No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams
Abstract
This paper deals with the detection of open cracks in beam structures that undergo transverse vibrations. The investigation is aimed at detecting the location of open cracks in damaged beams by minimizing measurement data and baseline information of the structure. The study is carried out by using the continuous wavelet transform (CWT). The application of this recent, but advanced, mathematical tool is initially presented through a theoretical background, which is believed to be valuable for bridging the gap between the CWT and previous existing techniques. It is shown how the possibility to efficiently identify localized damages by CWT comes up from the intrinsic capability of the wavelets to collect several mathematical tools in only one mathematical aspect: derivatives, convolution and appropriate smoothing of data are translated into the CWT. Simulations show how the redundancy of the CWT in the functional space is able to efficiently identify locations of open cracks in the presence of noisy or clean data. Indeed, the possibility to approach the problem by using different families of wavelets, for several available scales, allows a successful application of the characteristic microscopy of the wavelets. The technique may be promisingly applied to discrete vibrational data.
... Numerous further attempts were focused on the application of wavelet transforms (WTs) due to their high sensitivity to small disturbances in the modal curvatures. The continuous wavelet transform (CWT) was successfully used by Gentile and Messina [31] for detection and localization of open cracks in beams. This approach was confirmed by Douka et al. [32] in the similar study of identification of cracks in beams, which was additionally improved by proposing an intensity factor law for the quantification of the crack size. ...
... From the results presented in Figure 4, it is clearly visible that for the frequency ranges of 0-20 Hz and higher the damage localization and identification are not possible due to blurring the damage signature, while for the range of 0-10 Hz one can precisely localize the simulated damage and even identify its boundaries. Moreover, the boundary effect for this range of frequency does not appear, which is a great advantage of this approach with respect to WT-based damage identification procedures, where this effect can mask damage signatures near the boundaries as well as significantly decrease distinguishability of the detected damage, as it was reported in numerous studies [31,33,43,80]. One can observe that in the case of TFDs the frequency range is equivalent to the length of the support of the applied wavelet in the WT-based approaches in the light of the increasing the width of the distortion on the boundaries due to the boundary effect. ...
... The obtained results for the particular damage scenarios and modes are presented in Figure 5. From the results presented in Figure 4, it is clearly visible that for the frequency ranges of 0-20 Hz and higher the damage localization and identification are not possible due to blurring the damage signature, while for the range of 0-10 Hz one can precisely localize the simulated damage and even identify its boundaries. Moreover, the boundary effect for this range of frequency does not appear, which is a great advantage of this approach with respect to WT-based damage identification procedures, where this effect can mask damage signatures near the boundaries as well as significantly decrease distinguishability of the detected damage, as it was reported in numerous studies [31,33,43,80]. One can observe that in the case of TFDs the frequency range is equivalent to the length of the support of the applied wavelet in the WT-based approaches in the light of the increasing the width of the distortion on the boundaries due to the boundary effect. ...
The paper presents the novel method of damage identification and quantification in beams using the Wigner-Ville distribution (WVD). The presented non-parametric method is characterized by high sensitivity to a local stiffness decrease due to the presence of damage, comparable with the sensitivity of the wavelet-based approaches, however the lack of selection of the parameters of the algorithm, like wavelet type and its order, and the possibility of reduction of the boundary effect make this method advantageous with respect to the mentioned wavelet-based approaches. Moreover, the direct relation between the energy density resulting from the application of WVD to modal rotations make it possible to quantify damage in terms of its width and depth. The results obtained for the numerical modal rotations of a beam presented in this paper, simulating the results of non-destructive testing achievable with the shearography non-destructive testing method, confirm high accuracy in localization of a damage as well as quantification of its dimensions. It was shown that the WVD-based method is suitable for detection of damage represented by the stiffness decrease of 1% and can be identified and quantified with a high precision. The presented results of quantification allowed extracting information on damage width and depth.
... It must be said though, that when damage assessment is concerned, response-based methods fail to provide both the type and the severity of damage, whereby model-based methods become a useful option. Several vibration-based methods that rely only on recorded responses have been proposed in literature (Pandey et al 1991, Stubbs et al 1992, Ratcliffe 1997, Zhang & Aktan 1998, Wahab & De Roeck 1999, Ho & Ewins 2000, Pai & Jin 2000, Lu et al. 2002, Parloo et al 2003, Limongelli 2003, Gentile & Messina 2003, Dutta & Talukdar 2004, Zhang et al 2013, Surace et al 2014, Corrado et al. 2015. This paper will focus on methods that perform localization of damage through the detection of irregularities in the deflected shape of the structure. ...
... This means that all the available modes or of operational shapes should be included in the damage localization procedure in order to have accurate results. Several damage localization algorithms based on the detection of shape irregularities have been proposed in literature (Pandey et al 1991, Stubbs et al 1992, Ratcliffe 1997, Wahab & De Roeck 1999, Ho & Ewins 2000, Limongelli 2003, Gentile & Messina 2003, Zhang et al 2013, Surace et al 2014. Most of them rely on modal shapes, others on operational deflected shapes retrieved from Frequency Response Functions. ...
... ; Y x s is the signal transformed using the wavelet and s is a real positive number called 'dilation parameter' of the wavelet function. A specific derivative can be approached through a wavelet transform, by choosing appropriately the number of vanishing moments of a Gaussian wavelet (Gentile and Messina, 2003). As equation (10) shows, the signal transformed using the wavelet is proportional to the m-th derivative of the function: for m=2 the transformed signal is proportional to the curvature. ...
Several methods proposed in literature for the localization of stiffness losses rely on the detection of irregularities in the deflected shape of the structure. This requires accurate description of the deflected shape achievable through a high spatial resolution of sensors, high quality or measures and accounting for the approximations introduced by signal processing. In the first part of this paper a survey of vibration-based damage localization algorithms based on the detection of (changes of) irregularities in the deflected structural shape is reported. Most of these methods rely on damage parameters defined in terms of the local variations of curvature due to the direct relationship of this parameter with the variations of stiffness. Due to some drawbacks related to the estimation of curvature from noisy recorded responses, other methods have been proposed to detect local variations of the deformed shape without directly computing the curvature. Also, many of the methods proposed in literature have been validated only on numerical models, due to the scarce availability of experimental data recorded on damaged structures. Recently data recorded on benchmark structures have become available giving the opportunity to verify the capability of these methods for damage localization in real-world conditions. In the last part of the paper, a method for damage localization based on the detection of localized changes in the structural deformed shapes, the Interpolation Method, is applied to two benchmark structures. The first is the UCLA Factor Building whose response to several non-destructive earthquakes has been recorded by a dense network of sensors. The second is the 7th storey portion of building tested to collapse, using base inputs of increasing severities, on the USDS shaking table.
... Small changes in modal parameters due to damage are difficult to detect without post-processing to enhance the signals [8,23]. The CWT has been used for this purpose on mode shapes from measurements on beams and plates to detect damage and its location [10,[24][25][26] and on the displacement response of Euler-Bernoulli beams under various loading conditions [27]. The WT provides a means to analyze localized areas of a signal at different scales of time and space [28]. ...
... The performance of CWT applied to mode shapes is investigated numerically as a function of the location and depth of cracks for a beam and the effect of sampling intervals are found to be an important factor for wavelet analysis [37]. Successful applications of the wavelet analysis are mostly performed on numerical models with significant levels of damage [26,38]. However, CWT can fail to detect and indicate the location of damage and produce multiple false detections when the measurements are noisy and the level of damage is low [39,40]. ...
... In this study, in order to detect local anomalies of the mode shapes caused by damage, the Gaussian wavelet [64] was used as the basis of wavelet function for the CWT analysis (Fig. 10). The symmetric and oscillatory shapes of the Gaussian wavelet results in a better description of local features contained in the mode shapes [26,65]. A more detailed discussion about wavelets and damage detection using the CWT can be found in [24]. ...
This paper casts a response-based damage detection approach that combines different techniques for remote monitoring of structural health. An experimental study is carried out on a continuous steel I beam fabricated by the assembly of three portions joined together by bolt connections. Each beam portion has a 2mm width notch covering half of the beam depth. The test setup is designed in a fashion that not only it allows one to choose desired boundary conditions (simply supported, clamped-clamped), but it also allows different reversible damage scenarios by weakening/strengthening the stiffness of the beam at each joint. Data analysis is performed in two stages: Firstly, the Continuous Wavelet Transform (CWT) method is applied to identify abnormality of retrieved vibration mode shapes measured from specific damaged states at a few measuring points (16). However, it has been shown that the wavelet analysis quite often failed to identify slight perturbations in the mode shapes usually contaminated by noise at low level of damage severities. Secondly, the so-called Principal Component Analysis (PCA) is proposed as a comprehensive statistical study to improve the robustness of the structural health monitoring (SHM) by re-expressing the wavelet coefficients in terms of the most underlying variability. A high number of measurements (approximately 100) is conducted at each damage level to supply a large enough number of variables. The analysis has shown that the Principal Components (PCs) are oriented in a way that most significant features in the data are represented by the first few PCs with the largest variance, whereas random features caused by noise involve higher order PCs that contribute less to the overall variance. The results indicate that using the contribution of the CWT together with de-noising algorithm of the PCA may highly increase the applicability of SHM for damage decision making for beam-like structures in noisy conditions.
