Detection of a radial crack in annular structures using guided circumferential waves and continuous wavelet transform

Mechanical Systems and Signal Processing 30 (2012) 157–167

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Detection of a radial crack in annular structures using guided circumferential waves and continuous wavelet transform Yu Liu, Zheng Li n, Kezhuang Gong State Key Laboratory of Turbulence and Complex System & College of Engineering, Peking University, Beijing 100871, China

a r t i c l e i n f o

abstract

Article history: Received 28 December 2009 Received in revised form 16 December 2011 Accepted 17 January 2012 Available online 17 February 2012

A quantitative identification method for radial cracks based on the guided circumferential wave (GCW) and continuous wavelet transform (CWT) is proposed in this research. The Gabor wavelet is used to extract a proper frequency component of a beneficial wave mode for evaluation of crack from the highly dispersive and multimode waves. The GCW components with different frequencies are considered and their effectiveness in crack detection is compared. Both numerical simulations and experiments are conducted on a Plexiglas annulus with a radial crack. Results show that the crack location is determined with high precision using the CWT to extract an appropriate GCW component, and the 1st mode corresponding to the Rayleigh surface wave is more reliable in crack detection than other modes under investigation. Furthermore, a GCW component with too low a frequency involves overlapping of waves and insensitivity to damages while that with too high a frequency has small amplitude and is complicated with many wave modes. Thus a wave component which simultaneously has adequate amplitude and good sensitivity is proven to be effective for examination of crack. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Radial crack Guided circumferential wave Gabor wavelet PZT sensors

1. Introduction Detection of fatigue cracks in annular shaped or pipe-like structures is important to monitor the structural safety and has attracted lots of attention in recent years [1–3]. These cracks usually form on the inner surface and grow in the radial direction, thus called radial cracks. They are generally difficult to be inspected in practical application due to their hidden character (initiates from the inner surface) and the complex geometries of their host structures. Previous research [4–6] has shown that guided circumferential wave (GCW) is a promising technique because it can propagate a relatively far distance and interrogate the entire structure, including the inaccessible components. However, application of the GCW for evaluation of radial cracks mainly involves two difficulties. One difficulty is the inherent complexity of the waveform and structure of the GCW. The underlying principle of wave propagation has to be investigated before it is used to assess damages. After the initial work done by Viktorov et al. [7], Liu et al. [8,9] studied the steady-state and transient behaviors of the GCW in an elastic annulus. They developed the dispersion curves of the GCW and calculated the wave structures of the lower modes. Furthermore, Valle et al. [4,10] theoretically analyzed the dispersion curves of the GCW in a two-layered cylinder and then experimentally explored them using a broadband laser ultrasonic technique. Most recently, Gridin et al. [11] derived the dispersion relations of both SH

n

Corresponding author. Tel.: þ86 10 62754624. E-mail address: [email protected] (Z. Li).

0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2012.01.020

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and Lamb-type GCW modes in an annulus waveguide based on asymptotic techniques. These available studies reveal the nature of the GCW and provide guidance for implementation of radial crack detection. Another difficulty arises as to the interpretation of the dispersive and multi-mode GCW signals for extraction of damage features. The measured GCW signals are extremely complicated due to the interaction of the crack with multiple modes, mode conversion, reflection from boundaries and environmental noises. For a structural wave usually composed of several modes and a wide range of frequency components, a sensitive frequency component of a beneficial wave mode for crack identification needs to be selected. Generally, too low a frequency component will cause overlapping of waves and low detection sensitivity, while too high a frequency component becomes more complicated with many wave modes, making the damage more difficult to be detected [12–14]. For this reason, a robust signal processing technique is required, and the continuous wavelet transform (CWT) has been proven powerful because of its excellent time-frequency characteristics in processing of dispersive and multi-mode guided waves [15–17]. Available work on evaluation of radial cracks in annular-shaped and pipe-like structures is largely limited due to aforementioned difficulties, especially for experimental investigation. A relevant study was reported in Ref. [3], where the radial crack was interrogated by a conventional ultrasonic method; however, the location and size of the crack were not obtained. Thus further investigation for quantitative diagnosis of radial cracks is needed. The objective of this research is to propose a quantitative identification method for radial cracks based on the GCW and CWT. By considering GCW waves with different frequency contents and using the CWT to extract an effective frequency component of a beneficial wave mode, quantitative information of the radial crack is achieved and some guidance on inspection-frequency selection is suggested. Both numerical simulation and experimental investigation are conducted in this research. 2. Continuous wavelet transform (CWT) This section briefly describes the application of CWT for the analysis of dispersive waves (detailed information can be found in author’s previous work [17]). The Gabor wavelet is selected because it has the best time-frequency resolution in the family of continuous wavelets according to the Heisenberg uncertainty principle. Thus it is advantageous to accurately determine the arrival time of a specific frequency component, which is critical to quantitatively evaluate the crack. ^ ðoÞ are expressed as The Gabor wavelet cG(t) and its Fourier transform c G " # rffiffiffiffiffiffiffi 2 1 o0 ðo0 =gÞ 2 ffiffiffiffi t expðio0 tÞ cG ðtÞ ¼ p ð1Þ exp  4 2 p g pffiffiffiffiffiffi rffiffiffiffiffiffiffi 2p g

