 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| M.V. Ishkhanyan and A.V. Karapetyan, "Dynamics of a Homogeneous Ball on a Horizontal Plane with Sliding, Spinning, and Rolling Friction Taken into Account," Mech. Solids. 45 (2), 155-165 (2010) |
| Year |
2010 |
Volume |
45 |
Number |
2 |
Pages |
155-165 |
| DOI |
10.3103/S0025654410020019 |
| Title |
Dynamics of a Homogeneous Ball on a Horizontal Plane with Sliding, Spinning, and Rolling Friction Taken into Account |
| Author(s) |
M.V. Ishkhanyan (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia, margarita-ish@yandex.ru)
A.V. Karapetyan (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia, avkarapetyan@yandex.ru) |
| Abstract |
We analyze the dynamics of a homogeneous ball on a horizontal plane with friction of all kinds, namely, sliding, spinning, and rolling friction, taken into account. The qualitative-analytic study of the ball dynamics is supplemented with numerical experiments.
The problem on the motion of a homogeneous ball on a horizontal plane with friction was apparently first studied in 1758 by I. Euler (Leonard Euler's son) with sliding friction taken into account in the framework of the Coulomb model. I. Euler showed that the ball sliding ceases in finite time, after which the ball uniformly rolls along a fixed straight line and uniformly spins about the vertical. This result has long become classical and is described in many textbooks on theoretical mechanics.
In 1998, V. F. Zhuravlev considered the problem of motion of a homogeneous ball on a horizontal plane with sliding and spinning friction taken into account in the framework of the Contensou-Zhuravlev model [1, 2] and showed that the ball sliding and spinning cease simultaneously, after which the ball uniformly rolls along a fixed straight line. The Contensou-Zhuravlev theory was further developed in [3-7].
In the present paper, we consider the motion of a homogeneous ball on a horizontal plane with friction of all kinds taken into account in the framework of the model proposed in [8]. We show that, in one and the same time, both the sliding velocity and the angular velocity of the ball become zero. Our studies are based on the results obtained in [2], the properties of the friction model proposed in [8], and the method for qualitative analysis of dynamics of dissipative systems [9, 10]. The qualitative-analytic study is supplemented with numerical experiments. |
| Keywords |
dry friction, sliding, spinning, and rolling friction |
| References |
| 1. | P. Contensou,
"Couplage Entre Frottenment de Glissement et Frottenment de Pivotement Dans la Théorie de la Toupie,"
in Kreiselprobleme Gyrodynamics
(IUTAM Symp. Celerina, Berlin etc., Springer; Mir, Moscow, 1967),
pp. 201-216 (60-77). |
| 2. | V. Ph. Zhuravlev,
"The Model of Dry Friction in the Problem of the Rolling of Rigid Bodies,"
Prikl. Mat. Mekh.
62 (5), 762-767 (1998)
[J. Appl. Math. Mech. (Engl. Transl.)
62 (5), 705-710 (1998)]. |
| 3. | A. A. Kireenkov,
"A Method for the Calculation of the Force and Torque of Friction in a Combined Model of Dry Friction for Circular Contact Areas,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 51-56 (2003)
[Mech. Solids (Engl. Transl.)
38 (3), 39-43 (2003)]. |
| 4. | V. Ph. Zhuravlev,
"Friction Laws in the Case of Combination of Slip and Spin,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 4, 81-88 (2003)
[Mech. Solids (Engl. Transl.)
38 (4), 52-58 (2003)]. |
| 5. | V. Ph. Zhuravlev,
"Dynamics of a Heavy Homogeneous Ball on a Rough Plane,"
Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela,
No. 6, 3-9 (2006)
[Mech. Solids (Engl. Transl.)
41 (6), 1-5 (2006)]. |
| 6. | A. A. Kireenkov,
"Coupled Models of Sliding and Rolling Friction,"
Dokl. Ross. Akad. Nauk
419 (6), 759-762 (2008)
[Dokl. Phys. (Engl. Transl.)
53 (4), 233-236 (2008)]. |
| 7. | A. A. Kireenkov,
"Combined Model of Sliding and Rolling Friction in Dynamics of Bodies on a Rough Plane,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 116-131 (2008)
[Mech. Solids (Engl. Transl.)
43 (3), 412-425 (2008)]. |
| 8. | A. V. Karapetyan,
"A Two-Parameter Friction Model,"
Prikl. Mat. Mekh.
73 (4), 515-519 (2009)
[J. Appl. Math. Mech. (Engl. Transl.)
73 (4), 367-370 (2009)]. |
| 9. | A. V. Karapetyan,
"Global Qualitative Analysis of Tippe Top Dynamics,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 33-41 (2008)
[Mech. Solids (Engl. Transl.)
43 (3), 342-348 (2008)]. |
| 10. | A. V. Karapetyan,
"Qualitative Analysis od Dynamics of Dissipative Systems with Symmetry on the Basis of the Method of Generalized Smale Diagrams,"
in Contemporary Problems in Mathematics and Mechanics,
Vol. 2: Mechanics, No. 2
(Izd-vo MGU, Moscow, 2009),
pp. 192-200
[in Russian]. |
| 11. | M. V. Ishkhanyan,
"Dynamics of a Homogeneous Ball on the Surface with Friction,"
in Proc. Conference-Competition of Young Scientists of Institute of Mechanics, MSU
(Izd-vo MGU, Moscow, 2009),
pp. 99-105
[in Russian]. |
|
| Received |
18 March 2009 |
| Link to Fulltext |
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