American Institute of Mathematical Sciences

Advanced
December  2015, 8(6): 1047-1054. doi: 10.3934/dcdss.2015.8.1047

Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid

Elbaz I. Abouelmagd 1, , Juan L. G. Guirao 2, , Aatef Hobiny 3, and  Faris Alzahrani 3,

1. 

National Research Institute of Astronomy and Geophysics, Cairo
2. 

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203-Cartagena, Región de Murcia
3. 

Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah

Received  May 2015 Revised  August 2015 Published  December 2015

The main aim of the present work is to study the positions of the equilibria points and their stability in the frame work of satellite approximation. The significant implication is that the motion around these points is unstable in the linear sense. The principle of angular momentum conservation is used as a tool to reduce the degree of freedom of the dynamical systems of equations. The positions of the relative equilibria are explicitly found as well as necessary and sufficient conditions for stable motion in the linear sense are stated.
Keywords: Dumbbell satellite, stability, relative equilibria points..
Mathematics Subject Classification: Primary: 70F05, 37C27, 37J3.
Citation: Elbaz I. Abouelmagd, Juan L. G. Guirao, Aatef Hobiny, Faris Alzahrani. Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1047-1054. doi: 10.3934/dcdss.2015.8.1047
References:
[1]

E. I. Abouelmagd, Existence and stability of triangular points in the restricted three-body problem with numerical applications,, Astrophysics and Space Science, 342 (2012), 45.  doi: 10.1007/s10509-012-1162-y.  Google Scholar

[2]

E. I. Abouelmagd, M. E. Awad, E. M. A. Elzayat and I. A. Abbas, Reduction the secular solution to periodic solution in the generalized restricted three-body problem,, Astrophysics and Space Science, 350 (2014), 495.  doi: 10.1007/s10509-013-1756-z.  Google Scholar

[3]

E. I. Abouelmagd, J. L. G. Guirao and J. A. Vera, Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body,, Commun. Nonlinear Sci. Numer Simulation, 20 (2015), 1057.  doi: 10.1016/j.cnsns.2014.06.033.  Google Scholar

[4]

E. I. Abouelmagd, M. C. Alhothuali, J. L. G. Guirao and H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three朾ody problem,, Advances in Space Research, 55 (2015), 1660.  doi: 10.1016/j.asr.2014.12.030.  Google Scholar

[5]

G. Avanzini and M. Fedi, Effects of eccentricity of the reference orbit on multi-tethered satellite formations,, Acta Astronautica, 94 (2014), 338.  doi: 10.1016/j.actaastro.2013.03.019.  Google Scholar

[6]

A. A. Burov, I. I. Kosenko and H. Troger, On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator,, Mechanics of Solids, 47 (2012), 269.  doi: 10.3103/S0025654412030028.  Google Scholar

[7]

A. Celletti and V. Sidorenko, Some properties of the dumbbell satellite attitude dynamics,, Celest. Mech. Dyn. Astr., 101 (2008), 105.  doi: 10.1007/s10569-008-9122-0.  Google Scholar

[8]

K. Nakanishi, H. Kojima and T. Watanabe, Trajectories of in-plane periodic solutions of tethered satellite system projected on van der Pol planes,, Acta Astronautica, 68 (2011), 1024.  doi: 10.1016/j.actaastro.2010.09.014.  Google Scholar

[9]

J. A. Vera, On the periodic solutions of a rigid dumbbell satellite placed at L4 of the restricted three body problem,, International Journal of Non-Linear Mechanics, 51 (2013), 152.  doi: 10.1016/j.ijnonlinmec.2013.01.013.  Google Scholar

[10]

B. Wong and A. Misra, Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points,, Acta Astronautica, 63 (2008), 1178.  doi: 10.1016/j.actaastro.2008.06.022.  Google Scholar

[11]

W. Zhang, F. B. Gao and M. H. Yao, Periodic solutions and stability of a tethered satellite system,, Mechanics Research Communications, 44 (2012), 24.  doi: 10.1016/j.mechrescom.2012.05.004.  Google Scholar

show all references

References:
[1]

E. I. Abouelmagd, Existence and stability of triangular points in the restricted three-body problem with numerical applications,, Astrophysics and Space Science, 342 (2012), 45.  doi: 10.1007/s10509-012-1162-y.  Google Scholar

[2]

E. I. Abouelmagd, M. E. Awad, E. M. A. Elzayat and I. A. Abbas, Reduction the secular solution to periodic solution in the generalized restricted three-body problem,, Astrophysics and Space Science, 350 (2014), 495.  doi: 10.1007/s10509-013-1756-z.  Google Scholar

