phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2018 - 3 issue » pp. 373-394

Archive of Issues

Russia Moscow
Year
2018
Volume
28
Issue
3
Pages
373-394
<<
>>
Section Mechanics
Title On periodic motions of a symmetrical satellite in an orbit with small eccentricity in the case of multiple combinational resonance of the third and fourth orders
Author(-s) Safonov A.I.ab, Kholostova O.V.bc
Affiliations Research and Production Company “Infosystem-35”a, Moscow Aviation Instituteb, Moscow Institute of Physics and Technologyc
Abstract The motion of a near-autonomous time-periodic two-degree-of-freedom Hamiltonian system in the vicinity of a linearly stable trivial equilibrium is considered. The values of the problem parameters are supposed to be such that the system implements both a double combinational third-order resonance and a fourth-order resonance. The problem of existence and stability of resonant periodic motions of the system is considered. The study is carried out using as an example the problem of the motion of a dynamically symmetric satellite (a rigid body) relative to the center of mass in the central Newtonian gravitational field in an elliptical orbit with small eccentricity. The satellite's periodic motions generated from its stationary rotations in a circular orbit (hyperboloidal and conical precessions) for the resonant values of the parameters are considered as unperturbed ones. The normalization of the Hamiltonian functions of perturbed motion is performed, and the equilibrium positions of approximate (model) systems are determined. The corresponding resonant periodic motions of the satellite in the vicinity of these unperturbed motions are obtained by the Poincare method, and their geometric interpretation is given. The unstable periodic motions and the motions that are stable for the majority (in the sense of Lebesgue measure) of the initial conditions and formally stable are revealed.
Keywords Hamiltonian system, multiple resonance, stability, periodic motion, dynamically symmetrical satellite, hyperboloidal precession, conical precession
UDC 531.36
MSC 70H05, 70H14, 70H15, 70K45
DOI 10.20537/vm180308
Received 15 August 2018
Language Russian
Citation Safonov A.I., Kholostova O.V. On periodic motions of a symmetrical satellite in an orbit with small eccentricity in the case of multiple combinational resonance of the third and fourth orders, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 3, pp. 373-394.
References
  1. Korteweg D.J. Sur certaines vibrations d'ordre superieur et d'intensite anomale, vibrations de relation, dans les mecanismes a plusieurs degres de liberte, Archives Neerlandaises des Sciences Exactes et Naturelles, Serie 2, 1898, tome 1, pp. 229-260.
  2. Beth H. Les oscillations autour d'une position d'equilibre dans le cas d'existence d'une relation lineaire simple entre les nombres vibratoires, Archives Neerlandaises des Sciences Exactes et Naturelles, Serie 2, 1911, tome 15, pp. 246-283.
  3. Beth H.J.E. Les oscillations autour d'une position d'equilibre dans le cas d'existence d'une relation lineaire simple entre les nombres vibratoires (suite), Archives Neerlandaises des Sciences Exactes et Naturelles, Serie 3A (Sciences Exactes), 1912, tome 1, pp. 185-208.
  4. Markeev A.P. Tochki libratsii v nebesnoi mekhanike i kosmodinamike (Libration points in celestial mechanics and cosmodynamics), Moscow: Nauka, 1978, 312 p.
  5. Kunitsyn A.L., Markeev A.P. Stability in resonant cases, Itogi Nauki i Tekhniki. Ser. Obshchaya mekhanika, 1979, vol. 4, pp. 58-139 (in Russian).
  6. Markeev A.P. Stability in Hamiltonian systems, Nelineinaya Mekhanika, Moscow: Fizmatgiz, 2001, pp. 114-130 (in Russian).
  7. Markeyev A.P. Third-order resonance in a Hamiltonian system with one degree of freedom, Journal of Applied Mathematics and Mechanics, 1994, vol. 58, issue 5, pp. 793-804. DOI: 10.1016/0021-8928(94)90004-3
  8. Markeyev A.P. The behaviour of a non-linear Hamiltonian system with one degree of freedom at the boundary of a parametric resonance domain, Journal of Applied Mathematics and Mechanics, 1995, vol. 59, issue 4, pp. 541-551. DOI: 10.1016/0021-8928(95)00063-1
  9. Markeev A.P. Parametric resonance and nonlinear oscillations of a heavy rigid body in the neighborhood of its planar rotations, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1995, no. 5, pp. 34-44 (in Russian).
  10. Kholostova O.V. Parametric resonance in the problem of nonlinear oscillations of a satellite in an elliptic orbit, Cosmic Research, 1996, vol. 34, no. 3, pp. 288-292.
  11. Kholostova O.V. On motions of a Hamiltonian system with one degree of freedom under resonance in forced oscillations, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1996, no. 3, pp. 167-175 (in Russian).
  12. Kholostova O.V. The motion of a system close to Hamiltonian with one degree of freedom when there is resonance in forced vibrations, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, issue 3, pp. 399-406. DOI: 10.1016/S0021-8928(96)00050-0
  13. Markeyev A.P. The critical case of fourth-order resonance in a hamiltonian system with one degree of freedom, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, issue 3, pp. 355-361. DOI: 10.1016/S0021-8928(97)00045-2
  14. Kholostova O.V. The non-linear oscillations of a satellite with third-order resonance, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, issue 4, pp. 539-547. DOI: 10.1016/S0021-8928(97)00068-3
  15. Kholostova O.V. Non-linear oscillations of a hamiltonian system with one degree of freedom and fourth-order resonance, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, issue 6, pp. 883-892. DOI: 10.1016/S0021-8928(98)00113-0
  16. Kholostova O.V. On motions of a one-degree-of-freedom system close to a Hamiltonian system under resonance of the fourth order, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1999, no. 4, pp. 25-30 (in Russian).
