Vibration analysis of embedded carbon nanotubes using wave propagation approach

A vibration analysis of single- and double-walled carbon nanotubes as well as nanotubes embedded in an elastic matrix is presented using wave propagation approach. Approximate Flügge shell equations are proposed as the governing equations of vibration for the carbon nanotubes studied here. The double-walled nanotubes are assumed to be coupled together through the van der Waals force between the inner and outer nanotubes. For embedded carbon nanotubes, an elastic medium surrounding the nanotubes is described by a Winkler model. Effects of nanotube parameters and vibrational modes on the natural frequency are investigated and analyzed based on the proposed elastic continuum model.