Vibration analysis of nanobars with deformable boundary conditions in nonlocal elasticity

The free axial vibration response of nanorods with arbitrary boundary conditions is presented based on the nonlocal elasticity theory. First part of study, nonlocal elasticity theory, Fourier sine series and Stokes' transformation are briefly reviewed. Nanorods are modelled with attached springs at both ends. Fourier sine series and Stokes' transformation are used to determine the vibrational response. Several comparisons are made between the results of the presented method and previous works in the literature. Very good agreement is derived when enough terms are included in the Fourier sine series expansion. The second part of study, cracked nanorods are investigated. Crack is modelled by an attached spring. Axial deflection function is represented by two Fourier sine series. The general frequency determinant is constructed by applying Stokes' transformation to the boundary conditions. This determinant can also be used to determine the free vibration frequencies of a cracked nanorod with classical supporting conditions. The influence of the several parametres on the free vibration response is examined in some numerical examples. The calculated results indicate that the present analytic procedure provides a more straightforward than the other analytical methods.