PSI - Issue 54

J.V. Araújo dos Santos et al. / Procedia Structural Integrity 54 (2024) 575–584 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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al. (2016), Rucevskis et al. (2016), Yang et al. (2017), Oliveira et al. (2022), Uwayed et al. (2023)). The problem of identifying small damage or multiple damage is usually very difficult, as there is always noise in experimental measurements. In these cases, we need to apply numerical techniques which must have low noise sensitivity, so that this noise is not amplified in the process of differentiation. The result of this amplification is that the computed derivatives will have a non-smooth pattern, with several peaks that are not due to damage, leading to unsatisfactory identifications. The present approach to solve this problem, i.e. to reduce the amplification of noise, is the computation of derivatives with Savitzky-Golay smoothing and differentiation filters (Savitzky and Golay (1964)). We also propose a baseline-free method for the identification of slot edges, based on the norm of modal strains. The paper is structured as follows: The formulation of Savitzky-Golay smoothing and differentiation filters for one- and two-dimensional sets of data can be found in Section 2. Section 3 contains a description of the damage identification procedure, involving computations of modal strains and their norm. The results of damage identifications, depending on the mode shape and the type of Savitzky-Golay filter, are discussed in Section 4 after the description of the plate as well as the damage scenarios and cases considered. An analysis of the noise effects is also presented in this section. Finally, conclusions are drawn in Section 5. 2. Savitzky-Golay Smoothing and Differentiation Filters A one-dimensional Savitzky-Golay smoothing filter can be constructed by finding the coefficients of a polynomial of degree ( ) = ∑ =0 = 0 + 1 + ⋯ + −1 −1 + (2.1) such that the error between a 2 +1 points data set ( ) and the values of the polynomial ( ) = ∑ [ ( ) − ( )] 2 =− (2.2) is minimized in a least-squares sense. Savitzky and Golay (1964) proved that the least-square fitting of any data set with a polynomial and the computation of the smoothed values is equivalent to a discrete convolution ( )= ∑ ℎ +1, ( − ) =− (2.3) where the weighting factors ℎ +1, are independent of the data points ( ) . The definition of ℎ +1, and its computation relies on the solution of the normal equations of the least-squares problem which has a solution that can be written in matrix form as = (2.4) with = [ 0 1 … −1 ] , = [ ( − ) … ( −1 ) ( 0 ) ( 1 ) … ( )] (2.5) and is a ( +1)× (2 +1) matrix given by = ( ) −1 (2.6) where