Detection and Localization of Small Notches in Plates Using Lamb Waves
Brad M. Beadle*, Stefan Hurlebaus*, Laurence J. Jacobs**, Lothar Gaul*
*Institute A of Mechanics, University of Stuttgart
Allmandring 5 b, 70550 Stuttgart, Germany
[email protected]
**School of Civil and Environmental Engineering
Georgia Institute of Technology
Atlanta, Georgia 30332, USA
ABSTRACT
The interaction of the first antisymmetric (ao) Lamb wave in an aluminum plate with small surface notches (< 25%
of the plate thickness) is investigated. The study is motivated by the need for real-time, early detection of cracks
in plate-like structures. The experimental procedure uses a piezoelectric pinducer point source to generate and a
laser Doppler vibrometer to detect Lamb waves in the plate. A spectrogram of the measured plate motion is used
to isolate a nondispersive portion of the ao Lamb mode from the rest of the signal for use in the notch detection
algorithm. An efficient algorithm uses information about the notch reflected energy to locate the notch and to
calculate a reflection coefficient. An estimate of the smallest detectable notch is made based on the experimental
reflection coefficient vs. notch depth characteristic and the noise floor. A finite element model of the plate is used
to verify the experimental results.
1. Introduction
One approach to structural health monitoring is based on generating and detecting elastic waves in the structure
of interest. For plate-like structures, these waves are called Lamb waves. Lamb waves have received extensive
attention as a tool for active sensing diagnostics [1-7]. In a study by Pinto, et al. [8], Lamb waves were lasergenerated and detected in a perfect plate and a plate containing a surface notch having depths 25%-100% (notch
depth/plate thickness = 0.25-1.0). The notch was located via correlation of a series of slowness spectra which
were calculated from a time-frequency representation of the data. The signal processing employed is time
consuming and is therefore unsuitable for real-time structural health monitoring.
Similar to the approach employed in [8], the current study correlates time-frequency representations, specifically
the spectrograms, of measured data in order to locate the notch. However, this study concerns itself with the
detection of small (< 25%) surface notches. Additionally, only a small portion of the spectrograms are correlated
in this study, specifically that portion corresponding to a nondispersive region of the first antisymmetric (ao) Lamb
mode. The data processing is relatively fast and can be completed in real-time. Using the spectrogram, an
average reflection coefficient as a function of notch depth is then computed. Finally, the smallest experimentally
detectable notch is estimated based on the reflection coefficient vs. notch depth characteristic and the
experimental noise floor.
Described in section 2 is a finite element (FE) model of a notched plate in which Lamb waves are excited.
Presented in section 3 is the data processing method which uses the displacement-time history at the receiver
location on the plate to localize the notch and to compute an average reflection coefficient. In section 4, the
experimental setup is described, the experimental reflection coefficient vs. notch depth characteristic is compared
to the FE-calculated characteristic, and the smallest experimentally detectable notch is estimated.
2. Finite Element Model
Transient analysis of Lamb wave interaction with a surface notch in a plate is conducted using the commercially
available FE software package ANSYS. Depicted in figure 1 is the FE model of the notched plate, including the
plate geometry, discretization about the notch, and the load history. The dimensions and material properties
where chosen to match those for the experimental plate. The length dimension of 43 mm at the left side of the
plate was chosen as small as possible to expedite computational time, but such that no reflections from the fixed
end would be observed at the receiver location during the simulation period (0 – 60 µs). The plate is discretized
by 2-dimensional, plane-strain PLANE42 elements. A single element has 4 nodes, with each node having two
degrees of freedom, i.e. displacement in the x- and y- directions. The entire model contains approximately 30,000
nodes. As a general guideline, the element lengths should be less than 1/10 of the smallest wavelength of
interest, and the time step should be less than 1/10 of the shortest period of interest. Since the maximum
frequency of interest is 4 MHz in this study, the elements were chosen to have a side length less than 0.1 mm;
and, the time step was chosen to be 0.025 µs, with the out-of-plane displacement uy at the receiver location being
recorded every 0.1 µs. Simulations were performed for a perfect plate without a notch, and for the cases of a
plate having notch depths of h/w = 5%, 10%, 15%, 20%, and 25%. The various notch depths were realized
simply by removing the appropriate number of elements at the notch location. The elements composing the plate
all had identical shape so as to avoid the reflection of wave energy which occurs at nonuniform mesh locations.
Broadband Lamb wave excitation was realized using the impulsive ramp excitation at the source location, as
depicted in figure 1.
