The determination of propagation path lengths of dispersive flexural waves through structures

Journal

ofSoundand

Vibration (1981) 75(3). 453-457

LETTER

TO THE EDITOR

THE DETERMINATION OF PROPAGATION PATH LENGTHS OF DISPERSIVE FLEXURAL WAVES THROUGH STRUCTURES 1.

INTRODUCTION

In the past, path lengths and delay times of flexural waves between two points on a structure have been determined by using either cross-correlation, for example [l], or related functions [2]. The use of these methods is difficult in a system having only one or two propagation paths, since the cross-correlograms of transmitted and received signals are, of necessity, broad. A greater number of paths would increase the difficulty further. Recently, two independent works have shown how dispersed signals may be recompressed to give sharp arrivals provided the dispersion law is known. The first of these works [3], which is concerned with propagation paths through mechanical structures, differs slightly from the second [4], which is concerned with geological survey contexts, in that convolving the impulse responses of Booer et al. with the Fourier transform of the group velocity will give the impulse responses of Pavic and White. The authors present here experimental confirmation that these methods do work in a multipath situation and enable separate arrivals, that would otherwise be indistinguishable, to be resolved. 2.

PROPAGATION

THROUGH

A DISPERSIVE

MEDIUM

It is well known that the impulse response, h(t), at a point on a structure as a result of a signal input at some other point is the Fourier transform of the transfer function, H(w), between the two points. For discrete paths in a non-dispersive medium, h(t) will consist of delta functions in the time domain, each delta function being at a position corresponding to the time delay down the corresponding path. However, if the paths are dispersive then the arrivals will no longer be sharp, but spread out and for multiple paths, overlap may prevent discrimination of any of the arrivals. Almost simultaneously and independently Pavic and White, and Booer et al. realized that, provided the dispersion relation of the separate paths is known the dispersed impulse responses can be recompressed to give sharp arrivals again by remarkably simple processing. Their algorithms are not quite the same, but the connection between them is very close. Booer et al. effectively change the variable of the transfer function from w, the angular frequency, to k, the wavenumber, through the dispersion law of the paths. Whence Fourier transforming H(k) yields (theoretically) delta functions for discrete paths: i.e., the modified “impulse response” is given by

h(l) = [_I H(k(w))

eik’dk.

(1)

Note that changing the variable to wavenumber even puts the correct dimensions of length in the inverse transform domain. By contrast, Pavic and White stay in the frequency domain and derive from the complex conjugate transfer function H*(f), the quantity p(l; fl, f~): p(f;fi&)=& 2

1

f2 H*(f) exp (-ik(f)f) I ft

df.

(2)

453 0022-460X/81/070453+05

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@ 1981 Academic

Press Inc. (London)

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454

LETTER

TO

‘THE

EDITOR

Note that, apart from the slightly different notation (Pavic and White do not explicitly use the transfer function), equation (2) here corresponds to their equation (6). Now, at first sight the two methods, as exemplified by equations (1) and (21, appear dissimilar, but change of variable from frequency to wavenumber in equation (2) gives 1

~(1; kl, kz)=2x(f2_fl)

I

H*(k(w)

exp (-ikl))

gdk,

whereupon, apart from the scaling factor of 27r ( fz -f,) and the finite limits of integration, the only essential difference between equations (3) and (1) is that equation (3) has an additional dwldk, the group velocity, in the integrand. Hence, to obtain Pavic and White’s function, p, from Booer et af.‘s corrected impulse response, h(I), one convolves h(r) with the Fourier transform of the group velocity. Thus, theoretically, Pavic and White’s function will be broadened from that of Booer et al. by the convolution. Of course, in practice, dw/dk is slowly varying and the practical differences between the two methods will be very small. In the experiment described below, equation (1) is used for reduction of the data. 3. EXPERIMENTAL

