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We consider the inverse problem of identifying a single open crack in a longitudinally vibrating rod having non-uniform smooth profile. Without any a priori assumption on the smallness of the damage and assuming that the rod profile is symmetric with respect to the mid-point of the rod axis, we present a constructive diagnostic algorithm from minimal frequency data. We show that the crack can be uniquely identified, up to a symmetric position, from the first two positive natural frequencies of the rod under free-free end conditions. We also show that the non-uniqueness of the damage location can be removed by using as data the first positive resonant frequency of the free-free rod and the first antiresonant frequency of the driving-point frequency response evaluated at one end of the rod. The results of numerical simulations and of applications of the method to experimental data agree well with the theory. (C) 2015 Elsevier Ltd. All rights reserved.