Size dependent forced vibration of nanoplates with consideration of surface effects

In this article, an analytical method is presented to study the size dependent forced vibration of rectangular nanoplates under general external loading using a generalized form of Kirchhoff plate model. The effects of surface properties including surface elasticity, surface residual stresses and surface mass density are considered which are bases for size dependent behaviors due to increase in surface to volume ratios at smaller scales. At first, a complete discussion is given for size dependent natural frequencies which are then used in forced vibration analyses. It is shown that the surface properties compact the frequency spectrums of the nanoplates. Saving generality and using the superposition principle, closed form solution is derived for time response of nanoplates under general harmonic loads. As a result, some elliptic curves are obtained for which the surface properties will not change the time response of nanoplates when a point load is applied on any point of these curves. It is observed that, for actuations inside these ellipses, the surface effects reduce the vibration amplitude while increase it for actuations outside the curves. Sensitivity of the problem to the excitation frequencies is also studied and various examples are given to illustrate the trend of size dependencies.