  17. Kholostova O.V. On bifurcations and stability of resonance periodic motions of hamiltonian systems with one degree of freedom caused by degeneration of the hamiltonian, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 1, pp. 89-110. DOI: 10.20537/nd0601005
  18. Kholostova O.V. Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate, Journal of Applied Mathematics and Mechanics, 2006, vol. 70, issue 4, pp. 516-526. DOI: 10.1016/j.jappmathmech.2006.09.005
  19. Markeev A.P. On a multiple resonance in linear Hamiltonian systems, Doklady Physics, 2005, vol. 50, no. 5, pp. 278-282. DOI: 10.1134/1.1941506
  20. Markeyev A.P. Multiple parametric resonance in Hamiltonian systems, Journal of Applied Mathematics and Mechanics, 2006, vol. 70, issue 2, pp. 176-194. DOI: 10.1016/j.jappmathmech.2006.06.001
  21. Markeev A.P. Lineinye gamil'tonovy sistemy i nekotorye zadachi ob ustoichivosti dvizheniya sputnika otnositel'no tsentra mass (Linear Hamiltonian systems and some problems of stability of satellite motion relative to the center of mass), Moscow-Izhevsk: Institute of Computer Science, 2009, 396 p.
  22. Markeev A.P. On one special case of parametric resonance in problems of celestial mechanics, Astronomy Letters, 2005, vol. 31, no. 5, pp. 350-356. DOI: 10.1134/1.1922534
  23. Markeev A.P. Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass, Astronomy Letters, 2005, vol. 31, no. 9, pp. 627-633. DOI: 10.1134/1.2039974
  24. Kholostova O.V. On periodic motions of a nonautonomous Hamiltonian system in one case of multiple parametric resonance, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 477-504 (in Russian). DOI: 10.20537/nd1704003
  25. Kholostova O.V. Motions of a two-degree-of-freedom Hamiltonian system in the presence of multiple third-order resonances, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 2, pp. 267-288 (in Russian). DOI: 10.20537/nd1202005
  26. Kholostova O.V. The interaction of resonances of the third and fourth orders in a Hamiltonian two-degree-of-freedom system, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 671-683 (in Russian). DOI: 10.20537/nd1504004
  27. Kholostova O. Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances, International Journal of Non-Linear Mechanics, 2015, vol. 73, pp. 64-68. DOI: 10.1016/j.ijnonlinmec.2014.11.005
  28. Kholostova O.V. Zadachi dinamiki tverdykh tel s vibriruyushchim podvesom (Problems of dynamics of rigid bodies with vibrating suspension), Moscow-Izhevsk: Institute of Computer Science, 2016, 308 p.
  29. Safonov A.I., Kholostova O.V. On the periodic motions of a Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a double three-order resonance, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, no. 3, pp. 418-438 (in Russian). DOI: 10.20537/vm160310
  30. Kholostova O.V., Safonov A.I. On equilibrium positions bifurcations of Hamiltonian system in cases of double combined third order resonance, Trudy Moskovskogo Aviatsionnogo Instituta, 2018, no. 100. http://trudymai.ru/published.php?ID=93297
  31. Duboshin G.N. On the rotational motion of artificial celestial bodies, Byulleten' Instituta Teoreticheskoi Astronomii Akademii Nauk SSSR, 1960, no. 7, pp. 511-520 (in Russian).
  32. Chernous'ko F.L. On the stability of regular precession of a satellite, Journal of Applied Mathematics and Mechanics, 1964, vol. 28, issue 1, pp. 181-184. DOI: 10.1016/0021-8928(64)90145-5
  33. Markeev A.P. Resonant effects and stability of stationary rotations of a satellite, Kosmicheskie Issledovaniya, 1967, vol. 5, no. 3, pp. 365-375 (in Russian).
  34. Beletskii V.V. Dvizhenie sputnika otnositel'no tsentra mass v gravitatsionnom pole (Satellite's motion about center of mass in a gravitational field), Moscow: Moscow State University, 1975, 308 p.
  35. Sarychev V.A. Asymptotically stable stationary rotation of a satellite, Kosmicheskie Issledovaniya, 1965, vol. 3, no. 5, pp. 667-673 (in Russian).
  36. Beletskii V.V. Motion of an artificial satellite about its center of mass, Jerusalem: Israel Program for Scientific Translations, 1966.
  37. Markeev A.P. On the rotational motion of a dynamically symmetric satellite in an elliptical orbit, Kosmicheskie Issledovaniya, 1967, vol. 5, no. 4, pp. 530-539 (in Russian).
  38. Malkin I.G. Nekotorye zadachi teorii nelineinykh kolebanii (Some problems of the theory of nonlinear oscillations), Moscow: Gostekhizdat, 1956, 492 p.
  39. Glimm J. Formal stability of Hamiltonian systems, Communications of Pure and Applied Mathematics, 1964, no. 4, pp. 509-526.
Full text
/doc/issues-2018/18-03-08.pdf
<< Previous article
Next article >>