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December  2015, 8(6): 1035-1045. doi: 10.3934/dcdss.2015.8.1035

Dynamics of a tethered satellite with variable mass

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, Egypt
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, Saudi Arabia, Saudi Arabia

Received  May 2015 Revised  August 2015 Published  November 2015

In this paper, we provide an analytical study regarding the dynamics of a tethered satellite system, when the central gravitational field is generated by a variable mass object. We show that, in general, the equations of motion for the tethered satellite in the general case as well as in satellite approximation become different from the classical ones, provided that variable mass is considered. We also prove that these expressions could be reduced to the classical ones under the first Meshcherskii's law for variable mass. Moreover, we show that Meshcherskii's transformation is not valid for the dynamics of a dumbbell satellite system.
Keywords: dumbbell satellite, variable mass, Meshcherskii's transformation., Tethered satellite, Beletsky's equation.
Mathematics Subject Classification: Primary: 70E17, 70E20, 70E40; Secondary: 37C2.
Citation: 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
References:
[1]

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,, Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 1057.  doi: 10.1016/j.cnsns.2014.06.033.  Google Scholar

[2]

F. Austin, Nonlinear dynamics of free-rotating flexibly connected double-mass space station,, Journal of Spacecraft and Rockets, 2 (1965), 901.   Google Scholar

[3]

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

[4]

V. V. Beletsky and E. M. Levin, Dynamics of Space Tether Systems,, Univelt, (1993).   Google Scholar

[5]

V. V. Beletsky, Motion of an Artificial Satellite about its Center of Mass,, Israel Program for Scientific Translations, (1966).   Google Scholar

[6]

A. Burov and A. Dugain, Planar oscillations of a vibrating dumbbell-like body in a central field of forces,, Cosmic Research, 49 (2011), 353.  doi: 10.1134/S0010952511040010.  Google Scholar

[7]

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

[8]

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

[9]

V. Chobotov, Gravitational excitation of extensible dumbbell satellite,, Journal of Spacecraft and Rockets, 4 (1967), 1295.  doi: 10.2514/3.29074.  Google Scholar

[10]

G. Colombo, E. M. Gaposchkin, M. D. Grossi and G. C. Weiffenbach, The skyhook: A shuttle-borne tool for low-orbital-altitude research,, Meccanica, 10 (1975), 3.  doi: 10.1007/BF02148280.  Google Scholar

[11]

M. L. Cosmo and E. C. Lorenzini, Tethers in Space Handbook,, 3rd Ed., (1997).   Google Scholar

[12]

L. Gang, H. Jing, M. Guangfu and L. Chuanjiang, Nonlinear dynamics and station keeping control of a rotating tethered satellite system in halo orbits,, Chinese Journal of Aeronautics, 26 (2013), 1227.   Google Scholar

[13]

J. L. G. Guirao, J. A. Vera and B. A. Wade, On the periodic solutions of a rigid dumbbell satellite in a circular orbit,, Astrophys Space Sci, 346 (2013), 437.  doi: 10.1007/s10509-013-1456-8.  Google Scholar

[14]

F. C. Hurlbut and J. L. Potter, Tethered aerothermodynamic research needs,, Journal of Spacecraft and Rockets, 28 (1991), 50.  doi: 10.2514/6.1990-533.  Google Scholar

[15]

S. K. Jha and A. K. Shrivastava, Equations of motion of the elliptical restricted problem of three bodies with variable masses,, The Astronomical Journal, 121 (2001), 580.  doi: 10.1086/318006.  Google Scholar

[16]

A. J. Maciejewski, M. Przybylska, L. Simpson and W. Szumiński, Non-integrability of the dumbbell and point mass problem,, Celest. Mech. Dyn. Astr., 117 (2013), 315.  doi: 10.1007/s10569-013-9514-7.  Google Scholar

[17]

I. V. Meshcherskii, Works on the Mechanics of Bodies of Variable Mass,, GITTL, (1952).   Google Scholar

[18]

M. A. Munitsina, Relative equilibrium on the circular Keplerian orbit of the "Dumbbells-Load'' system with unilateral connections,, Automation and Remote Control, 68 (2007), 1476.  doi: 10.1134/S0005117907090020.  Google Scholar

[19]

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

[20]

D. D. Nixon, Dynamics of a spinning space station with a counterweight connected by multiple cables,, Journal of Spacecraft and Rockets, 9 (1972), 896.  doi: 10.2514/6.1972-172.  Google Scholar

[21]

M. Pasca and E. Lorenzini, Collection of martian atmospheric dust with a low altitude tethered probe,, in Spaceflight Mechanics 1991; Proceedings of the 1st AAS/AIAA Annual Spaceflight Mechanics Meeting, (1991), 1121.   Google Scholar

[22]

C. D. Pengelley, Preliminary survey of dynamic stability of cable-connected spinning space station,, Journal of Spacecraft and Rockets, 3 (1966), 1456.  doi: 10.2514/3.28677.  Google Scholar

[23]

