Nonstationary Stochastic Analysis of Fractional Viscoelastic Euler-Bernoulli Beams

In this work a dynamic analysis of a viscoelastic beam subjected to nonstationary stochastic load is led; the latter is modeled as the product of a white noise process and a deterministic modulating function. The considered beam is made up of a viscoelastic material, whose constitutive law involves linear fractional operators. The partial fractional differential equation governing the beam deflection, written according to the Euler-Bernoulli hypothesis, is solved by adopting a Galerkin approach; this involves the linear modes of the corresponding elastic beam and some generalized displacements. Accordingly, a set of uncoupled fractional differential equations for the generalized displacements is obtained. These equations are solved employing a proposed numerical approach that relies on the Grunwald-Letnikov discretization scheme; in this way, the statistics of the beam response are easily computed. Finally, the proposed numerical method is validated for the stationary case exploiting an analytical solution derived from a frequency domain approach.