A Probabilistic Approach to the Crack Identification in a Beam-like Structure Using Monitored Mode Shapes and Their Curvature Data with Uncertainty

Abstract

This work, which has been carried out for defects in aircraft structures diagnosis, is considering of measurement errors, different accelerometers positioning and numerical differentiation of the natural vibrations mode shapes as the sources of uncertainty, which affects on the crack identification results. A physical theory to solve the forward identification problem assumed the Timoshenko beam model with opened crack. With 1D cracked beam finite element (FE) model at the different position and depth of the crack we reconstruct the first 5 mode shapes, using vibration amplitudes measured on the fixed uniformly distributed points on the beam surface. All measured amplitudes are noisy by randomly distributed errors. Once these modes reconstructed, we determine their curvatures using central schemes for the first and second spatial finite differences. To obtain the probabilistic means for decision about the damage existence, the multiple numerical simulations of the FE models have been carried out for both intact and defected beams with different damage severity. Using results of these simulations we reconstruct the probability density functions for maximum difference between vibration amplitudes of intact beam and beam with known damage that allow us to separate the damaged and undamaged cases. Next, we calculate the empirical probability distributions, which allow to estimate a probability of the crack location and to distinguish the cases of presence and absence of defect. Then, we compare the sensitivity and robustness of crack parameters predictions by the mode shapes and modal curvatures analysis at the different instrumental precision using known and identified cracks’ characteristics. We established that damage identification using mode shape curvatures data is less reliable because increased noise caused by twofold numerical differentiation of amplitudes measured on the discrete set of points, whereas the mode shapes are more suitable for distinguish perfect intact and damaged state of a structure. Our results confirm that each vibration mode is most sensitive to the damage within the own specific intervals along the beam, and besides, such the sensitivity increases along with mode’s number.