Section
|
Mechanics
|
Title
|
New classes of particular solutions to one problem on gyrostat motion
|
Author(-s)
|
Zyza A.V.a,
Khomyak T.V.b,
Platonova E.S.a
|
Affiliations
|
Donetsk National Universitya,
National Technical University Dnipro Polytechnicb
|
Abstract
|
The paper studies the existence of two new classes of polynomial solutions to differential equations related to the problem of the gyrostat motion with a fixed point in the magnetic field, taking into account the Barnett–London effect. A common feature of the structure of these classes is that the functions that set the invariance relations for the unit vector components of the symmetry axis of the active force fields are either rational functions of the first component of the specified vector or of the auxiliary variable. Three new particular solutions to the polynomial classes under consideration are constructed. These solutions are described by the functions obtained by the inversion of hyperelliptic integrals. It has been proved that another constructed solution of the polynomial structures under study, for which the movement of the gyrostat has the property of precession, is a particular case of a known solution.
|
Keywords
|
Kirchhoff–Poisson equations, Barnett–London effect, gyrostat, polynomial solution, invariance relation
|
UDC
|
531.38
|
MSC
|
70E05, 70E17, 70E40
|
DOI
|
10.35634/vm220209
|
Received
|
2 March 2022
|
Language
|
Russian
|
Citation
|
Zyza A.V., Khomyak T.V., Platonova E.S. New classes of particular solutions to one problem on gyrostat motion, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 2, pp. 298-318.
|
References
|
- Kharlamov P.V. Lektsii po dinamike tverdogo tela (Lectures on solid dynamics), Novosibirsk: Novosibirsk State University, 1965.
- Gashenenko I.N., Gorr G.V., Kovalev A.M. Klassicheskie zadachi dinamiki tverdogo tela (Classical problems of solid dynamics), Kiev: Naukova Dumka, 2012.
- Gorr G.V., Maznev A.V. Dinamika girostata, imeyushchego nepodvizhnuyu tochku (Dynamics of a gyrostat with a fixed point), Donetsk: Donetsk National University, 2010.
- Thomson W. On the motion of rigid solids in a liquid circulating irrotationally through perforations in them or in any fixed solid, Proceedings of the Royal Society of Edinburgh, 1872, vol. 7, pp. 668-682. https://doi.org/10.1017/S0370164600042875
- Volterra V. Sur la théorie des variations des latitudes, Acta Mathematica, 1899, vol. 22, pp. 201-357. https://doi.org/10.1007/bf02417877
- 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: Gostekhteorizdat, 1949, pp. 31-152 (in Russian).
- Gray A. A treatise on gurostatics and rotational motion, New York: Dover, 1959.
- Levi-Civita T., Amaldi U. Lezioni di meccanica razionale. Volume secondo. Dinamica dei sistemi con un numero finito di gradi di libertá. Parte Seconda, Bologna: Nicola Zanichelli, 1927.
- Rumyantsev V.V. On orientation control and satellite stabilization by rotors, Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika, 1970, no. 2, pp. 83-96 (in Russian). https://zbmath.org/?q=an:0207.23402
- Kharlamov P.V. On the equations of motion of a system of solids, Mekhanika Tverdogo Tela. Issue 4. Respublikanskii mezhvedomstvennyi sbornik, Kiev: Naukova dumka, 1972, pp. 52-73 (in Russian).
- Kharlamov P.V. On the invariant relations of a system of differential equations, Mekhanika Tverdogo Tela. Issue 6. Respublikanskii mezhvedomstvennyi sbornik, Kiev: Naukova dumka, 1974, pp. 15-24 (in Russian).
