PARABOLIC MIRROR AND AIR-COUPLED TRANSDUCER FOR
MULTIMODAL PLATE WAVE DETECTION
Bernard Hostenand Michel Castaings
Laboratoire de Mecanique Physique, Bordeaux 1 University, UMR C.N.R.S. 5469, 351 cours de
la Liberation, 33405 Talence Cedex, France
ABSTRACT. A conventional contact piezoelectric transducer, excited by a broadband burst (chirp) that
covers its whole frequency bandwidth, is used as a transmitter to simultaneously generate several Lamb waves
in a plate. Throughout the propagation, these modes leak energy into the ambient air, producing bulk waves in
many directions. The association of a parabolic mirror and an air-coupled transducer (PMAT) allows these
waves to be received in air with a large angular aperture. By displacing the PMAT in a direction parallel to the
plate, but without changing its orientation like when standard air-coupled receivers are used, a series of
temporal waveforms are captured. Signal processing then allows the phase velocity of the several Lamb waves
to be measured in large wave number and frequency domains. These resulting data are used to identify the
moduli of elasticity for composite plates made of long glass fibers and polymer matrix.
INTRODUCTION
When using Lamb waves in NDT applications, it is usual to simplify the interpretation of
the waveforms related to the transmission or reflection of these waves by defects present in the
structure by generating a as pure as possible incident mode. In NDE applications, for instance
when recovering the mechanical properties of a material, it is common to launch Lamb waves
using the well-known solid edge, water-coupled or air-coupled techniques. In these methods,
the incident bulk wave is converted into Lamb modes. According to the angular aperture of the
incident field and to the Snell's law, these modes are either pure or limited in number.
In a recent QNDE meeting [1], a technique to identify the elasticity moduli of plates
made of composite materials, using plane air-coupled transducers, has been presented. The
advantage of this technique is that it is contact-less and single sided. However there is a
difficulty to apply the technique when the characteristics of the material are unknown. Indeed
to launch a Lamb mode in the plate, the transducers angles of orientation need to satisfy the
Snell's law condition. Since the material properties are unknown, the phase velocities of the
Lamb modes and so these angles are also unknown. Another difficulty is that as many modes
as possible need to be collected to be able to identify a unique solution of the material
properties. Then, the technique requires several scans at various angles [1].
In this paper, a new way to simultaneously launch and detect several Lamb modes is
presented. A conventional contact piezoelectric transducer, excited by a broadband burst
(chirp), is used as a transmitter. A parabolic mirror is then used to increase the angular aperture
of an air-coupled receiver. In consequence, the wave number range of the Lamb modes
propagating in the plate is very large. Adequate signal capture and processing then allow rapid
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/$20.00
1243
measurement of their phase velocities in a large frequency range. With this method, the elastic
properties
of the
bevelocities
estimatedineven
if the
materialrange.
is totally
measurement
ofplate
their can
phase
a large
frequency
Withunknown.
this method, the elastic
properties of the plate can be estimated even if the material is totally unknown.
AIR COUPLED TRANSDUCER AND PARABOLIC MIRROR (PMAT)
AIR COUPLED TRANSDUCER AND PARABOLIC MIRROR (PMAT)
There are many techniques to manufacture air-coupled transducers [2-6]. None of these
can employ
lenses technique
used air-coupled
for water-coupled
transducers,
due
the
Therethe
are classical
many techniques
to manufacture
transducers
[2-6]. None
of to
these
important
mismatch
betweenlenses
air and
solid acoustic
impedances.
The presence
of any
can employ
the classical
technique
used for
water-coupled
transducers,
due material
to the
mismatch
and solid the
acoustic
impedances.
presence
of any[7].
material
inimportant
transmission
modebetween
reducesairdrastically
energy
produced The
by the
transducer
More
in transmission
reduces
drastically
energy the
produced
by the
transducer
[7]. system
More
recently,
parabolicmode
mirrors
have been
used totheimprove
resolution
of surface
imaging
recently, parabolic mirrors have been used to improve the resolution of surface imaging system
[8,9].
