On the motion of a bead on a rough hoop freely rotating around a vertical diameter

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

We consider the problem of the motion of a heavy bead strung on a rough heavy hoop freely rotating around a vertical diameter. Non-isolated sets of steady state motions of the system are identified, and their bifurcation diagrams are constructed. The dependence of these solutions on an essential parameter of the problem—the constant of the cyclic integral—is studied. The results obtained are compared with the results obtained previously for the case when a rough hoop rotates around a vertical diameter with a constant angular velocity. Characteristic phase portraits are constructed for various combinations of system parameters.

Full Text

Restricted Access

About the authors

А. А. Burov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

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

V. I. Nikonov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Email: nikon_v@list.ru
Russian Federation, Moscow

Е. А. Nikonova

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences; Sirius University of Science and Technology Sirius Federal territory

Email: nikonova.ekaterina.a@gmail.com
Russian Federation, Moscow; Sochi

References

  1. Burov A.A. On bifurcations of relative equilibria of a heavy bead sliding with dry friction on a rotating circle // Acta Mechanica. 2010. V. 212. № 3–4. P. 349–354. https://doi.org/10.1007/s00707-009-0265-1
  2. Krementulo V.V. Stability of a gyroscope having a vertical axis of the outer ring with dry friction in the gimbal axes taken into account // J. Appl. Math. Mech. 1960. V. 24. № 3. P. 843–849.
  3. 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. P. 3765–3770.
  4. Leine R.I., van de Wouw N. Stability and convergence of mechanical systems with unilateral constraints. Lecture Notes in Applied and Computational Mechanics. Berlin: Springer, 2008. V. 36. 236 p.
  5. Leine R.I., van Campen D.H. Bifurcation phenomena in non-smooth dynamical systems // Eur. J. Mechanics A. Solids. 2006. V. 25. P. 595–616.
  6. Leine R.I. Bifurcations of equilibria in non-smooth continuous systems // Physica D. 2006. V. 223. P. 121–137.
  7. Ivanov A. Bifurcations in systems with friction: Basic models and methods // Regul. Chaotic Dyn. 2009. V. 14. № 6. P. 656–672.
  8. Ivanov A.P. Fundamentals of the theory of systems with friction. M.‒Izhevsk: SIC “Regular and chaotic dynamics”, Izhevsk Institute of Computer Science, 2011. 304 p. (in Russian).
  9. Karapetian A.V. Stability of stationary motions. M.: Editorial URSS, 1998. 168 p. (in Russian).
  10. Karapetian A.V. Stability and bifurcation of motions. M.: Publishing House of the Moscow University, 2020. 186 p. (in Russian).
  11. Chetaev N.G. Stability of motions. M.: Nauka. 1965. 176 p. (in Russian).
  12. Rumiantsev V.V. On the stability of steady motions // J. Appl. Math. Mech. 1966. V. 30. № 5. P. 1090–1103.
  13. Vozlinskii V.I. On the relations between the bifurcation of the equilibria of conservative systems and the stability distribution on the equilibria curve // J. Appl. Math. Mech. 1967. V. 31. № 2. P. 418–427.
  14. Vozlinskii V.I. On the stability of points of equilibrium branching // J. Appl. Math. Mech. 1978. V. 42. № 2. P. 270–279.
  15. Rubanovsky V.N., Samsonov V.A. Stability of stationary motions in examples and problems. M.: Nauka, 1988. 303 p. (in Russian).
  16. Burov A.A., Nikonov V.I. Motion of a heavy bead along a circular hoop rotating around an inclined axis // Int. J. Non-Linear Mech. 2021.V. 137. Article 103791. https://doi.org/10.1016/j.ijnonlinmec.2021.103791

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1. Bead on a hoop.

Download (6KB)
Indexing metadata
3. Fig. 2. Bifurcation diagrams in the absence of friction for different combinations of parameters: on the plane on the left; on the plane on the right.

Download (21KB)
Indexing metadata
4. Fig. 3. Regions of sign constancy of submodular expressions of inequality (3.4).

Download (10KB)
Indexing metadata
5. Fig. 4. Subregions of regions corresponding to solutions of inequality (3.4).

Download (38KB)
Indexing metadata
6. Fig. 5. Bifurcation diagram in the presence of friction for different combinations of parameters. Here , .

Download (29KB)
Indexing metadata
7. Fig. 6. Dependence on the friction coefficient: left, center, right.

Download (15KB)
Indexing metadata
8. Fig. 7. Phase portrait for pre-bifurcation combinations of parameters.

Download (48KB)
Indexing metadata
9. Fig. 8. Phase portrait for post-bifurcation combinations of parameters.

Download (38KB)
Indexing metadata
10. Fig. 9. Neighborhood of sets for post-bifurcation combinations of parameters.

Download (19KB)
Indexing metadata
11. Fig. 10. Neighborhood of sets for post-bifurcation combinations of parameters.

Download (20KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences