Identification of two cracks in a rod by minimal resonant and antiresonant frequency data

In this paper we consider the identification of two cracks of equal severity in a uniform free-free rod under longitudinal vibration. Each crack is simulated by a translational spring connecting the two adjacent segments of the rod and the cracks are considered to be small. We show that the inverse problem can be formulated and solved in terms of three frequency data only, corresponding to a suitable set of low resonant and antiresonant frequencies. Closed-form expressions of the damage parameters in terms of the measured frequency shifts are obtained. The paper improves existing results available in the literature, since the use of antiresonant frequencies allows to exclude all the symmetrical crack locations occurring when only natural frequency are used as data. The analysis also explains why the use of high frequency data introduces spurious damage locations in the inverse problem solution. Numerical simulations show that if accurate input data are available then damage identification leads to satisfactory results.