- Email: [email protected]
Characterization of nonlinear ultrasonic effects using the dynamic wavelet fingerprint technique
Journal of Sound and Vibration 389 (2017) 364–379
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Characterization of nonlinear ultrasonic effects using the dynamic wavelet fingerprint technique Hongtao Lv, Jingpin Jiao n, Xiangji Meng, Cunfu He, Bin Wu Department of Mechanical Engineering, Beijing University of Technology, Beijing, China
a r t i c l e in f o
abstract
Article history: Received 19 May 2016 Received in revised form 5 November 2016 Accepted 8 November 2016 Handling Editor: L.G. Tham Available online 16 November 2016
An improved dynamic wavelet fingerprint (DWFP) technique was developed to characterize nonlinear ultrasonic effects. The white area in the fingerprint was used as the nonlinear feature to quantify the degree of damage. The performance of different wavelet functions, the effect of scale factor and white subslice ratio on the nonlinear feature extraction were investigated, and the optimal wavelet function, scale factor and white subslice ratio for maximum damage sensitivity were determined. The proposed DWFP method was applied to the analysis of experimental signals obtained from nonlinear ultrasonic harmonic and wave-mixing experiments. It was demonstrated that the proposed DWFP method can be used to effectively extract nonlinear features from the experimental signals. Moreover, the proposed nonlinear fingerprint coefficient was sensitive to micro cracks and correlated well with the degree of damage. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Dynamic wavelet fingerprint Nonlinear ultrasound Micro crack Harmonic Wave-mixing Nonlinear fingerprint coefficient
1. Introduction Recent years have seen great research interest in various nondestructive testing methods based on nonlinear effects associated with ultrasonic wave propagation [1,2]. These effects include the generation of higher harmonics [3] and subharmonics [4], shifting of the resonance frequency [5], and mixed-frequency responses [6]. The primary reason for this interest is the strong nonlinearity of lattice anharmonicity of materials, microstructures of solids, defects, boundaries in crystal structure etc. Nonlinear ultrasonic methods have been used for damage evaluation in metallic structures [7] (even those with complex shapes [8]), composites [9], bones [10], soft tissues and biological media [11], concrete [12], soil and granular materials [1], and glass [2]. For all these applications of nonlinear ultrasonic, a quantity extracted from the measured response is used as a damage sensitive feature that indicates the presence of damage in a structure, which also called nonlinear ultrasonic feature. Actually, identifying features that can accurately distinguish a damaged structure from an undamaged one is the focus of most nondestructive testing [13–15]. For nonlinear ultrasonic methods, the existence of defects or microstructural damage often results in a frequency shift of the ultrasonic signal compared with the input signal. Because of the complex nature of nonlinear ultrasonic methods, it is generally difficult or unfeasible to assess nonlinear ultrasonic features directly from the raw ultrasonic signals. To extract damage-sensitive features from the measured ultrasonic signal, three general techniques have been proposed: Fourier spectral analysis [16–18], bispectral analysis [19–21], and waveform analysis [22]. n Correspondence to: Department of Mechanical Engineering, Beijing University of Technology, Ping Le Yuan 100#, Chaoyang District, Beijing 100124, China. E-mail addresses: [email protected] (H. Lv), [email protected] (J. Jiao), [email protected] (X. Meng), [email protected] (C. He), [email protected] (B. Wu).
http://dx.doi.org/10.1016/j.jsv.2016.11.009 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
365
Fourier spectral analysis is the most widely used method for nonlinear ultrasonic feature detection. From the calculated frequency spectrum, the amplitude response at any feature frequency (e.g., the fundamental harmonic, higher harmonic, subharmonic, or resonance frequency) can be obtained easily. Measurement of the well-known nonlinear parameter β in method of harmonic generation is a good example of the applications of Fourier spectral analysis [16–18]. Bispectral analysis is a third-order spectral analysis technique, which results in a frequency–frequency–amplitude relationship that reflects the coupling between signals at different frequencies. Because of its sensitivity to quadratic phase coupling, bispectral analysis is an attractive signal-analysis tool for detecting nonlinearities due to damage and has been widely used for nonlinearity measurements in various systems. Hillis et al. [19], Courtney et al. [20] and Jiao et al. [21] applied bispectrum analysis to the nonlinear response of modulated ultrasonic signals and proposed its application in crack and intergranular corrosion detection in metal structures. Unlike the two spectral analysis methods described above, waveform analysis extracts the characteristics of defects directly using wave-packet information in time domain. For example, Croxford et al. [22] applied the noncollinear mixing technique for nonlinear ultrasonic detection of plasticity and fatigue, whereby the nonlinear interaction of two shear waves and damage was indicated by the presence of longitudinal packet-waves in the waveform. The method of waveform analysis is simple and intuitive. Unfortunately, however, in many cases it is difficult or even impossible to detect significant variation in the waveform. The traditional techniques described above, whether in the time or frequency domain, are vulnerable to noise contamination and interference from undesired signals; hence, their effectiveness in the extraction and characterization of weak nonlinear responses is limited. The wavelet transform method is a multi-resolution analysis approach that is effective for the extraction of significant details and information in both time and frequency domains, and is widely used for processing transient and nonstationary signals [23–26]. The basic operation of wavelet transform involves dilation and translation operations, which allow multi-scale analysis of the signal. Therefore, this method can be used to effectively extract both time-domain and frequency-domain features of the inspected signal. To take advantage of the potential of the wavelet transform, a dynamic wavelet fingerprint (DWFP) technique [27–29] was developed from the wavelet transform algorithm for feature extraction from dynamic signals. In this technique, DWFPs are constructed on the spot from a projection of the continuous wavelet coefficients of the transient signal, thereby converting one-dimensional time traces into two-dimensional binary images. Several studies have shown that the DWFP technique is effective in characterizing linear features of ultrasonic signals in nondestructive testing [30–35]. Because the waveform features of interest are too subtle to identify in the time domain, Hinders and co-workers [28,35] investigated the usefulness of the DWFP technique to render guided wave-mode information in two-dimensional binary images. The use of wavelets allows both time- and frequency-domain features from the original signals to be retained, and allows image processing to be used to automatically extract features that correspond to the arrival times of the guided wave modes. Hou et al. [32,33] applied the DWFP technique using tomographic images to estimate the arrival times of multiple Lamb wave modes. The amount of white area in the DWFP images was used as a feature to distinguish false modes caused by noise and other interference from the true modes of interest. Hinders and Hou [31] applied the DWFP technique in the ultrasonic detection of suspected scattering objects in periodontal probing. The DWFP patterns were classified and mapped onto an inclination index curve. The performance of the algorithm was evaluated by comparing the ultrasonic probing results with those of full-mouth manual probing at the same sites. Until now, DWFP has only been used for the extraction of linear features with unequal frequency intervals/ or equal scale intervals. In the present study, an improved DWFP was applied to extract nonlinear ultrasonic features from measured signals with equal frequency intervals/ or unequal scale intervals. In Section II, a DWFP algorithm with equal frequency intervals is developed for the characterization of nonlinear effects in ultrasonic signals. In Section III, we investigate the effect of wavelet functions, the scale of the fingerprints to determine appropriate parameters in the wavelet transform, and the white subslice ratio during the slice-projection operation. The practical demonstration of the application of the algorithms to experimental ultrasonic signals is presented in Section IV.
2. DWFP technique for extraction of nonlinear ultrasonic features To extract the nonlinear features in the measured ultrasonic signals, a DWFP technique is proposed, as outlined in the flowchart in Fig. 1. The wavelet transform of the measured signals is first conducted with an equal frequency interval. Then, the wavelet scalogram is smoothed by a median filter and normalized by the wavelet scalogram at the fundamental frequency; a slice-projection operation is applied to the normalized wavelet scalogram in the frequency range of the nonlinear response. Finally, nonlinear features are extracted from the two-dimensional fingerprint image. 2.1. Wavelet transform with an equal frequency interval The continuous wavelet transform of a signal, x (t ), can be written as
WTx (a, b) =
1 a
+∞
∫−∞
⎛ t − b⎞ ⎟ dt x (t ) ψ * ⎜ ⎝ a ⎠
(1)
366
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 1. Flowchart of the DWFP technique for extraction of nonlinear ultrasonic features.