... Several families of wavelets are described in the literature; in this study, wavelets from the Gaussian wavelet family were used due to their high efficiency in the detection of singularities [31,35]. The family of Gaussian wavelets is based on the Gaussian function g(x) = C a e −x 2 , by taking the a-th derivative of g(x) [43]. ...
... The family of Gaussian wavelets is based on the Gaussian function g(x) = C a e −x 2 , by taking the a-th derivative of g(x) [43]. The first four wavelets from the Gaussian family have the following form [31]: ...
Externally bonded reinforcements are commonly and widely used in civil engineering objects made of concrete to increase the structure load capacity or to minimize the negative effects of long-term operation and possible defects. The quality of adhesive bonding between a strengthened structure and steel or composite elements is essential for effective reinforcement; therefore, there is a need for non-destructive diagnostics of adhesive joints. The aim of this paper is the detection of debonding defects in adhesive joints between concrete beams and steel plates using the modal analysis approach. The inspection was based on modal shapes and their further processing with the use of continuous wavelet transform (CWT) for precise debonding localization and imaging. The influence of the number of wavelet vanishing moments and the mode shape interpolation on damage imaging maps was studied. The results showed that the integrated modal analysis and wavelet transform could be successfully applied to determine the exact shape and position of the debonding in the adhesive joints of composite beams.
... The appropriateness of CWT in damage detection problems, in one-and two-dimensional structures has been confirmed by numerous research studies. Several results on the application of CWT for damage detection problems have been reported for one-dimensional structures, like cracks in beams [23][24][25][26], and twodimensional structures, like cracks and delaminations in plates [26][27][28][29][30][31][32]. In all these studies, CWT is adapted to baseline-free damage detection algorithms, which, undoubtedly, is the great advantage of WTs in such applications. ...
... This sudden change, in turn, results in difficulties to visualize the damage position, because the magnitude of the values of wavelet coefficients at the boundary are often higher than the resulting values of the coefficients representing damage signatures. Numerous studies on the boundary effect were performed in the last two decades [25,26,28,42], and numerous solutions to reduce this effect have been proposed. The early studies on the boundary effect reduction were based on its padding or substitution of some values at the boundary regions. ...
In this paper, a novel damage identification approach based on differences in modal rotation fields obtained from shearographic measurements of structures and post-processed with the 2D continuous wavelet transform-based algorithm is presented. It is shown that the baseline approach, which assumes analysis of differences in modal rotation fields, allows to improve the sensitivity of a method to damage, making it possible to identify smaller damage with respect to the baseline-free approach. Moreover, a new approach for the reduction of the boundary effect is presented, which allows for the improvement visualization of the resulting maps using the proposed damage identification method. The proposed method was tested with eight damage scenarios, which allowed to prove repeatability of the obtained results, as well as to define a detectability threshold which is on the level of 3.5% of the thickness reduction.
... In this paper, the performance of the approach based on the S-transform is tested and validated on 1D and 2D vibration data obtained during both numerical and experimental studies. In particular, the case studies presented in this paper, are focused on proving effectiveness of an identification of damage of various types and various structures, comparable to the mentioned WT-based algorithms, however the application of the Stransform allows to resign from selection of a basis function and completely reduces the boundary effect [53][54][55][56][57], which are the two main problems of WT-based algorithms in application, among others, to structural damage identification problems. The mentioned advantageous properties of the S-transform as well as the obtained promising results lead to further deep studies on its application in vibration-based structural damage identification problems. ...
... From the presented results, one can also notice that the S-transformbased algorithm does not produce any disturbances at boundaries, known in publications as the boundary effect, which is one of the serious problems in damage identification studies using e.g. WT [53][54][55][56][57]. This is an important advantage of the current approach which makes it possible to identify damage also near the clamped boundaries of a tested structure. ...
An improvement of vibration-based damage identification in structures attracted much attention over the last decades. Many of the developed algorithms to-date, like those based on wavelet transforms (WTs), reveal high sensitivity to damage, however suffer from numerous drawbacks, which include significant dependency of a damage detectability on selection of a basis function or the boundary effect. This paper presents a novel alternative approach for damage identification in beams and plates based on the S-transform, which combines numerous advantages of the short-time Fourier transform and an ability of multiresolution analysis of WTs. The efficiency of the proposed approach was tested and validated on 1D and 2D mode shapes of various composite structures and various types of damage obtained both from numerical simulations and experimental studies. The obtained results confirmed high sensitivity of the approach based on the S-transform to structural damage as well as revealed full reduction of the boundary effect.
... Any mother wavelet is continuous in both time and frequency domains and is the source of construction of other basis functions. Several research studies have confirmed the effectiveness of CWTs in 1D and 2D damage detection in beams (Gentile and Messina 2003;Douka et al. 2003;Loutridis et al. 2004;Rucka andWilde 2006), trusses (Mousavi et al. 2021) and plates Rucka and Wilde 2006;Huang et al. 2009;Fan and Qiao 2009;Janeliukstis et al. 2017;Yang and Oyadiji 2017;Zhou et al. 2018), respectively. CWTs have been successfully applied in all these studies, regardless of their potential lack of baseline information. ...
The aim of this paper is to find the proper families and mother wavelets for a successful localization of multiple damages in beams. The post-processing of differences in modal rotations, experimentally measured with shearography, and differences in modal curvatures of aluminium beams are carried out with 14 wavelet families and 84 mother wavelets. The damage identifications show that the best families of wavelets for the post-processing of modal data are the Shannon, frequency B-spline, and complex Morlet families. In order to select the modes that are more sensitive to damage, a novel parameter for the evaluation of the most changed mode is proposed. The boundary effect, which is often found in wavelet-based damage identifications, is also addressed in this paper. The results show that the post-processing of differences in modal curvatures yields better damage identifications than the post-processing of differences in modal rotations.
... Wang and Deng (1999) proposed that the wavelet transform be directly applied to spatially distributed structural response signals, such as surface profile, displacement, strain, or acceleration measurements. The CWT of the fundamental mode shape and its Lipschitz exponent was used to detect the damage location and extent in a beam by Hong et al. (2002), Gentile and Messina (2003), Douka et al. (2003), and Chang and Chen (2003). The wavelet packet transform (WPT) is an extension of the WT, which provides a complete level-by-level decomposition of signal. ...
... parameter (denotes window width) τ shifting parameter (denotes location of moving wavelet window)Gentile and Messina[24] studied the selection criteria for a wavelet basis function in the presence of noise and demonstrated the performance of Symlet and Gaussian wavelets. Complex Gaussian, complex wavelet and complex frequency B-spline wavelet functions were applied on time history responses from four stories of ASCE benchmark steel frame structures to study effect of wavelet function choice on calculating modal parameters. ...
Vibration based techniques are gaining popularity for damage detection, precise localization and damage quantification. These
techniques involve recording of the vibration signatures and their analysis for temporal and spectral characteristics to arrive at
conclusive statement. Among different damage identification techniques, most of the algorithms are model based approaches and
lacks standardization and accuracy. In this paper, feasibility of using output-only model-free wavelet based techniques for
damage identification of 6-storied scaled reinforced concrete (RC) building is studied. The vibration signals at different floor
levels of the RC building were acquired using wireless accelerometers. The vibration measurements were carried out for different
cases i.e. bare frame and varying mass at different floors. The signal discontinuity of the acceleration response of RC building
was extracted using complex continuous Gaussian wavelet transformation and analysed. The results show that wavelet
coefficients are directly influenced by the change in physical properties of structure and are able to detect damage to a reasonable extent.
... Figure 7.7 shows large angular wavelet coefficient values at the two dimensional extremities of the beans which semi to indicate strong singularities at these positions. Gentile and Messina (2003) elaborated that the appearance of these singularities are due to the boundary conditions imposed upon the beans. In addition. ...
p>The use of GFRP sandwich materials is increasingly becoming popular. However, during the manufacturing process internal and external defects could occur. A vibration based NDE approach, using an SHM system, offers the possibility of examining the GFRP sandwich structure while it is in service. The research presented in this thesis highlights the various aspects of implementing this SHM based approach by studying and making improvements to individual components of the SHM system, namely an embedded sensor network and intelligent signal processing algorithms. A beam configuration was chosen as the test platform of choice to implement the SHM system for the GFRP sandwich structure.
Considering the issue of embedding a sensor network within the GFRP sandwich structure, a modified vacuum infusion manufacturing technique was conceived which enabled a small network of optical FBG strain sensors to be successfully embedded between the skin and core of the GFRP sandwich beam structure.
The intelligent signal processing algorithm developed utilised the wavelet transform estimated Lipschitz exponent as a damage sensitive signal feature. It was first successfully demonstrated, in its spatially varying form, for characterising debonds in numerically simulated models of GFRP sandwich beams. Subsequent experimental investigation of manufactured GFRP sandwich beam specimens, containing embedded FBG strain sensors and subjected to random loading conditions, showed that key features in the singularity spectrum, which is the spread of Lipschitz exponents, and its associated moment generating function, not only allowed successful identification of a variety of defects, but they also enabled quantitative identification of their approximation position on the GFRP sandwich beam specimen.</p
... It is an excellent mathematical tool for analysing non-steady-state signals. It has been extensively applied to various structures for damage identification [24][25][26][27][28]. Hou et al. used the Daubechies wavelet to transform the acceleration response of the structural dynamic model with wavelet transform and identified the loss position according to the sudden signal point changes [29,30]. ...
