Skip to main content
SpringerLink
Log in

Satellite with periodical mass redistribution: relative equilibria and their stability

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

The dynamics of a spacecraft with variable mass distribution is considered, assuming a central Newtonian gravitational field. The equations of spatial attitude motion are derived within the framework of the so-called satellite approximation and examined via a Hamiltonian approach. The centre of mass of the satellite multibody system moves along an elliptic orbit of arbitrary eccentricity. For a prescribed attitude motion, control via the mass redistribution is determined. Various classes of relative equilibria are detected, and the respective control is obtained. The stability of the spacecraft spatial attitude motions is examined in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate texts in mathematics, vol. 60, p. 519. Springer, Berlin (1989)

    Google Scholar 

  • Atchonouglo, E., Vallée, C., Monnet, T., Fortuné, D.: Identification of the ten inertia parameters of a rigid body. JAMM 72, 22–25 (2008)

    MathSciNet  MATH  Google Scholar 

  • Beletskii, V.V.: Motion of an Artificial Satellite About its Center of Mass, p. 261. Israel Program for Scientific Translation, Jerusalem (1966)

  • Beletsky, V.V.: Essays on the Motion of Celestial Bodies, p. 392. Birkhaüser, Basel (2001)

    Book  Google Scholar 

  • Burov, A., Kosenko, I.: On plane oscillations of a body with a variable mass distribution in an elliptic orbit. In: Proceedings of the ENOC 2011, 24–29 July 2011, Rome, Italy (2011)

  • Burov, A.A.: Poisson variations in the problem of the stability of equilibria in rigid body mechanics. J. Appl. Math. Mech. 74(1), 68–75 (2010)

    Article  MathSciNet  Google Scholar 

  • Burov, A.A.: Oscillations of a vibrating dumbbell on an elliptic orbit. Dokl. Phys. 56, 182–185 (2011)

    Article  ADS  Google Scholar 

  • Burov, A.A., Chevallier, D.P.: Dynamics of affinely deformable bodies from the standpoint of theoretical mechanics and differential geometry. Rep. Math. Phys. 62, 283–321 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  • Burov, A.A., Kosenko, I.I.: Planar vibrations of a solid with variable mass distribution in an elliptic orbit. Dokl. Phys. 56, 548–552 (2011)

    Article  ADS  Google Scholar 

  • Burov, A., Kosenko, I.: On planar oscillations of a body with a variable mass distribution in an elliptic orbit. J. Mech. Eng. Sci. 225, 2288–2295 (2011)

    Article  Google Scholar 

  • Chetaev, N.G.: Theoretical Mechanics, p. 407. Mir, Moscow (2001)

    MATH  Google Scholar 

  • Chicone, C.: Ordinary Differential Equations with Applications. Springer, New York (2006)

    MATH  Google Scholar 

  • De Bra, D.B., Delp, R.H.: Satellite stability and natural frequencies in a circular orbit. J. Astron. Sci. 8(1), 14–17 (1961)

    Google Scholar 

  • Lichtenberg, A.J., Lieberman, M.A.: Regular and Stochastic Motion, p. 499. Springer, New York (1983)

    Book  Google Scholar 

  • Magnus, K.: Die Stabilitat partikularer Drehbewegungen von Satelliten beliebiger Form auf einer Kreisbahn. Ing.-Arch. 34, S.129 (1965)

    Article  Google Scholar 

  • Markeev, A.P.: Linear Hamiltonian Systems and Some Problems of Stability of Motion of a Satellite with Respect to a Center of Mass. Regular and Chaotic Dynamics, p. 396, Izhevsk (2009). (In Russian)

  • Markeev, A.P., Sokolskii, A.G.: On the problem of stability of relative equilibrium of a satellite in a circular orbit. Cosm. Res. 13(2), 119–125 (1975)

    ADS  Google Scholar 

  • Moser, J.K.: Lectures on Hamiltonian systems. Memoirs of the American Mathematical Society, Courant Institute of Mathematical Science, New York, 81, 1–60 (1968)

  • Moser, J.: Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics (AM-77). Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (1973)

    Google Scholar 

  • Pascal, M.: Sur le mouvement d’un triple bâtonnet dans un champ Newtonien. J. de Mécanique. 11, 147–160 (1972)

    ADS  MATH  Google Scholar 

  • Polyanskaya, I.P.: Oscillations of a satellite with compensating devices in an elliptical orbit. Cosm. Res. 20, 674–681 (1982)

    Google Scholar 

  • Rubanovsky, V.N., Samsonov, V.A.: Stability of Stationary Motions in Examples and Problems. Nauka, Moscow (1988). (In Russian)

    Google Scholar 

  • Schiehlen, W.: Über den Drallsatz für Satelliten mit im Innern bewegten Massen, Z. angew. Math. Mech., Bd.46, Sonderheft, S. T132–T134 (1966)

  • Schiehlen, W.: Über die Lagestabilisirung künstlicher Satelliten auf elliptischen Bahnen. Diss. Dokt.-Ing., 148 S. technische Hochschule Stuttgart, Stuttgart (1966)

  • Schiehlen, W., Kolbe, O.: Gravitationsstabilisierung von Satelliten auf elliptischen Bahnen. Arch. Appl. Mech. (Ing.-Arch.) 38, 389–399 (1969)

    MATH  Google Scholar 

  • Seiliger, D.N.: Mechanics of similarly varying system. In: Schulze, A. (ed.) Typography, p. 23. Typography of A. Schulze, Odessa (1890-91). (In Russian)

  • Sidorenko, V.V., Neustadt, A.I.: Investigation of the stability of long-periodic planar motion of a satellite in a circular orbit. Cosm. Res. 38, 307–321 (2000)

    Google Scholar 

  • Sławianowski, J.J.: The mechanics of the homogeneously-deformable body. Dynamical models with high symmetries. Z. Angew. Math. Mech. 62, 229–240 (1982)

    Article  Google Scholar 

  • Stoker, J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems, p. 294. Wiley, New York (1950)

    MATH  Google Scholar 

  • Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients, vol. 1, p. 386. Israel Program for Scientific translations. Jerusalem, London (1975)

Download references

Acknowledgements

The paper has been prepared with help from project EMaDeS (Centro-01-0145-FEDER-000017), the Portuguese Foundation for Science and Technology via the Centre for Mechanical and Aerospace Science and Technologies, C-MAST (POCI-01-0145-FEDER-007718) for the second author, RFBR Project No. 16-01-00625a for the first author, and RSF Project No. 14-21-00068 for the third author at MAI.

Author information

Authors and Affiliations

  1. Dorodnicyn Computing Centre of RAS, Moscow, Russia

    Alexander Burov & Ivan Kosenko

  2. National Research University High School of Economy, Moscow, Russia

    Alexander Burov

  3. University of Beira Interior, Covilhã, Portugal

    Anna Guerman

  4. Moscow Aviation Institute (National Research University), Moscow, Russia

    Ivan Kosenko

Authors
  1. Alexander Burov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Guerman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ivan Kosenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Kosenko.

Ethics declarations

Conflict of interest

The authors certify that they have no conflict of interest to declare.

Additional information

This article is part of the topical collection on Innovative methods for space threats: from their dynamics to interplanetary missions.

Guest Editors: Giovanni Federico Gronchi, Ugo Locatelli, Giuseppe Pucacco and Alessandra Celletti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burov, A., Guerman, A. & Kosenko, I. Satellite with periodical mass redistribution: relative equilibria and their stability. Celest Mech Dyn Astr 131, 1 (2019). https://doi.org/10.1007/s10569-018-9874-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10569-018-9874-0

Keywords

Search

Navigation