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An unsupervised learning algorithm for fatigue crack detection in waveguides
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2009 Smart Mater. Struct. 18 025016 (http://iopscience.iop.org/0964-1726/18/2/025016) View the table of contents for this issue, or go to the journal homepage for more
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IOP PUBLISHING
SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 18 (2009) 025016 (11pp)
doi:10.1088/0964-1726/18/2/025016
An unsupervised learning algorithm for fatigue crack detection in waveguides Piervincenzo Rizzo1,4 , Marcello Cammarata1, Debaditya Dutta2, Hoon Sohn3 and Kent Harries1,5 1 Department of Civil and Environmental Engineering, University of Pittsburgh, 949 Benedum Hall, 3700 O’Hara Street, Pittsburgh, PA 15261, USA 2 Department of Civil and Environmental Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA 3 Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea
E-mail:
[email protected]
Received 16 August 2008, in final form 9 December 2008 Published 22 January 2009 Online at stacks.iop.org/SMS/18/025016 Abstract Ultrasonic guided waves (UGWs) are a useful tool in structural health monitoring (SHM) applications that can benefit from built-in transduction, moderately large inspection ranges, and high sensitivity to small flaws. This paper describes an SHM method based on UGWs and outlier analysis devoted to the detection and quantification of fatigue cracks in structural waveguides. The method combines the advantages of UGWs with the outcomes of the discrete wavelet transform (DWT) to extract defect-sensitive features aimed at performing a multivariate diagnosis of damage. In particular, the DWT is exploited to generate a set of relevant wavelet coefficients to construct a uni-dimensional or multi-dimensional damage index vector. The vector is fed to an outlier analysis to detect anomalous structural states. The general framework presented in this paper is applied to the detection of fatigue cracks in a steel beam. The probing hardware consists of a National Instruments PXI platform that controls the generation and detection of the ultrasonic signals by means of piezoelectric transducers made of lead zirconate titanate. The effectiveness of the proposed approach to diagnose the presence of defects as small as a few per cent of the waveguide cross-sectional area is demonstrated. (Some figures in this article are in colour only in the electronic version)
ambient-vibration-based (Chondros et al 1998), and impedance-based methods (Giurgiutiu and Rogers 1997, Park et al 2003) have been proposed for the detection of cracks in steel structures. Recently, methods based on ultrasonic guided waves (UGWs) have gained popularity due to the capability of inspecting moderately large areas using a single probe attached or embedded in the structure while maintaining high sensitivity to small flaws. UGWs can travel relatively large distances with little attenuation and they offer the advantage of exploiting one or more of the phenomena associated with transmission, reflection, scattering, mode-conversion, and absorption of acoustic energy (Alleyne and Cawley 1996, Rose 1999, Staszewski 2003, Kundu 2004, Giurgiutiu 2005, Rizzo and Lanza di Scalea 2007). In the case of an engineering system with complex geometry, the UGW-based approach is sometimes hindered by the multiple diffraction
1. Introduction Steel structures are ubiquitous in mechanical, industrial, and civil engineering systems. Failure of these structures is often attributed to fatigue or fracture cracks that can develop at such locations as the flange–web junction of a welded plate bridge girder, a railway track, or the substructures of a power generation plant. In most cases cracks cannot be avoided; thus there is a need for non-destructive inspection (NDI) or structural health monitoring (SHM) techniques aimed at detecting structural deficiencies at early stages of deterioration. NDI techniques such as acoustic emission (Roberts and Talebzadehb 2003), eddy current (Sophian et al 2001), 4 Author to whom any correspondence should be addressed. 5 William Kepler Whiteford Faculty Fellow.
