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Home » Issues » Archive of Issues » 2021 - 1 issue » pp. 102-115

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Ukraine Donetsk
Year
2021
Volume
31
Issue
1
Pages
102-115
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Section Mechanics
Title An approach in studying gyrostat motion with variable gyrostatic moment
Author(-s) Gorr G.V.a
Affiliations Institute of Applied Mathematics and Mechanics, Donetska
Abstract The problem of the motion of a gyrostat with a fixed point and a variable gyrostatic moment under the action of gravity force is considered. A new method for integrating the equations of motion of a system consisting of a carrier body and three rotors that rotate around the main axes is proposed. The method can be attributed to the method of variation of the constant in the function for the gyrostatic moment, which linearly depends on the vector of vertical. In case of a constant multiplier, the gyrostatic moment satisfies the Poisson equation, and its variation is found from the integral of areas. The original equations have been reduced to a fifth-order system. New solutions of these equations are obtained in the case of a spherical mass distribution for the gyrostat and for the precessional motions of a carrier body. An explicit form of the gyrostatic moment is established for the case of three invariant relations.
Keywords gyrostat, gravity field, invariant relations, reduction of equations, spherical gyrostat
UDC 531.392
MSC 70E05, 70E17, 70E40
DOI 10.35634/vm210108
Received 28 May 2020
Language Russian
Citation Gorr G.V. An approach in studying gyrostat motion with variable gyrostatic moment, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 1, pp. 102-115.
References
  1. Thomson W. On the motion of rigid solids in a liquid circulating irrotationally through perforations in them or in any fixed solid, Proceeding of the Royal Society of Edinburg, 1872, vol. 7, pp. 668-682. https://doi.org/10.1017/S0370164600042875
  2. Volterra V. Sur la théorie des variations des latitudes, Acta Mathematica, 1899, vol. 22, pp. 201-357. https://doi.org/10.1007/BF02417877
  3. Zhukovskii N.E. On the motion of a solid body having cavities filled with a homogeneous dropping liquid, Sobranie sochinenii. Tom 1 (Collected works, vol. 1), Moscow: Gostekhizdat, 1949, pp. 31-152 (in Russian).
  4. Gray A. A treatise on gurostatics and rotational motion. Theory and applications, New York: Dover Publications, 1959.
  5. Rumyantsev V.V. About orientation control and satellite stabilization by rotors, Vestnik Moskovskogo Universiteta. Seriya I. Matematika, Mekhanika, 1970, no. 2, pp. 83-96 (in Russian). https://zbmath.org/?q=an:0207.23402
  6. Kharlamov P.V. On the equations of motion of a system of solids, Izvestiya Akademii Nauk SSSR. Mekhanika Tverdogo Tela, 1972, no. 4, pp. 52-73 (in Russian).
  7. Wittenburg J. Dynamics of systems of rigid bodies, Stuttgart: B.G. Teubner, 1977.
  8. Gorr G.V., Maznev A.V., Kotov G.A. Dvizhenie girostata s peremennym girostaticheskim momentom (The movement of the gyrostat with a variable gyrostatic moment), Donetsk: Institute of Applied Mathematics and Mechanics, 2017.
  9. Levy-Civita T., Amaldi U. Lezioni di Meccanica Razionale. Vol. 2. Dinamica dei sistemi con un numero finito di gradi di libertà. Part 2, Bologna: 1927. (In Italian).
  10. Borisov A.V., Mamaev I.S. Rigid body dynamics, De Gruyter, 2019. https://doi.org/10.1515/9783110544442
  11. Kane T.R., Fowler R.C. Equivalence of two gyrostatic stability problems, Journal of Applied Mechanics, 1970, vol. 37, issue 4, pp. 1146-1147. https://doi.org/10.1115/1.3408674
  12. Roberson R.E. The equivalence of two classical problems of free spinning gyrostats, Journal of Applied Mechanics, 1971, vol. 38, issue 3, pp. 1146-1147. https://doi.org/10.1115/1.3408879
  13. Aslanov V.S., Doroshin A.V. The motion of a system of coaxial bodies of variable mass, Journal of Applied Mathematics and Mechanics, 2004, vol. 68, issue 6, pp. 899-908. https://doi.org/10.1016/j.jappmathmech.2004.11.012
  14. Aslanov V.S., Doroshin A.V. Two cases of motion of an unbalanced gyrostat, Mechanics of Solids, 2006, vol. 41, issue 4, pp. 29-39. http://mtt.ipmnet.ru/en/Issues.php?y=2006&n=4&p=29
  15. Gorr G.V. On three invariant relations of the equations of motion of a body in a potential field of force, Mechanics of Solids, 2019, vol. 54, no. 2, pp. 234-244. https://doi.org/10.3103/S0025654419030105
  16. Gorr G.V. On three invariant of the equations of motion of a body in a potential field of force, Mechanics of Solid, 2019, vol. 54, suppl. 2, pp. S104-S114.
  17. Gorr G.V., Tkachenko D.N., Shchetinina E.K. Research on the motion of a body in a potential force field in the case of three invariant relations, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 3, pp. 327-342. https://doi.org/10.20537/nd190310
  18. Kowalewski N. Eine neue partikuäre Lösung der Differenzialgleichungen der Bewegung eines schweren starren Körpers um einen festen Punkt, Mathematische Annalen, 1908, vol. 65, issue 4, pp. 528-537. https://doi.org/10.1007/BF01451168
  19. Steklov V.A. A new particular solution of differential equations of motion of a heavy rigid body with a fixed point, Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lyubitelei Estestvoznaniya, 1899, vol. 10, issue 1, pp. 1-3 (in Russian).
  20. Goryachev D.N. A new particular solution to the problem of the motion of a heavy rigid body around a fixed point, Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lyubitelei Estestvoznaniya, 1899, vol. 10, issue 1, pp. 23-24 (in Russian).
  21. Kharlamov P.V. Polynomial solutions of the equations of motion of a body with a fixed point, Journal of Applied Mathematics and Mechanics, 1965, vol. 29, issue 1, pp. 26-35. https://doi.org/10.1016/0021-8928(65)90147-4
  22. Gorr G.V., Maznev A.V. Dinamika girostata, imeyushchego nepodvizhnuyu tochku (The dynamics of a gyrostat having a fixed point), Donetsk: Donetsk National University, 2010.
  23. Gorr G.V., Kudryashova L.V., Stepanova L.A. Klassicheskie zadachi dinamiki tverdogo tela. Razvitie i sovremennoe sostoyanie (Classical problems in the theory of solid bodies. Their development and current state), Kiev: Naukova Dumka, 1978.
  24. Gashenenko I.N., Gorr G.V., Kovalev A.M. Klassicheskie zadachi dinamiki tverdogo tela (Classical problems of the dynamics of a rigid body), Kiev: Naukova Dumka, 2012. https://zbmath.org/?q=an:1284.70002
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