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A method for acoustic source location in plate-type structures
Mechanical Systems and Signal Processing 93 (2017) 92–103
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
A method for acoustic source location in plate-type structures Amir Mostafapour, Saman Davoodi ⇑ Mechanical Engineering Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 30 September 2016 Received in revised form 18 January 2017 Accepted 4 February 2017
Keywords: Plate type structures Shannon entropy Wavelet transform Cross time frequency spectrum Dispersion curves
a b s t r a c t In this study an algorithm based on Shannon entropy, cross-time frequency spectrum (CTFS) and frequency varying velocities was proposed for structure health monitoring in two-layer plate. A linear array of two sensors is applied to capture the signals. By reducing the number of sensors we used a secondary pattern to get enough information for source locating. For this purpose a pattern for secondary points based on Shannon entropy and cost function was developed. Then to estimate the time delay of signals, cross-time frequency spectrum function was taken from captured signals. The time delay was calculated when CTFS function reached the maximum value. By taking short time Fourier transform of cross correlation function of captures signals and using dispersive curves, time delay and corresponding frequency dependent velocity are estimated. The experiments were carried out and the results showed high precision of presented algorithm. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In today’s economy, structures must remain in operation for long periods of time. Monitoring of corrosion, erosion and crack growth in these structures are becoming significant and must be accounted for in the decisions made regarding usage and maintenance of them. Acoustic emission is a technique which is being used increasingly in the field of structural integrity monitoring using fracture mechanics. This technique has been widely used in industries for detection of defects in pipe and plate type structures. Tobias [1] derived an exact solution for the configuration of three sensors in a plate. The location of the source on the plate is determined at the intersection points of two hyperbolas. Mostafapour et al. [2] used the theories of wavelet transform and cross-time frequency spectrum to locate AE source with frequency-varying wave velocity in plate type structures. They used rectangular array of four sensors on plate. Yang et al. [3] proposed an acoustic emission analysis method based on multiple signal classification method (MUSIC) to calculate the direction of arrival of the wave signal in plate type structures. They estimated the direction of arrival using the multiple signal classification and the time delay of the wave was gained using the continuous wavelet transform. Sedlak et al. [4] compared the first-arrival determination results for thin plate obtained from two-steps AIC picker with Kurz’s method, STA/LTA method and standard threshold crossing technique. Jiao et al. [5] proposed a technique based on the theory of modal acoustic emission for plate structure source location with one sensor. Based on dispersion characteristics of guided wave propagation, they isolated two modes in acoustic signal captured by one sensor. Jumaili et al. [6] presented an improved automatic delta T mapping technique using a clustering algorithm to automatically identify and select the highly correlated events at each grid point. They used minimum difference approach to determine the AE source location. Kundu et al. [7] proposed a technique to locate the acoustic source in large anisotropic plates with the help of six sensors without knowing the direction dependent velocity profile in the plate. ⇑ Corresponding author. E-mail address: [email protected] (S. Davoodi). http://dx.doi.org/10.1016/j.ymssp.2017.02.006 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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For isotropic plates the required number of sensors can be reduced from 6 to 4. Aljets et al. [8] developed an AE source location method for large plate-like structures using a combined time of flight and modal source location algorithm. They used three sensors in a triangular array with a sensor to sensor distance of just a few centimeters. By analyzing the A0 and S0 components of the signal, the arrival times and the distances can be evaluated using mode separation. Jiang et al. [9] used acoustic emission tomography based on simultaneous algebraic reconstruction technique which combined the traditional location algorithm with the SART algorithm using AE events as its signal sources. They examined two-mode damage source location in the Q235B steel plate to validate the algorithm effectiveness. Ernst et al. [10] presented an approach which required one sensor to identify and localize the source of acoustic emission in plates. They used the time reversal principle and dispersive nature of the flexural wave mode. The signal shape of the transverse velocity response contains information about the propagated paths of the incoming elastic waves. The information was made accessible by a numerical time reversal simulation. It was analyzed for an infinite Mindlin plate then by 3D FEM simulation. Surgeon and Wevers [11] investigated the modal nature of AE signals to locate AE source during tensile and bending tests. They used two different plate wave theories (simple and higher order plate wave theories) to calculate the propagation velocities of extensional and flexural modes and theoretically they estimated the arrival time differences between two modes. Mostafapour and Davoodi [12] proposed a method for continuous leakage source location with one sensor in gas-filled pipe and noisy environment based on wavelet analysis and modal location theory. By wavelet decomposition a frequency range was selected and flexural and extensional modes were analyzed. Masoumi and Ashory [13] used two approaches to identify damage in plate-type structures. The first one was to form uniform load surface by using mode shapes of damaged structure and then using 2-D wavelet transform for detecting damage. The second one was based on forming generalized flexibility matrix using mode shapes of damaged structure and then 2-D wavelet transform was applied. In this study a new algorithm based on using a linear array of two acoustic sensors for health monitoring of plates is developed. By reducing the number of sensors into two, first a pattern for secondary points was developed. In this pattern a database was gathered from Hsu-Nielsen source. Then space feature based on entropy of wavelet transform coefficient signals was extracted. A combination of cross time frequency spectrum with frequency dependent wave velocity was used to calculate the time delay and acoustic source location. The experiments were carried out to validate the proposed algorithm. 2. Experimental set-up In this study to validate the proposed method, some laboratory experiments were carried out on a two-layered thin plate with different acoustic impedances of layers. An aluminum plate with dimensions of 70 70 cm and 3 mm thickness was considered. A polyethylene layer of 1 mm thickness was hot coated on a surface of 70 60 cm on aluminum plate. The mechanical and acoustic properties of materials are shown in Table 1. Hsu-Nielsen source is an aid to simulate an acoustic emission event using the fracture of a brittle graphite lead in a suitable fitting. This test consists of breaking a 0.3 mm diameter pencil lead approximately, 3 mm from its tip by pressing it against the surface of the plate as shown in Fig. 1. To capture the acoustic signals, two R15a sensors were installed in a linear array as shown in Fig. 2. 3. Acoustic source location principle 3.1. Wavelet transform New aspects of image and signal processing over the past few years were introduced by the wavelet transform. Wavelet transform is as an effective method for acoustic signal analysis. Wavelet analysis defined as decomposing a signal into two parts of low frequency named as approximation and high frequency as detail. In discrete wavelet, details do not analyze again instead, in packet wavelets approximations and details are divided in two parts. In this study both discrete and packet wavelets are used. Wavelet transform of FðtÞ is as:
Z
CWTðf ; fÞ ¼
1
1
1 tf pffiffiffi FðtÞw dt f f
ð1Þ
which f ; f and w show frequency, time shift and the complex conjugate of wavelet, respectively [14]. Packet wavelet transform is shown as wij;k which i; j and u are modulation, dilation and translation parameters respectively:
wij;u ¼ 2j=2 wi ð2j t uÞ
ð2Þ
Table 1 Mechanical and acoustic properties. Material
Young’s modulus (GPa)
Shear modulus (GPa)
Poisson’s ratio
Density (g/cm3)
Acoustic impedance (g/cm2s * 105)
PE (HDPE) Al (AA5052)
0.7 70.3
0.31 25.9
0.46 0.33
0.95 2.68
2.93 17.8
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Fig. 1. Hsu-Nilsen source.
Fig. 2. Experimental set-up.
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95
n
In this equation i ¼ 1; 2; . . . ; j and n shows wavelet decomposition level. We can obtain wi as:
1 1 X t u w2i ðtÞ ¼ pffiffiffi hðuÞwi 2 2 u¼1 1 X 1 t u gðuÞwi w2iþ1 ðtÞ ¼ pffiffiffi 2 2 u¼1
ð3Þ
In these equations wi is main wavelet and gðuÞ and hðuÞ are discrete filters [14]. The components of signal packet wavelet in the special level can be obtained as: i
f j ðtÞ ¼
X
C ij;u wij;u ðtÞDt
ð4Þ
which C ij;u are the coefficients related to signal. According to packet wavelet frequency relation, frequency range of wholesale and retail components in j level can be obtained:
1 0; f s ð2j Þ 2 1 1 f s ð2j Þ; f s ð2ðj1Þ Þ 2 2
ð5Þ
The component energy at j level is expressed as:
Eij ¼
t 2 X i f j ðfÞ
ð6Þ
s¼t0
3.2. Secondary points algorithm The acoustic signals caused by pencil lead breaking were propagated through the plate and captured by acoustic sensors mounted on the plate. These signals can be modeled as:
Si ðtÞ ¼ S0i ðtÞ þ wi ðtÞ
ð7Þ
S0i ðtÞ
where is the original acoustic signal and wi ðtÞ is the noises (correlated and impulsive noises). The senor-source distance can be written as:
ri ¼ VðxÞ ti
ð8Þ
Fig. 3. Secondary line for k = 1.
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which VðxÞ is the propagation velocity and t i is the propagation time from acoustic source to sensors. Based on Toibas algorithm [1] for acoustic source location in plates, at least three sensors are required. In this study we used two sensors. According to Fig. 3, the pencil lead breaking positions on secondary line had the same distance to the sensors ðk ¼ r1 =r2 ¼ 1Þ, so the captured signals should have good correlation. By increasing the sensor-source distance and due to dispersion nature of medium, attenuation will increase and Shannon entropy will decrease. For this purpose by wavelet transform, the signals were split into an approximation and a detail and the Shannon entropy of details for each layer were calculated as:
X HðX j Þ ¼ Pd ln Pd d
kX j k2 ¼
Pd ¼ jX j;d j2 =kX j k
X 2 X j;d
ð9Þ
d
where X j;d is dth data for j-layer of detail. In this study we used 3-layer wavelet decomposition and each signal was shown with 4-member vector (AE[d1,d2,d3,a3]) which its member was Shannon entropy of details. The attenuation ratio in details is in related with sensor-source distance ratio (k) [15]. For better analysis we can calculate the cost function as [15]:
MicðnÞ ¼
HðX 2n Þ þ HðX 3n Þ P4 j¼1 HðX jn Þ
Micð1Þ Costðk; nÞ ¼ Micð2Þ
ð10Þ
where X 1n ; X 2n and X 3n are the first, second and third layer of wavelet transform details for sensor n. So the points which placed on a secondary line (k) have the same cost function. 3.3. Cross-time-frequency spectrum principle The amplitude and energy of acoustic signals caused by simulated source (Hsu-Nielsen) will decrease during propagation in plate-type structures due to dispersion, scattering and absorption. These signals were captured by AE sensors in time delay of s due to their different distance from source, so a robust similarity measure is necessary to estimate time delay. On the other hand in disperse medium the velocity of acoustic wave is frequency-dependent. This velocity can be determined by the frequency corresponding to the time delay in combination with the known dispersive curve [2]. To find simultaneously time-delay and its corresponding frequency information, the general form of time-frequency distribution can be used as [16]:
Cðt; xÞ ¼
1 4p 2
ZZZ
1 1 eihtihxþihm /ðh; sÞS m s S m þ s dm ds dh 2 2
ð11Þ
In this equation SðtÞ; S ðtÞ and /ðh; sÞ are acoustic signal, complex conjugate of signal and Kernel function, respectively. The spectrogram which is used for time-frequency analysis is as:
2 Z 1 eixs SðsÞqðt sÞds Psp ðt; xÞ ¼ pffiffiffiffiffiffiffi 2p
ð12Þ
The spectrogram of signal can be estimated by computing the squared magnitude of short time Fourier transform of signal. The cross time frequency spectrum of two acoustic signals can be calculated as [16]:
C x;y ðt; xÞ ¼
1 4p2
ZZZ
1 1 eihtisx /ðh; sÞx m s y m s dm ds dh 2 2
ð13Þ
If at given time, the short-time Fourier transform of signals have similar spectral content, the CTFS will show peaks at frequencies [2]. The time delay and frequency can be derived as:
½x0 ; s ¼ arg max C xy ðt; xÞ
ð14Þ
We used the obtained frequency to measure frequency-dependent signal velocity using dispersive curves. 4. Results In this study an algorithm based on Shannon entropy, wavelet based cross-time frequency spectrum and dispersive nature of waves was proposed for AE source locating. The algorithm is based on using two sensors. By reducing the number of sensors it is necessary to use secondary information to source locating. One of the methods to measure the signal information is Shannon entropy which satisfies reasonable guesses to define signal information. The secondary lines were developed on the plate. The position of AE source on an arbitrary line had the same distance from two sensors rr12 ¼ cte . By increasing the sensor-source distance and due to dispersion nature of medium, attenuation will increase and therefore the wave ener-
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Fig. 4. Secondary lines for different k.
gies and Shannon entropy will change. In Fig. 4 the secondary lines for different value of k are shown. For this purpose, first the signals caused by Hsu-Nielsen source were captured by two sensors with operational frequency range of 0.1–1 MHz. The preamplifiers (PAC 1220) with 40 dB gain were used to signal processing. Acoustic signals were acquired at 2 MHz sampling rate and therefore frequencies up to 1 MHz were considered. In this study Hsu-Nielsen simulated source was used to generate acoustic signals and we used the generated signals from this source to verify the location results of proposed algorithm. To find the relation between distance and Shannon entropy on a secondary line, wavelet transform was used. By wavelet transform, the signals were split into an approximation and a detail and the Shannon entropy of details for each layer can be calculated. The lower values of Shannon entropy have more predictable information of signals in compare with high values. By increasing the acoustic signal propagation distance the signal shape will scatter and the attenuation ratio is linked to sensor-source distance ratio rr12 . The acoustic source positions on an arbitrary line have the same sensor-source distance
Fig. 5. Details and approximation of 3-layer decomposition S = a3 + d3 + d2 + d1.
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ratio. On the other hand the ratio of Shannon entropy of lower frequency segments to the higher frequency ones indicates Mic value (Eq. (10)) for each sensor. So we can obtain a link between attenuation, sensor-source distance and Shannon entropy. We used Daubechies wavelet (dB4) and 3-layer wavelet decomposition. The decompositions of recorded signal (details and approximation) are shown in Fig. 5. According to Table 2 the cost function for points on a secondary line had a good consistent. As it mentioned previously, a linear array of two sensors was employed. The geometrical coordinates of one corner of aluminum plate were considered as ð0; 0Þ and so the positions of sensors are shown in Table 3. In two-layer plate, the acoustic signals will be more attenuated when pass from one layer to another due to the difference in acoustic impedances and a partial amount of energy will transmit when the acoustic waves reach the layers boundary [12]. The AE energy transmitted from one medium to another one depends on acoustic impedances. This portion of energy can be obtained as:
AEenergy ¼
q2 c2 q1 c1 q2 c2 þ q1 c1
ð15Þ
in which qi and ci are materials densities and acoustic emission propagation velocities, respectively. Acoustic waves propagated in the plate as guided waves. Dispersion phenomenon makes these waves in initial position distorted with the transmission process. Guided waves propagate in plate with an infinite number of modes for both symmetric and antisymmetric displacements. The normal way to describe the propagation characteristics is to use dispersion curves based on plate mode phase velocity as a function of the product of frequency time thickness. The group (C gr ) and phase (C p ) velocities of these waves can be obtained as:
C gr ¼
dx df
ð16Þ
x
Cp ¼
f
where x and f are circular frequency and wave number. In Fig. 6 group velocity dispersion curves is shown. As it shown for frequency ranges up to 600 kHz S0 mode (named extensional mode) propagated faster than A0 mode. This mode had a maximum group velocity of 4210 m/s per frequency of 410 kHz. But generally for ranges upper than 600 kHz, A0 mode (flexural mode) can travel faster. The generated acoustic signals propagated through the plate and captured by two sensors on it. The source-sensor distances can be calculated as:
r1 ¼ VðxÞ t 1 r2 ¼ VðxÞ t 2
!
1 r 2 t 1 t 2 ¼ rVð xÞ
r1 r2
ð17Þ
¼k
To estimate the source position, first by using wavelet transform and Shannon entropy (cost function), the value of k can be calculated. By finding (k), the number of variable parameters for AE source location is reduced and now we can use correlation technique (CTFS) for source location. As it mentioned the cost function of points on a secondary line is same. On the other hand to determine the time delay between two sensors and the propagation speed we should measure the signals similarity. Due to the acoustic source characteristics and nature of propagation, the maximum energy of the signal is carried out per a specific range. This range is important in source locating procedure. The energies of signals can be calculated by:
Z E¼
ti
t0
v 2 ðtÞdt
ð18Þ
In which v ðtÞ is the acoustic signal amplitude. So per some frequency ranges (due to different value of signal amplitude) the wave’s energies will be more than the other ranges. To estimate the power spectral density of signals, Fast Fourier Transform was taken from the captured signals. It is clear that we can find several peaks with different values of power spectral densities. In this condition the signal components which have distinguished effects on source locating will be selected. In our condition due to the acoustic source characteristics (Hsu-Nielsen source) and the propagation nature, the acoustic waves
Table 2 Secondary lines performance evaluation. Average cost function
Secondary line k
No. of auxiliary points
0.885 0.926 0.985 0.998 1.02 1.03 1.08 1.10 1.14
0.25 0.5 0.7 0.9 1 1.1 1.3 1.7 2
10 10 10 10 10 10 10 10 10
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A. Mostafapour, S. Davoodi / Mechanical Systems and Signal Processing 93 (2017) 92–103 Table 3 Acoustic source location results. Test no.
Sensor 1 pos. mm (x, y)
Sensor 2 pos. mm (x, y)
Hsu-Nielsen pos. mm (x, y)
Estimated source pos. mm (x, y)
Error% (x, y)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(100, 200) (100, 200) (100, 200) (100, 200) (200, 200) (200, 200) (200, 200) (200, 200) (300, 150) (300, 150) (300, 150) (300, 150) (400, 650) (400, 650) (400, 650) (400, 650) (450, 550) (450, 550) (450, 550) (450, 550)
(600, 200) (600, 200) (600, 200) (600, 200) (500, 200) (500, 200) (500, 200) (500, 200) (650, 150) (650, 150) (650, 150) (650, 150) (650, 650) (650, 650) (650, 650) (650, 650) (650, 550) (650, 550) (650, 550) (650, 550)
(200, 50) (300, 50) (400, 50) (500, 50) (300, 50) (350, 50) (450, 50) (250, 50) (400, 50) (450, 50) (550, 50) (600, 50) (450, 100) (550, 100) (600, 100) (600, 50) (500, 100) (550, 50) (600, 100) (600, 50)
(195, 48) (293, 49) (409, 52) (518, 51) (290, 50) (362, 52) (440, 52) (261, 48) (414, 50) (465, 53) (542, 49) (615, 49) (459, 103) (535, 102) (590, 97) (594, 50) (516, 97) (535, 52) (617, 104) (618, 48)
(2.5, 4) (2.3, 2) (2.2, 4) (3.6, 2) (3.3, 0) (3.4, 4) (2.2, 4) (4.4, 4) (3.5, 0) (3.3, 6) (1.6, 2) (2.5, 2) (2, 3) (2.7, 2) (1.7, 3) (1, 0) (3.2, 3) (2.7, 4) (2.8, 4) (2.9, 4)
Fig. 6. Group velocity dispersion curve.
Fig. 7. Frequency domain of captured signal.
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propagate in a narrow band. Fig. 7 shows the acoustic signal frequency domain. The generated signals are burst waves which propagated in a range of frequency of 0–200 kHz. The greatest amount of energy is carried per a frequency range of 135 kHz. These captured signals due to their different propagation distances, have a time delay which calculating of this parameter has an important role in location algorithm. To estimate the time delay of captured signals, the cross correlation of signals were calculated in time, frequency and timefrequency domains as shown in Fig. 8a–c and three dimension diagram was calculated as Fig. 8d. According to Eq. (14), the peak of cross correlation function represents the time delay (s). For test no 1, this value is 0.01 ms. Fast Fourier transform was taken from cross correlation of captured signals and cross spectrums were obtained (as shown in Fig. 8b). From the cross-time frequency spectrum, the peak frequency and corresponding peak time can be obtained. Based on Eq. (13), by taking short time Fourier transform of cross correlation for signals, time delay and corresponding frequency-dependent wave velocity were obtained (Fig. 8c). The real-time determined wave velocity can be obtained from the group speed of the acoustic wave under the peak frequency. It is clear that the frequency range of cross correlation is from 0 to 300 kHz, and the maximum amount of cross correlation FFT occurred per frequency of 125 kHz. Based on Fig. 6 for this frequency only S0 mode is propagated and dominant and its velocity can be obtained as 3650 m/s. The algorithm used in this study based on using two sensors contains the following steps to source location:
Fig. 8. Cross correlation of signals in (a) time domain, (b) frequency domain, (c) time-frequency domain, (d) three dimensional diagram.
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Developing of secondary lines on the plate based on Shannon entropy and cost function Measuring the cross time frequency spectrum (CTFS) of the captured signals and estimating the time delay and its corresponding frequency Calculating dispersion curves and obtaining the frequency dependent velocity Estimating source position Generally the cross correlation of two spatially captured acoustic waves is a burst signal in time domain as shown in Fig. 8a and the peak time represents the time delay between captured signals. From the time-frequency domain of crosscorrelation function, we can obtain the peak frequency ranges and the spread time. So the time delay and corresponding frequency information can be simultaneously estimated from the cross-time frequency spectrum. The real-time determined acoustic speed can be obtained from the group speed of the acoustic wave (S0 mode) showed in dispersion curves under the peak frequency. By changing the major frequency of AE source the values of time delay and frequency-varying group velocity of excited mode of acoustic signal will change based on cross-time frequency spectrum relations and so the location of source will be estimated based on new data for time delay and propagation velocity and there will be no effects on locating results precise. A MATLAB code based on Shannon entropy, wavelet transform and cross time frequency spectrum was developed and the AE source location results are shown in Table 3. The captured signals were split by wavelet transform and the lower values of Shannon entropy were considered due to the predictable information of signals. So each position of AE source on a secondary line propagates the signal with a unique Shannon entropy vector but the acoustic source position on an arbitrary secondary line have the same sensor-source distance ratio and based on Eq. (5) the cost function will be the same. In this case study by connecting the two sensors, the monitoring surface will be divided into two plane (sections). Generally by calculating cost function and CTFS two points will be obtained for an arbitrary secondary line as acoustic source position which one of them is virtual. The propagated signals from these points (as AE source locations) will captured by sensors with different Shannon entropy vectors and waveform (due to different values of multi-reflected waves which interfered with pure acoustic signals and different dispersion nature of medium). So each position of source on secondary line will propagate a unique signal with Shannon entropy vector. In the other words the calculated value of Mic (based on Eq. (5)) per these sources positions will be different but for both points, the cost function will be the same. So by comparing the captured
Fig. 9. The acoustic source locating flow chart.
Error 2.9%
Error 2.8%
Error 2.7%
Error 3.2%
Error 1%
Error 1.7%
Error 2.7%
Error 2.5%
Error 3.3%
Error 3.5%
Error 2.2%
Error 2.2% Error 4.4%
300
Error 3.3%
400
Error 2.3%
500
Error 2.5%
Acoustic Source Position
600
Error 3.4%
Error 3.6%
700
Error 2%
A. Mostafapour, S. Davoodi / Mechanical Systems and Signal Processing 93 (2017) 92–103
Error 1.6%
102
200 100 0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Test No. Hsu-Nielsen source X Pos.
Estimated source X Pos.
7
8
Error 3%
Error 3%
Error 4% Error 4%
6
Error 4%
5
Error 0%
Error 4%
4
Error 2%
Error 4%
3
Error 2%
Error 4%
2
Error 6%
Error 0%
1
Error 0%
Error 2%
60
Error 4%
80
Error 2%
100 Error 4%
Acoustic Source Position
120
Error 2%
Error 3%
Fig. 10. Acoustic emission source results in X direction.
40 20 0 9 10 11 12 13 14 15 16 17 18 19 20
Test No. Hsu-Nielsen source Y Pos.
Estimated source X Pos.
Fig. 11. Acoustic emission source results in Y direction.
Fig. 12. AE locating results for cross-correlation and proposed algorithm.
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entropy of Shannon and Mic value with our databases (which used for developing secondary lines) we can distinguish the real source position. The acoustic source locating flow chart for proposed algorithm is shown in Fig. 9. The results showed that the proposed technique was suitable for acoustic source locating and the maximum location errors of 6 percent were obtained. Also the acoustic source location results are shown in Figs. 10 and 11. To compare the results of proposed algorithm with correlation based methods, the experiments were carried out with four acoustic sensors. As it shown in Fig. 12 the average source locating error of this algorithm is less than common methods. On the other hand this algorithm is suitable for health monitoring of plate-type structures which have limited access with precise locating results. The proposed algorithm has some advantages in compare with other traditional methods as: – High locating precise – Less sensitivity to noises and unwanted signals – Suitable for health monitoring of structures with limited availability due to low number of sensors The disadvantages of this algorithm are as: – High volume of data and experiments to develop secondary lines on plate based on Shannon entropy – Placed the acoustic source in an area between two sensors 5. Conclusion In this paper AE source locating algorithm is proposed based on using two sensors. This algorithm is suitable for condition health monitoring of plate-type structures which have limited access. A linear array of two sensors is used for source location. By reducing the number of sensors we used a secondary pattern. For this purpose, first we proposed a secondary pattern based on Shannon entropy and cost function and then by cross time frequency spectrum and frequency varying velocity, the time delay for two captured signals and propagation velocity were calculated. Finally using the obtained information from secondary pattern, CTFS and dispersion curves, the acoustic source were estimated with a high precision. References [1] A. Toibas, Acoustic emission source location in two dimensions by an array of three sensors, Nondestruct. Test. 9 (1991) 2551–2556. [2] A. Mostafapour, S. Davoodi, M. Ghareaghaji, Acoustic emission source location in plates using wavelet analysis and cross time frequency spectrum, Ultrasonics 54 (2014) 2055–2062. [3] H. Yang, T. Shin, S. Lee, Source location in plates based on the multiple sensors array method and wavelet analysis, J. Mech. Sci. Technol. 28 (2014) 1–8. [4] P. Sedlak, Y. Hirose, M. Enoki, Acoustic emission localization in thin multi-layer plates using first-arrival determination, Mech. Syst. Signal Process. 36 (2013) 636–649. [5] J. Jiao, C. He, B. Wu, R. Fei, Application of wavelet transform on modal acoustic emission source location in thin plates with one sensor, Int. J. Pres. Ves. Pip. 81 (2004) 427–431. [6] S. Jumaili, M. Pearson, K. Holford, M. Eaton, R. Pullin, Acoustic emission source location in complex structures using full automatic delta T mapping technique, Mech. Syst. Signal Process. 72–73 (2016) 513–524. [7] T. Kundu, H. Nakatani, N. Tekeda, Acoustic source localization in anisotropic plates, Ultrasonics 52 (2012) 740–746. [8] D. Aljets, A. Chong, S. Wilcox, K. Holford, Acoustic emission source location on large plate-like structures using a local triangular sensor array, Mech. Syst. Signal Process. 30 (2012) 91–102. [9] Y. Jiang, F. Xu, B. Xu, Acoustic emission tomography based on simultaneous algebraic reconstruction technique to visualize the damage source location in Q235B steel plate, Mech. Syst. Signal Process. 64 (2015) 452–464. [10] R. Ernst, F. Zwimpfer, J. Dual, One sensor acoustic emission localization in plates, Ultrasonics 64 (2016) 139–150. [11] M. Surgeon, M. Wevers, One sensor linear location of acoustic emission events using plate wave theories, Mater. Sci. Eng. 265 (1999) 254–261. [12] A. Mostafapour, S. Davoodi, Continuous leakage location in noisy environment using modal and wavelet analysis with one sensor, Ultrasonics 62 (2015) 305–311. [13] M. Masoumi, M.R. Ashory, Damage identification in plate-type structures using 2-D spatial wavelet transform and flexibility-based methods, Lett. Fract. Micromech. 183 (2013) 259–266. [14] K.P. Soman, K.I. Ramachandran, Insight into Wavelets from Theory to Practice, second ed., Prentice-Hall of India Pvt. Limited, 2005. [15] D. Bianchi, E. Mayrhofer, M. Groschl, G. Betz, A. Vernes, Wavelet packet transform for detection of signal events in acoustic emission signals, Mech. Syst. Signal Process. 65 (2015) 441–451. [16] L. Cohen, Time-frequency distribution – a review, Proc. IEEE 77 (1989) 941–981.
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
A method for acoustic source location in plate-type structures Amir Mostafapour, Saman Davoodi ⇑ Mechanical Engineering Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 30 September 2016 Received in revised form 18 January 2017 Accepted 4 February 2017
Keywords: Plate type structures Shannon entropy Wavelet transform Cross time frequency spectrum Dispersion curves
a b s t r a c t In this study an algorithm based on Shannon entropy, cross-time frequency spectrum (CTFS) and frequency varying velocities was proposed for structure health monitoring in two-layer plate. A linear array of two sensors is applied to capture the signals. By reducing the number of sensors we used a secondary pattern to get enough information for source locating. For this purpose a pattern for secondary points based on Shannon entropy and cost function was developed. Then to estimate the time delay of signals, cross-time frequency spectrum function was taken from captured signals. The time delay was calculated when CTFS function reached the maximum value. By taking short time Fourier transform of cross correlation function of captures signals and using dispersive curves, time delay and corresponding frequency dependent velocity are estimated. The experiments were carried out and the results showed high precision of presented algorithm. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In today’s economy, structures must remain in operation for long periods of time. Monitoring of corrosion, erosion and crack growth in these structures are becoming significant and must be accounted for in the decisions made regarding usage and maintenance of them. Acoustic emission is a technique which is being used increasingly in the field of structural integrity monitoring using fracture mechanics. This technique has been widely used in industries for detection of defects in pipe and plate type structures. Tobias [1] derived an exact solution for the configuration of three sensors in a plate. The location of the source on the plate is determined at the intersection points of two hyperbolas. Mostafapour et al. [2] used the theories of wavelet transform and cross-time frequency spectrum to locate AE source with frequency-varying wave velocity in plate type structures. They used rectangular array of four sensors on plate. Yang et al. [3] proposed an acoustic emission analysis method based on multiple signal classification method (MUSIC) to calculate the direction of arrival of the wave signal in plate type structures. They estimated the direction of arrival using the multiple signal classification and the time delay of the wave was gained using the continuous wavelet transform. Sedlak et al. [4] compared the first-arrival determination results for thin plate obtained from two-steps AIC picker with Kurz’s method, STA/LTA method and standard threshold crossing technique. Jiao et al. [5] proposed a technique based on the theory of modal acoustic emission for plate structure source location with one sensor. Based on dispersion characteristics of guided wave propagation, they isolated two modes in acoustic signal captured by one sensor. Jumaili et al. [6] presented an improved automatic delta T mapping technique using a clustering algorithm to automatically identify and select the highly correlated events at each grid point. They used minimum difference approach to determine the AE source location. Kundu et al. [7] proposed a technique to locate the acoustic source in large anisotropic plates with the help of six sensors without knowing the direction dependent velocity profile in the plate. ⇑ Corresponding author. E-mail address: [email protected] (S. Davoodi). http://dx.doi.org/10.1016/j.ymssp.2017.02.006 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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For isotropic plates the required number of sensors can be reduced from 6 to 4. Aljets et al. [8] developed an AE source location method for large plate-like structures using a combined time of flight and modal source location algorithm. They used three sensors in a triangular array with a sensor to sensor distance of just a few centimeters. By analyzing the A0 and S0 components of the signal, the arrival times and the distances can be evaluated using mode separation. Jiang et al. [9] used acoustic emission tomography based on simultaneous algebraic reconstruction technique which combined the traditional location algorithm with the SART algorithm using AE events as its signal sources. They examined two-mode damage source location in the Q235B steel plate to validate the algorithm effectiveness. Ernst et al. [10] presented an approach which required one sensor to identify and localize the source of acoustic emission in plates. They used the time reversal principle and dispersive nature of the flexural wave mode. The signal shape of the transverse velocity response contains information about the propagated paths of the incoming elastic waves. The information was made accessible by a numerical time reversal simulation. It was analyzed for an infinite Mindlin plate then by 3D FEM simulation. Surgeon and Wevers [11] investigated the modal nature of AE signals to locate AE source during tensile and bending tests. They used two different plate wave theories (simple and higher order plate wave theories) to calculate the propagation velocities of extensional and flexural modes and theoretically they estimated the arrival time differences between two modes. Mostafapour and Davoodi [12] proposed a method for continuous leakage source location with one sensor in gas-filled pipe and noisy environment based on wavelet analysis and modal location theory. By wavelet decomposition a frequency range was selected and flexural and extensional modes were analyzed. Masoumi and Ashory [13] used two approaches to identify damage in plate-type structures. The first one was to form uniform load surface by using mode shapes of damaged structure and then using 2-D wavelet transform for detecting damage. The second one was based on forming generalized flexibility matrix using mode shapes of damaged structure and then 2-D wavelet transform was applied. In this study a new algorithm based on using a linear array of two acoustic sensors for health monitoring of plates is developed. By reducing the number of sensors into two, first a pattern for secondary points was developed. In this pattern a database was gathered from Hsu-Nielsen source. Then space feature based on entropy of wavelet transform coefficient signals was extracted. A combination of cross time frequency spectrum with frequency dependent wave velocity was used to calculate the time delay and acoustic source location. The experiments were carried out to validate the proposed algorithm. 2. Experimental set-up In this study to validate the proposed method, some laboratory experiments were carried out on a two-layered thin plate with different acoustic impedances of layers. An aluminum plate with dimensions of 70 70 cm and 3 mm thickness was considered. A polyethylene layer of 1 mm thickness was hot coated on a surface of 70 60 cm on aluminum plate. The mechanical and acoustic properties of materials are shown in Table 1. Hsu-Nielsen source is an aid to simulate an acoustic emission event using the fracture of a brittle graphite lead in a suitable fitting. This test consists of breaking a 0.3 mm diameter pencil lead approximately, 3 mm from its tip by pressing it against the surface of the plate as shown in Fig. 1. To capture the acoustic signals, two R15a sensors were installed in a linear array as shown in Fig. 2. 3. Acoustic source location principle 3.1. Wavelet transform New aspects of image and signal processing over the past few years were introduced by the wavelet transform. Wavelet transform is as an effective method for acoustic signal analysis. Wavelet analysis defined as decomposing a signal into two parts of low frequency named as approximation and high frequency as detail. In discrete wavelet, details do not analyze again instead, in packet wavelets approximations and details are divided in two parts. In this study both discrete and packet wavelets are used. Wavelet transform of FðtÞ is as:
Z
CWTðf ; fÞ ¼
1
1
1 tf pffiffiffi FðtÞw dt f f
ð1Þ
which f ; f and w show frequency, time shift and the complex conjugate of wavelet, respectively [14]. Packet wavelet transform is shown as wij;k which i; j and u are modulation, dilation and translation parameters respectively:
wij;u ¼ 2j=2 wi ð2j t uÞ
ð2Þ
Table 1 Mechanical and acoustic properties. Material
Young’s modulus (GPa)
Shear modulus (GPa)
Poisson’s ratio
Density (g/cm3)
Acoustic impedance (g/cm2s * 105)
PE (HDPE) Al (AA5052)
0.7 70.3
0.31 25.9
0.46 0.33
0.95 2.68
2.93 17.8
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Fig. 1. Hsu-Nilsen source.