[8,9]. The air-coupled transducer used in the present work is similar to the T17 transducer
transducer
in the present
is similar
to the
describedThe
in air-coupled
reference [10].
It is anused
electrostatic
planework
transducer,
made
of aT17
thintransducer
metalized
described
in
reference
[10].
It
is
an
electrostatic
plane
transducer,
made
of
a
thin
metalized
membrane (Polyethylene Terephtalate, thickness 5 µm) and a metallic back-plate.
A static
membrane
(Polyethylene
Terephtalate,
thickness
5 um) andThe
a metallic
back-plate.
staticby
voltage
of 100
volts is applied
between these
two electrodes.
back-plate
is made A
rough
voltage of 100
volts
applied
between
these
two electrodes.
is efficiency,
made roughtheir
by
sandblasting,
using
17isµm
in diameter
sand
grains.
To increaseThe
theback-plate
transducers
sandblasting,
using 17upum
grains. To
increase
the transducers
theiris
diameter
is increased
to in45diameter
mm. Ansand
advantage
of the
sandblasting
is that efficiency,
this technique
diameter
is increased
up to 45ofmm.
Antransducers.
advantage of the sandblasting is that this technique is
well
adapted
to the production
large
well adapted
to the production
large is
transducers.
An aluminum
parabolic of
mirror
machined and attached to the air-coupled transducer
aluminum
mirror any
is machined
and attached
the air-coupled
transducer
as shownAn
in Fig.
2. Theparabolic
mirror reflects
wave coming
from thetotransducer
towards
the focus
as
shown
in
Fig.
2.
The
mirror
reflects
any
wave
coming
from
the
transducer
towards
the focus
line of coordinates (p/2,0). The profile of the parabolic mirror is given by the equation:
line of coordinates (p/2,0). The profile of the parabolic mirror is given by the equation:
(1+ sinθ)
2
y = 2px and y = p (l+sinO)
y = 2px and y = p (cosθ)
(cosO)
(1)
0)
where the axis x, y and the angle θ are defined in Fig. 1. The distance y M2 − y M1 between the
where the axis x, y and the angle 6 are defined in Fig. 1. The distance y^ -yM1 between the
two
diameter a,
a, and
and the
the parameter
parameterpp
twoextremities
extremitiesofofthe
themirror
mirrormust
must be
be equal
equal to
to the
the transducer
transducer diameter
defines
the
minimum
and
maximum
angles.
In
this
realization,
with
p
=
94
mm,
the angle
angle
defines the minimum and maximum angles. In this realization, with p = 94 mm, the
range
is
around
20°.
range is around 20°.
DIRECTIVITY
DIRECTIVITYOF
OFTHE
THEPMAT
PMAT
The
the transducer
transducer alone
alone and
and then
thenthat
thatofofthe
the
Theefficiency
efficiency (acoustic
(acoustic pressure
pressure per
per volt)
volt) of
of the
PMAT
in reference
reference [10].
[10].Both
Bothtransducer
transducer
PMATwere
weremeasured
measuredwith
withthe
themembrane
membrane method
method presented
presented in
and
the efficiency
efficiency versus
versus the
the angle
angleofof
andPMAT
PMATwere
weresustained
sustained by
by aa goniometer
goniometer to
to measure
measure the
emission.
It
is
clear
in
Fig.
2,
that
the
angular
aperture
of
the
PMAT
is
much
larger
than
thatofof
emission. It is clear in Fig. 2, that the angular aperture of the PMAT is much larger than that
the
range is
is around
around20°,
20°,down
downtoto- –66dB.
dB.ItItisisalso
also
thetransducer.
transducer.As
Aspredicted,
predicted, the
the PMAT
PMAT angle
angle range
M2
Air coupled
Transducer
M
a
y2 =2px
M1
θ
x
p
F&O)
F ( ,0 )
2
FIGURE1. 1.Schematic
Schematicofofthe
theparabolic
parabolicmirror.
mirror.
