Nonlocal nonlinear free vibration of functionally graded nanobeams

In this paper, nonlinear free vibration of functionally graded (FG) nanobeams with immovable ends, i.e. simply supported-simply supported (SS) and simply supported-clamped (SC), is studied using the nonlocal elasticity within the frame work of Euler–Bernoulli beam theory with von kármán type nonlinearity. The material properties are assumed to change continuously through the thickness of the FG nanobeam according to a power-law distribution. The analytical solution for the nonlinear natural frequency is established using the method of multiple scale. The small scale effects on the linear/nonlinear nonlocal frequency to the linear/nonlinear classical frequency ratios (the linear/nonlinear frequency ratios) are examined for various parameters such as the FG nanobeam length, the FG nanobeam thickness to length ratio (the thickness ratio), the vibration amplitude to the radius of gyration ratio (the amplitude ratio), and the boundary condition. As a main result, it is observed that while the linear frequency ratios are independent of the gradient index, the nonlinear frequency ratios vary with the gradient index.