REPEATABILITY OF AN ABLATION SOURCE USING A TIMEFREQUENCY REPRESENTATION
Kritsakorn Luangvilai and Laurence J. Jacobs
School of Civil and Environmental Engineering
Georgia Institute of Technology
Atlanta GA 30332-0355 USA
ABSTRACT. Ultrasonic Lamb waves attenuate when propagating through viscoelastic plates.
Experimental attenuation values are usually obtained with transducers by comparing measured timedomain signals at two different locations. However, this method can measure only a few points on the
attenuation-frequency dispersion curves, and usually only for some specific mode. It has been shown
that multi-mode dispersion curves (plus their associated energy distribution) can be obtained by
operating on the transient, time-domain signals, generated and detected with laser ultrasonics. This
technique uses proper signal processing technique, namely, a time-frequency representation (TFR).
The current research uses a similar methodology for Lamb wave attenuation measurements which
provides a series of attenuation values for many modes. As a first step, the laser source repeatability
(both amplitude and frequency spectrum) must be verified. This research studies the variation of a
laser ablation source by considering the surface wave propagating on an isotropic half-space. A series
of experimental time-domain surface wave signals from different propagation distances are operated
on with a TFR and compared in, strictly, propagation-distance-independent energy slownessfrequency domain. The amplitude constant is defined for comparison purposes. Also, the proposed
signal processing technique is instituted in an error-free manner.
INTRODUCTION
In plate-like structures, ultrasonic Lamb wave attenuation is one of the important
characteristics of viscoelastic materials. This property is normally presented as a function
of frequency and individual modes, as shown in Figure 1. Note that these curves are the
imaginary parts of the wavenumbers which satisfy the complex dispersion relation at a
given frequency. The dispersion relation for viscoelastic materials is the same as that of
elastic materials, except that all wave speeds (longitudinal and shear) and wavenumber are
complex numbers. To experimentally obtain those curves, conventional single-source
generation, multi-receiver detection using transducers can be employed. Time-domain
signal decay between two locations can be used to calculate attenuation. However, this
method usually gives only a few points in a narrow frequency bandwidth and for only a
single mode, due to the limitations of transducers. This research proposes using laser
ultrasonic techniques to experimentally measure attenuation in viscoelastic plates; laser
ultrasonics enables the broadband generation and detection of multi-mode Lamb waves.
Moreover, the laser system is a non-contact, point-like system, which minimizes the
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
1509
source/receiver
source/receivervariation
variationdue
duetotoother
otherfactors
factorssuch
suchasastransducer
transducercoupling
couplingororpressure
pressure on
on
transducers,
transducers,and
andreduces
reducesdiffraction
diffractionand
andnear
nearfield
fieldeffects.
effects.
Previous
Previous research
research shows
shows that
that the
the use
use ofof laser
laser ultrasonics
ultrasonics with
with aa time-frequency
time-frequency
representation
representation (TFR)
(TFR) signal
signal processing
processing technique
technique isis very
very effective
effective inin obtaining
obtaining the
the
dispersion
dispersioncurves
curvesofofan
anisotropic
isotropicelastic
elasticplate
plate[1,2].
[1,2].These
Theseresults
resultsshow
showhow
howgroup
groupvelocity
velocity
changes
changeswith
withfrequency
frequency(in
(intwo
twodimensions),
dimensions),and
andare
arequalitatively
qualitativelyencouraging,
encouraging,providing
providing
ananexcellent
excellentmatch
matchwith
withnumerical
numericalsimulations.
simulations.However,
However,the
theTFR
TFRgives
givesthree-dimensional
three-dimensional
results;
results;the
thethird
thirddimension,
dimension,besides
besidestime
timeand
andfrequency,
frequency,ininprinciple,
principle,represents
represents energy
energy
content
contentthat
thatcorresponds
correspondstotoeach
eachtime-frequency
time-frequencycell.
cell.Since
Sincethis
thisenergy
energydimension
dimensionhas
hasnot
not
been
beenquantified,
quantified,this
thisresearch
research studies
studiesit,it,and
andtries
triestotouse
useitittotoextract
extractenergy
energyinformation
information
from
fromexperimental
experimentaltime
time signals.
signals.This
Thisenergy
energyinformation
information will
will then
then lead
lead toto attenuation
attenuation of
of
ultrasonic
ultrasonicLamb
Lambwave
wavemodes.
modes.