0964-1726/09/025016+11$30.00
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© 2009 IOP Publishing Ltd Printed in the UK
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Figure 1. Dimensions of the steel W 6 × 15 (SI: W 150 × 22.5) section: (a) beam span and cross section. (b) Elevation and plan views of the flange in the tension side.
configuration of the system is discordant or inconsistent with its baseline configuration. The baseline describes the normal operative conditions and usually consists of an existing set of data or patterns. The framework of outlier analysis is available in classical statistical textbooks (Barnett and Lewis 1994) and it has been proposed in several monitoring applications spanning from simulated lumped-mass systems (Worden 1997), composites (Worden et al 2000b, Sohn et al 2007), and bolt loosening (Park et al 2005), to damage detection in aircraft wings (Manson and Worden 2003), steel strands (Rizzo and Lanza di Scalea 2007, Rizzo et al 2007), and rails (Rizzo and Lanza di Scalea 2007). The present study extends the novel SHM scheme recently introduced (Rizzo and Lanza di Scalea 2007, and Rizzo et al 2007) to the identification of fatigue cracks in structural waveguides. In addition, the novelty of the work consists of a systematic investigation of the propagating modes and frequencies that perform best in terms of damage detection success rate. The scheme uses a set of defect-sensitive features to construct a uni-dimensional or multi-dimensional damage index vector. The vector is fed to an outlier analysis to detect anomalous structural states. This general framework is applied to the detection of fatigue cracks in a steel beam. The probing hardware consists of wafer piezoelectric transducers (PZTs) made of lead zirconate titanate materials used for both ultrasound generation and detection of chosen frequencies. These PZTs are suitable in SHM applications as they are easy to attach to the structure surface or to embed in the structure.
associated with the geometric boundaries (Masserey et al 2006). However, wave propagation in complex media can be modeled using various simulation techniques, including the finite difference equation (Lee and Staszewski 2003a, 2003b), boundary element method (Norris and Vemula 1995, Wang and Shen 1997), and local interactions simulation approach (LISA) (Lee and Staszewski 2003a, 2003b). With these methods it is possible to study the propagation of waves in the presence of cracks in complex structures. Experimental works showed, for instance, that the scattered near field contains valuable information for crack characterization (Liu et al 1991, Scala and Bowles 2000, Sohn and Krishnaswamy 2003, Masserey and Mazza 2007). UGWs and particularly Lamb waves have successfully been used to monitor the propagation of fatigue cracks in metallic structures (Cook and Berthelot 2001, Fromme and Sayir 2002a, 2002b, Fromme et al 2004, Leong et al 2005, Park et al 2006, Staszewski et al 2007, Kim and Sohn 2008, Dutta et al 2008). In the last few years the need for improvements in the fields of signal conditioning, feature extraction, and defect classification became evident in order to enhance the detection, sizing, location, and classification ability of UGW-based monitoring schemes (Rizzo and Lanza di Scalea 2006, 2007). Generally, these improvements consist of the evaluation of certain damage-sensitive features extracted from the time domain, the frequency domain, and/or the joint time–frequency domain of the ultrasonic signal. Once these features are extracted, they can be coupled to an automatic classification algorithm to detect the presence of anomalies (unsupervised learning class), or to determine the presence, size, and location of defects (supervised learning class). The first class offers the fundamental advantage that the information about damage does not need to be known a priori, contrary to the supervised learning class. This paper presents an unsupervised learning algorithm based on outlier analysis to monitor the onset, growth, and propagation of fatigue cracks by means of UGWs. The detection of small fatigue cracks is relevant for predicting the remaining life of structures and for developing an effective condition-based maintenance approach. The outlier analysis is a novel detection method that establishes whether a new
2. Experimental test bed and protocol To address fatigue crack initiation and growth, a test bed exhibiting these features is required. The test bed is simply a vehicle on which to perform the investigation, and thus a specimen having (a) predictable fatigue behavior; (b) at a reasonable scale; and (c) in a reasonably short time period is required. Statically determinant, three-point flexural fatigue loads were applied to a 2.74 m long W 6 × 15 (SI: W 150 × 22.5) steel beam conforming to ASTM A36 material specifications. The dimensions of the steel section are shown in figure 1. Two notches were cut into the bottom (tension) flange near the center of the beam span as shown in figure 1(b). These 2
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Figure 2. (a) Photograph of the part of the tension flange of the steel beam with PZT emitter and sensor S1 ; (b) schematic figure showing the transmitter and both sensors.
