Size dependent vibration of curved nanobeams and rings including surface energies

In this article, explicit solution is presented for size and geometry dependent free vibration of curved nanostructures with consideration of surface properties. Additional surface elasticity, surface residual stress and also surface mass density are included in the study to generalize the existing classical theories. Considering the general presentation of the problem and regardless of the boundary conditions, the deviations of actual dynamic characteristics from classical theories for various geometries are found as new results. The modified natural frequencies of the structure are thus obtained and presented as normalized with respect to their corresponding conventional values.