Rotation around a fixed point of a solid body with an ellipsoidal cavity completely filled with an inhomogeneous fluid

DOI: 10.34759/trd-2023-128-06

Аuthors

Temnov A. N.^*, Yan N. O.^**

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: antt45@mail.ru
**e-mail: yno64528@gmail.com

Abstract

In this paper, the equations of spherical motion of a solid body with a rotating inhomogeneous incompressible fluid filling a completely ellipsoidal cavity are obtained and investigated. The stability of rotation of a solid with an inhomogeneous fluid having a linear density distribution is considered. The purpose of this article is to study the effect of an inhomogeneous fluid on the stability of rotation of a solid with a fluid around the axis of dynamic symmetry. In the formulation of the problem, a solid body with an ellipsoidal cavity rotates at an angular velocity around a fixed point that coincides with the geometric center of the cavity, and an inhomogeneous ideal fluid completely filling this cavity performs a homogeneous vortex motion in it with the angular velocity of the fluid. To derive the equation of motion of the system under consideration, we use the Euler-Lagrange equations, which, under the action of potential forces. To solve the problem of the stability of the system’s motion, we use the second Lyapunov method, and construct the Lyapunov function using the Chetaev method. Sufficient conditions for the stability of the rotation of a solid body with a fluid around the vertical axis of dynamic symmetry are derived. The condition is obtained in the form of inequalities of the roots of quadratic forms corresponding to the perturbed motion of a body with a fluid. The obtained equations of motion make it possible to study the stability of stationary motions of the system in question in order to assess the effect of fluid stratification on the dynamics of the body.

Keywords:

inhomogeneous liquid, solid body, ellipsoidal cavity, uniform vortex motion, stability

References

1. Zhukovskii N.E. O dvizheniya tverdogo tela, imeyushchego polosti, napolnennye odnorodnoi kapel’noi zhidkost’yu (Dynamics of a rigid body with cavities partially filled with liquid), Moscow, Izd-vo MGTU im. N.E. Baumana, 2017, 137 p.
2. Th. Sloudsky. De la rotation de la terre supposee fluide a son interieur, Bulletin de la Societe Imperiale des naturalistes de Moscou, 1895, vol. IX, pp. 285–318.
3. Hough S.S. The oscillations of a rota ting ellipsoidal shell containing fluid, Philosophical Transactions of the Royal Society of London, A, 1895, vol. 186, part 1, pp. 469–506.
4. Poincare H. Sur la precession des corps deformables, Bulletin astronomique, 1910, vol. 27, pp. 321–356.
5. Sudenkov S.N. Kanonicheskie uravneniya dvizheniya tverdogo tela s vikhrevym zapolneniem (Canonical equations of motion of a rigid body with vortex filling), Donetsk, Institut prikladnoi matematiki i mekhaniki AN USSR, 1977, pp. 67–71.
6. Savchenko A.Ya., Ignatov A.L. Prikladnaya matematika i mekhanika, 1974, vol. 10, no. 8, pp. 107-111.
7. Ignat’ev A.O. Mekhanika tverdogo tela, 1977, no. 9, pp. 82–86.
8. Pozdnyakovich E.V. Mekhanika tverdogo tela, 1978, no. 10, pp. 49-54.
9. Ai Min Vin, Temnov A.N. Trudy MAI, 2015, no. 79. URL: http://trudymai.ru/eng/published.php?ID=55633
10. Pozhalostin A.A., Goncharov D.A. Trudy MAI, 2017, no. 95. URL: https://trudymai.ru/eng/published.php?ID=84412
11. Pak Songi, Grigor’ev V.G. Trudy MAI, 2021, no. 119. URL: https://trudymai.ru/eng/published.php?ID=159785. DOI: 34759/trd-2021-119-08
12. Dolzhanskii F.V. Izvestiya AN SSSR. Fizika atmosfery i okeana, 1977, no. 2, pp. 201-203.
13. Temnov A.N. Trudy MVTU im. N.E. Baumana, 1979, vol. 293, pp. 50–57.
14. Temnov A.N., Yan Naing Oo. Inzhenernyi zhurnal: nauka i innovatsii, 2022, no. 7, DOI: 10.18698/2308-6033-2022-7-2192
15. Temnov A.N. Izvestiya vuzov. Mashinostroenie, 1979, no. 7, pp. 149–151.
16. Yan Naing Oo. III mezhvuzovskaya konferentsiya aspirantov, soiskatelei i molodykh uchenykh «Nauka, tekhnologii i biznes»: sbornik trudov, Moscow, Izd-vo MGTU im. N.E. Baumana, 2021, pp. 209–222.
17. Mel’khior P. Zemnye prilivy (Earth Tides), Moscow, Mir, 1968, 482 p.
18. Lur’e A.I. Analiticheskaya mekhanika (Analytical mechanics), Moscow, Fizmatgiz, 1961, 824 p.
19. Moiseev N.N., Rumyantsev V.V. Dinamika tela s polostyami, soderzhashchimi zhidkost’ (Dynamics of a body with cavities containing liquid Moscow), Nauka, 1965, 439 p.
20. Merkin D.R. Vvedenie v teoriyu ustoichivosti dvizheniya (Introduction to the theory of motion stability), Moscow, Nauka, 1976, 320 p.

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