Stochastic analysis of small-scale beams with internal and external damping

Random flexural vibrations of small-scale Bernoulli–Euler beams with internal and external damping are investigated in the paper. Such a kind of problem occurs in design and optimization of structural components of smart miniaturized electromechanical systems. Internal damping is related to the viscoelastic stress–strain law describing the beam time-dependent behavior while the external damping represents the interaction between the structure and the surrounding viscous environment. In the proposed formulation, Boltzmann superposition integral and fractional-order viscoelasticity are exploited to capture the above mentioned non-conventional phenomena. Size effects caused by long range interactions arising at small-scale are also taken into account and modeled by means of an integral nonlocal formulation based on the stress-driven approach. Moreover, in order to reproduce stochastic vibrations of small-size devices, such as resonators and sensors, the external loading are modeled by taking into account their random nature. The relevant dynamic problem is thus described by a fractional-order stochastic partial differential equation in space and time equipped with standard and non-classical boundary conditions. A proper differential eigenanalysis for the nonlocal viscoelastic bending problem is performed to find the modal shapes and closed-form expressions of power spectral density and stationary variance of displacements are provided. The proposed methodology, accounting for size-effects, hereditariness behaviors, viscous properties of surrounding environment and randomness of external loadings, is suitable to model and capture the effective behavior of miniaturized devices. The presented theoretical findings and numerical outcomes show the influence of size-effects and viscoelastic parameters on the structural response.