Exact reconstruction of multiple concentrated damages on beams

In this paper, an exact procedure for the reconstruction of multiple concentrated damages on a straight beam is proposed. The concentrated damages are modelled as Dirac's delta distributions capturing the effect of concentrated stiffness reduction. The presented procedure requires the knowledge of vibration mode shape displacements together with the relevant natural frequency, for the reconstruction of each damage position and intensity. The exact solution of the inverse problem at hand has been pursued by exploiting the analytical structure of the explicit closed form expressions provided for the vibration mode shapes of beams in the presence of an arbitrary number of cracks. The proposed procedure is first presented under the hypothesis that the displacements of a vibration mode shape are known at the cracked cross-sections. In this case, explicit closed form expressions of the crack severities are formulated. A further simple reconstruction approach allows the evaluation of the exact positions and intensity of the concentrated damages, if displacements of two vibration mode shapes are known at a single cross-section between two consecutive cracks. The proposed reconstruction procedure is applied for the identification of multiple cracks on a free-free beam where measurements have been simulated by means of a finite element analysis.