 | | 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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| Yu. M. Urman, "Irreducible tensors and their applications in problems of dynamics of solids," Mech. Solids. 42 (6), 883-896 (2007) |
| Year |
2007 |
Volume |
42 |
Number |
6 |
Pages |
883-896 |
| Title |
Irreducible tensors and their applications in problems of dynamics of solids |
| Author(s) |
Yu. M. Urman (Nizhnii Novgorod State Pedagogical University, Ul’yanova 1, Nizhnii Novgorod, 603005, Russia, urman37@mail.ru) |
| Abstract |
One difficulty encountered in solving mechanical problems with complicated interaction is to express either the moments of forces or the force function via the phase variables of the problem. Here various transformations of coordinate systems are used, because interactions are determined by a relation between tensor variables one of which refers to the body and the other refers to the field. In this connection, the usual definition of a tensor in Cartesian coordinates is inconvenient because of the fact that the components of a tensor of rank l≥2 can be arranged as several linear combinations that behave differently under rotations of the coordinate system. Naturally, one needs to define tensors in such a way that their components and linear combinations of these be transformed in a unified manner under rotations of the coordinate system. This requirement is satisfied by irreducible tensors. The mathematical apparatus of irreducible tensors was created to satisfy the requirements of quantum mechanics and turned out to be rather universal. As far as the author knows, this apparatus was first used in mechanics by G. G. Denisov and the author of the present paper [1]. Using this apparatus, one can see the clear physical meaning of complicated interactions, express these interactions in invariant form, easily perform transformations from one coordinate system to another coordinate system turned relative to the first, consider rather complicated types of interactions writing them in compact form explicitly depending on the phase variables of the problem, easily use the symmetry of both the rigid body and the force field structure, and perform the averaging procedure for the entire object rather than componentwise. The present paper further develops the paper [1]. We present a brief introduction to the theory of irreducible tensors. We show that the force function of various interactions between a rigid body and a force field can be represented as the scalar product of irreducible tensors. We study general properties of evolution motions of a rigid body in axisymmetric and nonsymmetric force fields under the action of moments caused by various harmonics of the force function. |
| References |
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10 (6), 1-9 (1975)]. |
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56 (11), 2081-2086 (1986). |
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|
| Received |
20 January 2007 |
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