[3]

E. I. Abouelmagd, J. L. G. Guirao and J. A. Vera, Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body,, Commun. Nonlinear Sci. Numer Simulation, 20 (2015), 1057.  doi: 10.1016/j.cnsns.2014.06.033.  Google Scholar

[4]

E. I. Abouelmagd, M. C. Alhothuali, J. L. G. Guirao and H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three朾ody problem,, Advances in Space Research, 55 (2015), 1660.  doi: 10.1016/j.asr.2014.12.030.  Google Scholar

[5]

G. Avanzini and M. Fedi, Effects of eccentricity of the reference orbit on multi-tethered satellite formations,, Acta Astronautica, 94 (2014), 338.  doi: 10.1016/j.actaastro.2013.03.019.  Google Scholar

[6]

A. A. Burov, I. I. Kosenko and H. Troger, On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator,, Mechanics of Solids, 47 (2012), 269.  doi: 10.3103/S0025654412030028.  Google Scholar

[7]

A. Celletti and V. Sidorenko, Some properties of the dumbbell satellite attitude dynamics,, Celest. Mech. Dyn. Astr., 101 (2008), 105.  doi: 10.1007/s10569-008-9122-0.  Google Scholar

[8]

K. Nakanishi, H. Kojima and T. Watanabe, Trajectories of in-plane periodic solutions of tethered satellite system projected on van der Pol planes,, Acta Astronautica, 68 (2011), 1024.  doi: 10.1016/j.actaastro.2010.09.014.  Google Scholar

[9]

J. A. Vera, On the periodic solutions of a rigid dumbbell satellite placed at L4 of the restricted three body problem,, International Journal of Non-Linear Mechanics, 51 (2013), 152.  doi: 10.1016/j.ijnonlinmec.2013.01.013.  Google Scholar

[10]

B. Wong and A. Misra, Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points,, Acta Astronautica, 63 (2008), 1178.  doi: 10.1016/j.actaastro.2008.06.022.  Google Scholar

[11]

W. Zhang, F. B. Gao and M. H. Yao, Periodic solutions and stability of a tethered satellite system,, Mechanics Research Communications, 44 (2012), 24.  doi: 10.1016/j.mechrescom.2012.05.004.  Google Scholar

[1]

Jifeng Chu, Zaitao Liang, Pedro J. Torres, Zhe Zhou. Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2669-2685. doi: 10.3934/dcdsb.2017130
[2]

Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439
[3]

Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373
[4]

James Montaldi. Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry. Journal of Geometric Mechanics, 2014, 6 (2) : 237-260. doi: 10.3934/jgm.2014.6.237
[5]

Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35
[6]

Florian Rupp, Jürgen Scheurle. Classification of a class of relative equilibria in three body coulomb systems. Conference Publications, 2011, 2011 (Special) : 1254-1262. doi: 10.3934/proc.2011.2011.1254
[7]

David Rojas, Pedro J. Torres. Bifurcation of relative equilibria generated by a circular vortex path in a circular domain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 749-760. doi: 10.3934/dcdsb.2019265
[8]

Christine Bachoc, Gilles Zémor. Bounds for binary codes relative to pseudo-distances of $k$ points. Advances in Mathematics of Communications, 2010, 4 (4) : 547-565. doi: 10.3934/amc.2010.4.547
[9]

PaweŁ Hitczenko, Georgi S. Medvedev. Stability of equilibria of randomly perturbed maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 369-381. doi: 10.3934/dcdsb.2017017
[10]

Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101
[11]

Jifa Jiang, Lei Niu. On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 217-244. doi: 10.3934/dcds.2016.36.217
[12]

Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569
[13]

Dieter Schmidt, Lucas Valeriano. Nonlinear stability of stationary points in the problem of Robe. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1917-1936. doi: 10.3934/dcdsb.2016029
[14]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701
[15]

Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934/mbe.2007.4.567
[16]

William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics & Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485
[17]

Yacine Chitour, Frédéric Grognard, Georges Bastin. Equilibria and stability analysis of a branched metabolic network with feedback inhibition. Networks & Heterogeneous Media, 2006, 1 (1) : 219-239. doi: 10.3934/nhm.2006.1.219
[18]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-13. doi: 10.3934/dcdss.2020067
[19]

Elbaz I. Abouelmagd, Juan L. G. Guirao, Aatef Hobiny, Faris Alzahrani. Dynamics of a tethered satellite with variable mass. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1035-1045. doi: 10.3934/dcdss.2015.8.1035
[20]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences