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Phase correction improved multiple signal classification for impact source localization under varying temperature conditions
Journal Pre-proofs Phase correction improve d multiple signal classification for impact source localization under varying temperature conditions Zhenghao Zhang, Yongteng Zhong, Jiawei Xiang, Yongying Jiang PII: DOI: Reference:
S0263-2241(19)31238-2 https://doi.org/10.1016/j.measurement.2019.107374 MEASUR 107374
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
7 October 2019 16 November 2019 6 December 2019
Please cite this article as: Z. Zhang, Y. Zhong, J. Xiang, Y. Jiang, Phase correction improve d multiple signal classification for impact source localization under varying temperature conditions, Measurement (2019), doi: https:// doi.org/10.1016/j.measurement.2019.107374
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` Phase correction improved multiple signal classification for impact source localization under varying temperature conditions Zhenghao Zhang, Yongteng Zhong * , Jiawei Xiang and Yongying Jiang College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou 325035, P.R.China
(*Corresponding author. E-mail address: [email protected]) Abstract: Lamb waves are commonly used to locate low velocity impacts in composite structures. However, it is readily influenced by varying temperature conditions that the phase drift of uniform linear array signals will quite difference between steering vector and actual signal subspace. Therefore, the traditional multiple signal classification (MUSIC) method might fail to localize the impact sources. Consider the influence of varying temperatures on Lamb waves, a phase correction improved multiple signal classification (PC-MUSIC) method is proposed. The phase errors are calculated by comparing the varying temperature conditions with the normal work condition to obtain a correction matrix, which is employed to correct the steering vector. The impact source location represented by the angle and distance in polar coordinates is finally estimated by scanning the whole monitoring area using the corrected steering vector. The experiment investigations show that the present method has higher location estimation accuracy in composite structures than traditional MUSIC method for the varying temperature conditions. Keywords: Phase correction; Multiple Signal Classification; Impact source localization; Composite structures; Temperature conditions
1.
Introduction
Since Lamb waves can scan cover large areas on plate-like structures and have quite sensitivity to the impact, its application has been increased in structural health monitoring (SHM) of composite structures [1-5]. Some SHM systems based on Lamb waves are constructed to monitor the composite structural condition [6, 7]. Much impact source localization methods based on Lamb wave have been proposed to locate impact source of composite structures [8-11]. However, Lamb waves are extremely sensitive to the change of actual environment, which leads to current systems and methods failed to locate impact source [12-14]. Much research has concentrated on the temperature effects on composite structures and SHM system. Asadnia et al. reported the characterization of piezoelectric transducers (PZT) at high temperature (up to 390 ℃) and high pressure (up to 105 kPa) conditions, and founded that there was fluctuations on resonant frequency with increasing temperature [15]. Ha et al. presented the role of the adhesive layer on piezoelectric transducers-induced Lamb waves propagation in structures exposed to elevated temperatures [16]. Wu et al. established a thermal-vibration test system to carry out experimental research on the thermal modal of high-temperature-resistant composite wing structure [17]. Salamone et al. dealt with the health monitoring of fiber-reinforced composite panels using ultrasonic guided waves and flexible piezocomposite transducer patches in a changing temperature environment (from -40 ℃ to +60 ℃) [18]. Putkis et al. investigated the effect of temperature variation on guided wave propagation in anisotropic composite materials and presented a model to predict anisotropic velocity changes under varying thermal conditions [19]. Konstantinidis et al. researched the effect of temperature environment on the stability of SHM system over a long term [20]. On this basis, much temperature compensation methods based Lamb waves are developed for promoting the accuracy of impact source localization. Fendzi et al. presented a temperature compensation method used Hilbert transform and least square algorithm to reconstruct sensor signals at any arbitrary temperature [21]. Liu et al. proposed a baseline signal reconstruction method which used Hilbert transform and orthogonal matching pursuit to compensate the phase and amplitude of baseline signal [22]. Qiu et al. have done plenty of works on SHM of composite structures under time varying environment. An online updating Gaussian Mixture Model-based damage evaluation method was proposed to improve damage evaluation reliability under time varying conditions; a temperature compensation method based on adaptive filter adaptive liner neural network was presented to compensate the amplitude-change and phase-shift due to temperature [23, 24]. Wang et al. developed a baseline-free damage diagnosis technique based on time reversal theory to
reduce the influence of environmental and operational variability on damage scattering signals [25]. The authors developed uniform linear array-based MUSIC algorithm for impact monitoring. Even it has been successfully verified on complex composite structure, temperature effects were not considered. The research of MUSIC algorithm under varying temperature conditions is still lack. To deal with this problem, an improved MUSIC method based on phase correction is presented for impact source localization under temperature fields. In this method, the phase errors matrix is constructed by comparing normal temperature signals and other temperature signals, and further employed to correct the steering vector. Finally, the improved PC-MUSIC method could locate the impact source more accurately at any temperature. This paper is structured as follows: the experiment of temperature fields’ effect on sensor array signals analysis is given in section 2. In section 3, a PC-MUSIC method is presented for impact source localization of composite structures. The experimental investigations of impact source localization methods under temperature fields are performed in section 4. Finally, conclusion is given in section 5.
