Free longitudinal vibrations analysis of nanorods based on Eringen’s nonlocal theory with considering surface energy using wave approach

In this paper analysis of free longitudinal vibrations of continuous nanorod (nanotube) in cylindrical coordinate system, have been done by wave method. Therefore, first the axial force equation of nanorod is derived from its motion equation. Then by using of Eringen’s nonlocal elasticity theory, nonlocal axial force of nanorod is obtained. After deriving axial force equation, which includes surface energy parameters, equations of motion and axial force of nanorod in terms of positive-going and negative-going waves are obtained. By placing the spring at the end of the boundaries of the nanorod, wave propagation matrix is obtained under boundary conditions of clamped-clamped and clamped-free. Eventually by calculation the dimensionless frequency of nanorod for both of the mentioned boundary conditions, dimensionless frequency changes vs nonlocal parameter and also vs surface energy parameters as diagram have been investigated. In each diagram, first three modes of dimensionless natural frequency under difference poisson’s ratios are investigated