 | | 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: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| << Previous article | Volume 56, Issue 6 / 2021 | Next article >> |
| Zveryaev E.M., "Low Frequency Vibrations of a Long Elastic Strip," Mech. Solids. 56 (6), 980-995 (2021) |
| Year |
2021 |
Volume |
56 |
Number |
6 |
Pages |
980-995 |
| DOI |
10.3103/S0025654421060261 |
| Title |
Low Frequency Vibrations of a Long Elastic Strip |
| Author(s) |
Zveryaev E.M. (Keldysh Institute of Applied Mathematics, Moscow, 125047 Russia; Moscow Aviation Institute (National Research University), Moscow, 125993 Russia, zveriaev@mail.ru) |
| Abstract |
The application of the iterative Saint-Venant-Picard-Banach method is described on the example of constructing the solution of the system for differential equations of motion of the theory of elasticity with a small parameter for a long strip at perturbation frequencies commensurate with the frequencies of transverse vibrations of the beam. The system is transformed in such a way that the equations are integrated sequentially and without increasing in order. The unknowns are calculated using the first-order Picard operators so that the previously calculated unknowns are input for the next equation, etc. The integrals of all unknowns for problem that make it possible to fulfill all the boundary conditions on the long and short sides are found. The convergence of the solution is ensured using a small thin-walled parameter in accordance with the Banach’s contraction principle. Satisfaction of the boundary conditions at the long edges leads to two equations for slowly and two singular ones for rapidly varying components of the solution, which depend only on the longitudinal coordinate. It is shown that when these equations are reduced to one with the loss of a rapidly changing component, the Timoshenko equation is obtained. The presentation is illustrated by two examples of loading a strip with a transverse distributed load and a concentrated force. A technique for establishing orders of magnitude by a small parameter relative to the load is described. |
| Keywords |
semi-inverse method, iterations, contraction mapping principle, Timoshenko's equation, stresses at corners |
| Received |
14 September 2020 | Revised |
27 September 2020 | Accepted |
15 October 2020 |
| Link to Fulltext |
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