2. Experimental setups and sensor signals analysis 2.1 Experiment setups A series of experiments are carried out to investigate the temperature effect on uniform linear array (ULA) response signals. The thermostat and electric furnace are applied to simulate different temperature fields of composite structures in experiments. The experiment of thermostat setup is shown in Fig. 1 (a), including a thermostat, an epoxy laminate plate and an integrated structural health monitoring scanning system (ISHMS). The accuracy of thermostat is 1 ℃. The epoxy laminate plate is placed on shelf in the thermostat, and the wires through the hole of rubber plug to connect sensors and ISHMS as shown in Fig. 1 (b). The dimension of epoxy laminate plate is 40 cm 40 cm 0.2 cm and the ply sequence is [02/904/02]S, the thickness of each ply is 0.125 mm. The excitation sensor and sensing sensor array is controlled by ISHMS to obtain the response signals under temperature fields. A ULA bonded on the epoxy laminate plate surface consists of 6 PZTs, which are labeled as PZT1, PZT2,…, PZT6 respectively from the left to the right and the distance between adjacent PZT center is 1 cm. Another single PZT upon the array is labeled as the excitation sensor PZT0 to simulate the impact source, and the distance between PZT0 and PZT1 is 20 cm, as shown in Fig. 1 (c). Thermostat
Rubber plug (b) The connection
20 cm mmmmm mm
Excitation sensor
ISHMS Foam tape (a) Experimental setups
ULA
(c) The sensor arrangement
Fig. 1. Experimental setups of consistent temperature field
The experiment of inconsistent temperature field setup is shown in Fig. 2, including an ISHMS, an electric furnace, an
epoxy laminate plate, an infrared thermal imager system. The temperature range which can be observed by infrared thermal imager is from -40 ℃ to +12000 ℃, and the measurement accuracy is 1%. The resolution of the thermal imaging map is 680 480, and the thermal sensitivity of infrared thermal imager is 30 mk. The dimension of epoxy laminate plate is 50 cm 50 cm 0.2 cm. The electric furnace with 300 W power is used as the local heat source to simulate the inconstant temperature fields on the plate, and the heat the center of plate, and the heating zone is the center of the epoxy laminate plate. The plate is fixed with aluminum profile frame, and the ULA bonded on the plate consist of 7 PZTs. The sensor arrangements and systems parameters in ISHMS are identical with the frontal experiment, except the distance between PZT0 and PZT1 is 25 cm. Infrared thermal imager system
ISHMS
Electric furnace Fig. 2. Experimental setups of inconsistent temperature field
2.2
The change of signals induced by different temperature fields 1.5 Direct waves
1
1 Amplitude/V
Amplitude/V
0.5 0
0
-0.5
-0.5
-1
-1 -1.5
0.5
-1.5 0
1
2 3 Time/（ 10 4 s） (a) The signals of PZT1
4
5
1.4
1.6
1.8
2
2.2
2.4
2.6
Time/（ 10 4 s） (b) The direct wave changes
Fig. 3. The PZT1 signal changes due to consistent temperature field from -30 ℃ to 80 ℃
To study signals change induced by temperature, array response signals under different consistent temperature fields are compared. The environment temperature is set to be -30 ℃, -20 ℃,…, +80 ℃, and all experimental equipments including the plate, sensors, adhesive layer and wires are in normal operation under temperature fields. In parameter setting of ISHMS, the sampling rate is set to be 10 MHz and the excitation frequency is 50 kHz, and the sampling length is 5000 including 500 pre-trigger samples. The sensor array signals are obtained under normal environment temperature with 25 ℃. And then the ISHM acquires ULA signals when the time of thermostat has been kept for 10 minutes in a certain temperature to make sure temperatures of whole plate are consistent. The signals of PZT1 at different temperatures in the range of -30 ℃ to 80 ℃ are illustrated in Fig. 3 (a), and the direct waves of signals are shown in Fig. 3 (b). It can be seen that the changes of amplitude and phase in response signals are accompanied by the temperature
change. The response signal phase contains the position information of the impact signal, thus it is great important to locate the impact source position. The variation of peak point phases of direct waves with environment temperature is showed in Fig. 4. The times of peak point in direct waves of different sensors drift background congruously from -30 ℃ to 25 ℃. The times of peak point in PZT1 direct waves have a change suddenly from 25 ℃ to 30 ℃. And other PZT signals also have the sudden change in different short temperature range. Furthermore, the variation rules of different sensors’ direct wave phase with temperature change are almost similar. 2250 2200
Time/(10^-7)
2150 2100 2050 2000 1950 -30 -20 -10
0
10 20 30 40 Temperature/ ℃
50
60
70
80
Fig. 4. The phase changes in different signals under constant temperature field
(a) Normal temperature
(c) Highest temperature of 60.65 ℃
(b) Highest temperature of 39.8 ℃
(d) Highest temperature of 78.67 ℃
Fig. 5. The thermal picture of plate under inconsistent temperature fields
In order to simulate the inconstant temperature environment, the electric furnace is used to produce inconsistent temperature field on the whole epoxy laminate plate. The highest temperature of plate is measured by the infrared
thermal imager throughout the heating process to ensure the safety of plate. The normal temperature of plate is 21.87 ℃ and minimum temperature of ground is 19.31 ℃ as shown in Fig. 5 (a). The infrared imaging pictures of plate under varying temperature field are shown in Fig. 5 (a), (b) and (c). The highest temperature is from 21.87 ℃ to 78.67 ℃ and the minimum temperature almost unchanged. The signals of PZT1 at temperatures fields illustrated in Fig. 6 (a), and the direct waves of signals are shown in Fig. 6 (b). The variation of peak point phases of direct waves with temperature in inconstant temperature experiment is shown in Fig. 7. It can be seen that the phase of direct wave in different sensor signals have a good linear relationship versus temperature. All peak point phase of different sensors drift background at an identical rate with environmental temperature change. 2
Direct wave
1.5 1 Amplitude/V
Amplitude/V
1
0
0.5 0 -0.5 -1
-1
-1.5 -2
0
1
2
4
3
1.8
5
2
2.2
2.4
2.6
2.8
3
Time/（ 10 4 s）
Time/（ 10 4 s） (a) The signals of PZT1
(b) The direct wave changes
Fig. 6. The PZT1 signal changes due to inconsistent temperature field from 22 ℃ to 79 ℃ 2600
Time/(10^-7)
2550
2500
2450
2400 0 2350 20
30
40
50
60
70
80
Highest temperature/ ℃ Fig. 7. The phase changes in different signals under inconsistent temperature field
3.
PC-MUSIC method Because of the high accuracy and excellent imaging resolution, the application of MUSIC algorithm has been
increased in impact source localization for composite structures [26-29]. Suppose the wavelength of impact signal is , and an interval between adjacent sensors in piezoelectric sensor array is d, which less than or equal to / 2 . The number of PZTs in ULA is M , and the distance between impact source and response sensor PZT1 is r as shown in Fig. 8.
Impact
r
ri d
0
L
PZT1
PZTi
PZTM
Fig. 8. MUSIC algorithm used uniform linear array
The sensors in ULA are labeled as PZT1, PZT2,…, PZTM from the left to the right, and response signal of PZT1 is expressed as
x1 (t ) u (t ) exp( j 0 t jkr ) where u (t ) denotes the impact signal,
(1)
0 means the center frequency of impact signal, k 0 / c , c is the Lamb
wave velocity. The distance between impact source and PZTi is calculated as
ri r 2 d 2 (i 1) 2 2rd (i 1) cos
(2)
The impact signal arriving time difference between PZTi and PZT1 is defined as
i (r ri ) / c
(3)
Then response signal of PZTi in ULA can be denoted as
xi (t )
r x1 (t ) exp( j0 i ) ni (t ) , i 1,2, , M ri
(4)
where ni (t ) express the background noise. The steering vector is denoted as
a i ( r , )
r exp( j 0 i ) ri
(5)
For the whole uniform linear array, the response signals can be presented as
X(t ) A(r , ) x1 (t ) N(t )
(6)
where
X(t ) [x1 (t ), x 2 (t ), , x M (t )]T
A(r , ) [a1 (r , ), a 2 (r , ), , a M (r , )]T
N(t ) [n1 (t ), n 2 (t ), , n M (t )]T
The current 2D-MUSIC algorithm in ref. [25] is used to locate the distance and angle under normal environment. However, the phase shifts in response signals of ULA can lead to the steering vector unequal to actual signal subspace. Therefore the steering vector corresponding impact source position is hardly orthogonal to noise subspace, and the
accuracy of impact source position in spatial spectrum decreased. Thus the steering vector after corrected is used to deal with this problem. Suppose there are phase errors in steering vector of sensor array signals, and phase errors matrix can be denoted as
Γ( ) diag[1 ( ), 2 ( ), , M ( )]
(7)
The phase error in response signal of PZTi is
i ( ) exp[ j 0 (ˆi i )] exp( j 0 i ), i 1,2, , M
(8)
where ˆi is the actual time delay of impact signal arriving time difference between PZTi and PZT1, and i is the signal phase error of PZTi. The array steering vector after corrected is performed as
ˆ (r , ) Γ( ) A(r , ) A
(9)
By substituting Eq. (9) into Eq. (6), the actual impact source position can be estimated. The covariance matrix of array signal vector is
ˆ 1 XX H U Σ U H U Σ U H R S S S N N N N
(10)
where the superscript H denotes the conjugate transpose , U S denotes the signal subspace spanned by the eigenvector matrix corresponding to the largest eigenvalue, U N means the noise subspace spanned by the eigenvector matrix corresponding to those small eigenvalues, N is the sampling length. The spatial spectrum can be calculated as PMUSIC (r , )
1 H ˆ H ˆ A (r , )U N U N A (r , )
(11)
where the coordinate of peak point in spatial spectrum is the impact source position. The block diagram of the proposed PC-MUSIC method for locating impact source under temperature field is shown in Fig. 9. Array signals under normal temperature field
Array signals under different temperature fields
Response signals
Phase error matrix
Covariance matrix
Noise space
Corrected steering vector
Spatial spectrum
Impact source position Fig. 9. The block diagram of the present PC-MUSIC method
4. Impact source localization results and discussion 4.1 Experimental investigation under consistent temperature fields
From the Fig. 3 (b), it can be conducted that there is a holistic change of phase and amplitude in 5 peaks of direct wave, but not a mutation in some sampling points. The amplitude change has no effect on the phase change calculation, thus phase correction is adopted to improve accuracy of impact source localization. The highest peak’s phase difference between direct waves under different temperatures is used to represent the phase delay of response signals. The Table 1 shows the times of peak point in direct waves, and the time differences are calculated to structure the phase error matrix which is used to correct the steering vector. The impact source is estimated by using the steering vector corrected. Table 1. The direct wave’s highest peaks times of different sensors in constant temperature experiment Time/10-7s Temperature
PZT1
PZT2
PZT3
PZT4
PZT5
PZT6
25
2128
2115
2125
2135
2164
2190
-30
2032
2021
2027
2038
2061
2083
-20
2053
2040
2047
2058
2080
2106
-10
2072
2060
2066
2079
2102
2128
0
2091
2082
2088
2099
2121
2148
10
2112
2102
2107
2119
2146
2169
20
2118
2107
2114
2125
2154
2178
30
2137
2127
2133
2145
2172
2006
40
2157
2146
2154
2162
2196
2027
50
2177
2166
2174
1991
2218
2049
60
2015
2195
2201
2016
2247
2079
70
2051
2235
2046
2053
2088
2122
80
2082
2067
2075
2082
2117
2154
/℃
(21.8 cm, 92°)
0.8
1 Spatial spectrum
Spatial spectrum
1 (38.0 cm, 92°)
0.6 0.4 0.2 0 50 100 Angle/( o)
20 150
0
10
(a) 2D-MUSIC algorithm
30
40
Distance/cm
50
0.8 0.6 0.4 0.2 0 50 100 Angle/( o)
20 150
0
10
30
40
50
Distance/cm
(b) PC-MUSIC method
Fig. 10. The spatial spectrums of two methods under temperature field with 70 ℃
The comparison between 2D-MUSIC algorithm results and present method results in consistent temperature experiment is shown in this section. The excitation sensor position in polar coordinate is (20.0 cm, 90°). The monitoring area is of a distance from 0 to 50 cm and the direction from 0° to 180° with a step of 0.1 cm distance and 1° direction respectively. Fig. 10 (a) shows the spatial spectrum of 2D-MUSIC algorithm under consistent temperature field with 70 ℃, where the distance error is 18.0 cm and the angle error is 2°. Using the present PC-MUSIC method, the spatial spectrum obtained is shown as Fig. 10 (b), where the distance error is 1.8 cm and the angle error is 2°. The whole localization results using 2D-MUSIC algorithm under different consistent temperature fields are compared with that of PC-MUSIC method, listed in Table 2. In 2D-MUSIC algorithm results, the maximum error of distance estimation is 18.0 cm and the maximum error of angle estimation is 2°. When impact source localization used PC-MUSIC method, the
localization distance error is less than 3 cm and the localization angle error is less than 2°. Table 2. The localization results of constant temperature field experiment Parameters
2D-MUSIC results
2D-MUSIC errors
rˆ1 /cm
E r1 /cm
ˆ1 / °
E 1 / °
PC-MUSIC results
PC-MUSIC errors
rˆ2 /cm
ˆ2 / °
E r /cm 2
E 2 / °
Temperature /℃
4.2
25
20.2
92
0.2
2
20.2
92
0.2
2
-30
28.3
92
8.3
2
21.3
91
1.3
1
-20
33.0
91
13.0
1
22.7
92
2.7
2
-10
27.3
92
7.3
2
21.1
92
1.1
2
0
17.2
91
2.8
1
18.5
91
1.5
1
10
23.6
91
3.6
1
22.6
91
2.6
1
20
22.9
94
2.9
4
20.8
92
0.8
2
30
17.8
92
2.2
2
18.7
91
1.3
1
40
19.3
92
0.7
2
19.5
92
0.5
2
50
26.5
92
6.5
2
20.7
91
0.7
1
60
27.3
92
7.3
2
18.3
92
1.7
2
70
38.0
92
18.0
2
21.8
92
1.8
2
80
26.6
92
6.6
2
20.4
91
0.4
1
Experimental investigation under inconsistent temperature fields
The Table 3 shows the peak point times of direct waves in inconsistent temperature experiment. The phase correction matrix is calculated to correct the steering vector used the peak point phase change, further the impact position is estimated. The excitation sensor polar coordinate is (25.0 cm, 90°). Fig. 11(a) shows the spatial spectrum of 2D-MUSIC algorithm in inconsistent temperature experiment with 60 ℃ highest temperature, where the distance error is 16.4 cm and the angle error is 3°. The spatial spectrum of PC-MUSIC method under same temperature field is shown as Fig. 11 (b), where the distance error is 0.4 cm and the angle error is 1°. Table 4 lists the whole localization results under inconsistent temperature fields of 2D-MUSIC algorithm and PC-MUSIC method. The maximum error of distance estimation in 2D-MUSIC algorithm results is 16.4 cm and the maximum error of angle estimation is 5°. The maximum distance error decreases to be 1.2 cm and the maximum angle error decreases to be 3° by using PC-MUSIC method. The present PC-MUSIC method results have lower localization errors than that obtained from the MUSIC algorithm. Table 3. The direct wave’s highest peaks times of different sensors in inconsistent temperature experiment Time/10-7s PZT1
PZT2
PZT3
PZT4
PZT5
PZT6
PZT7
21.87
2389
2378
2382
2397
2417
2452
2481
30.02
2399
2389
2394
2409
2428
2457
2486
39.80
2411
2398
2406
2420
2441
2469
2501
50.02
2424
2413
2421
2437
2458
2486
2516
60.65
2440
2429
2439
2456
2476
2505
2539
70.01
2455
2448
2458
2475
2497
2527
2561
78.67
2474
2464
2477
2494
2518
2548
2583
Highest temperature/℃ /
1
1
0.8
0.8
Spatial spectrum
Spatial spectrum
(8.6 cm, 93°)
0.6 0.4 0.2 0 50 100 Angle/( o)
30
20 150
10
0
50
40
0.6
(24.6 cm, 91°)
0.4 0.2 0 50 100 Angle/( o)
Distance/cm
(a) 2D-MUSIC algorithm
30
20 150
10
0
40
50
Distance/cm
(b) PC-MUSIC method
Fig. 11. The spatial spectrums of two methods under inconsistent temperature field with 60 ℃ highest temperature Table 4. The localization results of varying temperature field experiment Parameters
2D-MUSIC results
rˆ1 /cm
ˆ1 / °
2D-MUSIC errors
PC-MUSIC results
PC-MUSIC errors
E r1 /cm
rˆ2 /cm
ˆ2 / °
E r 2 /cm
E 2 / °
Highest temperature /℃
5.
E 1 / °
21.87
25.7
93
0.7
3
25.7
93
0.7
3
29.54
21.8
93
3.2
3
24.3
93
0.7
1
39.80
17.2
93
7.8
3
24.4
93
0.6
3
49.86
16.8
92
8.2
2
25.9
93
0.9
3
60.65
8.6
93
16.4
3
24.6
91
0.4
1
70.77
33.6
94
8.6
4
23.8
93
1.2
3
78.67
15.4
95
9.6
5
25.7
93
0.7
3
Conclusions
The varying temperature effects of composite structures on Lamb waves are investigated that the phase of uniform linear array signals would be drifted. The 2D-MUSIC algorithm can not remain the orthogonality of steering vector and noise subspace for the phase drifts. Therefore, the accuracy of 2D-MUSIC algorithm for impact source localization is reduced accordingly. To solve this problem, a PC-MUSIC method is developed to localize the impact source of composite structures. Using the phase of normal work condition as baseline, the phase errors are calculated to generate a correction matrix to correct the steering vector. The impact source localization is finally estimated by 2D-MUSIC algorithm to deal with the corrected steering vector. Experimental investigations show that the present method has higher impact source localization accuracy. It is expected that the method can readability in real-world applications of more complex composite structures.
Acknowledgements The authors are grateful to the support from the National Natural Science Foundation of China (No. U1709208) and the Zhejiang Special Support Program for High-level Personnel Recruitment of China (No. 2018R52034).
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Highlights： PC-MUSIC method is presented for impact localization under temperature conditions Phase errors correction matrix is developed to correct the steering vector Experiment results in two varying temperature fields has higher
accuracy
[30]
Dear Editors: We are submitting the manuscript entitled “Phase correction improved multiple signal classification for impact source localization under varying temperature conditions” to the journal “Measurement”. All authors have seen the manuscript and approved to submit to your journal. The authors claim that the content of this paper has not been published or is under consideration for publication elsewhere. And if accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder.
