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Symbolic Investigation of the Dynamics of a System of Two Connected Bodies Moving Along a Circular Orbit

  • Sergey A. GutnikEmail author
  • Vasily A. Sarychev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)

Abstract

The dynamics of the system of two bodies, connected by a spherical hinge, that moves along a circular orbit under the action of gravitational torque is investigated. Computer algebra method based on the resultant approach was applied to reduce the satellite stationary motion system of algebraic equations to a single algebraic equation in one variable that determines all planar equilibrium configurations of the two–body system. Classification of domains with equal numbers of equilibrium solutions is carried out using algebraic methods for constructing discriminant hypersurfaces. Bifurcation curves in the space of system parameters that determine boundaries of domains with a fixed number of equilibria of the two–body system were obtained symbolically. Depending on the parameters of the problem, the number of equilibria was found by analyzing the real roots of the algebraic equations.

Keywords

Satellite-stabilizer system Gravitational torque Circular orbit Lagrange equations Algebraic equations Equilibrium orientation Computer algebra Discriminant hypersurface 

References

  1. 1.
    Sarychev, V.A.: Problems of orientation of satellites, Itogi Nauki i Tekhniki. Ser. Space Research, vol. 11. VINITI, Moscow (1978). (in Russian)Google Scholar
  2. 2.
    Sarychev, V.A.: Investigation of the dynamics of a gravitational stabilization system, Sb. Iskusstv. Sputniki Zemli (Collect. Artif. Earth Satellites). Izd. Akad. Nauk SSSR, Moscow, no. 16, pp. 10–33 (1963). (in Russian)Google Scholar
  3. 3.
    Sarychev, V.A.: Relative equilibrium orientations of two bodies connected by a spherical hinge on a circular orbit. Cosm. Res. 5, 360–364 (1967)Google Scholar
  4. 4.
    Sarychev, V.A.: Equilibria of two axisymmetric bodies connected by a spherical hinge in a circular orbit. Cosm. Res. 37(2), 176–181 (1999)Google Scholar
  5. 5.
    Gutnik, S.A., Sarychev, V.A.: Application of computer algebra methods to investigate the dynamics of the system of two connected bodies moving along a circular orbit. Program. Comput. Softw. 45(2), 51–57 (2019)CrossRefGoogle Scholar
  6. 6.
    Gutnik, S.A., Sarychev, V.A.: Symbolic-numerical methods of studying equilibrium positions of a gyrostat satellite. Program. Comput. Softw. 40(3), 143–150 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gutnik, S.A., Sarychev, V.A.: Application of computer algebra methods for investigation of stationary motions of a gyrostat satellite. Program. Comput. Softw. 43(2), 90–97 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gutnik, S.A.: Symbolic-numeric investigation of the aerodynamic forces influence on satellite dynamics. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 192–199. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-23568-9_15CrossRefGoogle Scholar
  9. 9.
    Gutnik, S.A., Sarychev, V.A.: A symbolic investigation of the influence of aerodynamic forces on satellite equilibria. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 243–254. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-45641-6_16CrossRefGoogle Scholar
  10. 10.
    Char, B.W., Geddes, K.O., Gonnet, G.H., Monagan, M.B., Watt, S.M.: Maple Reference Manual. Watcom Publications Limited, Waterloo (1992)zbMATHGoogle Scholar
  11. 11.
    Michels, D.L., Lyakhov, D.A., Gerdt, V.P., Hossain, Z., Riedel-Kruse, I.H., Weber, A.G.: On the general analytical solution of the kinematic cosserat equations. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 367–380. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-45641-6_24CrossRefGoogle Scholar
  12. 12.
    Chen, C., Maza, M.M.: Semi-algebraic description of the equilibria of dynamical systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 101–125. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-23568-9_9CrossRefGoogle Scholar
  13. 13.
    Meiman, N.N.: Some problems on the distribution of the zeros of polynomials. Uspekhi Mat. Nauk 34, 154–188 (1949). (in Russian)MathSciNetGoogle Scholar
  14. 14.
    Gantmacher, F.R.: The Theory of Matrices. Chelsea Publishing Company, New York (1959)zbMATHGoogle Scholar
  15. 15.
    Batkhin, A.B.: Parameterization of the discriminant set of a polynomial. Program. Comput. Softw. 42(2), 65–76 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    England, M., Errami, H., Grigoriev, D., Radulescu, O., Sturm, T., Weber, A.: Symbolic versus numerical computation and visualization of parameter regions for multistationarity of biological networks. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 93–108. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66320-3_8CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Moscow State Institute of International Relations (MGIMO University)MoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  3. 3.Keldysh Institute of Applied Mathematics (Russian Academy of Sciences)MoscowRussia

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