On the stability of nonlinear angular oscillations of a mechanical analogue of the motions of a rigid body with two fluids


Аuthors

Win K. 1, 2*, Temnov A. N.1**

1. Baumann Moscow State Technical University, 105005, Moscow, 2nd Baumanskaya St., b. 5, c. 1
2. Peter the Great St. Petersburg Polytechnic University, 29, Polytechnicheskaya str., St. Petersburg, 195251, Russia

*e-mail: win.c.latt@gmail.com
**e-mail: antt45@mail.ru

Abstract

The paper studies a mechanical system with a spherical pendulum that models nonlinear oscillations of the interface between two liquids that completely fill a movable cylindrical container. It is demonstrated that if the direction cosines that define the position of the spherical pendulum are chosen as generalized coordinates, the equations of motion of the equivalent mechanical system exactly correspond to the equations of the dynamics of a rigid body with two liquids in the approximation up to the second order of smallness. To analyze higher orders of smallness and compare the behavior of the mechanical model with the real system, it is necessary to involve amplitude-frequency dependences. The paper also derives numerical calculations of the linear and nonlinear coefficients of the equations of motion for different filling depths of each liquid. As a result, the amplitude-frequency characteristics and instability regions of forced angular oscillations of the interface between liquids in a cylindrical vessel and a mechanical model with a spherical pendulum corresponding to the liquid levels are constructed.

Keywords:

instability regions, parametric resonance, spherical bearing, pendulum, direction cosines, nonlinear coefficients, amplitude-frequency characteristics

References

  1. Mikishev. G. N., Rabinovich B. I. Dynamics of a rigid body with cavities partially filled with liquid. - M.: Mashinostroenie, 1968. 532 p.
  2. Bauer H. F. Nonlinear mechanical model for the description of propellant sloching.-“ AIAA Journ.”, 1966, vol. 4, N 9, p. 1662-1668
  3. Lukovsky I.A. On the study of the motion of a solid body with a liquid performing nonlinear oscillations. - "Applied Mechanics", 1967, v. 3, issue 6, pp. 119-127.
  4. Lukovskiy I.A. Introduction to nonlinear dynamics of a rigid body with cavities containing liquid; Ed. V.A. Trotsenko; Academy of Sciences of the Ukrainian SSR. Institute of Mathematics. – Kyiv: Nauk. Dumka, 1990. 296 p. – ISBN 5-12-001308-2.
  5. Narimanov G.S., Dokuchaev L.V., Lukovsky I.A. Nonlinear dynamics of an aircraft with liquid. Moscow, "Mashinostroenie", 1977. 208 p.
  6. Stolbetsov V.I., Fishkis V.M. On one mechanical model of a liquid performing considerable oscillations in a spherical cavity. - "Bulletin of the USSR Academy of Sciences, Mechanics of Liquids and Gases", 1968, No. 5, pp. 119-123.
  7. Moiseev G.A. Some issues of delinearization in the dynamics of complex oscillatory systems. - "Applied Mechanics", 1972, v. 8, issue. 11, pp. 88-96.
  8. Moiseev N.N., Rumyantsev V.V. Dynamics of a body with cavities containing liquid. M. Publishing House "Nauka", 1965, 441 p.
  9. Win Ko Ko, Temnov A.N. Theoretical investigation of the effects of vibrations of two immiscible liquids in a limited volume // Bulletin of Tomsk State University. Mathematics and mechanics. 2021, no. 69. DOI: 10.17223/19988621/69/8
  10. Win Ko Ko, Temnov A.N. Angular oscillations of a rigid body with a two-layer liquid near the main resonance // Proceedings of MAI. 2021. No. 119. DOI: 10.34759/trd-2021-119-03 
  11. Win Ko Ko, Temnov A. N. Oscillations of discretely stratified fluids in a cylindrical vessel and their mechanical analogues. Bulletin of Bauman Moscow State Technical University. Series “Natural Sciences”. No. 3. Pp. 57-69.
  12. Win Ko Ko, Temnov A. N. Amplitude-frequency characteristics and stability regions of a two-layer fluid under angular oscillations of a rigid body // Applied Mathematics and Mechanics. - 2023. - Vol. 87. - No. 6. - Pp. 995-1005. doi: 10.31857/S0032823523060103
  13. Win Ko Ko Oscillations of a multilayer fluid in cavities of stationary and moving bodies: dis. cand. Phys.-Math. Sciences: D 002.240.01. - Institute of Problems in Mechanics of the Russian Academy of Sciences, Moscow, 2018-157 p.
  14. Win Ko Ko, Temnov A.N. On the stability of nonlinear motions of a mechanical model of a body with two fluids // Proceedings of MAI. 2024. No. 139. URL: https://trudymai.ru/published.php?ID=183456
  15. Dokuchaev L. V. Nonlinear dynamics of an aircraft with deformable elements – M.: Mechanical Engineering. 1987. – 231 p.
  16. Limarchenko O.S. Nonlinear problems of fluid dynamics in non-cylindrical reservoirs. Kyiv, Adverta, 2017, 130 p.
  17. Zaika V.V., Maslennikov A.L. Synthesis of a control system for a spherical pendulum by compensating for nonlinearities // Fundamental foundations of mechanics. – 2019. – №4
  18. Liska R. Nonhydrostatic two-layer models of incompressible flow. Computers & Mathematics with Applications, 1995, vol. 29, no. 9, pp. 25–37.
  19. Choi W., Camassa R. Fully nonlinear internal waves in a two-fluid system. Journal of Fluid Mechanics, 1999, no. 396, pp. 1–36.
  20. Barannyk L.L., Papageorgiou D.T. Fully nonlinear gravity-capillary solitary waves in a two-fluid system of finite depth. Journal of Engineering Mathematics, 2002, vol. 42, pp. 321–339.
  21. Rocca M. La, Sciortino G., Adduce C., Boniforti M.A. Interfacial gravity waves in a two-fluid system. Fluid Dynamics Research, 2002, no. 30, pp. 31–66.
  22. Rocca M. La, Sciortino G., Adduce C., Boniforti M.A. Experimental and theoretical investigation on the sloshing of a two-liquid system with free surface. Physics of Fluids, 2005, no. 17, paper no. 062101.
  23. Camassa R., Hurley M.W., McLaughlin R.M., Passaggia P.-Y., Thomson C.F.C. Experimental investigation of nonlinear internal waves in deep water with miscible fluids. Journal of Ocean Engineering and Marine Energy, 2018, vol. 4, pp. 243–257.
  24. Blinkova A. Yu., Ivanov S. V., Kuznetsova E. L., Mogilevich L. I. Nonlinear waves in a viscoelastic cylindrical shell containing a viscous incompressible fluid and surrounded by an elastic medium // Electronic journal “Proceedings of MAI”. Volume №78.
  25. Grishanina T.V., Shklyarchuk F.N. Application of the compartment method to the calculation of oscillations of liquid-propellant launch vehicles. -MAI, 2017. 100 p.
  26. Pozhalostin A.A., Goncharov D.A. On parametric axisymmetric oscillations of a liquid in a cylindrical vessel // Electronic journal "Proceedings of the MAI". Volume №95.
  27. Pak Songi, Grigoriev V.G. Stability of thin-walled axisymmetric coaxial structures containing liquid under multifactor loads// Proceedings of MAI. Volume № 119. DOI: https://doi.org/10.34759/trd-2021-119-08

mai.ru — informational site MAI

Copyright © 2000-2026 by MAI

 Вход