 | | 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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| << Previous article | Volume 49, Issue 5 / 2014 | Next article >> |
| A.I. Moshinskii, "Equation of Longitudinal Vibrations of a Beam with Coefficients Periodically Depending on the Coordinate," Mech. Solids. 49 (5), 518-530 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
5 |
Pages |
518-530 |
| DOI |
10.3103/S0025654414050045 |
| Title |
Equation of Longitudinal Vibrations of a Beam with Coefficients Periodically Depending on the Coordinate |
| Author(s) |
A.I. Moshinskii (St. Petersburg Chemical-Pharmaceutical Academy, ul. Professora Popova 14, St. Peterburg, 197376 Russia, alex-moshinskij@yandex.ru) |
| Abstract |
The equation of small longitudinal vibrations of a beam is considered for the case in which the coefficients are periodic and have "gaps" (sharp decreases in the value) at some point in the space inside the periodicity interval. We suggest to describe the process locally in the domain of minimum of the coefficients by an equation of the boundary layer approximation. Attention is mainly paid to the analysis of this equation. A time hierarchy of simplified models for describing this process is established. |
| Keywords |
homogenization, asymptotics, model hierarchy, effective coefficients |
| References |
| 1. | N. S. Bakhvalov and G. P. Panasenko,
Homogenization of Processes in Periodic Media.
Mathematical Problems of Mechanics of Composite Materials
(Nauka, Moscow, 1984)
[in Russian]. |
| 2. | B. E. Pobedrya,
Mechanics of Composite Materials
(Izd-vo MGU, Moscow, 1984)
[in Russian]. |
| 3. | R. Christensen,
Introduction to Mechanics of Composite Materials
(Wiley, New York, 1979; Mir, Moscow, 1982). |
| 4. | A. N. Tikhonov and A. A. Samarskii,
Equations of Mathematical Physics
(Nauka, Moscow, 1977; Dover, New York, 1990). |
| 5. | B. E. Pobedrya,
"Principles of Computational Mechanics of Composites,"
Mekh. Komp. Mater.
32 (6), 729-746 (1996)
[Mech. Comp. Mater. (Engl. Transl.)
32 (6), 504-515 (1996)]. |
| 6. | A. I. Moshinskii,
"Effective Diffusion of a Dynamically Passive Impurity in Narrow Channels,"
Zh. Prikl. Mekh. Tekhn. Fiz.
28 (3), 62-71 (1987)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
28 (3), 374-382 (1987)]. |
| 7. | A. I. Moshinskii,
"Effective Diffusion of Multicomponent Mixtures in Multidimensional Flows,"
Zh. Prikl. Mekh. Tekhn. Fiz.
32 (4), 113-120 (1991)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
32 (4), 571-578 (1991)]. |
| 8. | N. N. Lebedev and I. P. Skal'skaya,
"Integral Expansion of an Arbitrary Function in Terms of Spherical Functions,"
Prikl. Mat. Mekh.
30 (2), 252-258 (1966)
[J. Appl. Math. Mech. (Engl. Transl.)
30 (2), 314-321 (1966)]. |
| 9. | N. N. Lebedev,
Special Functions and Their Applications
(Fizmatgiz, Moscow-Leningrad, 1963)
[in Russian]. |
| 10. | A. I. Moshinskii,
"Expansions of Arbitrary Functions in Integrals Using Spherical Functions,"
Differ. Uravn.
25 (4), 716-721 (1989)
[Differ. Equations (Engl. Transl.)
25 (4), 497-501 (1989)]. |
| 11. | N. N. Lebedev and I. P. Skal'skaya,
"Some Boundary Value Problems of Mathematical Physics and of the Theory
of Elasticity for a Hyperboloid of Revolution of One Sheet,"
Prikl. Mat. Mekh.
30 (5), 889-896 (1966)
[J. Appl. Math. Mech. (Engl. Transl.)
30 (5), 1052-1060 (1966)]. |
| 12. | H. Bateman and A. Erdélyi,
Higher Transcendental Functions. Hypergeometric Function. Legendre Functions
(McGraw-Hill, New York, 1955; Nauka, Moscow, 1973). |
| 13. | M. A. Lavrentiev and B. V. Shabat,
Methods of Theory of Functions of a Complex Variable
(Nauka, Moscow, 1973)
[in Russian]. |
| 14. | L. Brillouin and M. Parodi,
Wave Propagation in Periodic Structures
(McGraw Hill, New York, 1946; Izdat. Inostr. Lit., Moscow, 1959). |
| 15. | D. Ya. Barzokas, A. I. Zobnin, V. Z. Parton, and A. L. Shatalov,
"Heating of a Composite of Periodic Structure in a High-Frequency Electromagnetic Field,"
Zh. Prikl. Mekh. Tekhn. Fiz.
38 (1), 143-150 (1997)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
38 (1), 130-136 (1997)]. |
| 16. | N. S. Bakhvalov, K. Yu. Bogachev, and M. E. Eglit,
"Numerical Calculation of Effective Elastic Moduli for Incompressible Porous Material,"
Mekh. Komp. Mater.
32 (5), 579-587 (1996)
[Mech. Comp. Mater. (Engl. Transl.)
32 (5), 399-405 (1996)]. |
|
| Received |
25 July 2011 |
| Link to Fulltext |
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