# TIMOSHENKO EQUATION, VIOLATION OF CONTINUITY AND SOME APPLICATIONS

• I. T. Selezov Institute of Hydromechanics
Keywords: the Timoshenko equation, hyperbolicity, violation of continuity, elastic foundation, wavelength, frequency, Еuclidean space

### Abstract

We study the Timoshenko model of bending beam vibrations, that includes the beginning from a brief general consideration and the fast transition from n-dimensional Euclidean space to 4-dimensional space with respect to spatial coordinates and time. As a result, the Timoshenko equation is obtained on the basis of a mathematical approach, without a correction coefficient (shear coefficient) as a special case of a more general our extended refined equation. We investigate the problem of the effect of liquid, as a special case of an elastic base, on shear in Timoshenko elastic plate. It is shown that any media contacting with the plate reduce the shear effect. The violation of continuity is noted, which has not been considered previously. The works based on the Timoshenko model are presented for a beam on an elastic base. In the case of a rectangular change in the cross section, another matching problem immediately arises, connected with appearing reflected and transmitted waves. From the solvability of the problem for the phase velocity in the case of short wavelengths (high frequencies), the yield to the characteristic is studied and it is shown that in connection with the violation of continuity, the applicability of the classical theory takes place at wavelengths of more than 5 thicknesses. The problem of elastic plates floating on a liquid layer is studied in detail, using various theories. Variational formulations without taking into account the violation of continuity are considered and commented, the separation of variables in the Timoshenko equation is considered.

### References

1. Cauchy, A. L. (1828). Sur l’equilibre et le mouvement d’une lame solide. Exercices Math., No. 3. pp. 245 327.
2. Poisson, S. D. (1829). Mémoire sur l’équilibre et le mouvement des corps élastiques. Mém. Acad. Roy. Sci., No. 8. pp. 357 570.
3. Selezov, I. T. (2000). Degenerated hyperbolic approximation of the wave theory of elastic plates. Ser. Operator Theory. Advances and Applications. Differential Operators and Related Topics. Proc. of Mark Krein Int. Conf. (Ukraine, Odessa, 18–22 August 1997). Ba-sel/Switzerland: Birkhauser, No. 117. P. 339–354.
4. Selezov, I. T. (2018). Development and application of the Cauchy-Poisson method to layer elastodynamics and the Timoshenko equation. Cybernetics and Systems Analysis, Vol. 54, No.3, pp. 434-442. doi: https://doi.org/10.1007/s10559-018-0044-x.
5. Grigolyuk, E. I. & Selezov, I. T. (1973). Nonclassical theories of vibrations of bars, plates and shells. Advances in Sciences and Engineering. Mechanics of Deforming Solids, Moscow: Acad. Sci. USSR.
6. Dunford, N. & Schwartz, J. T. (1963). Linear operators. Part II. Spectral theory. Adjoint opera-tors in Hilbert space. New York London: Interscience Publishers.
7. Selezov, I. T. & Korsunsky, S. V. (1991). Transformation of plane waves overmoving, or elas-tic bottom inhomogeneity. Rozprawy Hydrotechniczne, No. 54, pp. 49–54.
8. Timoshenko, S. (1956). Strength of materials. Part 2. Advanced theory and problems. Third Edition. D. Van Nostrand Company, Inc. Princeton. New Jersey, Toronto, New York, London.
9. Achenbach, A. D., Keshava, S. P. & Herrmann, G. (1967). Free waves in a plate supported by a semi-infinite continuum. Trans. ASME, E34, N 4. pp. 397–404.
10. Yu, Y.-Y. (1960). Nonlinear flexural vibrations of sandwich plates. J. Aero/Space.Vol. 27, No. 4, pp. 272–282.
11. Lloid, J. R. & Miklowitz, J. (1962). Wave propagation in an elastic beam or plate on an elastic foundation. Trans. ASME, E29, No. 3. pp. 450–464.
12. Shubov, M. A. (2002). Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model. Mathematische Nachrichten, 241(1). pp. 125–162. doi: https://doi.org/10.1002/1522-2616(200207)241:1<125::AID-MANA125>3.0.CO;2-3.
13. Doetsch, G. (1956). Anleitung zum praktischen Gebrauch der Laplace-Transformation: R. Oldenbourg, Muenchen.
14. Krylov, V. I. & Skoblya, N. S. (1974). Methods of approximate Fourier transform and invert-ing Laplace transform. Moskow: Nauka (In Russian).
15. Barbashov, B. M. & Nesterenko, V. V. (1983). Continuous summetries in field theory. Fortsch. Phys, B 11, No. 10. pp. 535–567.
16. Nesterenko, V. V. (1989). Singular Lagrangians with higher derivatives. J. Phys. A. Math. Gen., Vol. 22, No. 10, pp. 1673–1687.
17. Nesterenko, V. V. (1993). To the theory of transverse vibrations of the Timoshenko beam. Appl. math and mekh., Vol. 57, No. 4. pp. 83–91 (In Russian).
18. Chervyakov, A. M. & Nesterenko, V. V. (1993). Is it possible to assign physical meaning to field theory with higher derivatives. Physical Review D., Vol. 48, No. 12. pp. 5811–5817.
19. Bakhvalov, N. S. & Eglit, M. E. (2005). On equations of higher order exactness describing vi-brations of thin plates (In Russian). Appl. math and mekh., Vol. 69 (4). pp. 656–675.
Published
2020-03-03
How to Cite
Selezov, I. T. (2020). TIMOSHENKO EQUATION, VIOLATION OF CONTINUITY AND SOME APPLICATIONS. Bulletin of Zaporizhzhia National University. Physical and Mathematical Sciences, (2), 150-157. Retrieved from http://journalsofznu.zp.ua/index.php/phys-math/article/view/265
Section
Articles