Figure 1. 2-D finite element model of a notched plate.
3. Signal Processing
Depicted in figure 2 is the FE-computed time history of the out-of-plane displacement uy at the receiver location
for the perfect plate case. A spectrogram of this time signal is shown to the immediate right. The spectrogram
was computed using the signal-processing toolbox in MATLAB. The advantage of the spectrogram is that
multimode, dispersive Lamb wave propagation can be readily analyzed. The spectrogram is constructed by first
chopping the time history into a series of overlapping windows, then taking the fast Fourier transform of each
window, and finally plotting the Fourier coefficient magnitudes as a function of the temporal window position. The
displacement-time history contains 601 points and was windowed using a Hanning window having a length of 60
data points, with an overlap of 58 data points between adjacent windows. This choice of windowing gives
adequate resolution along both the time and frequency axes. As discussed in [9], a short window gives good
resolution along the time axis, but poor resolution along the frequency axis; and, a long window gives good
resolution along the frequency axis, but poor resolution along the time axis. The first antisymmetric (ao) Lamb
mode and the first symmetric (so) Lamb mode are clearly present in the perfect plate spectrogram. These modes
correspond to Lamb waves which propagate directly from the source to the receiver. The occurrence of other
modes in the spectrogram correspond to higher-order Lamb modes generated at the source location and to
modes which reflect off one or more plate edges before arriving at the sensor location.
Figure 2. FE-computed displacement-time history and spectrogram depicting Lamb wave propagation in a perfect
plate (left) and the corresponding spectrogram for a 25% notched plate (right).
The right-hand spectrogram in figure 2 corresponds to the FE simulation of a plate which contains a 25% surface
notch. The presence of the notch is evidenced by the occurrence of energy patches in the notched plate
spectrogram which are not present in the perfect plate spectrogram. Of particular interest to this study is the
energy patch in the notched plate spectrogram centered at the location (1 MHz, 29 µs). This energy patch
corresponds to the energy from the source-generated ao Lamb mode which has been reflected by the notch and
which then propagates to the receiver as an ao Lamb mode. The temporal location of this energy patch is used to
locate the notch, and the energy content is used to compute a reflection coefficient.
Notch location is achieved by correlating the perfect plate spectrogram with the notched plate spectrogram. The
base region is depicted by a rectangular box drawn on the perfect plate spectrogram in figure 2. The rectangular
region of the spectrogram is encompassed by the frequency band fband = 0.6 – 1.75 MHz and the time band tband =
12-18 µs, and the region itself is denoted by Sperfect{fband , tband}. Because the energy in this region propagates
nondispersively, the group velocity cg of this energy partition can be calculated from
cg =
do
,
to
(1)
where do = 46 mm is the distance between source and receiver, and to is the time at which the energy (after being
averaged along the frequency axis) in the region {fband , tband} is maximum. The special correlation function is then
calculated,
Corr (t ) =
S perfect { f band , t band } ⊗ S notch { f band , t band + t} − S perfect { f band , t band + t}
S perfect { f band , t band } ⊗ S perfect { f band , t band + t}
.
(2)
Here, Sperfect{fband , tband + t} and Snotch{fband , tband + t} denote the portions of the perfect plate and notched plate
spectrograms, respectively, which are encompassed by the same rectangular box as with Sperfect{fband , tband}, with
the exception that the box center has been shifted by time t. The ⊗ operator denotes a matrix scalar product, that
is, given two equally-sized matrices A and B, then A ⊗ B =
Aij Bij . The correlation will achieve a maximum
∑∑
i
j
at the time corresponding to the roundtrip time it takes the ao mode to travel from the receiver, to the notch, and
back to the receiver. Accordingly, the estimated receiver-notch distance d(t) is given by
d (t ) =
t
cg
2
(3)
Depicted in figure 3 is a plot of the correlation Corr(t) as a function of receiver-notch distance d(t), as determined
from the FE simulation of a 25% notched plate. The correlation curve has minima at points a and d since the
denominator in equation 2 becomes large at the corresponding times. There is an extreme value at point a
because the denominator at this time is Sperfect{fband , tband} ⊗ Sperfect{fband , tband}, and the scalar product of a matrix
with itself is always large. There is an extreme value at point d occurring at a distance of d = 34 mm, because the
denominator is given by the scalar product of Sperfect{fband , tband} with a portion of the spectrogram having similar
energy content. This portion of the spectrogram, located in the region {0.6 – 1.75 MHz , 35-41µs} on the perfect
plate spectrogram in figure 2, corresponds to the source → right-hand plate edge → receiver path which the ao
mode takes. The right-hand plate edge basically behaves like a 100% crack in both the perfect plate and notched
plate cases. The denominator in equation 2 therefore filters out the effect which geometric boundaries have on
the correlation function.