the loading set-up shown in figure 4. The mid-span load was cycled from 4.5 to 40.5 kN resulting in a load range of 36 kN, which corresponded to a tensile stresses of 22.7 to 212.7 MPa or a stress range of 190 MPa at the notch root on the tension flange. Cycling was carried out at a rate of 1 Hz. The notch site was continuously monitored for crack initiation. Figure 5 shows the history of crack propagation in both west and east flanges of the steel beam. After approximately 9000 cycles, cracks at each notch root were identified by the crack gages and they could be observed visually. It is noted that the resolution of the crack gages is 0.25 mm; thus any crack smaller than this value is not captured. Following every few thousand cycles, the cyclic loading was paused and a static load of 22 kN (average of fatigue load range) was applied to the beam. Under this constant load, data from the PZTs were collected. A total of 13 acquisitions were made. The acquisition number (1–13) as a function of the number of load cycles is superimposed in figure 5.
notches served as fatigue crack initiators, and also helped to increase the stress at this section in order to accelerate the development of fatigue cracks. The notches were designed to have a theoretical fatigue life (to crack initiation) on the order of 40 000 cycles at an applied stress range of 190 MPa. The notches on either side of the web were the same to mitigate eccentric behavior of the section. Fatigue cracks were expected to form at the sharp root of each notch. Three PSI-5A4E type PZT-wafer transducers (1.0 cm × 1.0 cm × 0.051 cm) were mounted on the bottom flange of the beam. One transducer acted as a transmitter and was placed in between the two sensors which were located 50 cm apart (figure 2). The crack initiator fell between the transmitter and sensor S1 . A National Instruments (NI) PXI® unit running under LabVIEW® was employed for signal excitation, detection, and acquisition. Five-cycle 10 V peakto-peak (ppk) tonebursts modulated with a Hann window were used as excitation signals. A sweep of frequencies ranging between 50 and 300 kHz with a frequency step equal to 25 kHz was excited. Thus, a total of 13 frequencies were considered. The signals were sampled at 10 MHz, averaged ten times to increase the signal-to-noise ratio, and stored for post-processing analysis. Typical waveforms recorded from the pristine structure by both sensors S1 and S2 at 50, 200, and 300 kHz are shown in figure 3. In the assumption that the wave propagation in the bottom flange is identical to the propagation of Lamb waves in a flat 6.6 mm thick plate, the following modes can exist: the first symmetric S0 , the first antisymmetric A0 mode, and the second anti-symmetric A1 mode. The latter has a cut-off frequency of 250 kHz. The suite of instrumentation was completed by four electrical resistance crack propagation gages placed to monitor the onset and growth of the fatigue cracks. These gages were placed at both notch roots on the upper and lower surfaces of the flange (figure 2). To initiate and propagate the cracks, the beam was loaded in simple mid-span loading over a span length of 2.74 m using
3. Structural health monitoring algorithm The overall SHM algorithm implemented in this paper is illustrated in the flowchart of figure 6. To increase the statistical population, white Gaussian noise was added to the ultrasonic signal. This was repeated 19 times. Thus, for each frequency a total of 20 × 13 (the latter represents the number of acquisitions over the entire fatigue loading protocol) = 260 time histories were available. The noise was created by the MATLAB randn function that generates arrays of random numbers whose elements are normally distributed with zero mean and standard deviation equal to 1. The function was pre-multiplied by a factor that determines the noise level. A factor equal to 0.001 was considered to obtain a noise function representative background noise of the stored waveforms. Each time waveform was windowed in order to separate the S0 mode from the A0 mode, and for signals above 250 kHz, 3
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Figure 3. Typical waveforms recorded from the pristine structure by both sensors S1 (figures (a), (c), and (e)) and S2 (figures (b), (d), and (f)) at 50 kHz (figures (a) and (b)), 200 kHz (figures (c) and (d)), and 300 kHz (figures (e) and (f)).
Figure 4. Loading configuration for the experiment on the steel beam.
Figure 5. Propagation history of the crack emanating from the notch-tip of the west and east parts of the tension flange of the steel beam. The acquisition number history is superimposed.
to separate the A0 mode from the A1 mode. It should be pointed out that to increase the statistical population other solutions can be used by simply increasing the number of acquisitions, or by triggering the sensing hardware to the peak of the sinusoidal fatigue load (Gupta et al 2007).
