Abstract
This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field on an elliptical orbit. Equations of motion are derived in a Hamiltonian form, and two periodic solutions as well as the necessary conditions for their existence are acquired. By analyzing linearized equations of perturbed motions, Lyapunov instability domains and domains of stability in the first approximation are obtained. In addition, the third- and fourth-order resonances are investigated in linear stability domains. A constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. Then, a nonlinear stability analysis for two periodic solutions is performed in the third- and fourth-order resonance cases as well as in the nonresonance case.
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References
Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)
Celletti, A., Perozzi, E.: Celestial Mechanics: The Waltz of the Planets. Springer, New York (2007)
Colombo, G.: Rotational period of the planet mercury. Nature 208, 575–578 (1965)
Goldreich, P., Sciama, D.W.: An explanation of the frequent occurrence of commensurable mean motions in the solar system. Mon. Not. R. Astron. Soc. 130, 159–181 (1965)
Rambaux, N., Bois, E.: Theory of the mercury’s spin-orbit motion and analysis of its main librations. Astron. Astrophys. 413, 381–393 (2004)
D’Hoedt, S., Lemaitre, A.: The spin-orbit resonance of mercury: a hamiltonian approach. Proc. Int. Astron. Union 196, 263–270 (2004)
Beletskii, V.V.: On Satellite Libration. In: Beletskii, V.V. (ed.) Artificial Earth Satellites, pp. 13–31. Akad. Nauk SSSR, Moscow (1959)
Beletskii, V.V.: The Satellite Motion About Center of Mass. Publishing House Science, Moscow (1965)
Beletskii, V.V., Lavrovskii, E.K.: On the theory of the resonance rotation of Mercury. Soviet Astron. 52, 1299–1308 (1975)
Petryshyn, W.V., Yu, Zs.: On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit. Nonlinear Anal. Theory Methods Appl. 9, 969–975 (1985)
Bruno, A.D.: Families of periodic solutions to the Beletsky equation. Cosmic Res. 40, 274–295 (2002)
Chu, J.F., Liang, Z.T., Torres, P.J., Zhou, Z.: Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass. Discrete Contin. Dyn. Syst. Ser. B 22, 2669–2685 (2017)
Khentov, A.A.: On the stability in the first approximation of the earth artificial satellite rotation about the center of mass. Kosmich. Issled. 6, 793–795 (1968)
Markeev, A.P.: Stability of equilibrium states of Hamiltonian systems: a method of investigation. Mech. Sol. 39, 1–8 (2004)
Churkina, T.E.: The stability of periodic linear oscillations of a satellite about the direction of the major axis of an elliptic orbit. J. Appl. Math. Mech. 79, 426–431 (2015)
Churkina, T.E.: Stability of a planar resonance satellite motion under spatial perturbations. Mech. Sol. 42, 507–516 (2007)
Churkina, T.E.: Satellite rotation stability at a Mercurian type resonance. Mech. Sol. 49, 127–135 (2014)
Churkina, T.E., Stepanov, S.Y.: On the stability of periodic mercury-type rotations. Regul. Chaotic Dyn. 22, 851–864 (2017)
Khentov, A.A.: On rotational motion of a satellite. Kosmicheskiye Issled. 22, 130–131 (1984)
Markeev, A.P., Bardin, B.S.: A planar rotational motion of a satellite in an elliptic orbit. Cosmic Res. 32, 583–589 (1994)
Markeev, A.P.: To the problem of plane periodic rotations of a satellite in an elliptic orbit. Mech. Sol. 43, 400–411 (2008)
Bardin, B.S., Chekina, E.A., Chekin, A.M.: On the stability of a planar resonant rotation of a satellite in an elliptic orbit. Regul. Chaotic Dyn. 20, 63–73 (2015)
