Analysis of damped vibrations of thin bodies embedded into a fractional derivative viscoelastic medium

Abstract

AbstractDamped vibrations of elastic thin bodies, such as plates and circular cylindrical shells, embedded into a viscoelastic medium, the rheological features of which are described by fractional derivatives, are considered in the present article. Besides the forces of viscous friction, a thin body is subjected to the action of external forces dependent on the coordinates of the middle surface and time. The boundary conditions are proposed in such a way that the governing equations allow the Navier-type solution. The Laplace integral transform method and the method of expansion of all functions entering into the set of governing equations in terms of the eigenfunctions of the given problem are used as the methods of solution. It is shown that as a result of such a procedure, the systems of equations in the generalized coordinates could be reduced to infinite sets of uncoupled equations, each of which describes damped vibrations of a mechanical oscillator based on the fractional derivative Kelvin-Voigt model.