The steady-state response of size-dependent functionally graded nanobeams to subharmonic excitation

This study aims to investigate the nonlinear forced vibration of functionally graded (FG) nanobeams. It is assumed that material properties are gradually graded in the direction of thickness. Nonlocal nonlinear Euler–Bernoulli beam theory is used to derive nonlocal governing equations of motion. The linear eigenmodes of FG nanobeams are used to transform a partial differential equation of motion into a system of ordinary differential equations via the Galerkin method. The multiple scale method is used to find the governing equations of the steady-state responses of FG nanobeams excited by a distributed harmonic force with constant intensity. It is also assumed that the working frequency is close to three times greater than the lowest natural frequency. Based on the equation governing the linear natural frequencies of FG nanobeams, the influence of the small scale parameter, material composition, and stiffness of the foundation on the linear relationship among natural frequencies is studied. Results show that superharmonic response or a combination of resonances may occur as well as a subharmonic response depending on the power-law index and stiffness of the foundation. Then the governing equations of a steady-state response of FG nanobeams for four possible solutions are obtained depending on the value of the small scale parameter. It is shown that the simplest response of FG nanobeams is a subharmonic response or superharmonic response. The equations governing the frequency–response curves are obtained and the effects of the power-law index and small scale parameter on them are discussed.