Sincerely yours Zhenghao Zhang,Yongteng Zhong, Jiawei Xiang and Yongying Jiang
Yongteng Zhong College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou, 325035, P.R.China. E-mail: [email protected] [31]
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
[32]
S0263-2241(19)31238-2 https://doi.org/10.1016/j.measurement.2019.107374 MEASUR 107374
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
7 October 2019 16 November 2019 6 December 2019
Please cite this article as: Z. Zhang, Y. Zhong, J. Xiang, Y. Jiang, Phase correction improve d multiple signal classification for impact source localization under varying temperature conditions, Measurement (2019), doi: https:// doi.org/10.1016/j.measurement.2019.107374
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` Phase correction improved multiple signal classification for impact source localization under varying temperature conditions Zhenghao Zhang, Yongteng Zhong * , Jiawei Xiang and Yongying Jiang College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou 325035, P.R.China
(*Corresponding author. E-mail address: [email protected]) Abstract: Lamb waves are commonly used to locate low velocity impacts in composite structures. However, it is readily influenced by varying temperature conditions that the phase drift of uniform linear array signals will quite difference between steering vector and actual signal subspace. Therefore, the traditional multiple signal classification (MUSIC) method might fail to localize the impact sources. Consider the influence of varying temperatures on Lamb waves, a phase correction improved multiple signal classification (PC-MUSIC) method is proposed. The phase errors are calculated by comparing the varying temperature conditions with the normal work condition to obtain a correction matrix, which is employed to correct the steering vector. The impact source location represented by the angle and distance in polar coordinates is finally estimated by scanning the whole monitoring area using the corrected steering vector. The experiment investigations show that the present method has higher location estimation accuracy in composite structures than traditional MUSIC method for the varying temperature conditions. Keywords: Phase correction; Multiple Signal Classification; Impact source localization; Composite structures; Temperature conditions
1.
Introduction
Since Lamb waves can scan cover large areas on plate-like structures and have quite sensitivity to the impact, its application has been increased in structural health monitoring (SHM) of composite structures [1-5]. Some SHM systems based on Lamb waves are constructed to monitor the composite structural condition [6, 7]. Much impact source localization methods based on Lamb wave have been proposed to locate impact source of composite structures [8-11]. However, Lamb waves are extremely sensitive to the change of actual environment, which leads to current systems and methods failed to locate impact source [12-14]. Much research has concentrated on the temperature effects on composite structures and SHM system. Asadnia et al. reported the characterization of piezoelectric transducers (PZT) at high temperature (up to 390 ℃) and high pressure (up to 105 kPa) conditions, and founded that there was fluctuations on resonant frequency with increasing temperature [15]. Ha et al. presented the role of the adhesive layer on piezoelectric transducers-induced Lamb waves propagation in structures exposed to elevated temperatures [16]. Wu et al. established a thermal-vibration test system to carry out experimental research on the thermal modal of high-temperature-resistant composite wing structure [17]. Salamone et al. dealt with the health monitoring of fiber-reinforced composite panels using ultrasonic guided waves and flexible piezocomposite transducer patches in a changing temperature environment (from -40 ℃ to +60 ℃) [18]. Putkis et al. investigated the effect of temperature variation on guided wave propagation in anisotropic composite materials and presented a model to predict anisotropic velocity changes under varying thermal conditions [19]. Konstantinidis et al. researched the effect of temperature environment on the stability of SHM system over a long term [20]. On this basis, much temperature compensation methods based Lamb waves are developed for promoting the accuracy of impact source localization. Fendzi et al. presented a temperature compensation method used Hilbert transform and least square algorithm to reconstruct sensor signals at any arbitrary temperature [21]. Liu et al. proposed a baseline signal reconstruction method which used Hilbert transform and orthogonal matching pursuit to compensate the phase and amplitude of baseline signal [22]. Qiu et al. have done plenty of works on SHM of composite structures under time varying environment. An online updating Gaussian Mixture Model-based damage evaluation method was proposed to improve damage evaluation reliability under time varying conditions; a temperature compensation method based on adaptive filter adaptive liner neural network was presented to compensate the amplitude-change and phase-shift due to temperature [23, 24]. Wang et al. developed a baseline-free damage diagnosis technique based on time reversal theory to
reduce the influence of environmental and operational variability on damage scattering signals [25]. The authors developed uniform linear array-based MUSIC algorithm for impact monitoring. Even it has been successfully verified on complex composite structure, temperature effects were not considered. The research of MUSIC algorithm under varying temperature conditions is still lack. To deal with this problem, an improved MUSIC method based on phase correction is presented for impact source localization under temperature fields. In this method, the phase errors matrix is constructed by comparing normal temperature signals and other temperature signals, and further employed to correct the steering vector. Finally, the improved PC-MUSIC method could locate the impact source more accurately at any temperature. This paper is structured as follows: the experiment of temperature fields’ effect on sensor array signals analysis is given in section 2. In section 3, a PC-MUSIC method is presented for impact source localization of composite structures. The experimental investigations of impact source localization methods under temperature fields are performed in section 4. Finally, conclusion is given in section 5.
