On the Motion of a Point Particle on a Homogeneous Gravitating Ball with a Spherical Inclusion

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

A problem of motion of a point particle on a surface of a homogeneous gravitating ball with a spherical inclusion of a differing density is considered. It is assumed that the body rotates uniformly around its symmetry axis. It is supposed that "besides the gravitation force, the particle is subjected to dry friction.

The gravitational properties outside the ball are described. The dependence of existence, bifurcations, and stability of relative equilibria of the point particle on the body surface on the parameters of the problem is studied. The results are represented both analytically and as numerically obtained bifurcation diagrams.

Full Text

Restricted Access

About the authors

A. A. Burov

FRC CSC RAS

Author for correspondence.
Email: jtm@narod.ru
Russian Federation, Moscow

V. I. Nikonov

FRC CSC RAS

Email: nikon_v@list.ru
Russian Federation, Moscow

E. S. Shalimova

Lomonosov Moscow State University

Email: ekateryna-shalimova@yandex.ru
Russian Federation, Moscow

References

  1. Guibout V., Scheeres D.J. Stability of surface motion on a rotating ellipsoid // Celest. Mech.&Dyn. Astron., 2003, vol. 87, pp. 263–290.
  2. Bellerose J., Girard A., Scheeres D.J. Dynamics and Control of Surface Exploration. Robots on Asteroids // in: Optimization & Cooperative Control Strategies. LNCIS381. / Ed. by Hirsch M.J. et al. Heidelberg: Springer, 2009. pp. 135–150.
  3. Jiang Y., Zhang Y, Baoyin H. Surface motion relative to the irregular celestial bodies // Planetary&Space Sci., 2016, vol. 127, pp. 33–43.
  4. Tardivel S., Scheeres D. J., Michel P. et al. Contact motion on surface of asteroid // J. Spacecraft&Rockets, 2015, vol. 51, no. 6, pp. 1857–1871.
  5. Zhang Y., Zeng X., Circi C. et al. The motion of surface particles for the asteroid 101955 Bennu // Acta Astronaut., 2019, vol. 163, pt. B, pp. 3–10.
  6. Ryabova, G.O. Asteroid 1620 Geographos: I. Rotation. // Solar System Res., 2002, vol. 36, pp. 168–174.
  7. Van de Wouw N., Leine R.I. Stability of stationary sets in nonlinear systems with set-valued friction // Proc. 45th IEEE Conf. Decision and Control and European Control Conf. (CDC2006), San Diego, USA, 2006, pp. 3765–3770.
  8. Leine R.I., van de Wouw N. Stability and convergence of mechanical systems with unilateral constraints // Lecture Notes in Appl.&Comput. Mech., 2008, vol. 36, 236 p.
  9. Leine R.I., van Campen D.H. Bifurcation phenomena in non-smooth dynamical systems // Europ. J. Mech. A/Solids, 2006, vol. 25, pp. 595–616.
  10. Leine R.I. Bifurcations of equilibria in non-smooth continuous systems // Physica D, 2006, vol. 223, pp. 121–137.
  11. Ivanov A. Bifurcations in systems with friction: Basic models and methods // R&C Dyn., 2009, vol. 14, no. 6, pp. 656–672.
  12. Ivanov A.P. Fundamentals of the Theory of Systems with Friction. Moscow; Izhevsk: Izhevsk Inst. of Comput. Sci., 2011, 304 p. (in Russian)
  13. Burov A.A., Yakushev I.A. Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction // JAMM, 2014, vol. 78, iss. 5, pp. 460–467.
  14. Burov A.A., Shalimova E.S. On the motion of a heavy material point on a rotating sphere (dry friction case) // R&C Dyn., 2015, vol. 20, no. 3, pp. 225–233.
  15. Burov A.A., Shalimova E.S. Bifurcations of relative equilibria of a heavy bead on a rotating parabolic bowl with dry friction // Mech. of Solids, 2016, vol. 51, no. 4, pp. 395–405.
  16. Shalimova E.S. On the motion of a material point on a rotating sphere with dry friction (the case of the vertical axis) // Nelin. Din., 2016, vol. 12, no. 3, pp. 369–383. (in Russian)
  17. Burov A.A., Kosenko I.I., Shalimova E.S. Relative equilibria of a massive point on a uniformly rotating asteroid // Dokl. Phys., 2017, vol. 62, no. 7, pp. 359–362.
  18. Burov A.A., Nikonov V.I., Shalimova E.S. On the motion of a point particle on a homogeneous gravitating ball with a spherical cavity in the presence of dry friction // Mech. of Solids, 2021, vol. 56, no. 8, pp. 1587–1598.
  19. Burov A.A., Nikonov V.I., Shalimova E.S. On relative equilibria on the surface of a spherical cavity inside a uniformly rotating gravitating sphere // Mech. of Solids, 2022, vol, 57, no. 8, pp. 1862–1872.
  20. Burov A.A., Nikonov V.I. Relative equilibria of a heavy point on a uniformly rotating inclined plane // Mech. of Solids, 2023, vol. 58, no. 1, pp. 131–139.
  21. Burov A.A., Nikonov V.I. On the relative equilibria of a heavy bead on a uniformly rotating rough spoke // Mech. of Solids, 2023, vol. 58, no. 3, pp. 748–753.
  22. Beletsky V.V. Generalized restricted circular three-body problem as a model for dynamics of binary asteroids // Cosmic Res., 2007, vol. 45, pp. 408–416.
  23. Beletsky V.V., Rodnikov A.V. On evolution of libration points similar to Eulerian in the model problem of the binary-asteroids dynamics // J. Vibroengng., 2008, vol. 10, no. 4, pp. 550–556.
  24. Beletsky V.V., Rodnikov A.V. Stability of triangle libration points in generalized restricted circular three-body problem // Cosmic Res., 2008, vol. 46, pp. 40–48.
  25. Munitsyna M.A. Relative equilibria of a point in a gravity field of a symmetrical solid body // in: Problems of Investigating the Stability and Stabilization of Motion. Moscow: Computing Center of the RAS, 2009. pp. 14–19. (in Russian)
  26. Beletsky V.V., Rodnikov A.V. Coplanar libration points in the generalized restricted circular problem of three bodies // Nelin. Din., 2011, vol. 7, no. 3, pp. 569–576. (in Russian)
  27. Burov A.A., Nikonov V.I. Libration points inside a spherical cavity of a uniformly rotating gravitating ball // Rus. J. of Nonlin. Dyn., 2021, vol. 17, no. 4, pp. 413–427.
  28. Lidov M.L., Gordeeva Y.F. Effect of mascons on prediction of the motion of artificial lunar satellites // Cosmic Res., 1974, vol. 12, pp. 449.
  29. Lidov M.L., Lyakhova V.A., Solov’ev A.A. Semi-analytical calculation of the motion of an artificial satellite of the Moon // Cosmic Res., 1975, vol. 13, pp. 249–274.
  30. Lidov M.L., Gordeeva Yu.F. Influence of “mascons” on the determination of the coefficients of the gravitational field of the moon // in: New Methods of Satellite Geodesy, 1977, pp. 564–586.
  31. Pozharitsky G.K. Stability of equilibria for the systems with dry friction // JAMM, 1962, vol. 26, iss. 1, pp. 5–14.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1

Download (115KB)
Indexing metadata
3. Fig. 2

Download (145KB)
Indexing metadata
4. Fig. 3

Download (110KB)
Indexing metadata
5. Fig. 4

Download (138KB)
Indexing metadata
6. Fig. 5

Download (195KB)
Indexing metadata
7. Fig. 6

Download (161KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences