Damage Detection on a Beam with Multiple Cracks: A Simplified Method Based on Relative Frequency Shifts †

Identifying cracks in the incipient state is essential to prevent the failure of engineering structures. Detection methods relying on the analysis of the changes in modal parameters are widely used because of the advantages they present. In our previous research, we found that eigenfrequencies were capable of indicating the position and depth of damage when sufficient vibration modes were considered. The damage indicator we developed was based on the relative frequency shifts (RFS). To calculate the RFSs for various positions and depths of a crack, we established a mathematical relation that involved the squared modal curvatures in the healthy state and the deflection of the healthy and damaged beam under dead mass, respectively. In this study, we propose to calculate the RFS for beams with several cracks by applying the superposition principle. We demonstrate that this is possible if the cracks are far enough from each other. In fact, if the cracks are close to each other, the superposition method does not work and we distinguish two cases: (i) when the cracks affect the same beam face, the frequency drop is less than the sum of the individual frequency drops, and (ii) on the contrary, cracks on opposite sides cause a decrease in frequency, which is greater than the sum of the frequency drop due to individual damage. When the RFS curves are known, crack assessment becomes an optimization problem, the cost function being the distance between the measured RFSs and all possible RFSs for several vibration modes. Thus, the RFS constitutes a benchmark that characterizes damage using only the eigenfrequencies. We can accurately locate multiple cracks and estimate their severity through experiments and thus prove the reliability of the proposed method.