^ ðoÞ ¼ pffiffiffiffi c G 4

p

o0

" exp 

ðg=o0 Þ2 ðoo0 Þ2 2

# ð2Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where t is the time, o is the frequency, g ¼ p 2=ln 2 and o0 ¼2p are constants. Consider a dispersive wave that propagates along the x direction, Z þ1 1 AðoÞexp½iðotkxÞdo ð3Þ uðx,tÞ ¼ 2p 1 where k is the wave number and A is the amplitude. The CWT of u(x,t) using the Gabor wavelet as a mother wavelet is defined as [18] pffiffiffi Z þ 1 a ^ ðaoÞdo AðoÞexpðiobikxÞc ð4Þ Wuða,bÞ ¼ G 2p 1 where a and b are the scale and time shift parameters in the CWT, respectively. For the Gabor wavelet, the parameter a is inversely proportional to the frequency f, i.e., a¼1/f when o0 ¼2p is taken [17]. Therefore, the CWT enables to represent a wave signal on a time-frequency plane, from which detailed information of the wave (amplitude, frequency content, ^ ðaoÞ in Eq. (4) is localized around o ¼ o /a. Thus on the localized interval, arrival time, etc.) can be identified. Note that c 0 G the amplitude A(o) and group velocity cg ¼do/dk can be approximated by A(o) ¼A(o0/a) and cg ¼ Do/Dk ¼(o  o0/a)/ (k k0), respectively, where k0 is the wave number at frequency o0/a. Then the wave number k is expressed as k¼ k0 þ (o  o0/a)/cg. Substituting above approximations as well as Eq. (2) into Eq. (4), the magnitude of the wavelet coefficient is finally written as "   # rffiffiffiffiffiffiffi   1 o0  o0  ðo0 =gÞ2 x 2 ffiffiffiffipffiffiffi 9Wuða,bÞ9 ¼ p b ð5Þ A exp  4 cg a 2a2 p a g Eq. (5) shows that for a given parameter a (or 1/f), 9Wu(a,b)9 gets its maximum when b¼x/cg, i.e., the time parameter b equals the group velocity propagation time x/cg; furthermore, this maximum is proportional to the amplitude 9A(o0/a)9. Therefore, the peak of the magnitude of the wavelet coefficient indicates a frequency component at o0/a, propagating with a group velocity cg. On this basis, the amplitude and arrival time of any frequency component can be determined, and an appropriate inspection frequency can be selected for further diagnosis of radial cracks.