where WTx (a, b) are the wavelet coefficients, a is the scaling parameter, b is the time shift parameter, and * denotes the conjugate [25,36,37]. The factor 1/ a is used to ensure energy conservation for the transform. ψ ( t − b)/a is generated by
(
)
dilation and translation from the mother wavele ψ ( t ). Here, the relationship between the scale factor a and frequency can be described as
f=
fc a × TS
(2)
where fc is the center frequency of the mother wavelet in Hz, TS is the sampling period and f is the pseudo-frequency corresponding to the scale a , in Hz. The wavelet transform coefficients signify the similarity between the wavelet and the signal; that is, the larger the coefficient, the greater the similarity between the wavelet and signal. The high-frequency components in the signal can be seen at small scales, and the low-frequency components in the signal show up as largescale features. For traditional wavelet transforms, the scale is arithmetical. According to the correlation between the scale, a , and frequency, f , (shown in Eq. 2), the frequency resolution of the wavelet coefficients is varied in the frequency domain and is low at higher frequencies, which can lead to deteriorated frequency localization. To improve the location accuracy of the nonlinear components at high frequencies, the wavelet transform of the signal is conducted with an arithmetical frequency interval in this paper. The frequency sequence with the same interval is given by
{ fscale } =
{1, 2, 3 … a max } × fm amax
(3)
where amax is the maximum scale of the wavelet transform and fm is the maximum frequency considered in the nonlinear measurements. For example, fm should be greater than the harmonic frequency for the harmonic method and greater than the mixing frequency for the wave-mixing method. The scale sequence can then be obtained as follows:
{ ascale } =
fc
{ fscale } × TS
(4)
If the continuous wavelet transform of a signal is conducted using the scale sequence above, the wavelet coefficients at different frequencies will have identical frequency resolution. Then, the wavelet scalogram (time frequency representation) is defined as energy density spectrum [38]:
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
SG x (a, b) = WTx (a, b) WT x* (a, b) = WTx (a, b) 2
367
(5)
This wavelet scalogram can simultaneously keep track of the local time and frequency information in the signal. And it is capable of revealing hidden aspects of signal, which may possibly be missed by other techniques [39]. Besides, it can simplify the subsequent slice operation. 2.2. Definition of the nonlinear fingerprint coefficient Because of the influences of the environmental conditions, the variable parameters of the experimental system, and the properties of the sample, the measured signal inevitably contains interference, which disturbs the extraction of weak nonlinear effects. In this study, the wavelet scalogram is smoothed using a median filter, where the value in wavelet scalogram is replaced by the median of its neighborhood, to weaken the introduced noises. Similar to the definitions of the nonlinear parameters used in the harmonic and wave-mixing methods [16,22], the effects of waves at the fundamental frequency are taken into account, and the normalized wavelet scalogram can be obtained as follows:
SGxn (a, b) =
SG x (a, b) SG f1 SG f 2
(6)
where SGxn (a, b) indicates the normalized wavelet scalogram, and SG f1, SG f2 are the maximum wavelet scalograms at frequencies f1 , f2. For harmonic measurements, SG f1 = SG f2, which is the maximum wavelet scalogram at the fundamental (i.e., excitation) frequency; for wave-mixing measurements, SG f1 ≠ SG f2, which are the maximum wavelet scalograms at the two different excitation frequencies. A slice-projection operation is then applied to the normalized wavelet scalogram to generate a two-dimensional blackand-white fingerprint, as shown in Fig. 2. The three-dimensional normalized wavelet scalogram within the bandwidth of the nonlinear response (harmonic component or mixing component) is cut continuously with equal-thickness slices, and projected onto the time-frequency plane, which results in a two-dimensional black-and-white (0 and 1) image resembling a human fingerprint. The slice, with thickness H, contains two subslices. One subslice is black and the other is white; the thickness of the white slice is about η % (called white subslice ratio) of the total thickness. In this paper, H is one third of the peak of three-dimensional normalized wavelet scalogram within the bandwidth of the nonlinear response, obtained from the lowest degree of damage. It should be noted that there is low-level noise in the normalized wavelet scalogram sometimes; therefore, the projection corresponding to the bottom slice of the normalized wavelet scalogram is omitted from the DWFP image. As illustrated in Fig. 2, the number of ridges in the fingerprint image is dependent not only on the thickness of the slice but also on the peak of the normalized wavelet scalogram. Therefore, if the thickness of the slice is fixed, the number of ridges is directly proportional to the peak of the normalized wavelet scalogram. Accordingly, the amount of white area in the DWFP image can be used as the nonlinear feature, and the nonlinear fingerprint coefficient S can be defined as M
S=
N
∑ ∑ Iij
(7)
i=1 j=1
Fig. 2. Slice-projection operation to generate the two-dimensional black-and-white fingerprint.
368
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
where Iij indicates the pixel value (0 or 1) in row i and column j of the DWFP image, and M and N are the dimensions of the DWFP image. Because of the relationship between the normalized wavelet scalogram and the nonlinear ultrasonic features, it is possible to use the nonlinear fingerprint coefficient to quantitatively evaluate the early stages of damage in structures.
3. Effects influencing DWFP for the extraction of nonlinear features To optimize the performance of the proposed algorithm, it is necessary to carefully select its parameters, such as the type of wavelet function, the range of the scale and the ratio of white subslice, such that the nonlinear ultrasonic effect can be effectively extracted from the DWFP. In this section, using the finite-element model (FEM), the optimal wavelet function, the optimal scale and the optimal white subslice ratio were determined. 3.1. Finite-element model and typical results A two-dimensional FEM for simulating the nonlinear interaction between Lamb waves and a crack is represented in Fig. 3. The simulations were implemented using Abaqus/Explicit software. The parameters of the plate used in the simulation are listed in Table 1. In Fig. 4, the phase–velocity dispersion curves of the Lamb waves for a 1.7-mm-thick steel plate are shown. The dashed and solid lines are the dispersion curves of the fundamental and second-harmonic Lamb waves, respectively. Considering the condition for phase–velocity matching for the generation of the cumulative second-harmonic Lamb mode [40], the fundamental Lamb wave (S1 mode) and the double-frequency Lamb wave (S2 mode) were selected as mode pairs at an excitation frequency of about 2.2 MHz. The desired S1 mode was obtained by applying the same concentrated transient loads in x-direction at the upper left and lower left surfaces of the model (as shown in Fig. 3) [41]. The excitation was a Hamming-windowed tone-burst consisting of 10 cycles at a frequency of 2.2 MHz. The displacement components were recorded on the surface of the plate at a distance of 102 mm from the left end. A microcrack was positioned 100 mm away from the left end of the plate; its shape was modeled as an ellipse with a hard-contact frictionless surface. The major and minor axes of the ellipse determine the length and width of the crack, respectively. To investigate the dependence of nonlinear effects on the length of the microcrack, a series of simulations was performed on models with microcracks of identical width but different lengths. The width, w, was fixed at 10 nm, and the length, l, was varied (0, 0.6, 0.7, 0.8, 0.9, and 1 mm). A typical waveform obtained from a damaged plate with a 0.8-mm-long microcrack is shown in Fig. 5(a). The signal was transformed using the cmor7-7 wavelet with a scale factor of 1024, where the maximum analyzing frequency, fm , was 6 MHz. The three-dimensional distribution of the wavelet scalogram is shown in Fig. 5(b), and the normalized local wavelet scalogram within a harmonic frequency bandwidth is shown in Fig. 5(c). A thick contour-slice operation is applied to the local wavelet scalogram within the harmonic frequency bandwidth. A corresponding DWFP image is shown in Fig. 5(d). It can be seen that the DWFP image represents the nonlinear harmonic response in a two-dimensional time–frequency domain, and contains information pertaining to nonlinear effects. 3.2. Performance of different wavelet functions The performance of nine types of wavelet function (haar, cgau4, sym6, mexh, cmor1-1, cmor3-3, cmor5-5, cmor7-7, and cmor9-9) for the extraction of nonlinear effects was investigated. The simulated signals were processed using the proposed
Fig. 3. Depiction of the model and parameters used in the finite-element model.
Table 1 Parameters of the steel plate used in the model. Thickness
Length
Density
Elastic Modulus
Poisson's Ratio
1.7 mm
400 mm
7850 kg/m3
208.42 GPa
0.2959
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
369
Fig. 4. Phase-velocity dispersion curves for a 1.7-mm-thick steel plate.
Fig. 5. Simulated signal and wavelet scalograms processed using the proposed method with a scale factor of 1024. (a) Waveform; (b) wavelet scalogram; (c) normalized local harmonic scalogram; (d) DWFP image.
DWFP technique with a scale factor of 1024; representative results are shown in Fig. 6. It can be seen that there is significant difference between the DWFP images when the identical simulation signal is transformed with the different wavelet functions. As shown in Fig. 6(a)–(f) and (i), for the cases of haar, cgau4, sym6, mexh, cmor1-1, cmor3-3, and cmor9-9, there are no recognizable fingerprints in the two-dimensional time–frequency images. Moreover, the images do not provide any evidence of nonlinear effects at the center frequency of the harmonic components (4.4 MHz). However, for the case of cmor5-5 and cmor7-7, features of fingerprints are visible in the DWFP images, as shown in Fig. 6(g, h), and the center frequency of the fingerprints matches the center frequency of the harmonic components. A possible reason for the better performance of these two wavelets is that they are modulated in the time domain by a Gaussian-shaped time window,
370
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 6. DWFP images acquired using different wavelet functions: (a) haar; (b) mexh; (c) sym6; (d) cgau4; (e) cmor1-1; (f) cmor3-3; (g) cmor5-5; (h) cmor77; (i) cmor9-9.
which presents the optimal resolution simultaneously in the time and frequency domains [42]. In addition, the wave shapes in both cases are similar to that of the analyzed signal [43]. It can also be observed in Fig. 6(g) and Fig. 6(h) that the DWFP image obtained with cmor7-7 has more ridges than that obtained with cmor5-5. This can be attributed to the fact that cmor7-7 is more sensitive to the nonlinear component, which leads to a higher normalized wavelet scalogram. Moreover, the width of the DWFP image acquired using cmor7-7 reflects the duration of the waveform more accurately. Therefore, cmor7-7 was selected as the optimal mother wavelet for nonlinear ultrasonic signals analysis. 3.3. Effect of scale factor To determine the influence of scale range on the extraction of nonlinear effects, an identical signal was transformed using the cmor7-7 wavelet function over different scales (64, 128, 256, 512, 1024, and 2048) in the frequency range [0,6 MHz]; the representative results are shown in Fig. 7. There is a large difference between the DWFP images depending on the scale used: the larger the scale is, the higher the frequency resolution of the DWFP image and the clearer the fingerprints. For small scales (64, 128 and 256), it is difficult to identify the fingerprints because the DWFP images are blurred. For large scales (512, 1024 and 2048), however, the fingerprints can be recognized clearly. The reason for this phenomenon is that the proposed DWFP technique is based on the wavelet transform of signals with the same frequency resolution in a fixed frequency range [fmin , fmax ]. Consequently, the frequency interval in the wavelet scalogram is
Δf =
fmax − fmin a
(8)
The frequency interval Δf in Eq. (8) is inversely proportional to the scale factor, a , used in the wavelet transform. Therefore, with increasing scale factor, the frequency interval of the wavelet scalogram decreases and the resolution of the DWFP image becomes higher. However, the scale also affects the computation time of the wavelet transform: with
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
371
Fig. 7. DWFP images with different scale factors: (a) 64; (b) 128; (c) 256; (d) 512; (e) 1024; (f) 2048.
increasing scale factor, the computation time of wavelet transform increased. Therefore, by comprehensive consideration of the frequency resolution and computation time, a scale factor of 1024 was selected as the optimal value for the transformation of the nonlinear ultrasonic signals.
