The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam

We considered the problem on transversal oscillationsof two-layer straight bar, which is under the action of thelengthwise random forces. It is assumed that the layers of the barwere made of nonhomogenous continuously creeping material and thecorresponding modulus of elasticity and creeping fractional orderderivative of constitutive relation of each layer are continuousfunctions of the length coordinate and thickness coordinates.Partial fractional differential equation and particular solutionsfor the case of natural vibrations of the beam of creepingmaterial of a fractional derivative order constitutive relation inthe case of the influence of rotation inertia are derived. For thecase of natural creeping vibrations, eigenfunction and timefunction, for different examples of boundary conditions, aredetermined. By using the derived partial fractional differentialequation of the beam vibrations, the almost sure stochasticstability of the beam dynamic shapes, corresponding to the nth shape of the beam elastic form, forced by a bounded axially noiseexcitation, is investigated. By the use of S. T. Ariaratnam'sidea, as well as of the averaging method, the top Lyapunov exponentis evaluated asymptotically when the intensity of excitationprocess is small