In this reported work, the free axial vibration response of carbon nanotubes (CNTs) with arbitrary boundary conditions is studied based on the non-local elasticity theory. Using Fourier sine series together with Stokes'transformation, the general frequency determinant of CNTs is obtained. The main advantage of this method is its capability of dealing with rigid or restrained boundary conditions. Comparisons between the results of the presented method and previous works in the literature have been performed. Good agreement is obtained when enough terms are included in the Fourier series expansion. The effects of spring parameters on the vibration frequencies are discussed in detail. The proposed analytical method can be utilised for dynamic analyses of nanorods (CNTs) with arbitrary boundary conditions.