To
Toexperimentally
experimentallymeasure
measureattenuation
attenuationininviscoelastic
viscoelasticplates,
plates,the
theentire
entireprocess
processcan
can
be
bedivided
dividedinto
intothree
threestages.
stages.First,
First,source
sourcerepeatability
repeatabilityisisstudied
studiedand
andverified.
verified.Surface
Surfacewave
wave
signals
signalscan
canbe
beused
usedatatthis
thisstage.
stage.These
Thesesurface
surfacewaves
wavesare
arenon-dispersive
non-dispersiveand
andsingle-mode;
single-mode;
they
they are
are appropriate
appropriate toto be
be used
used asas benchmark
benchmark test
test signals.
signals. Second,
Second, the
the proposed
proposed signal
signal
processing
processing algorithm
algorithm isis tested
tested with
with non-stationary,
non-stationary, dispersive,
dispersive, and
and multi-moded
multi-moded signals.
signals.
Lamb
Lambwaves
waves inin an
an elastic
elastic plate
plate are
are fitted
fitted atat this
this stage,
stage,because
because the
the attenuation
attenuation of
of signal
signal
amplitudes
amplitudesisisknown
knowntotofollow
followthe
the 1/Vr
1/√rspreading
spreadingrule.
rule.Lastly,
Lastly,the
thesame
samealgorithm
algorithm can
canbe
be
applied
appliedtototime-domain
time-domain Lamb
Lamb wave
wave signals
signals from
from aa viscoelastic
viscoelastic plate.
plate. Attenuation
Attenuation in
in this
this
plate
plate can
can be
be obtained
obtained after
after accounting
accounting for
for geometric
geometric spreading.
spreading. This
This paper
paper presents
presents the
the
results
resultsofofthe
thefirst
firststage
stage--source
sourcerepeatability.
repeatability.
TIME-FREQUENCY
TIME-FREQUENCYREPRESENTATION
REPRESENTATION
The
The time-frequency
time-frequency representation
representation (TFR)
(TFR) shows
shows energy
energy localization
localization in
in both
both the
the
time
time and
and frequency
frequency domains.
domains. The
The TFR
TFR gives
gives information
information as
as to
to the
the time
time atat which
which each
each
frequency
frequencycomponent
componentarrives.
arrives.Among
Amongthe
themany
manycandidate
candidate TFRs,
TFRs, aa windowed
windowed Fourier
Fourier
100
90
80
Attenuation (Np/m)
70
60
50
40
30
20
10
0
0
0.2
0.4
0.6
0.8
11
1.2
0.8
1.2
Frequency
F re q u e n c y (M
(M Hz)
Hz)
FIGURE
FIGURE1:
1:Attenuation
Attenuationof
of Lamb
Lamb waves.
waves.
1510
1.4
1.6
1.8
2
transform or short-time Fourier transform (STFT) has proved to be appropriate for
transient, multi-mode Lamb wave signals, therefore an STFT will be used for transient
surface wave signals.
The concept of the STFT is that the time-domain signal is segmented into
overlapping portions. Each portion is multiplied by a window function with the same
length and then Fourier transformed. Each portion approximates the frequency distribution
at the center time in that duration. Mathematically, in discrete format, the STFT value can
be written as [3]:
(1)
where f[n] is the sampled time-domain signal, and g[n] is a symmetric window of unity
norm and length N.
In addition to the STFT, the reassignment method is used to refine the TFR's
resolution. The reassignment algorithm compresses the amplitude contours to their centers
of gravity, thus reducing the spread of amplitudes in the time- frequency map [4]. Note that
this reassignment algorithm does not improve the accuracy of the TFR. The reassignment
procedure has the advantages that it improves the readability of the TFR, and that it refines
the fixed time-frequency grid into a fixed slowness-frequency grid. The latter will be very
useful when comparing Lamb wave signals with different propagation distances, since a
fixed time- frequency grid will result in a variable slowness-frequency grid.