Structural health monitoring is extensively used in new and old structures. During the process of monitoring, a large amount of data is generated. The selection of the appropriate methodology for the analysis of these data constitutes a major problem for maintenance personnel. Therefore, the purpose of this study is to programme data according to the curvature mode and wavelet transform theory to achieve automatic identification of structural damage and explore its applicability. First, the damage model is established with finite element software, and the applicability of the theory is verified by analysing the wavelet coefficients before and after structural damage. Thereafter, a programme is written to achieve the automatic output of the structural damage location based on the finite element results and basic theory. Data fitting is then performed to estimate the degree of structural damage. To evaluate the effects of the practical application of the programme, experimental verification is conducted. The experimental results demonstrate that the programme can automatically output the damage location and avoid the occurrence of calculation errors.
... Yan et al. [21] used energy spectrum of signals analyzed with wavelet transform to evaluate the ability of detecting cracks in a honeycomb sandwich plate by using a natural frequency and dynamic response. Gentile and Messina [22] used CWT to detect opening of cracks in beam structures. In [23], the authors presented damage detection in concrete using Fast Fourier transform (FFT) and CWT. ...
This paper summarizes the results of research aimed at assessing cracks in reinforced concrete structures using embedded ultrasonic sensors. The diffuse ultrasonic waves were considered to evaluate the health status of the tested structures. There are different algorithms used to detect cracks in the structure, but most studies have been performed on benchmark reinforced concrete (RC) structures and in laboratory conditions. Since there were difficulties with the validity of damage detection in real structures in the presence of environmental changes and noises, the application of advanced signal processing methods was necessary. Therefore, the wavelet transform was applied to process ultrasonic signals acquired from multiple civil structures. It is shown that the ultrasonic sensors with an applied wavelet transform algorithm on collected signals can successfully detect cracks in the laboratory as well as in a real environment. Experimental results showed a perfect match for detecting damage and quasi-static load in the presence of environmental changes. The results were confirmed with other techniques. In addition, designing an extra filter for removing noises can be avoided by using the applied algorithms. The obtained results confirmed that diffuse ultrasonic sensor methodology with the proposed algorithm is useful and effective in monitoring real RC structures, and it is better than traditional techniques.
... Wavelet transform has raised lots of interest due to its intrinsic capability to do differentiation and noise reduction in the same procedure. Gentile & Messina (2003) applied continuous wavelet transform to discrete vibrational data to detect crack damage in beam structure. In this study, the wavelet transform was conducted based on Matlab code accomplished by Luo et al (2006). ...
Existing vibration-based approaches for damage detection are qualitative. This paper presents a novel two-step approach for characterisation of laminar damage, such as delamination in composites and thickness reduction in metallic structures due to corrosion damage. This new approach first employs a gapped smoothing method to determine the location from curvature data. The severity of damage is then determined in the second step using an inverse method by matching predictions of finite element analysis with deflection (or curvature) data pertinent to low-frequency vibrational response. An assessment of various approaches for computing curvature reveals that wavelet transform is a promising method for simultaneously removing noise and computing curvature. Numerical simulations show that this new two-step approach is capable of quantifying the size and severity of structural damage to be used as input for a residual strength assessment.
... In the application of wavelet analysis with discrete displacement data, the number of measurement points (space sampling) and measurement noise play a critical role (Quek et al., 2001;Hong et al., 2002;Gentile and Messina, 2003). The space sampling controls mainly the localization resolution and the presence of noise affects mainly the accuracy of the identification results since the wavelet coefficients are highly sensitive to local perturbations. ...
Wavelet analysis can be used in local damage detection due to its ability of revealing discontinuities induced by damage in the displacement field. This paper focuses on the application of wavelet analysis to detect and identify multiple damages using the static deflection of beams. The local damages are located by the wavelet maxima lines and their severity are evaluated from a damage index obtained from the wavelet coefficients along the corresponding maxima lines. A series of experimental tests were conducted to examine the performance of the methodology for multiple damage scenarios. The static deflections of the beam were measured by a Digital Image Correlation system. As an application, a l1 regularization based filter is adopted to diminish the measurement noise which is critical in the application of wavelet analysis. The paper shows the capability of using wavelet analysis for closely spaced notch-type damage detection. It also analyzes the limits of the method in estimating damage with relative small severity in the presence of severe ones.
... In the frequency domain, Time-invariant methods use Fourier analysis as the primary signalprocessing tool and time-invariant models to describe the structural behaviour. Damage features are usually defined in terms of modal parameters, mainly combinations of frequencies, modal shapes and their derivatives [13,14,15,16,17,18,19] or in terms of Operational Deformed Shape (ODS) retrieved from Frequency Response Functions (FRFs) [20,21,22,23,24,25]. Time-invariant methods that operate in the time domain use statistical tools to develop mathematical models of the structural response, based on the internal structure of the recorded data. ...
This paper is integral part of the Special Issue on “Existing Concrete Structures: Structural Health Monitoring and Testing for condition assessment.” It deals with vibration‐based methods (VBMs) for damage localization that approach the problem of structural integrity management through the analysis of the dynamic response of the structure under ambient or forced vibrations. In the last years, these methods received a widespread interest in the structural health monitoring (SHM) community due to the possibility to use them for continuous SHM and real time damage identification. The performance of these methods is commonly verified on numerical models or laboratory specimens that, by their nature, cannot reproduce all the sources of uncertainties found in practice. The availability of data recorded on a real benchmark, the S101 bridge in Austria, enabled the comparison of three well known vibration‐based time‐invariant methods for damage localization, namely, the curvature method, the interpolation error method, and the strain energy method. The bridge, built in the early 1960, is a typical example of a European highway bridge. Responses to ambient vibrations were recorded both in the undamaged and in several different damage scenarios artificially inflicted to the bridge. This paper reports the results of the application of the three mentioned methods of damage localization to this case study.
... Double cracked cantilever beam and bridge structures are analyzed by CWT (Continuous wavelet transform), results shown the proposed method is effective in single and multi-damage detection even in noisy environment [12] [6]. Some mode shapes are better sensitive in damage but not all mode shapes [13]. Developed a new method from beam damage index is proposed (MSDBDI) for plate structure [14]. ...
: In this paper tunable Q-factor wavelet transform is implemented into damage identification. Fixed-Fixed beam damage identification problem is demonstrated. Translation and Rotational mode shapes are used as an input signal, the TQWT algorithm depends Q-factor and asymptotic redundancy, when it matches with the oscillatory behavior of the input signal it is tuned. This method decomposes a signal into a high-Q-factor and low-Q-factor component, and it can be used to differentiate the damaged and undamaged mode shapes of the beam structure. TQWT coefficient is used as damage index to locate and quantify the damage. Proposed method evaluated experimentally and results shows TQWT algorithm has a potential to detect even a small damage (10% stiffness loss) present in the structure.
... Wavelet transform (WT) is very effective signal processing tool for detecting discontinuity in a noisy signal [11,12]. In literature, wavelet based signal processing techniques are broadly used for the detection of slope discontinuity in a beam elastic line [13][14][15][16][17]. One of the first applications of WT to detect crack in a beam was presented by Surace and Ruotolo [18]. ...
An optimistic attempt for the detection of a crack in a beam using discrete wavelet transform is proposed. Finite element model of the simply supported beam with a transverse open crack is considered. Crack in a beam causes a slope discontinuity in the elastic line. The deflection of the beam is used to locate the slope discontinuity using wavelet transform. The effect of measurement noise on the wavelet decomposition level is studied. An inexpensive digital image based technique is employed to obtain the deflection of the cracked beam for experimental investigations. White pixels due to dust particles are removed using connected component algorithm. A suitable edge detection technique is selected for finding the edged image on the basis of peak signal to noise ratio values. The deflection of the non-cracked edge of the beam is used as an input signal for wavelet transform to detect the crack location. Detection of a crack in the beam with different crack depths is investigated experimentally. The proposed method works well with numerically simulated as well as experimentally measured cracked beam deflections.
... For this purpose, the complex Morlet wavelet was utilized for analyzing the acceleration time histories. Gentile and Messina [6] used the CWT for open cracks identification in beam structures under transverse vibrations. Le and Argoul [7] used the CWT of the free vibration decay responses of a linear system for modal identification. ...
In this paper, acceleration responses of steel MRFs under different ground motions using incremental dynamic analysis (IDA) are used for nonlinear damage diagnosis. In the first step, auto-regressive moving-average with exogenous input (ARX) model, along with stabilization diagram, is utilized to assess the natural frequencies of MRFs. In the second step, complex Morlet (cmorfb-fc) wavelet-based refined damage-sensitive features (rDSFs), as new DSFs considering higher mode contributions and end-effect modifications, are proposed. Improved efficiency and accuracy of the proposed rDSFs are presented through benchmark steel MRFs. Results indicate that the damage pattern (DP) causing first plastic hinge formation and also the damaged story can be accurately recognized using the proposed wavelet-based rDSFs. In addition, the damage extent (severity) for each DP at each story can be assessed using the cumulative sum of wavelet energy at time-shift b (or sum of scalogram for each time) over the refined effective time of vibration (refined ETV) for each output acceleration when compared with that over the refined ETV for the input ground acceleration. Finally, results also indicate improvement over the DSFs existing in the technical literature.
