On the periodic solutions of a rigid dumbbell satellite in a circular orbit

Guirao, Juan Luis García; Vera, Juan A.; Wade, Bruce A.

doi:10.1007/s10509-013-1456-8

Cited by 6 publications

(9 citation statements)

References 7 publications

(15 reference statements)

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“…For doing this we shall use the averaging theory of dynamical systems, see Appendix for more details on it. We have been inspired by other works where these techniques have been used for studying other perturbed dynamics problems, see for instance [2,3,4,5,6,7,8,9,10].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…We start with the description of the different elements which appear in the statement of Theorem 4.1 for the particular case of the differential system (8). Thus, we have that Ω = R 4 , k = 2 and n = 4.…”

Section: Proof Of the Theorems 11 And 13mentioning

confidence: 99%

“…Then, by Theorem 4.1 we have that for every simple zero (X 0 * 1 , X 0 * 2 ) ∈ V of the system of nonlinear functions (4) we have a periodic solution (x 1 , x 2 , x 3 , x 4 )(t, ε) of system (8) such that…”

Section: Proof Of the Theorems 11 And 13mentioning

confidence: 99%

“…Going back through the change of coordinates (7) we get a periodic solution (x 1 , x 2 , x 3 , x 4 )(t, ε) of system (8) such that…”

Section: Proof Of the Theorems 11 And 13mentioning

confidence: 99%

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On the Periodic Auto–Oscillations of an Electric Circuit with Periodic Imperfections on Its Variables

Bustos¹,

Lpez

Martnez³

2013

Appl. Math. Inf. Sci.

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Abstract:The aim of the present paper is to study the periodic auto-oscillations of an electric circuit with periodic imperfections on its variables composed by three condensers, one of them without charge, and two bobbins. We model this system by the Lagrangian approach using the morphology of the Hill problem and the main tool used for proving the results is the averaging theory of dynamical systems.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Proof Of the Theorems 11 And 13mentioning

confidence: 99%

Section: Proof Of the Theorems 11 And 13mentioning

confidence: 99%

“…Going back through the change of coordinates (7) we get a periodic solution (x 1 , x 2 , x 3 , x 4 )(t, ε) of system (8) such that…”

Section: Proof Of the Theorems 11 And 13mentioning

confidence: 99%

See 2 more Smart Citations

On the Periodic Auto–Oscillations of an Electric Circuit with Periodic Imperfections on Its Variables

Bustos¹,

Lpez

Martnez³

2013

Appl. Math. Inf. Sci.

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“…Axis O X 1 of the moving frame is always parallel to the dumbbell features for tethered satellite systems. See section 5.9 in Celletti (2010) and some original articles and references therein (Celletti and Sidorenko 2008;Guirao et al 2013;Khan and Goel 2011;Schechter 1964;Vera 2013). Moreover dumbbell can be considered as a simplified model of a large spacecraft or an orbital cable system equipped with an elevator (see e.g.…”

Section: Figmentioning

confidence: 99%

Non-integrability of the dumbbell and point mass problem

Maciejewski

Przybylska

Simpson

et al. 2013

Celest Mech Dyn Astr

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This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. This problem is a simplification of a problem of a symmetric rigid body and a point mass, and has numerous applications in Celestial Mechanics and Astrodynamics. The non-integrability of this system is proven. This was achieved thanks to an analysis of variational equations along a certain particular solution and an investigation of their differential Galois group. Nowadays this approach is the most effective tool for study integrability of Hamiltonian and non-Hamiltonian systems.Keywords Three-body problem · Morales-Ramis theory · Differential Galois theory · Non-integrability Equations of motion, symmetries and reductionConsidered is the gravitational three-body problem with a single constraint. Three point masses, m 1 , m 2 and m 3 move in a plane under mutual gravitational interaction. Masses m 2 and m 3 are connected by a massless inflexible rod of length l > 0 to form a dumbbell. A pictorial description of the problem is given in Fig. 1. Various celestial objects are considered to possess such bimodal mass distribution because do not have enough gravitational force to form their shape into a spherical object, e.g. some asteroids such as (51) Nemausa and (216) Kleopatra; meteorites, especially large irons or nucleus of some comets e.g. Comet Borrelly (for detailed references see Povenmire 2002).The dumbbell satellite has attracted the attention of scientists since the middle of 20th century because it is suitable for an investigation of the general properties of the rigid body motion in a gravity field and provides important

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On the dynamics of planar oscillations for a dumbbell satellite in $$\varvec{J_{2}}$$ J 2 problem

2015

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In the present paper, we study regular and chaotic dynamics from planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter J 2 . We theoretically show the existence of chaotic oscillations provided that the eccentricity becomes arbitrarily small, and the parameter J 2 is of the same order of magnitude as the eccentricity. This is carried out by applying the so-called Melnikov method. Finally, for arbitrarily chosen values for the parameters involved in such a problem, we study the transition from regular to chaotic oscillations for a dumbbell satellite via the analysis of chaotic maps and Poincaré surfaces of section, respectively.

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On the periodic solutions of a rigid dumbbell satellite in a circular orbit

Abstract: Abstract. The aim of the present paper is to provide sufficient conditions for the existence of periodic solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in a circular orbit.

Cited by 6 publications

References 7 publications

On the Periodic Auto–Oscillations of an Electric Circuit with Periodic Imperfections on Its Variables

On the Periodic Auto–Oscillations of an Electric Circuit with Periodic Imperfections on Its Variables

Non-integrability of the dumbbell and point mass problem

On the dynamics of planar oscillations for a dumbbell satellite in $$\varvec{J_{2}}$$ J 2 problem

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