2. Experimental setups and sensor signals analysis 2.1 Experiment setups A series of experiments are carried out to investigate the temperature effect on uniform linear array (ULA) response signals. The thermostat and electric furnace are applied to simulate different temperature fields of composite structures in experiments. The experiment of thermostat setup is shown in Fig. 1 (a), including a thermostat, an epoxy laminate plate and an integrated structural health monitoring scanning system (ISHMS). The accuracy of thermostat is 1 ℃. The epoxy laminate plate is placed on shelf in the thermostat, and the wires through the hole of rubber plug to connect sensors and ISHMS as shown in Fig. 1 (b). The dimension of epoxy laminate plate is 40 cm 40 cm 0.2 cm and the ply sequence is [02/904/02]S, the thickness of each ply is 0.125 mm. The excitation sensor and sensing sensor array is controlled by ISHMS to obtain the response signals under temperature fields. A ULA bonded on the epoxy laminate plate surface consists of 6 PZTs, which are labeled as PZT1, PZT2,…, PZT6 respectively from the left to the right and the distance between adjacent PZT center is 1 cm. Another single PZT upon the array is labeled as the excitation sensor PZT0 to simulate the impact source, and the distance between PZT0 and PZT1 is 20 cm, as shown in Fig. 1 (c). Thermostat
Rubber plug (b) The connection
20 cm mmmmm mm
Excitation sensor
ISHMS Foam tape (a) Experimental setups
ULA
(c) The sensor arrangement
Fig. 1. Experimental setups of consistent temperature field
The experiment of inconsistent temperature field setup is shown in Fig. 2, including an ISHMS, an electric furnace, an
epoxy laminate plate, an infrared thermal imager system. The temperature range which can be observed by infrared thermal imager is from -40 ℃ to +12000 ℃, and the measurement accuracy is 1%. The resolution of the thermal imaging map is 680 480, and the thermal sensitivity of infrared thermal imager is 30 mk. The dimension of epoxy laminate plate is 50 cm 50 cm 0.2 cm. The electric furnace with 300 W power is used as the local heat source to simulate the inconstant temperature fields on the plate, and the heat the center of plate, and the heating zone is the center of the epoxy laminate plate. The plate is fixed with aluminum profile frame, and the ULA bonded on the plate consist of 7 PZTs. The sensor arrangements and systems parameters in ISHMS are identical with the frontal experiment, except the distance between PZT0 and PZT1 is 25 cm. Infrared thermal imager system
ISHMS
Electric furnace Fig. 2. Experimental setups of inconsistent temperature field
2.2
The change of signals induced by different temperature fields 1.5 Direct waves
1
1 Amplitude/V
Amplitude/V
0.5 0
0
-0.5
-0.5
-1
-1 -1.5
0.5
-1.5 0
1
2 3 Time/（ 10 4 s） (a) The signals of PZT1
4
5
1.4
1.6
1.8
2
2.2
2.4
2.6
Time/（ 10 4 s） (b) The direct wave changes
Fig. 3. The PZT1 signal changes due to consistent temperature field from -30 ℃ to 80 ℃
To study signals change induced by temperature, array response signals under different consistent temperature fields are compared. The environment temperature is set to be -30 ℃, -20 ℃,…, +80 ℃, and all experimental equipments including the plate, sensors, adhesive layer and wires are in normal operation under temperature fields. In parameter setting of ISHMS, the sampling rate is set to be 10 MHz and the excitation frequency is 50 kHz, and the sampling length is 5000 including 500 pre-trigger samples. The sensor array signals are obtained under normal environment temperature with 25 ℃. And then the ISHM acquires ULA signals when the time of thermostat has been kept for 10 minutes in a certain temperature to make sure temperatures of whole plate are consistent. The signals of PZT1 at different temperatures in the range of -30 ℃ to 80 ℃ are illustrated in Fig. 3 (a), and the direct waves of signals are shown in Fig. 3 (b). It can be seen that the changes of amplitude and phase in response signals are accompanied by the temperature
change. The response signal phase contains the position information of the impact signal, thus it is great important to locate the impact source position. The variation of peak point phases of direct waves with environment temperature is showed in Fig. 4. The times of peak point in direct waves of different sensors drift background congruously from -30 ℃ to 25 ℃. The times of peak point in PZT1 direct waves have a change suddenly from 25 ℃ to 30 ℃. And other PZT signals also have the sudden change in different short temperature range. Furthermore, the variation rules of different sensors’ direct wave phase with temperature change are almost similar. 2250 2200
Time/(10^-7)
2150 2100 2050 2000 1950 -30 -20 -10
0
10 20 30 40 Temperature/ ℃
50
60
70
80
Fig. 4. The phase changes in different signals under constant temperature field
(a) Normal temperature
(c) Highest temperature of 60.65 ℃
(b) Highest temperature of 39.8 ℃
(d) Highest temperature of 78.67 ℃
Fig. 5. The thermal picture of plate under inconsistent temperature fields
In order to simulate the inconstant temperature environment, the electric furnace is used to produce inconsistent temperature field on the whole epoxy laminate plate. The highest temperature of plate is measured by the infrared
thermal imager throughout the heating process to ensure the safety of plate. The normal temperature of plate is 21.87 ℃ and minimum temperature of ground is 19.31 ℃ as shown in Fig. 5 (a). The infrared imaging pictures of plate under varying temperature field are shown in Fig. 5 (a), (b) and (c). The highest temperature is from 21.87 ℃ to 78.67 ℃ and the minimum temperature almost unchanged. The signals of PZT1 at temperatures fields illustrated in Fig. 6 (a), and the direct waves of signals are shown in Fig. 6 (b). The variation of peak point phases of direct waves with temperature in inconstant temperature experiment is shown in Fig. 7. It can be seen that the phase of direct wave in different sensor signals have a good linear relationship versus temperature. All peak point phase of different sensors drift background at an identical rate with environmental temperature change. 2
Direct wave
1.5 1 Amplitude/V
Amplitude/V
1
0
0.5 0 -0.5 -1
-1
-1.5 -2
0
1
2
4
3
1.8
5
2
2.2
2.4
2.6
2.8
3
Time/（ 10 4 s）
Time/（ 10 4 s） (a) The signals of PZT1
(b) The direct wave changes
Fig. 6. The PZT1 signal changes due to inconsistent temperature field from 22 ℃ to 79 ℃ 2600
Time/(10^-7)
2550
2500
2450
2400 0 2350 20
30
40
50
60
70
80
Highest temperature/ ℃ Fig. 7. The phase changes in different signals under inconsistent temperature field
3.
PC-MUSIC method Because of the high accuracy and excellent imaging resolution, the application of MUSIC algorithm has been
increased in impact source localization for composite structures [26-29]. Suppose the wavelength of impact signal is , and an interval between adjacent sensors in piezoelectric sensor array is d, which less than or equal to / 2 . The number of PZTs in ULA is M , and the distance between impact source and response sensor PZT1 is r as shown in Fig. 8.