FIGURE
1244
0
0
noticeable that the efficiency of the PMAT is about half that of the transducer alone. This
lower level of efficiency does not permit, with the actual technology and for the application
described in this paper, the use of two PMAT for emission and reception. For this reason, in
the following, a contact PZT transducer is used for generating the Lamb modes in the plate,
with a high level of energy.
CHIRP EXCITATION
Since one purpose of this work is the simultaneous generation of several Lamb modes, it
is important to excite the transmitter in the largest frequency domain compatible with the
bandwidth of the transducers. The frequency domain of the air-coupled transducer is around
100-500 Khz, at -15 dB. To precisely control the frequency domain, the emitter is exited with
a chirp calculated in a computer, sent to an arbitrary function generator and amplified by a
power amplifier. This chirp can be made as long as necessary in order to input sufficient
energy at each frequency component.
Figure 3 presents the temporal shape of the chirp and the corresponding frequency
spectrum.
3
Pressure (Pa/V)
2.2 -
0.75 _ .
0
5
15
10
20
25
35
30
FIGURE 2. Directivity at 180 KHz for transducer alone (•••) and PMAT (—).
Amplitude (A.U.)
Amplitude (A.U.)
-0.5 _
Frequency (MHz)
'0
20
40
60
80
0
Time in uS
0.1
0.2
0.3
0.4
0.5
FIGURE 3. Chirp waveform sent to the arbitrary function generator to feed the transmitter,
a) Time representation b) Frequency representation.
1245
0.6
0.7
0.8
PMAT
Air coupled
receiver
PMAT
Piezoelectric
transmitter
Air coupled
receiver
Piezoelectric
transmitter
Lamb modes in composite plate
Lamb modes in composite plate
FIGURE 4. Set up for the generation and the reception of several Lamb modes in a plate.
FIGURE 4. Set up for the generation and the reception of several Lamb modes in a plate.
PHASE
OFofMULTI-LAMB
FIGUREVELOCITY
4. Set up for theMEASUREMENT
generation and the reception
several Lamb modesMODES
in a plate.
PHASE
VELOCITY
OF MULTI-LAMB
MULTI-LAMB
MODES
PHASE
VELOCITY
MEASUREMENT
OF
MODES
Figure
4 presentsMEASUREMENT
the rig used to simultaneously
generate
and receive several plate
modes.Figure
A piezoelectric
transducer
is
placed
in
contact
with
the
plate.
It hasseveral
a frequency
44 presents
the
to simultaneously
simultaneously generate
generate and
and
receive
plate
Figure
presents
theofrig
rigtheused
used
to
receive
several
plate
bandwidth
similar
than
that
air-coupled
transducer.
The
PMAT,
used
as
a
receiver,
is
modes.
A
piezoelectric
transducer
is
placed
in
contact
with
the
plate.
It
has
a
frequency
modes. A
piezoelectric
transducer
is placed
contact
with of
thecontact
plate. Itforhasthisa frequency
sustained
and
translated
by
aofmotorized
table.intransducer.
The
absence
scanningisis
bandwidth
similar
than
that
the
air-coupled
The
PMAT,
used
as
a
receiver,
bandwidthtosimilar
than that of the
air-coupled transducer.
The PMAT,
used as awith
receiver,
is
mandatory
reach
reproducible
measurements.
is compatible
industrial
sustained
and
translated
by
aa motorized
table. This
The arrangement
absence of
of contact
contact
for this
this scanning
scanning
is
sustained
and
translated
by
motorized
table.
