Analysis of damped vibrations of linear viscoelastic plates with damping modeled with fractional derivatives

Abstract

Damped vibrations of a rectangular viscoelastic plate, whose dynamic behaviour is described by a set of three linear equations in three mutually orthogonal displacements of points of its median surface, are considered. Damping features of the plate are determined by fractional derivatives with respect to time. Viscosity is referred to modal character. The Laplace integral transform method is employed as a method of solution, which is followed by the expansion of the desired functions in series with respect to eigenfunctions of the problem. However, unlike in the traditional approach, when rationalization of a characteristic equation with fractional powers is carried out during the transition from image to pre-image, here the nonrationalized characteristic equation is solved by the method suggested by the authors. The solution is obtained in the form of the sum of two terms, one of which governs the drift of the system's equilibrium position and is defined by the quasi-static processes of creep occurring in the system, and the other term describes damped vibrations around the equilibrium position and is determined by the systems's inertia and energy dissipation. The influence of viscosity on the solution is shown, and the time dependence of the plate points displacements is analyzed.