Simultaneous excitation system for efficient guided wave structural health monitoring

Mechanical Systems and Signal Processing 95 (2017) 506–523

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Simultaneous excitation system for efficient guided wave structural health monitoring Jiadong Hua a,b, Jennifer E. Michaels b,⇑, Xin Chen b, Jing Lin a a b

Department of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA

a r t i c l e

i n f o

Article history: Received 25 October 2016 Received in revised form 21 March 2017 Accepted 24 March 2017

Keywords: Lamb waves Simultaneous excitation Dispersion-compensated filtering

a b s t r a c t Many structural health monitoring systems utilize guided wave transducer arrays for defect detection and localization. Signals are usually acquired using the ‘‘pitch-catch” method whereby each transducer is excited in turn and the response is received by the remaining transducers. When extensive signal averaging is performed, the data acquisition process can be quite time-consuming, especially for metallic components that require a low repetition rate to allow signals to die out. Such a long data acquisition time is particularly problematic if environmental and operational conditions are changing while data are being acquired. To reduce the total data acquisition time, proposed here is a methodology whereby multiple transmitters are simultaneously triggered, and each transmitter is driven with a unique excitation. The simultaneously transmitted waves are captured by one or more receivers, and their responses are processed by dispersion-compensated filtering to extract the response from each individual transmitter. The excitation sequences are constructed by concatenating a series of chirps whose start and stop frequencies are randomly selected from a specified range. The process is optimized using a Monte-Carlo approach to select sequences with impulse-like autocorrelations and relatively flat crosscorrelations. The efficacy of the proposed methodology is evaluated by several metrics and is experimentally demonstrated with sparse array imaging of simulated damage. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Lamb waves offer potential for structural health monitoring (SHM) because of their capability of propagation over a relatively long distance with low attenuation while maintaining sensitivity to changes in structural characteristics [1,2]. A common strategy for Lamb wave SHM is to use arrays of transducers together with the ‘‘pitch-catch” data acquisition method; i.e., exciting each transducer sequentially and recording responses by the remaining ones either simultaneously or separately [3–5]. However, in both cases the data acquisition process can be time consuming since multiple transmission-reception cycles are needed to loop through all of the transmitters. Additional switching time may also be required depending upon the specific hardware implementation. A further increase in data acquisition time occurs when extensive signal averaging is used to achieve a high signal-to-noise ratio (SNR). For example, the data acquisition time of an eight-element sparse transducer array can be several minutes for a pulse repetition time of 100 ms and 50 averages for each reception. Such a long data recording time can severely limit the efficiency of the Lamb wave SHM system and cause problems for field applications. ⇑ Corresponding author. E-mail address: [email protected] (J.E. Michaels). http://dx.doi.org/10.1016/j.ymssp.2017.03.036 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

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A major adverse effect of a long data acquisition time on Lamb wave SHM is signal mismatch if environmental and operational changes occur during the acquisition process. Even in the laboratory small temperature changes are possible in the span of a few minutes, and it is well-known that these small changes have a significant effect on Lamb wave signals [6,7]. Load changes are also possible, such as in the case of aircraft maneuvers during normal operations, and these changes also affect Lamb wave signals [5,8]. Various methods have been proposed to compensate for such changes, but it has been shown that improved performance is obtained when all signals are acquired under the same environmental conditions, particularly applied load [9]. A straightforward approach that can minimize the data recording time is to excite all transmitters simultaneously, which has been applied to ultrasonic ranging systems [10,11]. The major issue arising from the simultaneous excitations is ultrasonic crosstalk; i.e., signals from different transmitters overlap with each other at reception. The most promising methods to minimize crosstalk are those assigning a unique excitation for each transmitter and applying matched filtering at reception [10–13]. Therefore, the key is to select a series of excitation signals that have good correlation performance; i.e., a sharp autocorrelation and a flat cross-correlation. Signals that have been commonly used include linear and non-linear chirp sequences and pseudo-random binary sequences such as Barker codes, Golay codes, Gold codes, and chaotic sequences [12,13]. In this paper, we apply the simultaneous, coded excitation method to a spatially distributed array of transducers that generate and detect Lamb waves. This problem is similar to the ultrasonic ranging problem in that transducers are bandlimited, but it is significantly different in that the crosstalk is much more severe and dispersion prohibits direct application of matched filtering. Since the Lamb wave sensors are omnidirectional and operate in pitch-catch mode, all transmissions are present in all received signals in approximately equal strengths. Thus, what is considered ‘‘signal” versus ‘‘crosstalk” depends upon which transmitter is being considered during analysis. In contrast, ultrasonic ranging system transducers are directional (albeit with significant beam spread) and each channel of interest uses a transmitter and receiver that are essentially co-located (comparable to pulse-echo mode), resulting in an inherently larger signal of interest compared to the crosstalk coming from the other transmitters. Another issue when applying simultaneous excitations to a guided wave transducer array is that from a practical standpoint there must be dedicated transmitters and receivers. For reception to occur on a transmitting transducer, a T-R (transmit-receive) switch must be used to protect the receive amplifier during the transmission, which effectively limits the length of the excitation to the time prior to the first signal arrival. For typical systems this time could be as short as tens of microseconds, which would essentially prohibit coded excitations of useful lengths (i.e., hundreds of microseconds). Thus, for a system with N transducers, rather than being able to acquire N(N
1)/2 unique signals as is customary with separate excitations, the N transducers must be split into NT transmitters and (N
NT) receivers, yielding a reduced total of NT(N
NT) possible signals. Proposed here is a method of constructing multiple excitation signals by concatenating a number of linear chirps. Each chirp has its start and stop frequencies randomly selected within a certain frequency range, which ensures that the excitation lies within the desired bandwidth. The selection of those start and stop frequencies of chirps is optimized using a Monte-Carlo approach with an objective function defined to maximize the main lobe of the autocorrelations while minimizing the side lobe level of the auto-correlation and the peak values of the cross-correlations. On reception, a dispersion-compensated filtering approach as described in our previous work [14,15] is adopted here as a consequence of the dispersive nature of Lamb waves. Performance is evaluated by comparing signals to those obtained via separate excitations as well as by comparing delay-andsum images constructed by residual signals before and after introduction of simulated damage. This paper is an extension of results previously reported by the authors [16]. A similar approach was described by De Marchi et al. [17] for guided wavefield imaging using a laser vibrometer but with different excitations and processing methods. The rest of this paper is organized as follows. Section 2 introduces the theoretical background of the Lamb wave multitransducer simultaneous excitation system, including a review of dispersion-compensated filtering and details of the excitation sequence construction. Section 3 quantifies the parameter choice during excitation construction by simulations. Section 4 describes two experiments using an 8-element transducer array on an aluminum plate. Section 5 demonstrates and discusses the performance of the proposed method, and Section 6 contains concluding remarks. 2. Theory 2.1. Dispersion-compensated matched filtering Conventional matched filtering refers to the cross-correlation between the measured signal, w(t), and the excitation signal (a.k.a. the template), s(t),