Impact
r
ri d
0
L
PZT1
PZTi
PZTM
Fig. 8. MUSIC algorithm used uniform linear array
The sensors in ULA are labeled as PZT1, PZT2,…, PZTM from the left to the right, and response signal of PZT1 is expressed as
x1 (t ) u (t ) exp( j 0 t jkr ) where u (t ) denotes the impact signal,
(1)
0 means the center frequency of impact signal, k 0 / c , c is the Lamb
wave velocity. The distance between impact source and PZTi is calculated as
ri r 2 d 2 (i 1) 2 2rd (i 1) cos
(2)
The impact signal arriving time difference between PZTi and PZT1 is defined as
i (r ri ) / c
(3)
Then response signal of PZTi in ULA can be denoted as
xi (t )
r x1 (t ) exp( j0 i ) ni (t ) , i 1,2, , M ri
(4)
where ni (t ) express the background noise. The steering vector is denoted as
a i ( r , )
r exp( j 0 i ) ri
(5)
For the whole uniform linear array, the response signals can be presented as
X(t ) A(r , ) x1 (t ) N(t )
(6)
where
X(t ) [x1 (t ), x 2 (t ), , x M (t )]T
A(r , ) [a1 (r , ), a 2 (r , ), , a M (r , )]T
N(t ) [n1 (t ), n 2 (t ), , n M (t )]T
The current 2D-MUSIC algorithm in ref. [25] is used to locate the distance and angle under normal environment. However, the phase shifts in response signals of ULA can lead to the steering vector unequal to actual signal subspace. Therefore the steering vector corresponding impact source position is hardly orthogonal to noise subspace, and the
accuracy of impact source position in spatial spectrum decreased. Thus the steering vector after corrected is used to deal with this problem. Suppose there are phase errors in steering vector of sensor array signals, and phase errors matrix can be denoted as
Γ( ) diag[1 ( ), 2 ( ), , M ( )]
(7)
The phase error in response signal of PZTi is
i ( ) exp[ j 0 (ˆi i )] exp( j 0 i ), i 1,2, , M
(8)
where ˆi is the actual time delay of impact signal arriving time difference between PZTi and PZT1, and i is the signal phase error of PZTi. The array steering vector after corrected is performed as
ˆ (r , ) Γ( ) A(r , ) A
(9)
By substituting Eq. (9) into Eq. (6), the actual impact source position can be estimated. The covariance matrix of array signal vector is
ˆ 1 XX H U Σ U H U Σ U H R S S S N N N N
(10)
where the superscript H denotes the conjugate transpose , U S denotes the signal subspace spanned by the eigenvector matrix corresponding to the largest eigenvalue, U N means the noise subspace spanned by the eigenvector matrix corresponding to those small eigenvalues, N is the sampling length. The spatial spectrum can be calculated as PMUSIC (r , )
1 H ˆ H ˆ A (r , )U N U N A (r , )
(11)
where the coordinate of peak point in spatial spectrum is the impact source position. The block diagram of the proposed PC-MUSIC method for locating impact source under temperature field is shown in Fig. 9. Array signals under normal temperature field
Array signals under different temperature fields
Response signals
Phase error matrix
Covariance matrix
Noise space
Corrected steering vector
Spatial spectrum
Impact source position Fig. 9. The block diagram of the present PC-MUSIC method
4. Impact source localization results and discussion 4.1 Experimental investigation under consistent temperature fields
From the Fig. 3 (b), it can be conducted that there is a holistic change of phase and amplitude in 5 peaks of direct wave, but not a mutation in some sampling points. The amplitude change has no effect on the phase change calculation, thus phase correction is adopted to improve accuracy of impact source localization. The highest peak’s phase difference between direct waves under different temperatures is used to represent the phase delay of response signals. The Table 1 shows the times of peak point in direct waves, and the time differences are calculated to structure the phase error matrix which is used to correct the steering vector. The impact source is estimated by using the steering vector corrected. Table 1. The direct wave’s highest peaks times of different sensors in constant temperature experiment Time/10-7s Temperature
PZT1
PZT2
PZT3
PZT4
PZT5
PZT6
25
2128
2115
2125
2135
2164
2190
-30
2032
2021
2027
2038
2061
2083
-20
2053
2040
2047
2058
2080
2106
-10
2072
2060
2066
2079
2102
2128
0
2091
2082
2088
2099
2121
2148
10
2112
2102
2107
2119
2146
2169
20
2118
2107
2114
2125
2154
2178
30
2137
2127
2133
2145
2172
2006
40
2157
2146
2154
2162
2196
2027
50
2177
2166
2174
1991
2218
2049
60
2015
2195
2201
2016
2247
2079
70
2051
2235
2046
2053
2088
2122
80
2082
2067
2075
2082
2117
2154
/℃
(21.8 cm, 92°)
0.8
1 Spatial spectrum
Spatial spectrum
1 (38.0 cm, 92°)
0.6 0.4 0.2 0 50 100 Angle/( o)
20 150
0
10
(a) 2D-MUSIC algorithm
30
40
Distance/cm
50
0.8 0.6 0.4 0.2 0 50 100 Angle/( o)
20 150
0
10
30
40
50
Distance/cm
(b) PC-MUSIC method
Fig. 10. The spatial spectrums of two methods under temperature field with 70 ℃
The comparison between 2D-MUSIC algorithm results and present method results in consistent temperature experiment is shown in this section. The excitation sensor position in polar coordinate is (20.0 cm, 90°). The monitoring area is of a distance from 0 to 50 cm and the direction from 0° to 180° with a step of 0.1 cm distance and 1° direction respectively. Fig. 10 (a) shows the spatial spectrum of 2D-MUSIC algorithm under consistent temperature field with 70 ℃, where the distance error is 18.0 cm and the angle error is 2°. Using the present PC-MUSIC method, the spatial spectrum obtained is shown as Fig. 10 (b), where the distance error is 1.8 cm and the angle error is 2°. The whole localization results using 2D-MUSIC algorithm under different consistent temperature fields are compared with that of PC-MUSIC method, listed in Table 2. In 2D-MUSIC algorithm results, the maximum error of distance estimation is 18.0 cm and the maximum error of angle estimation is 2°. When impact source localization used PC-MUSIC method, the