Fig. 2. Experimental set-up.
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95
n
In this equation i ¼ 1; 2; . . . ; j and n shows wavelet decomposition level. We can obtain wi as:
1 1 X t u w2i ðtÞ ¼ pffiffiffi hðuÞwi 2 2 u¼1 1 X 1 t u gðuÞwi w2iþ1 ðtÞ ¼ pffiffiffi 2 2 u¼1
ð3Þ
In these equations wi is main wavelet and gðuÞ and hðuÞ are discrete filters [14]. The components of signal packet wavelet in the special level can be obtained as: i
f j ðtÞ ¼
X
C ij;u wij;u ðtÞDt
ð4Þ
which C ij;u are the coefficients related to signal. According to packet wavelet frequency relation, frequency range of wholesale and retail components in j level can be obtained:
1 0; f s ð2j Þ 2 1 1 f s ð2j Þ; f s ð2ðj1Þ Þ 2 2
ð5Þ
The component energy at j level is expressed as:
Eij ¼
t 2 X i f j ðfÞ
ð6Þ
s¼t0
3.2. Secondary points algorithm The acoustic signals caused by pencil lead breaking were propagated through the plate and captured by acoustic sensors mounted on the plate. These signals can be modeled as:
Si ðtÞ ¼ S0i ðtÞ þ wi ðtÞ
ð7Þ
S0i ðtÞ
where is the original acoustic signal and wi ðtÞ is the noises (correlated and impulsive noises). The senor-source distance can be written as:
ri ¼ VðxÞ ti
ð8Þ
Fig. 3. Secondary line for k = 1.
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which VðxÞ is the propagation velocity and t i is the propagation time from acoustic source to sensors. Based on Toibas algorithm [1] for acoustic source location in plates, at least three sensors are required. In this study we used two sensors. According to Fig. 3, the pencil lead breaking positions on secondary line had the same distance to the sensors ðk ¼ r1 =r2 ¼ 1Þ, so the captured signals should have good correlation. By increasing the sensor-source distance and due to dispersion nature of medium, attenuation will increase and Shannon entropy will decrease. For this purpose by wavelet transform, the signals were split into an approximation and a detail and the Shannon entropy of details for each layer were calculated as:
X HðX j Þ ¼ Pd ln Pd d
kX j k2 ¼
Pd ¼ jX j;d j2 =kX j k
X 2 X j;d
ð9Þ
d
where X j;d is dth data for j-layer of detail. In this study we used 3-layer wavelet decomposition and each signal was shown with 4-member vector (AE[d1,d2,d3,a3]) which its member was Shannon entropy of details. The attenuation ratio in details is in related with sensor-source distance ratio (k) [15]. For better analysis we can calculate the cost function as [15]:
MicðnÞ ¼
HðX 2n Þ þ HðX 3n Þ P4 j¼1 HðX jn Þ
Micð1Þ Costðk; nÞ ¼ Micð2Þ
ð10Þ
where X 1n ; X 2n and X 3n are the first, second and third layer of wavelet transform details for sensor n. So the points which placed on a secondary line (k) have the same cost function. 3.3. Cross-time-frequency spectrum principle The amplitude and energy of acoustic signals caused by simulated source (Hsu-Nielsen) will decrease during propagation in plate-type structures due to dispersion, scattering and absorption. These signals were captured by AE sensors in time delay of s due to their different distance from source, so a robust similarity measure is necessary to estimate time delay. On the other hand in disperse medium the velocity of acoustic wave is frequency-dependent. This velocity can be determined by the frequency corresponding to the time delay in combination with the known dispersive curve [2]. To find simultaneously time-delay and its corresponding frequency information, the general form of time-frequency distribution can be used as [16]:
Cðt; xÞ ¼
1 4p 2
ZZZ
1 1 eihtihxþihm /ðh; sÞS m s S m þ s dm ds dh 2 2
ð11Þ
In this equation SðtÞ; S ðtÞ and /ðh; sÞ are acoustic signal, complex conjugate of signal and Kernel function, respectively. The spectrogram which is used for time-frequency analysis is as:
2 Z 1 eixs SðsÞqðt sÞds Psp ðt; xÞ ¼ pffiffiffiffiffiffiffi 2p
ð12Þ
The spectrogram of signal can be estimated by computing the squared magnitude of short time Fourier transform of signal. The cross time frequency spectrum of two acoustic signals can be calculated as [16]:
C x;y ðt; xÞ ¼
1 4p2
ZZZ
1 1 eihtisx /ðh; sÞx m s y m s dm ds dh 2 2
ð13Þ
If at given time, the short-time Fourier transform of signals have similar spectral content, the CTFS will show peaks at frequencies [2]. The time delay and frequency can be derived as:
½x0 ; s ¼ arg max C xy ðt; xÞ
ð14Þ
We used the obtained frequency to measure frequency-dependent signal velocity using dispersive curves. 4. Results In this study an algorithm based on Shannon entropy, wavelet based cross-time frequency spectrum and dispersive nature of waves was proposed for AE source locating. The algorithm is based on using two sensors. By reducing the number of sensors it is necessary to use secondary information to source locating. One of the methods to measure the signal information is Shannon entropy which satisfies reasonable guesses to define signal information. The secondary lines were developed on the plate. The position of AE source on an arbitrary line had the same distance from two sensors rr12 ¼ cte . By increasing the sensor-source distance and due to dispersion nature of medium, attenuation will increase and therefore the wave ener-
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97
Fig. 4. Secondary lines for different k.
gies and Shannon entropy will change. In Fig. 4 the secondary lines for different value of k are shown. For this purpose, first the signals caused by Hsu-Nielsen source were captured by two sensors with operational frequency range of 0.1–1 MHz. The preamplifiers (PAC 1220) with 40 dB gain were used to signal processing. Acoustic signals were acquired at 2 MHz sampling rate and therefore frequencies up to 1 MHz were considered. In this study Hsu-Nielsen simulated source was used to generate acoustic signals and we used the generated signals from this source to verify the location results of proposed algorithm. To find the relation between distance and Shannon entropy on a secondary line, wavelet transform was used. By wavelet transform, the signals were split into an approximation and a detail and the Shannon entropy of details for each layer can be calculated. The lower values of Shannon entropy have more predictable information of signals in compare with high values. By increasing the acoustic signal propagation distance the signal shape will scatter and the attenuation ratio is linked to sensor-source distance ratio rr12 . The acoustic source positions on an arbitrary line have the same sensor-source distance
Fig. 5. Details and approximation of 3-layer decomposition S = a3 + d3 + d2 + d1.