Bardin, B.S., Chekina, E.A.: On the stability of resonant rotation of a symmetric satellite in an elliptical orbit. Regul. Chaotic Dyn. 21, 377–389 (2016)
Bardin, B.S., 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, 808–823 (2017)
Liang, Z.T., Liao, F.F.: Periodic solutions for a dumbbell satellite equation. Nonlinear Dyn. 95, 2469–2476 (2019)
Celletti, A., Sidorenko, V.: Some properties of the dumbbell satellite attitude dynamics. Celest Mech. Dyn. Astron. 101, 105–126 (2008)
Koh, D., Flashner, H.: Global analysis of gravity gradient satellite’s pitch motion in an elliptic orbit. J. Comput. Nonlinear Dyn. (2015). https://doi.org/10.1115/1.4029621
Amel’kin, N.I.: The equilibrium positions of a satellite carrying a three-degree-of-freedom powered gyroscope in a central gravitational field. J. Appl. Math. Mech. 77, 181–189 (2013)
Amelkin, N.I., Kholoshchak, V.V.: Stability of the steady rotations of a satellite with internal damping in a central gravitational field. J. Appl. Math. Mech. 81, 85–94 (2017)
Amelkin, N.I., Kholoshchak, V.V.: Steady rotations of a satellite with internal elastic and dissipative forces. J. Appl. Math. Mech. 81, 431–441 (2017)
Yue, B.Z., Ahmad, S., Song, X.J.: Casimir method for attitude stability analysis of liquid-filled spacecraft. Sci. Sin. Phys. Mech. Astron. 43, 401–406 (2013)
Yan, Y.L., Yue, B.Z.: Analytical method for the attitude stability of partically liquid filled spacecraft with flexible appendage. Acta Mech. Sin. 33, 208–218 (2017)
Robe, T.R., Kane, T.R.: Dynamics of an elastic satellite-I. Int. J. Solids Struct. 3, 333–352 (1967)
Wittenburg, J.: Permanente Drehungen zweier durch ein Kugelgelenk gekoppelter, starrer Körper. Acta Mech. 19, 215–226 (1974)
Rimrott, F.P.J., Janabi-Sharifi, F.: A torque-free flexible model gyro. J. Appl. Mech. Mar. 59, 7–15 (1992)
Liu, Y.Z.: The stability of the permanent rotation of a free multibody system. Acta Mech. 79, 43–51 (1989)
Liu, Y.Z., Rimrott, F.P.J.: On the permanent rotation of a torque-free two-body system with a flexible connection. J. Appl. Mech. Mar. 61, 199–202 (1994)
Chernous’ko, F.L.: On the motion of a satellite about its center of mass under the action of gravitational moments. J. Appl. Math. Mech. 27, 708–722 (1963)
Beletsky, V.V.: Spacecraft Attitude Motion in Gravity Field Moscow State Univ., Moscow (1975)
Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55, 531–773 (1992)
Markeyev, A.P.: A constructive algorithm for the normalization of a periodic hamiltonian. J. Appl. Math. Mech. 69, 323–337 (2005)
Markeev, A.P.: Libration Points in Celestial Mechanics and Space Dynamics. Nauka, Moscow (1978)
Meyer, K., Hall, G.R.: Introduction to Hamiltonian Dynamical System and the N-Body Problem. Springer, Cham (2017)
Gustavson, F.G.: On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71, 670–686 (1966)
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Springer, Berlin (1971)
Moser, J.: New aspects in the theory of stability of Hamiltonian systems. Commun. Pure Appl. Math. 11, 81–114 (1958)
Glimm, J.: Formal stability of hamiltonian systems. Commun. Pure Appl. Math. 17, 509–526 (1963)
Vidal, C.: Stability of equilibrium positions of hamiltonian systems. Qual. Th. Dyn. Syst. 7, 253–294 (2008)
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Zhong, X., Zhao, J., Yu, K. et al. On the stability of periodic motions of a two-body system with flexible connection in an elliptical orbit. Nonlinear Dyn 104, 3479–3496 (2021). https://doi.org/10.1007/s11071-021-06516-x
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DOI: https://doi.org/10.1007/s11071-021-06516-x