K. A. Polzin, E. Y. Choueiri, P. Gurfil and N. J. Kasdin, Plasma propulsion options for multiple terrestrial planet finder architectures,, Journal of Spacecraft and Rockets, 39 (2002), 347.  doi: 10.2514/2.3833.  Google Scholar

[24]

M. B. Quadrelli, Dynamics and control of novel orbiting formations with internal dynamics,, Journal of the Astronautical Sciences, 51 (2003), 319.   Google Scholar

[25]

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

[26]

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, J. L. G. Guirao and J. A. Vera, Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body,, Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 1057.  doi: 10.1016/j.cnsns.2014.06.033.  Google Scholar

[2]

F. Austin, Nonlinear dynamics of free-rotating flexibly connected double-mass space station,, Journal of Spacecraft and Rockets, 2 (1965), 901.   Google Scholar

[3]

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

[4]

V. V. Beletsky and E. M. Levin, Dynamics of Space Tether Systems,, Univelt, (1993).   Google Scholar

[5]

V. V. Beletsky, Motion of an Artificial Satellite about its Center of Mass,, Israel Program for Scientific Translations, (1966).   Google Scholar

[6]

A. Burov and A. Dugain, Planar oscillations of a vibrating dumbbell-like body in a central field of forces,, Cosmic Research, 49 (2011), 353.  doi: 10.1134/S0010952511040010.  Google Scholar

[7]

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

[8]

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

[9]

V. Chobotov, Gravitational excitation of extensible dumbbell satellite,, Journal of Spacecraft and Rockets, 4 (1967), 1295.  doi: 10.2514/3.29074.  Google Scholar

[10]

G. Colombo, E. M. Gaposchkin, M. D. Grossi and G. C. Weiffenbach, The skyhook: A shuttle-borne tool for low-orbital-altitude research,, Meccanica, 10 (1975), 3.  doi: 10.1007/BF02148280.  Google Scholar

[11]

M. L. Cosmo and E. C. Lorenzini, Tethers in Space Handbook,, 3rd Ed., (1997).   Google Scholar

[12]

L. Gang, H. Jing, M. Guangfu and L. Chuanjiang, Nonlinear dynamics and station keeping control of a rotating tethered satellite system in halo orbits,, Chinese Journal of Aeronautics, 26 (2013), 1227.   Google Scholar

[13]

J. L. G. Guirao, J. A. Vera and B. A. Wade, On the periodic solutions of a rigid dumbbell satellite in a circular orbit,, Astrophys Space Sci, 346 (2013), 437.  doi: 10.1007/s10509-013-1456-8.  Google Scholar

[14]

F. C. Hurlbut and J. L. Potter, Tethered aerothermodynamic research needs,, Journal of Spacecraft and Rockets, 28 (1991), 50.  doi: 10.2514/6.1990-533.  Google Scholar

[15]

S. K. Jha and A. K. Shrivastava, Equations of motion of the elliptical restricted problem of three bodies with variable masses,, The Astronomical Journal, 121 (2001), 580.  doi: 10.1086/318006.  Google Scholar

[16]

A. J. Maciejewski, M. Przybylska, L. Simpson and W. Szumiński, Non-integrability of the dumbbell and point mass problem,, Celest. Mech. Dyn. Astr., 117 (2013), 315.  doi: 10.1007/s10569-013-9514-7.  Google Scholar

[17]

I. V. Meshcherskii, Works on the Mechanics of Bodies of Variable Mass,, GITTL, (1952).   Google Scholar

[18]

M. A. Munitsina, Relative equilibrium on the circular Keplerian orbit of the "Dumbbells-Load'' system with unilateral connections,, Automation and Remote Control, 68 (2007), 1476.  doi: 10.1134/S0005117907090020.  Google Scholar

[19]

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

[20]

D. D. Nixon, Dynamics of a spinning space station with a counterweight connected by multiple cables,, Journal of Spacecraft and Rockets, 9 (1972), 896.  doi: 10.2514/6.1972-172.  Google Scholar

[21]

M. Pasca and E. Lorenzini, Collection of martian atmospheric dust with a low altitude tethered probe,, in Spaceflight Mechanics 1991; Proceedings of the 1st AAS/AIAA Annual Spaceflight Mechanics Meeting, (1991), 1121.   Google Scholar

[22]

C. D. Pengelley, Preliminary survey of dynamic stability of cable-connected spinning space station,, Journal of Spacecraft and Rockets, 3 (1966), 1456.  doi: 10.2514/3.28677.  Google Scholar

[23]

K. A. Polzin, E. Y. Choueiri, P. Gurfil and N. J. Kasdin, Plasma propulsion options for multiple terrestrial planet finder architectures,, Journal of Spacecraft and Rockets, 39 (2002), 347.  doi: 10.2514/2.3833.  Google Scholar

[24]

M. B. Quadrelli, Dynamics and control of novel orbiting formations with internal dynamics,, Journal of the Astronautical Sciences, 51 (2003), 319.   Google Scholar

[25]

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

[26]

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

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