PSI - Issue 37

Andrzej Katunin et al. / Procedia Structural Integrity 37 (2022) 292–298 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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2.2. Standardized damage index The majority of DI concept based damage identification methods require the baseline data of a healthy structure for analysis of the changes in dynamic parameters. In practice, the baseline data is rarely available therefore various approaches have been proposed for the estimation of a state of a healthy structure. In this study, the DI approach based on smoothing polynomials was selected for processing of the mode shape data. The approach is based on the assumption that the curvature mode shape of a healthy structure has a smooth surface and thus it can be obtained by means of a regression analysis from the data of a damaged structure. The DI is calculated as the absolute difference between the measured curvature mode shape and the generated smoothed surface representing the healthy structure: , = |( 2 2 ) ( 2 , ) − ( ) ( 2 , ) | + |( 2 2 ) ( 2 , ) − ( ) ( 2 , ) |, (1) where m is a mode number, is the measured transverse displacement of the structure, , are smoothed curvature mode shape surfaces, and s and p are numbers of grid point in x and y direction, respectively. The curvature mode shapes are calculated by using the central difference approximation from the measured mode shapes, while the smoothed surfaces are generated by means of a locally weighted linear least-squares fitting (Rucevskis et al., 2016). The summarized DI for all five modes considered in this study is defined as follows: ( , ) = ∑ ( , , ) 2 ( ) 2 =1 , (2) For the evaluation of the obtained damage indices the statistical hypothesis testing approach (Bayissa et al., 2008) is employed. To classify damaged and healthy elements the obtained DIs are standardized at first: ( , ) = ( , )− , (3) where DI and DI are the mean and the standard deviation of DIs, respectively. The decision on whether the element is damaged is established based on the level of significance for the presence of damage (90 %, 95 % and 99 %) used in the hypothesis test which can be determined from a pre-assigned probability threshold (the corresponding damage threshold values are 1.28, 2, and 3). In the present study, the threshold value for the 90% confidence level was selected to classify damaged elements, and accordingly the values of SDIs lower than the threshold value of 1.28 were set to zero in SDI plots. 2.3. Curvelet transform CT is a kind of multiscale transform introduced by Candès et al. (2006), which has numerous advantages over the popular wavelet transforms, namely, directional selectivity, lack of the boundary effect in the resulting sets of coefficients, and excellent filtration performance. According to these properties, CT was chosen to process the acquired mode shapes. Originally, basis functions (curvelets) in CT are defined in polar coordinates (see (Candès et al., 2006) for more details), however, for applications in the Cartesian coordinate systems it was represented in ℝ 2 . The curvelet coefficients (∙) are defined as (Candès et al., 2006): ( , , ) = 4 1 2 ∫ ̂ ( ) ̃ ( ) ( 〈2 − 1 , 2 − ⁄2 2 , 〉) , (4) where ̂ is the analyzed signal in the frequency domain, is the shear matrix, is the frequency, ̃ ( ) is the frequency window in the Cartesian coordinate system, given by: