PSI - Issue 64

Paul Winkler et al. / Procedia Structural Integrity 64 (2024) 1264–1270 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 1a: Test setup with sensors

Fig. 1b: Crack mapping

3. Modal analysis The measured values recorded with the 24 accelerometers arranged in a grid and a sampling rate of 2048 Hz were used to identify a series of natural oscillations in a frequency range up to 1000 Hz with the associated modal parameters using the Stochastic Subspace Identification (SSI) method (Peeters and De Rock 2001). Several of the vibrations had eigenfrequencies that were close together. As a result of the increasing damage, the eigenfrequencies in particular have changed. As the degree of these shifts was different for different eigenfrequencies, there were sometimes changes in the order of the natural oscillations between individual damage levels. To be able to follow the development of the modal parameters this order can be achieved by comparing the identified mode shapes using the Modal Assurance Criterion (MAC) (Allemang and Brown 1982). The MAC specifies the direction cosine between two respective eigenvectors under consideration. Accordingly, the values lie between 0 and 1 or 0 and 100%. The MAC values are usually plotted in a matrix in which the rows and columns correspond to the two systems being compared. In the case of the solutions to the eigenvalue problem for an undamped multi-degree-of-freedom system, i.e. when two identical solutions with mutually orthogonal eigenvectors are compared with each other, a unit matrix is obtained. If the eigenmodes calculated for a numerical model are compared with those identified from an experiment, perfect orthogonality is not to be expected, but the MAC matrix should still have approximately the structure of a diagonal matrix whose values on the main diagonal should be close to 1 or 100%. Such a MAC matrix is shown in Fig. 2a. There, the eigenmodes identified from the tests after two different load levels were compared with each other. The diagonal structure is clearly recognisable, with clusters of similar mode shapes at relatively closely spaced eigenfrequencies in some places around the main diagonal. Once the eigenmodes from different tests have been correctly assigned, the development of the modal parameters with increasing load or increasing degree of damage can be visualised graphically. This comparison is shown in Fig. 2b for the eigenfrequencies identified for different load levels. All eigenfrequencies were scaled to the respective value after the first load level, as the boundary conditions often change during initial loading, so that a reference to the state before initial loading can lead to misinterpretations. Fig. 4 shows a general trend of decreasing eigenfrequencies with increasing damage. This overall trend corresponds well with the assumption that increasing crack formation is associated with an increasing reduction in stiffness. For some modes the eigenfrequency is raising with increasing damage for the lower load steps. At the beginning of the test we had problems to keep the bearing conditions of the structure constant after reassembling the crossbeams for