Ultrasonic guided wave techniques have great potential for structural health monitoring applications. Appropriate mode and frequency selection is the basis for achieving optimised damage monitoring performance. In this paper, several important guided wave mode attributes are introduced in addition to the commonly used phase velocity and group velocity dispersion curves while using the general corrosion problem as an example. We first derive a simple and generic wave excitability function based on the theory of normal mode expansion and the reciprocity theorem. A sensitivity dispersion curve is formulated based on the group velocity dispersion curve. Both excitability and sensitivity dispersion curves are verified with finite element simulations. Finally, a goodness dispersion curve concept is introduced to evaluate the tradeoffs between multiple mode selection objectives based on the wave velocity, excitability and sensitivity.

1. E Excitability
F Force vector
G Goodness function
Pnn Integrated power flow in the x1 direction
S Sensitivity
αi Ratio of wave number
λ,μ Lame Constants
ρ Density
σij Stress components
ω Radial frequency
1.0 INTRODUCTION
Real time condition assessment will become the next major technical
advancement in the maintenance of aircraft and aerospace vehicles.
When the structural health monitoring (SHM) system is effective, early
damage detection will help to prevent catastrophic failure. On the other
hand, the service life of aging aircraft can be prolonged based on its
actual condition.
After a few decades of research and development, ultrasonic guided
wave technology developed for nondestructive evaluation has demon-
strated its potential in SHM. This is mostly due to its capability of
detecting both surface and internal damage over a comparably long
range. As a result, a sparse sensor array can be used to monitor a large
area. In the past few years, the theory of guided wave propagation,
especially the phase velocity dispersion curve and group velocity
dispersion curves, has been applied to SHM research. Some concepts of
mode selection are discussed in previous literature for different types of
defect detection (Ghosh 1998, Kundu 1998, Rose 1999) However, only
a few works are reported in terms of optimised mode selection and trans-
ducer design in a quantitative sense (Wilcox 2001, Gao 2007). The
ABSTRACT
Ultrasonic guided wave techniques have great potential for structural
health monitoring applications. Appropriate mode and frequency
selection is the basis for achieving optimised damage monitoring perfor-
mance. In this paper, several important guided wave mode attributes are
introduced in addition to the commonly used phase velocity and group
velocity dispersion curves while using the general corrosion problem as
an example. We first derive a simple and generic wave excitability
function based on the theory of normal mode expansion and the
reciprocity theorem. A sensitivity dispersion curve is formulated based
on the group velocity dispersion curve. Both excitability and sensitivity
dispersion curves are verified with finite element simulations. Finally, a
goodness dispersion curve concept is introduced to evaluate the tradeoffs
between multiple mode selection objectives based on the wave velocity,
excitability and sensitivity.
NOMENCLATURE:
an(x1) Amplitude of nth wave mode as a function of x1
cp Phase velocity
cg Group velocity
f Frequency
h Thickness
k Wave number
t Time
ui Displacement components
ui Acceleration components
v Velocity Components
v Velocity vector
v* Complex conjugate of velocity vector
xi Coordinate directions
Bi Weighting factor of each partial wave
THE AERONAUTICAL JOURNAL JANUARY 2010 VOLUME 114 NO 1151 49
Paper No. 3343. Manuscript received 3 September 2008, accepted 28 December 2008.
Goodness dispersion curves for
ultrasonic guided wave based SHM:
a sample problem in corrosion monitoring
H. Gao
hgao@innerspec.com
Research and Development, Innerspec Technologies
Lynchburg, Viginia
USA
J. L. Rose
krp12@engr.psu.edu
Department of Engineering Science and Mechanics
Penn State University
Pennsylvania
USA
…

2. idea of NME is to express the actual wave field as a superposition of
orthogonal guided wave mode solutions. The general theory of NME in
an elastic and piezoelectric plate is described in a textbook by Auld (Auld
1990). Ditri and Rose used the NME technique for guided wave
excitation in isotropic plates and pipes (Ditri et al 1992; Ditri et al 1994).
