On sticking zone of a box with internal oscillator on a horizontal plane


Аuthors

Kovalev N. V.*, Baikov A. E.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: nick_scr@mail.ru
**e-mail: alexander@baikov.org

Abstract

Problems of the theory of mechanical systems with dry friction are popular at present among researches and find many applications while studying, for example, motion of mechanisms consisting of a solid body and moving masses in it. Discontinuity of the right hand sides of the equations of motion creates many problems while studying, and classical theory of differential equations does not work for the equations of movement with dry friction.

The first paragraph of the article considers very simple and popular model of the system with dry friction, i.e. a box clenched by springs on the conveyor belt. A dry friction force, subordinate to the Amonton’s-Coulomb law, acted between the box and the belt. A sticking zone was plotted, and periodic movements and limit cycle of the box on the conveyor belt were obtained.

The article regards translational movement of a box with internal oscillator along the horizontal plane. An Amonton’s-Coulomb dry force acted between the box and the plane. Equations of movement of the box with internal oscillator were obtained it the second paragraph of the article. These equations concede general solution in every domain of continuity. The slipping zone conditions when the static friction forces act on the box were obtained.

The third paragraph of the article validates mathematical correctness of the model under consideration. This correctness is based on the concept of Philipov’s solution of the ODE system with discontinuity of the right hand sides. The model determinateness follows out from Filipov’s theorem on existence and uniqueness-in-the-future solution of Cauchy problem.

The final motions of the system are studied in the last paragraph. The box always falls into the sticking zone after several iterations.

Keywords:

dry friction, Amonton’s-Coulomb law, sticking zone, Filippov solution, ultimate motions

References

  1. Ramsey G and Rahnejat H. Fundamentals of tribology, London, Imperial College Press, 2008, 391 p.

  2. Chernous'ko F.L. Doklady RAN, 2005, no. 1, pp. 56 - 60.

  3. Chernous’ko F.L. Analysis and Optimization of the Motion of a Body Controlled by Means of Movable Internal Mass, Journal of Applied Mathematics and Mechanics, 2006, vol. 70, no. 6, pp. 819 - 842.

  4. Chernous'ko F.L. Doklady Akademii nauk, 2016, vol. 470, no. 4, pp. 406 – 410.

  5. Bardin B.S., Panev A.S. Trudy MAI, 2015, no. 84, available at: http://trudymai.ru/eng/published.php?ID=62995

  6. Bardin B.S. and Panev A.S. On Dynamics of a Rigid Body Moving on a Horizontal Plane by Means of Motion of an Internal Particle, Vibroengineering Procedia, 2016, vol. 8, pp. 135 - 141.

  7. Panev A.S. Trudy MAI, 2018, no. 98, available at: http://trudymai.ru/eng/published.php?ID=90072

  8. Bardin B.S., Panev A.S. On the Motion of a Rigid Body with an Internal Moving Point Mass on a Horizontal Plane, AIP Conference Proceedings, 2018, vol. 1959, no. 1. DOI: 10.1063/1.5034582

  9. Bardin B.S., Panev A.S. On the Motion of a Body with a Moving internal Mass on a Rough Horizontal Plane, Rus. Journal Nonlinear Dynamics, 2018, vol. 14, no. 4, pp. 519 - 542. DOI: https://doi.org/10.20537/nd180407

  10. Sobolev N.A., Sorokin K.S. Experimental Investigation of a Model of a Vibration-Driven Robot with Rotating Masses, Journal of Computer and Systems Sciences International, 2007, vol. 46, no. 5, pp. 826 - 835.

  11. Bolotnik N.N., Figurina T.Yu. Optimal Control of the Rectilinear Motion of a Rigid Body on a Rough Plane by Means of the Motion of Two Internal Masses, Journal of Applied Mathematics and Mechanics, 2008, vol. 72, no. 2, pp. 126 - 135.

  12. Volkova L.Yu., Yatsun S.F. Simulation of the Plane Controlled Motion of a Three-Mass Vibration System, Journal of Computer and Systems Sciences International, 2012, vol. 51, no. 6, pp. 859 - 878.

  13. Xiong Zhan, Jian Xu. Locomotion analysis of a vibration-driven system with three acceleration-controlled internal masses, Advances in Mechanical Engineering, 2015, vol. 7, no. 3, available at: http://journals.sagepub.com/doi/pdf/10.1177/1687814015573766

  14. Ivanov A.P., Sakharov A.V. Nelineinaya dinamika, 2012, vol. 8, no. 4, pp. 763 - 772.

  15. Andronov A.A., Vitt A.A., Khaikin S.E. Teoriya kolebanii (Theory of Oscillations), Moscow, Nauka, 1981, 914 p.

  16. Ivanov A.P. Osnovy teorii sistem s treniem (Fundamentals of Theory of Systems with Friction), Moscow-Izhevsk, NITs “Regulyarnaya i khaoticheskaya dinamika”, Institut komp'yuternykh issledovanii, 2011, 304 p.

  17. Andronov V.V., Zhuravlev V.F. Sukhoe trenie v zadachakh mekhaniki - Moscow-Izhevsk: NITs “Regulyarnaya i khaoticheskaya dinamika”, Institut komp'yuternykh issledovanii, 2010, 184 p.

  18. Filippov A.F. Matematicheskii sbornik, 1960, vol. 51 (93), no. 1, pp. 99 - 128.

  19. Filippov A.F. Differential Equations with Discontinuous Righthand Side, Mathematics and Its Applications, 1988, no. 18, pp. 1 - 2.

  20. Kodington E.A., Levinson N. Teoriya obyknovennykh differentsial'nykh uravnenii (Theory of ordinary differential equations), Moscow, URSS, 2010, 472 p.


Download

mai.ru — informational site MAI

Copyright © 2000-2022 by MAI

Вход