3.1. Discrete wavelet transform Ultrasonic signals were processed through the DWT, which decomposes the original time-domain signal by computing 4
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Figure 6. Flowchart of the proposed algorithm for damage detection based on DWT feature extraction and outlier analysis.
shown that few wavelet coefficients are effective to extract features that are sensitive to damage. In the present study the eight largest wavelet coefficient moduli from the decomposition level associated with the frequency of the generated tonebursts were retained.
its correlation with a short-duration wave called the mother wavelet that is flexible in time and in frequency. DWT processing consists of two main parts: decomposition and reconstruction. The decomposition phase transforms the function into wavelet coefficients following hierarchical steps (levels) of different time and frequency resolution. Each level contains the signal information in both the time and the frequency domains over a certain frequency bandwidth. The filtering outputs are then downsampled. De-noising of the original signal can be achieved if only a few wavelet coefficients, representative of the signal, of one or more levels are retained and the remaining coefficients, related to noise, are discarded. In the reconstruction process, the coefficients are upsampled to regain their original number of points and then passed through reconstruction filters. The reconstruction filters are closely related but not equal to those of the decomposition. In the present study, the Daubechies mother wavelet of order 40 (db40) was considered. In previous publications (Rizzo and Lanza di Scalea 2005, 2006, 2007) it was demonstrated that the db40 is suitable to de-noise and compress narrowband ultrasonic signals. Moreover it was
3.2. Feature extraction As shown in the flowchart of figure 6, several UGWbased statistical features were extracted from the unprocessed waveform, the wavelet coefficient vector, the reconstructed signal, and the frequency spectrum of the reconstructed signal (frequency domain). These features were used to compute a damage index (DI) able to enhance the detection of the defect. The index is defined as the ratio between a certain feature FS2 calculated from the signal detected by sensor S2 , over the same feature FS1 extracted from sensor S1 : DI = 5
FS2 . FS1
(1)
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As the notch is located in the section of the beam between the actuator and the sensor S1 , the value of the DI is expected to be equal to 1 when no defects are present, i.e. when the structure is pristine. The damage indices were then considered to form a multidimensional DI vector for the unsupervised learning algorithm based on the outlier analysis.
The normal probability plot (Nair et al 2006) was computed to verify the Gaussian distribution of the baseline data, i.e. to verify that the values of the DI were normally distributed and independent. In the present application the data should be centered on the value 1, which identifies the structure in its pristine conditions. Any drift from unity is due to small variations of the PZTs’ sensitivities and the difference of the wave’s travel path from the transmitter to the two sensors. In the circumstance that the baseline data do not comply with the normality distribution the threshold values can be still estimated (Sohn et al 2005, Park and Sohn 2006).
3.3. Outlier analysis An outlier is a datum that appears inconsistent with a set of data, the baseline that describes the normal condition of the structure under investigation. Ideally the baseline should include typical variations in environmental or operative conditions (e.g. temperature, humidity, and loads) of the structure (Worden et al 2002, Sohn et al 2002). In the analysis of one-dimensional elements, the detection of outliers is a straightforward process based on the determination of the discordancy between the one-dimensional datum and the baseline. One of the most common discordancy tests is based on the deviation statistics, z ζ , defined as (Barnett and Lewis 1994) ¯ |x ζ − x| zζ = (2) σ where x ζ is the potential outlier, and x¯ and σ are the mean and the standard deviation of the baseline, respectively. The mean and standard deviation can be calculated with or without the potential outlier, depending upon whether inclusive or exclusive measures are preferred. A set of p -dimensional (multivariate) data consists of n observations in p variables. The discordancy test equivalent to equation (2) is expressed by the Mahalanobis squared distance (MSD), Dζ , which is a non-negative scalar defined as
Dζ = ({x ζ } − {x}) ¯ T · [K ]−1 · ({x ζ } − {x}) ¯
4. Experimental results 4.1. Univariate analysis This section presents the outlier analysis results when the features were considered separately. Figure 7 shows the discordancy values as a function of the sample number. The damage indices computed from three statistical features extracted from the wavelet coefficient vector were considered. These features are the root mean square (RMS), the ppk, and the K -factor defined as: n 1 RMS = x2 ppk = max(x i ) − min(x i ) n i=1 i
K -factor = max(x i ) × RMS.