Previous studies of wave propagation enabled us to generate guided
wave phase velocity dispersion curves, group velocity dispersion curves,
and wave structure of a given mode. The previous study of wave
excitation enabled us to simulate the resulting wave field from a
prescribed excitation sources. However, the factors of the source contri-
bution and the mode characteristics were not isolated and explained in a
simple manner. As a result, there is still no direct answer to the following
questions. What is the excitability of the guided wave modes? What is
the sensitivity of guided waves to general corrosion damage? What are
the best modes to be used in this application? How are we going to
design transducers to effectively excite these modes? These are the
questions that we are trying to answer in this paper by way of the devel-
opment of a ‘goodness’ dispersion curve concept.
In Section 2, we start with a brief review of the matrix method for
dispersion curve and wave structure calculations. Then, an excitability
dispersion curve is derived from the normal mode expansion theory. A
sensitivity dispersion curve is derived from group velocity dispersion
curves. In Section 3, we verified our definitions of excitability dispersion
curve and sensitivity dispersion curve using finite element simulation. In
Section 4, a goodness dispersion curve is defined from the excitability
and sensitivity dispersion curves using our multi-feature optimisation
software. Finally, the optimised guided wave mode candidates are
recommended for general corrosion and erosion damage monitoring.
2.0 GUIDED WAVE MODE ANALYSIS
2.1 Dispersion curve and wave structure
Figure 1 shows the model of wave propagation in an aluminum
plate. The wave equation is
Here, ρ is the density of the material, λ and μ are the two Lame
Constants. u is the displacement and i, j are the coordinate indices.
For Lamb waves in a plate, the wave field can be expressed as a
linear combination of four partial waves.
purpose of this study is to push the frontier of guided wave mode analysis
and optimised mode selection by introducing a new concept of a
‘goodness’ dispersion curve. As a sample problem, the problem of
monitoring the thickness reduction introduced by general corrosion or
sand erosion in aircraft plate like structures is addressed in this paper.
Our research on guided wave mechanics is based on the achievements
of many great scientists in the past. Theoretical studies of ultrasonic
guided waves were undertaken when Lord Rayleigh and Lamb studied
surface acoustic waves (Rayleigh 1885) and waves in plates (Lamb
1917). Helmholtz decomposition and partial wave theory has been
widely used and well addressed in the textbooks and review articles
(Achenbach 1973, Graff 1973, Auld 1990, Rose 1999, Lowe 1995,
Chimenti 1997). Transfer matrix and global matrix methods were
developed to study the waves in multilayered structures including fiber
reinforced composites (Thomson 1950; Haskell 1953, Knopoff 1964,
Kundu 1985, Nayfeh 1995, Hosten et al 1993; Rokhlin et al 2002).
Besides the analytical models, a semi-analytical finite element (SAFE)
method was also used to simulate guided wave propagation (Lagasse
1973, Huang et al 1984 Garvric 1995 , Hayashi 2003, Matt et al 2005;
Bartoli et al 2006, Gao 2007).
Analytical methods for solving ultrasonic guided wave excitation
problems generally fall into two categories: one is based on an integral
transform method and the other on a normal mode expansion technique.
The integral transform method is discussed in Rose (Rose 1999) for shear
horizontal guided wave excitation in an isotropic plate. Recently,
Giurgiutiu (Giurgiutiu 2005) used the integral transform method to study
Lamb wave excitation in an aluminum plate. The frequency tuning effect
of guided wave excitation is also investigated. Raghavan et al 2005
extended the integral transform method to a three dimensional analysis of
guided wave excitation in isotropic plates. Mal et al studied the wave
excitation phenomenon in unidirectional and cross-ply composites from a
localised source, especially to simulate the process of impact and
acoustic emission effects using the global matrix method and simplified
models using plate theory (Mal 2002; Banerjee et al 2005). Although the
integral transform method can be used to analyse wave excitation from
localised sources, the inverse process of the integral transform is usually
very difficult. In addition, the formulation of the integral transform
method is usually very cumbersome. The normal mode expansion
(NME) technique is based on a reciprocity relation in elasticity. The basic
50 THE AERONAUTICAL JOURNAL JANUARY 2010
Figure 1. Sketch of wave propagation model.
