Axial vibration of non-uniform and non-homogeneous nanorods based on nonlocal elasticity theory

An elastic rod model is developed to study the small scale effect on axial vibration of non-uniform and non-homogeneous nanorods by using the theory of nonlocal elasticity. The finite element method is adopted to obtain the numerical solutions to the proposed model. Based on the present study, it can be concluded that the nonlocal frequency is less than the local (classical) frequency due to the effect of small length scale. Besides, increasing the nonlocal scale coefficient tends to decrease the frequency of the non-uniform and non-homogeneous nanorods. Furthermore, the nonlocal effects decrease with the increase of the non-uniform and non-homogeneous nanorods length and eventually disappear when the length exceeds a certain value. Finally, it is noted that the nonlocal effects are more noticeable for higher modes and stiffer structure. Furthermore, it can be concluded that the relative difference in frequency ratio between non-uniform and non-homogeneous nanorods and uniform nanorods converges to zero as the nanorod length increases, and the relative difference in frequency ratio is more pronounced for clamped–clamped boundary condition than clamped-free one.