GEOMETRIC SPREADING OF SURFACE WAVES
Laser excitation on the surface of semi-infinite medium (or half-space) naturally
generates surface waves. For a given excitation function of source, the transient wave
propagation problem, which is a boundary- value problem, can be solved conventionally by
integral transform techniques [5]. Since every function can be decomposed into the
superposition of time-harmonic functions by the use of a Fourier series, it is reasonable to
consider the time-harmonic solution (or solve the problem in frequency domain). As a
result, the problem requires only a single transformation in space. In the present case, when
the point excitation is the normally applied stress, time-harmonic solutions (stresses and
displacements) for isotropic elastic half-space can be written in the form:
a(r,z,t) = A(r,zy(kr-'a)
(2)
where a is any field quantity, A is a complex amplitude, and cylindrical coordinates are
employed - r is distance from the source, z is the distance into the medium, and t is time.
Following Miller and Pursey [6], the amplitude on the surface, A can be separated as:
A(r,0)=§r
(3)
Vr
where AQ is a constant amplitude. From (3), clearly, the magnitudes of any quantities
decrease as 1/Vr.
EXPERIMENTAL PROCEDURE
This study uses an aluminum half-space specimen. A Q-switched, Nd:YAG laser,
with the pulse energy that generates ultrasonic waves in ablation regime, is used to generate
1511
ultrasonic
normally applied
applied stress
stress source,
source, which
which
ultrasonic waves.
waves. The
The ablation
ablation source
source behaves
behaves like
like aa normally
naturally
generates
a
surface
wave.
On
the
detection
side,
a
heterodyne
interferometer
is
naturally generates a surface wave. On the detection side, a heterodyne interferometer is
used
to
detect
the
out-of-plane
motions
at
the
specimen
surface.
The
measured
time-domain
used to detect the out-of-plane motions at the specimen surface. The measured time-domain
signals,
mm every
every 1.5
signals, for
for propagation
propagation distances
distances ranging
ranging from
from 30
30 to
to 79.5
79.5 mm
1.5 mm,
mm, are
are digitally
digitally
recorded
recorded by
by an
an oscilloscope,
oscilloscope, and
and transferred
transferred to
to aa personal
personal computer.
computer.
The
the STFT.
STFT. The
The resulting
resulting TFR
TFR is
is then
then
The time-domain
time-domain signals
signals are
are operated
operated on
on with
with the
reassigned.
independent of
of propagation
propagation distance,
distance, the
the time
time axis
axis is
is
reassigned. In
In order
order to
to make
make the
the TFR
TFR independent
normalized
by dividing
dividing by
by propagation
propagation distance.
distance. The
The TFR
TFR
normalized to
to energy
energy slowness
slowness (Sl
(Slee)) axis
axis by
now
domain, which
which in
in this
this
now presents
presents the
the dispersion
dispersion curve
curve in
in energy
energy slowness-frequency
slowness-frequency domain,
case,
is
expected
to
consist
of
a
single,
non-dispersive
mode
propagating
with
the
surface
case, is expected to consist of a single, non-dispersive mode propagating with the surface
wave
defined as
as the
the real
real part
part
wave slowness.
slowness. From
From the
the reassigned
reassigned STFT,
STFT, the
the amplitude
amplitude constants,
constants, defined
of
and then
then plotted
plotted against
against the
the
0 in
of AAo
in (2),
(2), at
at some
some typical
typical frequencies
frequencies are
are calculated,
calculated, and
propagation
window of
of 385-point
385-point length
length
propagation distance,
distance, r.r. Note
Note that
that in
in this
this study,
study, aa rectangular
rectangular window
and
frequency of
and overlapping
overlapping of
of 192-point
192-point length
length (from
(from the
the sampling
sampling frequency
of 100
100 MHz)
MHz) is
is used
used in
in
the
the STFT
STFT algorithm.
algorithm.
To
any possible
possible errors
introduced by
by the
the
To make
make sure
sure that
that the
the results
results are
are free
free from
from any
errors introduced
processing,
the whole
whole part
part of
of the
the surface
surface wave
wave
processing, the
the cases
cases when
when the
the window
window does
does not
not cover
cover the
signal
the variation
variation of
of laser
laser ablation
ablation
signal are
are excluded.
excluded. The
The variation
variation in
in the
the remaining
remaining results
results is
is the
source
source only.
only.
