Nonlinear free vibrations of functionally graded nanobeams with surface effects

Nonlinear free vibration of functionally graded nanobeams is studied in this paper within the framework of Euler–Bernoulli beam model including the von Kármán geometric nonlinearity. It is assumed that material properties follow power law distributions through thickness direction. A second order nonlinear ordinary differential equation with quadratic and cubic nonlinear terms is obtained by using the free vibration modes of the corresponding linear problem. The direct numerical integration method is employed to find the natural frequencies of FG nanobeams with different boundary conditions. Numerical results demonstrate the normalized natural frequencies in different nanobeam dimensions, vibration amplitudes and volume fraction indices of FG material. Finally, the surface effects on the phase plane trajectory of fundamental mode time-dependent function have been shown.