Stochastic dynamic analysis of fractional viscoelastic systems

Abstract

A method is presented to compute the non-stationary response of single-degree-of-freedom structural systems with fractional damping. Based on an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion is reverted to a set of coupled linear equations involving additional half oscillators, the number of which depends on the discretization of the fractional derivative operator. In this context, it is shown that such a set of oscillators can be given a proper fractal representation, with a Mandelbrot dimension depending on the fractional derivative order a. It is then seen that the response second-order statistics of the derived set of coupled linear equations can be built, in a closed form, for stochastic inputs of relevant interest in engineering practice. For this a preliminary eigenvector expansion shall be pursued. The method applies for fractional damping of arbitrary order a (0£ a £1). Results are compared to Monte Carlo simulation data obtained based on a standard discretization of the Caputo’s fractional derivative.