3.4. Effect of white subslice ratio Similarly, to investigate the effect of white subslice ratio on the extraction of nonlinear effects, the same signal was transformed using the cmor7-7 wavelet function at scale of 1024, and sliced at different white subslice ratio (from 0.1 to 0.9 with a step 0.1). The typical results are shown in Fig. 8. It can be seen that the ratio of white sublice has great effect on the ridge of fingerprint. In particular, along with the increase of white subslice ratio comes an increase in the width of white ridge. When the white subslice ratio is too small (0.1 and 0.2), the DWFP images are fuzzy because of the narrow ridges; on the other hand, when the white subslice ratio is too large (0.6, 0.7, 0.8 and 0.9), the fingerprint is difficult to discern due to the wide ridges. Clearly distinct fingerprints can be obtained at an intermediate white subslice ratio. In this paper, a white subslice ratio 0.3 was selected as the optimal value for the transformation of the nonlinear ultrasonic signals.
3.5. Results for microcracks with different lengths The DWFP technique was applied for the analysis of nonlinear signals obtained from the FEM with microcracks of different lengths. The DWFP images corresponding to the harmonic components for different microcracks are shown in Fig. 9. These fingerprint images differ depending on the crack length, and there is some association between the fingerprint features and the length of cracks. In addition, the amount of white area and the number of ridges in the DWFP images increase with increasing microcrack length. The magnitude of the nonlinear fingerprint coefficient (calculated with Eq. (7)) is shown in Fig. 10 as a function of crack length. The nonlinear fingerprint coefficient increases monotonically with the length of the microcrack, demonstrating that the proposed DWFP technique can be used to quantitatively evaluate microcrack in structure.
4. Application of the DWFP technique To demonstrate the effectiveness of the proposed DWFP technique for nonlinear feature extraction, the proposed DWFP technique was applied to the analysis of two typical kinds of nonlinear ultrasonic signals obtained from experiments of harmonic Lamb waves and experiments of Lamb waves mixing.
372
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 8. DWFP images with different white subslice ratio: (a) 0.1; (b) 0.2; (c) 0.3; (d) 0.4; (e) 0.5; (f) 0.6; (g) 0.7; (h) 0.8; (i) 0.9.
Fig. 9. DWFP images corresponding to the harmonic components for micro cracks of different lengths: (a) 0 mm; (b) 0.6 mm; (c) 0.7 mm; (d) 0.8 mm; (e) 0.9 mm; (f) 1 mm.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
373
Fig. 10. Relationship between the nonlinear fingerprint coefficient and the micro crack length.
4.1. Experimental system A schematic of the nonlinear acoustic measurement setup is shown in Fig. 11, in which the RAM 5000 system is the core component. During each experiment, Lamb waves are generated and detected using contact wedge transducers, and the distance d between the two transducers is 15 cm. For nonlinear ultrasonic harmonic and wave-mixing experiments, the central frequencies of the two transducers (one used as the transmitter T, and another used as the receiver R) are different, and then the two transducers (T and R) are fixed to Plexiglas-wedge with different angle, which detailed in the following two subsections. Nonlinear Lamb-wave measurements were conducted on dog-bone-shaped steel specimens, as shown in Fig. 12. In each sample, a notch of 2-mm length and 1-mm width was machined in the middle of the lateral side. Then, one specimen was kept pristine to be used as a reference, and all other specimens were fatigued under different numbers of cycles to introduce micro cracks of different lengths. A typical optical micrograph, shown in Fig. 13, shows plates with cracks of various sizes. The micro cracks have irregular interfaces, and their widths vary over a considerable range. The widths and lengths of the micro cracks are provided in Table 2. Experiments were conducted on the specimens described above using harmonic and mixing-wave methods, and the measured signals were processed using the proposed DWFP method, as described in the following two subsections. 4.2. Harmonic measurements For the experimental generation of harmonic Lamb waves, according to the condition of phase-velocity matching depicted in Fig. 4, a narrow-band longitudinal piezoelectric transducer with a central frequency of 2.25 MHz was used as the transmitter and a broad-band longitudinal piezoelectric transducer with a central frequency of 5 MHz was used as the receiver. According to the Snell theorem, Lamb mode pairs can be excited and received via a Plexiglas-wedge with an oblique angle of 27.1°.
Fig. 11. Schematic diagram of the experimental system.
374
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 12. Schematic diagram of the test specimen and a photograph of the micro crack.
Fig. 13. Micrographs of micro cracks of various lengths in steel plates: (a) 10 mm; (b) 15 mm; (c) 20 mm. Table 2 Width and length of the three micro cracks analyzed. Specimen number
1
2
3
4
Length Width
– –
10 mm 6.4 mm
15 mm 35.5 mm
20 mm 96.1 mm
Considering the frequency response of the experiment system, the excitation frequency was determined by a frequencysweep experiment on an undamaged specimen to generate a significant harmonic nonlinear response [40]. In Fig. 14, the amplitude–frequency dependences of the fundamental and second-harmonic Lamb waves are shown over the range of 0.2– 4 MHz. It can be seen that both the fundamental and second-harmonic waves have maximum amplitudes at 2 MHz. To obtain a significant nonlinear response, the excitation frequency of the sinusoidal component was selected at this frequency. Accordingly, the excitation signal was a Hamming-windowed tone-burst consisting of 10 cycles at a frequency of 2 MHz.
Fig. 14. Measured amplitude of the fundamental and second-harmonics Lamb waves versus excitation frequency.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
375
Fig. 15. Typical waveforms for (a) an undamaged specimen and (b) a specimen with a 10-mm-long micro crack.
The harmonic experiments were conducted on the four specimens. Typical waveforms received from the different samples are shown in Fig. 15. From these waveforms, it is difficult to discern whether there is harmonic generation induced by nonlinearity. DWFP images corresponding to the harmonic components of the signals measured from samples with micro cracks of different lengths are shown in Fig. 16. It can be seen that the number of ridges and the amount of white area in the images increase with increasing micro crack length. The dependence of the calculated nonlinear fingerprint coefficient on the crack length is shown in Fig. 17. For each specimen, the harmonic measurement was repeated five times at the same position, and the error bars in the figure represent the standard deviation of these measurements. It is clear that the nonlinear fingerprint coefficient increases with increasing crack length and thus the degree of damage in the plate. 4.3. Wave-mixing measurements It is well known that, at lower frequencies, the first symmetric mode (S0) is the fastest mode and is almost non-dispersive, which makes it promising in nondestructive testing. In this subsection, experiments of Lamb-wave mixing were conducted on the four different specimens using the S0 mode at low frequencies. Two commercial piezoelectric transducers
Fig. 16. DWFP images corresponding to the harmonic components for micro cracks of different lengths: (a) 0 mm; (b) 10 mm; (c) 15 mm; (d) 20 mm.
376
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 17. Relationship between the nonlinear fingerprint coefficient and the micro crack length.
with central frequencies of 0.5 MHz and 1 MHz were used as the transmitter, T and receiver, R, respectively. The S0 mode can be excited and received via a Plexiglas-wedge with an oblique angle of 30°. To investigate the effect of the driving frequency on the performance of wave mixing, frequency-sweep experiments were firstly conducted on an undamaged specimen to obtain a significant mixed-frequency nonlinear response. Driving frequency f1 was fixed ( f1 ¼0.45 MHz) and driving frequency f2 was varied in the range of 0.5–0.66 MHz. The amplitude of the sidebands at the sum frequencies generated by wave mixing was measured with the RAM 5000 system. The amplitudefrequency dependence of the sidebands measured in the specimen without a microcrack is shown in Fig. 18. It is noted that there is a resonance point at the driving frequency of 0.6 MHz (as shown in the black dashed line). To obtain significant interaction between the two waves and the microcrack, wave-mixing experiments were conducted at two driving frequencies: 0.45 MHz and 0.6 MHz. A tone burst of 50 V generated by the RAM-5000 system was fed into the transmitting transducer. The tone burst was composed of two sinusoidal components at the two driving frequencies and had a duration of 45 ms. Representative waveforms received from the different samples are shown in Fig. 19. It is difficult to discern whether there is sum-frequency generation induced by nonlinearity. DWFP images corresponding to the sum-frequency components for signals measured from samples with different microcrack lengths are shown in Fig. 20. The number of ridges and the amount of white area in the images increase with microcrack length. The dependence of the calculated nonlinear fingerprint coefficients on the microcrack length is shown in Fig. 21. For each specimen, the wave-mixing measurement was repeated five times at the same position, and the error bars in the figure represent the standard deviation of these measurements. Similar to the result obtained for the harmonic experiments, the proposed nonlinear fingerprint coefficient increases monotonically with microcrack length and thus the degree of damage in the plate. Despite the identical trends in the sensitivity of the two nonlinear fingerprint coefficients (harmonic component and wave-mixing component) with respect to the micro crack length, there is a slight difference is their absolute sensitivities. The possible reason can be explained as follows. It is well known that the nonlinear ultrasonic effects induced by damage are strongly dependent on detection parameters, such as frequency and amplitude. Since there is a great difference between the wave-mixing measurements and harmonic measurements in experimental conditions, such as frequency and amplitude,
Fig. 18. Amplitude-frequency dependence of sidebands at the sum frequency detected in the sample without micro cracks when f1 is 0.45 MHz.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
377
Fig. 19. Typical waveforms for (a) an undamaged specimen and (b) a specimen with a 10-mm-long micro crack.
Fig. 20. DWFP images corresponding to the mixing components for micro cracks of different length: (a) 0 mm; (b) 10 mm; (c) 15 mm; (d) 20 mm.
the measured nonlinear responses by two methods are inevitably different. As a result of the frequency dependence of nonlinear ultrasonic effects at different damage degree, there is a slight difference in absolute sensitivities for the wavemixing measurements and harmonic measurements. Nevertheless, the experimental results show that the proposed dynamic wavelet transform technique is effective for extracting nonlinear effects in measured signals, and the proposed nonlinear fingerprint coefficient correlates well with the degree of damage.
