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

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 asimilar formulation as vibrations and thus, it can be used to describethe vibration behavior. Boundaries reflect the propagating waves after incident. Firstly, the governing equation of nanorods longitudinal vibration based on the Eringen’s nonlocal elasticitytheory 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 amplitudesof 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 exactanalytical solution for arbitrary boundary conditions natural frequencies is derived. To validate this approach, the exact solutions of special boundary conditions cases (clamped-clamped andclamped-free) are used. At the end, the effect of nonlocal parameter on the natural frequencies and boundary stiffness for arbitrary boundary condition is discussed.

Keywords:

Nanorod vibration , Wave propagation , Arbitrary boundary , Nonlocal elasticity theory , Small-scale effect