Figure 2. Dispersion curves of a 2mm thick aluminum plate. (a) phase velocity (b) group velocity.
(a) (b)
. . . (1)
. . . (2.2)
. . . (2.1)

3. 2.2 Guided wave excitability
Based on the theory of reciprocity and normal mode expansion, a
guided wave excitability dispersion curve is introduced in this
section. The normal mode expansion technique is the process of
expressing an arbitrary wave field as a combination of the modal
solutions.
The emphasis of the following work is to determine the coeffi-
cient an(x1) for a given loading situation. In order to do this, we will
need the reciprocity relation
Substituting Equation (5) into Equation (6), and integrate over the
thickness of the plate, we obtain Equation (7).
Here, Pnn is the entire power flow in the x1 direction, fsn is related
to surface loading, and fvn is the body loading. In general, the loading
terms are functions of x1. Equation (8) shows the detailed expres-
sions for the surface loading and the body loading respectively.
In structural health monitoring, the wave excitation can be
achieved with an embedded transducer or a surface mounted trans-
ducer. For the case of an embedded transducer, the terms of fvn is
effective; and for the case of a surface mounted transducer, the terms
of fsn is effective.
Equation (7) is a partial differential equation of the function an(x1),
which can be solved using a standard integration technique.
Assuming that a distributed load is within the range of [L2, L2] the
wave propagation in the positive x1 direction must have zero
amplitude at the left side of the source. Equation (9) is the solution
for rightward propagating wave modes.
an (x1) = 0 x1 ≤ L1
Here, αi, i = 1,2,3,4 , are ratios of wave number in the x3 direction
versus the x1 direction for each partial wave. Their values are;
The stress free boundary conditions are σ33 = 0 and σ31 = 0 at x3 = 0,
and x3 = h. Substituting Equation (2) into the boundary conditions,
we obtain the following:
The phase velocity dispersion curve is the set of solutions between f and
cp that makes the determinant of the coefficient matrix zero. After that,
group velocity dispersion curves can be calculated using Equation (4).
Figure 2 shows the phase velocity dispersion curve and group
velocity dispersion curves for the 2mm thick aluminum plate. The
first six mode lines are also marked in the figure.
The difference of guided wave modes over the same curve is best
illustrated by the wave structure difference from point to point over the
curve, which is the wave field distribution along the thickness
direction. Figure 3 shows two examples for the S0 mode: one is at
500kHz and the other at 1,000kHz. For the 500kHz S0 mode, the
displacement field is predominantly in plane. However, for the
1,000kHz S0 mode, the out of plane displacement is dominant on the
surfaces. This difference in wave structure will lead to differences in
the excitation behaviour and sensitivity to damages. Therefore, when a
statement is made about a mode, it is highly recommended to include
both the mode line and frequency for all guided wave applications.
GAO AND ROSE GOODNESS DISPERSION CURVES FOR ULTRASONIC GUIDED WAVE BASED SHM: A SAMPLE POBLEM IN... 51
Figure 3. Wave structure of the S0 wave mode at 500kHz and 1,000kHz for a 2mm thick aluminum plate.
(a) (b)
. . . (2.3)
. . . (2.4)
. . . (6)
. . . (8.1)
. . . (8.2)
. . . (9.1)
. . . (7)
. . . (5.2)
. . . (5.1)
. . . (4)
. . . (3)

4. 52 THE AERONAUTICAL JOURNAL JANUARY 2010
Figure 4. Excitability dispersion curves of guided wave modes in a 2mm thick aluminum plate.
(a) (b)
Figure 5. Guided wave monitoring scheme for general corrosion.
Figure 6. Time of flight sensitivity dispersion curve of guided wave modes in a 2mm thick aluminum plate.

5. The entire wave field is a summation of all the leftward propa-
gating waves and the rightward propagating waves. In a strict sense,
a complete solution includes propagating wave modes as well as
evanescent wave modes. The wave number of an evanescent mode is
either pure imaginary or complex. This imaginary part of the wave
number will introduce an exponentially decaying factor in the wave
field in the propagation direction. Typically, the attenuation factor is
large enough that the amplitude of the evanescent wave modes
decays in less than a wavelength. Therefore, for practical purpose,
the evanescent waves can be neglected outside the source region.
Our next step is to correlate the solution of an with the properties
of guided wave modes and the excitation source. For the case of top
surface mounted transducers, the solution of an can be reduced to;
The two contribution factors of an are from the mode and
from the excitation source. It is now clear that
When the position is outside the source region, an(x1)is a harmonic
wave function of x1 with amplitude as given in Equation (10). This means
that the amplitude of a given mode is a constant outside the source region.
Similarly, for leftward propagating waves, n < 0, the solutions are
in Equation (11).
a–n(x1) = 0 X1 ≥ L2
GAO AND ROSE GOODNESS DISPERSION CURVES FOR ULTRASONIC GUIDED WAVE BASED SHM: A SAMPLE POBLEM IN... 53
Figure 7. Sample receiving signals from a spike pulse excitation. (a) sample RF waveforms (b) Image of stacked RF waveform.
(a) (b)
Figure 8. Comparison of dispersion curves obtained from numerical
simulation and theoretical modal analysis. The theoretical curves are
shown in dotted line superimposed on the numerical simulation results.
Figure 9. Comparison of the amplitude spectrum of
V2 from modal analysis and numerical simulation.
. . . (9.2)
. . . (9.3)
. . . (11.3)
. . . (10)
. . . (12)
. . . (11.1)
. . . (11.2)

6. The sensitivity dispersion curve of the 2mm aluminum plate is
plotted in Fig. 6.
From this figure, it is clear that the sensitivity of the guided
wave modes varies with frequency. At the low frequency region of
the S0 mode, the waves are very insensitive. However, when the
frequency increases, there is a very sensitive region, where the
time of flight sensitivity can be as much as 0·8 micro seconds per
square mm cross-sectional area loss. When the frequency further
increases, the sensitivity value is negative, which means the time
of flight in the corroded case is less than that of the good situation.
Sensitivity peaks are also observed for the A1, A2, S1, and S2
modes.
3.0 FINITE ELEMENT SIMULATION
3.1 Phase velocity dispersion curve
A finite element model is used to verify the definition of
excitability with a spike pulse excitation at a localised region
using ABAQUS. Figure 7 shows the receiving signals when
excitation force is normal to the surface.
A two dimensional Fourier transform algorithm is used to
reveal the frequency-phase velocity domain distribution of the
signal. The dispersion curves obtained from the numerical
simulation matches very well with the theoretical analysis. See
Fig. 8.
3.2 Excitability dispersion curve
Figure 9 also shows that some modes have strong amplitudes and
some are weak. Since the excitation spectrum of a spike pulse
localised in a small region is uniform in the frequency and phase
velocity domain, the amplitude variation is predominantly due to
the difference in the excitability of the guided wave modes.
According to modal analysis, the amplitude spectrum of the V2
component should be proportional to the square of vn2, the
normalised particle velocity at the surface.
the excitability of a wave mode from surface loading is directly
related to normalised particle motion velocity. Therefore, we define
three excitability functions for the excitation in three orthogonal
directions.
Here, v1, v2, v3, are the components of particle motion velocity
calculated from wave structure analysis. Since E2 is zero for all
Lamb waves, the two other excitability dispersion curves , for the
2mm thick aluminum plate are shown in Fig. 4.
2.3 Guided wave sensitivity to general corrosion
When aluminum plates in flight structures are attacked by general
chemical corrosion or sand erosion, there will be a small amount of
metal loss on the surface. As illustrated in Fig. 5, a pair of a trans-
mitter and a receiver is used to detect the metal loss. Different from
a vertical crack defect, the general corrosion defect will not produce
a significant amount of reflection. However, the transmission time
will be affected by the damage since the thickness reduction changes
the group velocity of wave propagation. In this section, we will build
up a quantitative relation between time of flight variation and the
severity of corrosion or erosion metal loss.
