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On the Motion of a Point Particle on a Homogeneous Gravitating Ball with a Spherical Cavity in the Presence of Dry Friction

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Abstract—

The problem of motion of a point particle on the surface of a homogeneous gravitating ball with a spherical cavity is considered. It is assumed that the body rotates uniformly around its axis of symmetry. At the same time, it is assumed that a particle located on the outer or inner (inside the cavity) surface of the body, in addition to the force of gravity, is affected by dry friction. The gravitational properties inside the cavity and outside the ball are described. The dependence of the existence, bifurcations, and stability of the relative equilibria of a point particle on the outer or inner body surface on the parameters of the problem is studied. The results obtained both analytically and numerically are presented in the form of bifurcation diagrams. The research is motivated by the possible existence of cavities or mass concentrations (mascons) in large and small celestial bodies.

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REFERENCES

  1. V. Guibout and D. J. Scheeres, “Stability of surface motion on a rotating ellipsoid,” Celestial Mech. Dyn. Astron. 87, 263–290 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  2. J. Bellerose, A. Girard, and D. J. Scheeres, “Dynamics and control of surface exploration. robots on asteroids,” in Optimization & Cooperative Control Strategies. LNCIS 381, Ed. by M. J. Hirsch, (Springer, Berlin, 2009), pp. 135–150.

    MATH  Google Scholar 

  3. Y. Jiang, Y. Zhang, and H. Baoyin, “Surface motion relative to the irregular celestial bodies,” Planet. Space Sci. 127, 33–43 (2016).

    Article  ADS  Google Scholar 

  4. S. Tardivel, D. J. Scheeres, P. Michel, et al., “Contact motion on surface of asteroid,” J. Spacecraft Rockets 51 (6), 1857–1871 (2015).

    Article  ADS  Google Scholar 

  5. Y. Zhang, X. Zeng, C. Circi, et al., “The motion of surface particles for the asteroid 101955 Bennu,” Acta Astron. 163, 3–10 (2019).

    Article  Google Scholar 

  6. N. Van de Wouw and R. I. Leine, “Stability of stationary sets in nonlinear systems with set-valued friction,” in Proc. 45th IEEE Conf. Decision and Control (San Diego, 2006), pp. 4271–4276.

  7. R. I. Leine and N. van de Wouw, “Stability properties of equilibrium wets of nonlinear mechanical systems with dry friction and impact,” Nonlinear Dyn. 51 (4), 551–583 (2008).

    Article  Google Scholar 

  8. R. I. Leine and N. van de Wouw, Stability and Convergence of Mechanical Systems with Unilateral Constraints (Springer, Berlin, 2008), Vol. 36.

    Book  Google Scholar 

  9. R. I. Leine and D. H. van Campen, “Bifurcation phenomena in non-smooth dynamical systems,” Eur. J. Mech. A. Solids 25, 595–616 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  10. R. I. Leine, “Bifurcations of equilibria in non-smooth continuous systems,” Phys. D (Amsterdam) 223, 121–137 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  11. A. Ivanov, “Bifurcations in systems with friction: basic models and methods,” Regular Chaotic Dyn. 14 (6), 656–672 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  12. A. P. Ivanov, Fundamentals of the Theory of Systems with Friction (Regular&Chaotic Dynamics, Moscow, 2011) [in Russian].

    Google Scholar 

  13. A. A. Burov and E. S. Shalimova, “On the motion of a heavy material point on a rotating sphere (dry friction case),” Regular Chaotic Dyn. 20 (3), 225–233 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  14. A. A. Burov and E. S. Shalimova, “Bifurcations of relative equilibria of a heavy bead on a rotating parabolic bowl with dry friction,” Mech. Solids 51 (4), 395–305 (2016). https://doi.org/10.3103/S0025654416040038

    Article  ADS  Google Scholar 

  15. E. S. Shalimova, “On the motion of a material point on a rotating sphere with dry friction (the case of the vertical axis),” Nelineinaya Din. 12 (3), 369–383 (2016).

