Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 46236, 18 pages
doi:10.1155/MPE/2006/46236
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.
Abstract
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.