... Detection of changes in modal parameters due to low levels of damage can be improved with post-processing of vibration data (Chen et al., 1995). In particular, the Wavelet Transform (WT) has been used to extract more detailed information from modal shapes (Gentile and Messina, 2003;Rucka and Wilde, 2006;Solís et al., 2013). The WT provides a means to characterize local features in signals at different scales of time and space (Wang and Deng, 1999). ...
Numerous procedures have been developed for the detection and the localization of damage in structures based on changes in the dynamic or static response of structures. Among these, procedures based on wavelet analysis of mode shapes appear to offer a superior performance especially for low levels of damage. In order to evaluate the relative merit of these approaches, criteria based on statistical and probabilistic performance are evaluated as a function of damage level for an experimental beam. These measures include the probability of detection, the probability of false alarms, and the safety index. The safety index used in this application is for the beam under pure bending, which provides a uniform criteria for damage at any location along the beam. The experimental data is obtained on a steel beam where the level of damage is controlled at two locations along its length. A total of 16 equally spaced accelerometers are deployed along the length of the beam. The dynamic properties used to illustrate the proposed procedure are changes in the frequency of the first mode of vibration of the beam and changes in the wavelet coefficients for the first mode of vibration. The data obtained for 5 increasing level of damage at two locations is used to derive prediction equations for the dynamic properties and to estimate the probability of detection and of false alarms as a function of damage level. The results indicate that the procedure based on wavelets is more efficient than the one based on natural frequencies in detecting and localizing low levels of damage. The results also indicate that a monitoring strategy based on wavelets can detect damage before structural safety is significantly compromised while maintaining low probabilities of false alarms.
... For the case of Wavelet Analysis methods (DWA, CWT and WPS) a mother wavelet needed to be specified. Comparison of several studies carried out with different mother wavelets (see [11] and [12]) showed that good results were found for the Gauss 4 mother wavelet for the CWT method in the scale 2. ...
In this document, the performance of damage detection methods used for the evaluation of bridge structures was determined. To do that, these methods were applied to the experimental dynamic parameters obtained from cracked steel I beams. Different damage scenarios were simulated in the steel I beams through saw-cuts perpendicular to the longitudinal axis of the beams. The damage detection methods were evaluated under different damage scenarios tried to represent fatigue damage in bridge structures. Ambient vibration tests, before and after damage, were performed on analyzed beams and the obtained dynamic parameters were used for the damage detection procedure. For all the evaluated scenarios, only the first three mode shapes were taken into account. Results indicated high possibility of damage detection when the severity of damage increase, damage is close to a measuring point and far away from an inflexion point and/or the boundary conditions. It was found that Level I methods (just detection) are not confidence to detect damage. On the other hand, level II methods (location) had good performance for the most severe damage scenarios. It was found out that Wavelet based methods are the best choice for their application to bridge structures.
... The performance of WTs depends on the scale selected for decomposition of the signal. Gentile and Messina (2003) stated that an appropriate wavelet and scale selection is an efficient approach when data is noisy. They concluded that the finest scale is not the best choice when the data is noisy. ...
Wavelet transforms (WTs) have gained popularity due to their ability to identify singu-larities by decomposing mode shapes of structures. In VBDD, the support condition of a structure influences structural responses and modal properties. In fact, the structural responses and modal properties are a lot more sensitive to changing boundary conditions than to crack and fatigue damage, resulting in inaccurate damage detection results. Therefore, in this study, sensitivity tests to estimate a suitable distance range which allows damage detection by imposing single support damage are carried out. The estimated appropriate distance is then applied to detect damage at multiple supports. This involved the applicability of response acceleration of plate structures to support assessment by applying continuous wavelet transform (CWT) and discrete wavelet transform (DWT). The damage cases have been introduced by releasing bolts at specified fixed supports of the plate to simulate the damage. The response accelerations of the rectangular plate at points close to the supports were measured and decomposed using CWT and DWT to assess the structural integrity of each support. The results showed that an appropriate distance range was necessary for accurate damage detection, and both, CWT and DWT could provide reliable outputs. However, the first-and fourth-level detail coefficients of DWT failed to indicate damage in some cases. A more detailed investigation of the effect of different wavelet scale ranges on damage detection using CWT demonstrated that the accuracy of damage detection increased as the scale decreased.
... Despite being of fundamental importance in various fields of study, numerical differentiation (ND) is known to be an ill-posed problem, meaning that small perturbations lead to large error in the approximate solution. In this context, several works have reported the potential offered by the CWT to derivative calculation [18][19][20] , which originates from its intrinsic capability to combine smoothing with differentiation, as briefly explained in what follows. It should be mentioned first that the CWT itself, and a closely related transform known as the S-transform (ST), are well-studied IF estimators [21,22] , but here, the CWT will be used for a completely different purpose, which is derivative approximation. ...
A new method for nonparametric and adaptive instantaneous frequency (IF) estimation of monocomponent signals based on time-frequency distributions (TFDs) is presented. This method uses an estimate of the IF second-order derivative to approximate the width of the TFD observation window associated with the estimation least mean squared error (MSE), which was previously derived in a closed-form expression. The derivative estimate is obtained in two steps. First, a preliminary TFD is computed and its local maxima are used as a rough estimate of the IF trajectory. Thence, the continuous wavelet transform (CWT) is introduced for calculations of high-order derivatives. The proposed method is evaluated and compared with state-of-the-art algorithms based on the intersection of confidence intervals (ICI) rule. Numerical results demonstrate that the proposed method achieves a significant improvement in estimation accuracy.
... Ambient-vibration measurements provide non-destructive testing data that can be obtained quickly at a reasonable expense. Although subject to much debate (Behmanesh & Moaveni, 2015;Gentile & Messina, 2003;Moaveni, Conte, & Hemez, 2009), modal parameters derived from ambient vibrations, and more particularly fundamental frequencies, have been shown to be indicators of structural damage (Astorga, Guéguen, & Kashima, 2018;Calvi, Pinho, & Crowley, 2006;Clinton, Bradford, Heaton, & Favela, 2006;Katsanos, Sextos, & Elnashai, 2014;Michel, Zapico, Lestuzzi, Molina, & Weber, 2011;Mucciarelli et al., 2004;Vidal, Navarro, Aranda, & Enomoto, 2013). ...
After a damaging earthquake, assessment of the residual seismic capacity is required for large parts of the building stock. Increased vulnerability of structures together with the threat of immediate aftershocks call for rapid and objective decision making. Structural identification has the potential to reduce parameter-value uncertainties of physics-based models through interpreting measurement data. Significant amounts of uncertainty are associated with the non-linear behaviour of structures during extreme events such as earthquakes. Therefore, a structural identification methodology that accommodates multiple sources of systematic modelling uncertainties is used. Error-domain model falsification (EDMF) enables structural identification through combining damage grades observed by visual inspection with fundamental frequencies that are derived from ambient vibrations. Parametric uncertainties of a hysteretic model are reduced with the two information sources in order to extrapolate the vulnerability of the building regarding future earthquakes. The applicability of the methodology is shown using measurements made on a mixed reinforced-concrete unreinforced-masonry building tested on a shaking table. Based on nonlinear time-history analyses involving single-degree-of-freedom models, EDMF leads to more precise, yet robust, vulnerability predictions of earthquake-damaged buildings when compared with prediction ranges that are obtained without data interpretation.
... To examine the damage-induced local characteristics without baseline data, advanced signal processing methods are commonly used, exemplified by methods like wavelet analysis or fractal dimension analysis [14][15][16]. Gentile and Messina [17] studied the Gaussian wavelet transforms in localizing open cracks of beams and concluded that high-order Gaussian derivative wavelets were more sensitive to damage. Cao and Qiao [18] employed the stationary wavelet transform to improve the noise robustness of mode shapes and applied continuous wavelet transform to localize the damage. ...
Damage localization in plate-type structures has been widely investigated by exploring the structural characteristic deflection shapes (CDS's) or their spatial derivatives. Despite the substantial advances in this kind of methods, several key issues still need to be addressed to boost their efficiency for practical applications. This study considers three essential problems: susceptibility to measurement noise, absence of baseline-data on pristine structures, and selection of measurement sampling interval and that of the parameters to be used in the de-noising techniques for more accurate damage localization. To tackle these problems, a novel baseline-free adaptive damage localization approach is proposed, which combines the robust Principal Component Analysis (PCA) with Gaussian smoothing. A damage localization evaluator is defined to determine both the spatial sampling interval of the CDS's and the scale parameter of Gaussian smoothing to warrant a better damage localization. Moreover, effects of the measurement noise and numerical errors due to the use of the finite difference scheme on the estimate of the CDS derivatives are quantified. Finally, the feasibility and the effectiveness of the proposed method are verified both numerically and experimentally by using a cantilever plate with a small damage zone. It is found that the second-order spatial derivative of the CDS's is able to provide the best damage localization results among the first four order spatial derivatives of the CDS's.
... [8] temel titreşim modunun sürekli dalgacık dönüşümünü kullanarak kiriş üzerindeki çatlağın konumu ve büyüklüğünü belirlemiştir. Benzer bir çalışma Gentile ve Messina [9] tarafından yapılmıştır. Sürekli dalgacık dönüşümünün çatlak konumu ve büyüklüğünün gürültülü ve temiz data üzerinden tespit edebildiğini göstermişlerdir. ...
