On a particle motion along the leier fixed in a precessing rigid body

A. V. Rodnikov

Russian journal of nonlinear dynamics, Tome 7 (2011) no. 2, pp. 295-311.

Résumé

We study a particle motion along a cable with ends fixed in a precessed rigid body. Such cable called «the leier» is a model of space elevator for a dynamically symmetric asteroid. (The Dutch term «leier» means the rope with fixed ends). In this paper we find two integrable cases of the particle motion equations (for zero and right nutation angle) Phase portraits for integrable situations are built taking into account conditions of motion with the tense cable and assuming the body gravitation is close to gravitational field of two equal point masses that are in the axis of dynamical symmetry. Using «the Generalized Restricted Circular Problem of Three Bodies» by V. V. Beletsky, we study the particle equilibria on the leier in the plane containing the body mass center and being perpendicular to the precession axis for all possible nutation angles. Some facts on these equilibria stability are formulated.

Détail

Keywords: space elevator, space tether system, asteroid, unilateral constraint, problem of three bodies.

Bibliographie

[1] Dal V., Tolkovyi slovar zhivago Velikorusskago yazyka, v. 2, Izdanie M. O. Volfa, CPb.-M., 1881, 776 pp.

[2] Beletskii V. V., Dvizhenie iskusstvennogo sputnika Zemli otnositelno tsentra mass, Nauka, M., 1965, 416 pp.

[3] Sebekhei V., Teoriya orbit: Ogranichennaya zadacha trekh tel, Nauka, M., 1982, 656 pp.

[4] Beletskii V. V., Novikova E. T., “Ob otnositelnom dvizhenii svyazki dvukh tel na orbite”, Kosmicheskie issledovaniya, 7:3 (1969), 377–384

[5] Beletskii V. V., “Ob otnositelnom dvizhenii svyazki dvukh tel na orbite: 2”, Kosmicheskie issledovaniya, 7:6 (1969), 827–840

[6] Markeev A. P., Tochki libratsii v nebesnoi mekhanike i kosmodinamike, Nauka, M., 1978, 312 pp.

[7] Kosenko I. I., “O tochkakh libratsii vblizi gravitiruyuschego vraschayuschegosya trekhosnogo ellipsoida”, PMM, 45:1 (1981), 26–33 | MR | Zbl

[8] Kosenko I. I., “Tochki libratsii v zadache o trekhosnom gravitiruyuschem ellipsoide: Geometriya oblasti ustoichivosti”, Kosmicheskie issledovaniya, 19:2 (1981), 200–209

[9] Ivanov A. P., “Ob ustoichivosti v sistemakh s neuderzhivayuschimi svyazyami”, PMM, 48:5 (1984), 725–732 | MR

[10] Kosenko I. I., “Non-linear analysis of the stability of the libration points of a triaxial ellipsoid”, J. Appl. Math. Mech., 49:1 (1985), 17–24 | DOI | MR

[11] Ivanov V. A., Sitarskii Yu. S., Dinamika poleta sistemy gibko svyazannykh kosmicheskikh ob'ektov, Mashinostroenie, M., 1986, 244 pp.

[12] Beletskii V. V., Levin E. M., Dinamika kosmicheskikh trosovykh sistem, Nauka, M., 1990, 336 pp.

[13] Wang L.-Sh., Cheng Sh.-F., “Dynamics of two spring-connected masses in orbit”, Celestial Mech. Dynam. Astronom., 63:3-4 (1995/96), 289–312 | MR

[14] Scheeres D. J., Ostro S. J., Hudson R. S., Werner R. A., “Orbits close to asteroid 4769 Castalia”, Icarus, 121 (1996), 67–87 | DOI

[15] Ivanov A. P., Dinamika sistem s mekhanicheskimi soudareniyami, Mezhdunarodnaya programma obrazovaniya, M., 1997, 336 pp.

