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2013
Impact Factor

    Boris Bardin

    Boris Bardin
    Volokolamskoe sh. 4, Moscow, 125993, Russia
    Moscow Aviation Institute (National Research University)

    Professor of Moscow Aviation Institute (National Research University), Doctor of Physics and Mathematics
    Chief of Department of Theoretical Physics, superviser of speciality "Application of mathematical methods in the problems of aerospace engineering" at MAI

    Publications:

    Bardin B. S., Chekina E. A.
    Abstract
    We consider satellite motion about its center of mass in a circle orbit. We study the problem of orbital stability for planar pendulum-like oscillations of the satellite. It is supposed that the satellite is a rigid body whose mass geometry is that of a plate. We assume that on the unperturbed motion the middle or minor inertia axis of the satellite lies in the orbit plane, i.e., the plane of the satellite-plate is perpendicular to the plane of the orbit.
    In this paper we perform a nonlinear analysis of the orbital stability of planar pendulum-like oscillations of a satellite-plate for previously unexplored parameter values corresponding to the boundaries of regions of stability in the first approximation, where the essential type resonances take place. It is proved that on the mentioned boundaries the planar pendulum-like oscillations are formally orbital stable or orbitally stable in third approximation.
    Keywords: Hamiltonian system, normal form, planar periodic motion, resonance, satellite, orbital stability
    Citation: Bardin B. S., Chekina E. A.,  On the stability of planar oscillations of a satellite-plate in the case of essential type resonance, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp.  465–476
    DOI:10.20537/nd1704002
    Bardin B. S., Chekina E.
    Abstract
    We deal with the problem of stability for a resonant rotation of a satellite. It is supposed that the satellite is a rigid body whose center of mass moves in an elliptic orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the resonant rotation with respect to planar perturbations has been performed in detail earlier. In this paper we investigate the stability of the resonant rotation with respect to both planar and spatial perturbations for a nonsymmetric satellite. For small values of the eccentricity we have obtained boundaries of instability domains (parametric resonance domains) in an analytic form. For arbitrary eccentricity values we numerically construct domains of stability in linear approximation. Outside the above stability domains the resonant rotation is unstable in the sense of Lyapunov.
    Keywords: Hamiltonian system, resonant periodic motion, parametric resonance, satellite, stability
    Citation: Bardin B. S., Chekina E.,  On the stability of a resonant rotation of a satellite in an elliptic orbit, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp.  619–632
    DOI:10.20537/nd1604006
    Bardin B. S., Savin A. A.
    Abstract
    We deal with the problem of orbital stability of planar periodic motions of a heavy rigid body with a fixed point. We suppose that the mass center of the body is located in the equatorial plane of the inertia ellipsoid. Unperturbed motions represent oscillations or rotations of the body around a principal axis, keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of perturbed motion are obtained in Hamiltonian form. Domains of orbital instability are established by means of linear analysis. Outside of the above domains nonlinear study is performed. The nonlinear stability problem is reduced to a stability problem of a fixed point of symplectic map generated by the equations of perturbed motion. Coefficients of the above map are obtained numerically. By analyzing of the coefficients mentioned rigorous results on orbital stability or instability are obtained.

    In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities the problem of orbital stability is studied analytically.
    Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action–angel variables, orbital stability
    Citation: Bardin B. S., Savin A. A.,  On orbital stability pendulum-like oscillations and rotation of symmetric rigid body with a fixed point, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp.  249-266
    DOI:10.20537/nd1202004
    Bardin B. S.
    Abstract
    We deal with the problem of orbital stability of pendulum like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev—Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the base of a nonlinear analysis.

    In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities we studied the problem analytically. In general case we reduce the problem to the stability study of fixed point of the symplectic map generated by equations of perturbed motion. We calculate coefficients of the symplectic map numerically. By analyzing of the coefficients mentioned we establish orbital stability or instability of the unperturbed motion. The results of the study are represented in the form of stability diagram.
    Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, KAM theory
    Citation: Bardin B. S.,  On orbital stability of pendulum like motions of a rigid body in the Bobylev-Steklov case, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp.  535-550
    DOI:10.20537/nd0904006
    Bardin B. S.
    Abstract
    We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3:1. We study nonlinear conditionally-periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally-periodic. By using the KAM theory methods we show that the most of conditionally-periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that became not conditionally-periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.
    Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, KAM theory
    Citation: Bardin B. S.,  On nonlinear oscillations of Hamiltonian system in case of fourth order resonance, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp.  57-74
    DOI:10.20537/nd0701004

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