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ratio. On the other hand the ratio of Shannon entropy of lower frequency segments to the higher frequency ones indicates Mic value (Eq. (10)) for each sensor. So we can obtain a link between attenuation, sensor-source distance and Shannon entropy. We used Daubechies wavelet (dB4) and 3-layer wavelet decomposition. The decompositions of recorded signal (details and approximation) are shown in Fig. 5. According to Table 2 the cost function for points on a secondary line had a good consistent. As it mentioned previously, a linear array of two sensors was employed. The geometrical coordinates of one corner of aluminum plate were considered as ð0; 0Þ and so the positions of sensors are shown in Table 3. In two-layer plate, the acoustic signals will be more attenuated when pass from one layer to another due to the difference in acoustic impedances and a partial amount of energy will transmit when the acoustic waves reach the layers boundary [12]. The AE energy transmitted from one medium to another one depends on acoustic impedances. This portion of energy can be obtained as:
AEenergy ¼
q2 c2 q1 c1 q2 c2 þ q1 c1
ð15Þ
in which qi and ci are materials densities and acoustic emission propagation velocities, respectively. Acoustic waves propagated in the plate as guided waves. Dispersion phenomenon makes these waves in initial position distorted with the transmission process. Guided waves propagate in plate with an infinite number of modes for both symmetric and antisymmetric displacements. The normal way to describe the propagation characteristics is to use dispersion curves based on plate mode phase velocity as a function of the product of frequency time thickness. The group (C gr ) and phase (C p ) velocities of these waves can be obtained as:
C gr ¼
dx df
ð16Þ
x
Cp ¼
f
where x and f are circular frequency and wave number. In Fig. 6 group velocity dispersion curves is shown. As it shown for frequency ranges up to 600 kHz S0 mode (named extensional mode) propagated faster than A0 mode. This mode had a maximum group velocity of 4210 m/s per frequency of 410 kHz. But generally for ranges upper than 600 kHz, A0 mode (flexural mode) can travel faster. The generated acoustic signals propagated through the plate and captured by two sensors on it. The source-sensor distances can be calculated as:
r1 ¼ VðxÞ t 1 r2 ¼ VðxÞ t 2
!
1 r 2 t 1 t 2 ¼ rVð xÞ
r1 r2
ð17Þ
¼k
To estimate the source position, first by using wavelet transform and Shannon entropy (cost function), the value of k can be calculated. By finding (k), the number of variable parameters for AE source location is reduced and now we can use correlation technique (CTFS) for source location. As it mentioned the cost function of points on a secondary line is same. On the other hand to determine the time delay between two sensors and the propagation speed we should measure the signals similarity. Due to the acoustic source characteristics and nature of propagation, the maximum energy of the signal is carried out per a specific range. This range is important in source locating procedure. The energies of signals can be calculated by:
Z E¼
ti
t0
v 2 ðtÞdt
ð18Þ
In which v ðtÞ is the acoustic signal amplitude. So per some frequency ranges (due to different value of signal amplitude) the wave’s energies will be more than the other ranges. To estimate the power spectral density of signals, Fast Fourier Transform was taken from the captured signals. It is clear that we can find several peaks with different values of power spectral densities. In this condition the signal components which have distinguished effects on source locating will be selected. In our condition due to the acoustic source characteristics (Hsu-Nielsen source) and the propagation nature, the acoustic waves
Table 2 Secondary lines performance evaluation. Average cost function
Secondary line k
No. of auxiliary points
0.885 0.926 0.985 0.998 1.02 1.03 1.08 1.10 1.14
0.25 0.5 0.7 0.9 1 1.1 1.3 1.7 2
10 10 10 10 10 10 10 10 10
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A. Mostafapour, S. Davoodi / Mechanical Systems and Signal Processing 93 (2017) 92–103 Table 3 Acoustic source location results. Test no.
Sensor 1 pos. mm (x, y)
Sensor 2 pos. mm (x, y)
Hsu-Nielsen pos. mm (x, y)
Estimated source pos. mm (x, y)
Error% (x, y)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(100, 200) (100, 200) (100, 200) (100, 200) (200, 200) (200, 200) (200, 200) (200, 200) (300, 150) (300, 150) (300, 150) (300, 150) (400, 650) (400, 650) (400, 650) (400, 650) (450, 550) (450, 550) (450, 550) (450, 550)
(600, 200) (600, 200) (600, 200) (600, 200) (500, 200) (500, 200) (500, 200) (500, 200) (650, 150) (650, 150) (650, 150) (650, 150) (650, 650) (650, 650) (650, 650) (650, 650) (650, 550) (650, 550) (650, 550) (650, 550)
(200, 50) (300, 50) (400, 50) (500, 50) (300, 50) (350, 50) (450, 50) (250, 50) (400, 50) (450, 50) (550, 50) (600, 50) (450, 100) (550, 100) (600, 100) (600, 50) (500, 100) (550, 50) (600, 100) (600, 50)
(195, 48) (293, 49) (409, 52) (518, 51) (290, 50) (362, 52) (440, 52) (261, 48) (414, 50) (465, 53) (542, 49) (615, 49) (459, 103) (535, 102) (590, 97) (594, 50) (516, 97) (535, 52) (617, 104) (618, 48)
(2.5, 4) (2.3, 2) (2.2, 4) (3.6, 2) (3.3, 0) (3.4, 4) (2.2, 4) (4.4, 4) (3.5, 0) (3.3, 6) (1.6, 2) (2.5, 2) (2, 3) (2.7, 2) (1.7, 3) (1, 0) (3.2, 3) (2.7, 4) (2.8, 4) (2.9, 4)
Fig. 6. Group velocity dispersion curve.
Fig. 7. Frequency domain of captured signal.
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propagate in a narrow band. Fig. 7 shows the acoustic signal frequency domain. The generated signals are burst waves which propagated in a range of frequency of 0–200 kHz. The greatest amount of energy is carried per a frequency range of 135 kHz. These captured signals due to their different propagation distances, have a time delay which calculating of this parameter has an important role in location algorithm. To estimate the time delay of captured signals, the cross correlation of signals were calculated in time, frequency and timefrequency domains as shown in Fig. 8a–c and three dimension diagram was calculated as Fig. 8d. According to Eq. (14), the peak of cross correlation function represents the time delay (s). For test no 1, this value is 0.01 ms. Fast Fourier transform was taken from cross correlation of captured signals and cross spectrums were obtained (as shown in Fig. 8b). From the cross-time frequency spectrum, the peak frequency and corresponding peak time can be obtained. Based on Eq. (13), by taking short time Fourier transform of cross correlation for signals, time delay and corresponding frequency-dependent wave velocity were obtained (Fig. 8c). The real-time determined wave velocity can be obtained from the group speed of the acoustic wave under the peak frequency. It is clear that the frequency range of cross correlation is from 0 to 300 kHz, and the maximum amount of cross correlation FFT occurred per frequency of 125 kHz. Based on Fig. 6 for this frequency only S0 mode is propagated and dominant and its velocity can be obtained as 3650 m/s. The algorithm used in this study based on using two sensors contains the following steps to source location:
Fig. 8. Cross correlation of signals in (a) time domain, (b) frequency domain, (c) time-frequency domain, (d) three dimensional diagram.