Assume that the group velocity of a guided wave mode for the
plate before corrosion is v, and the distance between the transmitter
and the receiver is L. The estimated time of flight of the wave mode
from transmitter to receiver is
The difference in time of flight with and without corrosion is;
When the relative thickness reduction is small, is not a function
of h(x),
From here we can see that the time of flight shift is proportional to
the cross-sectional area loss of the corrosion. The sensitivity of the
guided wave mode to corrosion/erosion defects can be defined as
54 THE AERONAUTICAL JOURNAL JANUARY 2010
(a) (b)
Figure 10. Influence of damage intensity on the through transmission guided wave signal. (a) Sample signals, (b) linear fit of the time of flight.
. . . (13)
. . . (14)
. . . (15)
. . . (18)
. . . (16)
. . . (17)

7. and sensitivity are considered in this paper. Comparing the sensitivity
dispersion curve and the guided wave excitability curve, we observed
that for the modes at the sensitivity peak, the E1 excitability is signifi-
cantly larger than E3. This means that in order to get a strong and
sensitive signal, the transducer is preferred to have dominantly normal
loading. As an illustration of the goodness mode selection concept,
three mode selection rules used in the optimisation are group velocity
Cg, excitability E3 and sensitivity S defined in Equations (4), (13) and
(17) respectively. In the optimisation process, the value of E3 and S are
first mapped to a (0 1) space using Equation (20).
Here, gcg, gE3 and gS are the individual goodness functions for
excitability and sensitivity. 1, 0·6 and 0·8 are the threshold values. In
the program, there are several ways of defining the final goodness
function. As an example, the goodness function for this problem is
defined as
G = gcg × gE3 × gS
The result of the goodness function is plotted in Fig. 11. The
recommended mode candidates are circled on the dispersion curve.
A transducer designed to excite these modes will be sensitive to
general corrosion damage and also have adequate signal amplitude.
5.0 CONCLUSION
Optimised mode selection and transducer design are the basis for
optimised performance in guided wave based structural health
monitoring. The study presented in this paper is focused on the
process of guided wave mode selection using a goodness dispersion
curve concept to evaluate trade-offs between multiple design objec-
tives. The sample problem used to illustrate this concept is the
monitoring of metal loss due to general corrosion and sand erosion
The information of the mode amplitudes is also extracted from
Fig. 8 and compared with the expectation from the normal mode
expansion theory. A good match between the results proved our
definition of the excitability function.
The place that has a comparably larger difference is A0 and S0 at
the high frequency region. This is because the dispersion curves of
the two modes are very close. The result in the numerical simulation
is a combination of these two modes.
3.3 Sensitivity dispersion curve
Sensitivity dispersion curves are now validated with a finite element
simulation. A gradual triangular general corrosion defect is
simulated over a length of 200mm. The maximum defect depth
increased from 0·04mm to 0·2mm of the plate thickness. Therefore,
the cross-sectional area of the corrosion is from 4mm2
to 20mm2
.
Figure 10(a) shows the received signals for three situations when the
excitation is the A1 mode at 2MHz. Figure 10(b) shows the time of
flight extracted from the signals. The linear fit indicates that the rate
of time of flight shift is 0·38μs/mm2
. The minus sign is due to the
fact that ΔA is negative.
Sensitivity values of several other modes are also verified with
finite element simulation. Table 1 illustrates the results of sensitivity
from the theoretical expectation in section 2 and finite element
simulation. The match between the theory and the simulation is very
good.
Table 1
Comparison of sensitivity value obtained from
theoretical study and numerical simulation
Test Mode Frequency Theoretical Simulated
Mode Order (kHz) Sensitivity Sensitivity
1 1 350 –0·018 –0·019
2 2 650 0·036 0·033
3 3 2,000 0·36 0·38
4 4 2,830 0·47 0·53
4.0 GOODNESS DISPERSION CURVE: AN
OVERALL EVALUATION
The process of SHM system design optimisation involves the
considerations of wave mode selection, transducer design, sensor
networking topology, and system level mechanical and electronic
optimisations (Gao 2007). A good wave mode selection is the basis
for the final success of the monitoring system. The key attributes of
each guided wave mode defined in the previous sections revealed the
nature of guided wave excitation, propagation, and interaction with
damages. The next step is to select wave mode candidates for this
application. If we say that wave mechanics study is basic science,
the definition of an overall goodness function is a process of
engineering. The nature of each application determines the critical
features to be considered. In addition, the designer might also have
preference in the decision of critical feature and threshold selection.