[16] Tethers in space handbook, Smithsonian Astrophysical Observatory for NASA Marshall Space Flight Center, Grant NAG8-1160 monitored by C. C. Rupp., 3rd ed., eds. M. L. Cosmo, E. C. Lorenzini, 1997, 241 pp.

[17] Scheeres D. J., Ostro S. J., Hudson R. S., DeJong E. M., Suzuki S., “Dynamics of orbits close to asteroid 4179 Toutatis”, Icarus, 132 (1998), 53–79 | DOI

[18] Scheeres D. J., Williams B. G., Miller J. K., “Evaluation of the dynamic environment of an asteroid: Applications to 433 Eros”, J. Guid. Contr. Dynam., 23:3 (2000), 466–475 | DOI

[19] Krupa M., Schagerl M., Steindl A., Troger H., “Stability of relative equilibria: 1. Comparison of four methods”, Meccanica, 35 (2001), 325–351 | DOI | MR

[20] Krupa M., Steindl A., Troger H., “Stability of relative equilibria: 2. Dumbell satellites”, Meccanica, 35 (2001), 353–371 | DOI | MR

[21] Scheeres D. J., “Stability of binary asteroids”, Icarus, 159 (2002), 271–283 | DOI

[22] Beletskii V. V., “Nekotorye zadachi dinamiki dvoinykh asteroidov”, Sovremennye problemy mekhaniki i fiziki kosmosa, eds. V. S. Avduevskii, A. V. Kolesnichenko, Fizmatlit, M., 2003, 27–40

[23] Burov A. A., “O suschestvovanii i ustoichivosti ravnovesii mekhanicheskikh sistem so svyazyami, realizuemymi bolshimi potentsialnymi silami”, PMM, 67:2 (2003), 222–230 | MR | Zbl

[24] Rodnikov A. V., “O dvizhenii gruza po torosu, zakreplennomu na gantelevidnom kosmicheskom apparate”, Kosmicheskie issledovaniya, 42:4 (2004), 444–448

[25] Koon W. S., Marsden J. E., Ross S., Lo M., Scheeres D. J., “Geometric mechanics and the dynamics of asteroid pairs”, Ann. New York Acad. Sci., 1017 (2004), 11–38 | DOI

[26] Vasilkova O. O., “Three-dimensional periodic motion in the vicinity of the equilibrium points of an asteroid”, Astron. Astrophys., 430:2 (2005), 713–723 | DOI | Zbl

[27] Cendra H., Marsden J. E., “Geometric mechanics and the dynamics of asteroid pairs”, Dyn. Syst., 20:1 (2005), 3–21 | DOI | MR | Zbl

[28] Scheeres D. J., Bellerose J., “The restricted Hill full 4-body problem: Application to spacecraft motion about binary asteroids”, Dyn. Syst., 20:1 (2005), 23–44 | DOI | MR | Zbl

[29] Gabern F., Koon W. S., Marsden J. E., “Spacecraft dynamics near a binary asteroid”, Discrete Contin. Dyn. Syst., 2005, suppl., 297–306 | MR | Zbl

[30] Beletskii V. V., Ivanov M. B., Otstavnov E. I., “Modelnaya zadacha o kosmicheskom lifte”, Kosmicheskie issledovaniya, 43:2 (2005), 157–160 | MR

[31] Rodnikov A. V., “O polozheniyakh ravnovesiya gruza na trose, zakreplennom na gantelevidnoi kosmicheskoi stantsii, dvizhuscheisya po krugovoi geotsentricheskoi orbite”, Kosmicheskie issledovaniya, 44:1 (2006), 62–72

[32] Kosenko I. I., Stepanov S. Ya., “Ustoichivost polozhenii otnositelnogo ravnovesiya orbitalnoi svyazki s uchetom udarnykh vzaimodeistvii: Neogranichennaya zadacha”, Izv. RAN. MTT, 2006, no. 4, 86–96

