Ultrasonics 38 (2000) 620–623www.elsevier.nl/locate/ultras

Determination of plate source, detector separation from one signal

Stephen Holland a,*, Tadej Kosel b, Richard Weaver c, Wolfgang Sachse aa Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853-1503, USA

b Faculty of Mechanical Engineering, University of Ljubljana, 1001 Ljubljana, Sloveniac Department of Theoretical and Applied Mechanics, University of Illinois at Champaign-Urbana, Urbana, IL 61801, USA

Abstract

We address the problem of locating a transient source, such as an acoustic emission source, in a plate. We apply time–frequencyanalysis to the signals detected at a receiver. These highly dispersive and complex waveforms are measured for source–receiverseparations ranging from 40 to 180 plate thicknesses and at frequencies such that 10 to 20 Rayleigh–Lamb branches are included.Reassigned, smoothed, pseudo-Wigner–Ville distributions are generated that exhibit the expected sharp ridges in the time–frequency plane, lying along the predicted frequency–time-of-arrival relations. The source–receiver separation can be determinedfrom such plots. © 2000 Elsevier Science B.V. All rights reserved.

Keywords: Acoustic emission; Plate modes; Source–detector separation; Source location; Spectrogram; Time–frequency analysis; Ultrasound;Wave dispersion; Wigner–Ville distribution

1. Introduction evolution results from differences in speeds of propaga-tion of various wave modes present in a signal, theirdamping, and the effects of geometrically induced waveAcoustic emission measurements are often used to

determine the location of an acoustic source in a struc- dispersion. The arrival times of dispersive modes can bedisplayed in time–frequency space. We compare in thisture. Conventional source location algorithms rely on

the measurement of the arrival time of one or more paper the measured arrival times for dispersive modeswith values calculated using an elastodynamic model.wave modes at a number of sensors, and the inversion

of the relative arrival time data to determine the location The free parameters in the model include t0, the instantof source activation and d, the source–detector separa-and time of activation of the source. Only one type of

information, one signal arrival time, is extracted from a tion. These are adjusted such that the model optimallymatches the measured data.waveform and, with triangulation, the source can be

located, provided that the speed of propagation of thesignal is known. It also possible to identify in a waveformtwo features whose speed of propagation is known and

2. Time–frequency analysis of waveformsdifferent. This forms the basis of an earlier patent [4].In contrast, we describe here a method which utilizes

Fig. 1 shows the measurement geometry and a samplethe entire waveform, viewed in the time–frequencysynthetic waveform from a normal force step-excitationdomain, to determine both the time of signal activationin a steel plate 10 mm thick, with a source–receiveras well as the source–detector separation. The extractionseparation of 1.80 m. The first step in processing is toof this distance from a single detected signal will permitcompute a time–frequency distribution. Such a distribu-the use of fewer transducers in source locationtion localizes a signal in both time and frequencymeasurements.domains. That is, it provides simultaneous time reso-Our method relies on the identification of guidedlution and inversely proportional frequency resolution.modes propagating in a plate. The generated modes areFor example, a gated burst has position in both timegoverned by the characteristics of the source. Theirand frequency, but the conventional time and frequencyrepresentations present only one aspect. A time–fre-* Corresponding author. Fax: +1-607-255-9179.

E-mail address: sdh4@cornell.edu (S. Holland) quency distribution combines both time and frequency

0041-624X/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S0041-624X ( 99 ) 00206-1

621S. Holland et al. / Ultrasonics 38 (2000) 620–623

Fig. 1. Measurement geometry and sample waveform (d=180h, steel ).

information into a single representation. There are a by:large number of possible time–frequency distributions,

Y(t, f )=g(t)1thowever, we will focus only on the two which are most

often used. These are the spectrogram and the Wigner– GP−2

2h(t)[x(t+t/2)x1(t−t/2)] e−j2pft dtH. (3)Ville distribution.

The spectrogram is based on the short time FourierHere 1

tis defined as a convolution with respect to timetransform (STFT) with a sliding window. One example

t. The function g(t) is the smoothing function in timeis:and h(t) restricts the range of the integral in t.Restricting the range in t is equivalent to smoothing in

Y(t, f )=KP−2

2x(t) e(t−t)2/T2 e−j2pft dtK2 . (1) frequency. The SPWVD reduces to the conventional

Wigner–Ville distribution when h(t)=1 and g(t)=d(t).Eq. (1) gives the definition of a spectrogram with a The SPWVD is the Wigner–Ville distribution we usedGaussian window function of half-width T. It is the to obtain the results described here [1].power spectrum of a signal which corresponds to the A reassignment algorithm is used to sharpen thesquared magnitude of the Fourier transform of the features in the time–frequency distribution. The reas-windowed signal. signment algorithm moves energy ‘uphill’ in time–fre-

The Wigner–Ville distribution provides increased quency–amplitude space, thus sharpening the image.Auger et al. [1] provide a detailed discussion of theresolution relative to the spectrogram, but it has interfer-reassignment process. Fig. 2 depicts the reassignedence terms. The definition of the Wigner–Ville distribu-Wigner–Ville distribution of the signal from Fig. 1.tion is given by the following equation:

Y(t, f )=P−2

2x(t+t/2)x1(t−t/2) e−j2pft dt. (2) 3. Matching the time–frequency distribution with

computed curvesWhen t is near zero, x(t+t/2) and x1(t−t/2) are

The v vs. k dispersion curves depicted in Fig. 3(a)coherent and their product contributes to the integral.for the allowable plate modes can be readily calculatedHowever, when t is large, x(t+t/2) and x1(t−t/2) have

incoherent phases and thus average to zero. The Wigner–Ville distribution can be thought of as being a squaredFourier transform centered about a point.

