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In this paper we present a theoretical analysis of free axial vibrations of rods described in terms of the fractional continuum mechanics. This formulation is able to capture the size effect arising in different problems of solid mechanics. The natural frequencies and modal shapes for clamped rods are obtained and the effects of the derivative order, as well as the corresponding length-scale parameters in the fractional model are discussed. The results are compared with those derived from the Eringen non-local elasticity theory, which is an approach widely used to model structures showing size effects.