Issue 49

E.U.L. Palechor et alii, Frattura ed Integrità Strutturale, 49 (2019) 614-629; DOI: 10.3221/IGF-ESIS.49.56 625   0 x dx       (4) In the Wavelet transform, the signal f(x) is weighted by a function of variable “x”. The respective function is given by [25]:   1/2 , * a b x b x a a            (5) The functions   a,b ψ x are called wavelets or mother wavelet functions. The functions of the Fourier transform by windows usually oscillate and decay rapidly. In contrast to the functions   a,b ψ x , the number of oscillations remains constant with window changing. This means that a wavelet is "stretched" or "dilated" along the space represented by the vales of x along the x-axis. For Fourier transform (FT) with windows, the window size remains constant while the number of oscillations changes. This principle is illustrated in Fig. 20. (a) FT Functions (b) FT Windowed FT Functions (c) Wavelets Transform Functions. Figure 20 : Comparison signals: Fourier transform and Wavelet transform [25]. The calculation of the wavelet coefficients in each possible scale generates a good amount of data. To minimize this task, only a subset of scales and positions are chosen. The chosen scales and positions are based on powers of two, called dyadic scales, which results in a much more efficient and fast analysis. This analysis is called the Discrete Wavelet Transform (DWT) [25]. For this purpose, the scale is defined as j a 2  and the translation or displacement j b k2  where (j,k) ϵ Z and Z is the conjunct of an integer number. Using in these parameters the DWT is defined as follow [28]:         /2 , , 2 2 j j j k j k DWT f x x k dx f x x dx              (6) During the choice of the mother wavelet in the application of the DWT, approximately, one hundred functions were tested so that the best function for the identification of damages [10, 29] could be selected. The selected functions are rbio2.6, bior6.8, sym6 and db5. R ESULTS Case 1 n Fig. 21 the graphs of the data corresponding to the first frequency of the beam vs mass position is shown along the 26 nodes of the beam (discretization made at every 20 cm). The added mass used was 3.266 kg, positioned on each node. I