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Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 46236, 18 pages

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

Faculty of Mechanical Engineering, University of Niš, 18000 Niš, Serbia and Montenegro, Mathematical Institute SANU, Belgrade 11001, Serbia and Montenegro

Received 25 January 2005; Accepted 17 March 2005

Copyright © 2006 Katica (Stevanović) Hedrih. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We considered the problem on transversal oscillations of two-layer straight bar, which is under the action of the lengthwise random forces. It is assumed that the layers of the bar were made of nonhomogenous continuously creeping material and the corresponding modulus of elasticity and creeping fractional order derivative of constitutive relation of each layer are continuous functions of the length coordinate and thickness coordinates. Partial fractional differential equation and particular solutions for the case of natural vibrations of the beam of creeping material of a fractional derivative order constitutive relation in the case of the influence of rotation inertia are derived. For the case of natural creeping vibrations, eigenfunction and time function, for different examples of boundary conditions, are determined. By using the derived partial fractional differential equation of the beam vibrations, the almost sure stochastic stability of the beam dynamic shapes, corresponding to the nth shape of the beam elastic form, forced by a bounded axially noise excitation, is investigated. By the use of S. T. Ariaratnam's idea, as well as of the averaging method, the top Lyapunov exponent is evaluated asymptotically when the intensity of excitation process is small.