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3. Numerical simulation of radial crack detection This section numerically investigates the radial crack detection using FEM analysis. Transient GCW in a Plexiglas annulus subjected to different impulse loads are calculated with the program ABAQUS. The two-dimensional model is shown in Fig. 1 and its material and geometry properties are listed in Table 1. Excited by an impulse load F(t) on the outer surface of the model with an angle of 301 to the radial direction, transient GCW will propagate along the circumference direction (clockwise and counterclockwise). For clarity, the clockwise GCW is used in this work. A radial crack of a depth d ¼2 mm (as shown in Fig. 1) with an angle of 401 with respect to the line OA is considered. When the GCW encounters the crack, it will be reflected back to point A and transmitted to point B. Thus two points A and B on the outer surface of the annulus are selected to measure the incident and crack-reflected GCWs. For calculating the GCW, the mesh size of the annulus model is set as 0.5 mm  0.5 mm and the time step is 10  7 s. The group velocity curves of the GCW in a Plexiglas annulus with stress-free boundary conditions and the same properties with those listed in Table 1 are shown in Fig. 2 (calculated with the theory in Ref. [8]). It is seen that in the low frequency range, the wave field is relatively simple with less modes (for example, there are only two modes below 158 kHz), which is beneficial to crack detection; however, a low frequency GCW component is insensitive to small damages because of its relatively large wavelength and the overlapping of wave modes. In contrast, a high frequency GCW component ensures a good sensitivity to small damages while it contains much more wave modes (for example, four modes exist above 310 kHz), which increases the difficulty of signal interpretation. Therefore, a proper inspection frequency should be selected in order to effectively evaluate the radial crack. To examine the effectiveness of different frequency components of the GCW, GCW fields excited by two different exciting loads F(t) are considered, as shown in Fig. 3. The two loads have different impulse duration Dt (defined as the time interval between the two points with 50% of the maximum amplitude) and frequency contents (obtained with the Fourier transform). The load II has a longer impulse duration (Dt ¼36 ms) than the load I (Dt ¼2 ms) and its energy concentrates in a lower frequency range below 40 kHz. In contrast, the load I is sharper in the time domain while it has a much wider range of frequency, containing higher frequency components. 3.1. Radial crack detection in load I case Fig. 4 shows the GCW signals (calculated tangential strains at points A and B) excited by the Load I for the intact (d¼ 0 mm) and damaged (d¼ 2 mm) models. An obvious difference can be found in the marked rectangular area in Fig. 4(a) between the two models, which demonstrates the disturbance of the crack on the GCW. Differences also exist at point B (in Fig. 4(b)) between the two models, i.e., the signal of the damaged model is slightly delayed in time and has smaller

Fig. 1. Plexiglas annulus.

Table 1 Material and geometry properties. Density r (kg/m3) 1170

Yong’s modulus E (Pa) 9

4.965  10

Poisson ratio n

Inner radius r1 (mm)

Outer radius r2 (mm)

Z ¼r1/r2

0.3275

46.21

49.76

0.9286

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2500 2 nd mode

Group velocity (m/s)

2000

1500

3 rd mode st

1 mode

1000

500

0

0

100

200

300 400 Frequency (kHz)

500

600

Fig. 2. Group velocity curves of the GCW in a Plexiglas annulus.

4000 Load I

Load I

Load II

Amplitude (N*s)

Amplitude (N)

100

Δt

50

0 0

50

100

Load II

3000 2000 1000 0

150

0

200 400 Frequency (kHz)

Time (μs)

600

Fig. 3. Two cases of load F(t): time and frequency contents.

x 10

3

−3

−3

crack disturbance

d=0mm

d=0mm 2

d=2mm

1

Strain

Strain

2

0 −1 −2

x 10

3

d=2mm

1 0 −1

0

50

100 Time (µs)

150

−2

0

50

100

150

Time (µs)

Fig. 4. GCW signals at (a) point A and (b) point B for the models without (d¼ 0 mm) and with (d ¼2 mm) a radial crack.

amplitudes than that of the intact model. These differences imply the presence of the crack; however, quantitative information (location and size of the crack, etc.) cannot be directly obtained from the complicated time-domain signals. As mentioned in Fig. 3, the Load I has a wide frequency domain, thus GCW signals excited by it contain rich frequency information. To select an appropriate frequency component for crack detection, the multi-frequency CWT analysis is first conducted. Fig. 5 shows the CWT results on a frequency range [0 kHz, 250 kHz] of GCW signals at point A for the two models, where the color bar specifies the magnitude of the wavelet coefficient. The first peaks around 50 ms in both Fig. 5(a) and (b) indicate the incident waves arriving at point A. By comparing the results of the two models, an additional peak is found around 100 ms in the damaged model (denoted by the arrow in Fig. 5(b)), which demonstrates the crack-reflected wave. Interestingly, this peak shows up only in a certain frequency area but not the whole frequency range. One can see that when the frequency is lower than about 70 kHz, the crack-reflected wave cannot be found due to the