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101
Developing of secondary lines on the plate based on Shannon entropy and cost function Measuring the cross time frequency spectrum (CTFS) of the captured signals and estimating the time delay and its corresponding frequency Calculating dispersion curves and obtaining the frequency dependent velocity Estimating source position Generally the cross correlation of two spatially captured acoustic waves is a burst signal in time domain as shown in Fig. 8a and the peak time represents the time delay between captured signals. From the time-frequency domain of crosscorrelation function, we can obtain the peak frequency ranges and the spread time. So the time delay and corresponding frequency information can be simultaneously estimated from the cross-time frequency spectrum. The real-time determined acoustic speed can be obtained from the group speed of the acoustic wave (S0 mode) showed in dispersion curves under the peak frequency. By changing the major frequency of AE source the values of time delay and frequency-varying group velocity of excited mode of acoustic signal will change based on cross-time frequency spectrum relations and so the location of source will be estimated based on new data for time delay and propagation velocity and there will be no effects on locating results precise. A MATLAB code based on Shannon entropy, wavelet transform and cross time frequency spectrum was developed and the AE source location results are shown in Table 3. The captured signals were split by wavelet transform and the lower values of Shannon entropy were considered due to the predictable information of signals. So each position of AE source on a secondary line propagates the signal with a unique Shannon entropy vector but the acoustic source position on an arbitrary secondary line have the same sensor-source distance ratio and based on Eq. (5) the cost function will be the same. In this case study by connecting the two sensors, the monitoring surface will be divided into two plane (sections). Generally by calculating cost function and CTFS two points will be obtained for an arbitrary secondary line as acoustic source position which one of them is virtual. The propagated signals from these points (as AE source locations) will captured by sensors with different Shannon entropy vectors and waveform (due to different values of multi-reflected waves which interfered with pure acoustic signals and different dispersion nature of medium). So each position of source on secondary line will propagate a unique signal with Shannon entropy vector. In the other words the calculated value of Mic (based on Eq. (5)) per these sources positions will be different but for both points, the cost function will be the same. So by comparing the captured
Fig. 9. The acoustic source locating flow chart.
Error 2.9%
Error 2.8%
Error 2.7%
Error 3.2%
Error 1%
Error 1.7%
Error 2.7%
Error 2.5%
Error 3.3%
Error 3.5%
Error 2.2%
Error 2.2% Error 4.4%
300
Error 3.3%
400
Error 2.3%
500
Error 2.5%
Acoustic Source Position
600
Error 3.4%
Error 3.6%
700
Error 2%
A. Mostafapour, S. Davoodi / Mechanical Systems and Signal Processing 93 (2017) 92–103
Error 1.6%
102
200 100 0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Test No. Hsu-Nielsen source X Pos.
Estimated source X Pos.
7
8
Error 3%
Error 3%
Error 4% Error 4%
6
Error 4%
5
Error 0%
Error 4%
4
Error 2%
Error 4%
3
Error 2%
Error 4%
2
Error 6%
Error 0%
1
Error 0%
Error 2%
60
Error 4%
80
Error 2%
100 Error 4%
Acoustic Source Position
120
Error 2%
Error 3%
Fig. 10. Acoustic emission source results in X direction.
40 20 0 9 10 11 12 13 14 15 16 17 18 19 20
Test No. Hsu-Nielsen source Y Pos.
Estimated source X Pos.
Fig. 11. Acoustic emission source results in Y direction.
Fig. 12. AE locating results for cross-correlation and proposed algorithm.
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entropy of Shannon and Mic value with our databases (which used for developing secondary lines) we can distinguish the real source position. The acoustic source locating flow chart for proposed algorithm is shown in Fig. 9. The results showed that the proposed technique was suitable for acoustic source locating and the maximum location errors of 6 percent were obtained. Also the acoustic source location results are shown in Figs. 10 and 11. To compare the results of proposed algorithm with correlation based methods, the experiments were carried out with four acoustic sensors. As it shown in Fig. 12 the average source locating error of this algorithm is less than common methods. On the other hand this algorithm is suitable for health monitoring of plate-type structures which have limited access with precise locating results. The proposed algorithm has some advantages in compare with other traditional methods as: – High locating precise – Less sensitivity to noises and unwanted signals – Suitable for health monitoring of structures with limited availability due to low number of sensors The disadvantages of this algorithm are as: – High volume of data and experiments to develop secondary lines on plate based on Shannon entropy – Placed the acoustic source in an area between two sensors 5. Conclusion In this paper AE source locating algorithm is proposed based on using two sensors. This algorithm is suitable for condition health monitoring of plate-type structures which have limited access. A linear array of two sensors is used for source location. By reducing the number of sensors we used a secondary pattern. For this purpose, first we proposed a secondary pattern based on Shannon entropy and cost function and then by cross time frequency spectrum and frequency varying velocity, the time delay for two captured signals and propagation velocity were calculated. Finally using the obtained information from secondary pattern, CTFS and dispersion curves, the acoustic source were estimated with a high precision. References [1] A. Toibas, Acoustic emission source location in two dimensions by an array of three sensors, Nondestruct. Test. 9 (1991) 2551–2556. [2] A. Mostafapour, S. Davoodi, M. Ghareaghaji, Acoustic emission source location in plates using wavelet analysis and cross time frequency spectrum, Ultrasonics 54 (2014) 2055–2062. [3] H. Yang, T. Shin, S. Lee, Source location in plates based on the multiple sensors array method and wavelet analysis, J. Mech. Sci. Technol. 28 (2014) 1–8. [4] P. Sedlak, Y. Hirose, M. Enoki, Acoustic emission localization in thin multi-layer plates using first-arrival determination, Mech. Syst. Signal Process. 36 (2013) 636–649. [5] J. Jiao, C. He, B. Wu, R. Fei, Application of wavelet transform on modal acoustic emission source location in thin plates with one sensor, Int. J. Pres. Ves. Pip. 81 (2004) 427–431. [6] S. Jumaili, M. Pearson, K. Holford, M. Eaton, R. Pullin, Acoustic emission source location in complex structures using full automatic delta T mapping technique, Mech. Syst. Signal Process. 72–73 (2016) 513–524. [7] T. Kundu, H. Nakatani, N. Tekeda, Acoustic source localization in anisotropic plates, Ultrasonics 52 (2012) 740–746. [8] D. Aljets, A. Chong, S. Wilcox, K. Holford, Acoustic emission source location on large plate-like structures using a local triangular sensor array, Mech. Syst. Signal Process. 30 (2012) 91–102. [9] Y. Jiang, F. Xu, B. Xu, Acoustic emission tomography based on simultaneous algebraic reconstruction technique to visualize the damage source location in Q235B steel plate, Mech. Syst. Signal Process. 64 (2015) 452–464. [10] R. Ernst, F. Zwimpfer, J. Dual, One sensor acoustic emission localization in plates, Ultrasonics 64 (2016) 139–150. [11] M. Surgeon, M. Wevers, One sensor linear location of acoustic emission events using plate wave theories, Mater. Sci. Eng. 265 (1999) 254–261. [12] A. Mostafapour, S. Davoodi, Continuous leakage location in noisy environment using modal and wavelet analysis with one sensor, Ultrasonics 62 (2015) 305–311. [13] M. Masoumi, M.R. Ashory, Damage identification in plate-type structures using 2-D spatial wavelet transform and flexibility-based methods, Lett. Fract. Micromech. 183 (2013) 259–266. [14] K.P. Soman, K.I. Ramachandran, Insight into Wavelets from Theory to Practice, second ed., Prentice-Hall of India Pvt. Limited, 2005. [15] D. Bianchi, E. Mayrhofer, M. Groschl, G. Betz, A. Vernes, Wavelet packet transform for detection of signal events in acoustic emission signals, Mech. Syst. Signal Process. 65 (2015) 441–451. [16] L. Cohen, Time-frequency distribution – a review, Proc. IEEE 77 (1989) 941–981.