In this study, newly improved damage detection algorithms of the recent researches have been investigated in time-frequency domain analysis. It is known that effective damage assessment can be made on accelerations, velocities and displacements as kinematic information and the derived energy consumption through dynamic behavior of a structural system in view of structural health studies. The wavelet transforms are applied to the kinematic parameters inferred from analytical models which do not have stationary and ergodic characteristics and results are used to determine the damage potentials of the structural members and nonlinearity if exist. Cases in which the continuous wavelet transform and wavelet packet transform are applied show great potential for an effective damage detection. It is believed that these methods should be used in a complementary way to effectively and correctly assess the health condition of an engineering structure.
... Wang and Deng [29] used the approach of defect localization utilizing spatial wavelets. Gentile [30] investigated the application of continuous wavelet transforms on the dynamic discrete data, for damage detection of beam elements. Youliang and Aiqun [31] used wavelet packet transform (WPT) for the structural health monitoring of longspan suspension bridges. ...
In this paper, a structural model updating technique is presented based on the wavelet analysis of the structural responses. The sensitivity of the wavelet coefficients of the dynamic response with respect to the structural parameters is evaluated using the incomplete measured responses. To achieve an accurate sensitivity equation, measured data are incorporated in mathematical formulation. The least-square algorithm with the appropriate weighting algorithm is used for solving the over-determined system of equations. The proposed method is applied numerically to the simulated data of a 2D truss model and a 3D frame model. The results show the great promise of the wavelet transform via the proposed sensitivity-based approach for structural model updating and damage detection.
... Damages close to the beginning and end of the structure can be masked due to the high values of the coefficients near the edges. Different approaches to this problem have been proposed in the literature, like advanced mathematical methods in Messina and Gentile [37] or more simple ones like smooth extensions of the signal from beyond the boundaries (Rucka and Wilde [29], Solis et al. [2]). By equation (2) it is possible to conclude that higher scales of CWT use more points to evaluate the signal, hence, the length of the extension of the signal depends on the scales used in the transform. ...
Different approaches on the detection of damages based on dynamic measurement of structures have appeared in the last decades. They were based, amongst others, on changes in natural frequencies, modal curvatures, strain energy or flexibility. Wavelet analysis has also been used to detect the abnormalities on modal shapes induced by damages. However the majority of previous work was made with non-corrupted by noise signals. Moreover, the damage influence for each mode shape was studied separately. This paper proposes a new methodology based on combined modal wavelet transform strategy to cope with noisy signals, while at the same time, able to extract the relevant information from each mode shape. The proposed methodology will be then compared with the most frequently used and wide-studied methods from the bibliography. To evaluate the performance of each method, their capacity to detect and localize damage will be analyzed in different cases. The comparison will be done by simulating the oscillations of a cantilever steel beam with and without defect as a numerical case. The proposed methodology proved to outperform classical methods in terms of noisy signals.
... Consequently, neither the lower mode forms nor any of the higher single modes can be considered as useful for detecting the damage by default. They also stated that accessible modes or practical reflective forms should be reviewed [15]. In a numerical study by Castro et al, the quality of the damage detection was studied based on the order of vibrational modes, which formulated the modes order effect on damage detection. ...
... Hong et al. [8] identified damage in beam in terms of Lipschitz exponent by applying Mexican hat wavelet transform to decompose the beam's fundamental mode shape. Later on, Gentile and Messina [9] determined damage characteristics in a beam using continuous wavelet transform (CWT) to analyze vibrational deformation modes. Fan and Qiao [10] detected damage in a plate model by applying a 2-D CWT to decompose the mode shape. ...
Mode shape is a reliable modal property that provides information regarding the health status of structures. The decomposition of mode shape by using wavelet transform (WT) has provided researchers information needed to identify damage in structures. However, WT decomposition of mode shape differences has shown to provide damage information. In this paper, a comparative study was presented on WT decomposition of mode shape and mode shape difference to detect damage in plate structures. Since reliable higher-order mode shapes can be very difficult to obtain, this study utilized the first mode. Numerical models of square plates with different boundary conditions (one, two and four sides fixed), were applied to evaluate the capabilities of mode shape and mode shape difference to detect damage. Damage was imposed at different locations in the plate models by reducing thickness at such locations. The damage detectability of both methods was analysed and compared when noisy data was applied. The results showed that WT decomposition of mode shape difference was not affected by border distortion, and capable of detecting damage at all locations. However, border distortion problem affected detection of damage at some locations when mode shape was decomposed.
... In the absence of a damage feature bank and physics-based model of structures, damage can be detected by comparing the damage features of damaged structures with baseline data of heathy structures [18][19][20]. Even when the baseline data of healthy structures is not available, structural damage identification in the form of detection, localisation and relative severity quantification can still be achieved by measuring the deviations from some properties of healthy structures such as normal distribution of probability density function under random excitation and smoothness of mode shapes for geometrically uniform and material-isotropic structures [21][22][23][24]. ...
Damage-induced local singularities in structural characteristic deflection shapes (CDS’s) are widely used in non-model-based damage localisation. Despite substantial advances in this kind of methods, several issues must be addressed to boost their efficiency and practical applications. This study deals with two essential problems of CDS-based damage localisation: the noise robustness of CDS estimation and the criterion to properly weight damage information of several CDS’s. On the first problem, it is well known that CDS estimation is vulnerably compromised by various uncertainties such as measurement noise and computational errors, which will decrease the accuracy and increase the difficulties in damage localisation. A modified common eigenvector analysis (CEA) is proposed based on a bank of digital filters and a joint approximate diagonalisation technique, which statistically estimates the CDS’s as the common eigenvectors of a set of covariance matrices. On the second problem, a new robust damage index (DI) is proposed, which is comprised of damage-caused local shape distortions of several CDS’s weighted by their participation factors in the covariance matrix at zero-time delay. The advantage of doing this is to include fair contributions from changes of all CDS’s concerned and the proposed DI provides a measure of damage-induced changes in the covariance matrix. Then a numerical study is presented to demonstrate the noise robustness of the modified CEA method over proper orthogonal decomposition and second-order blind identification in CDS estimation. Moreover, a comparison of the proposed DI over some traditional damage localisation methods is conducted based on an experimental study. The results of numerical and experimental studies demonstrate that the proposed CDS estimation method is more robust to noise and the proposed DI is highly accurate for multi-damage localisation.
... In this paper, the continuous wavelet in order to diagnose damages in pipes. The application and efficiency of the wavelet transform (WT) in structural health monitoring (SHM) and in a variety of engineering applications has been studied in [23][24][25] The mathematical background of this transform has been presented completely n the following, a brief mathematical discussion about the continuous presented. ...
... The wavelet coefficients of the wavelet function are related to the derivatives of the input signal of the same order as the number of vanishing moments of the wavelet function [23,24,25]. Therefore, by selecting 2 vanishing moments, the CWT coefficients of mode shapes differences give information about the change in the second derivatives of mode shapes (modal curvatures) which is a well-known sensitive feature for damage detection. ...
This paper uses the Continuous Wavelet Transform Analysis on mode shapes for damage identification. The wavelet analysis is applied to the difference in the mode shapes between a healthy and a damaged state. The paper also includes a novel methodology for estimating the level of noise of the experimental mode shapes based on a standard Signal to Noise Ratio (SNR). The estimated SNRs are used for identifying and making emphasis on the less noisy data. Moreover, a mass attached to the structure is considered to enhance the sensitivity of the structure to damage. Modal analysis is performed for different positions of the mass along the beam. The results obtained for all the positions of the mass are combined so an averaging process is implicitly applied. The paper presents the results from an experimental test of a cantilever steel beam with different severity levels of damage at the same location. The results show that the use of the attached mass reduces the effect of noise and increases the sensitivity to damage. Little damage can be identified with the proposed methodology even using a small number of sensors and only the first five bending modes.
... Douka et al [5] proposed intensity factor for estimating the crack depth. Gentile and Messina [6] analyzed beam structure mathematically with Gaussian wavelet; results ...
The objective of this work focuses on damage detection in a beam structure using wavelet transform. The wavelet transform is a powerful tool for detecting the damages in beam and plate like structures. The data used for damage detection are transverse nodal displacement values. The different types of continuous wavelet transforms like Daubechies, Symlets, Coifl ets, Gaussian, Dmeyer, Morlet, Biorsplines, and Reversebior are used. The rectangular beam is modeled numerically. The modal analysis is performed in a beam with a fi xed-fi xed boundary condition. The natural frequency and mode shapes of different damaged cases are examined. Damage is simulated by reducing young's modulus value percentage of the element. The difference data of damaged and undamaged mode shape data's are used. Mode sensitivity and wavelet selection procedure examined for beam structure. The absolute wavelet coeffi cient data's are plotted for damage localization and quantifi cation. From this work it is proved the Symmetrical wavelet type is best for damage identifi cation using translational spatial data.
... The calculation of nth order derivative can be achieved by only one transform procedure, instead of repeated transforms (Shao et al. 2000). Wavelet transform has been introduced in various aspects of civil engineering, such as detection of open cracks in damaged beams (Gentile and Messina 2003;Messina 2004), evaluation of the seismic performance of the structure (Das and Gupta 2008). Wavelet transform has been adopted to investigate various traffic related issues such as automatic detection of freeway incidents Ghosh-Dastidar and Adeli 2003;Adeli 2002, 2003;Adeli 2000, 2001;Zheng et al. 2011aZheng et al. , 2011b, traffic features around freeway work zones (Adeli and Ghosh-Dastidar 2004;Ghosh-Dastidar and Adeli 2006), traffic flow forecasting (Boto-Giralda et al. 2010;Jiang and Adeli 2005;Xie et al. 2007), and traffic pattern recognition (Jiang and Adeli 2004). ...
