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In this paper we consider one of the basic inverse problems in damage detection based on natural frequency data, namely the identification of a single open crack in a uniform simply supported beam from measurement of the first and the second natural frequency. It is commonly accepted in the literature that the knowledge of this set of spectral data allows for the unique determination of the severity and the position (up to symmetry) of the damage. However, in spite of the fact that many numerical evidences are in support of this property, the result is rigorously proved only when the severity of the crack is small. In this paper we definitely show, by means of an original constructive method, that the above result holds true for any level of crack severity.