In the present paper, the free vibration response of single-walled carbon nanotubes (SWCNTs) is investigated. The governing equations of motion are derived using a variation approach and the free vibration frequencies are obtained employing two different formulations. In the first part of the paper, the case of the cantilever nanotube with concentrated mass at its free end, in the presence of nonlocal effects, is considered and the Hamilton principle is reformulated, in order to find the equation of motion and the boundary conditions; it turns out that they are the same limit conditions obtained by Reddy and Pang, using a direct approach. In the second one, instead, by employing two different approaches two approximate formulas are deduced the first one is derived by applying the Rayleigh Principle, as defined to Meirovitch, whereas the second approximate formula is derived by a formulation given in energy terms. Numerical examples end the paper and some comparisons with existing results are offered. Comparisons of the present numerical results with those from the open literature show an excellent agreement.