Z Rsw ðtÞ ¼

þ1


1 þ1

Z ¼


1

sðsÞwðt þ sÞds sðs
tÞwðsÞds:

ð1Þ

This process is also referred to as pulse compression in non-dispersive wave propagation scenarios used in radar, sonar, and bulk wave ultrasonics, where the measured signal w(t) can be thought of as the sum of scaled and shifted versions of the

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excitation s(t). However, when the medium is strongly dispersive, as is the case for Lamb wave propagation, the shape of the measured signal no longer matches that of the excitation signal, and thus conventional matched filtering is no longer effective. To address this issue, we apply a dispersion-compensated matched filtering method described in our previous work [14,15]. The measured signal is correlated with dispersed versions of the excitation signal, which change as a function of propagation distance. The dispersed excitation signal is computed in the frequency domain using theoretical dispersion curves,

Dðx; xÞ ¼ SðxÞe
jkðxÞx :

ð2Þ

Here S(x) is the Fourier transform of s(t), k(x) is the theoretical dispersion curve of the propagating Lamb wave mode, and x is the propagation distance. Dispersion-compensated filtering then becomes,

~ sw ðxÞ ¼ R

Z

þ1

ð3Þ

wðtÞdðx; tÞdt;
1

~ sw ðxÞ is the dispersion-compensated crosswhere d(x,t) is the inverse Fourier transform of D(x,x) and the spatial function R correlation of w(t) and s(t). By applying the general form of Parseval’s Theorem to Eq. (3), the dispersion-compensated cross-correlation can be expressed in the frequency domain as,

~ sw ðxÞ ¼ 1 R 2p

Z

þ1

Z


1 þ1

¼ 21p


1

WðxÞD ðx; xÞdx

ð4Þ

WðxÞS ðxÞejkðxÞx dx:

This expression is very similar to that for conventional cross-correlation but with the exponential term being ejkðxÞx rather than ejxt . The dispersion-compensated auto-correlation is the case for which w(t) = s(t), and is

~ ss ðxÞ ¼ 1 R 2p

Z

þ1
1

SðxÞS ðxÞejkðxÞx dx:

ð5Þ

Now consider the case whereby the measured signal corresponds to an excitation signal s1(t) after it has propagated a distance of ^ x but it is correlated with a different excitation signal s2(t). The dispersion-compensated cross-correlation becomes,

~ 12 ðxÞ ¼ 1 R þ1 S1 ðxÞe
jkðxÞ^x S ðxÞejkðxÞx dx R 2 2p
1 R þ1 ¼ 21p
1 S1 ðxÞS2 ðxÞejkðxÞðx
^xÞ dx:

ð6Þ

Eq. (6) demonstrates the spatial shifting property of the dispersion-compensated cross-correlation; that is, if a signal propagates a distance of ^ x according to the dispersion curve k(x), its dispersion-compensated cross-correlation also shifts a distance of ^ x. In practice, the propagation distance x is discretized to a set of P values of interest and the time t is discretized to a set of L values as per the sampling frequency. Therefore, the dispersed signals d(x,t) can be written in the form of an L  P matrix,

2

dðx1 ; t 1 Þ dðx2 ; t 1 Þ    dðxP ; t1 Þ

3

6 dðx1 ; t 2 Þ dðx2 ; t 2 Þ    dðxP ; t2 Þ 7 7 6 7: D¼6 .. .. .. .. 7 6 5 4 . . . . dðx1 ; tL Þ dðx2 ; t L Þ    dðxp ; tL Þ

ð7Þ

The measured signal w(t) can be expressed as a column vector,

2

3 wðt1 Þ 6 wðt2 Þ 7 6 7 7 W¼6 6 .. 7; 4 . 5

ð8Þ

wðtL Þ and dispersion-compensated filtering then becomes the multiplication of the vector and the matrix,

M ¼ WT D:

ð9Þ

The end result is that w(t), a dispersive signal in time, is mapped to m(x), a non-dispersive signal in space. Dispersioncompensated matched filtering is similar to other dispersion compensation methods (e.g., [18,19]) but its similarity to conventional matched filtering makes it more intuitive and generally easier to understand.

