Wavelet-based damage identification methods in space domain

Liew and Wang (1998) and Wang and Deng (1999) use for the first time wavelet transform to analyse numerical and experimental, static and dynamic, space domain structural responses of simple cracked beams, and to identify damage. They highlight that WA is capable of identifying the abrupt variation in beam deflection due to damage through a local abnormality of the wavelet coefficients at that position. Subsequently many authors studied and showed the effectiveness and versatility of wavelet analysis as tool to detect, localize and quantify damage in generic structural deflections. Hereafter some of the most interesting works in terms of methodology, results and applications are presented. Focusing on the Gaussian wavelets, Gentile and Messina (2003) discuss in a numerical-theoretical way the CWT features of derivation, convolution and smoothing of noisy data. Due to CWT limitation in the presence of noise (in fact CWT behaves as a high-pass filter at the fine scales and looses details at the large scales), they highlight the need of a trade-off for the scales in detecting damage. Moreover, due to CWT redundancy regarding the free choice of scales, the authors recommend the use of continuous WT rather than discrete WT. By analysing different cracked beam modeshapes, the authors notice that the sensitivity in damage detection with respect to crack location depends on the modeshape local curvature in the damaged area. Furthermore, Messina (2004), dealing with transversal beam vibrations in both the non-transformed and Fourier transformed domains, discuss the ability of CWT in conjunction with differential operators to act as frequency filter and therefore to reduce undesired high frequency noise.

Loutridis et al. (2004) analyse through CWT both the analytical and the experimental fundamental vibration mode of a double-cracked cantilever beam by using the 4th order Symlet wavelet. In addition to the task of locating the crack positions, they propose an intensity factor as indicator of crack size. Through a numerical and an experimental study, Wang and Wu (2011) detect the location of a delamination in a beam structure under static loading with a spatial wavelet transform using Gabor wavelet. A barely invisible perturbation in the deflection profile of the delaminated beam at the two delamination edges owing to the curvature discontinuity is discerned through the WA.

Pakrashi et al. (2007) present a detailed numerical and experimental study regarding the issue of efficient and robust calibration of position and extent of

damage in structures. It is observed that wavelet analysis on the mode or the static deflected shape of a structure can successfully identify the presence and the location of the damage even at significant noise levels using the 4th order Coiflet wavelet. Partial windowing of the deflected shapes and consequent wavelet analysis of the segments is found to improve the localization. As far as the damage quantification is concerned, while a wavelet based calibration of damage is found to be inconsistent and unstable due to noise, the authors propose a wavelet-kurtosis based calibration technique which is more robust and consistent. Montanari et al. (2013) consider a FE model of a cracked fiber-reinforced beam in order to analyse its static deflection. The fiber effect in the crack opening response is taken into account through a bridged crack model. Damage detection and calibration are studied in the presence of synthetic noise by varying the crack depth and the fiber yielding condition. When a large proportion of the fibers have yielded, even with constant crack depth, the damage location becomes easier through wavelet analysis. The kurtosis and wavelet-kurtosis techniques for damage severity calibration, exposed in (Pakrashi et al., 2007), are compared. While the kurtosis index seems to well describe damage severity when most of the fibers are yielded, the wavelet-kurtosis technique is seen to be insensitive to damage severity both when few fibers have yielded and when most of them have yielded. Again Pakrashi et al. (2009) statistically deal with the identification of the existence, location, and extent of an open crack from the first fundamental modeshape of a simply supported beam by using CWT with the 4th order Coiflet wavelet. The problem of false alarm and its significant reduction by the use of multiple measurements are illustrated.

Rucka and Wilde (2006b) analyse numerically and experimentally the first modeshape of a plexiglass cracked cantilever by the one-dimensional CWT and the first modeshape of a clamped steel plate with a central defect by the two- dimensional CWT. The 4th order Gaussian wavelet and the reverse biorthogonal 5.5 wavelet, having both four vanishing moments, are used to analyse the beam and the plate, respectively. The problem of damage detection in plates is tackle also by Huang et al. (2009) which develop a 2D CWT algorithm for SHM. The feasible and accuracy of the method in locating the damage positions and in qualitatively assessing the damage severity for both dense and sparse sensor networks is numerically demonstrated.

Zhong and Oyadiji (2011a) compare Stationary Wavelet Transform (SWT) and the Discrete Wavelet Transform (DWT) as tools for small crack detection in beam-like structures. The first four mode shapes of damaged simply supported beams, obtained numerically and experimentally, are analysed. Although crack information can be obtained from the detail coefficients of the SWT or of the DWT of mode shapes, due to the fact that DWT is a down-sampling algorithm whereas SWT is an up-sampling one, the SWT provides better crack identification than DWT, especially when the crack is relatively small and noise

is relevant. Cao and Qiao (2008) propose and validate numerically and experimentally, under relatively high noise environment, a novel methodology, so-called Integrated Wavelet Transform (IWT), for damage detection in structural vibration mode shapes. The IWT algorithm has the merit of integrating the SWT, to extract pure damage information by eliminating random noise and regular interferences, and the CWT to reveal abnormality from the extracted damage. Moreover, a guideline for rationally choosing the optimal mother wavelet for effective damage identification is provided. Reverse biorthogonal 2.2 wavelet is chosen as the optimal mother wavelet for both the wavelet transforms. Gökdağ and Kopmaz (2009) develop a new wavelet-based damage detection approach based on the assumption that a damaged mode shape of a beam is approximately composed of an undamaged mode and other contributors such as variations induced by measurement and local damage. Through DWT, assuming a suitable wavelet function and decomposition level, a proper approximation function to be used as undamaged mode is extracted from the damaged one. In this way, a reliable damage index is defined taking the difference of the CWT coefficients of the damaged mode and those of the approximation function. The method is tested and validate numerically and experimentally.

Rucka (2011) investigates both experimentally and numerically the behavior of the CWT damage detection technique in analyzing the first eight mode shapes of a cantilever beam with damage in the form of notch of depth 20%, 10% and 5% of the beam height. The analysis is performed using the Gaussian wavelets of 4th, 6th and 8th order, having respectively 4, 6 and 8 vanishing moments. The experimental results highlight that damage detection by the wavelet analysis is more effective on higher measured mode shapes and using wavelets with smaller numbers of vanishing moments. Since higher modes contain more regions in which the curvature is null and consequently there is less sensitivity to damage, a reliable damage localization needs at least two modes. In the work of Gianniccaro et al. (2009), based on Gaussian continuous wavelet transforms which behave as differentiator filters with easily controlled lowpass cutting frequencies, a new global index, aimed at identifying damaged places through wavelet analysis of dynamical shapes, is introduced. The appeal of the new damage index is to overcome the need of the undamaged condition of the structure and to bring in an unique formulation the information related to several dynamical shapes. In this way the method makes the analyst free from choosing a specific mode, thus allowing a straightforward multimodal analysis. Moreover, the global index also provides the possibility of reducing the long wave signals which, to a certain extent, can hide damaged places.

Umesha et al. (2009) present a method for crack detection and quantification in beams based on Symlet wavelet analysis. The static deflection is measured at a particular point for various locations of a point load along the beam. This deflection profile is used as the input signal for wavelet analysis. Due to variation

in deflection at some points, compared to their adjacent points, peaks in the wavelet coefficients are observed. Since these peak points can be related to damage, sensor points or supports, to locate the real damage position, the false indicators of damage are eliminated by performing wavelet analysis of the deflection profile measured at another point. A generalized curve, considering the variations of damage size and location, intensity of load, flexural rigidity and beam length, is proposed to quantify the damage.

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