Geodesic
One-axis equilibrium orientations to an attracting center of a~symmetric prolate orbital gyrostat with an elasctic beam
Sibirskij žurnal industrialʹnoj matematiki, Tome 20 (2017) no. 3, pp. 92-100.

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We study the motion of a symmetric prolate stationary gyrostat along a Keplerian circular orbit in a Newtonian central field of forces in the restricted formulation. An elastic beam is clamped by one end in the body of the gyrostat along its axis of symmetry. The beam has a point mass at the free end. The inextensible elastic beam (which is, for simplicity, of constant circular cross-section) performs infinitesimal space vibrations in the process of the motion of the system. Moreover, we neglect the terms nonlinear with repect to the displacements of the points of the beam in the tensor of inertia of the system. We consider the following (so-called semi-inverse) problem: Under what kinetic moment of the gyrostat among its relative equilibria (the states of rest in the orbital coordinate system) is an arbitrary coordinate axis defined in the coordinate system associated with the gyrostat collinear to the local vertical? In the discretization of the problem, we give the values of the Lagrange coordinates defining the deformation of the beam in these equilibria and the value of the gyrostatic moment guaranteeing the presence of the equilibrium.
Keywords: orbital prolate symmetric gyrostat, circular orbit, central Newtonian force field, elastic beam
Mots-clés : point mass, one-axis orientation. 
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S. V. Chaikin. One-axis equilibrium orientations to an attracting center of a~symmetric prolate orbital gyrostat with an elasctic beam. Sibirskij žurnal industrialʹnoj matematiki, Tome 20 (2017) no. 3, pp. 92-100. http://geodesic.mathdoc.fr/item/SJIM_2017_20_3_a9/

[1] Beletskii V. V., Dvizhenie iskusstvennogo sputnika otnositelno tsentra mass, Nauka, M., 1965

[2] Chaikin S. V., “Vliyanie sil inertsii i gravitatsii na deformatsiyu uprugogo sterzhnya, zaschemlennogo v korpuse orbitalnogo girostata”, Sib. zhurn. industr. matematiki, 18:4 (2015), 88–97 | MR | Zbl

[3] Demin V. G., Markov Yu. G., Minyaev I. S., “O dvizhenii sputnika, nesuschego vyazkoupruguyu shtangu s gruzom na kontse, na krugovoi orbite”, Kosm. issledovaniya, 26:3 (1988), 366–373

[4] Hunt J. W., Ray J. C., Flexible booms, momentum wheels, and subtle gravity-gradient instabilities, AIAA Papers, 92-1673, 1992, 9 pp.

[5] Beletskii V. V., Chaikin S. V., “Uchet peremescheniya tsentra mass girostata s uprugim sterzhnem pri analize ustoichivosti semeistva ego ravnovesii”, Vestn. MGU. Ser. 1. Matematika. Mekhanika, 2006, no. 1, 42–47 | MR | Zbl

[6] Lebedev N. N., Skalskaya I. P., Uflyand Ya. S., Sbornik zadach po matematicheskoi fizike, Gostekhizdat, M., 1955

[7] Meirovitch L., “Liapunov stability fnalysis of hybrid dynamical systems in the neighborhood of nontrivial equilibrium”, AIAA J., 12:7 (1974), 889–898 | DOI | MR | Zbl

[8] Chaikin S. V., “Ustoichivost netrivialnykh otnositelnykh ravnovesii girostata s uprugim sterzhnem s massoi na kontse”, Mekhanika tverdogo tela, 2005, no. 35, 189–198 | MR

[9] Chaikin S. V., “Ustoichivost semeistva netrivialnykh ravnovesnykh orientatsii na prityagivayuschii tsentr girostata s uprugim sterzhnem”, Prikl. matematika i mekhanika, 68:6 (2004), 971–983 | MR | Zbl

[10] Nabiullin M. K., Statsionarnye dvizheniya i ustoichivost uprugikh sputnikov, Nauka, Novosibirsk, 1990 | MR

[11] Routh E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, Macmillan, London, 1884 | MR

[12] Vilke V. G., Analiticheskie i kachestvennye metody mekhaniki sistem s beskonechnym chislom stepenei svobody, Izd-vo MGU, M., 1986 | MR

[13] Korn G., Korn T., Spravochnik po matematike, Nauka 1977, M.