Investigation of the inertia of the lateral motions effect on free axial vibration of nanorods using nonlocal Rayleigh theory

In this paper, free axial vibration of nanorods is investigated by focusing on the inertia of the lateral motions effects. To this end, Rayleigh and nonlocal theories considering the inertia of the lateral motions and the small scale effects, respectively, are used. Then, by implementing the Hamilton’s principle nonlocal governing equation of motion and boundary conditions are derived. Since using nonlocal elasticity causes that the 2-order local governing equation is changed to the 4-order nonlocal governing equation while number of boundary condition remains constant (one boundary condition at each end of nanorod), the governing equation is solved using Rayleigh-Ritz method. In Rayleigh-Ritz method a suitable shape function for the problem should be selected. The shape function must at least satisfy the geometrical boundary conditions. In the present study, orthogonal polynomials are selected as shape functions then they are normalized by using the Gram-Schmidt process for more rapid convergence.
Then, the first five axial natural frequencies of nanorod with clamped-clamped and clamped-free end conditions are obtained. In the next step, effects of various parameters like length of nanorod, diameter of nanorod and nonlocal parameter value on natural frequencies are investigated. Results of the present study can be useful in more accurate design of nano-electro-mechanical systems in which nanotubes are used.