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) structural systems with fractional derivative damping, subjected to stationary and non-stationary inputs. Based on a few manipulations involving 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 degrees of freedom, the number of which depends on the discretization of the fractional derivative operator. As a result of the proposed variable transformation and discretization, the stochastic analysis 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 infinite order like the original one. Specifically, for inputs of most relevant engineering interest, it is seen that the response second-order statistics can be readily obtained in a closed form, to be implemented in any symbolic package. The method applies for fractional damping of arbitrary order α(0≤α≤1). The results are compared to Monte Carlo simulation data.

► A method for computing the stochastic response of fractionally-damped SDOF. ► We consider both stationary and non-stationary inputs. ► The system response is reverted to a first order Markov process based on a proper variable transformation. ► Examples show accurate results as compared to Monte Carlo simulation data.