Exact analytical approach for free longitudinal vibration of nanorods based on nonlocal elasticity theory from wave standpoint

Article history: Received 20 September 2016 Received in revised form 11 May 2017 Accepted 14 May 2017 Available online 10 June 2017 In this paper, free longitudinal vibration of nanorods is investigated from the wave viewpoint. The Eringen’s nonlocal elasticity theory is used for nanorods modelling. Wave propagation in a medium has a similar formulation as vibrations and thus, it can be used to describe the vibration behavior. Boundaries reflect the propagating waves after incident. Firstly, the governing equation of nanorods longitudinal vibration based on the Eringen’s nonlocal elasticity theory is derived. Secondly, the propagation matrix for nanorod waveguide is derived and then the reflection matrix for spring boundary condition is calculated. The relations between amplitudes of propagation and reflection waves in the waveguide dominant are then combined in a matrix form format to set up a laconic efficient method for free axial vibration analysis of nanorods. The exact analytical solution for arbitrary boundary conditions natural frequencies is derived. To validate this approach, the exact solutions of special boundary conditions cases (clamped-clamped and clamped-free) are used. At the end, the effect of nonlocal parameter on the natural frequencies and boundary stiffness for arbitrary boundary condition is discussed. © 2017 Iranian Society of Acoustics and Vibration, All rights reserved.