# TIMOSHENKO EQUATION, VIOLATION OF CONTINUITY AND SOME APPLICATIONS

### 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.

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*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