(4)
Figures 7(a), (c) and (e) refer to the propagation of the first anti-symmetric mode at 50 kHz, 225 kHz, and 300 kHz, respectively. Figures 7(b), (d) and (f) refer to the propagation of the S0 mode at 150 kHz, 225 kHz, and 300 kHz, respectively. The values of the respective thresholds are superimposed. At low frequencies (figures 7(a) and (b)) a large number of inliers, i.e. false negative indications, are visible. In figures 7(c) and (d), clear steps are visible at almost every 20 sample numbers. These steps denote the progressive increase of the crack size. Figure 7(c) also shows outliers for samples 61–80. These data are associated to acquisition 4 at 5000 cycles. According to the crack gage reading (figure 5) the beam was still pristine at that instance. Thus, these outliers are classified as false positives, although they may represent crack initiation and propagation smaller than the 0.25 mm resolution of the crack gages used in the experiment. Both A0 and S0 modes showed outliers in the sample range 81–260, i.e. 100% detection rate. On observing figures 7(c) and (d), the damage index from each feature possesses a different sensitivity (represented by the rate of change of the discordancy plot) to the presence of damage. Qualitatively, the discordancy increases with increasing defect size; however, it can be seen that the sensitivity to damage is larger for the K -factor-based damage index applied to the A0 mode (figure 7(c)) and the RMS-based damage index applied to the S0 mode (figure 7(d)). It must be noted that the step-like behavior of the discordancy between two adjacent acquisitions is not always visible. The difference in size between such defects is so small that may lay in a few wavelet coefficients of lower amplitude
(3)
where {x ζ } is the potential outlier vector, {x} ¯ is the mean vector of the baseline, [K ] is the covariance matrix of the baseline, and T symbolizes the transpose operation. Both vectors {x ζ } and {x} ¯ are p-dimensional, whereas [K ] is a square matrix of order p . As in the univariate case, the baseline mean vector and covariance matrix can be inclusive or exclusive. In the present study, as the potential outliers are always known a priori, both z ζ and Dζ are calculated exclusively without contaminating the statistics of the baseline data. A new datum is an outlier if the corresponding value of z ζ or Dζ falls above a set threshold. When the available number of baseline data is limited, it is common practice to compute the threshold based on a Monte Carlo simulation (Guttormsson et al 1999, Worden et al 2000a). However, in the present study a different approach was adopted. The baseline was computed from the first 60 (=3 acquisitions × (1 original + 19 noise corrupted)) time histories that, according to figure 5, were representative of the pristine structure at load cycles 0, 1250, and 2500, respectively. Once the values of z ζ and Dζ of the baseline distribution were determined, the threshold value was taken as the usual value of 3σ equal to 99.73% of the Gaussian confidence limit. 6
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0
Figure 7. Discordancy test for the features damage index root mean square, peak-to-peak, and K -factor from the wavelet coefficient vector calculated for (a) the A0 mode at 50 kHz, (b) the S0 mode at 150 kHz variance, (c) the A0 mode at 225 kHz, (d) the S0 mode at 225 kHz, (e) the A0 mode at 300 kHz, and (f) the S0 mode at 300 kHz. Continuous line: RMS; dashed line: peak-to-peak; dotted line: K -factor. Table 1. Univariate analysis: percentage of outliers detected by using the damage index based on the root mean square (RMS), peak-to-peak (ppk) amplitude, and K -factor (K) of the vector of the wavelet (w) coefficients. The relative wavelength λ (= cph / f where cph is the wave phase velocity) is indicated.
