Nonlinear random vibration of functionally graded nanobeams based on the nonlocal strain gradient theory

In this work, the nonlinear random vibration of functionally graded (FG) nanobeams resting on a viscoelastic foundation is investigated using the nonlocal strain gradient theory and the regulated equivalent linearization method (RELM). The material properties of the FG nanobeam are assumed to change continuously in the direction of thickness according to a simple power-law rule. The governing equation of motion for the FG nanobeam is derived by using Hamilton’s principle and then is converted into an ordinary nonlinear differential equation by the Galerkin method. The mean-square value of the response of the FG nanobeam is obtained. Comparing the obtained results with the published results for the homogeneous and non-homogeneous beams showed the accuracy of the obtained solutions. In addition, the use of the RELM gives accurate mean-square values of the response over a large range of the radius of gyration. Effects of various parameters on the mean value of the response of the FG nanobeam are investigated. It is observed that the mean-square value of the response of the FG nanobeam increases by increasing the power-law exponent, the nonlocal parameter, and the spectral density; and by reducing the material length scale parameter and the coefficients of the viscoelastic foundation.