Abstract
Trajectory data collected using the video image processing techniques are prone to noise. Trajectory data extracted using TRAZER, the commercial video image processing software contain the noise associated with the false detection besides the white noise. This paper proposes a method based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) to smooth such trajectory data. In this approach, trajectory data are decomposed into a finite number of intrinsic modes (IMF) and a unique residue is computed to obtain each mode. This monotonic residue gives the smoothed trajectory. The instantaneous speeds of the vehicles are then estimated using the method of continuous wavelet transforms (CWT), discrete wavelet transforms (DWT), and numerical differentiation. Internal consistency analyses show that the Wavelet Transforms (WT) methods are effective in reducing the noise amplification of the speed profile. It was also observed that the corrections applied on trajectory data have a significant effect on macroscopic traffic relations.
... We heuristically selected values that lead to the best discrimination between PDP and HSC. Wavelet differentiator filters are an alternative to FIR filters used in this study 52,53 . We found that the described FIR filter can be replaced by the one based on a continuous wavelet transform with a Gaussian mother function and scale parameter a = 10 (for a sampling frequency 50 Hz) without affecting the values of the parameters analyzed in this study. ...
Posturography is routinely used to qualitatively assess one of the cardinal symptoms of Parkinson's disease -- postural instability. While most measures of balance control are derived from displacement of the center of pressure there is evidence that such control is more likely to be velocity-based. We performed static posturographic tests (eyes open and eyes closed) during quiet standing in narrow stance for n=30 patients with Parkinson's disease (PDP) in the ON state and compared the results with those of n=30 age-matched senior controls (HSC) and n=60 young controls (HYC). We used differentiator filters to generate time series of low-frequency fluctuations of sway velocity and calculated their Lempel-Ziv complexity (LZC). With eyes closed, the mediolateral LZC of HSC 0.21 (0.02) was significantly higher than those of HYC 0.19 (0.02) and PDP 0.18 (002). Thus aging and PD have opposite effects on mediolateral LZC which strongly differentiates between HSC and PDP (92% sensitivity and 87% specificity).
The present study is concerned with detecting a crack in the beam numerically and experimentally using maximal overlapping discrete wavelet transform (MODWT). Slope discontinuity in the beam elastic line can reveal the presence of a crack in a beam. The MODWT is used for locating the cracks in a beam. MODWT coefficients are compared with discrete wavelet transform coefficients at different noise levels. For the experimental validation, high-resolution measurements of the cantilever beam deflection are obtained using photographic measurements. The algorithm is tested for the detection of cracks away and near the ends of a beam. Crack detection near the end of the beam is difficult due to border distortion. Border distortion suppresses the signal in that region. Isomorphism is utilized to shrink the border distortion zone. It helps in the better visualization of the spike due to the crack located away from the ends. Moving averaging is applied to the MODWT-based results for better localization of the spike near the crack position.
Rapid post-earthquake damage assessment forms a critical element of resilience, ensuring a prompt and functional recovery of the built environment. Monitoring-based approaches have the potential to significantly improve upon current visual inspection-based condition assessment that is slow and potentially subjective. The large variety of sensing solutions that has become available at affordable cost in recent years allows the engineering community to envision permanent-monitoring applications even in conventional low-to-mid-rise buildings. When combined with adequate structural health monitoring (SHM) techniques, sensor data recorded during earthquakes have the potential to provide automated near-real-time identification of earthquake damage. Near-real time building assessment relies on the tracking of damage-sensitive features (DSFs) that can be directly and rapidly derived from dynamic monitoring data and scaled with damage. We here offer a comprehensive review of such damage-sensitive features in an effort to formally assess the capacity of such data-driven indicators to detect, localize and quantify the presence of nonlinearity in seismic-induced structural response. We employ both a parametric analysis on a simulated model and real data from shake-table tests to investigate the strengths and limitations of purely data-driven approaches, which typically involve a comparison against a healthy reference state. We present an array of damage-sensitive features which are found to be robust with respect to noise, to reliably detect and scale with nonlinearity, and to carry potential to localize the occurrence of nonlinear behavior in conventional structures undergoing earthquakes.
This paper presents a damage identification method for beam-type structures based on the wavelet analysis. The damage location is predicted by a Damage Locating Index (DLI) based on a combined identification of local maxima from the scalograms of mode shapes. A new Damage Severity Index (DSI) based on the Lipschitz exponent is formulated such that it is not dependent from the damage locations, boundary conditions, and mode order, as long as a high enough number of measuring points is considered. This study focuses on its application using relatively few measurement points and on the consistency of the DSI. Results from a beam with double-notch damage are analyzed. The results show that the method is applicable with a sparse measurement grid and the DSI Reference values from a static analysis can be referred for damage evaluation using modal displacements. The practical limits of the method are also discussed.
The primary objective of this chapter is to develop output-only modal identification and structural damage detection. Identification of multidegree of freedom (MDOF) linear time invariant (LTI) and linear time variant (LTV—due to damage) systems based on time–frequency (TF) techniques—such as short-time Fourier transform (STFT), empirical mode decomposition (EMD), and wavelets—and also a newly merging blind source separation (BSS) technique is discussed. STFT, EMD, and wavelet methods developed to date are reviewed in sufficient detail. In addition, a Hilbert transform (HT) approach to determine frequency and damping is also presented. In this chapter, STFT, EMD, HT, and wavelet techniques are developed for decomposition of free-vibration response of MDOF systems into their modal components. Once the modal components are obtained, each one is processed using HT to obtain the modal frequency and damping ratios. In addition, the ratio of modal components at different degrees of freedom facilitates determination of mode shape. In cases with output-only modal identification using ambient/random response the random decrement technique is used to obtain free-vibration response.
The advantage of TF techniques is that they are signal-based; hence, they can be used for output-only modal identification. A three degree of freedom 1:10 scale model test structure is used to validate the proposed output-only modal identification techniques based on STFT, EMD, HT, and wavelets. Both measured free-vibration and forced-vibration (white noise) responses are considered. The second objective of this chapter is to show the relative ease with which the TF techniques can be used for modal identification and their potential for real-world applications wherein output-only identification is essential. Recorded ambient vibration data processed using techniques such as random decrement technique can be used to obtain the free-vibration response, so that further processing using TF-based modal identification can be performed.
هدف اصلی این پروژه یافتن بهترین روش برای تشخیص زودهنگام ترک های خستگی در سیستم روتور - یاتاقان دیسک می باشد. برای بررسی اثر ترک تنفسی بر سیستم روتور، ابتدا سیستم موردنظر با استفاده از روش المان محدود مدلسازی شده است. در ادامه ماتریس های ضرایب معادله حرکت استخراج شده اند و پس از آن مدلسازی ترک و رفتار تنفسی آن انجام شده اند. در مرحله بعد با روش عددی "هوبولت" پاسخ های سیستم روتور در دو حالت کارکرد پایدار و شروع به کار (سیگنال گذرا)همچنین برای عمق های مختلف تَرک در شفت بدست آمده اند. برای بررسی پاسخ سیستم از نمودارهای سیگنال زمانی، تبدیل فوریه، تبدیل فوریه زمان کوتاه، انرژی ضرایب ویولت گسسته و نهایتاً نمودار انرژی ضرایب تبدیل ویولت پیوسته بهره برده شده است. با بررسی روش های فوق، نشانه های وجود ترک تنفسی و ترک کاملاً باز در سیستم روتور یاتاقان دیسک .معرفی شده اند
هدف اصلی این پروژه یافتن بهترین روش برای تشخیص زودهنگام ترکهای خستگی در سیستم روتور -
یاتاقان دیسک میباشد. برای بررسی اثر ترک تنفسی بر سیستم روتور، ابتدا سیستم موردنظر با استفاده -
از روش المان محدود مدلسازی شده است. در ادامه ماتریسهای ضرایب معادله حرکت استخراج شدهاند
و پس از آن مدلسازی ترک و رفتار تنفسی آن انجام شدهاند. در مرحله بعد با روش عددی "هوبولت"
پاسخهای سیستم روتور در دو حالت کارکرد پایدار و شروع به کار )سیگنال گذرا( همچنین برای عمق-
های مختلف تَرک در شفت بدست آمدهاند. برای بررسی پاسخ سیستم از نمودارهای سیگنال زمانی،
تبدیل فوریه، تبدیل فوریه زمان کوتاه، انرژی ضرایب ویولت گسسته و نهایتاً نمودار انرژی ضرایب تبدیل
ویولت پیوسته بهره برده شده است. با بررسی روشهای فوق، نشانههای وجود ترک تنفسی و ترک کاملاً
باز در سیستم روتور یاتاقان دیسک معرفی شدهاند.
