 | | 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 and D. M. Klimov, "On the dynamics of the Thompson top (tippe top) on the plane with real dry friction," Mech. Solids. 40 (6), 117-127 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
6 |
Pages |
117-127 |
| Title |
On the dynamics of the Thompson top (tippe top) on the plane with real dry friction |
| Author(s) |
V. Ph. Zhuravlev (Moscow)
D. M. Klimov (Moscow) |
| Abstract |
There is an extensive literature on Thompson top dynamics. A rather
detailed bibliography can be found in [1].
The most important aspect in the construction of a mathematical model of
the tippe top is the choice of the character of interaction between the top
and the supporting plane at the point of contact. The no-slip condition is
most often used for bodies rolling on rigid surfaces, which results in a
nonholonomic statement of problem.
Contensou [2] observed that this setting is inadequate. In particular, he
showed that if one applies Coulomb's dry friction law in differential form
with subsequent integration over the contact area, then, for sufficiently
large angular spin velocities, the resultant vector of dry sliding friction
forces bears resemblance to viscous friction.
A complete model of dry friction in the contact (including not only the
resultant force but also the resultant torque) without restrictions on the
sliding kinematics was constructed in [3]. This model is used in the
present paper, which studies local problems (stability of stationary
rotations) as well as the global behavior of the top, including the
complete inversion phenomenon surprising curious observers. |
| References |
| 1. | A. P. Markeev, Dynamics of a Body in Contact with a Solid Surface
[in Russian], Nauka, Moscow, 1992. |
| 2. | P. Contensou, "Couplage entre frottement de glissement et
frottement de pivotement dans la théorie de la toupie," in Gyroscopy
Problems [in Russian], pp. 66-77, Mir, Moscow, 1967. |
| 3. | V. Ph. Zhuravlev, "The dry friction model in the problem of
rolling of rigid bodies," PMM [Applied Mathematics and Mechanics],
Vol. 62, No. 5, pp. 762-767, 1998. |
| 4. | V. Ph. Zhuravlev, "Friction laws in the case of combination of
slip and spin," Izv. RAN. MTT [Mechanics of Solids], No. 4, pp. 81-88,
2003. |
| 5. | G. A. Baker and P. Graves-Morris, Padé Approximants [Russian
translation], Mir, Moscow, 1986. |
| 6. | V. Ph. Zhuravlev and A. A. Kireenkov, "Padé expansions in the
two-dimensional model of Coulomb friction," Izv. RAN. MTT [Mechanics of
Solids], No. 2, pp. 3-14, 2005. |
| 7. | V. Ph. Zhuravlev, Fundamentals of Theoretical Mechanics [in
Russian], Nauka, Moscow, 2001. |
| 8. | Yu. G. Martynenko, "Spin-up of a gyroscope with contactless
suspended rotor bearings," Izv. AN SSSR. MTT [Mechanics of Solids], No. 5,
pp. 35-40, 1973. |
| 9. | A. V. Karapetyan, Stability of Stationary Motions [in Russian],
Editorial URSS, Moscow, 1988. |
| 10. | V. V. Rumyantsev, "On the stability of motion of gyrostats of
certain form," PMM [Applied Mathematics and Mechanics], Vol. 22, No. 3,
pp. 5-19, 1958. |
| 11. | V. A. Samsonov, "Branching and some properties of nonlinear
mechanical systems," in Nonlinear Mechanics [in Russian], pp. 321-361,
Fizmatlit, Moscow, 2001. |
| 12. | R. I. Leine and Ch. Glocker, "A set-valued force law for spatial
Coulomb-Contensou friction," Europ. J. Mech., Vol. 22, pp. 193-216,
2003. |
| 13. | V. Ph. Zhuravlev and D. M. Klimov, Applied Methods in Vibration
Theory [in Russian], Nauka, Moscow, 1988. |
|
| Received |
17 March 2005 |
| << Previous article | Volume 40, Issue 6 / 2005 | Next article >> |
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