2), 179–186 (1991)
Vitins, M.: Keplerian motion and gyration. Celest. Mech. 17(2), 173–192 (1978)
Whittaker, E., McCrae, S.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge Mathematical Library). Cambridge University Press, Cambridge (1988). https://doi.org/10.1017/CBO9780511608797
Winter, O.C., Murray, C.D.: Resonance and chaos: I. First-order interior resonances. Astron. Astrophys. 319, 290–304 (1997)
Zlatoustov, V.A., Okhotsimsky, D.E., Sarychev, V.A., Torzhevsky, A.P.: Investigation of a satellite oscillations in the plane of an elliptic orbit. Cosm. Res. 2(5), 657–666 (1964)
Acknowledgements
This research described in this paper was carried out at and supported by the Department of Aerospace Engineering, Texas A&M University, College Station, TX. Prof. John Junkins is acknowledged for suggesting the osculating surface that led to the development of Binet–Poincaré sections of this paper. The work in this paper is partially supported by the National Science Foundation under Award No. NSF CMMI-1634590. It is also partially supported by the U.S. Air Force Office of Scientific Research (FA9550-15-1-0313 and FA9550-17-1-0088).
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Eapen, R.T., Majji, M. & Alfriend, K.T. Attitude dynamics of a rigid body in Keplerian motion. Celest Mech Dyn Astr 133, 2 (2021). https://doi.org/10.1007/s10569-020-10000-w
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DOI: https://doi.org/10.1007/s10569-020-10000-w