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2.2. Dispersion-compensated matched filtering with simultaneous excitations Consider an array of N + 1 transducers, where N transducers act as transmitters and the remaining one is the receiver. Let hi(t) represent the received signal when the ith transmitter is excited by an impulse and Hi(x) the corresponding frequency domain representation, also known as the transfer function. If all N transmitters are excited simultaneously with signals s1 ðtÞ; s2 ðtÞ; . . . ; sN ðtÞ, and linear superposition holds, the response w(t) (or W(x) in the frequency domain) recorded by the receiver is of the form,

wðtÞ ¼ w1 ðtÞ þ w2 ðtÞ þ    þ wN ðtÞ N X si ðtÞ  hi ðtÞ ¼

ð10Þ

i¼1 N X WðxÞ ¼ ½Si ðxÞHi ðxÞ: i¼1

Here the asterisk denotes convolution and Si(x) is the Fourier transform of the ith excitation signal si(t). Written in vector form, the received signal becomes,

2

w1 ðt 1 Þ

3

2

w2 ðt1 Þ

3

2

wN ðt 1 Þ

3

6 w1 ðt 2 Þ 7 6 w2 ðt2 Þ 7 6 wN ðt 2 Þ 7 7 6 7 7 6 6 7þ6 7 þ  þ 6 7 W ¼6 . . 6 . 7 6 . 7 6 .. 7 4 . 5 4 . 5 4 . 5 w1 ðt L Þ

w2 ðtL Þ

ð11Þ

wN ðt L Þ

¼ W1 þ W2 þ    þ WN : To extract the response of the ith transmitter from the received signal, dispersion-compensated matched filtering is applied using dispersed versions of the ith excitation signal, and is calculated as,

Mi ¼ WT Di ¼ WTi Di þ

N X

WTj Di :

ð12Þ

j¼1; j–i

Here, Di is a dictionary containing multiple dispersed versions of the ith excitation signal as a function of propagation distance, which is obtained as per Eqs. (2) and (7). If the received signal contains a single arrival from each transmitter, then P T Wj Di is the sum of all the (shifted) crossWTi Di contains the (shifted) auto-correlation of the ith excitation signal and correlations between the ith excitation and the other excitation signals. Good correlation characteristics of the excitation signals; i.e., a sharp auto-correlation and a flat cross-correlation with the other excitations, will enable the recognition of a specific excitation in the presence of crosstalk. 2.3. Construction of unique excitation sequences Excitation signals in a simultaneous excitation system are designed to obtain good correlation characteristics. The basic idea proposed here is to construct each signal by concatenating a number of linear chirps, each of which has start and stop frequencies selected within a specified frequency range. The selection of those start and stop frequencies is optimized via a Monte-Carlo approach with the objective function defined as the ratio of the main lobe level of the auto-correlation to the maximum value among the main lobe levels of the cross-correlations together with the side lobe level of the autocorrelation. This type of excitation is selected in an attempt to use all of the available bandwidth associated with a particular guided wave mode. A linear chirp can be defined as,

  pðf s
f e Þt2 ; yðt; f s ; f e ; T c Þ ¼ sin 2pf s t þ Tc

0 6 t 6 Tc;

ð13Þ

where fs and fe are the start and stop frequencies, and Tc is the chirp duration. By concatenating a series of such chirps with different start and stop frequencies, unique sequences can be constructed for simultaneous excitations. Although the durations of each chirp signal could be different, they are taken to be the same in this study. Therefore, an excitation sequence constructed using chirps can be represented as,

 8  y t; f 1s ; f 1e ; KT ; 0 6 t < KT > >  >  > T T T > 6 t < 2T < y t
K ; f 2s ; f 2e ; K ; K K sðtÞ ¼ . . > > . >  > > : y t
ðK
1ÞT ; f ; f ; T ; ðK
1ÞT 6 t 6 T; Ks Ke K K K

ð14Þ

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where f1s, f2s,. . .,fKs and f1e, f2e,. . ., fKe are randomly selected from the range [fmin, fmax]. K is the number of chirps in one excitation sequence and T is the sequence duration. For clarity, each constructed excitation sequence is called a chirplet sequence in this paper. Now consider a total number of N chirplet sequences (one for each transmitter) with each sequence described as per Eq. (14). Dispersion-compensated matched filtering is applied at reception to extract the compressed response of each individual excitation; i.e., the main lobe of its auto-correlation. A natural optimization is to select the set of chirplet sequences that has the largest auto-correlation peak and the smallest cross-correlation peak. Therefore, an objective function Fi for the ith excitation is defined as,

Fi ¼

Rml ii ; sl maxj–i ðRii ; Ri1 ; Ri2 ;    ; Rij ;    ; RiN Þ

ð15Þ

sl where Rml ii is the main-lobe peak of the ith auto-correlation, Rii is the side-lobe peak of the ith auto-correlation, Rij is the peak of the cross-correlation between the ith and jth excitations signals, and N is the number of transmitters as mentioned previously. Here, both the auto- and cross- correlations are converted to envelope format by the Hilbert transform. By taking the geometric mean, the performance of all transmitters is taken into account and the overall objective function F becomes,

vffiffiffiffiffiffiffiffiffiffi uN u Y N F ¼ t Fi:

ð16Þ

i¼1

By using the geometric mean rather than the arithmetic mean, one very small Fi will significantly reduce F, whereas it would not for the arithmetic mean. Therefore, the existence of any small Fi is prevented by maximizing the geometric mean F. The selection of suitable chirplet sequences is optimized through a Monte-Carlo approach in which a large number of chirplet sequences are constructed using randomly generated chirps and evaluated per the objective function F. The detailed procedure is presented in the following five steps. 1. Construct M (M  N) chirplet sequences as per Eq. (14), in which T, K, fmin, and fmax are pre-determined. 2. Calculate M auto-correlation functions and M(M
1)/2 cross-correlation functions. The M main lobe peak values of the auto-correlations are stored in vector Q1, the M side lobe peaks of the auto-correlations are stored in vector Q2, and the M(M
1)/2 peaks of the cross-correlations are stored in matrix Q3, ml ml Q 1 ¼ ½Rml 11 ; R22 ; . . . ; RMM ;

ð17Þ

Q 2 ¼ ½Rsl11 ; Rsl22 ; :::; RslMM ; 2

R12 6 0 6 Q3 ¼ 6 6 .. 4 . 0

ð18Þ

R13



R1M

R23 ...

 .. .

R2M .. .