5. Conclusions In this study, an improved DWFP technique was proposed for characterization of nonlinear ultrasonic effects. To facilitate characterization of the nonlinear components, a DWFP algorithm was developed with equal intervals. The amount of white area in the fingerprint image was used to characterize the nonlinear ultrasonic effects. The performance of different wavelet functions, the effect of different scale factors and different white subslice ratio on the extraction of nonlinear features from
378
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 21. Relationship between the nonlinear fingerprint coefficient and the micro crack length.
simulated signals was investigated. It has been shown that the nonlinear effects can be effectively extracted when DWFP is conducted using a cmor7-7 wavelet with a scale factor of 1024 at white subslice ratio 0.3. Nonlinear ultrasonic experiments including the measurement of harmonics and wave mixing were conducted in plates to detect micro cracks. The proposed DWFP method was demonstrated to be able to effectively extract the nonlinear features from the experimental signals. Moreover, the nonlinear features were sensitive to micro cracks, and correlated well with the degree of damage in the plates. Because the nonlinear features are extracted from the dynamic fingerprint, which is related to the time-frequency distribution of the measured signals, the nonlinear feature in this technique contains nonlinear information in both the time and frequency domains. This technique therefore has great potential for nonlinear measurements.
Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant Nos. 11572010, 11272017 and 11132002); the National Key Research and Development Program of China (2016YFF0203002).
References [1] K.Y. Jhang, Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: a review, Int. J. Precis. Eng. Manuf. 10 (1) (2009) 123–135. [2] D. Broda, W.J. Staszewski, A. Martowicz, T. Uhl, V.V. Silberschmidt, Modelling of nonlinear crack wave interactions for damage detection based on ultrasound – a review, J. Sound Vib. 333 (4) (2014) 1097–1118. [3] L.A. Ostrovsky, P.A. Johnson, Dynamic nonlinear elasticity in geomaterials, Riv. del nuovo cimento 24 (7) (2001) 1–46. [4] L.A. Ostrovsky, S.N. Gurbatov, J.N. Didenkulov, Nonlinear acoustics in Nizhni Novgorod (A review), Acoust. Phys. 51 (2) (2005) 114–127. [5] P.A. Johnson, B. Zinszner, P.N.J. Rasolofosaon, Resonance and elastic nonlinear phenomena in rock, J. Geophys. Res. 101 (B5) (1996) 11533–11564. [6] G.L. Jones, D.R. Kobett, Interaction of elastic waves in an isotropic solid, J. Acoust. Soc. Am. 35 (1) (1963) 5–10. [7] H. Ogi, M. Hirao, S. Aoki, Noncontact monitoring of surface-wave nonlinearity for predicting the remaining life of fatigued steels, J. Appl. Phys. 90 (1) (2001) 438–442. [8] K. Sheppard, A. Zagrai, D. Donskoy, A non-linear acoustic, vibro-modulation technique for the detection and monitoring of contact-type defects, including those associated with corrosion, Corros. Rev. 25 (1–2) (2007) 81–96. [9] M. Meo, U. Polimeno, G. Zumpano, Detecting damage in composite material using nonlinear elastic wave spectroscopy methods, Appl. Compos. Mater. 15 (3) (2008) 115–126. [10] M. Muller, D. Mitton, M. Talmant, P. Johnsonc, P. Laugiera, Nonlinear ultrasound can detect accumulated damage in human bone, J. Biomech. 41 (5) (2008) 1062–1068. [11] C.W. Connor, K. Hynynen, Bio-acoustic thermal lensing and nonlinear propagation in focused ultrasound surgery using large focal spots: a parametric study, Phys. Med. Biol. 47 (11) (2002) 1911–1928. [12] C. Payan, V. Garnier, J. Moysan, P.A. Johnson, Applying nonlinear resonant ultrasound spectroscopy to improving thermal damage assessment in concrete, J. Acoust. Soc. Am. 121 (4) (2007) 125–130. [13] I. Gurov, M. Volynsky, E. Vorobeva, Dynamic wavefront evaluation in phase shifting interferometry based on recurrence fringe processing, Proc. Int. Conf. Adv. Phase Meas. Methods Opt. Imaging 1236 (2010) 479–485. [14] C.A. Miller, M.K. Hinders, Classification of flaw severity using pattern recognition for guided wave-based structural health monitoring, Ultrasonics 54 (1) (2014) 247–258. [15] R. Carandente, P. Cawley, A method to estimate the size of corrosion patches with guided waves in pipes, AIP Conf. Proc.-Am. Inst. Phys. 1430 (2012) 1929–1936. [16] K.Y. Jhang, K.C. Kim, Evaluation of material degradation using nonlinear acoustic effect, Ultrasonics 37 (1) (1999) 39–44. [17] M.X. Deng, Analysis of second-harmonic generation of Lamb waves propagating in layered planar structures with imperfect interfaces, Appl. Phys. Lett.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
379
88 (22) (2006) 221902. [18] C. Pruell, J.Y. Kim, J. Qu, L. Jacobs, Evaluation of plasticity driven material damage using Lamb waves, Appl. Phys. Lett. 91 (23) (2007) 231911. [19] C.L. Nikias, M.R. Raghuveer, Bispectrum estimation: a digital signal processing framework, Proc. IEEE 75 (7) (1987) 869–891. [20] C.R.P. Courtney, B.W. Drinkwater, S.A. Neild, P.D. Wilcox, Factors affecting the ultrasonic intermodulation crack detection technique using bispectral analysis, NDT E Int. 41 (3) (2008) 223–234. [21] J.P. Jiao, J.J. Sun, G.H. Li, B. W, C.F. He, Evaluation of the intergranular corrosion in austenitic stainless steel using collinear wave mixing method, NDT E Int. 69 (2015) 1–8. [22] A.J. Croxford, P.D. Wilcox, B.W. Drinkwater, P.B. Nagy, The use of non-collinear mixing for nonlinear ultrasonic detection of plasticity and fatigue, J. Acoust. Soc. Am. 126 (5) (2009) 117–122. [23] Jl Chen, Z.P. Li, J. Pan, G.G. Chen, Y.Y. Zi, J. Yuan, B.Q. Chen, Z.J. He, Wavelet transform based on inner product in fault diagnosis of rotating machinery: a review, Mech. Syst. Signal Process. 70 (2016) 1–35. [24] F. Moreau, D. Gibert, M. Holschneider, G. Saracco, Wavelet analysis of potential fields, Inverse Probl. 13 (1) (1997) 165–178. [25] S. Mallat, A Wavelet Tour of Signal Processing, Academic press, Paris, 1999. [26] J.P. Jiao, W.H. Liu, J. Zhang, Q. Zhang, C.F. He, B. Wu, Time–frequency analysis for ultrasonic measurement of liquid-layer thickness, Mech. Syst. Signal Process. 35 (1–2) (2013) 69–83. [27] C.A. Miller, M.K. Hinders, Classification of flaw severity using pattern recognition for guided wave-based structural health monitoring, Ultrasonics 54 (1) (2014) 247–258. [28] M.K. Hinders, J.P. Bingham, Lamb wave pipe coating disbond detection using the dynamic wavelet fingerprinting technique, Rev. Quant. Nondestruct. Eval. 29 (2010) 615–623. [29] J. Bingham, M. Hinders, Lamb wave characterization of corrosion-thinning in aircraft stringers: experiment and three-dimensional simulation, J. Acoust. Soc. Am. 126 (1) (2009) 103–113. [30] J. Bingham, M. Hinders, A. Friedman, Lamb wave detection of limpet mines on ship hulls, Ultrasonics 49 (8) (2009) 706–722. [31] J.D. Hou, S.T. Rose, M.K. Hinders, Ultrasonic periodontal probing based on the dynamic wavelet fingerprint, EURASIP J. Adv. Signal Process. 2005 (7) (2005) 1137–1146. [32] J.D. Hou, K.R. Leonard, M.K. Hinders, Automatic multi-mode Lamb wave arrival time extraction for improved tomographic reconstruction, Inverse Probl. 20 (6) (2004) 1873. [33] J.D. Hou, M.K. Hinders, Dynamic wavelet fingerprint identification of ultrasound signals, Mater. Eval. 60 (9) (2002) 1089–1093. [34] J. Bingham, M. Hinders, A. Friedman, Lamb wave detection of limpet mines on ship hulls, Ultrasonics 49 (8) (2009) 706–722. [35] J.D. Hou, S.T. Rose, M.K. Hinders, Ultrasonic periodontal probing based on the dynamic wavelet fingerprint, EURASIP J. Adv. Signal Process. (7) (2005) 1137–1146. [36] S. Mallat, W.L. Hwang, Singularity detection and processing with wavelets, IEEE Trans. Inf. Theory 38 (2) (1992) 617–643. [37] C.A. Zala, I. Barrodale, K.I. McRae, High Resolution Deconvolution of Ultrasonic Traces, Signal Processing and Pattern Recognition in Nondestructive Evaluation of Materials, 44, Springer Berlin, Heidelberg, 1988, 101–108. [38] Z. Peng, F. Chu, Y. He, Vibration signal analysis and feature extraction based on reassigned wavelet scalogram, J. Sound Vib. 253 (5) (2002) 1087–1100. [39] S.K. Yadav, S. Banerjee, T. Kundu, On suitability of feature extraction techniques for local damage detection, Proc. SPIE - Int. Soc. Opt. Eng. 7983 (2011). [40] M.X. Deng, J. Yang, Characterization of elastic anisotropy of a solid plate using nonlinear Lamb wave approach, J. Sound Vib. 308 (1) (2007) 201–211. [41] X. Wan, Q. Zhang, G.H. Xu, P.W. Tse, Numerical simulation of nonlinear lamb waves used in a thin plate for detecting buried micro-cracks, Sensors 14 (5) (2014) 8528–8546. [42] K.C. Gryllias, I.A. Antoniadis, Estimation of the instantaneous rotation speed using complex shifted Morlet wavelets, Mech. Syst. Signal Process. 38 (1) (2013) 78–95. [43] Y.Y. He, Y. Liu, Experimental research into time–frequency characteristics of cavitation noise using wavelet scalogram, Appl. Acoust. 72 (10) (2011) 721–731.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Characterization of nonlinear ultrasonic effects using the dynamic wavelet fingerprint technique Hongtao Lv, Jingpin Jiao n, Xiangji Meng, Cunfu He, Bin Wu Department of Mechanical Engineering, Beijing University of Technology, Beijing, China
a r t i c l e in f o
abstract