The
absence
for
is
constraint
since
the emitter
could measurements.
be for instanceThis
an imbedded
transducer
in the with
structure.
mandatory
to
reach
reproducible
arrangement
is
compatible
industrial
mandatory to reach reproducible measurements. This arrangement is compatible with industrial
is represented
by
the function
The set of the
waveformscould
captured
each an
position
x2 transducer
constraint
be for
forfor
instance
imbedded
thestructure.
structure.
constraint since
since the emitter
emitter could be
instance
an imbedded
transducer ininthe
One
example
of
the
intricate
waveforms
captured
by
the
PMAT
is
shown
infunction
Fig. 5. In
s(t, x 2 ).The
represented by
by the
the
function
each position
position xx22 isis represented
The set
set of
of waveforms
waveforms captured
captured for
for each
that
situation,
the
classical
signal
processing
[11]
that
transforms
the
space/time
representation
waveforms captured
capturedby
bythe
thePMAT
PMATisisshown
shownininFig.
Fig.5.5.InIn
ss(t,x
(t, x 22)).. One
One example
example of
of the
the intricate
intricate waveforms
to the wave-number/frequency
representation
useful to representation
separate
all the
s(that
t,
x 2situation,
S(ν, k ) is very
)situation,
the
[11]
that transforms
transforms
the space/time
space/time
representation
that
the classical
classical signal
signal processing
processing
[11] that
the
modes
in the waveforms. Thisrepresentation
is done by using
2D-Fourier
transform:
to the wave-number/frequency
is very
very useful
useful
to separate
separateall
allthe
the
ss(t,x
S(ν,the
k ) is
(t, x 2contained
to
2)) to the wave-number/frequency representation S(v,k)
+∞
+∞
modes
contained
in
the
waveforms.
This
is
done
by
using
the
2D-Fourier
transform:
modes
contained
in
the
waveforms.
This
is
done
by
using
the
2D-Fourier
transform:
S(ν, k ) =
s(t, x )exp(−iωt )dt exp(−ikx )dx
(2)
ò (ò
(
−∞+∞ −∞+∞
)
)
2
2
2
S(ν, k ) = ò|^J%(t,x
x 2 )exp(−iωt )dt exp(−ikx2)dx
(2)
(2)
ò−∞ s(t,2ν)exp(-icot)dt\xp(-ikx
2 )dx22 S(ν,k) give the wave numbers
−∞
, the maxima in the function
k of
ForS(v,k)=
each frequency
νv,, the
in Then,
the function
function
S(ν,k) give
give the
the wave
wave
numbers
kofof
For
the modes
that
are
generated
in maxima
the plate.
it is straightforward
to
deduce
the kphase
For each
each frequency
frequency
the
maxima
in
the
S(v,k)
numbers
the
that
generated
plate. Then,
Then, itit is
is straightforward
straightforward toto deduce
deduce the
the phase
phase
the modes
modes
thattheare
are
generated
inω the
the
plate.
C p =in
velocities
from
formula
k , where ω = 2πν .
=ω
,
where
ω
=
2πν
.
velocities
velocities from
from the
the formula
formula C
Cp =
%,
where
co
=
2iw.
k
1
Amplitude (V)
1
'Amplitude
(V)
Amplitude (V)
0.5
0.5
0.5
0
0
-0.5
-0.5
-0.5
-1
-1
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
Time (mS)
Time
Time (mS)
(mS)
0.7
0.7
0.7
0.8
0.8
0.8
FIGURE
5. 5.Waveform
to
several
Lamb
modes propagating
propagatingininthe
the
FIGURE
Waveformcaptured
capturedwith
withthe
thePMAT
PMATcorresponding
correspondingto
to several
several Lamb
Lamb modes
modes
FIGURE
5. Waveform
captured
with
the
PMAT
corresponding
propagating in
the
Perspex
plate
and
leaking
ininair.
Perspex
plate
and
leaking
air.
Perspex plate and leaking in air.