SET-UP

AND

PROCEDURE

In order to test the validity of equation (1) to determine path lengths in a simple multipath situation, a steel bar of thickness O-015 m and length 1.72 m was used. Flexural waves were generated at one end by an electrodynamic shaker and the resulting signal was received at the other end by a piezoelectric accelerometer. Since multiple reflections occur at the free ends of the bar, it is a multipath system. By driving the shaker with rapid sine wave sweeps the complex transfer function was determined by using Hewlett-Packard 5420A signal processing equipment. The dispersion relation for flexural waves along a uniform bar of thickness t is, according to elementary theory, with neglect of rotary inertia and shear effects, k = (2rf”2)/(Et2/12p)1’4, 1

300

-400 l

I

(4) I

I

1

1

I 4 25.000

I

0.0 L(m)

Figure

1. The function

h(l) in arbitrary

units plotted

against

length

(m) for

square root dispersion law.

LETTER

TO

THE

455

EDITOR

where E is Young’s modulus and p is the density. For the bar used, (Et’/12~)“~ was 4.70 m sWif2.By using the dispersion relation (4) the transfer function mapped into the wavenumber domain and its corresponding spatial impulse response, h(l), was obtained. The result is shown in Figure 1. The figure shows paths corresponding to l-7 m, 5.1 m. and 8.5 m. The observed amplitude density function also contains a high frequency modulation, the reason for which is not understood at present but is thought to be due to the reflectivity at the ends of the bar being a complex function of frequency. However, the modulation can be removed by a Hilbert transform on h(I) to obtain its imaginary component. The modulus of the resulting complex quantity gives the envelope of h(I). The result of this is illustrated in Figure 2 and shows the path lengths more clearly. It is possible

1

4501

50

25.000

0.0

Figure

2. The envelope

of h(l) plotted

against

Time

Figure

3. The impulse

response

I(m) for square

root dispersion

(5)

function

(dispersion

not corrected).

law.

456

LETTER

TO

THE

EDITOR

to see at least six arrivals in the figure. This contrasts with the envelope obtained without considering flexural dispersion as shown in Figure 3, in which only the first three arrivals are well separated. Although the square root dispersion law separates the peaks of h(l), the peaks do become broader for the longer paths indicating that the dispersion law of equation (5) is not quite correct. In fact, it is possible to derive a better dispersion law, in this particular case. This law is found to be k = const. xP.“~. If this dispersion relation is used, sharper peaks in the function h(Z) are obtained (see Figure 4) and it is possible to discern seven arrivals across the whole of the window. Undoubtedly, more could be made visible by extending the window. However, this would require a greater number of points for the FFT which is not available on the current equipment.

18.000 1 (m)

Figure 4. The variation of h(l), for k = const. x?‘*~ dispersion law with length (m).

4. DISCUSSION The results presented here demonstrate that the method of Booer et al. and of Pavic and White works well in the simple case of flexural waves propagating in a single simple beam and offers hope for application to more complex structures. ACKNOWLEDGMENTS

This work has been carried out with the support of Procurement Executive, Ministry of Defence. The authors would also like to thank Dr A. K. Booer for his advice and guidance during the course of the work. The Plessey Company Limited, Plessey Marine Research Unit, Wilkinthroop House, Templecombe, Somerset, England

P. R. BRAZIER-SMITH-~ D. BUTLER J. R. HALSTEAD

(Received 6 March 1980, and in revised form 27 November 1980) t Now with Topexpress Limited, 1 Portugal Place, Cambridge CB5 8AF, England.

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457

REFERENCES Cross1. P. H. WHITE 1969 Journal of the Acoustical Society of America 45, 1118-1128. correlation in structural systems: dispersion and non-dispersion waves. 2. P. J. HOLMES 1974 Journal of Sound and Vibration 35, 253-297. The experimental characterization of wave propagation systems: I. Non-dispersive waves in lumped systems. II. Continuous systems and the effects of dispersion. 3. G. PAVIC and R. G. WHITE 1977 Acustica 38, 76-80. On the determination of transmission path importance in dispersive systems. 4. A. K. BOOER, J. CHAMBERS and I.M. MASON 1977 Electronic Letters 13, 453-455. Fast numerical algorithm for the recompression of dispersed time signals.