a sender ⎯
⎯→ sender
so
ao
b sender ⎯⎯→
notch ⎯⎯→
receiver
ao
ao
c sender ⎯⎯→
notch ⎯⎯→
receiver
d
ao
ao
sender ⎯⎯→
edge ⎯⎯→
receiver
so
ao
ao
e sender ⎯⎯→
edge ⎯⎯→
notch ⎯⎯→
receiver
f
ao
ao
ao
sender ⎯⎯→
edge ⎯⎯→
notch ⎯⎯→
receiver
Figure 3. Correlation vs. receiver-notch distance determined from FE simulation of a plate having a 25% notch.
The correlation characteristic in figure 3 achieves a maximum at the distance d = 21.3 mm (point c). This
calculated distance corresponds to the distance between the receiver and the notch, and agrees fairly well with
the actual distance of do = 20 mm. There is an extreme value at point c because the numerator at this time is
given by the matrix scalar product of Sperfect(fband , tband) with a region of the notched plate spectrogram centered at
the location (1 MHz, 29 µs) having similar energy content (see figure 2). The subtraction operation performed in
the numerator in equation 2 serves the purpose of further isolating characteristics which are present in the
notched plate spectrogram but not in the perfect plate spectrogram. The Lamb wave paths associated with each
extreme on the correlation curve are summarized in figure 3. For example, the maximum at point b corresponds
to an so Lamb mode which emanates from the source location, propagates to the notch, is mode converted at the
notch, and finally propagates to the receiver location as an ao Lamb mode. The so Lamb modes arriving at the
receiver location influence the correlation curve minimally, since the associated out-of-plane displacement is small
compared to that for the ao Lamb modes.
An average reflection coefficient for the ao mode can be readily computed using the spectrograms depicted in
figure 2 as
1
⎡ S notch{ f band , tband + t m } − S perfect{ f band , tband + t m } ⊗ S notch{ f band , tband + t m } − S perfect{ f band , tband + t m } ⎤ 2
⎥ ,
R=⎢
Snotch{ f band , tband } ⊗ S notch{ f band , tband }
⎢⎣
⎥⎦
(4)
where tm is the time at which the correlation curve in figure 3 achieves a maximum (point c). In the conventional
sense, a reflection coefficient is given by the ratio of the reflected amplitude to the incident amplitude for a
continuous, harmonic wave. The waveforms in the current situation are transient waveforms consisting of many
frequencies, with each frequency having its own reflection coefficient. The reflection coefficient given by equation
4 represents an energy-based average of the reflection coefficients in the frequency band fband. Depicted in figure
4 is a plot of the FE-calculated reflection coefficients as a function of notch depth. A quadratic least-squares fit
(solid curve) has been drawn through the calculated points (x’s).
Figure 4. FE-calculated (x’s with least-squares quadratic fit) and experimental (o’s) reflection coefficients as a
function of notch depth.
4. Experiment
Depicted in figure 5 is the experimental setup for sending and receiving Lamb waves in a 1 mm thick aluminum
plate. The dimensions of the plate are shown in the drawing. The plate rests on top of a bubble-wrap foundation,
so as to minimize the influence from the underlying structure. A function generator (Tabor 8551) is configured to
output a single harmonic burst having a 16 Vpk amplitude, a 1 MHz frequency, and a 50 Hz repetition rate. The
repetition rate was selected so that motion in the plate falls below the noise level before the subsequent burst
cycle begins. The voltage output from the function generator is amplified by a factor of ~12 using a power
amplifier (Siemens 10W) and is used to drive a small-diameter, ultrasonic transducer known as a ‘pinducer’
(Valpey Fisher VP-1093, 2 mm diameter, 33 mm length). A thin layer of ultrasonic couplant (Krautkrämer Hürth
ZG-F) is applied between the pinducer and the plate to improve the transmission of acoustic energy into the plate.
A commercially available Polytec laser Doppler vibrometer (LDV) (OFV 3001 Controller, OFV 353 Sensorhead)
configured for broadband operation (frequency < 1.5 MHz) is used to measure the out-of-plane surface velocity at
the receiver location. The output from the LDV is sent to an oscilloscope (Tektronix TDS340A) where it is digitally
stored for future processing. The oscilloscope is configured to average the signal 250 times in order to improve
the signal-to-noise ratio (SNR).
