PSI - Issue 37

Deniss Mironovs et al. / Procedia Structural Integrity 37 (2022) 410–416 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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by FE program ANSYS in orthogonal way (see Fig. 2c), so 12 th order polynomial does not fit this vector and does not represent subtle changes in the mode shape. This calls for another type of mode shape formation. What is more, the damage is located symmetrically in the center of the beam axis, so the delamination has little influence on the torsional strain. One might argue with the decision to approximate mode shapes, however, this is dictated by properties of multivariate Gaussian mathematics, i.e. covariance matrix is non-invertible if > (see Eq. (4)). It is possible, however, to implement single variable Gaussian distribution for each feature and multiply these probabilities. Multiple variables distribution product has large scale (sometimes up to 10 15 ) due to large data set, which complicates computations. This is overcome by taking logarithm from probabilities, thus giving scale of tens and hundreds (for example see Fig. 3c).

Delamination region

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Fig. 2. a) Damaged state 4 th mode; b) Damaged state 5 th torsional mode; c) Two mode shape vectors differently formed (X and Y directions combined); d) Comparison between interpolated 4 th mode shape vectors (only X direction plotted). As a workaround for this issue, probabilities of individual parameters (frequency or mode shape) can be compared between samples, see Fig. 3(a) for example. Probability histogram of test samples ( ) for 1 st mode is shown in Fig. 3(b), which clearly shows distinction between healthy and damaged states. More optimal representation of results is seen in Fig. 3(c), as ( ) is plotted against frequency axis. The evaluation of state based only on modal frequency changes would be incorrect, as for some cases damaged state frequency lies within healthy state frequency distribution. Multivariable data set, formed as a combined frequency and mode shape matrix Y , allows to include all possible changes in the analysis, thus giving a sample with completely different probability (Fig. 3(c)). It would be advantageous to preprocess mode shape vectors and obtain single scalar, for example, by using Modal Assurance Criterion (Pastor et al (2012)), or a relatively short vector, which describe slightest changes in original mode shapes vectors. This way one can also fully utilize multivariate Gaussian distribution computation, which captures correlations between features automatically, as opposed to single variable Gaussian distribution.