ANDE Course- Guided Waves Part II
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Transcript of ANDE Course- Guided Waves Part II
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ANDE Course: Guided Waves Part II
September 17, 2009
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Question on Energy Partitioning
Distribution of displ6acement and energy in dilatational, shear and surface waves from a harmonic normal load
on a half-space for = 0.25
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Overview
• Recap of Plate Guided Waves
• Guided Waves in Cylindrical Rods
• Guided Waves in Cylindrical Shells
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Possible Guided Wave “Modes”
• Propagating waves
• Attenuating waves
• Non-propagating waves
• Steady state response
• Transient response
Frequency domain response and Transient response can be very different!
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Method of Partial Waves sincos ikxikzAe
sincos ikxikzBe
sincoscos )( ikxikzikz eBeAe
down going wave
up going wave
Superposition of partial waves
Assume rigid boundary conditions at z = 0 and z = h (i.e., vertical component of displacement must vanish)
)1:(cos2
,1 21cos2 1 ni
n
hikenote
h
ncfgetweefrom
sin
1ccP
1
2
2
1 2;
1
hq
q
n
ccpn
Cut-off frequencies
Dispersion relation
h
c0
h
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Guided Waves in Plates
Dispersion curves for a traction-free isotropic Al plate Dispersion curves for symmetric (solid lines) and antisymmetric (dashed lines) SH modes in
an isotropic plate with free boundaries
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Dispersion
at 100 mm
A0 Mode: Freq : 13.6 kHz Vph = 995 m/s Vgr = 1846 m /s Excitation: 3 cycle Gaussian pulse
at 1000 mm
at 2000 mm
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Higher order modes in a Plate
Frequency: 2.25 MHz Transducer dia: 25 mm Wedge angle: 56 deg Wedge base: 83.5 mm Excitation: 3 cycle Hanning Pulse
Thickness: 7.1 mm Material: Aluminum
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Method of Potentials
ufuu 2)()2(
uω 2
1
uuu 2
ufωu 2)()2(
u),(
0
tF r
2
2
2
2 1
tcL
2
2
2
2 1
tcT
Equation of motion (isotropic elastic medium):
Rotation vector
dilatational rotational
Generalised form:
Helmholtz decomposition
Scalar potential
Vector potential
additional conditions
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Basic Equations and Solution Forms
,01
integer,0)(
)()(
2
22
2
2
2
2
2
2
2
fr
nfk
cdr
df
rdr
fd
ngnd
gd
egrf
L
kzti
02
02
0
2
2
22
2
2
2
22
2
2
22
2
T
r
r
T
rr
z
T
z
crr
crr
c
)(
1
2
2
22)(
1
)(
2
2
22)(
cos
sin)(
cos
sin)(
cos
sin)(
,sin
cos)(
kzti
n
T
kzti
nr
kzti
nz
L
kzti
n
en
nqrCJ
kc
qen
nqrCJ
en
nqrBJ
kc
pen
nprAJ
rz
zr
zr
rr
r
rzu
rzru
zrru
1)(1
1
1
r
u
r
r
u
z
u
r
u
z
u
r
u
r
u
r
zrrz
rzrrrr
2
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Guided Waves in Rods: Torsional Modes I
ikztierfu )( 0
11 2
2
22
2
fk
cf
rr
f
rr
f
T
2
2
2
1 );()( kc
qqrAJrfT
0)(
0 1
arar
rr
qrJ
rr
u
r
)(2)( 10 qaJqaqaJ Frequency equation
...,418.8,136.5aqn
0
1122
2
frr
f
rr
fBrrf )(
dispersivenonqBreu
cqwithdispersiveqerqAJu
zciti
Tn
offcut
nn
ikzti
n
T
0
0)(
)/(
1
;0qa
Displacements for these modes
Dispersion relation (given a root qn)
22 )()()/( aqkaca nT
Roots
special root
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Guided Waves in Rods: Torsional Modes II
Displacement pattern for the first two modes
Dispersion curves for the first three torsional modes.
Ec 0
ka
ka
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Guided Waves in Rods: Longitudinal Modes I
0)()(4)()()()()()(2
01
2
10
22
11
22 qaJpapqJkqaJpaJkqqaJpaJkqa
p
)(
)(
2
)()(
)()(
1
1
2
222
2
)(
00
2
2
)(
11
2
2
paJ
qaJ
k
kq
p
q
B
A
where
eqrqJprikJB
ABu
eqrikJprpJB
ABu
tkzi
z
tkzi
r
‘Pochhammer-Chree’ frequency equation
Displacements for these modes
2
2
0
)0(
02
1;kaE
cE
c
0;)(
)(
)(
1
)(
0
zr
kzti
kzti
eqrCJ
eprAJ
Low frequency limit: >> a
High frequency limit: << a
cP of L(0,1) cR cP of L(0,n) cT n=2,3,4…
(n = 0 case)
arrzrr ,0Boundary conditions:
2
2
2
2
2
2
kc
q
kc
p
T
L
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Guided Waves in Rods: Longitudinal Modes II
Phase velocity curves for a 10 mm dia fused quartz waveguide.
Illustration of frequency dependence of normal stress. The normalized normal
stress is plotted at different frequencies.