0

3 7 7 7: 7 5

ð19Þ

   RðM
1ÞM

3. Systematically select N chirplet sequences from the M constructions, and evaluate the overall correlation performance using the objective function F. Here, F can be efficiently calculated by picking the appropriate elements in Q1, Q2, and Q3. 4. Repeat Step 3 until the objective functions for all C NM combinations are calculated. 5. The combination of N chirplet sequences with the best performance is selected for the N excitations. It should be noted that the auto- and cross-correlation values in Eqs. (17)–(19) are computed using conventional correlations for computational expediency; this issue is further addressed in Section 3.3. As an example, these steps were executed using the following parameters: N = 4, M = 100, T = 500 ls, K = 10, and [fmin, fmax] = [50, 200] kHz. Table 1 lists the optimized chirplet frequencies that were obtained. 3. Simulations A number of simulations were performed to motivate parameter selection, consider other excitation sequences, and compare conventional to dispersion-compensated correlations. First, a parametric study was conducted to determine the best parameters for the chirplet sequences. Next, sequences were constructed based upon binary phase shift keying (BPSK) and compared to the chirplet sequences. Finally, dispersion-compensated correlations were compared to conventional correlations that were computed directly and after propagation with dispersion. Since an exhaustive search of all possible sequence parameters is not feasible, here we explicitly concentrate on rather short excitations (less than 1000 ls). Although long sequences would undoubtedly improve performance, the focus of the

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J. Hua et al. / Mechanical Systems and Signal Processing 95 (2017) 506–523 Table 1 Optimized chirplet frequencies for a 4-transmitter simultaneous excitation system. Frequencies f1s (kHz) f1e (kHz) f2s (kHz) f2e (kHz) f3s (kHz) f3e (kHz) f4s (kHz) f4e (kHz) f5s (kHz) f5e (kHz) f6s (kHz) f6e (kHz) f7s (kHz) f7e (kHz) f8s (kHz) f8e (kHz) f9s (kHz) f9e (kHz) f10s (kHz) f10e (kHz)

s1(t)

s2(t)

s3(t)

s4(t)

186.4 139.0 137.2 180.1 175.1 71.7 58.0 181.6 112.9 72.1 83.8 96.1 164.5 97.0 53.1 174.5 198.8 196.9 140.8 136.9

116.1 88.9 138.1 96.8 173.5 99.4 133.9 186.0 58.6 151.9 156.2 192.3 193.6 172.4 179.6 75.1 99.1 166.0 55.2 55.8

144.4 92.1 73.4 128.5 88.9 61.9 56.8 61.2 122.0 130.8 179.4 93.9 50.6 131.6 142.8 105.9 120.0 172.7 62.5 60.1

140.3 149.5 116.8 71.8 136.9 198.7 104.4 128.6 102.7 52.0 114.9 87.2 149.1 109.8 187.0 111.8 113.0 72.1 124.9 107.9

paper is primarily efficient data acquisition and long excitations increase the acquisition time. The dispersion curve and frequency range consistent with the experimental setup described in Section 4 are used for the simulations. 3.1. Choice of parameters for excitation construction The construction of chirplet sequences depends on the parameters K (number of chirps in one sequence), N (number of simultaneous excitations), T (sequence duration), M (number of constructions in Step 1) and [fmin, fmax] (chirplet frequency range). A large M, which is fixed at 100 in this paper, ensures enough candidates from the different combinations of C NM for the Monte-Carlo approach. The frequency range [fmin, fmax] is dictated by the system bandwidth and the specimen propagation characteristics, and is usually selected to match a desired guided wave mode. Here the frequency range [fmin, fmax] was selected as [50, 200] kHz, which is consistent with the dominant A0 mode in the experiments. The effects of parameters K, N, and T on the objective function F in Eq. (16) are quantified in this subsection; a larger F indicates higher SNR and better channel separation. To investigate the effect of K on F, its value was taken as 2, 5, 8, 10, 16, 20, 25, and 40 while N = 2 and T = 500 ls. Recall that F is calculated using conventional cross-correlations, so dispersion is not explicitly taken into account. As shown in Fig. 1, an unsuitable K results in a smaller F and reduced signal quality. The reason can be explained as follows. If K is too small, every sequence consists of a small number of random chirps. As a result, the constructed N sequences have similar shapes, which results in higher cross-correlation values between them. On the other side, if K is too large, the duration of each chirp is very small given that the total time T is fixed. The ratio of the auto-correlation main-lobe to self-noise (i.e., side-lobe content) decreases as the chirp duration decreases. In summary, a compromise K is needed in the construction procedure, and is selected to be 10 in this paper. To investigate the effects of N and T on F, K was fixed at 10 while N was varied from 2 to 6 with an increment of 1 and T from 200 ls to 800 ls with an increment of 200 ls. Fig. 2 shows the change in F associated with the changes of N and T. It can

Objective-function F

5.5 5 4.5 4 3.5 3

2

5

8 10

16

20

25

K Fig. 1. The objective function (F) versus the number of chirplets (K).

40

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J. Hua et al. / Mechanical Systems and Signal Processing 95 (2017) 506–523

6

Objective-function F

5.5 5 4.5

N=2 N=3 N=4 N=5 N=6

4 3.5 3

200

400

600

800

Fig. 2. The objective function (F) versus the excitation duration (T) for various numbers of transmitters (N).