Figure 5. Experimental setup for sending and receiving Lamb waves in a notched plate.
Depicted in figure 6 is the measured out-of-plane displacement and associated spectrogram at the receiver
location for the perfect plate. Also shown is a spectrogram of the measured out-of-plane displacement for the
24% notched plate. The spectrograms were computed using the same windowing as was applied in the FE case.
To allow direct comparison with the FE computations, the LDV-measured velocity has been integrated to yield the
depicted displacement-time history. The presence of a notch is evidenced by the energy patch which is centered
at (1 MHz, 31 µs) in the notched plate spectrogram, but which is not present in the perfect plate spectrogram.
The correlation curve depicted in figure 7 is constructed from the experimental spectrograms using equation 2. A
frequency band of fband = 0.6 – 1.75 MHz and a time band of tband = 15-21µs, as depicted by the rectangular box on
the perfect plate spectrogram, were used for calculations. The correlation characteristic achieves a maximum at
the distance d = 16.8 mm, which agrees only marginally with the actual distance of do = 20 mm. The discrepancy
is attributable to the non-impulsive excitation from the pinducer. Instead of producing a single burst, the pinducer
tended to ring, thereby outputting several harmonic cycles. This leads to a low estimate for the ao mode group
velocity (and therefore for the notch location) since the arrival of the energy maximum at the receiver location is
delayed. Despite the discrepancy in notch localization, the experimental correlation curve contains all the
characteristic extrema which are present in the correlation curve based on the FE calculations (see figure 3).
The reflection coefficients are experimentally determined for the notch depth cases h/w = 12%, 17%, and 24%.
Calculations were based on equation 4. However, the experimental values were multiplied by the factor (86 / 46)
to adjust for cylindrical geometric spreading in the plate. The experimentally determined reflection coefficients are
plotted in figure 4 along with those determined from FE calculations. Agreement between experiment and
simulation is good. An experimental signal-to-noise ratio of SNR=33 (1.5 MHz receiving bandwidth) has been
estimated, where the signal is assumed to be the incident ao mode arriving at the receiver location at a time of
t ~ 18 µs in figure 6. A corresponding noise floor given by 1/SNR has been drawn in figure 4. The intersection of
the noise floor with the FE-calculated characteristic occurs at a notch depth of ~10%. This value represents the
minimum detectable experimental notch depth for the experimental setup under evaluation. For this estimate, it
has been assumed that the FE-calculated characteristic approximates the trend in the experimental data, and the
geometric spreading between the receiver and the notch is negligible.
Figure 6. Experimental displacement-time history and spectrogram depicting Lamb wave propagation in a perfect
plate (left) and corresponding spectrogram for a 24% notched plate (right).
Figure 7. Experimental correlation vs. receiver-notch distance characteristic for a plate having a 24% notch.
5. Conclusion
An efficient signal processing method for Lamb wave-based notch detection and localization in a plate is
described. Specifically, a spectrogram was used to isolate a nondispersive portion of the ao Lamb mode from the
rest of the signal for use in the notch localization algorithm. Additionally, using the spectrogram, an average
reflection coefficient was calculated. The experimentally-determined reflection coefficient vs. notch depth
characteristic agreed well with that obtained from a FE model. Finally, for the experimental setup under
evaluation, the minimum detectable notch depth was estimated to be 10%. The results of this study can be used
for the real-time, early detection of cracks in plate-like structures. Eventually, the crack depth could also be
estimated using a measured reflection coefficient and a pre-determined reflection coefficient vs. notch depth
characteristic.
In future investigations, the experiment will be repeated for the case of broadband laser excitation. The use of
narrowband pinducer excitation in this work led to an error in the experimentally determined notch location. Since
laser-based excitation is expensive and unwieldy in practice, an additional aim will be the development of a costeffective, mechanical means of broadband excitation. In [10] for example, a piezoelectric source and a
complementary digital processing methodology were used to obtain a time-frequency representation of Lamb
wave propagation in a plate similar to that achievable through broadband laser excitation. Finally, the
experimental approach will be optimized so as to allow for the detection of the smallest possible notch. According
to [11], Lamb waves may be used to find notches when the wavelength-to-notch depth ratio is < 40. Based on
this estimate, it should be possible to detect notches as small as 8% in the current plate using the described
methodology.
Acknowledgement
Partial support to Brad M. Beadle provided by the Deutscher Akademischer Austauschdienst (DAAD) is gratefully
acknowledged.
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