[33] Rodnikov A. V., “O suschestvovanii bezudarnykh dvizhenii po leernoi svyazi, zakreplennoi na protyazhennom kosmicheskom apparate”, Kosmicheskie issledovaniya, 44:6 (2006), 553–560

[34] Rodnikov A. V., “The algoritms for capture of the space garbage using «leier constraint»”, Regul. Chaotic Dyn., 11:4, 483–489 | DOI | MR | Zbl

[35] Alpatov A. P., Beletskii V. V., Dranovskii V. I., Zakrzhevskii A. E., Pirozhenko A. V., Troger G., Khoroshilov V. S., Dinamika kosmicheskikh sistem s trosovymi i sharnirnymi soedineniyami, RKhD, M.–Izhevsk, 2006, 560 pp.

[36] Gabern F., Koon W. S., Marsden J. E., “Parking a spacecraft near an asteroid pair”, J. Guid. Contr. Dynam., 29:3 (2006), 544–553 | DOI

[37] Beletskii V. V., “Obobschennaya ogranichennaya krugovaya zadacha trekh tel kak model dinamiki dvoinykh asteroidov”, Kosmicheskie issledovaniya, 45:6 (2007), 435–442

[38] Beletskii V. V., Regulyarnye i khaoticheskie dvizheniya tverdykh tel, RKhD, M.–Izhevsk, 2007, 132 pp.

[39] Munitsyna M. A., “Otnositelnye ravnovesiya sistemy «gantel-gruz» s odnostoronnimi svyazyami na krugovoi keplerovskoi orbite”, Avtomatika i telemekhanika, 2007, no. 9, 9–15

[40] Burov A. A., Guerman A. D., “Steady motions of a tetrahedral satellite with tethered elements”, Proc. of the 6th European Nonlinear Dynamics Conference (ENOC 2008), 2008 http://lib.physcon.ru//?item=1753

[41] Rodnikov A. V., “Rotations of a dumbbell equipped with «the leier constraint»”, Journal of Vibroengineering, 10:4, 557–561

[42] Beletskii V. V., Rodnikov A. V., “Ob ustoichivosti treugolnykh tochek libratsii v obobschennoi ogranichennoi krugovoi zadache trekh tel”, Kosmicheskie issledovaniya, 46:1 (2008), 42–50 | Zbl

[43] Beletsky V. V., Rodnikov A. V., “On evolution of libration points similar to Eulerian in the model problem of the binary-asteroids dynamics”, Journal of Vibroengineering, 10:4 (2008), 550–556

[44] Buchin V., Burov A., Troger H., “A dumb-bell satellite with a cabin: Existence and stability of relative equilibria”, Proc. of the 6th European Nonlinear Dynamics Conference (ENOC 2008), 2008 http://lib.physcon.ru/?item=1749

[45] Rodnikov A. V., “On systems with «leier constraint» in the central Newtonian force field”, Proc. of the 6th European Nonlinear Dynamics Conference (ENOC 2008), 2008 http://lib.physcon.ru/?item=1729

[46] Beletskii V. V., Ocherki o dvizhenii kosmicheskikh tel, 3-e izd., URSS, M., 2009, 432 pp.

[47] Fahnestock E. G., Scheeres D. J., “Binary asteroid orbit expansion due to continued YORP spin-up of the primary and primary surface particle motion”, Icarus, 201:1 (2009), 135–152 | DOI

[48] Rodnikov A. V., “On dynamics of a dumbbell satellite with a small load on the leier”, Physics and Control (Catania, Italy, sept. 1–4, 2009)), 2009 http://lib.physcon.ru/?item=2028

[49] Rodnikov A. V., “O vliyanii leernoi svyazi na dvizhenie gantelevidnogo tela v tsentralnom nyutonovskom silovom pole”, Nelineinaya dinamika, 5:4 (2009), 519–533 | Zbl

[50] Vasilkova O. O., “Stability criterion for a light binary attracted by a heavy body”, Astronomy Lett., 36:3 (2010), 227–230 | DOI