Feature
A0 mode f = 50 kHz (λ = 30 mm)
S0 mode f = 150 kHz (λ = 34 mm)
A0 mode f = 225 kHz (λ = 12 mm)
S0 mode f = 225 kHz (λ = 22 mm)
A0 mode f = 300 kHz (λ = 8.7 mm)
S0 mode f = 300 kHz (λ = 16 mm)
RMS w ppk w Kw
35% 8.9% 13%
26% 23% 26%
100% 100% 100%
100% 86% 86%
71% 91% 76%
83% 91% 94%
that do not affect the extreme values of the wavelet coefficient vector. By applying the same routine to the time histories centered at 300 kHz, the results in figures 7(e) and (f) were obtained. The detection performance of the algorithm was not as satisfactory as for the 225 kHz case. This is most likely associated with the proximity of the group velocity of both S0 and A0 modes, which complicates the analysis of each mode considered separately. The detection rate of the features presented in figure 7 is summarized in table 1. The limited performance of the
algorithm at frequencies below 150 kHz can be partially attributed to the wavelength. It is generally acknowledged that UGWs having wavelengths equal to λ have low sensitivity to crack sizes smaller than λ/2. The quantitative results of the discordancy tests applied to the damage index associated with the three features and the two modes discussed in figure 7 are presented in figure 8. The figure shows the percentage of correct outliers detected as a function of the propagating frequency. As the symmetric mode was detected only above 150 kHz, the abscissas of the two figures differ. By comparing both histograms and the 7
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(a)
(b)
(c)
Figure 8. Damage detection rate performance of the discordancy test as a function of the propagating wave frequency for the features damage index root mean square, peak-to-peak, and K -factor from the wavelet coefficient vector calculated for (a) the A0 mode and (b) the S0 mode.
(d)
three features the RMS-based damage index extracted from the wavelet coefficient vector provided the highest detection rate. 225 kHz and the 250 kHz are the optimal frequencies. Table 2 summarizes the results of the univariate outlier analysis performed over the entire frequency spectrum by computing the damage index from the statistical features indicated in the flowchart of figure 6. Tables 2(a) and (b) refer to the propagation of the A0 mode and S0 mode, respectively. It is evident that the proper selection of the propagating mode and frequency, as well as of the feature and the analysis domain (time, frequency, joint time–frequency), is pivotal for the sensitivity of the hardware/probing system. 4.2. Multivariate analysis
Figure 9. Mahalanobis squared distance for the baseline (undamaged) and damaged data corrupted with noise for (a) the A0 mode propagating at 150 kHz, (b) the S0 mode propagating at 150 kHz, (c) the A0 mode propagating at 225 kHz, and (d) the S0 mode propagating at 225 kHz. The results from the 9D DI vector, and from two 2D DI vectors are presented. The horizontal lines represent the respective thresholds. Continuous line: 9D DI vector; dotted line: var-based and RMS-based DI of the reconstructed signal; dashed line: ppk-based and K -factor-based DI of the wavelet coefficient vector.
The features considered separately in the previous section were subsequently used simultaneously to construct a multidimensional damage index vector for the outlier analysis. The exclusive MSD for each of the 260 samples was calculated using equation (3). The purpose of combining features was to increase the sensitivity to damage compared to the singlefeature analysis. Nine features extracted from the time domain were considered. These features were the variance (var), RMS, and K -factor from the unprocessed signal, the RMS, ppk, and K -factor from the wavelet coefficient vector, and the var, RMS, and K -factor from the reconstructed signal. However the use of all may not be necessary and the selection of all features may degrade the detection performance. To investigate this aspect, a parametric analysis was carried out. All of the features were considered ranging from all combinations of two-dimensional (2D) DI vectors to the
