This article presents the solution for free vibration of nanobeams based on Eringen nonlocal elasticity theory and Timoshenko beam theory. The small scale effect is considered in the first theory, and the transverse shear deformation effects as well as rotary inertia are taken into account in the latter one. Through variational formulation and the Hamilton principle, the governing differential equations of free vibration of the nonlocal Timoshenko beam and the boundary conditions are derived. The obtained equations are solved by the differential transformation method (DTM) for various frequency modes of the beams with different end conditions. In addition, the effects of slenderness and on vibration behavior are presented. It is revealed that the slenderness affects the vibration characteristics slightly whilst the small scale plays a significant role in the vibration behavior of the nanobeam.

In the present paper, the differential transformation method is employed to develop a semianalytical solution for free transverse vibration of single-walled carbon nanotube (SWCNT) with arbitrary boundary conditions. The small scale effect is taken into consideration via Eringen's nonlocal elasticity theory while the transverse shear deformation effects and rotary inertia are taken into account in presented Timoshenko beam theory. Through variational formulation and the Hamilton's principle the governing differential equations and the boundary conditions are derived and then solved by a semi-analytical method called differential transformation method (DTM) for various frequency modes of beams and different edge conditions. Comparisons made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the presented approach. The effects of several parameters such as transverse shear deformation effects, slenderness ratios, boundary conditions and small scale on vibration characteristics of SWCNT are examined. The present study illustrates that the vibration characteristics of an SWCNT are strongly dependent on the small scale parameters.