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Fig. 5. Multi-frequency CWT analysis of GCW signals at point A for (a) d ¼ 0 mm and (b) d ¼ 2 mm.

x 10 I1

d=2mm

6 R1 4

x 10

8 d=0mm

I2 R2

2

0

Magnitude of Wavelet Coef.(s)

Magnitude of Wavelet Coef.(s)

8

d=0mm d=2mm

I1

6

4

I2

2

0 0

50

100 Time (µs)

150

0

50

100 Time (µs)

150

Fig. 6. Wave component of 100 kHz at (a) point A and (b) point B for d ¼ 0 mm and d ¼ 2 mm.

overlapping of waves. As the frequency increases, the crack-reflected peak is observed and it becomes sharper in the time domain (which is more beneficial to accurately examine the crack). However, when the frequency continues increasing above about 170 kHz, the crack-reflected peak becomes very weak with small amplitude. This low-amplitude wave component is easy to be distorted by noises (other unwanted waves and environmental noises) in real application. Thus neither a too low nor a too high frequency component is favorable for crack detection. For this reason, an appropriate GCW component at 100 kHz which is sharper in the time domain and simultaneously has adequate amplitude is chosen. The chosen GCW components at 100 kHz at points A and B for both models are extracted and plotted in Fig. 6. With the knowledge of dispersion curves of the GCW shown in Fig. 1, two wave modes exist at this frequency, namely the 1st mode and 2nd mode. The 2nd mode has a bigger group velocity (cg2 ¼1956 m/s) than the 1st mode (cg1 ¼1179 m/s). For results at point A in Fig. 6(a), the first two peaks for the intact and damaged models are almost the same, indicating the incident waves of the 2nd and 1st modes, denoted by I2 and I1, respectively. The amplitude of I2 is much smaller than that of I1. Similarly, the first two peaks for the intact model shown in Fig. 6(b) are the incident waves of the two modes arriving at point B. Knowing these incident wave peaks, the group velocities of the 2nd and 1st modes can be determined as cg2 ¼1874.4 m/s and cg1 ¼1289.8 m/s, respectively, through a ToF (time of flight) method cg ¼lAB/Dt, where tAB is the ToF of the incident waves at points A and B and lAB is the distance between the two points. Note that the measured group velocities are very close to the theoretical values. With the known group velocity, the crack can be located if the crack-reflected wave is extracted. A big difference is observed between 60 ms and 110 ms in Fig. 6(a), where two additional peaks appear at about 70 ms and 100 ms, respectively, termed as R2 and R1, demonstrating the crack-reflected waves of the 2nd and 1st modes. The R1 peak has big amplitude while the R2 peak is very small and unobvious, thus difficult to be utilized for crack localization. This suggests a solution for selection of the wave mode, i.e., the 1st mode is more reliable to evaluate the crack. In fact, the 1st mode is asymptotic to the Rayleigh surface wave as the frequency increases. It is nondispersive in the higher frequency range (as shown in Fig. 2) and is especially effective for crack inspection of the thin-walled annulus structures under investigation. Then, the crack location y, defined as the angle with respect to the line OA in Fig. 1, can be identified using the 1st mode through

