ADVANCES IN MODELING ULTRASONIC NOISE INDUCED BY
MACHINING ROUGHNESS: DEVELOPMENT AND VALIDATION
OF COMPUTATIONALLY EFFICIENT APPROXIMATIONS
R. B. Thompson, Y. Guo, and F. J. Margetan
Center for Nondestructive Evaluation
Iowa State University
Ames, Iowa 50011, US A
ABSTRACT. In papers given in previous years in this conference, a rigorous solution has been
presented and experimentally verified for the 2-D problem of an ultrasonic wave backscattered
from a periodically rough surface. While predicting experiment very well, that theory has the
limitation that it is very computationally intensive and that it only can be applied to the problem of
planar interfaces. Practical applications, however, call for solutions that can be evaluated more
rapidly and that can be generalized to the case in which the interface is curved. Such a theory is
presented in this paper. The backscattering problem is formulated as a scattering process in which
the strength of the scattering from individual ridges is viewed as an unknown. By summing over
all scattering events and comparing to the rigorous solution, these unknown scattering strengths
can be determined. In this paper, we present the theory, compare it to the rigorous solution, and
use it to make numerical predictions for the effects of curvature on the backscattered noise.
INTRODUCTION
Backscattered noise from rough surfaces can limit the detection of near-surface
flaws and interfere with grain noise signals that might be used to characterize the
microstructure [1]. For periodic roughnesses, such as those induced by machining, the
surface noise has a narrow bandwidth, and the resulting slow decay makes this a
particularly difficult case to deal with.
We have previously described a two-dimensional theory to predict the time
domain waveforms created by such roughness [2,3]. With the exception of neglecting
the scattering of upward propagating reflected waves of off adjacent grooves, an
excellent approximation for small ratios of the height to period of the surface roughness,
the theory is essentially exact. The basic approach starts with a previous time harmonic
solution for the scattering of an incident plane wave from a surface with periodic
roughness [4]. Integrating over the angular spectrum of plane waves that define the
incident beam, using Auld's electromechanical reciprocity relation [5] to compute the
electrical signal that would be observed at the transducer's port, and integrating over the
spectral components that define the incident pulse leads to a prediction of the time
domain waveform reflected from the surface. Comparison to experiment has shown this
approach to be able to provide accurate prediction of the noise.
Out of this fairly complex analysis comes a simple physical idea. A key process
is the excitation of a Scholte wave when the illuminating fields strike the rough surface.
When the wavelength of that Scholte wave equals the period of the surface, a resonance
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
125
develops
backscattered noise
noise and
and leads
leads to
to its
its narrow
narrow band,
band,
develops that
that greatly
greatly enhances
enhances the
the backscattered
slowly
slowly decaying
decaying nature.
nature.
Despite
have motivated
motivated the
Despite the
the successes
successes of
of that
that theory,
theory, two
two limitations
limitations have
the present
present
work.
First,
the
calculation
is
rather
time
consuming,
taking
several
hours
on
work. First, the calculation is rather time consuming, taking several hours on aa typical
typical
PC.
for an
an incident
incident plane
plane wave
wave is
is
PC. This
This is
is aa consequence
consequence of
of the
the fact
fact that
that the
the core
core solution
solution for
based
based on
on aa series
series expansion
expansion in
in terms
terms of
of diffraction
diffraction orders.
orders. Integrals
Integrals over
over aa spatial
spatial
frequency,
must be
be evaluated.
evaluated. In
In addition,
frequency, aa spatial
spatial coordinate,
coordinate, and
and aa temporal
temporal frequency
frequency must
addition,
the
the first
first step
step assumes
assumes that
that the
the surface,
surface, in
in the
the absence
absence of
of roughness,
roughness, is
is flat
flat and
and there
there is
is not
not
an
case of
of practical
practical interest.
an obvious
obvious way
way to
to extend
extend that
that solution
solution to
to curved
curved surfaces,
surfaces, aa case
interest. In
In
this
promise of
this paper,
paper, we
we describe
describe an
an approximate
approximate approach
approach that
that shows
shows promise
of removing
removing those
those
deficiencies.
deficiencies.
APPROACH
APPROACH
Our
problem as
as aa series
series of
of scattering
processes.
Our basic
basic approach
approach is
is to
to consider
consider the
the problem
scattering processes.
Motivated
waves as
key physical
physical
Motivated by
by the
the identification
identification of
of the
the generation
generation of
of Scholte
Scholte waves
as the
the key
ingredient,
of the
the backscattered
backscattered noise
noise as
ingredient, we
we seek
seek to
to describe
describe the
the generation
generation of
as is
is
schematically
schematically illustrated
illustrated in
in Figure
Figure 1.
1 . When
When the
the incident
incident wave
wave strikes
strikes one
one of
of the
the ridges,
ridges,
some
propagating to
to the
the left
left and
right along
some of
of the
the energy
energy is
is scattered
scattered into
into Scholte
Scholte waves
waves propagating
and right
along
the
ridges, some
some of
of their
their energy
energy is
is scattered
the surface.
surface. When
When these
these waves
waves strike
strike other
other ridges,
scattered into
into
bulk
bulk waves
waves reradiating
reradiating into
into the
the fluid.
fluid. The
The observed
observed surface
surface noise
noise is
is to
to be
be computed
computed by
by
summing
summing over
over these
these processes.
processes.
The
The authors
authors are
are not
not aware
aware of
of solutions
solutions in
in the
the literature
literature to the problem of
of
scattering
scattering from
from an
an individual
individual ridge,
ridge, although
although we
we recognize
recognize that
that this
this is
is aa canonical
canonical
studied in a portion of the literature that we have not yet
solution that may well have been studied
examined. In the absence
absence of
of such
such knowledge, we have adopted the following strategy.
develop analytical
analytical expressions
expressions for the scattered signal in terms of the unknown
We first develop
"scattering amplitudes”
amplitudes" of
of the
the elementary
elementary events described above. We then evaluate the
“scattering
strengths of
of the
the scattering
scattering processes through comparison to the previously
unknown strengths
"exact" theory [2,3].
[2,3]. At this level of approximation, we use a scalar model to
developed “exact”
describe the events.
rz xx
/ Scholte
Scholte wave
wave reradiates
reradiates as bulk
bulk wave
!&• gf
Incident _ ,.„„„. Jt /
wave
|I /
A
If /
|| /
\ |t^^~~~~~ Directly
Directly baekscattered
backscattered wave
wave
\ |j
iiziziz^A ^zzzzzz^: A
Ridge
f
/
/\~f
*
2
/ \ I!h
Water
Water
j ..........
/ 7 ................. i
Λ
Solid
Mode-converted
Mode-converted Scholte
Scholte wave
wave
FIGURE 1.
1. Schematic
Schematic representation
representation of
of the
the theoretical
theoretical approach
approach for a planar surface with roughness.
FIGURE
126
THEORETICAL FORMULATION FOR FLAT SURFACE WITH ROUGHNESS
Figure 2 illustrates the basic processes in more detail and defines the quantities in
the theory that represent the scattering processes. As shown in part a, the radiation
pattern of the transducer in a plane a distance d away from its surface is represented by
the function c/)(s,d,t}, where s is the coordinate transverse to the direction of
propagation. This quantity can be obtained from a beam model. A simple rotation of
coordinates leads to the expression for the incident pressure pl on the rough surface that
is shown in the figure.
When these fields strike the "i-th" ridge, as illustrated in Figure 2b, there will a
signal scattered back into the fluid, whose strength is related to that of the incident
pressure by a scattering quantity "B". In addition, Scholte waves will be excited that
propagate to the left and right along the surface, with amplitudes related to the incident
pressure by the scattering quantities S" and S+, respectively. The Scholte waves will
propagate along the surface, reradiating some of their energy back into the fluid as they
pass over each ridge. If one examines the scattering back into the fluid that takes place at
the "j-th" ridge, the total incident Scholte waves from the left, s-, and the right, s+, can be
described as the sums of the waves excited by all ridges to the left and the right
respectively. These incident Scholte waves will radiate bulk waves back into the fluid
whose amplitudes are given by M"s" and M+s+ respectively. The scattering parameters
B, S1, and M* will be functions of the incident angle 0', the frequency f, and the
detailed ridge geometry represented here by a parameter h. If we neglect attenuation of
the Scholte waves during propagation along the rough surface, the total scattered field
(SF) from the jth ridge may be written as
^ Transducer
,,. Incident plane Y\S^Cl^lj
(a)
Ridge
/\
A
A
?z (x9 o, t) = (f)(x cos 0,d + x sin 9, t)
Solid
M-
jth Ridge
(c)
FIGURE 2. Schematic representation of the scattering model:
a. Incident fields
b. Fields scattered back into the fluid at the "i-th" ridge by a unit amplitude incident wave
c. Fields scattered back into the fluid at the "j-th" ridge. The scattering parameters M" and
M+ relate the incident Scholte wave amplitudes s" and s+ to the strength of the wave
radiated into the fluid.
127
t<J
(i)
where vSch and vw are the speeds of a Scholte wave and a plane wave in water,
respectively. Prediction of the output electrical signal Vs observed when the transducer
is operated in pulse/echo mode is completed by summing the product of these scattered
fields and incident fields over the incident plane defined in Fig. la. This procedure is
motivated by experience with more exact calculations based on the electromechanical
reciprocity relation of Auld [5]. Thus:
(2)
COMPARISON TO EXACT THEORY
Calibrating the Scattering Parameters
There are five adjustable scattering parameters in this theory, B, S", S+, M", and
M . However, examination of Eq.(l) shows that they only occur in certain combinations
so that the number of parameters can be reduced to three by defining G" = S"M" and G+ =
S+M+, and retaining B as the third parameter. This set can be further reduced to two at
normal incidence, since G" = G+ by symmetry.
For selected cases, optimal values of B and G have been determined by
comparing the simple theory to the more exact numerical results. Consider the case in
which the rough surface has a simple sinusoidal profile described by a period A and
peak-to-peak height 2h. Near the primary resonance frequency we have parameterized
the dependence on the ridge height h through the equations
+
B-a-bhm
G - chn
(3)
(4)
Table 1 presents the values of the fitting parameters {a,b,c,m,n} that were obtained. For
the case of oblique incidence, we retained the assumption G" = G+, although it is
recognized that this is not rigorously correct.
Numerical Results
Figures 3 and 4 present a series of numerical results in which the predictions of
the exact and simplified theories are compared using the Table 1 parameters for the cases
of normal and oblique incidence. The material is brass and the transducer has a center
128
frequency
19.3°° corresponding
corresponding to
to the
the
frequency of 5 MHz. The angle of oblique incidence is 19.3
generation
be quite
quite
generation of a 45°
45° transverse wave in the solid. The agreement is believed to be
encouraging
encouraging in light of the many approximations made in the simplified analysis. It will
be
be noted
noted that
that the
the "exact"
“exact” results tend to decay more rapidly than those predicted by the
simplified
simplified theory. It is quite likely that inclusion of an attenuation term to describe the
loss
would be
be able
able to
to
loss of energy of the Scholte wave as it is scattered by successive ridges would
reduce this
this difference.
difference.
reduce
TABLE
micrometers.)
TABLE 1. Values of the parameters defining the scattering processes. ((A
Λ measured in micrometers.)
θ =0
D
D
193
θ=
= 19.3
10-22
aa
1.0965xx 101.0965
b
–6
4.845 x 10
10-*
8.179xlO8.179 x 10 –33
3.614x
10-–66
3.614 x 10
7
–7
cc
-6.7464 x 10 -
-2.0444 x 10 -–77
m
m
1.6
2.2143 − Λ × 1.4286 × 10 −33
2.2143-Axl.4286xlO"
nn
2.2143 − Λ × 1.4286 × 10 3
2.2143-Axl.4286xlO"
2.2143 − Λ × 1.4286 × 10 −33
2.2143-Axl.4286xlO"
−3
when Λ measured in micrometers.
0.3
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£ 0.0
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Time (microseconds)
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Time (microseconds)
(microseconds) '
Time
5fi
FIGURE 3.
3. Comparison
Comparison of
FIGURE
of the
the exact
exact and
and simplified
simplified theories
theories for
for normal
normal incidence
incidence on
on aa sinusoidally-rough
sinusoidally-rough
surface.
surface.
a. A
Λ=
a.
= 250
250 µm,
Jim, hh == 20
20 µm.
fim.
b. A
Λ=
= 350
µm, hh =
µm.
b.
350|im,
= 35
35|im.
c. A
Λ=
= 479
479 µm,
c.
(im, hh == 47
47 µm.
(im.
129
1
2
3
4
5
Time (microseconds)
6
7
2
3
4
5
6
Time (microseconds)
7
8
(b)
(a)
FIGURE 4. Comparison of the exact and simplified theories for oblique incidence on a sinusoidally-rough
surface.
a. A = 250 Jim, h = 20 |im.
b. A = 350jim, h = 35|im.
FIGURE 5. Schematic representation of the theoretical approach for a curved surface with roughness.
EXTENSION OF THEORY TO CURVED SURFACES
We have also completed a preliminary examination of the effects of surface
curvature. Figure 5 shows the geometry and Equation (5) is a generalization of Equation
(1) to this case. In these results, it is assumed that the central ray of the beam strikes the
surface normal to its tangent plane.
t•(
r e
0i9t-
({ i~°i}
J
l)
v
Sch
+B(h,f)pl\0J,t-
(5)
130
PREDICTIONS
PREDICTIONSOF
OF THE
THE EFFECTS
EFFECTS OF
OF CURVATURE
CURVATURE
Numerical
of curvature
curvature of
of the
the
Numerical results
results predicting
predicting the
the affect
affect of
of varying
varying the
the radius
radius of
interface
that were
were examined
examined
interfaceare
are shown
shown in
in Figure
Figure 66 for
for the
the same
same roughness
roughness parameters
parameters that
for
decreasing the
the radius
radius of
of
forthe
theplanar
planarinterface
interface in
in Figure
Figure 3.
3. The
The major
major result
result is
is that
that the
the decreasing
curvature
of
the
interface
decreases
the
level
of
the
slowly
decaying
tail
of
the
front
curvature of the interface decreases the level of the slowly decaying tail of the front
surface
essentially varying
varying over
over the
the
surface signal.
signal. This
This isis because
because the
the angle
angle of
of incidence
incidence is
is essentially
rough
surface,
changing
the
local
resonance
condition
and
decreasing
the
degree
of
rough surface, changing the local resonance condition and decreasing the degree of
constructive
constructiveinterference.
interference.
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(microseconds)
Time
44
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(c)
(c)
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11
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Time (microseconds)
(microseconds)
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55
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^yy^M^**^
Time (microseconds)
Time (microseconds)
FIGURE6.6. Use
Useofofthe
the simplified
simplified model
model to
to study
study the
the effects
effects of
of interface
FIGURE
interface curvature
curvature on
on roughness-induced
roughness-induced
noise.
noise.
250 Jim,
µm, hh == 20
20 jxm.
µm.
a.a. AΛ==250
350Jim,
µm, hh == 35
35 urn.
µm.
b.b. AΛ==350
479jim,
µm,hh==47
47 pm.
µm.
c.c. AΛ==479
CONCLUSIONS
CONCLUSIONS
simplified model
model for
for predicting
predicting time-domain
time-domain signals
AA simplified
signals resulting
resulting from
from surface
surface
roughness
has
been
introduced.
Its
predictions
have
been
found
to
be
in
roughness has been introduced. Its predictions have been found to be in reasonable
reasonable
agreement with
with aa more
more exact
exact theory
theory when
when the
the scattering
scattering parameters
parameters are
agreement
are empirically
empirically
determined. In
In our
our software
software embodiment
embodiment of
of the
the two
two approaches,
approaches, the
determined.
the simplified
simplified model
model isis
more
than
500
times
faster
than
the
exact
model,
a
result
that
is
believed
to
more than 500 times faster than the exact model, a result that is believed to be
be ofof
importance
to
prospective
industrial
users.
The
ability
to
extend
the
simplified
model
importance to prospective industrial users. The ability to extend the simplified model toto
131
curved surfaces has also been demonstrated, and predictions of the influence of curvature
on the noise have been made.
ACKNOWLEDGEMENTS
This work was supported by the NSF Industrial/University Cooperative Research
Program in Nondestructive Evaluation.
REFERENCES
1. Y. Guo, R. B. Thompson, and F. J. Margetan, "Simultaneous Measurement of Grain
Size and Shape from Ultrasonic Backscatiering Measurements Made from a Single
Surface", Review of Progress in Quantitative Nondestructive Evaluation 22, D. O.
Thompson and D. E. Chimenti, Eds., (AIP, New York, in press).
2. Y. Guo, F. J. Margetan, and R. B. Thompson, "Effects of Surface Roughness on
Ultrasonic Backscattered Noise", Review of Progress in Quantitative Nondestructive
Evaluation 20B, D. O. Thompson and D. E. Chimenti, Eds., (AIP, New York, 2001),
pp. 1306-1313.
3. Y. Guo, R. B. Thompson, and F. J. Margetan, "Verification if Time Domain Theories
for the Reflection of Ultrasonic Waves from Periodically Rough Surfaces", Review
of Progress in Quantitative Nondestructive Evaluation 21 A, D. O. Thompson and D.
E. Chimenti, Eds., (AIP, New York, 2002), pp. 75-82.
4. J. M. Claeys, O. Leroy, A. Jungman, and L. Adler, "Diffraction of Ultrasonic Waves
from Periodically Rough Liquid-solid Surface," J. Appl. Phys. 54, 5657-5662 (1983).
5. B. A. Auld, "General Electromechanical Reciprocity Relations Applied to the
Calculation of Elastic Wave Scattering Coefficients," Wave Motion 1, 3 (1979).
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