Abstract:
We study the Lyapunov stability problem of the resonant rotation of a rigid body satellite about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the satellite completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the above resonance rotation was started in [4, 6]. In the present paper, rigorous stability conclusions in the previously unstudied range of parameter values are obtained. In particular, new intervals of stability are found for eccentricity values close to 1. In addition, some special cases are studied where the stability analysis should take into account terms of degree not less than six in the expansion of the Hamiltonian of the perturbed motion. Using the technique described in [7, 8], explicit formulae are obtained, allowing one to verify the stability criterion of a time-periodic Hamiltonian system with one degree of freedom in the special cases mentioned.
Keywords:
Hamiltonian system, symplectic map, normal form, resonance, satellite, stability.
The work was carried out under the grant of the Russian Scientific Foundation (project No 14-21-00068) at the Moscow Aviation Institute (National Research University).
Citation:
Boris S. Bardin, Evgeniya A. Chekina, Alexander M. Chekin, “On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit”, Regul. Chaotic Dyn., 20:1 (2015), 63–73
\Bibitem{BarCheChe15}
\by Boris S. Bardin, Evgeniya A. Chekina, Alexander M. Chekin
\paper On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit
\jour Regul. Chaotic Dyn.
\yr 2015
\vol 20
\issue 1
\pages 63--73
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This publication is cited in the following 21 articles:
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Xue Zhong, Jie Zhao, Lunhu Hu, Kaiping Yu, Hexi Baoyin, “Periodic attitude motions of an axisymmetric spacecraft in an elliptical orbit near the hyperbolic precession”, Applied Mathematical Modelling, 2024, 115845
José Laudelino de Menezes Neto, “Investigation of the nonlinear stability of a pendulum with variable length in elliptic orbit”, DCDS-B, 29:4 (2024), 1611
Xue Zhong, Jie Zhao, Kaiping Yu, Minqiang Xu, “Stability Analysis of Resonant Rotation of a Gyrostat in an
Elliptic Orbit Under Third- and Fourth-Order Resonances”, Regul. Chaotic Dyn., 28:2 (2023), 162–190
B. S. Bardin, B. A. Maksimov, “On the Orbital Stability of Pendulum Periodic Motions of a Heavy Rigid Body with a Fixed Point, the Main Moments of Inertia of which are in the Ratio 1 : 4 : 1”, Prikladnaâ matematika i mehanika, 87:5 (2023), 784
Jie Zhao, Xue Zhong, Kaiping Yu, Minqiang Xu, “Effect of gyroscopic moments on the attitude stability of a satellite in an elliptical orbit”, Nonlinear Dyn, 111:16 (2023), 14957
B. S. Bardin, B. A. Maksimov, “On the Orbital Stability of Pendulum Periodic Motions of a Heavy Rigid Body with a Fixed Point in the Case of Principal Inertia Moments Ratio 1 : 4 : 1”, Mech. Solids, 58:8 (2023), 2894
B. S. Bardin, E. A. Chekina, A. M. Chekin, “On the Orbital Stability of Pendulum Oscillations
of a Dynamically Symmetric Satellite”, Rus. J. Nonlin. Dyn., 18:4 (2022), 589–607
B. S. Bardin, E. A. Chekina, “On the Orbital Stability of Pendulum-like Oscillations
of a Heavy Rigid Body with a Fixed Point in the
Bobylev – Steklov Case”, Rus. J. Nonlin. Dyn., 17:4 (2021), 453–464
Zhong X. Zhao J. Yu K. Xu M., “On the Stability of Periodic Motions of a Two-Body System With Flexible Connection in An Elliptical Orbit”, Nonlinear Dyn., 104:4 (2021), 3479–3496
B S Bardin, “Local coordinates in problem of the orbital stability of pendulum-like oscillations of a heavy rigid body in the Bobylev–Steklov case”, J. Phys.: Conf. Ser., 1925:1 (2021), 012016
T Churkina, “On stability of planar periodic motions of a satellite in vicinity of resonant rotation”, IOP Conf. Ser.: Mater. Sci. Eng., 927:1 (2020), 012004
Z. Liang, F. Liao, “Periodic solutions for a dumbbell satellite equation”, Nonlinear Dyn., 95:3 (2019), 2469–2476
B. S. Bardin, E. A. Chekina, “On the constructive algorithm for stability investigation of an equilibrium point of a periodic Hamiltonian system with two degrees of freedom in first-order resonance case”, Mech. Sol., 53:2 (2018), S15–S25
B. S. Bardin, A. N. Avdushkin, “Stability analysis of an equilibrium position in the photogravitational Sitnikov problem”, Eighth Polyakhov's Reading, AIP Conf. Proc., 1959, eds. E. Kustova, G. Leonov, N. Morosov, M. Yushkov, M. Mekhonoshina, Amer. Inst. Phys., 2018, 040002
B. S. Bardin, E. A. Chekina, “On orbital stability of planar oscillations of a satellite in a circular orbit on the boundary of the parametric resonance”, Eighth Polyakhov's Reading, AIP Conf. Proc., 1959, eds. E. Kustova, G. Leonov, N. Morosov, M. Yushkov, M. Mekhonoshina, Amer. Inst. Phys., 2018, 040003
Boris S. Bardin, Evgeniya A. Chekina, “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
Tatyana E. Churkina, Sergey Y. Stepanov, “On the Stability of Periodic Mercury-type Rotations”, Regul. Chaotic Dyn., 22:7 (2017), 851–864
J. Chu, Z. Liang, P. J. Torres, Zh. Zhou, “Existence and stability of periodic oscillations of a rigid Dumbbell satellite around its center of mass”, Discrete Contin. Dyn. Syst.-Ser. B, 22:7 (2017), 2669–2685
B. S. Bardin, E. A. Chekina, “Ob ustoichivosti rezonansnogo vrascheniya sputnika na ellipticheskoi orbite”, Nelineinaya dinam., 12:4 (2016), 619–632
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