Article history: Received 19 May 2016 Received in revised form 5 November 2016 Accepted 8 November 2016 Handling Editor: L.G. Tham Available online 16 November 2016
An improved dynamic wavelet fingerprint (DWFP) technique was developed to characterize nonlinear ultrasonic effects. The white area in the fingerprint was used as the nonlinear feature to quantify the degree of damage. The performance of different wavelet functions, the effect of scale factor and white subslice ratio on the nonlinear feature extraction were investigated, and the optimal wavelet function, scale factor and white subslice ratio for maximum damage sensitivity were determined. The proposed DWFP method was applied to the analysis of experimental signals obtained from nonlinear ultrasonic harmonic and wave-mixing experiments. It was demonstrated that the proposed DWFP method can be used to effectively extract nonlinear features from the experimental signals. Moreover, the proposed nonlinear fingerprint coefficient was sensitive to micro cracks and correlated well with the degree of damage. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Dynamic wavelet fingerprint Nonlinear ultrasound Micro crack Harmonic Wave-mixing Nonlinear fingerprint coefficient
1. Introduction Recent years have seen great research interest in various nondestructive testing methods based on nonlinear effects associated with ultrasonic wave propagation [1,2]. These effects include the generation of higher harmonics [3] and subharmonics [4], shifting of the resonance frequency [5], and mixed-frequency responses [6]. The primary reason for this interest is the strong nonlinearity of lattice anharmonicity of materials, microstructures of solids, defects, boundaries in crystal structure etc. Nonlinear ultrasonic methods have been used for damage evaluation in metallic structures [7] (even those with complex shapes [8]), composites [9], bones [10], soft tissues and biological media [11], concrete [12], soil and granular materials [1], and glass [2]. For all these applications of nonlinear ultrasonic, a quantity extracted from the measured response is used as a damage sensitive feature that indicates the presence of damage in a structure, which also called nonlinear ultrasonic feature. Actually, identifying features that can accurately distinguish a damaged structure from an undamaged one is the focus of most nondestructive testing [13–15]. For nonlinear ultrasonic methods, the existence of defects or microstructural damage often results in a frequency shift of the ultrasonic signal compared with the input signal. Because of the complex nature of nonlinear ultrasonic methods, it is generally difficult or unfeasible to assess nonlinear ultrasonic features directly from the raw ultrasonic signals. To extract damage-sensitive features from the measured ultrasonic signal, three general techniques have been proposed: Fourier spectral analysis [16–18], bispectral analysis [19–21], and waveform analysis [22]. n Correspondence to: Department of Mechanical Engineering, Beijing University of Technology, Ping Le Yuan 100#, Chaoyang District, Beijing 100124, China. E-mail addresses: [email protected] (H. Lv), [email protected] (J. Jiao), [email protected] (X. Meng), [email protected] (C. He), [email protected] (B. Wu).
http://dx.doi.org/10.1016/j.jsv.2016.11.009 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
365
Fourier spectral analysis is the most widely used method for nonlinear ultrasonic feature detection. From the calculated frequency spectrum, the amplitude response at any feature frequency (e.g., the fundamental harmonic, higher harmonic, subharmonic, or resonance frequency) can be obtained easily. Measurement of the well-known nonlinear parameter β in method of harmonic generation is a good example of the applications of Fourier spectral analysis [16–18]. Bispectral analysis is a third-order spectral analysis technique, which results in a frequency–frequency–amplitude relationship that reflects the coupling between signals at different frequencies. Because of its sensitivity to quadratic phase coupling, bispectral analysis is an attractive signal-analysis tool for detecting nonlinearities due to damage and has been widely used for nonlinearity measurements in various systems. Hillis et al. [19], Courtney et al. [20] and Jiao et al. [21] applied bispectrum analysis to the nonlinear response of modulated ultrasonic signals and proposed its application in crack and intergranular corrosion detection in metal structures. Unlike the two spectral analysis methods described above, waveform analysis extracts the characteristics of defects directly using wave-packet information in time domain. For example, Croxford et al. [22] applied the noncollinear mixing technique for nonlinear ultrasonic detection of plasticity and fatigue, whereby the nonlinear interaction of two shear waves and damage was indicated by the presence of longitudinal packet-waves in the waveform. The method of waveform analysis is simple and intuitive. Unfortunately, however, in many cases it is difficult or even impossible to detect significant variation in the waveform. The traditional techniques described above, whether in the time or frequency domain, are vulnerable to noise contamination and interference from undesired signals; hence, their effectiveness in the extraction and characterization of weak nonlinear responses is limited. The wavelet transform method is a multi-resolution analysis approach that is effective for the extraction of significant details and information in both time and frequency domains, and is widely used for processing transient and nonstationary signals [23–26]. The basic operation of wavelet transform involves dilation and translation operations, which allow multi-scale analysis of the signal. Therefore, this method can be used to effectively extract both time-domain and frequency-domain features of the inspected signal. To take advantage of the potential of the wavelet transform, a dynamic wavelet fingerprint (DWFP) technique [27–29] was developed from the wavelet transform algorithm for feature extraction from dynamic signals. In this technique, DWFPs are constructed on the spot from a projection of the continuous wavelet coefficients of the transient signal, thereby converting one-dimensional time traces into two-dimensional binary images. Several studies have shown that the DWFP technique is effective in characterizing linear features of ultrasonic signals in nondestructive testing [30–35]. Because the waveform features of interest are too subtle to identify in the time domain, Hinders and co-workers [28,35] investigated the usefulness of the DWFP technique to render guided wave-mode information in two-dimensional binary images. The use of wavelets allows both time- and frequency-domain features from the original signals to be retained, and allows image processing to be used to automatically extract features that correspond to the arrival times of the guided wave modes. Hou et al. [32,33] applied the DWFP technique using tomographic images to estimate the arrival times of multiple Lamb wave modes. The amount of white area in the DWFP images was used as a feature to distinguish false modes caused by noise and other interference from the true modes of interest. Hinders and Hou [31] applied the DWFP technique in the ultrasonic detection of suspected scattering objects in periodontal probing. The DWFP patterns were classified and mapped onto an inclination index curve. The performance of the algorithm was evaluated by comparing the ultrasonic probing results with those of full-mouth manual probing at the same sites. Until now, DWFP has only been used for the extraction of linear features with unequal frequency intervals/ or equal scale intervals. In the present study, an improved DWFP was applied to extract nonlinear ultrasonic features from measured signals with equal frequency intervals/ or unequal scale intervals. In Section II, a DWFP algorithm with equal frequency intervals is developed for the characterization of nonlinear effects in ultrasonic signals. In Section III, we investigate the effect of wavelet functions, the scale of the fingerprints to determine appropriate parameters in the wavelet transform, and the white subslice ratio during the slice-projection operation. The practical demonstration of the application of the algorithms to experimental ultrasonic signals is presented in Section IV.
2. DWFP technique for extraction of nonlinear ultrasonic features To extract the nonlinear features in the measured ultrasonic signals, a DWFP technique is proposed, as outlined in the flowchart in Fig. 1. The wavelet transform of the measured signals is first conducted with an equal frequency interval. Then, the wavelet scalogram is smoothed by a median filter and normalized by the wavelet scalogram at the fundamental frequency; a slice-projection operation is applied to the normalized wavelet scalogram in the frequency range of the nonlinear response. Finally, nonlinear features are extracted from the two-dimensional fingerprint image. 2.1. Wavelet transform with an equal frequency interval The continuous wavelet transform of a signal, x (t ), can be written as
WTx (a, b) =
1 a
+∞
∫−∞
⎛ t − b⎞ ⎟ dt x (t ) ψ * ⎜ ⎝ a ⎠
(1)
366
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 1. Flowchart of the DWFP technique for extraction of nonlinear ultrasonic features.
