Exact solution for natural frequencies of double-walled carbon nanotubes clamped at both ends

We report a study of the natural frequencies of the double-walled carbon nanotubes clamped at both ends. An exact solution is provided and compared with solutions reported in the literature. In contrast to earlier investigations, an analytical criterion is derived to establish the behavior of the roots of the characteristic equation. An approximate Bubnov–Galerkin solution is also obtained to compare natural frequencies at the lower end of the spectrum.