y¼

cg1 DT 3603  2pr 2 2

ð6Þ

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where DT is the ToF between the incident wave and crack-reflected wave at point A. The crack location is determined as 40.241 with an error of 0.62% to the actual location 401. Note that a difference can also be found in waves at point B, as shown in Fig. 6(b); an additional peak (the second peak for d ¼2 mm) exists between the 2nd and 1st modes for the damaged model compared with the intact model. This implies the mode conversion phenomenon due to interaction of the GCW with the crack. An examination of the arrival time of this peak shows that it is the converted 1st mode from the 2nd mode after interacting with the crack. Similar phenomenon can be seen in the experiments which will be discussed later. It should be noted that the signal at point B is mainly used to calculate the group velocity in this work. 3.2. Radial crack detection in load II case The same CWT procedure is used to interpret the GCW signals excited by the Load II, i.e., a multi-frequency CWT analysis for selection of an effective GCW component, followed by extraction of the chosen frequency component for crack localization. Thus the main attention here is paid to examine the different performances of these two load cases. Unlike the Load I, the Load II has a very narrow frequency range below 40 kHz. Thus GCW signals excited by the Load II have lower frequency components than those excited by the Load I, as shown in Fig. 7. A direct observation is that wave signals in Fig. 7 exhibit fewer fluctuations than those in Fig. 4 and only a slight variation of the signals between intact and damaged models is found. The low frequency character makes the wave insensitive to cracks. Similarly, an appropriate GCW component at 34 kHz of the two models is extracted and shown in Fig. 8(a) and (b) through the multi-frequency CWT analysis. As seen in Fig. 2, only the 1st and 2nd modes exist at 34 kHz, with a group velocity of cg1 ¼1023 m/s and cg2 ¼ 2028 m/s, respectively. Thus the first peak for each model in Fig. 8(a) corresponds to the incident wave of the 1st mode arriving at point A. The incident waves of the 2nd mode, however, are not seen because of the overlapping of these two modes. The wave component of 34 kHz has a larger wavelength than that of 100 kHz, thus its wave peak in the time domain is wider. In this case, the 2nd mode which has much smaller amplitude (as shown in Fig. 6) is hidden in the wide peak of the 1st mode. After the waves propagate a longer distance, the incident waves of the two

x 10

1

x 10

1 d=0mm

0 −0.5 −1

d=2mm

0.5 Strain

0.5 Strain

d=0mm

d=2mm

0 −0.5

0

100

200

−1

300

0

100

Time (µs)

200

300

Time (µs)

Fig. 7. GCW signals at (a) point A and (b) point B for the models without (d ¼0 mm) and with (d ¼2 mm) a radial crack.

−4

−4

d=0mm d=2mm

3 2 1 0

x 10 d=0mm

4

Magnitude of Wavelet Coef.(s)

I1

4

−4

x 10 Magnitude of Wavelet Coef.(s)

Magnitude of Wavelet Coef.(s)

x 10

d=2mm 3

I1 I2

2 1 0

0

100 200 Time (µs)

300

difference signal

1

R1

0.5

0 0

100 200 Time (µs)

300

0

100 200 Time (µs)

Fig. 8. Wave component of 34 kHz at (a) point A (b) point B and of (c) difference signal at point A between the intact and damaged models.

