Stability and Bifurcation of Resonance Periodic Motions of a Symmetric Satellite
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
STABILITY AND BIFURCATION OF RESONANCE PERIODIC MOTIONS OF A SYMMETRIC SATELLITE E. A. Sukhov Moscow Aviation Institute (National Research University) 4, Volokokamskoe Shosse, Moscow 125993, Russia [email protected]
UDC 531.01+521.13
We consider families of periodic motions emanating from the hyperboloidal precession of a satellite in the case of the third or fourth order resonance, as well as in the nonresonance case. We construct bifurcation diagrams for periodic motions in the case of small deviations of the energy integral constant from its value on the hyperboloidal precession. We study the orbital stability in the linear approximation. Bibliography: 6 titles. Illustrations: 3 figures.
1
Introduction
We consider a rigid-body satellite whose center of mass moves in the central gravitational field of forces in a circular orbit. To describe the satellite motion about the center of mass, we introduce an orbital reference frame OXY Z and a mobile reference frame Oxyz. We direct the OZ–, OX–, OY –axes along the radius vector of the center of mass of the satellite, transversal to the orbit and normal to the orbit plane. The Ox–, Oy–, Oz–axes are directed along the principal central axes of inertia of the satellite, and the moments of inertia are denoted by J1 , J2 , J3 . We assume that the satellite is dynamically symmetric (J1 = J2 ). The correspondence between the orbital and connected coordinates is given by the Euler angles ψ, θ, ϕ. The equations of motion of a dynamically symmetric satellite can be written in the canonical form with the Hamiltonian [1] H=
p2ψ
2 sin2 θ
+
p2θ γ cos θ − + cos ψ cot θ pψ 2 sin2 θ
1 cos ψ 1 + δ cos2 θ, − sin ψpψ + γ 2 cot2 θ + γ 2 sin θ 2
(1.1)
where pψ and pθ are the dimensionless impulses corresponding to the coordinates ψ and θ. The coordinate ϕ is cyclic and the corresponding impulse pϕ retains constant value, pϕ = J3 r0 , where r0 is the projection of the absolute angular velocity of the satellite to the Oz-axis of dynamical Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 139-148. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0155
155
symmetry. The dimensionless parameters γ and δ (|δ| 3) are introduced by the equalities γ = αβ and δ = 3 (α − 1), where α = J3 /J1 , β = r0 /ω0 , ω0 is the angular velocity of the radius vector of the center of mass. The true anomaly ν = ω0 t is an independent variable. The equations of motion with the Hamiltonian (1.1) have the partial solution θ0 = π/2, cos ψ0 = −γ, pθ0 = sin ψ0 , pψ0 = 0 for |γ| 1. The solution describes the hyperboloidal precession of the satellite for which the dynamical symmetry axis of the satellite lies in the plane perpendicular to the radius vector of the center of mass and forms the angle π − ψ0 with the normal to the orbit plane. For δ > 0 the hyperboloidal precession is stable in the sense of Lyapunov [2], and periodic motions in its neighborhood are of two types: short-period motions with a
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