RESULTS
RESULTS AND
AND DISCUSSION
DISCUSSION
A
by the
the
A typical
typical time-domain
time-domain surface
surface wave
wave signal
signal and
and its
its spectrum
spectrum calculated
calculated by
normal
high signal-to-noise
normal Fourier
Fourier transform
transform isis shown
shown in
in Figure
Figure 2.
2. This
This plot
plot shows
shows the
the high
signal-to-noise
ratio
typical aa TFR
TFR in
in
ratio and
and broad
broad frequency
frequency bandwidth
bandwidth of
of the
the optical
optical system.
system. Figure
Figure 33 shows
shows typical
the
the energy
energy slowness-frequency
slowness-frequency domain,
domain, and
and its
its cross-section.
cross-section. This
This plot shows that energy
localization
localization arrives
arrives with
with the
the slowness
slowness of
of 0.00033-0.00034
0.00033-0.00034 s/m which corresponds to the
surface
surface wave
wave slowness
slowness in
in aluminum.
aluminum. To
To calculate
calculate the amplitude constant at some specific
specific
frequency, the
the cross-sectional
cross-sectional area
area at
at that
that frequency is computed and multiplied by the
frequency,
square root
root of
of the
the propagation
propagation distance.
distance. The
The amplitude
amplitude constants are then plotted versus
square
propagation distances
distances for
for some
some specific
specific frequencies
frequencies as shown in figure 4 and 5. Most of the
propagation
errors in
in the
the amplitude
amplitude constants
constants (about
(about 94%
94% of
of the
the results) are in the order of 7%. A large
errors
0.03
0.02
0.01
0
|
f-0.01
I
-0.02
-0.03
-0.04
-0.05
10
11
12
13
14
15
16
17
19
20
2
3
4
5
6
7
Frequency (MHz)
(MHz)
Frequency
Time(µs)
(MS)
Time
FIGURE2:2:Typical
Typicaltime-domain
time-domainand
andfrequency
frequencyspectrum
spectrumofofsurface
surfacewave
wavesignals.
signals.
FIGURE
1512
variation (larger than 10%) is observed at high frequencies and a large propagation
variationThe
(larger
than
is observed
at high frequencies and a large propagation
distance.
reasons
are 10%)
discussed
as follows.
distance.
The reasons
as follows.signal, the major differences in shape from
Looking
closelyare
at discussed
each time-domain
Looking
time-domain
signal,the
themain
majorportion
differences
shape peaks
from
one signal
to theclosely
others atis each
in the
trail following
(the in
smaller
one signal
the others
is inThat
thepart
trailoffollowing
main portion
(the smaller
following
twotopositive
peaks).
the signalthe
is higher
in frequency
than thepeaks
main
following
two positive
peaks).ofThat
of the
signal
is higher
in frequency
the main
part.
As a result,
the spectrum
the part
surface
wave
portion
is naturally
more than
sensitive,
i.e.
part.
As
a
result,
the
spectrum
of
the
surface
wave
portion
is
naturally
more
sensitive,
i.e.
more varied, from measurement to measurement, at higher frequencies than at lower
more varied,
from
measurement
to measurement,
at higher
frequencieswindow
than at(which
lower
frequencies,
when
compared
to the center
frequency. Since
the rectangular
frequencies,
when
compared
to
the
center
frequency.
Since
the
rectangular
window
(which
results in the largest sidelobes) is used, this variation at high frequencies is then more
results in the
is distances
used, this (and
variation
high frequencies
is then more
pronounced.
Forlargest
large sidelobes)
propagation
their atassociated
lower signal-to-noise
pronounced.
large propagation
distances
associated
signal-to-noise
ratio),
the mainFor
characteristics
of the waves
are (and
moretheir
easily
distorted lower
by other
factors such
the main in
characteristics
of inside
the waves
are the
more
easily and
distorted
othersystem.
factorsThese
such
asratio),
imperfections
material (both
and on
surface)
noise by
in the
as
imperfections
in
material
(both
inside
and
on
the
surface)
and
noise
in
the
system.
These
distortions will introduce some variation to the results especially, again, for the high
distortions
will introduce some variation to the results especially, again, for the high
frequency
components.
frequency components.
