Analytical Solution for Free Longitudinal Vibration of Nanorods based on Nonlocal elasticity theory by Wave approach

In this paper, free longitudinal vibration of nanorods is investigated from wave viewpoint. Nanorods are modelled using nonlocal elasticity theory. Vibrations can be considered as travelling waves along structures. Waves propagate in a waveguide and reflect at boundaries. Firstly propagation and reflection matrices for spring boundary condition are derived. Secondly, these matrices are combined to provide a concise and systematic approach to free axial vibration analyses of nanorods. Exactly driven solution for clamped-clamped and clampedfree boundary conditions is used to validate this approach. Finally small-scale effect on natural frequencies is discussed.