Free and forced nonlinear vibrations of Bi-Directional functionally graded Euler–Bernoulli porous beams

Abstract

In this study, the free and forced vibrations of a bi-directional functionally graded porous beam are investigated. The governing equations of motion are derived by Hamilton’s principle, and the reduced temporal equation of motion with cubic and quintic nonlinear terms is obtained using the Galerkin approach. Analytical solutions for the nonlinear natural frequencies in addition to the primary resonance response curves are established by the method of multiple scales. The effects of the axial and transverse functionally graded indexes, initial amplitude, porosity parameter, and the elastic and mass density ratios on the nonlinear frequencies and the forced responses are examined.