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

Dynamics of a tethered satellite with variable mass

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.
Citation: Elbaz I. Abouelmagd, Juan L. G. Guirao, Aatef Hobiny, Faris Alzahrani. Dynamics of a tethered satellite with variable mass. Discrete and 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), 1057-1069. doi: 10.1016/j.cnsns.2014.06.033.

[2]

F. Austin, Nonlinear dynamics of free-rotating flexibly connected double-mass space station, Journal of Spacecraft and Rockets, 2 (1965), 901-906.

[3]

G. Avanzini and M. Fedi, Effects of eccentricity of the reference orbit on multi-tethered satellite formations, Acta Astronautica, 94 (2014), 338-350. doi: 10.1016/j.actaastro.2013.03.019.

[4]

V. V. Beletsky and E. M. Levin, Dynamics of Space Tether Systems, Univelt, San Diego, 1993.

[5]

V. V. Beletsky, Motion of an Artificial Satellite about its Center of Mass, Israel Program for Scientific Translations, Jerusalem, 1966.

[6]

A. Burov and A. Dugain, Planar oscillations of a vibrating dumbbell-like body in a central field of forces, Cosmic Research, 49 (2011), 353-359. doi: 10.1134/S0010952511040010.

[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), 269-284. doi: 10.3103/S0025654412030028.

[8]

A. Celletti and V. Sidorenko, Some properties of the dumbbell satellite attitude dynamics, Celest. Mech. Dyn. Astr., 101 (2008), 105-126. doi: 10.1007/s10569-008-9122-0.

[9]

V. Chobotov, Gravitational excitation of extensible dumbbell satellite, Journal of Spacecraft and Rockets, 4 (1967), 1295-1300. doi: 10.2514/3.29074.

[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), 3-20. doi: 10.1007/BF02148280.

[11]

M. L. Cosmo and E. C. Lorenzini, Tethers in Space Handbook, 3rd Ed., NASA Marshall Space Flight Center, Huntsville, 1997.

[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), 1227-1237.

[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), 437-442. doi: 10.1007/s10509-013-1456-8.

[14]

F. C. Hurlbut and J. L. Potter, Tethered aerothermodynamic research needs, Journal of Spacecraft and Rockets, 28 (1991), 50-57. doi: 10.2514/6.1990-533.

[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), 580-583. doi: 10.1086/318006.

[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), 315-330. doi: 10.1007/s10569-013-9514-7.

[17]

I. V. Meshcherskii, Works on the Mechanics of Bodies of Variable Mass, GITTL, Moscow, 1952.

[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), 1476-1481. doi: 10.1134/S0005117907090020.

[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), 1024-1030. doi: 10.1016/j.actaastro.2010.09.014.

[20]

D. D. Nixon, Dynamics of a spinning space station with a counterweight connected by multiple cables, Journal of Spacecraft and Rockets, 9 (1972), 896-902. doi: 10.2514/6.1972-172.

[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, Houston, TX, 1991, 1121-1139.

[22]

C. D. Pengelley, Preliminary survey of dynamic stability of cable-connected spinning space station, Journal of Spacecraft and Rockets, 3 (1966), 1456-1462. doi: 10.2514/3.28677.

[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), 347-356. doi: 10.2514/2.3833.

[24]

M. B. Quadrelli, Dynamics and control of novel orbiting formations with internal dynamics, Journal of the Astronautical Sciences, 51 (2003), 319-337.

[25]

B. Wong and A. Misra, Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points, Acta Astronautica, 63 (2008), 1178-1187. doi: 10.1016/j.actaastro.2008.06.022.

[26]

W. Zhang, F. B. Gao and M. H. Yao, Periodic solutions and stability of a tethered satellite system, Mechanics Research Communications, 44 (2012), 24-29. doi: 10.1016/j.mechrescom.2012.05.004.

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), 1057-1069. doi: 10.1016/j.cnsns.2014.06.033.

[2]

F. Austin, Nonlinear dynamics of free-rotating flexibly connected double-mass space station, Journal of Spacecraft and Rockets, 2 (1965), 901-906.

[3]

G. Avanzini and M. Fedi, Effects of eccentricity of the reference orbit on multi-tethered satellite formations, Acta Astronautica, 94 (2014), 338-350. doi: 10.1016/j.actaastro.2013.03.019.

[4]

V. V. Beletsky and E. M. Levin, Dynamics of Space Tether Systems, Univelt, San Diego, 1993.

[5]

V. V. Beletsky, Motion of an Artificial Satellite about its Center of Mass, Israel Program for Scientific Translations, Jerusalem, 1966.

[6]

A. Burov and A. Dugain, Planar oscillations of a vibrating dumbbell-like body in a central field of forces, Cosmic Research, 49 (2011), 353-359. doi: 10.1134/S0010952511040010.

[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), 269-284. doi: 10.3103/S0025654412030028.

