Issue 52

J. Akbari et alii, Frattura ed Integrità Strutturale, 52 (2020) 269-280; DOI: 10.3221/IGF-ESIS.52.21 271 For this purpose, the mathematical models of two-beam structures with different boundary conditions, including pined- pined and clamped-clamped, were established in MATLAB [19] by the finite-element modeling, using Bernoulli beam elements. Two different signal-processing techniques are employed for crack location detection in different damage scenarios by application of mode shape curvature in the absence and presence of noise in the data. S IGNAL P ROCESSING T ECHNIQUES n this paper damage detection of beams with different boundary conditions using discrete wavelet transforms (DWT), Teager-energy operator (TEO) and the combination of them have been explored. The reason for applying the mentioned methods is due to the capabilities of the methods for damage detection, especially in the presence of noisy conditions. Discrete Wavelet Transforms (DWT) Wavelet functions are composed of basis functions that have the capability of synthesizing the signal in time (location) and frequency (scale) domain. In wavelet analysis, the mother wavelet function is defined as Eqn. (1)   a,b 1 t-b ψ t = ψ . a a       (1) Mother wavelet in addition to time t, is described with a,b parameters in which they are the scaling and transformation parameters, respectively. Transformation of continuous wavelet for an arbitrary function, f(t) , is written as Eqn.(2)     * 1 t - b CWT a,b = f t . ψ dt , a a        (2) where J J a=2 , b=2 k . Parameters J,k are the indexes of the synthesizing level of an arbitrary signal. By substitution of these parameters in Eqn.(2), Eqn.(3) could be written as follows.     -J + * -J 2 - D . WT(J,k)=2 f t . ψ 2 t-k dt    (3) According to Eqn.(3), two types of filter is imposed on the signal. The first one is low- pass filter imparted as a scaling function or father wavelet that shows the approximation of a signal, and the second one is a high-pass frequency filter that accounts for the signal components in the higher frequencies (signal details). The coefficients of approximation and details are calculated as Eqn.(4)         + j - + j - cA k = f(t) θ x dx , cD k = f(t) ψ x dx ,       (4) where θ (x) is the scaling function or the father wavelet and ψ (x) is a mother wavelet [20]. The relation between mother and father wavelets could be written as Eqn.(5)     m m m d θ x ψ x ( 1) dx   (5) In this paper, for signal processing and damage detection, the Daubechies and Symlet wavelets [19] with different scaling have been implemented. I