Wavelet based crack detection has certain limitations such as, border distortion near the ends of a signal and the requirement of high measurement resolution. Border distortion restricts the crack detection near the ends of a beam. Detecting crack near the fixed end is critical as it faces the highest bending stresses. High measurement resolution is required for better localization of cracks and also to shrink the border distortion zone. In the present work, the effect of measurement resolution and wavelet scale on the border distortion is studied. Suitable wavelet scale is selected based on crack localization and smoothness of wavelet coefficients. The effect of measurement resolution on signal extension using isomorphism is presented. Photographic method is used to obtain the high resolution measurement of the beam deflection. The algorithm is tested for the detection of a crack in the border distortion zone. Also, the detection of two simultaneous cracks in the beam is presented
This chapter attempts to outline the pipeline failures and their impact on safety and environmental risks besides causing blockages and leaks in pipe networks. Furthermore, non-destructive testing techniques for detecting leakage and blockage are studied. They are visual detection approach, fluid odorant approach, mass balance approach, real-time transient model-based approach, pressure deviation approach, supervisory control and data acquisition system based on leakage identification approaches, pipeline hydrotest approach, magnetic and ultrasonic pipe detection approaches, acoustic emission inspection and monitoring system, wave warning system, pulse-echo flaw detector approach and acoustic wave reflectometry. Eventually, the stationary wavelet transform method is presented for structural health monitoring.
Introduction
Structural damage and its extent can be detected by vibration-based techniques to avoid failure or to minimize maintenance. Among different damage identification techniques, modal curvature approaches are widely researched and applied one. On the contrary, wavelet transformation (WT) methods are gaining popularity in damage identification of real life buildings because of their suitability for non-stationary signals and non-dependency on baseline data. This paper presents a novel approach utilizing complex continuous wavelet to effectively locate change in physical properties of reinforced concrete (RC) buildings by virtue of variation in frequency and mode shapes due to small change in mass and stiffness.
Methods
In this paper, the effect of variation of mass and stiffness of a building on the modal parameters is established analytically using theequation of motion for a multi-degree freedom system under forced vibration condition. A 3-D finite element model was developed for predicting the modal frequencies and mode shapes of the scaled down six storey RC building and the effect of addition of mass on a particular level of structure on the modal parameters was studied. Further, acceleration time histories were recorded with variation in mass on 3rd story of building using wireless tri-axial accelerometers and the time histories were processed to arrive at Curvature Damage Factor and wavelet coefficients for identification of the additional load on the particular floor.
Results
Vibration responses from all floors of RC building in ambient and loaded conditions were analyzed for frequency response spectra (FRS). Mode shapes were drawn for unloaded case and loaded cases. It was observed that the modal frequency of building decreases with the increase in mass at floors. It is observed that CDF approach could detect the change in mass in both numerical and experimental results. However, CDF algorithm could not detect the addition of load in case 1, 2 and 3, i.e. when load was less than 25 kg, i.e. only 2.6% of floor mass (960 kg). The acquired data for the above stated load cases were analyzed using complex Gaussian ‘cgau5’ wavelet in MATLAB toolbox to determine the singularity in the signal in terms of wavelet coefficient modulus. It is observed that the WT approach is able to precisely locate the change in physical parameters of the RC model building. However, it is seen that additional load could not be detected in case where only 9 kg, i.e. 0.93% of the total floor mass, was placed on 3rd floor.
Conclusions
From the research work, it is observed that CDF technique is inefficient in damage detection and always demand prior baseline information, which is usually difficult to obtain in practice. However, the wavelet transform-based approach more accurately detects the location of change without relying on intact state vibration data.
This paper provides an overview of methods to detect, locate, and characterize damage in structural and mechanical systems by examining changes in measured vibration response. Research in vibration-based damage identification has been rapidly expanding over the last few years. The basic idea behind this technology is that modal parameters (notably frequencies, mode shapes, and modal damping) are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties will cause detectable changes in the modal properties. The motivation for the development of this technology is presented. The methods are categorized according to various criteria such as the level of damage detection provided, model-based vs. non-model-based methods and linear vs. nonlinear methods. The methods are also described in general terms including difficulties associated with their implementation and their fidelity. Past, current and future-planned applications of this technology to actual engineering systems are summarized. The paper concludes with a discussion of critical issues for future research in the area of vibration-based damage identification.
A wide spectrum of investigations devoted to the determination of natural frequencies and mode shapes of beams with an open crack are presented in the literature. However, as is well known, an open crack is a fairly crude model of a fatigue crack. The study of the dynamic characteristics of a beam with a closing crack is the main aim of the present paper. The analytical approach which enables one to determine the effect of crack parameters (crack magnitude and location) on different dynamic characteristics of a cantilever Bernoulli–Euler beam with a closing edge transverse crack is performed. Natural frequencies, mode shapes and distortion of time functions describing wave shapes of displacement, acceleration and strain of different cross-sections of a beam are considered as dynamic characteristics to be investigated. The general solution of the problem is derived from the synthesis of particular solutions obtained for the crack-free beam and for the beam with an open crack. The possibility of origination of several modes of vibrations during crack opening is taken into account as well as the peculiarity of strain distribution in the vicinity of a crack. It is shown that analytically predicted relationships between the dynamic characteristics of a cracked beam and crack parameters are well-founded. The analytical approach makes it possible to solve the inverse problem of damage diagnostics with sufficient accuracy for practical purposes.
Applications of the wavelet transform in solid mechanics are presented herein. The analysis problem is addressed first where the objective is to derive an adaptive wavelet-based method for the static and dynamic analysis of structures. This is done via a collocation scheme. The damage identification problem is investigated next making use of the space-localized properties of the wavelet transform: the regularity of the solution is detected by looking at the amplitude of the coefficients of the wavelet decomposition of the response. Open problems are finally outlined that will be the object of future work.
The mathematical characterization of singularities with Lipschitz
exponents is reviewed. Theorems that estimate local Lipschitz exponents
of functions from the evolution across scales of their wavelet transform
are reviewed. It is then proven that the local maxima of the wavelet
transform modulus detect the locations of irregular structures and
provide numerical procedures to compute their Lipschitz exponents. The
wavelet transform of singularities with fast oscillations has a
particular behavior that is studied separately. The local frequency of
such oscillations is measured from the wavelet transform modulus maxima.
It has been shown numerically that one- and two-dimensional signals can
be reconstructed, with a good approximation, from the local maxima of
their wavelet transform modulus. As an application, an algorithm is
developed that removes white noises from signals by analyzing the
evolution of the wavelet transform maxima across scales. In two
dimensions, the wavelet transform maxima indicate the location of edges
in images
A survey of the technology of modal testing, a new method for describing the vibration properties of a structure by constructing mathematical models based on test data rather than using conventional theoretical analysis. Shows how to build a detailed mathematical model of a test structure and analyze and modify the structure to improve its dynamics. Covers techniques for measuring the mode, shapes, and frequencies of practical structures from turbine blades to suspension bridges.
When a structure is subjected to damage, its dynamic response changes, characterized by shifts in the eigenvalues and modifications of the eigenvectors. Considerable effort has been put into investigating the relationship between the damage location, the damage size and the corresponding changes in the eigenparameters. In most cases, emphasis has been on using the shift in eigenvalues as a means of determining the damage location, and the information derived from the change in eigenvectors has largely remained obscure. In this paper a systematic study is presented of the relationship between damage location, damage size and the changes in the eigenvalues and eigenvectors of a cantilever when subjected to damage. A finite element model of a uniform cross sectioned cantilever was chosen to provide data for analysis. The changes in the eigenvalues and eigenvectors are shown to follow a definite trend in relation to the location and extent of damage.
An experimental nondestructive vibration-based technique for locating a delamination in a composite beam is presented. The method operates on the fundamental displacement eigenvector, which is converted to a curvature mode shape. The application of a unique, gapped smoothing damage detection method to the curvature yields a damage index that locates the delamination, irrespective of its position along the beam or depth within the beam. The procedure can operate solely on data obtained from the damaged structure. Models or data from the undamaged structure are specifically not required or used during the analysis. The procedure reported here is highly satisfactory for experimental data, where small variations in the measurement cause false features in other published curvature-based damage detection methods. The damage location method is demonstrated with a finite element model of a composite beam, where a delamination is modeled by relaxing connectivity between elements at the desired location of the delamination. The results of an experimental investigation of a composite beam with a manufactured delamination are also presented. When the gapped smoothing method was used on the experimental modal data, the delamination was successfully located.
This paper examines the sensitivity of wavelet technique in the detection of cracks in beam structures. Specifically, the effects of different crack characteristics, boundary conditions, and wavelet functions employed are investigated. Crack characteristics studied include the length, orientation and width of slit. The two different boundary conditions considered are simply supported and fixed end support, and the two types of wavelets compared in this study are the Haar and Gabor wavelets. The results show that the wavelet transform is a useful tool in detection of cracks in beam structures. The dimension of the crack projected along the longitudinal direction can be deduced from the analysis. The method is sensitive to the curvature of the deflection profile and is a function of the support condition. For detection of discrete cracks, Haar wavelets exhibit superior performance.
During the last 20 years, the development of experimental modal analysis techniques has facilitated the accurate measurement of modal parameters in many types of structure. Alongside this work, several methods have been developed to detect structural damage by using location-dependent changes in the modal parameters. The paper extends the authors' work on a correlation coefficient termed the Multiple Damage Location Assurance Criterion (MDLAC) by introducing two methods of estimating the size of defects in a structure. Their effectiveness is illustrated using numerical data for two truss structures and both location and sizing algorithms are validated experimentally using a three-beam test structure. The paper also introduces a means of improving the computational efficiency of the damage location algorithm. The MDLAC approach offers the practical attraction of only requiring measurements of the changes in a few of the structure's natural frequencies between the undamaged states and is shown to provide good predictions of both the location and absolute size of damage at one or more sites.