be seen that F increases with increasing T but decreases with increasing N. Although increasing T results in a larger F, longer sequences are not considered because of their negative effects on data acquisition speed and data storage. Similarly, the choice of N is also a compromise between signal quality and data acquisition speed. Further discussion can be found in Section 5.2. Considering only the objective function F, the results in Fig. 2 can be used as a guide for parameter selection. For example, if there are four simultaneous excitations and the objective function F must be larger than 4, the sequence duration must be at least 400 ls. 3.2. Comparison with BPSK modulation The constructed chirplet sequence can be regarded as a K-bit code with each bit modulated by a chirp of random start and stop frequencies. In this subsection, a different modulation scheme, namely binary phase shift keying (BPSK), is considered for comparison. Each bit in BPSK is modulated by either a 0° or 180° version of the same signal, typically a tone burst. Here this signal is a chirp with specified start and stop frequencies as shown in Fig. 3, and the BPSK sequences are constructed by randomly combining K such 0° or 180° chirps. Each chirp is constructed with fmin = 50 kHz and fmax = 200 kHz so that the resulting frequency range is comparable to that of the chirplet sequences. For a given K, there are a total of 2K possible BPSK sequences. For N transmitters, the best N sequences can be selected from these 2K possibilities as per the five steps in Section 2.3. As previously mentioned, the objective function F depends on the choice of parameters. For a fair comparison to the chirplet sequences, the BPSK sequences were obtained using the same parameters: N = 2, T = 500 ls, and K = 10. The objective function F for the best 10 BPSK sequences is 3.33, which is considerably lower than the value of 5.29 that was obtained for the chirplet sequences. It is an interesting comparison because the number of BPSK sequences evaluated is 1024 (K = 10), whereas for the chirplet sequences, only 100 sequences are considered (M = 100). Unlike the random chirplet sequences in which the cross-correlations between each individual chirplet are potentially small in amplitude, the cross-correlations between the two out-of-phase ‘‘bits” in BPSK have the same

(a)

(b)

Fig. 3. Chirps at relative phases of (a) 0° and (b) 180°.

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amplitude as the auto-correlations, although reversed in sign. The additional spectral and temporal complexity of the random chirplets results in improved correlation characteristics as compared to BPSK using a single chirp. By increasing the bit number K, the BPSK sequences will undoubtedly generate higher values of F, although the results presented here for the K = 10 case indicate that even for high values of K, BPSK may not outperform chirplet sequences of the same length. However, when K is very large, it is unrealistic to search all 2K constructions to find the optimal excitations, and the length of the excitations could become prohibitive in practice. As an alternative, existing binary codes such as Golay and Gold codes could be readily used as was done in [19] rather than conducting an exhaustive search as was done here. Other types of sequences could also be considered (e.g., simple frequency modulation of bursts), which may perform better than BPSK with simple chirps. 3.3. Correlation characteristics in the presence of dispersion

2500

2500

2000

2000

1500

1500

1000

1000

Amplitude

Amplitude

The chirplet sequences in Section 2.3 are optimized based on their correlations in the time domain, while the correlations after dispersion-compensated matched filtering in Eq. (12) are in the spatial domain. Although the dispersion effect is compensated in Eq. (12), the shapes of conventional and dispersion-compensated correlations are not exactly the same. The Monte-Carlo approach was based upon conventional correlations instead of dispersion-compensated correlations for computational expediency based on the premise that they have similar correlation characteristics. In particular, the underlying assumption is that if the conventional correlations have the desired characteristics (i.e., sharp auto-correlations and flat cross-correlations), the compensated ones will as well. However, such an assumption needs further validation. In addition, if the dispersion cannot be compensated for some reason (e.g., theoretical dispersion curves are not available), the presence of dispersion will strongly influence the auto- and cross-correlations. The influence of dispersion on correlation characteristics is discussed here. Numerical simulations were performed to both validate the assumption and investigate the influence of dispersion on conventional correlations. In these simulations, several chirplet sequences were randomly generated using the parameters T = 500 ls, K = 10, and [fmin, fmax] = [50, 200] kHz. These signals were numerically propagated as the A0 mode in a 3.18-mmthick aluminum plate for distances of 0.15 m and 0.3 m. The conventional auto- and cross-correlations were obtained by

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(b)

Amplitude

Amplitude

(a)

600

600

700

800

0

0.1

0.2

0.3

0.4

0.5

Distance (m)

(c)

(d)

Fig. 4. Various auto-correlations examples. (a) Conventional with no dispersion, (b) conventional but with dispersion and a propagation distance of 0.15 m, (c) conventional but with dispersion and a propagation distance of 0.3 m, and (d) dispersion-compensated after a propagation distance of 0.3 m.

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directly correlating the excitation sequences with each other as per Eq. (1). The correlations between the excitation sequences and the responses after dispersive propagation, which are referred to as dispersive correlations, were also obtained as per Eq. (1). The dispersion-compensated correlations were obtained by dispersion-compensated matched filtering of the propagation responses as per Eq. (3). Only the 0.3 m responses were used since the spatial shifting property of Eq. (6) shows that the shapes of the dispersion-compensated cross-correlations are the same for different propagation distances. As an example, Fig. 4(a) shows the conventional auto-correlation, Fig. 4(b) shows the corresponding dispersive autocorrelation for the 0.15 m propagation distance, Fig. 4(c) shows the dispersive auto-correlation for the 0.3 m propagation distance, and Fig. 4(d) shows the dispersion-compensated auto-correlation, which is in the spatial domain. A comparison of auto-correlation main-lobe peaks and side-lobe maximum amplitudes for 15 different chirplet sequences are shown in Fig. 5(a) and (b), respectively. The main-lobe peaks are identical for the conventional and compensated auto-correlations, as can be seen from Eq. (5) by setting x = 0, but the peaks of the dispersive auto-correlations are much lower. As expected, the increased signal distortion with distance results in a lower peak for the larger propagation distance. The side-lobe maximum amplitudes are similar for all four cases. The ratio of the main-lobe peak to the side-lobe maximum amplitude is shown in Fig. 5(c) with the values for the dispersive cases being lower than either the conventional or compensated values. In addition, the increased effect of dispersion caused by the longer propagation distance lowers this ratio much more than for the shorter distance. Fig. 5(d) shows the peak cross-correlation amplitudes for 15 signal pairs (each pair consists of two randomly generated chirplet sequences), and it can be seen that they are similar for all four cases. From these results it can be seen that the conventional and dispersion-compensated correlations follow similar, although not identical, trends. Thus it is reasonable to expect that the sequence construction process of Section 2.3, which uses conventional rather than dispersion-compensated correlations, will yield reasonable, although likely non-optimal, sequences. It is also interesting to note that the presence of dispersion reduces the sharpness of the auto-correlation as evaluated by the ratio of the peak amplitude to the maximum side-lobe amplitude as seen in Fig. 5(c), while having much less influence on the cross-correlations shown in Fig. 5(d).

