A consistent approach to characterize random vibrations of nanobeams

Abstract

While there is a numerous amount of research on the random vibrations of classical structures in the literature, merely some limited attempts exist to formulate the random vibrations of nano-sized structures. This study deals with the random vibration of elastic beams at the ultra-small scale. The mixture unified gradient theory of elasticity is invoked to appropriately take into consideration the size-effect. Within the introduced augmented elasticity framework, the nanoscopic effects associated with the strain gradient theory and the stress gradient theory are consistently integrated into the classical elasticity theory. The corresponding boundary-value problem of dynamic equilibrium is, accordingly, established. The exact closed-form solution of the space-time correlation function and the mean-square of the transverse displacements of the mixture unified gradient elastic beam with simply supported ends is obtained for the space-time white noise excitation. The spatial variation of the mean-square value of the transverse displacements of the elastic nanobeam in terms of gradient length-scale parameters is graphically illustrated and thoroughly discussed. A consistent approach to characterize random vibrations of elastic nanobeams within the context of the mixture unified gradient elasticity theory is presented providing important insight into the random vibration analysis of nanostructures.