300

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163

modes are separated at point B, as shown in Fig. 8(b). This again validates that the 1st mode is more reliable in radial crack detection under investigation, and its group velocity is determined as cg1 ¼978.3 m/s. Unlike the results of the Load I case, there are no clear crack-reflected waves seen in Fig. 8(a) by comparing results of the two models. Thus it is more difficult to locate the crack in this case. A baseline subtraction method [14] is introduced, which assumes that the influence of the crack can be purely extracted by subtracting the response of the intact model from that of the damage model. The 34 kHz component of the difference signal at point A of the two models is shown in Fig. 8(c) and the peak at about 203 ms is identified as the crack-reflected wave of the 1st mode, denoted as R1. Then the crack location is obtained as 41.341 according to Eq. (6), with an error of 3.35% to the actual location 401. To summarize, the crack location is successfully determined in both cases by extracting an appropriate GCW component using the Gabor wavelet. However, GCW signal with different frequency contents performs differently in examination of the crack. In Load I case, the interaction of the GCW with the crack is well revealed and the crack is accurately located with an effective GCW component at 100 kHz. In contrast, for Load II case where the excited waves contain lower frequency components, the crack is more difficult to be located and the error of the predicted location is bigger. Generally, the low frequency components usually involve overlapping of waves and insensitivity to small damages, while the high frequency components suffer from its low energy and high attenuation. Moreover, one can see from the dispersion curves of the GCW that waves in the lower frequency range (below 40 kHz) are more dispersive than those around 100 kHz. With a high dispersion, the group velocity of a specific mode changes sharply with a small variation of frequency, making it difficult to accurately identify the ToF needed for crack localization. Therefore, the dispersion characters should also be considered for selection of a proper GCW component. Naturally, a frequency component that has relatively large amplitude and less dispersion should be selected. 4. Experimental investigation of radial crack detection To verify the proposed method, the experimental investigation is carried out in this section. Similarly to the numerical simulation, the GCW with different frequency contents is considered and the same CWT procedure is used to analyze the measured GCW signals. 4.1. Case I: GCW with lower frequency components To excite a GCW with lower frequency contents, a drop-weight measurement was conducted on a Plexiglas annulus. The specimen has the same material and radii listed in Table 1, but it is three-dimensional and has a length of 0.9 cm. As shown in Fig. 1, the GCW was excited using a steel ball (20 mm in diameter) falling from a height of 1.2 m, with an angle of 301 to the radial direction. A pre-crack of 1 mm width and d mm depth on the inner surface of the specimen was fabricated, where d¼ 1 mm, 2 mm and 3 mm were considered. Two strain gauges, connected to an ultra-dynamic strain amplifier (SDY2107), were glued at points A and B to measure the GCW signals (tangential dynamic strains). The measured signals are sampled at 200 MHz by an oscilloscope (LeCroy@ LC574AL). The measured signals at point A for the intact and damaged specimens are shown in Fig. 9(a). Their frequency contents can be seen from the multi-frequency CWT analysis illustrated in Fig. 9(b), where only the result of the intact specimen is plotted for clarity. One can see that the energy of the excited GCW mainly concentrates on a low frequency range below 50 kHz, which is similar to the Load II case in the numerical simulation. The GCW components at 34 kHz at points A and B for the intact and damaged specimens are extracted in Fig. 10, by selecting an appropriate inspection frequency using the multi-frequency CWT analysis. The first peak of each specimen in Fig. 10(a) and (b) corresponds to the incident wave of the 1st mode at point A and point B, respectively. The incident wave of the 2nd peak is not found due to the overlapping of waves. Thus the 1st mode is experimentally proven to be reliable for crack diagnosis and its group velocity can be obtained from its arrival times.

Fig. 9. Representative results of (a) measured GCW signals at point A for d ¼ 0 mm, d¼ 1 mm and d ¼ 2 mm (b) multi-frequency CWT analysis of the signals at point A for d¼ 0 mm.

I

d=0mm d=1mm d=2mm

1

0.01

R 0.005

0

0

1

100

200

Magnitude of Wavelet Coef.(s)

Y. Liu et al. / Mechanical Systems and Signal Processing 30 (2012) 157–167

Magnitude of Wavelet Coef.(s)

164

I 0.01

d=0mm d=1mm d=2mm

1

0.005

300

0

0

100

Time (µs)

200

300

Time (µs)

Fig. 10. Wave component of 34 kHz at (a) point A and (b) point B for d¼ 0 mm, d ¼1 mm and d ¼ 2 mm.

Table 2 Estimated crack location and reflection ratio. Specimen

Relative crack depth (%)

Crack location y

Relative error (%)

Reflection ratio (%)

d ¼1 mm d ¼2 mm

28.17 56.34

43.051 40.451

7.630 1.120

16.40 29.58

Fig. 11. Annulus specimen and PZT configuration.

The crack-reflected wave of the 1st mode can be determined through the comparison of intact and damaged specimens in Fig. 10(a). An additional peak appears in the damaged samples around 140 ms and its amplitude becomes bigger as the crack depth d increases. Thus this peak demonstrates the crack-reflected peak R1. Then the crack location is obtained according to Eq. (6) and listed in Table 2, where the relative crack depth is the crack depth d normalized by the thickness of the annulus h¼r2 r1, the reflection ratio is the ratio of crack-reflected amplitude and incident amplitude. The relative error of the predicted crack location is less than 8% and it becomes smaller as the crack depth increases. Furthermore, the value of the reflection ratio becomes larger with the increasing crack depth. This implies a potential approach to trace the damage extent from the change of reflection ratio. However, more investigation is needed to quantitatively evaluate of crack size. 4.2. Case II: GCW with higher frequency components To excite a GCW with higher frequency components that is similar to the Load I case in numerical simulation, a highvelocity impact that can produce a similar load to Load I is needed. However, it is not easy to be implemented and may cause local damage in the specimen. Here the piezoelectric (PZT) sensors are used to excite and receive GCW signals because of their potential capability in online structural health monitoring [19,20]. As shown in Fig. 11, the specimen has an inner radius of 46.65 mm, an outer radius of 49.29 mm, a length of 10.7 mm and the same material properties with those listed in Table 1. Two PZT wafers, 8 mm in diameter and 1 mm in thickness, were attached on the outer surface of the specimen. A 5.5-cycle sinusoid toneburst signal with a central frequency of 259 kHz modulated by a Hanning window, was generated by an arbitrary waveform generator (DG1022), and then fed into PZTA to excite the GCW. The GCW signals were

