ANDE Course: Guided Waves Part II

September 17, 2009

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

Overview

• Recap of Plate Guided Waves

• Guided Waves in Cylindrical Rods

• Guided Waves in Cylindrical Shells

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!

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

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

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

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

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

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

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

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

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

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.

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

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.

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

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)

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.

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

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

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

Circumferential Guided wave Simulation in pipes

A-scan for 3mm radial holes for different depth

20%

80%

60%

40%

100%

0%

Reverberations within the wedge

360 degree traveled wave Gates

Energy plots of holes from 1.5 mm to 9 mm diameter

20% 40% 60% 80%

Calculation of size of the defects

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

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

Long Range Guided Waves Dispersion Curves for pipes

Guided Mode Types

F(1,1)

L(0,1)

T(0,1)

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

)

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

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.

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.

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

PIMS Example Application

• 24” buried line in tank farm

• PIMS installed on buried section beneath instrument

• Connection box on yellow post