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Plate .vs. Rod
)(
0
2
0
1
1
2
22
)(
11
1
1
2
22
)()()(
)(
2
)()()(
)(
2
kzti
TTL
L
TT
L
Tz
kzti
TTL
L
TT
Tr
erkJkrkJakJ
akJk
k
kkCu
erkJkkrkJakJ
akJk
k
kkiCu
)(
11
22
)(
1
22
)cos()cos()sin(
)sin(
2
)sin()sin()sin(
)sin(
2
kzti
TTL
L
T
L
Tz
kzti
TL
L
TTr
exkDkxkDak
ak
k
kku
exkkxkDak
ak
k
kkiu
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Observation of Pochhammer-Chree Modes
A. D. Puckett and M.L. Peterson, Acoustic Research Letters Online 6(4), pp 268-273, 2005
Comparison of experimental and analytical signal for a 1 MHz pulse excitation propagated through a 0.2 inch long
25 mm dia fused quartz rod.
Trailing pulses are a result of the superposition of multiple propagating modes. For any broadband signal many modes are excited, and each individual arrival will be the superposition of many modes.
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Guided Waves in Rods: Flexural Modes
ka
ka
Ec 0
)(
2
2
)(
)(
)()(
)()()(
)()()(
kzti
TnT
Ln
kzti
TnTnLn
kztiTnTnLn
erkJk
kiBrkikAJCW
erkJr
rkJr
BnrkJ
r
nACV
er
rkJnrkJ
rBrkJ
rACU
Displacements for these modes
)(
)(
)(
cos)(
sin)(
cos)(
kzti
z
kzti
kzti
r
enrWu
enrVu
enrUu
with
End-face shear transduction at high frequencies
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Shells: Circumferential guided waves
Dispersion plots of circumferential SH wave in Al as a function of h/d. ct = 3040 m/s
h = 1 mm, d = 6 mm h = 1 mm, d = 15mm h = 1 mm, d = 15mm
Boundary condition:
Signals at 5 MHz
H. Nishino and K. Yoshida, Acoust. Sci. & Tech. 27, 6 (2006)
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Circumferential Lamb Modes in a Shell
For small h/a, shell plate For small a/h, shell rod
Shell thickness: h = b-a
Dispersion curve comparison between hollow-cylinder guided waves and Lamb waves. D/T represents the pipe diameter/wall thickness ratio.
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Higher order modes in a Pipe
Frequency: 2.25 MHz Transducer dia: 25 mm Wedge angle: 56 deg Wedge base: 83.5 mm Excitation: 3 cycle Hanning Pulse
ID: 74.5 mm OD: 84.5 mm Material: Mild steel
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Torsional guided waves along shell axis
2
2
2
11
)(
);()()(
0;)(
kc
krkBNrkAJrU
uuerUu
T
TTT
zr
kzti
baratr
u
r,0
0.0 0.5 1.0 1.5 2.00.0
2.0
4.0
6.0
8.0
10.0
Frequency (MHz)V
ph (
m/m
s)
0.0 0.5 1.0 1.5 2.00.0
2.0
4.0
6.0
8.0
10.0
12.0
Frequency (MHz)
Vph (
m/m
s)
Dispersion plot for a 3 mm thick Al plate Dispersion plot for a 3 mm thick Al pipe (outer dia: 300mm )
Boundary condition
Displacement function
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Energy: Pipe Inspector Need:
Refineries, Chemical, Fertilizer, and
Power Plants have in accessible pipe
support regions in pipelines that are
most prone to corrosion.
Solution:
Guided Ultrasonic Waves are generated
in the accessible regions and propagate
circumferentially to inaccessible
regions to detect and quantify
corrosion.
Unique
•Capable of detecting and imaging 1 mm
pitting corrosion.
•The Shell capability is 15 mm or higher.
•Commercialisation underway.
Energy Response
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Circumferential Guided wave Simulation in pipes
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A-scan for 3mm radial holes for different depth
20%
80%
60%
40%
100%
0%
Reverberations within the wedge
360 degree traveled wave Gates
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Energy plots of holes from 1.5 mm to 9 mm diameter
20% 40% 60% 80%
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Calculation of size of the defects
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Long Range GW Benefits
Guided Wave screening offers:
• High productivity – kms per day
• Access required only at remote locations
• Carried out with pipe on-line
• Sub-Sea equipment available
• 100% coverage except for Flanges, Tees and other large features
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Guided Waves in Pipes
• Guided waves travel along the pipe and are reflected from changes in the cross-section
• Amplitude of the reflection depends on the total change in the pipe wall cross-section
structure
transducer
guided wave
defect
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Long Range Guided Waves Dispersion Curves for pipes
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Guided Mode Types
F(1,1)
L(0,1)
T(0,1)
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Test achieving 80m one direction range
0 20 40 60 80 0.0
0.2
0.4
0.6
0.8
Distance (m)
Am
p (
mV
)
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Corrosion at entrance to sleeved road crossing
-30.0 -20.0 -10.0 0.0 10.0 0.0
0.2
0.4
0.6
0.8
1.0
Distance (m)
Am
p (
mV
)
+F1 +F2 +F3 +F4 -F1 -F2 -F3 -F4
corrosion
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Structural Health Monitoring?
• Provides a simple means of repeating guided wave inspection of a pipeline over an extended period of time.
• Sealed in a polyurethane mould to give lifetime protection.
PIMS is a transducer ring permanently attached to the pipe under interrogation.
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Motivation for SHM
• The time and cost incurred in the inspection of many pipelines is dominated by the access costs.
• The installation needs to be done once.
• Repeat testing is then a simple matter of attaching transducer cables at a conveniently placed location and retesting.
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Features
• Standard weather proof box is uniquely serial numbered
• Programmed with all test parameters during installation
• Custom connectors can be used such as for sub sea use
• Re-testing is a simple plug in and collect
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PIMS Example Application
• 24” buried line in tank farm
• PIMS installed on buried section beneath instrument
• Connection box on yellow post