1246
Perspex Plate
The stress-strain relation for orthotropic solids involves nine independent moduli of
elasticity C^. In each plane of symmetry, the number is reduced to four. For instance in a plane
defined by the direction xi, normal to the plate, and the direction x 2 , in the plate and along the
scan, the stress-strain relation is given by:
C12
(3)
o
o
Although the Perspex is almost isotropic, the measurements of the moduli of elasticity in
this plane P12 (X}X2) with traditional immersion/transmission techniques reveal a small
anisotropy, as shown in Table 1. From these values, the dispersion curves are computed and
plotted for a 4 mm thick Perspex plate (Fig. 6). The plot of the velocity curves (Fig. 6a) shows
that with a slight tilt of the PMAT towards the negative angles, it is possible to measure
negative phase velocities of a specific mode. This mode is called S_2 to refer to the negative
values of its phase velocity. In the literature, it is sometimes referred as Sl and plotted with a
positive phase velocity [12]. Therefore its group velocity, defined by the formula Vg = ^^ is
negative. This is incompatible with the present experiment that measures only modes with
positive group velocity. With the representation as S_2, the group velocity is positive (Fig. 7).
Figure 6 also presents the comparison between the computed dispersion curves and the
measurements. Both phase velocities and phase slowness (inverse of velocities) are shown. It is
clear that although the phase velocity representation exhibits high measured values for modes
S2 and S_2, it masks the comparison between experiments and predictions for other modes.
However, the slowness representation displays quite well the results for the whole set of
modes.
Figure 6b also shows that the angular range of the modes radiated in the coupling
medium and captured with the PMAT is in the range of the PMAT directivity (~ -2..18 °).
Composite Material
The measurements of multi-Lamb modes are now done after propagation in a 6.5 mm
thick plate made of 12 layers ([0°,90°]6) of glass fibers and polyester matrix. The
characteristics of this industrial composite material are a priori unknown.
Figure 8 shows the results of the measurements with just one scan. Again the angular
range, around 20° (-4..16°), corresponds to the directivity of the PMAT. Then these
measurements are used to identify the best estimation of the moduli of elasticity, following the
algorithm defined in reference [1]. The uniqueness of the solution is reinforced since the
measurements are now spread over large angular and frequency domains and a larger number
of modes. The results of the inverse problem are given at Table 2 and used to plot the
dispersion curves for comparison with the measurements, as shown in Fig. 8.
TABLE 1. Elastic properties of the 4 mm Perspex plate.
_Thickness
_ _ _ _ _(mm)
_ _ . _-[ 4 -
-
^ ! Density
^ . (g/cm3]
^ ^
1.2
__
__
1247
a)
a)
Phase Velocity (mm/µS)
Phase Velocity (mm/[jS)
20
20 -
S
A
2
1
10
10
S
1
S
0
0
A
0
-10
-10
S
-2
-20
-20
Frequency
(MHz)
Frequency
(MHz)
0
b)
b)
0.1
0.1
0.2
0.2
Phase Slowness
Slowness (uS/mm)
(µS/mm)
1 r-
0.3
0.3
0.4
0.4
0.50.5
0.60.6
Angle (°)
-i 2020
A
0
0.8
16
S
A
0
0.6
12
1
0.4 0.4
8
S
0.2
0.2
S
1
4
2
0
0
S
-2
Frequency (MHz)
-0.2
-0.2
0
0.1
0.2
0.3
0.4
FIGURE 6. Dispersion curves for the 4 mm Perspex plate; a) Velocity, b) Slowness
FIGURE 6. Dispersion curves for the 4 mm Perspex plate; a) Velocity, b) Slowness
Predicted curves (—) and experimental data (xxx).
Predicted curves () and experimental data (xxx).