We developed a general purpose guided wave mode selection
program to evaluate the trade-offs between each of the design objec-
tives. This software program takes the guided wave mode features as
inputs. The user can define mode selection rules and the program
provides optimised mode selection recommendations.
The process of the multi-feature optimisation algorithm is
summarised in Equation (19).
G = F(X). . . . (19)
Here, G is the goodness function. X is the feature vector, and F are
the mode selection rules. A variety of guided wave features can be
selected to define the rules. Phase velocity, group velocity, excitability
GAO AND ROSE GOODNESS DISPERSION CURVES FOR ULTRASONIC GUIDED WAVE BASED SHM: A SAMPLE POBLEM IN... 55
Figure 11. Goodness dispersion curve plotted with phase velocity
dispersion curve. The circled regions are the recommended guided
wave mode candidates for general corrosion monitoring.
. . . (21)
. . . (20.1)
. . . (20.2)
. . . (20.1)

8. 10. GHOSH, T., KUNDU, T. and KARPUR, P. Efficient use of Lamb modes for
detecting defects in large plates, Ultrasonics, 1998, 36, pp 791-801.
11. GIURGIUTIU, V. Tuned lamb wave excitation and detection with
Piezoelectric wafer active sensors for structural health monitoring. J
Intelligent Material Systems and Structures, 16, 2005, pp 291-305.
12. GRAFF, K. Elastic Waves in Solids, New York, USA, Oxford University
Press, 1973.
13. HASKELL, N.A. Dispersion of surface waves in multilayered media. Bull
Seismol Soc Am, 43, pp 17-34, 1953.
14. HAYASHI, T., SONG, W.J. and ROSE, J.L. Guided wave dispersion curves
for a bar with an arbitrary cross-section, a rod and rail example,
Ultrasonics, 2003, 41, pp 175-183.
15. HOSTEN, B. and CASTAINGS, M. Transfer matrix of multilayered
absorbing and anisotropic media. Measurements and simulations of
ultrasonic wave propagation through composite materials. J Acoust Soc
Am, 94, (3), pp 1488-1495.
16. HUANG, K.H. and DONG, S.B. Propagating waves and edge vibrations in
anisotropic composite cylinders, J Sound and Vibration, 1984, 96, (3),
pp 363-379.
17. KNOPOFF, L. Matrix method for elastic wave problems, Bull Seismol
Soc Am, 1964, 54, pp 431-438.
18. KUNDU T and MAL A.K. Elastic waves in multilayered solid due to a
dislocation source, Wave Motion, 1985, 7, (5), pp 459-471.
19. KUNDU, T., MAJI, A., GHOSH, T. and MASLOV, K. detection of kissing
bonds by Lamb waves, Ultrasonics, 1998, 35, pp 573-580.
20. LAGASSE, P.E. Higher-order finite element analysis of topographic
guides supporting elastic surface waves, J Acoustic Soc Am, 1973, 53,
(4), pp 114-128.
21. LAMB, H. Waves in elastic plates. Proc Roy Soc A, 93, 1917, pp 114-
128.
22. LIN, M. and CHANG, F.K. The manufacturing of composite structures
with a built-in network of piezoceramics, Composite Science and
Technology, 2002, 62, pp 919-939.
23. LOVE, A.E.H. Some Problems of Geodynamics, Cambridge University
Press, 1911.
24. LOWE, M.J.S. Matrix techniques for modeling ultrasonic waves in
multilayered media. IEEE Transactions on Ultrasonics, Ferroelectrics,
and Frequency Control, 1995, 42, (2), pp 525-541.
25. MATT, H. and BARTOLI, I. et al, Ultrasonic guided wave monitoring of
composite wing skin-to-spar bonded joints in aerospace structures. J
Acoust Soc Am, 2005, 118, (4), pp 2240-2252.
26. NAYFEH, A.H. Wave propagation in layered anisotropic media.
Amsterdam, Lausanne, New York, Oxford, Shannon, Tokyo, Elsevier.