    Article  MathSciNet  Google Scholar 

  16. A. A. Burov, I. I. Kosenko, and E. S. Shalimova, “Relative equilibria of a massive point on a uniformly rotating asteroid,” Dokl. Phys. 62 (7), 359–362 (2017).

    Article  ADS  Google Scholar 

  17. V. V. Beletsky, “Generalized restricted circular three-body problem as a model for dynamics of binary asteroids,” Cosmic Res. 45 (6), 408–416 (2007).

    Article  ADS  Google Scholar 

  18. V. V. Beletsky and A. V. Rodnikov, “Stability of triangle libration points in generalized restricted circular three-body problem,” Cosmic Res. 46 (1), 40–48 (2008).

    Article  ADS  Google Scholar 

  19. M. A. Munitsyna, “Relative equilibria of a point in a gravity field of a symmetrical rigid body,” in Problems of Investigating the Stability and Stabilization of Motion (Dorodnicyn Computing Centre, RAS, Moscow, 2009), pp. 14–19 [in Russian].

  20. N. I. Koshkin and M. G. Shirkevich, Handbook of Elementary Physics (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

  21. O. Lahayne, B. Pichler, R. Reihsner, et al., “Rubber friction on ice: experiments and modeling,” Tribol. Lett. 62, Art. No. 17, 1–19 (2016).

    Google Scholar 

  22. A. Burov, “Small letter σ and problems with absolute values,” Kvant, No. 1, 36–38 (2012).

    Google Scholar 

  23. L. E. Gendenstein, L. A. Kirik, and I. M. Gel’fgat, Solving Key Problems in Physics for Primary School. Grades 7–9 (Ileksa, Moscow, 2016).

    Google Scholar 

  24. A. A. Burov, “On bifurcations of relative equilibria of a heavy bead sliding with dry friction on a rotating circle,” Acta Mech. 212 (3-4), 349–354 (2010).

    Article  Google Scholar 

  25. G. K. Pozharitskii, “Stability of equilibria for the systems with dry friction,” Prikl. Mat. Mekh. 26 (1), 5–14 (1962).

    Google Scholar 

  26. A. A. Burov and I. A. Yakushev, “Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction,” J. Appl. Math. Mech. 78 (5), 460–467 (2014).

    Article  MathSciNet  Google Scholar 

  27. Automatic Interplanetary Stations “Phobos-1, 2,” Lavochkin Scientific and Production Association. https://www.laspace.ru/projects/planets/fobos/.

  28. V. V. Beletsky and O. P. Salimova, “Hill’s problem as a dynamical billiard,” Regular Chaotic Dyn. 1 (2), 47–58 (1996).

    Google Scholar 

  29. V. V. Beletsky and O. P. Salimova, “Hill’s problem as a dynamical billiard,” in Proc. IUTAM Symp. on Interaction between Dynamics and Control in Advanced Mechanical Systems (Springer,  New York, 1997), pp. 27–34.

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Funding

This research is partially supported by the program of the President of the Russian Federation for the federal support of young Russian scientists, candidates of science (project no. MK-1712.2019.1) and by the Russian Foundation for Basic Research (project no. 18-01-00335).

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Authors and Affiliations

  1. Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, Moscow, Russia

    A. A. Burov & V. I. Nikonov

  2. National Research University Higher School of Economics, Moscow, Russia

    A. A. Burov & V. I. Nikonov

  3. Moscow State University, Moscow, Russia

    E. S. Shalimova

Authors
  1. A. A. Burov
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  2. V. I. Nikonov
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  3. E. S. Shalimova
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Corresponding authors

Correspondence to A. A. Burov, V. I. Nikonov or E. S. Shalimova.

Additional information

Translated by N. Petrov

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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. Solids 56, 1587–1598 (2021). https://doi.org/10.3103/S0025654421080045

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