Identification of a crack in a rod based on changes in a pair of natural frequencies

This paper deals with the identification of a single crack in a vibrating rod based on the knowledge of the damage-induced shifts in a pair of natural frequencies. The crack is simulated by an equivalent linear spring connecting the two segments of the bar. The analysis is based on an explicit expression of the frequency sensitivity to damage and enables non-uniform bars under general boundary conditions to be considered. The inverse problem is generally "ill-posed", because even if the system is not symmetrical, cracks in different locations can still produce identical changes in a pair of natural frequencies. In spite of this, it is found that there are certain situations concerning uniform rods in which the effects of the non-uniqueness of the solution may be considerably reduced by means of a careful choice of the data. The theoretical results are confirmed by a comparison with dynamic measurements on steel rods with a crack. Some of the results are also valid for cracked beams in bending.