On spacial motions of an orbital tethered system

A. V. Rodnikov ; P. S. Krasil'nikov

Russian journal of nonlinear dynamics, Tome 13 (2017) no. 4, pp. 505-518.

Résumé

We study motions of a particle along a rope with ends fixed to an extended rigid body whose center of mass traces out a circular orbit in the central Newtonian force field. (Such a rope is called a tether.) We assume that the tether realizes an ideal unilateral constraint. We derive particle motion equations on the surface of the ellipsoid, which restricts the particle motion, and conditions that guarantee such motions. We also study the existence and stability of relative equilibria of the particle with respect to the orbital frame of reference. We prove stability of the integral manifold of the particle motions in the plane of the orbit. We note that small-amplitude librations near this manifold can be described by approximate equations that can be reduced to Riccati’s equation. We establish that generally the spacial motions of the particle are chaotic for initial conditions from some vicinity of the separatrix motion in the plane of the orbit and are regular in other cases. We also note that chaotic motions usually lead to a situation where the particle comes off the constraint, in other words, to motions inside the above-mentioned ellipsoid.

Détail

Keywords: space tethered system, unilateral constraint, tether
Mots-clés : chaos, Riccati equation.

Bibliographie

[1] Alpatov A. P., Beletsky V. V., Dranovskii V. I., Khoroshilov V. S., Pirozhenko A. V., Troger H., Zakrzhevskii A. E., Dynamics of tethered space systems, CRC Press, Boca Raton, Fla., 2010, 245 pp.

[2] Beletsky V. V., “On relative motion of two bodies couple on the orbit: 2”, Kosmicheskie Issledovaniya, 7:6 (1969), 827–840 (Russian)

[3] Beletsky V. V., Reguläre und chaotische Bewegung starrer Körper, Teubner, Leipzig, 1995, 156 pp.

[4] Beletskii V. V., Essays on the motion of celestial bodies, Birkhäuser, Basel, 2001, 372 pp.

[5] Beletsky V. V., Levin E. M., Dynamics of space tether systems, Univelt, San Diego, Calif., 1993, vii, 499 pp.

[6] Beletsky V. V., Novikova E. T., “On relative motion of two body couple on the orbit”, Kosmicheskie Issledovaniya, 7:3 (1969), 377–384 (Russian)

[7] Beletskii V. V., Kugushev E. I., Starostin E. L., “Free manifolds of dynamic billiards”, Regul. Chaotic Dyn., 2:3–4 (1997), 62–71 (Russian)

[8] Borisov A. V., Mamaev I. S., Dynamics of a rigid body: Hamiltonian methods, integrability, chaos, 2nd ed., R Dynamics, Institute of Computer Science, Izhevsk, 2005, 576 pp. (Russian)

[9] Burov A. A., “The existence and stability of the equilibria of mechanical systems with constraints produced by large potential forces”, J. Appl. Math. Mech., 67:2 (2003), 193–200

[10] Ivanov A. P., Dynamics of systems with mechanical collisions, Int. Programm of Education, Moscow, 1997 (Russian)

[11] Kozlov V. V., Treshchev D. V., Billiards: A genetic introduction to the dynamics of systems with impacts, Transl. Math. Monogr., 89, AMS, Providence, R.I., 1991, viii+171 pp.

[12] Rodnikov A. V., “Equilibrium positions of a weight on a cable fixed to a dumbbell-shaped space station moving along a circular geocentric orbit”, Cosmic Research, 44:1 (2006), 58–68

[13] Rodnikov A. V., “Existence of nonimpact motions along a wire rope fixed to an extended spacecraft”, Cosmic Research, 44:6 (2006), 532–539

[14] Rodnikov A. V., “On a particle motion along the leier fixed in a precessing rigid body”, Nelin. Dinam., 7:2 (2011), 295–311 (Russian)

[15] Beletsky V. V., Kasatkin G. V., Starostin E. L., “The pendulum as a dynamical billiard”, Chaos Solitons Fractals, 7:8 (1996), 1145–1178

[16] Beletsky V. V., Pankova D. V., “Connected bodies in the orbit as dynamic billiard”, Regul. Chaotic Dyn., 1:1 (1996), 87–103

[17] Burov A. A., Guerman A. D., Kosenko I. I., Ferraz A., Nikonov V. I., “Dynamics of space elevator on asteroid”, Space — The Gateway for Mankind's Future: Proc. of the 66th Internat. Astronautical Congr. (IAC'2015, Jerusalem, Israel, Oct 2015), v. 12, 9496–9504

[18] Burov A. A., Kosenko I. I., Guerman A. D., “Uniform rotations of tethered system connected to a Moon surface”, Acta Astronaut., 116 (2015), 349–354

[19] Burov A. A., Kosenko I. I., Troger H., “On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator”, Mech. Solids, 47:3 (2012), 269–284

[20] Lichtenberg A. J., Lieberman M. A., Regular and chaotic dynamics, Appl. Math. Sci., 38, 2nd ed., Springer, New York, 1992, 692 pp.

[21] Lanoix E. L.-M., Misra A. K., “Near-Earth asteroid missions using tether sling shot assist”, J. Spacecraft Rockets, 37:4 (2000), 475–480

[22] Misra A. K., “Dynamics of a tether attached to an asteroid”, 7th Internat. Workshop and Advanced School «Spaceflight Dynamics and Control» }, 2012 {\tt http://www.aerospace.ubi.pt/workshop2012

[23] Pearson J., “Anchored lunar satellites for cislunar transportation and communication”, J. Astronaut. Sci., 27:1 (1979), 39–62

[24] Peláez J., Bombardelli C., Scheeres D. J., “Dynamics of a tethered observatory at Jupiter”, J. Guid. Control Dyn., 35:1 (2012), 195–207

[25] Rodnikov A. V., “The algorithms for capture of the space garbage using «leier constraint»”, Regul. Chaotic Dyn., 11:4 (2006), 483–489

[26] Rodnikov A. V., “Rotations of a dumbbell equipped with the «leier constraint»”, J. Vibroeng., 10:4 (2008), 557–561

[27] Tethers in space handbook, 3rd ed., eds. M. L. Cosmo, E. C. Lorenzini, NASA Marshall Space Flight Center, Huntsville, Ala., 1997, 274 pp.