where WTx (a, b) are the wavelet coefficients, a is the scaling parameter, b is the time shift parameter, and * denotes the conjugate [25,36,37]. The factor 1/ a is used to ensure energy conservation for the transform. ψ ( t − b)/a is generated by
(
)
dilation and translation from the mother wavele ψ ( t ). Here, the relationship between the scale factor a and frequency can be described as
f=
fc a × TS
(2)
where fc is the center frequency of the mother wavelet in Hz, TS is the sampling period and f is the pseudo-frequency corresponding to the scale a , in Hz. The wavelet transform coefficients signify the similarity between the wavelet and the signal; that is, the larger the coefficient, the greater the similarity between the wavelet and signal. The high-frequency components in the signal can be seen at small scales, and the low-frequency components in the signal show up as largescale features. For traditional wavelet transforms, the scale is arithmetical. According to the correlation between the scale, a , and frequency, f , (shown in Eq. 2), the frequency resolution of the wavelet coefficients is varied in the frequency domain and is low at higher frequencies, which can lead to deteriorated frequency localization. To improve the location accuracy of the nonlinear components at high frequencies, the wavelet transform of the signal is conducted with an arithmetical frequency interval in this paper. The frequency sequence with the same interval is given by
{ fscale } =
{1, 2, 3 … a max } × fm amax
(3)
where amax is the maximum scale of the wavelet transform and fm is the maximum frequency considered in the nonlinear measurements. For example, fm should be greater than the harmonic frequency for the harmonic method and greater than the mixing frequency for the wave-mixing method. The scale sequence can then be obtained as follows:
{ ascale } =
fc
{ fscale } × TS
(4)
If the continuous wavelet transform of a signal is conducted using the scale sequence above, the wavelet coefficients at different frequencies will have identical frequency resolution. Then, the wavelet scalogram (time frequency representation) is defined as energy density spectrum [38]:
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
SG x (a, b) = WTx (a, b) WT x* (a, b) = WTx (a, b) 2
367
(5)
This wavelet scalogram can simultaneously keep track of the local time and frequency information in the signal. And it is capable of revealing hidden aspects of signal, which may possibly be missed by other techniques [39]. Besides, it can simplify the subsequent slice operation. 2.2. Definition of the nonlinear fingerprint coefficient Because of the influences of the environmental conditions, the variable parameters of the experimental system, and the properties of the sample, the measured signal inevitably contains interference, which disturbs the extraction of weak nonlinear effects. In this study, the wavelet scalogram is smoothed using a median filter, where the value in wavelet scalogram is replaced by the median of its neighborhood, to weaken the introduced noises. Similar to the definitions of the nonlinear parameters used in the harmonic and wave-mixing methods [16,22], the effects of waves at the fundamental frequency are taken into account, and the normalized wavelet scalogram can be obtained as follows:
SGxn (a, b) =
SG x (a, b) SG f1 SG f 2
(6)
where SGxn (a, b) indicates the normalized wavelet scalogram, and SG f1, SG f2 are the maximum wavelet scalograms at frequencies f1 , f2. For harmonic measurements, SG f1 = SG f2, which is the maximum wavelet scalogram at the fundamental (i.e., excitation) frequency; for wave-mixing measurements, SG f1 ≠ SG f2, which are the maximum wavelet scalograms at the two different excitation frequencies. A slice-projection operation is then applied to the normalized wavelet scalogram to generate a two-dimensional blackand-white fingerprint, as shown in Fig. 2. The three-dimensional normalized wavelet scalogram within the bandwidth of the nonlinear response (harmonic component or mixing component) is cut continuously with equal-thickness slices, and projected onto the time-frequency plane, which results in a two-dimensional black-and-white (0 and 1) image resembling a human fingerprint. The slice, with thickness H, contains two subslices. One subslice is black and the other is white; the thickness of the white slice is about η % (called white subslice ratio) of the total thickness. In this paper, H is one third of the peak of three-dimensional normalized wavelet scalogram within the bandwidth of the nonlinear response, obtained from the lowest degree of damage. It should be noted that there is low-level noise in the normalized wavelet scalogram sometimes; therefore, the projection corresponding to the bottom slice of the normalized wavelet scalogram is omitted from the DWFP image. As illustrated in Fig. 2, the number of ridges in the fingerprint image is dependent not only on the thickness of the slice but also on the peak of the normalized wavelet scalogram. Therefore, if the thickness of the slice is fixed, the number of ridges is directly proportional to the peak of the normalized wavelet scalogram. Accordingly, the amount of white area in the DWFP image can be used as the nonlinear feature, and the nonlinear fingerprint coefficient S can be defined as M
S=
N
∑ ∑ Iij
(7)
i=1 j=1
Fig. 2. Slice-projection operation to generate the two-dimensional black-and-white fingerprint.
368
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
where Iij indicates the pixel value (0 or 1) in row i and column j of the DWFP image, and M and N are the dimensions of the DWFP image. Because of the relationship between the normalized wavelet scalogram and the nonlinear ultrasonic features, it is possible to use the nonlinear fingerprint coefficient to quantitatively evaluate the early stages of damage in structures.
3. Effects influencing DWFP for the extraction of nonlinear features To optimize the performance of the proposed algorithm, it is necessary to carefully select its parameters, such as the type of wavelet function, the range of the scale and the ratio of white subslice, such that the nonlinear ultrasonic effect can be effectively extracted from the DWFP. In this section, using the finite-element model (FEM), the optimal wavelet function, the optimal scale and the optimal white subslice ratio were determined. 3.1. Finite-element model and typical results A two-dimensional FEM for simulating the nonlinear interaction between Lamb waves and a crack is represented in Fig. 3. The simulations were implemented using Abaqus/Explicit software. The parameters of the plate used in the simulation are listed in Table 1. In Fig. 4, the phase–velocity dispersion curves of the Lamb waves for a 1.7-mm-thick steel plate are shown. The dashed and solid lines are the dispersion curves of the fundamental and second-harmonic Lamb waves, respectively. Considering the condition for phase–velocity matching for the generation of the cumulative second-harmonic Lamb mode [40], the fundamental Lamb wave (S1 mode) and the double-frequency Lamb wave (S2 mode) were selected as mode pairs at an excitation frequency of about 2.2 MHz. The desired S1 mode was obtained by applying the same concentrated transient loads in x-direction at the upper left and lower left surfaces of the model (as shown in Fig. 3) [41]. The excitation was a Hamming-windowed tone-burst consisting of 10 cycles at a frequency of 2.2 MHz. The displacement components were recorded on the surface of the plate at a distance of 102 mm from the left end. A microcrack was positioned 100 mm away from the left end of the plate; its shape was modeled as an ellipse with a hard-contact frictionless surface. The major and minor axes of the ellipse determine the length and width of the crack, respectively. To investigate the dependence of nonlinear effects on the length of the microcrack, a series of simulations was performed on models with microcracks of identical width but different lengths. The width, w, was fixed at 10 nm, and the length, l, was varied (0, 0.6, 0.7, 0.8, 0.9, and 1 mm). A typical waveform obtained from a damaged plate with a 0.8-mm-long microcrack is shown in Fig. 5(a). The signal was transformed using the cmor7-7 wavelet with a scale factor of 1024, where the maximum analyzing frequency, fm , was 6 MHz. The three-dimensional distribution of the wavelet scalogram is shown in Fig. 5(b), and the normalized local wavelet scalogram within a harmonic frequency bandwidth is shown in Fig. 5(c). A thick contour-slice operation is applied to the local wavelet scalogram within the harmonic frequency bandwidth. A corresponding DWFP image is shown in Fig. 5(d). It can be seen that the DWFP image represents the nonlinear harmonic response in a two-dimensional time–frequency domain, and contains information pertaining to nonlinear effects. 3.2. Performance of different wavelet functions The performance of nine types of wavelet function (haar, cgau4, sym6, mexh, cmor1-1, cmor3-3, cmor5-5, cmor7-7, and cmor9-9) for the extraction of nonlinear effects was investigated. The simulated signals were processed using the proposed
Fig. 3. Depiction of the model and parameters used in the finite-element model.
Table 1 Parameters of the steel plate used in the model. Thickness
Length
Density
Elastic Modulus
Poisson's Ratio
1.7 mm
400 mm
7850 kg/m3
208.42 GPa
0.2959
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
369
Fig. 4. Phase-velocity dispersion curves for a 1.7-mm-thick steel plate.
Fig. 5. Simulated signal and wavelet scalograms processed using the proposed method with a scale factor of 1024. (a) Waveform; (b) wavelet scalogram; (c) normalized local harmonic scalogram; (d) DWFP image.
DWFP technique with a scale factor of 1024; representative results are shown in Fig. 6. It can be seen that there is significant difference between the DWFP images when the identical simulation signal is transformed with the different wavelet functions. As shown in Fig. 6(a)–(f) and (i), for the cases of haar, cgau4, sym6, mexh, cmor1-1, cmor3-3, and cmor9-9, there are no recognizable fingerprints in the two-dimensional time–frequency images. Moreover, the images do not provide any evidence of nonlinear effects at the center frequency of the harmonic components (4.4 MHz). However, for the case of cmor5-5 and cmor7-7, features of fingerprints are visible in the DWFP images, as shown in Fig. 6(g, h), and the center frequency of the fingerprints matches the center frequency of the harmonic components. A possible reason for the better performance of these two wavelets is that they are modulated in the time domain by a Gaussian-shaped time window,
370
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 6. DWFP images acquired using different wavelet functions: (a) haar; (b) mexh; (c) sym6; (d) cgau4; (e) cmor1-1; (f) cmor3-3; (g) cmor5-5; (h) cmor77; (i) cmor9-9.
which presents the optimal resolution simultaneously in the time and frequency domains [42]. In addition, the wave shapes in both cases are similar to that of the analyzed signal [43]. It can also be observed in Fig. 6(g) and Fig. 6(h) that the DWFP image obtained with cmor7-7 has more ridges than that obtained with cmor5-5. This can be attributed to the fact that cmor7-7 is more sensitive to the nonlinear component, which leads to a higher normalized wavelet scalogram. Moreover, the width of the DWFP image acquired using cmor7-7 reflects the duration of the waveform more accurately. Therefore, cmor7-7 was selected as the optimal mother wavelet for nonlinear ultrasonic signals analysis. 3.3. Effect of scale factor To determine the influence of scale range on the extraction of nonlinear effects, an identical signal was transformed using the cmor7-7 wavelet function over different scales (64, 128, 256, 512, 1024, and 2048) in the frequency range [0,6 MHz]; the representative results are shown in Fig. 7. There is a large difference between the DWFP images depending on the scale used: the larger the scale is, the higher the frequency resolution of the DWFP image and the clearer the fingerprints. For small scales (64, 128 and 256), it is difficult to identify the fingerprints because the DWFP images are blurred. For large scales (512, 1024 and 2048), however, the fingerprints can be recognized clearly. The reason for this phenomenon is that the proposed DWFP technique is based on the wavelet transform of signals with the same frequency resolution in a fixed frequency range [fmin , fmax ]. Consequently, the frequency interval in the wavelet scalogram is
Δf =
fmax − fmin a
(8)
The frequency interval Δf in Eq. (8) is inversely proportional to the scale factor, a , used in the wavelet transform. Therefore, with increasing scale factor, the frequency interval of the wavelet scalogram decreases and the resolution of the DWFP image becomes higher. However, the scale also affects the computation time of the wavelet transform: with
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
371
Fig. 7. DWFP images with different scale factors: (a) 64; (b) 128; (c) 256; (d) 512; (e) 1024; (f) 2048.
increasing scale factor, the computation time of wavelet transform increased. Therefore, by comprehensive consideration of the frequency resolution and computation time, a scale factor of 1024 was selected as the optimal value for the transformation of the nonlinear ultrasonic signals.
