Fundamental natural frequencies of double-walled carbon nanotubes

This study deals with evaluation of fundamental natural frequencies of double-walled carbon nanotubes under various boundary conditions. The Bubnov–Galerkin and Petrov–Galerkin methods are applied to derive the expressions for natural frequencies. Apparently for the first time in the literature explicit expressions are obtained for the natural frequencies. These can be useful for the designer to estimate the fundamental frequency in each of two series.