This paper presents a method based on wavelet transform to detect the existence and location of structural damage. The system output is processed with wavelet transform and the result is compared with that of the undamaged system. It is found that minor localized damage, which generates vibration signals almost undistinguishable from those of the undamaged system by conventional analysis methods in time and frequency domains, can induce significant changes in the wavelet coefficients of the vibration signals. Furthermore, the maximum difference between wavelet coefficients of the damaged and original systems occurs at the location of the damage.
The effect of damage on the dynamic properties of structures such as beams and plates has previously been studied. Most of the work considers the variation of natural frequencies, although some also considers the change in mode shape. However, little work considers using mode shape data for locating damage. This paper presents a technique for identifying and pinpointing structural damage in a beam. The procedure operates solely on mode shape data from the damaged structure, and does not require a priori knowledge of the undamaged structure. When damage is relatively severe, a finite difference approximation of Laplace's differential operator to the mode shape successfully identifies the location of the damage. When damage is less severe, it is shown that further processing of the Laplacian significantly increase the sensitivity of the procedure, and in some cases enables the location of less than 0.5% damage to be found. The procedure is best suited to the mode shape obtained from the fundamental natural frequency. The procedure is demonstrated using a finite element model of a beam. Experimental verification includes the location of crack damage in a steel beam, and a delamination in a glass reinforced plastic beam.
The vibration parameters of a structure are typically derived from acquired time domain signals, or from frequency domain functions that are computed from acquired time domain signals. For example, the modal parameters of a structure can be obtained by curve fitting a set of frequency response functions (FRFs), or by curve fitting a set of (time domain) impulse response functions. Similarly, the operating deflection shapes of a structure can be obtained either from a set of time domain responses, or from a set of frequency domain responses. Two of the most commonly asked questions about vibration are: 1. What is the deformation (deflection shape) of a machine or structure under a particular operating condition? 2. How much is the machine or structure actually moving at certain points? Time domain responses can be used to answer both of these questions, for linear as well as non-linear vibration. On the other hand, frequency domain responses can be used to answer these questions for specific frequencies. NOMENCLATURE
This work is an in-depth study of a boundary effect detection (BED) method for pinpointing locations of small damages in beams using operational deflection shapes (ODSs) measured by a scanning laser vibrometer. The BED method requires no model or historical data for locating structural damage. It works by decomposing a measured ODS into central and boundary-layer solutions using a sliding-window least-squares curve-fitting technique. For high-order ODSs of an intact beam, boundary-layer solutions are non-zero only at structural boundaries. For a damaged beam, its boundary-layer solutions are non-zero at the original boundaries and damage locations because damage introduces new boundaries. At a damage location, the boundary-layer solution of slope changes sign, and the boundary-layer solution of displacement peaks up or dimples down. The theoretical background is shown in detail. Noise and different types of damage are simulated to show how they affect damage locating curves. Experiments are performed on several different beams with different types of damage, including surface slots, edge slots, surface holes, internal holes, and fatigue cracks. Experimental results show that this damage detection method is sensitive and reliable for locating small damages in beams.
Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. The book clearly presents the standard representations with Fourier, wavelet and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications. Features: * Balances presentation of the mathematics with applications to signal processing * Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox * Companion website for instructors and selected solutions and code available for students New in this edition * Sparse signal representations in dictionaries * Compressive sensing, super-resolution and source separation * Geometric image processing with curvelets and bandlets * Wavelets for computer graphics with lifting on surfaces * Time-frequency audio processing and denoising * Image compression with JPEG-2000 * New and updated exercises A Wavelet Tour of Signal Processing: The Sparse Way, third edition, is an invaluable resource for researchers and R&D engineers wishing to apply the theory in fields such as image processing, video processing and compression, bio-sensing, medical imaging, machine vision and communications engineering. Stephane Mallat is Professor in Applied Mathematics at École Polytechnique, Paris, France. From 1986 to 1996 he was a Professor at the Courant Institute of Mathematical Sciences at New York University, and between 2001 and 2007, he co-founded and became CEO of an image processing semiconductor company. Includes all the latest developments since the book was published in 1999, including its application to JPEG 2000 and MPEG-4 Algorithms and numerical examples are implemented in Wavelab, a MATLAB toolbox Balances presentation of the mathematics with applications to signal processing.
Damage detection by the wavelet transform of the fundamental vibration mode receives much attention recently. Many investigations report successful applications of the wavelet transform in damage detection, but most of them appear to lack theoretical justifications. The objective of this contribution is to show the effectiveness of the wavelet transform by means of its capability to estimate the Lipschitz exponent. It is also addressed that the magnitude of the Lipschitz exponent can be used as a useful indicator of the damage extent. As a specific example, damaged beams are investigated both numerically and experimentally. The continuous wavelet transform (CWT) by a Mexican hat wavelet having two vanishing moments is utilized for the estimation of the Lipschitz exponent. The analysis by the CWT also gives a guideline to choose appropriate discrete wavelet transforms.
This paper discusses a structural damage detection technique based on wavelet analysis of spatially distributed structural response measurements. The premise of the technique is that damage (e.g. cracks) in a structure will cause structural response perturbations at damage sites. Such local perturbations, although they may not be apparent from the measured total response data, are often discernible from component wavelets. The viability of this new technique is demonstrated with two examples: one based on numerically simulated deflection responses of a uniform beam containing a short transverse crack under both static and dynamic loading conditions, and the other based on smooth analytical crack-tip displacement fields. In each of these examples, the deflection or displacement response is analyzed with the wavelet transform, and the presence of the crack is detected by a sudden change in the spatial variation of the transformed response. This damage detection technique may serve the purpose of structural health monitoring in situations where spatially distributed measurements of structural response in regions of critical concern can be made with, for example, networks of distributed sensors, optical fibers, computer vision and area scanning techniques. It appears that this new technique does not require any analysis of the complete structure in question, nor any knowledge of the material properties and prior stress states of the structure.
A damage in a structure alters its dynamic characteristics. The change is characterized by changes in the eigenparameters, i.e., natural frequency, damping values and the mode shapes associated with each natural frequency. Considerable effort has been spent in obtaining a relationship between the changes in eigenparameters, the damage location and the damage size. Most of the emphasis has been on using the changes in the natural frequencies and the damping values to determine the location and the size of the damage. In this paper a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure. By using a cantilever and a simply supported analytical beam model, it is shown here that the absolute changes in the curvature mode shapes are localized in the region of damage and hence can be used to detect damage in a structure. The changes in the curvature mode shapes increase with increasing size of damage. This information can be used to obtain the amount of damage in the structure. Finite element analysis was used to obtain the displacement mode shapes of the two models. By using a central difference approximation, curvature mode shapes were then calculated from the displacement mode shapes.
This book is intended to serve as a text for introductory courses in
computational fluid mechanics and heat transfer for advanced
undergraduates and/or first-year graduate students. The first part of
the book presents basic concepts and provides an introduction to the
fundamentals of finite-difference methods, while the second part is
devoted to applications involving the equations of fluid mechanics and
heat transfer. A description is given of the application of
finite-difference methods to selected model equations, taking into
account the wave equation, heat equation, Laplace's equation, Burgers'
equation (inviscid), and Burgers' equation (viscous). Numerical methods
for inviscid flow equations are considered along with governing
equations of fluid mechanics and heat transfer, numerical methods for
boundary-layer type equations, numerical methods for the 'parabolized'
Navier-Stokes equations, numerical methods for the Navier-Stokes
equations, and aspects of grid generation.
Detection of cracks by only measured mode shapes in damaged conditions
- A Gentile
- A Messina
Gentile, A., Messina, A., 2002. Detection of cracks by only measured mode shapes in damaged conditions. In: 3rd International
Conference on Identification in Engineering Systems, Swansea, Wales.
Wavelet Toolbox for use with MATLAB â Wavelet analysis of structures: statics, dynamics and damage identification
- M Misiti
- Y Misiti
- G Oppenheim
- J M Poggi
Misiti, M., Misiti, Y., Oppenheim, G., Poggi, J.M., 1996. Wavelet Toolbox for use with MATLAB â. The MathWorks Inc., Natick. Naldi, G., Venini, P., 1997. Wavelet analysis of structures: statics, dynamics and damage identification. Meccanica 32, 223–230.
Operational deflection shapes: background, measurement and applications
- Døssing
Døssing, O., Staker, C.H., 1987. Operational deflection shapes: background, measurement and applications. In: 5th International
Modal Analysis Conference, London, UK, pp. 1372-1378.
- K R Otnes
- L Enochson
Otnes, K.R., Enochson, L., 1978. In: Applied Time Series Analysis, vol. 1. John Wiley and Sons, New York.
- A Gentile
- A Messina
A. Gentile, A. Messina / International Journal of Solids and Structures 40 (2003) 295–315
A simplified approach to the numerical and experimental modelling of the dynamics of a cracked beam
- J Chance
- G R Thomlinson
- K Worden
Chance, J., Thomlinson, G.R., Worden, K., 1994. A simplified approach to the numerical and experimental modelling of the dynamics
of a cracked beam. In: 12th International Modal Analysis Conference, Honolulu, USA, pp. 778-785.
- I Daubechies
Daubechies, I., 1992. Ten Lectures on Wavelets. SIAM, Philadelphia, PA.
Numerical Recipes in C
- W H Press
- S A Teukolsky
- W T Vetterling
- B P Flannery
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in C. Cambridge University Press,
Cambridge.
Singularity detection and processing with wavelets
- Mallat