27. RAGHAVAN, A. and CESNIK, C.E.S. Finite-dimensional pieozoelectric
transducer modeling for guided wave based structural health
monitoring. Smart Materials and Structures, 2005, 14, pp 1448-1461.
28. RAYLEIGH, L. On waves propagating along the plane surface of an
elastic solid. Proc London Math. Soc, 17, 1885, pp 4-11.
29. ROKHLIN, S.I. and WANG, L. Ultrasonic waves in layered anisotropic
media: characterization of multidirectional composites, Int J Solids and
Structures, 2002, 39, pp 5529-5545.
30. ROSE, J.L. Ultrasonic Waves in Solid Media, Cambridge University
Press, Cambridge, UK, 1999.
31. THOMSON, W.T. Transmission of elastic waves through a stratified solid
material, J Applied Physics, 1950, 21, pp 89-93.
32 WILCOX, P.D. and LOWE, M.J.S. et al Mode and transducer selection for
long range Lamb wave inspection, J Intelligent Material Systems and
Structures, 2001, 12, pp 553-565.
in aircraft structures.
Beyond the review of general guided wave phase velocity, group
velocity, and wave structure analysis, this paper presents our new
definitions of guided wave excitability and sensitivity dispersion
curves. The result shows that the excitability of a guided wave mode
is directly related to the particle motion velocity of a given mode at
the excitation location. Therefore, for surface mounted transducers,
the excitability can be defined as the normalised particle velocity at
the excitation surface. These dispersion curves are verified with
finite element simulations combined with a 2D FFT signal
processing technique.
A guided wave sensitivity dispersion curve is very useful in the
selection of appropriate guided wave modes for a particular appli-
cation. Many sensitivity dispersion curves can be defined based on
the type of damage to be monitored. In this paper, we presented our
first sensitivity dispersion curve definition for general corrosion or
erosion damage. The result indicates that the guided wave modes
with a large group velocity slope and a small group velocity value
are the most sensitive modes. On the contrary, the most commonly
used A0 and S0 modes below the first cutoff frequency are not the
most sensitive ones.
Guided wave mode candidates are recommended from the
goodness dispersion curve after evaluating the group velocity,
excitability, and sensitivity dispersion curves. These include the S0
mode around 1MHz, A1 mode around 2MHz, S0 mode around
2·8MHz, and A2 mode around 3·8MHz for the 2mm thick aluminum
plate. The same concept can be extended to more complex structures
and other damage monitoring applications.
REFERENCES
1. ACHENBACH, J. D. Wave Propagation in Elastic Solids, Amsterdam,
North-Holland, 1973.
2. AULD, B.A. Acoustic Fields and Waves in Solids, 1990, Malabar,
Florida, USA, Krieger Publishing Company,
3. BANERJEE, S., PROSSER, W. and MAL, A.K. Calculation of the response
of a composite plate to localized dynamic surface loads using a new
wave number integral method, ASME J Applied Mechanics, 2005, 72,
pp 18-24.
4. CHIMENTI, D.E. Guided waves in plates and their use in material charac-
terization, Appl Mech Rev, 50, (5), 1997, pp 247-284.
5. DITRI, J.J. and J.L. ROSE Excitation of Guided Elastic Wave Modes in
Hollow Cylinders by Applied Surface Tractions, J Appied Physics, 72,
(7), 1992, pp 2589-2597.
6. DITRI, J.J. and ROSE, J.L. Excitation of guided waves in generally
anisotropic layers using finite source, ASME J Applied Mechanics,
1994, 61, (2), pp 330-338.
7 GAVRIC, L. Computation of propagating waves in free rail using finite
element techniques, J Sound and Vibration, 1995, 185, pp 531-543.
8. GAO, H and ROSE, J.L. Multifeature optimization of guided wave modes
for structural health monitoring of composites, Materials Evaluation,
2007, 65, (10), pp 1035-1041.
9. GAO, H. Ultrasonic Guided Wave Mechanics for Composite Material
Structural Health Monitoring, Engineering Science and Mechanics.
University Park, The Pennsylvania State University, PhD dissertation, 2007.
56 THE AERONAUTICAL JOURNAL JANUARY 2010