Random vibrations of axisymmetric viscoelastic nonlocal plates

The flexural vibrations of small-scale plates subjected to stochastic input are investigated. The dynamics of this problem is studied under the kinematic assumptions of the Kirchhoff axisymmetric model. This research is relevant to the design of bidimensional nano- and micro-structures used as energy harvesters, sensors, actuators, wave converters, components of transistors, bioinspired devices, small-scale robots, and similar applications. For these miniaturized structures, often made from unconventional materials, classical local elastic continuum theories fall short in providing accurate models. Therefore, their mechanical behavior is analyzed by considering two crucial aspects: viscoelasticity and nonlocality. The constitutive law employed is based on a stress-driven integral nonlocal model combined with fractional-order viscoelasticity. This approach captures both hereditary and size-dependent effects. The external dynamic loads are modeled by incorporating their stochastic nature to provide a more realistic representation of the system’s conditions. Under these assumptions, the dynamics of this complex system in terms of time-dependent transversal displacement field is governed by a stochastic partial integro-differential equation incorporating fractional differential operators. Solving such an equation is computationally challenging. For this reason, a semi analytical procedure to solve this problem by means of a modal decomposition procedure, evaluating the steadystate response and providing results in terms of power spectra is proposed. Presented results show how geometry, nonlocal parameter and viscoelastic coefficients influence the mechanical response and the main frequencies of the structure. Numerical and theoretical outcomes can help in the design of sophisticated small-scale bi-dimensional devices.