Nonlinear random vibrations of functionally graded porous nanobeams using equivalent linearization method

Abstract

In the present study, the nonlinear random vibration of a porous functionally graded nanobeam supported by a viscoelastic foundation has been investigated by making use of the statistical linearization method for the first time. Nonlocal Euler-Bernoulli beam theory with von Kármán type nonlinearity is utilized to derive the governing equation of the motion of the FG porous nanobeam. Two types of porosity distribution have been considered for the FG beam. To discretize the governing equation Galerkin's method is applied. The mean square value of the response of the resulted Duffing's equation in the presence of a random input with nonzero mean value has been extracted using statistical linearization method. The results of the present study are compared with the analytical results of the FPK approach and the results of the perturbation method for the classical limit case of the homogeneous beam. It is revealed through detailed numerical analysis that the statistical linearization method (in contrast to the perturbation method) gives reliable results and there is an excellent agreement between its results and the analytical results. The effect of various parameters on the mean value of the response is also investigated numerically and illustrated graphically. It is found that the mean square value of vibrations increases when porosity, power-law index, nonlocal parameter, or the mean value of the input increases.