Stationary and non-stationary stochastic response of linear fractional viscoelastic systems

A method is presented to compute the stochastic response of single-degree-of-freedom (SDOF) structuralsystems with fractional derivative damping, subjected to stationary and non-stationary inputs. Based ona few manipulations involving an appropriate change of variable and a discretization of the fractionalderivative operator, the equation of motion is reverted to a set of coupled linear equations involvingadditional degrees of freedom, the number of which depends on the discretization of the fractionalderivative operator. As a result of the proposed variable transformation and discretization, the stochasticanalysis becomes very straightforward and simple since, based on standard rules of stochastic calculus,it is possible to handle a system featuring Markov response processes of first order and not of infiniteorder like the original one. Specifically, for inputs of most relevant engineering interest, it is seen thatthe response second-order statistics can be readily obtained in a closed form, to be implemented in anysymbolic package. The method applies for fractional damping of arbitrary order α (0 ≤ α ≤ 1). Theresults are compared to Monte Carlo simulation data.