- 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
- 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 (in Russian). https://doi.org/10.35634/vm210108
- 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, p. 327-342. https://doi.org/10.20537/nd190310
- Ol'shanskii V.Yu. Linear invariant relations of Kirchhoff's equations, Journal of Applied Mathematics and Mechanics, 2015, vol. 79, issue 4, pp. 334-349. https://doi.org/10.1016/j.jappmathmech.2016.01.004
- Ol'shanskii V.Yu. Construction of linear invariant relations of Kirchhoff equations, Mechanics of Solids, 2019, vol. 54, no. 1, pp. 70-80. https://doi.org/10.3103/S0025654419010060
- Doroshin A.V. Analytical solutions for dynamics of dual-spin spacecraft and gyrostat-satellites under magnetic attitude control in omega-regimes, International Journal of Non-Linear Mechanics, 2017, vol. 96, pp. 64-74. https://doi.org/10.1016/j.ijnonlinmec.2017.08.004
- Doroshin A.V. Regimes of regular and chaotic motion of gyrostats in the central gravity field, Communications in Nonlinear Science and Numerical Simulation, 2019, vol. 69, pp. 416-431. https://doi.org/10.1016/j.cnsns.2018.10.004
- Aslanov V.S. A note on the “Exact solutions for angular motion of coaxial bodies and attitude dynamics of gyrostat-satellites”, International Journal of Non-Linear Mechanics, 2014, vol. 58, pp. 305-306. https://doi.org/10.1016/j.ijnonlinmec.2013.10.007
- Yehia H.M., El-kenani H.N. Effect of the gravity and magnetic field to find regular precessions of a satellite-gyrostat with principal axes on a circular orbit, Journal of Applied and Computational Mechanics, 2021, vol. 7, issue 4, pp. 2120-2128. https://doi.org/10.22055/jacm.2021.37913.3113
- El-Sabaa F.M., Amer T.S., Sallam A.A., Abady I.M. Modeling and analysis of the nonlinear rotatory motion of an electromagnetic gyrostat, Alexandria Engineering Journal, 2021, vol. 61, issue 2, pp. 1625-1641. https://doi.org/10.1016/j.aej.2021.06.066
- Yehia H.M. New solvable problems in the dynamics of a rigid body about a fixed point in a potential field, Mechanics Research Communications, 2014, vol. 57, pp. 44-48. https://doi.org/10.1016/j.mechrescom.2014.02.005
- Yehia H.M., Saleh E., Megahid S.F. New solutions of classical problems in rigid body dynamics, Mechanics Research Communications, 2015, vol. 69, pp. 40-44. https://doi.org/10.1016/j.mechrescom.2015.05.007
- Yehia H.M., Elmandouh A.A. A new conditional integrable case in the dynamics of a rigid body-gyrostat, Mechanics Research Communications, 2016, vol. 78, part A, pp. 25-27. https://doi.org/10.1016/j.mechrescom.2016.09.007
- Amer T.S., Amer W.S. Substantial condition for the fourth first integral of the rigid body problem, Mathematics and Mechanics of Solids, 2018, vol. 23, issue 8, pp. 1237-1246. https://doi.org/10.1177/1081286517716733
- Amer T.S., Farag A.M., Amer W.S. The dynamical motion of a rigid body for the case of ellipsoid inertia close to ellipsoid of rotation, Mechanics Research Communications, 2020, vol. 108, 103583. https://doi.org/10.1016/j.mechrescom.2020.103583
- Doroshin A.V. Exact solutions for angular motion of coaxial bodies and attitude dynamics of gyrostat-satellites, International Journal of Non-Linear Mechanics, 2013, vol. 50, pp. 68-74. https://doi.org/10.1016/j.ijnonlinmec.2012.10.012
- Aslanov V., Yudintsev V. Dynamics and chaos control of gyrostat satellite, Chaos, Solitons and Fractals, 2012, vol. 45, issues 9-10, pp. 1100-1107. https://doi.org/10.1016/j.chaos.2012.06.008
- Barnett S.J. Gyromagnetic and electron-inertia effects, Reviews of Modern Physics, 1935, vol. 7, issue 2, pp. 129-166. https://doi.org/10.1103/revmodphys.7.129
- London F. Superfluids. Vol. 1. Macroscopic theory of superconductivity, New York: Wiley, 1950.
- Kittel Ch. Introduction to solid state physics, New York: Wiley, 2005.
- Ziglin S.L. Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I, Functional Analysis and Its Applications, 1982, vol. 16, issue 3, pp. 181-189. https://doi.org/10.1007/BF01081586
- Samsonov V.A. On the rotation of a solid in a magnetic field, Izvestiya Akademii Nauk SSSR. Mekhanika Tverdogo Tela, 1984, no. 6, pp. 32-34 (in Russian).
- Kozlov V.V. To the problem of rotation of a solid body in a magnetic field, Izvestiya Akademii Nauk SSSR. Mekhanika Tverdogo Tela, 1985, no. 6, pp. 28-33 (in Russian).
- Kharlamov P.V. The current state and prospects for the development of the classical problems of solid state dynamics, Mekhanika Tverdogo Tela, 2000, issue 30, pp. 1-12 (in Russian).
- Klen F., Sommerfeld A. Über die Theorie des Kreisels. Heft 4, Leipzig: Teubner, 1965.
- Zyza A.V. Computer studies of polynomial solutions for gyrostat dynamics, Komp'yuternye Issledovaniya i Modelirovanie, 2018, vol. 10, no. 1, pp. 7-25 (in Russian). https://doi.org/10.20537/2076-7633-2018-10-1-7-25
- Zyza A.V. On generalized N. Kovalevski equations in two problems of rigid body dynamics, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 1, pp. 73-83 (in Russian). https://doi.org/10.20537/vm190107
- Zyza A.V., Khomyak T.V. On one case of integrability of the motion equations of a rigid body in magnetic field, Vestnik Donetskogo Natsional'nogo Universiteta. Seriya A: Estestvennye Nayki, 2012, issue 2, pp. 31-35 (in Russian).
- Borisov A.V., Tsiganov A.V. The motion of a nonholonomic Chaplygin sphere in a magnetic field, the Grioli problem, and the Barnett-London effect, Doklady Physics, 2020, vol. 65, issue 3, pp. 90-93. https://doi.org/10.1134/s1028335820030052
|
Full text
|
|