On Orbital Stability of Pendulum-like Satellite Rotations at the Boundaries of Stability Regions


    2019, Vol. 15, no. 4, pp.  415-428

    Author(s): Bardin B. S., Chekina E. A.

    The motion of a rigid body satellite about its center of mass is considered. The problem of the orbital stability of planar pendulum-like rotations of the satellite is investigated. It is assumed that the satellite moves in a circular orbit and its geometry of mass corresponds to a plate. In unperturbed motion the minor axis of the inertia ellipsoid lies in the orbital plane.
    A nonlinear analysis of the orbital stability for previously unexplored values of parameters corresponding to the boundaries of the stability regions is carried out. The study is based on the normal form technique. In the special case of fast rotations a normalization procedure is performed analytically. In the general case the coefficients of normal form are calculated numerically. It is shown that in the case under consideration the planar rotations of the satellite are mainly unstable, and only on one of the boundary curves there is a segment where the formal orbital stability takes place.
    Keywords: satellite, rotations, orbital stability, Hamiltonian system, symplectic map, normal form, combinational resonance, resonance of essential type
    Citation: Bardin B. S., Chekina E. A., On Orbital Stability of Pendulum-like Satellite Rotations at the Boundaries of Stability Regions, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp.  415-428
    DOI:10.20537/nd190403


    Download File
    PDF, 308.9 Kb

    References
    [1] Beletsky, V. V., Motion of a Satellite about Its Center of Mass in the Gravitational Field, MGU, Moscow, 1975, 308 pp. (Russian)  adsnasa
    [2] Markeev, A. P., “Stability of Plane Oscillations and Rotations of a Satellite in a Circular Orbit”, Cosmic Research, 13:3 (1975), 285–298  adsnasa; Kosmicheskie Issledovaniya, 13:3 (1975), 322–336 (Russian)  mathscinet  adsnasa
    [3] Akulenko, L. D., Nesterov, S. V., and Shmatkov, A. M., “Generalized Parametric Oscillations of Mechanical Systems”, J. Appl. Math. Mech., 63:5 (1999), 705–713  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 63:5 (1999), 746–756 (Russian)  mathscinet  zmath
    [4] Markeev, A. P. and Bardin, B. S., “On the Stability of Planar Oscillations and Rotations of a Satellite in a Circular Orbit”, Celest. Mech. Dynam. Astronom., 85:1 (2003), 51–66  crossref  mathscinet  zmath  adsnasa
    [5] Kane, T. R. and Shippy, D. J., “Attitude Stability of a Spinning Unsymmetrical Satellite in a Circular Orbit”, J. Astrounaut. Sci., 10:4 (1963), 114–119
    [6] Kane, T. R., “Attitude Stability of Earth-Pointing Satellites”, AIAA J., 3:4 (1965), 726–731  crossref  adsnasa
    [7] Meirovitch, L. and Wallace, F., “Attitude Instability Regions of a Spinning Unsymmetrical Satellite in a Circular Orbit”, J. Astrounaut. Sci., 14:3 (1967), 123–133
    [8] Markeev, A. P. and Sokolskii, A. G., “Investigation into the Stability of Plane Periodic Motions of a Satellite in a Circular Orbit”, Mech. Solids, 12:4 (1977), 39–48  adsnasa; Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 12:4 (1977), 46–57 (Russian)  adsnasa
    [9] Kholostova, O. V., “Linear Analysis of Stability the Planar Oscillations of a Satellite Being a Plate in a Circular Orbit”, Nelin. Dinam., 1:2 (2005), 181–190 (Russian)  mathnet  crossref
    [10] Markeev, A. P., “Stability of Planar Rotations of a Satellite in a Circular Orbit”, Mech. Solids, 41:4 (2006), 46–63; Izv. Akad. Nauk. Mekh. Tverd. Tela, 2006, no. 4, 63–85 (Russian)
    [11] Bardin, B. S. and Checkin, A. M., “About Orbital Stability of Plane Rotations for a Plate Satellite Travelling in a Circular Orbit”, Vestn. MAI, 14:2 (2007), 23–36 (Russian)
    [12] Bardin, B. S. and Chekin, A. M., “Orbital Stability of Planar Oscillations of a Satellite in a Circular Orbit”, Cosmic Research, 46:3 (2008), 273–282  crossref  adsnasa  elib; Kosmicheskie Issledovaniya, 46:3 (2008), 279–288 (Russian)
    [13] Bardin, B. S. and Chekina, E. A., “On the Stability of Planar Oscillations of a Satellite-Plate in the Case of Essential Type Resonance”, Nelin. Dinam., 13:4 (2017), 465–476 (Russian)  mathnet  crossref  mathscinet  zmath
    [14] Bardin, B. S. and Chekina, E. A., “On Orbital Stability of Planar Oscillations of a Satellite in a Circular Orbit on the Boundary of the Parametric Resonance”, AIP Conf. Proc., 1959:1 (2018), 040003  crossref  mathscinet
    [15] Bardin, B. S. and Chekina, E. A., “On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Case of Combinational Resonance”, Regul. Chaotic Dyn., 24:2 (2019), 127–144  mathnet  crossref  mathscinet  zmath  adsnasa
    [16] Lyapunov, A. M., The General Problem of the Stability of Motion, Fracis & Taylor, London, 1992, x+270 pp.  mathscinet  zmath
    [17] Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Appl. Math. Sci., 8, Springer, New York, 1972, ix, 369 pp.  crossref  mathscinet  zmath
    [18] Markeev, A. P., Libration Points in Celestial Mechanics and Space Dynamics, Nauka, Moscow, 1978, 312 pp. (Russian)  adsnasa
    [19] Ivanov, A. P. and Sokol'skii, A. G., “On the Stability of a Nonautonomous Hamiltonian System under a Parametric Resonance of Essential Type”, J. Appl. Math. Mech., 44:6 (1980), 687–691  crossref  mathscinet; Prikl. Mat. Mekh., 44:6 (1980), 963–970 (Russian)  mathscinet  zmath
    [20] Ivanov, A. P. and Sokol'skii, A. G., “On the Stability of a Nonautonomous Hamiltonian System under Second-Order Resonance”, J. Appl. Math. Mech., 44:5 (1980), 574–581  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 44:5 (1980), 811–822 (Russian)  mathscinet  zmath
    [21] Birkhoff, G. D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., 9, AMS, Providence, R.I., 1966, 305 pp.  mathscinet
    [22] Markeyev, A. P., “A Constructive Algorithm for the Normalization of a Periodic Hamiltonian”, J. Appl. Math. Mech., 69:3 (2005), 323–337  crossref  mathscinet; Prikl. Mat. Mekh., 69:3 (2005), 355–371 (Russian)  mathscinet
    [23] Bardin, B. S. and Chekina, E. A., “On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Second-Order Resonance Case”, Regul. Chaotic Dyn., 22:7 (2017), 808–823  mathnet  crossref  mathscinet  zmath  adsnasa
    [24] Bardin, B. S. and Chekina, E. A., “On the Constructive Algorithm for Stability Investigation of an Equilibrium Position of a Periodic Hamiltonian System with Two Degrees of Freedom in the First Order Resonance Case”, J. Appl. Math. Mech., 82:4 (2018), 20–32  mathscinet; Prikl. Mat. Mekh., 82:4 (2018), 414–426 (Russian)



    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License