An analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects

The nonlinear free vibration of nanobeams with considering surface effects (surface elasticity, tension and density) is studied using Euler–Bernoulli beam theory including the von kármán geometric nonlinearity. The component of plane stress, σzzσzz, is assumed to vary linearly through the beam thickness and satisfy the balance conditions between nanobeam bulk and its surfaces. Accordingly, surface density is introduced into the governing equation of the nonlinear free vibration of nanobeams. It is seen that the effect of surface density is independent of amplitude ratio. In addition, it is observed that in lower modes, surface density has insignificant effects on the variation of the natural frequency versus mode number, whereas this is not the case in higher modes where the surface density causes the normalized natural frequencies of the nanobeams to increase drastically. Moreover, it is shown that the effect of the surface density on the variation of the natural frequency of the nanobeam versus the thickness ratio decreases consistently with the increase of the mode number.