Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He's variational method

Nonlinear free vibration of axially functionally graded (AFG) Euler-Bernoulli microbeams with immovable ends is studied by using the modified couple stress theory. The nonlinearity of the problem stems from the von-Kármán's nonlinear strain-displacement relationships. Elasticity modulus and mass density of the microbeam vary continuously in the axial direction according to a simple power-law form. The nonlinear governing partial differential equation and the associated boundary conditions are derived by Hamilton's principle. By using Galerkin's approach, the nonlinear governing partial differential equation is reduced to a nonlinear ordinary differential equation. He's variational method is utilized to obtain the approximate closed form solution of the nonlinear ordinary governing equation. Pinned-pinned and clamped-clamped boundary conditions are considered. The influences of the length scale parameters, material variation, vibration amplitude, and boundary conditions on vibration responses are examined in detail.