 | | 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: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
|
| In English (Mech. Solids): | | 3212 |
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| << Previous article | Volume 48, Issue 6 / 2013 | Next article >> |
| A.P. Markeev, "On the Evolution of Rotation of a Solid under Inelastic Collisions with a Plane," Mech. Solids. 48 (6), 603-612 (2013) |
| Year |
2013 |
Volume |
48 |
Number |
6 |
Pages |
603-612 |
| DOI |
10.3103/S0025654413060010 |
| Title |
On the Evolution of Rotation of a Solid under Inelastic Collisions with a Plane |
| Author(s) |
A.P. Markeev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia, markeev@ipmnet.ru) |
| Abstract |
The motion of a solid in a homogeneous gravity field under inelastic collisions with an immovable absolutely smooth horizontal plane is considered. The body is a homogeneous ellipsoid of revolution. There exists a motion in which the ellipsoid symmetry axis is directed along a fixed vertical, the ellipsoid itself rotates about this axis at a constant angular velocity, and the ellipsoid bounce height over the plane decreases from impact to impact because of the collisions. We study the motion of the ellipsoid in a small neighborhood of the motion corresponding to this infinite-impact process.
The main goal is to compute the angle between the ellipsoid symmetry axis and the vertical at the discrete time instants corresponding to the collisions. The problem is solved in the first (linear) approximation. The analysis is based on the canonical transformation method used earlier in [1] to solve problems with absolutely elastic collisions.
There are quite a few studies dealing with infinite-impact processes (e.g., see the monographs [2, 3]). A method for continuous representation of systems with inelastic collisions was proposed in [4] and efficiently used in [3-5] when analyzing specific mechanical systems. |
| Keywords |
solid, impact, canonical transformation, stability |
| References |
| 1. | A. P. Markeev,
"Investigation of the Stability of the Periodic Motion
of a Rigid Body
when There Are Collisions with a Horizontal Plane,"
Prikl. Mat. Mekh.
58 (3), 71-81 (1994)
[J. Appl. Math. Mech. (Engl. Transl.)
58 (3), 445-456 (1994)]. |
| 2. | R. F. Nagaev,
Mechanical Processes with Repeated Decaying Collisions
(Nauka, Moscow, 1985)
[in Russian]. |
| 3. | A. P. Ivanov,
Dynamics of Systems with Mechanical Collisions
(International Educational Program, Moscow, 1997)
[in Russian]. |
| 4. | A. P. Ivanov,
"Analytical Methods in the Theory of Vibro-Impact Systems,"
Prikl. Mat. Mekh.
57 (2), 5-21 (1993)
[J. Appl. Math. Mech. (Engl. Transl.)
57 (2), 221-236 (1993)]. |
| 5. | A. P. Ivanov,
"Unique Form of Equations of Motion of a Heavy Solid
on a Horizontal Support,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 73-79 (1993)
[Mech. Solids (Engl. Transl.)]. |
| 6. | P. Appel,
Theoretical Mechanics, Vol. 2
(Fizmatgiz, Moscow, 1960)
[in Russian]. |
| 7. | W. D. MacMillan,
Dynamics of Rigid Bodies
(McGraw-Hill, New York, 1936; Izd-vo Inostr. Lit., Moscow, 1951). |
| 8. | A. P. Markeev,
Dynamics of a Solid Contacting with a Solid Surface
(Fizmatlit, Moscow, 1992)
[in Russian]. |
| 9. | A. P. Markeev,
Theoretical Mechanics
(NITs "Regular and Chaotic Dynamics," Izhevsk, 2001)
[in Russian]. |
| 10. | V. V. Stepanov,
A Course of Differential Equations
(Fizmatlit, Moscow, 1959)
[in Russian]. |
| 11. | A. P. Markeev,
"On the Stability of a Solid Rotating around the Vertical
and Colliding with a Horizontal Plane,"
Prikl. Mat. Mekh.
48 (3), 363-369 (1984)
[J. Appl. Math. Mech. (Engl. Transl.)
48 (3), 260-265 (1984)]. |
| 12. | I. G. Malkin,
Theory of Stability of Motion
(Nauka, Moscow, 1966)
[in Russian]. |
|
| Received |
06 April 2013 |
| Link to Fulltext |
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