The Problem of Rolling a Ball on a Surface of Revolution and its Numerical Investigationю

Аuthors

Kosenko I. I.^*, Kuleshov A. S.^**, Shishkov A. A.^***

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russia

*e-mail: kosenkoii@gmail.com
**e-mail: kuleshov@mech.math.msu.su
***e-mail: shish-tula@yandex.ru

Abstract

When creating models of various technical devices and processes, mechanical systems with nonholonomic constraints are often used. As an example, the classical work of M.V. Keldysh [1] is worth mentioning, in which the nonholonomic mechanical model that described the shimmy effect, i.e. the phenomenon of intense angular self-oscillations of the aircraft front wheels [2] was proposed. That is why further studies of the classical problems of nonholonomic mechanics, which now can be studied with the state-of-the-art mathematical, symbolic and numerical methods, present interest.

One of these classical problems of nonholonomic dynamics is the problem of a heavy homogeneous ball rolling without sliding along the perfectly rough surface of revolution. This problem has been considered at the end of the 19th - beginning of the 20th century in the works of E. J. Routh and F. Noether [3, 4]. Moreover, in these works, the surface on which the geometric center of the ball is located during movement, rather than the supporting surface along which the ball is rolling, was considered given. The article demonstrates how the ball motion equations can be reduced to the system of equations written in Cauchy form, i.e. reduce the problem of the ball motion to the Cauchy problem solving by defining the surface, along which the ball is moving, in an explicit form. Coefficients of the corresponding equations will depend on characteristics of the surface, along which the center of the ball moves, its principal curvatures and the Lame coefficients. If the equations of motion of a certain mechanical system are written in Cauchy form, then it is convenient to perform numerical analysis of the system under study. Thus, numerical analysis of the equations of motion of a rolling ball along the surface of revolution was performed by the MAPLE mathematical software. The numerically obtained results confirm the analytically proven statements about the ball motion.

Keywords:

Nonholonomic System, Homogeneous Ball, Surface of Revolution

References

1. Keldysh M.V. Shimmi perednego kolesa trekhkolesnogo shassi (Shimmy front wheel tricycle undercarriage), Moscow, Byuro novoi tekhniki NKAP, 1945, 34 p.

2. Zagordan A.A. Trudy MAI, 2011, no. 47. URL: https://trudymai.ru/eng/published.php?ID=26675

3. Routh E.J. The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject. Cambridge: Cambridge University Press, 2013.

4. Noether F. Uber rollende Bewegung einer Kugel auf Rotationsflachen, Leipzig, Teubner, 1909.

5. Kuleshov A.S., Solomina D.V. Problemy informatiki, 2021, no. 1, pp. 15–24.

6. Kuleshov A.S., Solomina D.V. Vestnik Sankt Peterburgskogo universiteta. Matematika. Mekhanika. Astronomiya, 2021, vol. 8, no. 4, pp. 653-660.

7. Kuleshov A.S., Solomina D.V. Application of the Kovacic Algorithm to the Problem of Rolling of a Heavy Homogeneous Ball on a Surface of Revolution, Communications in Computer and Information Science, 2021, vol. 1413, pp. 169–177. DOI: 10.1007/978-3-030-78759-2_15

8. Kuleshov A.S., Shishkov A.A. Trudy XXIII Mezhdunarodnoi konferentsii «Matematicheskoe modelirovanie i superkomp'yuternye tekhnologii», Nizhnii Novgorod, Izdatel'stvo Nizhegorodskogo universiteta, 2023, pp. 93-97.

9. Borisov A.V., Mamaev I.S., Kilin A.A. The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics, Regular & Chaotic Dynamics, 2002, vol. 7, issue 2, pp. 201-219. DOI: 10.1070/RD2002v007n02ABEH000205

10. Borisov A.V., Mamaev I.S., Bizyaev I.A. The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regular & Chaotic Dynamics, 2013, vol. 18, issue 3, pp. 277-328. DOI: 10.1134/S1560354713030064

11. Fasso F., Sansonetto N. Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints, Regular & Chaotic Dynamics, 2015, vol. 20, issue 4, pp. 449–462. DOI: 10.1134/S1560354715040048

12. Borisov A.V., Mamaev I.S. Symmetries and Reduction in Nonholonomic Mechanics, Regular & Chaotic Dynamics, 2015, vol. 20, issue 5, pp. 553-604. DOI: 10.1134/S1560354715050044

13. Fasso F., Sansonetto N. On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art, Regular & Chaotic Dynamics, 2022, vol. 27, issue 4, pp. 409-423. DOI: 10.1134/S1560354722040025

14. Kosenko I.I., Kuznesova L.V., Nikolaev A.V., Kuznetsov L.Yu., Oleinik A.V. Modelirovanie i virtual'noe prototipirovanie (Modeling and virtual prototyping: a tutorial), Moscow, INFRA-M, 2012, 176 p.

15. Kosenko I.I., Kuznetsova L.V., Nikolaev A.V., Kuznetsov L.Yu., Oleinik A.V. Proektirovanie i 3D – modelirovanie v sredakh CATIA V5, ANSYS i Dymola 7.3. (Projecting and 3D – Modeling in CATIA V5, ANSYS and Dymola 7.3 tools: a tutorial), Moscow, INFRA–M, 2020, 183 p.

16. Bardin B.S., Savin A.A. Trudy MAI, 2016, no. 85. URL: https://trudymai.ru/eng/published.php?ID=65212

17. Bardin B.S., Chekina E.A. Trudy MAI, 2016, no. 89. URL: https://trudymai.ru/eng/published.php?ID=72568

18. Safonov A.I. Trudy MAI, 2022, no. 126. URL: https://trudymai.ru/eng/published.php?ID=168988. DOI: 10.34759/trd-2022-126-02

19. Kholostova O.V., Safonov A.I. Trudy MAI, 2018, no. 100. URL: https://trudymai.ru/eng/published.php?ID=93292

20. Sokolov S.V. Trudy MAI, 2017, no. 95. URL: https://trudymai.ru/eng/published.php?ID=84387

21. Sokolov S.V. Trudy MAI, 2018, no. 100. URL: https://trudymai.ru/eng/published.php?ID=93532

22. Barkin M.Yu. Trudy MAI, 2014, no. 72. URL: https://trudymai.ru/eng/published.php?ID=47336

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