 | | 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 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| V. Ph. Zhuravlev, "Dynamics of a heavy homogeneous ball on a rough plane," Mech. Solids. 41 (6), 1-5 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
6 |
Pages |
1-5 |
| Title |
Dynamics of a heavy homogeneous ball on a rough plane |
| Author(s) |
V. Ph. Zhuravlev (Moscow) |
| Abstract |
The problem on a homogeneous ball rolling on a rough surface was studied in
numerous papers [1-6]. The most important aspect of this problem is the
concept of force interaction between the ball and the plane at the point of
contact. The authors of the cited papers believe themselves to be using the
Coulomb dry friction model just because they assume that the force at the
point of contact is directed opposite to the relative slip velocity and
does not depend on the modulus of the latter.
In reality, such a model is a speculative extrapolation of the Coulomb law
to conditions for which it was neither originally stated nor verified
experimentally. However, there are experiments refuting the possibility of
such a generalization of the model to cases in which pure slip is
accompanied with spin [7].
Dry friction effects, which are well known for extended contact regions
under complex slip kinematics [8, 9], also occur in the case of point
contacts. In the present paper, we study the rolling problem for a
homogeneous ball by using the Coulomb dry friction model in differential
form with subsequent integration over the contact region. The contact
region is determined according to the Hertz contact stress theory, and then
we pass to the limit of a point contact.
The equations of motion of a heavy ball with this friction model admit
exact integration and analysis, which shows that, by applying the dry
friction law consistently, one arrives at results contradicting those
obtained by using the simplified version of this law. |
| References |
| 1. | P. Appell, Theoretical Mechanics. Volume 2 [Russian translation],
Fizmatgiz, Moscow, 1960. |
| 2. | G. Coriolis, Mathematical Theory of Phenomena in the Game of
Billiards [Russian translation], Gostekhizdat, Moscow, 1956. |
| 3. | P. Painlevé, Lectures on Friction [Russian translation],
Gostekhizdat, Moscow, 1954. |
| 4. | E. Routh, A Treatise on the Dynamics of a System of Rigid Bodies. Volume 2
[Russian translation], Nauka, Moscow, 1983. |
| 5. | H. Resal, "Sur un point de la théorie mathématique des effets
du jeu de billard," C. R. Acad. Sci., Vol. 94, No. 3, pp. 1548-1551,
1882. |
| 6. | A. P. Markeev, Dynamics of a Body in Contact with a Rigid Surface
[in Russian], Nauka, Moscow, 1992. |
| 7. | 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,
pp. 201-216, 1963. |
| 8. | N. E. Zhukovskii, "Equilibrium condition for a rigid body that is
supported by a fixed plane along some area and can move along this plane
with friction," Collected Papers. Volume 1, pp. 339-354, Gostekhizdat,
Moscow, Leningrad, 1949. |
| 9. | V. V. Andronov,
"Mechanical systems with transformed dry friction,"
Izv. RAN. MTT [Mechanics of Solids], No. 1, pp. 40-49, 1988. |
| 10. | F. L. Chernousko,
"Equilibrium conditions for a body on a rough plane,"
Izv. RAN. MTT [Mechanics of Solids], No. 6, pp. 6-17, 1988. |
| 11. | G. M. Rozenblat, Dynamical Systems with Friction, NITs "Regular
and Chaotic Dynamics," Moscow, Izhevsk, 2005. |
| 12. | L. D. Landau and E. M. Lifshits, Theoretical Physics. Volume 7. Elasticity [in
Russian], Nauka, Moscow, 1965. |
| 13. | V. F. Zhuravlev, "On the dry friction model in the rolling
problem for rigid bodies," PMM [Applied Mathematics and Mechanics],
Vol. 62, No. 5, pp. 762-767, 1998. |
|
| Received |
03 March 2006 |
| << Previous article | Volume 41, Issue 6 / 2006 | Next article >> |
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