Issue 48

R. S. Y. R. C. Silva et alii, Frattura ed Integrità Strutturale, 48 (2019) 693-705; DOI: 10.3221/IGF-ESIS.48.65 695 Taking into account   , 0 Ψ a b x , the mathematical definition of a Continuous Wavelet Transform (CWT) applied to the displacement signal u(x) generates a coefficient , a b C that can be expressed in Eq. (3). Note that the CWT coefficient, , a b C , can be interpreted as the sum of all u(x) along a x-axis, multiplying u(x) by   , 0 Ψ a b x shifted and scaled. Therefore, the CWT coefficient, , a b C , can be defined as:       , , , Ψ a b a b o a b o C C x u x x dx      (3) These wavelet coefficients are very sensitive to discontinuities and singularities present in the analyzed signal. Considering this property, it was found that damage due to a sudden loss of stiffness can be detected through mode shapes with wavelet coefficients that achieve large amplitudes like a spike or an impulse at the damage location. This spike of the CWT coefficient is due to damage and is clearer in the finest scales of the CWT. This procedure is the basis of CWT damage detection [4]. P ROPOSED METHODOLOGY he proposed methodology is based on the use of three well-known techniques; Interpolation, Regularization and Continuous Wavelet Transform. Interpolation functions expand the information data available at a few points to many other points, thereby helping damage detection. In this paper researchers will note that, despite the differences between real data and simulated data; the proposed methodology, leads, in the cases analyzed, to the correct detection of the damage. Such methodology can be applied to mode shapes of bridge structures using experimental data or computationally simulated numerical data too. Another advantage is that it is not necessary to have the reference mode shapes of the undamaged or original structure. The methodology is divided into six main steps is shown in Fig. 1. Figure 1 : Flowchart of the proposed methodology. In this paper, either experimental tests or numerical simulation are used to illustrate the proposed methodology. The main steps represented in Fig. 1 are described as follows:  STEP-1: Experimental Tests or Numerical Simulation: The first step consists of carrying out real experimental tests or numerical simulation of the damaged structure.  STEP-2: Mode Shapes: From the previous step, mode shapes of the damaged structure are found. The mode shapes are the basic data that will be used in the next steps. T