Structural damage is often thought as a decay of the mechanical properties in a portion of the structure and it is represented by a localized decrease of stiffness. Accordingly, a common way to solve inverse problems posed in structural diagnostics is to determine the change in the stiffness coefficient caused by the damage such that the frequency variations are closest (in some least square sense) to those found experimentally. An error function which measures the distance between experimental and analytical frequency values is minimized via gradient-type methods and the stiffness distribution of a chosen reference configuration is iteratively updated, under some a priori assumptions on the coefficients to be identified. In dealing with real situations only a finite amount of data is available and the presence of many optimal solutions of the minimum problem cannot be excluded. Moreover, general properties of the variational methods for damage detection are rarely discussed and important questions are still partially unsolved even for diagnostic applications in bending vibrating beams.

Taking these aspects into account, a new approach to damage detection in beam structures based on frequency data is proposed in this paper. Assuming that the damage configuration is a perturbation of the undamaged one and that the linear mass density remains unchanged, it is shown that natural frequency shifts and antiresonant frequency shifts induced by the damage contain information on certain generalized Fourier coefficients of the unknown stiffness variation. In particular, the unknown stiffness coefficient is represented on a family of functions given (in the case of bending vibrations) by the square of the second order derivatives of the normalized eigenmodes of the beam. This choice follows rather naturally from the linearization of the Taylor's series development of the eigenvalues in the neighborhood of the undamaged beam, and it is used in the iterative procedure of identification.

The basic aspects of the method are discussed making reference to cracked steel beams either under axial or bending vibrations. As a general conclusion, it emerges that the technique gives a satisfactory identification of the damage, provided that frequency and antiresonant frequency shifts induced by the damage are larger than modelling/measurement errors. Experimental results showed that, in the inverse problem solution, noise and modelling errors on antiresonances are usually amplified strongly with respect to cases in which frequency data is used.

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