Issue 49

E.U.L. Palechor et alii, Frattura ed Integrità Strutturale, 49 (2019) 614-629; DOI: 10.3221/IGF-ESIS.49.56 624 Figure 19 : Spectrum Frequency. After the identification of the first frequency peak, a method was applied to correct the main lobe. This method is essentially based on data from spectral lines that are around the frequency peak and perform a more detailed approximation of the maximum value that lobe reaches. Consequently, a better resulting in a greater approximation of the true frequency of the structure is obtained. This is a very important step in this research, due to the small variation in the frequency generated by the additional mass and the decrease of the stiffness in the damaged places [23]. Thus, the frequency value can be determined for a resolution greater than that given by the FFT, performing a weighted average of the frequencies around a peak detected in the spectrum frequency (Eqn. (2)) [24].     Estimated Frequency j m i j m j m i j m Power i i f Power i           s f f N   (2) where: j is the frequency peak. Power (i) represents the value of the amplitude of the spectral line after the application of the FRF. Once the corrected frequencies are obtained, these values are plotted and interpolated, to obtain a vector with a larger number of data and thus calculate the wavelet coefficients. D ISCRETE W AVELET T RANSFORM . avelets have been widely used to analyze time domain signals. For Wavelet analysis of spatial domain signals, we can simply substitute time for a spatial coordinate “x”. Therefore f(x) is the function to be used in the wavelet transform. The function f(x) may be modes of vibration or displacements due to static load [25]. Similar to the Fast Fourier Transform (FFT) for windows, the one-dimensional Wavelet Transform projects a signal into a two- dimensional space. The Wavelet Transform of the signal f(x) can be, mathematically, defined as [26]:     1/2 * , f x b W a b a f x dx a                (3) where   * ψ .  indicates the complex conjugate of   ψ . . It is also assumed that the mean value of the function   ψ x disappears: W