The Wigner–Ville distribution is quadratic in x, so ifx is a sum (a+b), the Wigner–Ville distribution of xcontains an interference term 2ab in addition to thedesired quantity (a2+b2). These interference terms resultin an increased noise level of the Wigner–Ville distribu-tion relative to the spectrogram. In practice, these inter-ference terms can be dramatically reduced by smoothingin time and frequency. The result is the smoothed-pseudo

Fig. 2. Enhanced time–frequency distribution (reassigned).Wigner–Ville distribution (SPWVD), which is defined

622 S. Holland et al. / Ultrasonics 38 (2000) 620–623

Fig. 3. Dispersion, group velocity, and time–frequency arrival curves for steel; source–receiver separation of 180h.

given the material properties [2,3]. The group velocity 4. Results: synthetic and measured waveformsshown in Fig. 3(b) is determined as a function of

The analysis of synthetic waveforms provides a best-frequency from the dispersion relation by dv/dk. Thecase demonstration of what can be accomplished usingarrival times of various modes are determined fromsuch time–frequency analysis. Synthetic waveforms cor-d/vg. A sample result is shown in Fig. 3(c). The com-

puted arrival curves can then be visually matched withthe ridges in the measured time–frequency distributionshown in Fig. 3(d).

To effect the match, two parameters are selectivelyvaried: the source–detector separation d and the timeshift t0. The adjustment of the d parameter is made sothat the spacing in time of the different computedarrivals matches the measured data. The time shift t0 isdetermined by shifting the position of the calculatedarrivals so that they best match the time–frequencydistribution. The lowest anti-symmetric mode (denotedby A0), the lowest symmetric mode (denoted by S0),and the Rayleigh modes, along with the arrival tails,usually provide the best anchors for an accurate match.This is depicted in Fig. 4. The number of modes whichare easily identifiable in a representation varies greatlyand depends on source and sensor characteristics as well Fig. 4. Wave modes useful for matching ridges with computed

arrival curves.as the coupling of the sensor to the structure.

623S. Holland et al. / Ultrasonics 38 (2000) 620–623

level is likely the (2ab) interference terms in thedistribution.

By following the matching procedure outlined above,we have found that we can solve the inverse problem ofdetermining source–receiver offset from a single mea-sured signal. Fig. 5 shows a sample result. The source–detector separation d is most easily determined from thetemporal maxima and minima of the arrival curves inthe time–frequency plane. It is at these points that thesignal is particularly concentrated in time; it is also atthese points that modes suddenly appear or suddenlydisappear [3]. Unfortunately, the number and qualityof such identifiable arrivals is sometimes limited. Thecomplexity of the matching procedure precludes simpleautomation. We have found that we can obtain accuratemeasures of the source–receiver separation by manuallyFig. 5. Sample inverse problem in which source–receiver separationmatching calculated arrival curves to the entire time–was determined.frequency distribution.

responding to a normal force and the normal displace-ments were generated for given material properties and

5. Conclusionssource–receiver separations. With an infinite signal-to-noise ratio, such signals represent the ideal case. Not

We have demonstrated a method for the time–fre-surprisingly, the proposed time–frequency analysisquency analysis of plate modes. This method has beenworks better at large source–receiver separations. Theapplied to the inverse problem of determining theincreased temporal spacing of arrivals at larger separa-source–detector separation. We have demonstrated thattions enhances the delineation between the differentthis separation can be determined from a single signal.modes in time–frequency space.

To demonstrate the utility of this approach, wecarried out a number of experiments using a capillary-

Referencesbreak on the surface of a glass plate as a normal force,step-source. Although the experimental data exhibit less

[1] F. Auger, P. Flandrin, O. Lemoine, P. Goncalves, Time–Frequencydetail and fewer arrivals than the corresponding syn-Toolbox for Matlab, http://crttsn.univ-nantes.fr/auger/tftb.html,

thetic data, the time–frequency distribution of the 1998.detected signals exhibited sufficient detail for solving the [2] K.F. Graff, Wave Motion in Elastic Solids, Dover Publications,

New York, 1991.inverse source–receiver separation problem.[3] R.L. Weaver, Y.H. Pao, Axisymmetric elastic waves excited by aWe have also compared the results obtained using

point source in a plate, Trans. ASME J. Appl. Mech. 49 (4)the Wigner–Ville distribution and the spectrogram. We(1982) 821–836.

found that the Wigner–Ville distribution has a higher [4] W. Sachse, S. Sancar, Acoustic emission source location on plate-background noise level, although it provides a slightly like structures using a small array of transducers, US Patent

No. 4,592,034, filed 15 November 1982, issued 27 May 1986.sharper image. The source of this background noise