3.4. Effect of white subslice ratio Similarly, to investigate the effect of white subslice ratio on the extraction of nonlinear effects, the same signal was transformed using the cmor7-7 wavelet function at scale of 1024, and sliced at different white subslice ratio (from 0.1 to 0.9 with a step 0.1). The typical results are shown in Fig. 8. It can be seen that the ratio of white sublice has great effect on the ridge of fingerprint. In particular, along with the increase of white subslice ratio comes an increase in the width of white ridge. When the white subslice ratio is too small (0.1 and 0.2), the DWFP images are fuzzy because of the narrow ridges; on the other hand, when the white subslice ratio is too large (0.6, 0.7, 0.8 and 0.9), the fingerprint is difficult to discern due to the wide ridges. Clearly distinct fingerprints can be obtained at an intermediate white subslice ratio. In this paper, a white subslice ratio 0.3 was selected as the optimal value for the transformation of the nonlinear ultrasonic signals.
3.5. Results for microcracks with different lengths The DWFP technique was applied for the analysis of nonlinear signals obtained from the FEM with microcracks of different lengths. The DWFP images corresponding to the harmonic components for different microcracks are shown in Fig. 9. These fingerprint images differ depending on the crack length, and there is some association between the fingerprint features and the length of cracks. In addition, the amount of white area and the number of ridges in the DWFP images increase with increasing microcrack length. The magnitude of the nonlinear fingerprint coefficient (calculated with Eq. (7)) is shown in Fig. 10 as a function of crack length. The nonlinear fingerprint coefficient increases monotonically with the length of the microcrack, demonstrating that the proposed DWFP technique can be used to quantitatively evaluate microcrack in structure.
4. Application of the DWFP technique To demonstrate the effectiveness of the proposed DWFP technique for nonlinear feature extraction, the proposed DWFP technique was applied to the analysis of two typical kinds of nonlinear ultrasonic signals obtained from experiments of harmonic Lamb waves and experiments of Lamb waves mixing.
372
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 8. DWFP images with different white subslice ratio: (a) 0.1; (b) 0.2; (c) 0.3; (d) 0.4; (e) 0.5; (f) 0.6; (g) 0.7; (h) 0.8; (i) 0.9.
Fig. 9. DWFP images corresponding to the harmonic components for micro cracks of different lengths: (a) 0 mm; (b) 0.6 mm; (c) 0.7 mm; (d) 0.8 mm; (e) 0.9 mm; (f) 1 mm.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
373
Fig. 10. Relationship between the nonlinear fingerprint coefficient and the micro crack length.
4.1. Experimental system A schematic of the nonlinear acoustic measurement setup is shown in Fig. 11, in which the RAM 5000 system is the core component. During each experiment, Lamb waves are generated and detected using contact wedge transducers, and the distance d between the two transducers is 15 cm. For nonlinear ultrasonic harmonic and wave-mixing experiments, the central frequencies of the two transducers (one used as the transmitter T, and another used as the receiver R) are different, and then the two transducers (T and R) are fixed to Plexiglas-wedge with different angle, which detailed in the following two subsections. Nonlinear Lamb-wave measurements were conducted on dog-bone-shaped steel specimens, as shown in Fig. 12. In each sample, a notch of 2-mm length and 1-mm width was machined in the middle of the lateral side. Then, one specimen was kept pristine to be used as a reference, and all other specimens were fatigued under different numbers of cycles to introduce micro cracks of different lengths. A typical optical micrograph, shown in Fig. 13, shows plates with cracks of various sizes. The micro cracks have irregular interfaces, and their widths vary over a considerable range. The widths and lengths of the micro cracks are provided in Table 2. Experiments were conducted on the specimens described above using harmonic and mixing-wave methods, and the measured signals were processed using the proposed DWFP method, as described in the following two subsections. 4.2. Harmonic measurements For the experimental generation of harmonic Lamb waves, according to the condition of phase-velocity matching depicted in Fig. 4, a narrow-band longitudinal piezoelectric transducer with a central frequency of 2.25 MHz was used as the transmitter and a broad-band longitudinal piezoelectric transducer with a central frequency of 5 MHz was used as the receiver. According to the Snell theorem, Lamb mode pairs can be excited and received via a Plexiglas-wedge with an oblique angle of 27.1°.
Fig. 11. Schematic diagram of the experimental system.
374
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 12. Schematic diagram of the test specimen and a photograph of the micro crack.
Fig. 13. Micrographs of micro cracks of various lengths in steel plates: (a) 10 mm; (b) 15 mm; (c) 20 mm. Table 2 Width and length of the three micro cracks analyzed. Specimen number
1
2
3
4
Length Width
– –
10 mm 6.4 mm
15 mm 35.5 mm
20 mm 96.1 mm
Considering the frequency response of the experiment system, the excitation frequency was determined by a frequencysweep experiment on an undamaged specimen to generate a significant harmonic nonlinear response [40]. In Fig. 14, the amplitude–frequency dependences of the fundamental and second-harmonic Lamb waves are shown over the range of 0.2– 4 MHz. It can be seen that both the fundamental and second-harmonic waves have maximum amplitudes at 2 MHz. To obtain a significant nonlinear response, the excitation frequency of the sinusoidal component was selected at this frequency. Accordingly, the excitation signal was a Hamming-windowed tone-burst consisting of 10 cycles at a frequency of 2 MHz.
Fig. 14. Measured amplitude of the fundamental and second-harmonics Lamb waves versus excitation frequency.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
375
Fig. 15. Typical waveforms for (a) an undamaged specimen and (b) a specimen with a 10-mm-long micro crack.
The harmonic experiments were conducted on the four specimens. Typical waveforms received from the different samples are shown in Fig. 15. From these waveforms, it is difficult to discern whether there is harmonic generation induced by nonlinearity. DWFP images corresponding to the harmonic components of the signals measured from samples with micro cracks of different lengths are shown in Fig. 16. It can be seen that the number of ridges and the amount of white area in the images increase with increasing micro crack length. The dependence of the calculated nonlinear fingerprint coefficient on the crack length is shown in Fig. 17. For each specimen, the harmonic measurement was repeated five times at the same position, and the error bars in the figure represent the standard deviation of these measurements. It is clear that the nonlinear fingerprint coefficient increases with increasing crack length and thus the degree of damage in the plate. 4.3. Wave-mixing measurements It is well known that, at lower frequencies, the first symmetric mode (S0) is the fastest mode and is almost non-dispersive, which makes it promising in nondestructive testing. In this subsection, experiments of Lamb-wave mixing were conducted on the four different specimens using the S0 mode at low frequencies. Two commercial piezoelectric transducers
Fig. 16. DWFP images corresponding to the harmonic components for micro cracks of different lengths: (a) 0 mm; (b) 10 mm; (c) 15 mm; (d) 20 mm.
376
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 17. Relationship between the nonlinear fingerprint coefficient and the micro crack length.
with central frequencies of 0.5 MHz and 1 MHz were used as the transmitter, T and receiver, R, respectively. The S0 mode can be excited and received via a Plexiglas-wedge with an oblique angle of 30°. To investigate the effect of the driving frequency on the performance of wave mixing, frequency-sweep experiments were firstly conducted on an undamaged specimen to obtain a significant mixed-frequency nonlinear response. Driving frequency f1 was fixed ( f1 ¼0.45 MHz) and driving frequency f2 was varied in the range of 0.5–0.66 MHz. The amplitude of the sidebands at the sum frequencies generated by wave mixing was measured with the RAM 5000 system. The amplitudefrequency dependence of the sidebands measured in the specimen without a microcrack is shown in Fig. 18. It is noted that there is a resonance point at the driving frequency of 0.6 MHz (as shown in the black dashed line). To obtain significant interaction between the two waves and the microcrack, wave-mixing experiments were conducted at two driving frequencies: 0.45 MHz and 0.6 MHz. A tone burst of 50 V generated by the RAM-5000 system was fed into the transmitting transducer. The tone burst was composed of two sinusoidal components at the two driving frequencies and had a duration of 45 ms. Representative waveforms received from the different samples are shown in Fig. 19. It is difficult to discern whether there is sum-frequency generation induced by nonlinearity. DWFP images corresponding to the sum-frequency components for signals measured from samples with different microcrack lengths are shown in Fig. 20. The number of ridges and the amount of white area in the images increase with microcrack length. The dependence of the calculated nonlinear fingerprint coefficients on the microcrack length is shown in Fig. 21. For each specimen, the wave-mixing measurement was repeated five times at the same position, and the error bars in the figure represent the standard deviation of these measurements. Similar to the result obtained for the harmonic experiments, the proposed nonlinear fingerprint coefficient increases monotonically with microcrack length and thus the degree of damage in the plate. Despite the identical trends in the sensitivity of the two nonlinear fingerprint coefficients (harmonic component and wave-mixing component) with respect to the micro crack length, there is a slight difference is their absolute sensitivities. The possible reason can be explained as follows. It is well known that the nonlinear ultrasonic effects induced by damage are strongly dependent on detection parameters, such as frequency and amplitude. Since there is a great difference between the wave-mixing measurements and harmonic measurements in experimental conditions, such as frequency and amplitude,
Fig. 18. Amplitude-frequency dependence of sidebands at the sum frequency detected in the sample without micro cracks when f1 is 0.45 MHz.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
377
Fig. 19. Typical waveforms for (a) an undamaged specimen and (b) a specimen with a 10-mm-long micro crack.
Fig. 20. DWFP images corresponding to the mixing components for micro cracks of different length: (a) 0 mm; (b) 10 mm; (c) 15 mm; (d) 20 mm.
the measured nonlinear responses by two methods are inevitably different. As a result of the frequency dependence of nonlinear ultrasonic effects at different damage degree, there is a slight difference in absolute sensitivities for the wavemixing measurements and harmonic measurements. Nevertheless, the experimental results show that the proposed dynamic wavelet transform technique is effective for extracting nonlinear effects in measured signals, and the proposed nonlinear fingerprint coefficient correlates well with the degree of damage.