1248
0.5
0.6
-4
22
GroupVelocity (mm/µS)
_ GroupVelocity
GroupVelocity (mm/µS)
(mm/uS)
SS
S11
1.5
1.5
1.5
SS
2
2
1
1
0.5
0.5
0.5
S
S-2
-2
0
0
0.32I
0.32
0.32
0.36
0.4
0.36 0.4
0.36
0.4
Frequency (MHz)
Frequency
Frequency(MHz)
(MHz)
0.44
0.48
0.44
0.44
0.48 0.48
FIGURE 7. Group velocity for the S1,S2 and S-2 Lamb modes in the 4 mm Perspex plate.
FIGURE
Groupvelocity
velocityfor
forthe
theS8^82
and S_2 Lamb modes in the 4 mm Perspex plate.
FIGURE
7.7.Group
1,S2 and S-2 Lamb modes in the 4 mm Perspex plate.
Phase Slowness(µS/mm)
Phase Slowness(uS/mm)
Slowness(µS/mm)
1 Phase
1
A
1
0
A
0
0.8
0.8
0.8
Angle (°)
Angle (°)
Angle (°)20
20
20
15
15
15
0.6
0.6
0.6
1
10
A
0
1
S
0
5
0.2
0.2
0.2
S
0
5
S
2
1
S
0
S
2
1
0
0
S
-2
-0.2
-0.2
-0.2
10
10
A
S
0.4
0.4
0.4
Frequency
Frequency(MHz)
(MHz)
S
-2
0
0.05
0.05
0.1
0.1
0.15
0.15 0.2
0.2
0
0.05
0.1
0.15
0.2
-5
Frequency
(MHz)
0.25
0.3
0.35
0.4
0.25
0.3
0.35-5
0.25
0.3
0.35
FIGURE
FIGURE8.8.Dispersion
Dispersioncurves
curvesforforthe
theglass
glassfibers
fibers- polyester
- polyestermatrix
matrixcomposite
compositematerial.
material.
FIGURE 8. Dispersion curves for the glass fibers - polyester matrix composite material.
TABLE
TABLE2.2.Elastic
Elasticproperties
propertiesofofthe
theglass
glassfibers
fibers/polyester
/polyestermatrix
matrixcomposite
compositematerial.
material.
TABLE 2. Elastic
properties
glass fibers /polyester
Thickness
(mm)of the6.5
Densitymatrix
(g/cm3)composite
1.82 material.
^^—-——^^QLiSl^l-JLl^^
-f
C11
(GPa)
13.3
C12
(GPa)
7,8
Cll(GPa)
|13.3
jC12(GPa)
17,8
Thickness
(mm)
6.5
Density
(g/cm3) 3.2
C22
31
C66
(GPa)
C22(GPa)
(GPa)'"""""'""
'11T~~~~~~|'C66(aPa)
' 1.82
'
C11 (GPa)
13.3
C12 (GPa)
7,8
C22 (GPa)
31
C66 (GPa)
3.2
1249
0.4
0.4
CONCLUSIONS
A parabolic mirror has been used to increase the angular aperture of an air-coupled
transducer, thus allowing several plate modes radiating energy in air to be simultaneously
detected.
This device reduces the time of phase velocity measurements and considerably enhances
the process to measure the elastic properties of unknown materials. This technique is very
promising for NDE applications that aim to follow the variations of material properties due for
instance to humidity or mechanical constraints.
REFERENCES
1.
B.Hosten, M.Castaings, H.Tretout and H.Voillaume, "Identification of composite
materials elastic moduli from lamb wave velocities measured with single sided,
contactless ultrasonic method ", Review of Progress in Quantitative Nondestructive
Evaluation Vol. 20, ed. by D. O. Thompson and D. E. Chimenti 1023-1030 (2001).
2. D.Schindel, D.Hutchins, L.Zou, and M.Sayer, "Capacitance devices for the generation of
air-borne ultrasonic fields", IEEE Ultrasonic Symposium proceedings Tucson-USA (2)
(1992).
3.
M.I. Haller and B.T. Khuri-Yakub, "Micromachined 1-3 composites for ultrasonic air
transducers", Rev. Sci. Instrum. 65, 6 (1994).
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