2500

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Fig. 5. Values computed from various conventional and dispersion-compensated auto- and cross-correlations. (a) Auto-correlation main-lobe peak, (b) auto-correlation maximum side-lobe amplitude, (c) ratio of auto-correlation main-lobe peak to maximum side-lobe amplitude, and (d) cross-correlation maximum amplitude.

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4. Experiments Two experiments were conducted on a 6061-T6 aluminum plate with dimensions of 610 mm  610 mm  3.18 mm. A circular array of eight piezoelectric discs, which are 7 mm in diameter, 0.5 mm in thickness, radially polarized, and have a resonance frequency of 300 kHz, was bonded to the plate with two-component epoxy. To simulate damage, magnets were attached to the plate center. Fig. 6 shows the schematic diagram indicating the locations of the transducer array and the simulated damage, where the origin of the coordinate was set to be the lower left corner of the plate. The coordinates of the PZT elements and the damage are listed in Table 2. The first experiment was designed to verify the linear superposition assumption made in Section 2.2. First, PZT #1 and PZT #5 were excited simultaneously by two waveform generators with two chirplet sequences obtained using the method described in Section 2.3. The sequence length T was 500 ls and K, the number of chirps in each sequence, was 10. The minimum and maximum frequencies, fmin and fmax, were selected to be 50 kHz and 200 kHz, respectively, to match the frequency range in which the A0 fundamental Lamb wave mode was preferentially transmitted and received. The responses were recorded by the remaining six transducers and digitized using a National Instrument PXIe-5122 14-bit digitizer at a sampling frequency of 20 MHz. For each acquisition, 20 waveforms were averaged to improve the signal-to-noise ratio. Then, the same two transducers PZT #1 and PZT #5 were excited individually with their respective sequences, which were identical to those used in the simultaneous excitation case. The responses were recorded using the remaining transducers. The second experiment was designed to examine the effectiveness of the simultaneous excitation system for defect detection and localization. After validating the linear superposition assumption, a synthetic approach was taken to obtain the equivalent responses from simultaneous excitations. That is, each transducer was excited in turn using its own chirplet sequence and the responses at each receiver were summed to obtain those that would be obtained from simultaneous excitations. Several sets of optimal sequences were generated for excitations using different parameters (i.e., different values of N and T). Note that the frequency domain deconvolution method introduced in [20] was used here to further improve the data

#8 #7

610 mm

Simulated Damage

#6

#1 #2

#5 #3

#4

610 mm Fig. 6. Diagram of the aluminum plate specimen showing the transducer locations and simulated damage.

Table 2 Coordinates of the transducers and simulated damage. Description

x (mm)

y (mm)

PZT #1 PZT #2 PZT #3 PZT #4 PZT #5 PZT #6 PZT #7 PZT #8 Damage

181.0 191.0 250.5 349.5 401.5 429.0 379.5 290.0 305.0

303.0 256.5 193.5 190.0 227.0 303.0 404.0 428.0 305.0

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acquisition efficiency for this study. The excitation for all transducers when excited individually was actually a very broadband signal rather than the specific chirplet sequences, and the responses from the chirplet sequences were obtained as per deconvolution as described in [20]. The broadband signal used in this study was a linear chirp signal sweeping from 50 to 500 kHz with a duration of 200 ls. 5. Results and discussion 5.1. Validation of linear superposition assumption An example of the optimal chirplet sequence is shown in Fig. 7(a) and its corresponding auto-correlation is shown in Fig. 7(b). Fig. 7(b) clearly shows a dominant main lobe in the auto-correlation curve. In the 2-transmitter simultaneous excitation system described in the first experiment, all the receivers captured the responses from the two excitations at first. As an example, the thick gray line in Fig. 8(a) shows the response recorded by PZT #7. For comparison, the summation of responses recorded by PZT #7 from two separate excitations is shown using the thin black line. It is clear that the two waveforms are virtually identical. Fig. 8(b) shows the much smaller residual signals for both excitation cases, which were obtained by subtracting the baseline signal from the signal taken with the magnets attached to the plate. Although differences exist between the two waveforms because the variable background noise and the residual signals have similar amplitudes, a comparison of waveform details in Fig. 8(c) shows that they are still almost identical. In summary, Fig. 8 demonstrates that signals from simultaneous excitations are essentially identical to signals obtained by summing signals from individual excitations and thus validates the linear superposition assumption. A further validation is to compare the signals obtained after dispersion-compensated filtering since these signals have much higher resolution and will be used for defect detection. The residual signals shown in Fig. 8(b) are cross-correlated with dispersed versions of chirplet sequences of PZT #1 and PZT #5, respectively. Here, only the two fundamental modes, A0 and S0, are generated and A0 is dominant as a result of the 50 kHz to 200 kHz frequency range selection. Therefore, the theoretical dispersion curve of the A0 mode is used in the calculation of dispersed versions of excitation; i.e., the D matrix in Eq. (7). Fig. 9(a) shows the result from dispersion-compensated filtering using the chirplet sequence of PZT #1. The thick gray line is the result from simultaneous excitations and the thin black line is that from the summation of responses from separate excitations. Both results are normalized to unity amplitude for comparison. Similarly, Fig. 9(b) shows the dispersioncompensated filtering results using the chirplet sequence of PZT #5. Both results in Fig. 9(a) and (b) are identical and they demonstrate the validity of the linear superposition assumption as well. 5.2. Experimental quantification of signal quality The simulation results of Section 3 predict that fewer transmitters and longer excitation durations will produce less crosstalk noise. In this section this noise is quantified experimentally by considering different numbers of transmitters and excitation durations.

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(b) Fig. 7. An example of a constructed chirplet sequence. (a) Time domain waveform, and (b) its autocorrelation.

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(a)

(b)

(c) Fig. 8. Comparison of measured responses from a simultaneous excitation system with two transmitters versus the summation of responses from a separate excitation system. (a) Signals before baseline subtraction, (b) residual signals after baseline subtraction, and (c) zoomed residual signals.