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165

Fig. 12. Representative results of (a) measured GCW signal for d ¼0 mm and (b) multi-frequency CWT analysis of the measured signal.

Magnitude of Wavelet coef.(s)

7 d=0mm

I

1

6 5

d=1mm d=2mm

I

2

4 I

3

3

R1

R31 2

R21

1 0

0

50

100

150

Time (µs) Fig. 13. Wave component of 290 kHz for d ¼0 mm, d ¼ 1 mm and d¼ 2 mm.

received by PZTB and sampled at 200 MHz by an oscilloscope (LeCroy@LC574AL). A radial crack with a increasing depth d ¼1 mm, 2 mm and 3 mm was pre-fabricated on the inner surface with an angle of 301 with respect to the line O-PZTB. Fig. 12(a) shows a representative signal of the measured GCW signal for the intact specimen. The frequency content of the signal can be seen from its multi-frequency CWT analysis shown in Fig. 12(b). Excited by a toneburst signal with a central frequency of 259 kHz, the energy of the GCW mainly spreads between 200 kHz and 300 kHz. Note that three modes exist on this frequency range according to the dispersion curves shown in Fig. 2, namely the 3rd, 1st and 2nd modes. The incident peaks of the three modes are overlapped in the time domain as shown in Fig. 12(a). The GCW component at 290 kHz for the intact and damaged specimens is extracted in Fig. 13. The incident waves of the 3rd, 1st and 2nd modes (denoted as I3, I1 and I2, respectively) are separated in the short propagation distance between PZTA and PZTB, thanks to the excellent time-frequency characteristic of the Gabor wavelet. Then, their group velocities are obtained from their arriving times for the intact specimen, as cg3 ¼1327 m/s, cg1 ¼923.8 m/s and cg2 ¼679.5 m/s, respectively. Identification of the crack-reflected peak is much more complicated in this case since three modes exist and their group velocities are relatively close. The crack-reflected waves can be found around 80 ms, 100 ms and 120 ms (labeled as R31, R1 and R21) in Fig. 13, whose amplitudes become bigger with the increasing crack depth. However, it should be mentioned that these peaks are not the pure crack-reflected waves but influenced by inherent peaks exiting in the intact specimen (the two small peaks after the three incident peaks I3, I1 and I2). The inherent peaks in the intact specimen may come from the boundary reflections (three-dimensional specimen) and experimental noises. The meaning of R31, R1 and R21 peaks can be identified by analyzing their arrival times. Taking d¼2 mm for example, according to the group velocity on the dispersion curves, the crack-reflected wave of the 1st mode should arrive at 102.4 ms. Therefore, the peak R1 at 100.7 ms corresponds to the crack-reflected wave of the 1st mode. Similarly, the peak R31 (at 80.19 ms) is identified as the converted 1st mode from the 3rd mode, and the peak R21 is the converted 1st mode from the 2nd mode. In accordance with above mode selection, the R1 peak is again used to locate the crack, and the crack location is obtained as 28.071 for d ¼1 mm and 28.581 for d ¼2 mm, with an error as 6.43% and 4.73%, respectively (listed in Table 3). Similar to the lower frequency case, the error of the crack location decreases while the reflection ratio increases as the crack depth becomes larger.