single combination of the nine-dimensional (9D) DI vector. A total of 502 cases were analyzed. Figure 10 shows the results for the A0 (figures 9(a) and (c)) and the S0 (figures 9(b) and (d)) modes propagating at 150 kHz (figures 9(a) and (b)) and 225 kHz (figures 9(c) and (d)). To compare the performance of the multivariate algorithm, the MSD associated with the single 9D DI vector and with two 2D DI vectors are plotted. Specifically, the 2D vectors are formed by the var-based damage index and the RMS-based damage index from the reconstructed signals, and by the ppk-based damage index and the K -based damage index from the wavelet coefficient vector. Overall, the A0 mode outperforms the S0 mode and the frequency of 225 kHz is better than 150 kHz. The results associated with the propagation of the 225 kHz toneburst demonstrate that the selection of a large-dimensional 8
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Table 2. Univariate analysis: percentage of outliers detected by using the damage index based on the statistical features considered in this study and extracted from the time, frequency (fast Fourier transform), and the joint time–frequency domain (wavelet). (a) Results from the propagation of the first anti-symmetric (A 0 ) mode in the frequency range 50–300 kHz. (b) Results from the propagation of the first anti-symmetric (S0 ) mode in the frequency range 150–300 kHz. Var, variance; Max: maximum; CF: crest factor; K : K -factor, AFFT: area under the FFT. Original signal Freq. (kHz) Var RMS Max ppk CF K
Wavelet coefficient vector Var RMS Max ppk CF K
50 75 100 125 150 175 200 225 250 275 300
62 70 77 80 64 100 78 100 100 100 100
62 73 77 79 61 100 78 100 100 100 100
3 2 0 0 26 55 8 30 23 29 78 74 53 42 55 56 99 100 100 100 100 100
5 0 18 32 2 56 20 92 81 65 16
13 23 58 57 66 78 56 100 100 100 100
4 36 0 33 0 68 2 34 1 14 3 49 11 56 1 100 99 100 26 99 89 71
150 175 200 225 250 275 300
45 55 100 100 100 100 100
45 55 100 100 100 100 100
17 20 27 36 87 97 100 100 99 100 100 100 96 98
4 24 1 31 0 23 2
26 44 99 100 100 100 100
14 26 44 55 26 78 48 100 55 100 57 36 67 83
9 9 0 0 20 22 2 0 11 13 63 80 27 27 100 100 100 100 94 92 73 91 23 30 51 53 56 41 83
24 45 73 86 93 46 91
Reconstructed signal Var RMS Max ppk CF K
2 13 28 28 10 1 0 0 2 44 49 49 17 10 39 38 7 14 0 0 10 69 52 48 53 36 57 57 9 100 100 100 0 100 100 100 17 97 99 99 36 76 74 71 4 27 29 26 14 44 48 46 9 67 81 80 0 86 100 100 0 100 100 100 2 48 42 36 2 84 84 83
8 0 19 5 0 33 23 93 76 31 69
Original
Reconstructed
AFFT RMS AFFT RMS
8 0 14 7 0 27 23 89 76 46 81
1 10 2 1 0 0 5 49 28 4 9 78 1 0 84 2 35 100 9 43 99 4 100 100 24 100 100 1 88 100 47 73 74
63 73 77 79 61 100 78 100 100 100 100
3 0 28 26 0 62 34 100 92 98 61
28 0 49 38 0 48 57 100 100 99 71
33 29 28 28 63 58 99 99 100 100 31 25 55 59
4 33 19 11 38 44 0 73 99 1 100 100 52 100 100 0 38 100 1 69 100
44 56 100 100 100 100 100
31 57 100 100 100 23 81
26 46 80 100 100 36 83
The values of the MSD at sample numbers 61–80 in figures 10(c) and (d) are classified as outliers. This response validates the hypothesis formulated in section 4.1, and therefore the algorithm is able to detect surface defects or small crack initiation prior to the crack gage. It should be noted that the larger is the stepwise behavior of the MSD values the better is the algorithm’s ability to identify minimal variation of the crack’s size. It was also noted but is not discussed here that when the propagating wave mode is below 175 kHz, the algorithm performance degrades both in term of damage detection rate and damage size sensitivity. 4.3. Discussion A comprehensive study was conducted to understand why certain features outperform others. To carry out this investigation, three statistical features extracted from the A0 mode propagating at 200 kHz were considered: the RMS from the original (o) unprocessed signal (RMS o), the ppk of the wavelet (w) coefficient vector (ppk w), and the RMS of the signal frequency (RMS f) domain. The latter is computed by considering the frequency spectrum centered at 200 kHz and comprised between 150 and 250 kHz. Figure 10(a) shows the values of the respective damage index computed for the baseline data by applying equation (1). It can be seen that the damage indices associated with the ppk show larger scattering than the other two features. The close value of the RMS-based damage indices should not surprise, as they are both related to the same signal characteristic: the signal energy. As the data have little dispersion the standard deviation is small, and the values of discordancy computed from equation (2) are high. This is shown in figure 10(b), where
Figure 10. (a) Damage index for the baseline data; (b) discordancy test for the feature damage index considered in (a). Baseline (undamaged) and damaged data; (c) Mahalanobis squared distance for the baseline (undamaged) and damaged data associated with 2D DI vectors composed by the combination of features in (a).