5. Conclusions In this study, an improved DWFP technique was proposed for characterization of nonlinear ultrasonic effects. To facilitate characterization of the nonlinear components, a DWFP algorithm was developed with equal intervals. The amount of white area in the fingerprint image was used to characterize the nonlinear ultrasonic effects. The performance of different wavelet functions, the effect of different scale factors and different white subslice ratio on the extraction of nonlinear features from
378
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
Fig. 21. Relationship between the nonlinear fingerprint coefficient and the micro crack length.
simulated signals was investigated. It has been shown that the nonlinear effects can be effectively extracted when DWFP is conducted using a cmor7-7 wavelet with a scale factor of 1024 at white subslice ratio 0.3. Nonlinear ultrasonic experiments including the measurement of harmonics and wave mixing were conducted in plates to detect micro cracks. The proposed DWFP method was demonstrated to be able to effectively extract the nonlinear features from the experimental signals. Moreover, the nonlinear features were sensitive to micro cracks, and correlated well with the degree of damage in the plates. Because the nonlinear features are extracted from the dynamic fingerprint, which is related to the time-frequency distribution of the measured signals, the nonlinear feature in this technique contains nonlinear information in both the time and frequency domains. This technique therefore has great potential for nonlinear measurements.
Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant Nos. 11572010, 11272017 and 11132002); the National Key Research and Development Program of China (2016YFF0203002).
References [1] K.Y. Jhang, Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: a review, Int. J. Precis. Eng. Manuf. 10 (1) (2009) 123–135. [2] D. Broda, W.J. Staszewski, A. Martowicz, T. Uhl, V.V. Silberschmidt, Modelling of nonlinear crack wave interactions for damage detection based on ultrasound – a review, J. Sound Vib. 333 (4) (2014) 1097–1118. [3] L.A. Ostrovsky, P.A. Johnson, Dynamic nonlinear elasticity in geomaterials, Riv. del nuovo cimento 24 (7) (2001) 1–46. [4] L.A. Ostrovsky, S.N. Gurbatov, J.N. Didenkulov, Nonlinear acoustics in Nizhni Novgorod (A review), Acoust. Phys. 51 (2) (2005) 114–127. [5] P.A. Johnson, B. Zinszner, P.N.J. Rasolofosaon, Resonance and elastic nonlinear phenomena in rock, J. Geophys. Res. 101 (B5) (1996) 11533–11564. [6] G.L. Jones, D.R. Kobett, Interaction of elastic waves in an isotropic solid, J. Acoust. Soc. Am. 35 (1) (1963) 5–10. [7] H. Ogi, M. Hirao, S. Aoki, Noncontact monitoring of surface-wave nonlinearity for predicting the remaining life of fatigued steels, J. Appl. Phys. 90 (1) (2001) 438–442. [8] K. Sheppard, A. Zagrai, D. Donskoy, A non-linear acoustic, vibro-modulation technique for the detection and monitoring of contact-type defects, including those associated with corrosion, Corros. Rev. 25 (1–2) (2007) 81–96. [9] M. Meo, U. Polimeno, G. Zumpano, Detecting damage in composite material using nonlinear elastic wave spectroscopy methods, Appl. Compos. Mater. 15 (3) (2008) 115–126. [10] M. Muller, D. Mitton, M. Talmant, P. Johnsonc, P. Laugiera, Nonlinear ultrasound can detect accumulated damage in human bone, J. Biomech. 41 (5) (2008) 1062–1068. [11] C.W. Connor, K. Hynynen, Bio-acoustic thermal lensing and nonlinear propagation in focused ultrasound surgery using large focal spots: a parametric study, Phys. Med. Biol. 47 (11) (2002) 1911–1928. [12] C. Payan, V. Garnier, J. Moysan, P.A. Johnson, Applying nonlinear resonant ultrasound spectroscopy to improving thermal damage assessment in concrete, J. Acoust. Soc. Am. 121 (4) (2007) 125–130. [13] I. Gurov, M. Volynsky, E. Vorobeva, Dynamic wavefront evaluation in phase shifting interferometry based on recurrence fringe processing, Proc. Int. Conf. Adv. Phase Meas. Methods Opt. Imaging 1236 (2010) 479–485. [14] C.A. Miller, M.K. Hinders, Classification of flaw severity using pattern recognition for guided wave-based structural health monitoring, Ultrasonics 54 (1) (2014) 247–258. [15] R. Carandente, P. Cawley, A method to estimate the size of corrosion patches with guided waves in pipes, AIP Conf. Proc.-Am. Inst. Phys. 1430 (2012) 1929–1936. [16] K.Y. Jhang, K.C. Kim, Evaluation of material degradation using nonlinear acoustic effect, Ultrasonics 37 (1) (1999) 39–44. [17] M.X. Deng, Analysis of second-harmonic generation of Lamb waves propagating in layered planar structures with imperfect interfaces, Appl. Phys. Lett.
H. Lv et al. / Journal of Sound and Vibration 389 (2017) 364–379
379
88 (22) (2006) 221902. [18] C. Pruell, J.Y. Kim, J. Qu, L. Jacobs, Evaluation of plasticity driven material damage using Lamb waves, Appl. Phys. Lett. 91 (23) (2007) 231911. [19] C.L. Nikias, M.R. Raghuveer, Bispectrum estimation: a digital signal processing framework, Proc. IEEE 75 (7) (1987) 869–891. [20] C.R.P. Courtney, B.W. Drinkwater, S.A. Neild, P.D. Wilcox, Factors affecting the ultrasonic intermodulation crack detection technique using bispectral analysis, NDT E Int. 41 (3) (2008) 223–234. [21] J.P. Jiao, J.J. Sun, G.H. Li, B. W, C.F. He, Evaluation of the intergranular corrosion in austenitic stainless steel using collinear wave mixing method, NDT E Int. 69 (2015) 1–8. [22] A.J. Croxford, P.D. Wilcox, B.W. Drinkwater, P.B. Nagy, The use of non-collinear mixing for nonlinear ultrasonic detection of plasticity and fatigue, J. Acoust. Soc. Am. 126 (5) (2009) 117–122. [23] Jl Chen, Z.P. Li, J. Pan, G.G. Chen, Y.Y. Zi, J. Yuan, B.Q. Chen, Z.J. He, Wavelet transform based on inner product in fault diagnosis of rotating machinery: a review, Mech. Syst. Signal Process. 70 (2016) 1–35. [24] F. Moreau, D. Gibert, M. Holschneider, G. Saracco, Wavelet analysis of potential fields, Inverse Probl. 13 (1) (1997) 165–178. [25] S. Mallat, A Wavelet Tour of Signal Processing, Academic press, Paris, 1999. [26] J.P. Jiao, W.H. Liu, J. Zhang, Q. Zhang, C.F. He, B. Wu, Time–frequency analysis for ultrasonic measurement of liquid-layer thickness, Mech. Syst. Signal Process. 35 (1–2) (2013) 69–83. [27] C.A. Miller, M.K. Hinders, Classification of flaw severity using pattern recognition for guided wave-based structural health monitoring, Ultrasonics 54 (1) (2014) 247–258. [28] M.K. Hinders, J.P. Bingham, Lamb wave pipe coating disbond detection using the dynamic wavelet fingerprinting technique, Rev. Quant. Nondestruct. Eval. 29 (2010) 615–623. [29] J. Bingham, M. Hinders, Lamb wave characterization of corrosion-thinning in aircraft stringers: experiment and three-dimensional simulation, J. Acoust. Soc. Am. 126 (1) (2009) 103–113. [30] J. Bingham, M. Hinders, A. Friedman, Lamb wave detection of limpet mines on ship hulls, Ultrasonics 49 (8) (2009) 706–722. [31] J.D. Hou, S.T. Rose, M.K. Hinders, Ultrasonic periodontal probing based on the dynamic wavelet fingerprint, EURASIP J. Adv. Signal Process. 2005 (7) (2005) 1137–1146. [32] J.D. Hou, K.R. Leonard, M.K. Hinders, Automatic multi-mode Lamb wave arrival time extraction for improved tomographic reconstruction, Inverse Probl. 20 (6) (2004) 1873. [33] J.D. Hou, M.K. Hinders, Dynamic wavelet fingerprint identification of ultrasound signals, Mater. Eval. 60 (9) (2002) 1089–1093. [34] J. Bingham, M. Hinders, A. Friedman, Lamb wave detection of limpet mines on ship hulls, Ultrasonics 49 (8) (2009) 706–722. [35] J.D. Hou, S.T. Rose, M.K. Hinders, Ultrasonic periodontal probing based on the dynamic wavelet fingerprint, EURASIP J. Adv. Signal Process. (7) (2005) 1137–1146. [36] S. Mallat, W.L. Hwang, Singularity detection and processing with wavelets, IEEE Trans. Inf. Theory 38 (2) (1992) 617–643. [37] C.A. Zala, I. Barrodale, K.I. McRae, High Resolution Deconvolution of Ultrasonic Traces, Signal Processing and Pattern Recognition in Nondestructive Evaluation of Materials, 44, Springer Berlin, Heidelberg, 1988, 101–108. [38] Z. Peng, F. Chu, Y. He, Vibration signal analysis and feature extraction based on reassigned wavelet scalogram, J. Sound Vib. 253 (5) (2002) 1087–1100. [39] S.K. Yadav, S. Banerjee, T. Kundu, On suitability of feature extraction techniques for local damage detection, Proc. SPIE - Int. Soc. Opt. Eng. 7983 (2011). [40] M.X. Deng, J. Yang, Characterization of elastic anisotropy of a solid plate using nonlinear Lamb wave approach, J. Sound Vib. 308 (1) (2007) 201–211. [41] X. Wan, Q. Zhang, G.H. Xu, P.W. Tse, Numerical simulation of nonlinear lamb waves used in a thin plate for detecting buried micro-cracks, Sensors 14 (5) (2014) 8528–8546. [42] K.C. Gryllias, I.A. Antoniadis, Estimation of the instantaneous rotation speed using complex shifted Morlet wavelets, Mech. Syst. Signal Process. 38 (1) (2013) 78–95. [43] Y.Y. He, Y. Liu, Experimental research into time–frequency characteristics of cavitation noise using wavelet scalogram, Appl. Acoust. 72 (10) (2011) 721–731.