CONCLUSION
CONCLUSION
This research uses the STFT technique to verify the 1/√r geometric spreading of the
This
the The
STFT
technique
the lA/r
geometric
of the
amplitudes
ofresearch
surface uses
waves.
variation
of to
theverify
amplitude
constants
is spreading
the variation
of
amplitudes
of
surface
waves.
The
variation
of
the
amplitude
constants
is
the
variation
of
laser ablation source that is due to the complicated ablation mechanism. The results show
laser ablation source that is due to the complicated ablation mechanism. The results show
that the variation is about 7%, provided that the signal-to-noise ratio is sufficiently large as
that the variation is about 7%, provided that the signal-to-noise ratio is sufficiently large as
presented in this research. This condition emphasizes the careful design of the experiments.
presented in this research. This condition emphasizes the careful design of the experiments.
Besides that, in order to use the TFR to correctly extract the energy (in terms of amplitude)
Besides that, in order to use the TFR to correctly extract the energy (in terms of amplitude)
distribution, the parameters of the algorithm are critical and need to be properly selected. In
distribution, the parameters of the algorithm are critical and need to be properly selected. In
the results shown here, both the repeatability of the source and the processing procedure
the results shown here, both the repeatability of the source and the processing procedure
support the idea of using this methodology to experimentally measure attenuation of Lamb
support the idea of using this methodology to experimentally measure attenuation of Lamb
waves
wavesininviscoelastic
viscoelasticplates.
plates.
-4
x 10
5 x 10
5
4.5
4.5
4
4
Energy slowness (s/m)
3.5
^ 3.5
I
3
3
2.5
2.5
2
1.5
1.5
11
0.5
0.5
00
0
Magnitude
Magnitude
1
2
33
44
55
66
Frequency (MHz)
7
8
9
Frequency
Frequency (MHz)
(MHz)
FIGURE3:3:Energy
Energyslowness-frequency
slowness-frequencydispersion
dispersioncurve
curveand
anditsitscross-section
cross-sectionofofthe
thesurface
surfacewave.
wave.
FIGURE
1513
10
10
1MHs
•M.
2
1.5
A
1
" ' • • * * . ^*N- .» **^
1j5
1
V-V ..'«..*• ->..?••
0.5
**
n
°0
20
40
00
80
"0
20
r(mm)
xHl"*
1j4>MHz
x lOf*
2.5
2J5
2
2
1-B
1jB
1
1
as
O.fl
n
ao
40
r(mm)
40
80
80
r(mm)
eg
1.6 MHz
*„** ». * ***<
•
•. #**
n
w
2G
40
BO
60
r(mm)
FIGURE 4: Experimental amplitude constants vs propagation distance at 0.6, 1.0, 1.4, 1.8 MHz.
2.5
2.5
2
•
* *"
4
k'
2
m
*
•»"
•'
1
1
0.5
0-5
I
20
40
W
°(»
00
20
40
60
«0
i-(mm)
jt^
X|Q4
3 MHZ
2-5
2.5
2
2
1.B
>4MHZ
1.B
*•-*.-.> v:^,/
1
""v—^ -
1
0.5
0.5
n
n
20
40
60
BO
20
r(mm)
40
60
80
T(mm)
FIGURE 5: Experimental amplitude constants vs propagation distance at 2.2, 2.6, 3.0, 3.4 MHz.
1514
ACKNOWLEDGEMENTS
This research is partially supported by the National Science Foundation under grant
number CMS-021283.
REFERENCES
1. Niethammer, M., Jacobs, L., Qu, J., Jarzynski, J., J. Acous. Soc. Am. 109(5), 1841-1847
(2001).
2. Valle, C. Niethammer, M., Qu, J, Jacobs, L., J. Acous. Soc. Am. 110(3), 1282-1290
(2001).
3. Mallat, S., A Wavelet Tour of Signal Processing, 2nd edition, Academic Press, Great
Britain, 1999, pp.67-68.
4. Auger, F. and Flandrin, P., IEEE Trans. Sig. Proc. 43(5), 1068-1089 (1995).
5. Achenbach, J. D., Wave Propagation in Elastic Solids, Elsevier, Netherlands, 1973,
pp.262-325.
6. Miller, G. F. and Pursey, H., Proc. R. Soc. A233, 521-541 (1954).
1515