DI vector is not essential to improve the performance the algorithm. In fact, by observing figures 10(c) and (d), the values of the MSD associated with the 9D vector are close to the values obtained from one of the two 2D vectors. 9
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data for the outlier analysis was created by adding Gaussian random noise to the ultrasonic measurements. The importance of the selection of the proper propagating mode and frequency as well as of the signal features and algorithm to enhance the damage detection performance of the probing hardware was studied and demonstrated. It was demonstrated that the first anti-symmetric mode A0 propagating at 225 kHz is optimal. It was shown that combining multiple features in a multivariate analysis substantially improves the performance in terms of sensitivity to defect detection. By exploiting the stepwise response of the Mahalanobis squared distance it is possible to discriminate and quantify defects of various sizes. By combining as few as two features, the improvement in defect detection and discrimination was of several orders of magnitude compared to that of either feature considered individually. The SHM paradigm presented in this paper is applicable to many structural components having waveguide geometry (e.g. plates, rods, pipes) which lend themselves to guided ultrasonic wave propagation. Depending on the specific application, the ultrasonic configuration (whether pulseecho or pitch-catch) and the discrete wavelet transform decomposition levels, as well as the features considered in the computation of the damage index, may change. Finally, the experimental results suggest that whenever the acquisitions are made in a low noise environment, the use of the DWT may not be necessary to enhance the performance of the unsupervised algorithm. An ongoing study is currently focusing on the use of principal component analysis to reduce the dimensionality of the problem, while future research may evaluate the effect of different mother wavelets on the algorithm performance.
the discordancy test for the baseline and the damaged data are presented. The values of the discordancy associated with the RMS o and the RMS FFT o show higher sensitivity with respect to the increase of defect size compared to the ppk w. An analogous consideration can be extended to the multivariate analysis. For this analysis we combined the above features into three 2D vectors. The MSD for the undamaged and damaged data are shown in figure 10(c). As in figure 10(b), a stepwise trend is observed. However, the multivariate analysis outperforms any of the univariate results since no false negative indications were given, and the values of the MSD related to damage are several orders of magnitude larger. The vector including the RMS o and the RMS FFT o outperforms the other two vectors. This behavior can be linked to the elements k j k of the covariance matrix [K ] of equation (3). These elements are defined as
k jk =
n 1 (x i, j − x)(x ¯ i,k − x) ¯ n i=1
( j, k = 1, 2)
(5)
where n is the number of baseline data (60 in this case) and x i, j is the j -based damage index calculated for the i th sample datum. When the baseline data’s damage indices have low scatter (little dispersion) the factors in equation (5) are smaller than 1. Consequently, the elements of the inverse of the covariance matrix are much greater than 1. In addition, if two indices were closely related, two rows (or columns) of the baseline’s covariance matrix would be almost identical and therefore close to the condition of singularity. This will determine a further increase in the values of the MSD. Finally, it can be argued that to optimize the outlier analysis in terms of damage size sensitivity, it is important to select and combine baseline data features that possess low dispersion. The above considerations justify the experimental evidence that a higher dimensionality of the multivariate analysis does not necessarily improve the algorithm performance in terms of outlier detection and damage size sensitivity.
Acknowledgments The support of the University of Pittsburgh to Mr Cammarata through start-up funding available to the first author is acknowledged. The third and fourth authors acknowledge the support of the Korea Science and Engineering Foundation (M20703000015-07N0300-01510) and the Korea Research Foundation (D00462). Experimental work was conducted in the Watkins Haggart Structural Engineering Laboratory at the University of Pittsburgh.
5. Conclusions This paper presents an automated crack detection technique for metallic waveguides using agile PZTs. Specifically, the study focused on the detection of the onset and propagation of a fatigue crack induced on a steel structural beam. The technique is based on ultrasonic guided waves, discrete wavelet transform, and outlier analysis. Ultrasonic waves were generated through a PZT-wafer connected to an NI PXI unit used to excite a sinusoidally modulated tonebursts ranging from 50 to 300 kHz. The propagating waves, detected by a pair of PZT-wafer sensors, were digitized through the same NI PXI unit. The time waveforms were processed with the discrete wavelet transform to de-noise the signals and to generate a set of relevant damage-sensitive features used to construct a uni- or multi-dimensional damage index. The damage index was fed to an unsupervised learning algorithm based on outlier analysis aimed at detecting anomalous conditions of the structure and in particular the presence of fatigue cracks. The population of
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