[8]

A. Celletti and V. Sidorenko, Some properties of the dumbbell satellite attitude dynamics, Celest. Mech. Dyn. Astr., 101 (2008), 105-126. doi: 10.1007/s10569-008-9122-0.

[9]

V. Chobotov, Gravitational excitation of extensible dumbbell satellite, Journal of Spacecraft and Rockets, 4 (1967), 1295-1300. doi: 10.2514/3.29074.

[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), 3-20. doi: 10.1007/BF02148280.

[11]

M. L. Cosmo and E. C. Lorenzini, Tethers in Space Handbook, 3rd Ed., NASA Marshall Space Flight Center, Huntsville, 1997.

[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), 1227-1237.

[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), 437-442. doi: 10.1007/s10509-013-1456-8.

[14]

F. C. Hurlbut and J. L. Potter, Tethered aerothermodynamic research needs, Journal of Spacecraft and Rockets, 28 (1991), 50-57. doi: 10.2514/6.1990-533.

[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), 580-583. doi: 10.1086/318006.

[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), 315-330. doi: 10.1007/s10569-013-9514-7.

[17]

I. V. Meshcherskii, Works on the Mechanics of Bodies of Variable Mass, GITTL, Moscow, 1952.

[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), 1476-1481. doi: 10.1134/S0005117907090020.

[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), 1024-1030. doi: 10.1016/j.actaastro.2010.09.014.

[20]

D. D. Nixon, Dynamics of a spinning space station with a counterweight connected by multiple cables, Journal of Spacecraft and Rockets, 9 (1972), 896-902. doi: 10.2514/6.1972-172.

[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, Houston, TX, 1991, 1121-1139.

[22]

C. D. Pengelley, Preliminary survey of dynamic stability of cable-connected spinning space station, Journal of Spacecraft and Rockets, 3 (1966), 1456-1462. doi: 10.2514/3.28677.

[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), 347-356. doi: 10.2514/2.3833.

[24]

M. B. Quadrelli, Dynamics and control of novel orbiting formations with internal dynamics, Journal of the Astronautical Sciences, 51 (2003), 319-337.

[25]

B. Wong and A. Misra, Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points, Acta Astronautica, 63 (2008), 1178-1187. doi: 10.1016/j.actaastro.2008.06.022.

[26]

W. Zhang, F. B. Gao and M. H. Yao, Periodic solutions and stability of a tethered satellite system, Mechanics Research Communications, 44 (2012), 24-29. doi: 10.1016/j.mechrescom.2012.05.004.

[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 and Continuous Dynamical Systems - B, 2017, 22 (7) : 2669-2685. doi: 10.3934/dcdsb.2017130

[2]

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 and Continuous Dynamical Systems - S, 2015, 8 (6) : 1047-1054. doi: 10.3934/dcdss.2015.8.1047

[3]

Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303

[4]

Alexandr A. Zevin, Mark A. Pinsky. Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 293-297. doi: 10.3934/dcds.2000.6.293

[5]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319

[6]

Karol Mikula, Jozef Urbán, Michal Kollár, Martin Ambroz, Ivan Jarolímek, Jozef Šibík, Mária Šibíková. Semi-automatic segmentation of NATURA 2000 habitats in Sentinel-2 satellite images by evolving open curves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1033-1046. doi: 10.3934/dcdss.2020231

[7]

Shengyang Jia, Lei Deng, Quanwu Zhao, Yunkai Chen. An adaptive large neighborhood search heuristic for multi-commodity two-echelon vehicle routing problem with satellite synchronization. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2021225

[8]

Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451

[9]

Eduard Feireisl, Josef Málek, Antonín Novotný. Navier's slip and incompressible limits in domains with variable bottoms. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 427-460. doi: 10.3934/dcdss.2008.1.427

[10]

Xian Zhang, Vinesh Nishawala, Martin Ostoja-Starzewski. Anti-plane shear Lamb's problem on random mass density fields with fractal and Hurst effects. Evolution Equations and Control Theory, 2019, 8 (1) : 231-246. doi: 10.3934/eect.2019013

[11]

Kevin Zumbrun. L resolvent bounds for steady Boltzmann's Equation. Kinetic and Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048

[12]

Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058

[13]

Szandra Beretka, Gabriella Vas. Stable periodic solutions for Nazarenko's equation. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3257-3281. doi: 10.3934/cpaa.2020144

[14]

Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423

[15]

J. Leonel Rocha, Sandra M. Aleixo. Dynamical analysis in growth models: Blumberg's equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 783-795. doi: 10.3934/dcdsb.2013.18.783

[16]

S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660

[17]

Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 769-772. doi: 10.3934/dcdss.2015.8.769

[18]

Laurent Desvillettes, Clément Mouhot, Cédric Villani. Celebrating Cercignani's conjecture for the Boltzmann equation. Kinetic and Related Models, 2011, 4 (1) : 277-294. doi: 10.3934/krm.2011.4.277

[19]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607

[20]

Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic and Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (2)

[Back to Top]