5.2.1. Number of transmitters N By comparing Eqs. (9) and (12), it can be seen that the signal in an N-transmitter simultaneous excitation system has N
1 cross-correlation interference terms after dispersion-compensated filtering, resulting in a lower SNR than the corresponding signal from a separate excitation system. Fig. 10 shows a comparison of dispersion-compensated filtering results between the two excitation systems in a 4-transmitter setting. Fig. 10(a), (b), (c), and (d) shows the signals when PZTs #1, #2, #5, and #6 are transmitters, respectively, and PZT #3 is the receiver. It can be clearly seen in all four figures that the signals from the two excitation systems are different. To quantify the SNR of the signals from simultaneous excitations, the SNR as a function of the number of transmitters (N) is defined as,

SNRðNÞ ¼ 20 log

RMSðmsep Þ ; RMSðmsim
msep Þ

ð20Þ

where msim and msep are the signals after dispersion-compensated filtering in the simultaneous and separate excitation systems, respectively, and RMS refers to ‘‘root mean square.” Fig. 11 shows SNR as a function of N where each symbol corresponds to a specific transducer pair; the number of transducer pairs for N = 2, 3, 4, 5 and 6 is 12, 15, 16, 15, and 12, respectively. It can be seen that the SNR generally decreases as the number of transmitters N increases, which is consistent with the decrease in the objective function with N as shown in Fig. 2. For each value of N, there is a broad distribution of SNR values, which is caused by the specific experimental configuration considered here. Interfering signal components can result from multiple sources including direct arrivals, reflections from

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measured signal summed signal

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(b) Fig. 9. Comparison of dispersive matcher filtering from a simultaneous excitation system and a separate excitation system after summing the separate responses. Processed responses using the excitation signal from (a) PZT #1, and (b) PZT #5.

geometry (e.g., edges), and lack of modal purity, and the variation in SNR for each N indicates that each signal is affected differently. 5.2.2. Excitation duration T To investigate the effect of excitation duration T on signal quality, it is varied from 200 ls to 800 ls with an increment of 200 ls for the case of two transmitters (N = 2). To quantitatively compare the SNR of signals with different excitation durations, SNR(T) is calculated by comparing signals to those with T = 800 ls:

SNRðTÞ ¼ 20 log

RMSðm800ls Þ ; RMSðmT
m800ls Þ

ð21Þ

where mT is the signal after dispersion-compensated filtering in the simultaneous excitation system when the excitation duration is T and m800 ls is the corresponding signal when the duration is 800 ls. Fig. 12 shows that SNR(T) increases slightly as the excitation duration T increases for almost all of the signals, which is generally consistent with the simulation result of Fig. 2. However, the small effect on signal quality as measured by Eq. (21) is not as pronounced as might be expected by the clear increase in F with T in Fig. 2. As was the case in Fig. 11, there is also a broad distribution in SNR for each excitation length. 5.3. Evaluation of system effectiveness for damage detection The specific problem considered here is in situ damage detection, and the effectiveness of simultaneous excitations for this application is evaluated by applying the delay-and-sum (DAS) imaging method to residual signals (differenced signals before and after introduction of damage). Using the approach described in Section 4, responses from an N-transmitter simultaneous excitation system are obtained from the measured responses resulting from separate excitations. These responses are then processed by dispersion-compensated matched filtering after baseline subtraction, resulting in signals that are a function of distance rather than time. These residual signals in the distance domain are then input to the DAS imaging algorithm to generate images of the artificial damage (the magnet stack). Of interest is whether damage can be successfully identified and localized in these images for different combinations of N and the excitation duration T (specifically N = 2, 4, 6 and T = 200, 500, 800 ls).

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(d) Fig. 10. Comparison of dispersion-compensated matched filtering results for simultaneous and separate excitations when using four transmitters (N = 4) and PZT #3 as the receiver. Transmitting with (a) PZT #1, (b) PZT #2, (c) PZT #5, and (d) PZT #6.

To evaluate the performance of the simultaneous excitation method, a DAS image is first obtained using separate excitations with four transmitters to obtain a basis for comparison. The signal for the separate excitation is a normal linear chirp

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SNR(N) (dB)

10

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−5 2

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1

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0 −1 −2 −3 200

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signal, rather than the chirplet sequence, for all transmitters. To ensure that the comparison is valid between the two excitation systems, the normal linear chirp has its frequency sweeping from 50 kHz to 200 kHz over 500 ls (the same bandwidth and length). The DAS image value at location (x,y) is calculated as the sum of the dispersion-compensated filtering results delayed by the appropriate distance shifts [15,21],