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Table 3 Estimated crack location and reflection ratio. Specimen

Relative crack depth (%)

Crack location y

Relative error (%)

Reflection ratio (%)

d ¼1 mm d ¼2 mm

37.88 75.76

28.071 28.581

 6.43  4.73

17.04 24.19

By examining the GCWs excited by two commonly used techniques, results demonstrate that the proposed method is feasible in crack localization and the CWT is powerful in extraction of an appropriate GCW component from the dispersive GCW signals. Meanwhile, the two different frequency components work differently in damage detection. In the lower frequency case, the 2nd mode is hidden in the 1st mode because of the wave overlapping; while in the higher frequency case, the three wave modes are separated and their interaction with the crack can be well revealed. 5. Conclusions This paper aims to propose a quantitative identification method for radial cracks in annular structures based on the GCW and CWT. The Gabor wavelet is used to analyze the highly dispersive GCW for extraction of an appropriate frequency component of a beneficial wave mode to evaluate the crack. GCW components with lower and higher frequencies are extracted and their effectiveness in crack detection is compared. Both numerical simulation and experimental investigation are carried out on a Plexiglas annulus. The conclusions can be drawn as follows: 1. The crack location is successfully determined with a good precision using the proposed method. The Gabor wavelet contributes to the success by effectively extracting an appropriate GCW component of a beneficial wave mode from the multi-mode and dispersive GCW signals. 2. By examining GCWs with different frequency contents, a frequency component that simultaneously has adequate amplitude and sensitivity to damages is favorable for evaluation of cracks. Too low a frequency component involves difficulties of wave overlapping and insensitivity to small damages, while too high a frequency component is complicated with many modes, making it difficult to interpret the wave signals. 3. The 1st GCW mode corresponding to the Rayleigh surface wave is proven beneficial to radial crack inspection for the thin-walled annular structure of interest. It is less dispersive and has higher energy than the other modes under investigation, thus advantageous to extract damage feature and crack localization. It should be noted that the capability of the 1st mode needs to be examined when a thick-walled structure with damages on its inner surface is considered, since the 1st mode has high attenuation along the thickness. 4. The amplitude of the crack-reflected wave becomes larger as the crack depth increases. This implies an approach to qualitatively evaluate the damage extent by tracing the crack-reflected amplitude. However, more investigation is needed to quantitatively determine of the crack size. To conclude, results in this paper numerically and experimentally show that the proposed method is feasible and reliable in quantitative identification of radial cracks.

Acknowledgments This research was supported by the National Natural Science Foundation of China under Grant no. 10672002. The authors would like to thank the Ph.D. candidate Guiyun Gao in our lab for her help with the measurements. Special thanks go to Doctor Jin-Yeon Kim in Georgia Institute of Technology for his suggestions and help with revising the manuscript. References [1] C. Valle, M. Niethammer, J. Qu, L.J. Jacobs, Crack characterization using guided circumferential waves, J. Acoust. Soc. Am. 110 (2001) 1282–1290. [2] P.B. Nagy, M.B. Blodgett, M. Golis, Weep hole inspection by circumferential creeping waves, NDT&E Int. 27 (1994) 131–142. [3] J. Qu, Y.H. Berthelot, L.J. Jacobs, Crack detection in thick annular components using ultrasonic guided waves, Proceedings of the Institution of Mechanical Engineers, Part C, vol. 214, 2000, pp. 1163–1171. [4] C. Valle, J. Qu, L.J. Jacobs, Guided circumferential waves in layered cylinders, Int. J. Eng. Sci. 37 (1999) 1369–1387. [5] L. Satyarnarayan, J. Chandrasekaran, B. Maxfield, K. Balasubramaniam, Circumferential higher order guided wave modes for the detection and sizing of cracks and pinholes in pipe support regions, NDT&E Int. 41 (2008) 32–43. [6] Y.-M. Cheong, D.-H. Lee, H.-K. Jung, Ultrasonic guided wave parameters for detection of axial cracks in feeder pipes of PHWR nuclear power plants, Ultrasonics 42 (2004) 883–888. [7] I.A. Viktorov, Rayleigh and Lamb waves-physical theory and applications, New York, USA, 1967. [8] G. Liu, J. Qu, Guided circumferential waves in a circular annulus, J. Appl. Mech. 65 (1998) 424–430. [9] G. Liu, J. Qu, Transient wave propagation in a circular annulus subjected to transient excitation on its outer surface, J. Acoust. Soc. Am. 104 (1998) 1210–1220.

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