localization distance error is less than 3 cm and the localization angle error is less than 2°. Table 2. The localization results of constant temperature field experiment Parameters
2D-MUSIC results
2D-MUSIC errors
rˆ1 /cm
E r1 /cm
ˆ1 / °
E 1 / °
PC-MUSIC results
PC-MUSIC errors
rˆ2 /cm
ˆ2 / °
E r /cm 2
E 2 / °
Temperature /℃
4.2
25
20.2
92
0.2
2
20.2
92
0.2
2
-30
28.3
92
8.3
2
21.3
91
1.3
1
-20
33.0
91
13.0
1
22.7
92
2.7
2
-10
27.3
92
7.3
2
21.1
92
1.1
2
0
17.2
91
2.8
1
18.5
91
1.5
1
10
23.6
91
3.6
1
22.6
91
2.6
1
20
22.9
94
2.9
4
20.8
92
0.8
2
30
17.8
92
2.2
2
18.7
91
1.3
1
40
19.3
92
0.7
2
19.5
92
0.5
2
50
26.5
92
6.5
2
20.7
91
0.7
1
60
27.3
92
7.3
2
18.3
92
1.7
2
70
38.0
92
18.0
2
21.8
92
1.8
2
80
26.6
92
6.6
2
20.4
91
0.4
1
Experimental investigation under inconsistent temperature fields
The Table 3 shows the peak point times of direct waves in inconsistent temperature experiment. The phase correction matrix is calculated to correct the steering vector used the peak point phase change, further the impact position is estimated. The excitation sensor polar coordinate is (25.0 cm, 90°). Fig. 11(a) shows the spatial spectrum of 2D-MUSIC algorithm in inconsistent temperature experiment with 60 ℃ highest temperature, where the distance error is 16.4 cm and the angle error is 3°. The spatial spectrum of PC-MUSIC method under same temperature field is shown as Fig. 11 (b), where the distance error is 0.4 cm and the angle error is 1°. Table 4 lists the whole localization results under inconsistent temperature fields of 2D-MUSIC algorithm and PC-MUSIC method. The maximum error of distance estimation in 2D-MUSIC algorithm results is 16.4 cm and the maximum error of angle estimation is 5°. The maximum distance error decreases to be 1.2 cm and the maximum angle error decreases to be 3° by using PC-MUSIC method. The present PC-MUSIC method results have lower localization errors than that obtained from the MUSIC algorithm. Table 3. The direct wave’s highest peaks times of different sensors in inconsistent temperature experiment Time/10-7s PZT1
PZT2
PZT3
PZT4
PZT5
PZT6
PZT7
21.87
2389
2378
2382
2397
2417
2452
2481
30.02
2399
2389
2394
2409
2428
2457
2486
39.80
2411
2398
2406
2420
2441
2469
2501
50.02
2424
2413
2421
2437
2458
2486
2516
60.65
2440
2429
2439
2456
2476
2505
2539
70.01
2455
2448
2458
2475
2497
2527
2561
78.67
2474
2464
2477
2494
2518
2548
2583
Highest temperature/℃ /
1
1
0.8
0.8
Spatial spectrum
Spatial spectrum
(8.6 cm, 93°)
0.6 0.4 0.2 0 50 100 Angle/( o)
30
20 150
10
0
50
40
0.6
(24.6 cm, 91°)
0.4 0.2 0 50 100 Angle/( o)
Distance/cm
(a) 2D-MUSIC algorithm
30
20 150
10
0
40
50
Distance/cm
(b) PC-MUSIC method
Fig. 11. The spatial spectrums of two methods under inconsistent temperature field with 60 ℃ highest temperature Table 4. The localization results of varying temperature field experiment Parameters
2D-MUSIC results
rˆ1 /cm
ˆ1 / °
2D-MUSIC errors
PC-MUSIC results
PC-MUSIC errors
E r1 /cm
rˆ2 /cm
ˆ2 / °
E r 2 /cm
E 2 / °
Highest temperature /℃
5.
E 1 / °
21.87
25.7
93
0.7
3
25.7
93
0.7
3
29.54
21.8
93
3.2
3
24.3
93
0.7
1
39.80
17.2
93
7.8
3
24.4
93
0.6
3
49.86
16.8
92
8.2
2
25.9
93
0.9
3
60.65
8.6
93
16.4
3
24.6
91
0.4
1
70.77
33.6
94
8.6
4
23.8
93
1.2
3
78.67
15.4
95
9.6
5
25.7
93
0.7
3
Conclusions
The varying temperature effects of composite structures on Lamb waves are investigated that the phase of uniform linear array signals would be drifted. The 2D-MUSIC algorithm can not remain the orthogonality of steering vector and noise subspace for the phase drifts. Therefore, the accuracy of 2D-MUSIC algorithm for impact source localization is reduced accordingly. To solve this problem, a PC-MUSIC method is developed to localize the impact source of composite structures. Using the phase of normal work condition as baseline, the phase errors are calculated to generate a correction matrix to correct the steering vector. The impact source localization is finally estimated by 2D-MUSIC algorithm to deal with the corrected steering vector. Experimental investigations show that the present method has higher impact source localization accuracy. It is expected that the method can readability in real-world applications of more complex composite structures.
Acknowledgements The authors are grateful to the support from the National Natural Science Foundation of China (No. U1709208) and the Zhejiang Special Support Program for High-level Personnel Recruitment of China (No. 2018R52034).
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Highlights： PC-MUSIC method is presented for impact localization under temperature conditions Phase errors correction matrix is developed to correct the steering vector Experiment results in two varying temperature fields has higher
accuracy
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Dear Editors: We are submitting the manuscript entitled “Phase correction improved multiple signal classification for impact source localization under varying temperature conditions” to the journal “Measurement”. All authors have seen the manuscript and approved to submit to your journal. The authors claim that the content of this paper has not been published or is under consideration for publication elsewhere. And if accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder.
Sincerely yours Zhenghao Zhang,Yongteng Zhong, Jiawei Xiang and Yongying Jiang
Yongteng Zhong College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou, 325035, P.R.China. E-mail: [email protected] [31]
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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