Vðx; yÞ ¼

XX i

 ij ðd
dijxy Þ; m

ð22Þ

j

 ij is the envelope of the residual signal of transducer pair ij after dispersion-compensated filtering, dxy is the propwhere m agation distance of the wave packet travelling from the ith transmitter, through the position (x,y), and back to the jth receiver, and V(x,y) is the image pixel value. Fig. 13 shows DAS images obtained from the two excitation systems using different values of N and T. All the images are normalized by their maximum values and are shown on a 20 dB scale. Transmitters are marked by the ‘4’ symbols, receivers by the ‘h’ symbols, and the simulated damage by a ‘+’ symbol. Fig. 13(a) is the image obtained using separate excitations with N = 4 and T = 500 ls. The damage is well-localized and the level of artifacts is typical of a good-quality delay-andsum image. With four transmitters and four receivers, there are a total of 16 distinct signals, which is the maximum possible with eight transducers and dedicated transmitters and receivers. Thus, this is the best quality image using separate excitations. The rest of the images are generated using simultaneous excitations with the following parameters: (b) N = 2 and T = 500 ls, (c) N = 4 and T = 500 ls, (d) N = 6 and T = 500 ls, (e) N = 2 and T = 200 ls, and (f) N = 2 and T = 800 ls. From the simultaneous excitation images in Fig. 13(b)–(f), it can be seen that the damage at the center position is unambiguously identified and accurately located for all parameters selections. However, from the comparison between the images of Figs. 13(a) and 10(b)–(f), it can be seen that the artifacts in the simultaneous excitation images are larger than those from separate excitations. In particular and as previously discussed, the values of N and T affect signal quality and thus imaging performance. A comparison of Fig. 13(b)–(d) shows that imaging artifacts increase as N increases. Both the N = 2 and the N = 6 cases use 12 signals, whereas the N = 4 case uses 16 signals. It is no surprise that the N = 2 case clearly outperforms the N = 6 case, and it is interesting that the increased number of signals for the N = 4 case does not offset the increased noise resulting from more transmitters. The comparison between Fig. 13(b), (e) and (f), which were obtained with N = 2 and T = 500 ls, 200 ls, and 800 ls, respectively, shows that the image with the longer excitation duration has a reduction in imaging artifacts. Both phenomena are consistent with the simulation results of Fig. 2 that predict improved signal quality as N decreases and T increases. Even though the SNR results of Fig. 12 do not indicate a significant change in signal quality with T, the DAS imaging results are markedly improved with larger values of T. ij

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Fig. 13. Delay-and-sum images generated with different scenarios. (a) Separate excitations, N = 4, T = 500 ls. Simultaneous excitations, (b) N = 2, T = 500 ls, (c) N = 4, T = 500 ls, (d) N = 6, T = 500 ls, (e) N = 2, T = 200 ls, and (f) N = 2, T = 800 ls.

6. Concluding remarks A multi-channel Lamb wave simultaneous excitation system has been proposed in this paper with the goal of reducing the data acquisition time for an array system with dedicated transmitters and receivers. The effectiveness of the system for damage detection and localization has been demonstrated experimentally. The significance of this work is not only reducing the data acquisition time but also ensuring that all signals are acquired under identical environmental and operational conditions, which is an important consideration for SHM systems. The main downside of using this system is that

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the improved acquisition speed comes at the cost of degrading the signal SNR and subsequent imaging performance. In practice, the compromise between acquisition speed and signal SNR can be adjusted by controlling the number of transmitters and the length of the excitations. The reduction in data acquisition time depends upon the basis for comparison. Considering an array with N transducers, there are a total of N(N
1)/2 unique pitch-catch transducer pairs. If a separate excitation system is not multiplexed (that is, each transducer has its own digitizer), then all possible pairs can be acquired in N transmit cycles; a multiplexed system requires a separate excitation for each signal for a total of N(N
1)/2 transmit cycles. A simultaneous excitation system requires only one transmit cycle but has dedicated transmitters and receivers, which means that it cannot acquire all possible N(N
1)/2 unique signals. If there are NT dedicated transmitters, then there are a total of NT(N
NT) possible signals. The corresponding separate excitation system will require either NT transmit cycles for a non-multiplexed system, or NT(N
NT) transmit cycles for a multiplexed system. Thus, a system with simultaneous excitations improves the acquisition time by a factor ranging from as little as NT to as much as N(N
1)/2, depending upon the configuration of the separate excitation system to which it is being compared. An additional factor to consider is that the simultaneous excitation system does not have to wait for reverberations within the specimen to die out during the acquisition process. If no signal averaging is performed, all data can be acquired from a single set of simultaneous excitations. Thus, the time window over which environmental and operational conditions should remain stable is equal to the data recording window and does not depend upon the maximum repetition frequency. The recording window must be sufficiently longer than the excitation time for waves that are generated at the end of the excitation to propagate over the region of interest. Increasing the excitation time significantly (e.g., up to tens or even hundreds of milliseconds) may be quite reasonable from the standpoint of environmental and operational stability, but could be problematic from a data acquisition, transfer, storage, and processing standpoint. Although these results are promising, there is much room for future work to further quantify and improve the performance of the simultaneous excitation methodology. Dispersion-compensated correlations should be used for the objective function to see if that yields any improvement in performance. In this current work, the length of the excitation was limited to 800 ls, which admittedly is somewhat arbitrary but was short enough to allow for optimization of the proposed chirplet sequences. Using longer excitations should increase performance but at the expense of a longer transmit cycle and an increased data handling burden. This tradeoff can only be evaluated by first quantifying performance versus excitation length for propagation environments of interest. Other types of excitation signals should also be considered such as sequences of frequency-modulated bursts or binary encoding schemes using, for example, up-chirps and down-chirps rather than a 180° phase shift as was considered here. Quantification of performance also needs further investigation. Three methods were considered here: (1) the objective function F, (2) the SNR as compared to reference signals, and (3) qualitative assessment of imaging performance. Although they followed the same trends of increasing performance with decreasing numbers of transmitters and increasing excitation duration, only the objective function is independent of the specimen geometry. Additional experiments and simulations using a wider variety of propagation environments should be considered to develop a more detailed assessment of performance. Acknowledgements Mr. Jiadong Hua is supported by the National Natural Science Foundation of China (Grant Nos. 51505365 and 51421004) and the China Scholarship Council. References [1] J.L. Rose, Ultrasonic Waves in Solid Media, Cambridge University, Cambridge, 1999. [2] D.N. Alleyne, P. Cawley, The interaction of Lamb waves with defects, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39 (3) (1992) 381–397. [3] P.D. Wilcox, Omni-directional guided wave transducer arrays for the rapid inspection of large areas of plate structures, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50 (6) (2003) 699–709. [4] L. Yu, G. Bottai-Santoni, V. Giurgiutiu, Shear lag solution for tuning ultrasonic piezoelectric wafer active sensors with applications to Lamb wave array imaging, Int. J. Eng. Sci. 48 (10) (2010) 848–861. [5] X. Chen, J.E. Michaels, S.J. Lee, T.E. Michaels, Load-differential imaging for detection and localization of fatigue cracks using Lamb waves, NDT and E Int. 51 (2012) 142–149. 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