TEL AVIV UNIVERSITY
The Iby and Aladar Fleischman Faculty of Engineering
The Zandman-Slaner School of Graduate Studies
By
Eyal Arad (Dery)
Under the supervision of
Prof. Moshe Tur
1

2
Deals with the detection of propagating ultrasonic waves in plates, used for damage detection.
The detection is performed using a fiber optic sensor (specifically, a Fiber Bragg
Grating sensor) bonded to or embedded in the plate.

3
• General Introduction:
– Structural Health Monitoring and Non-Destructive Testing
– Lamb Waves in NDT
– Fiber Bragg Grating Sensors
• The goals of this work
• Analysis and results
– Analytical solutions
– Numerical solutions and comparison
– Tangentially bonded FBG
– Setup and Experimental Results
• Effects and implications
– Angular Dependence
– Rosette Calculations
• Summary of Findings
• Future Work

4
General Introduction
Structural Health Monitoring (SHM) and Non-
Destructive Testing (NDT)
• SHM- the process of damage identification (detection, location, classification and severity of damage) and prognosis
• SHM Goal- increase reliability, improve safety, enable light weight design and reduce maintenance costs
• NDT- an active approach of SHM
• Several NDT techniques exist, among them is Ultrasonic Testing
• Many Ultrasonic Testing techniques for plates utilizes Lamb Waves in a Pulse- Echo method ( damage= another source )
• Usually, both transducer and sensor are piezoelectric elements

5
Lamb Waves Implementations in NDT
• Lamb waves are Ultrasonic (mechanic) waves propagating in a thin plate (thickness<<wavelength)
• Important characteristics for NDT:
– Low attenuation over long distances
– Velocity depends on the frequency (could be dispersive)
– Creates strain changes that can be detected

6
Lamb Waves Implementations in NDT
• Some examples for suggested implementations in the aerospace field:
– Qing (
Smart Materials and Structures, v.14 2005
)
– Kojima (
Hitachi Cable Review, v. 23, 2004
)

Lamb Waves Propagation
• In infinite material 3 independent modes of displacement exist
• In thin plates the x and y displacements are coupled (boundary conditions) and move together
• Two types of modes exist:
– Symmetric waves
(around x)
7
– Antisymmetric waves

Lamb Waves Propagation
8
• Plane wave (infinite plate)
– Symmetric waves
(displacement) u x
 i
 
B cos
 y
 
C cos
 y e u y


 
B sin
 y
 
C sin
 y
 i
  x
  t

 e i
  x
  t
 tan
 b tan
 b
 
4

2


2
 
2

2
Where ξ is the wave number ω/v ph material
’ s constants
, and α,β are proportional to the
– Antisymmetric waves u x
 i
 
A sin
 y
 
D sin
 y u y

 
A cos
 y
 
D cos
 y tan tan
 b
 b
 


2  
2
2

4

2
 e i
  x
  t

 e i
  x
  t

α,β

9
Lamb Waves Propagation
• Cylindrical Lamb wave
– In the area close to the transducer
– Symmetric case u r

AH
1 u z
 
AH
0 r X

' 2

 
' 2

'
2

'
 
'
2

 cosh

'

 sinh z


2  
2

2
'
2

' z


2
2


2

'
2 cosh cosh


' b
' b cosh

' z

 e i
 t sinh sinh


' b
' b sinh

' z

 e i
 t
• H
0 and H
1 are Hankel Function of zero and first kind.
– For the antisymmetric solution it is only necessary to interchange sinh and cosh

Lamb Waves Propagation
• Dispersion relations (
Vph(f)
):
• Lamb wave modes tan
 b tan
 b tan
 b tan
 b








4

2
2




2
2
4

2
2

2
2

10
• The selected working mode is A
0

• Permanent, periodic perturbation of the refractive index
• λ
B
=2n eff
Λ
11
• Reflection curve
• Measuring Ultrasound according to:
 
B

0 .
79

B

R

P r, opt
R
0

(

)


 
B


R
 
P in, opt
R(

)
R
0
8
16
12
4
0
1549.0
Δλ
B neutral with strain
1549.2
P out,opt
1549.4
λ
B
1549.6
1549.8
1550.0

12
– Directional Sensitivity
– Small Size
– Fast Response – up to several MHz
– Ability to Embed inside Composites
– EMI, RFI Immunity
– Ability to Multiplex (several sensors on the same fiber)

13
SHM & NDT concept and goal
Lamb waves and their importance to NDT
FBG principle and advantages in NDT

• To build an analytical model for a pulse of propagating Lamb wave, in order to validate a Finite Element (numerical) model, for applying on complex cases which cannot be solved analytically.
wavefront
• To analyze the behavior of the detected ultrasonic signal at close range to the transducer, where the wave is cylindrical.
PZT
14
• Extending published plane wave analysis, to analyze the effect of close range sensing on the angular dependence of FBGs and on angle-to-source calculations.
Incident wave y, ε yy
ε
2
θ
ε
1
FBG
ε
A
FBG x, ε xx arbitrary direction

15
Lamb Wave Solutions for a Pulse Input
• Input: single period sine function pulse u x x
• Plane wave A
0
:
– x=0

 x x


0 ,
0 , y y

 b , t b ,




 signal
F ( signal )
 i
 
A sin
 b
 
D sin
 b

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
2.3


A y

( x i


 sin
0 ,
F ( signal y
 b

 b )

G

) sin
A
 
 b
 cos
 y
 
G cos
 y

5
4
3
Using inverse Fourier transform to convert to the time domain
2
1
0
2.4
2.5
time
2.6
2.7
2.8
x 10
4 -
-1
1.5
2 2.5
t
3
A
0 plane wave displacements u x
(blue) and u x=0 and y=b y
3.5
4 x 10
4
(green) vs. time at

watch
Lamb Wave Solutions for a Pulse Input
6000
• For all x (
Plane wave A
– Dispersion relation-
A
0
0
) is dispersive u x
( x , y u y
( x , y
 b ,

)
 b ,

)
 u x
( x
 u y
( x

0 , y

0 , y
 b ,

) e i
 x
 b ,

) e i
 x
5000
4000
3000
2000
1000
0.5
1 1.5
2 2.5
f[Hz]
Dispersion relation for A
0
3 3.5
4
(blue) and S
0
4.5
x 10
5
(black)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
2.5
3 3.5
4 4.5
5 5.5
6 u x displacements of A
0 t x 10
-4 at distances of 0 (blue), 10(green), 20 (red) and 30cm (cyan)
16
3
2
1
0
-1
-2
5
4
2 3 4 t
5 6 x 10
-4 u y displacements of A
0 at distances of 0 (blue), 10(green), 20 (red) and 30cm (cyan)

watch
Lamb Wave Solutions for a Pulse Input
1
• Cylindrical Lamb wave
0.8
u r
( r r
( r
 r
0
, z
 r
0
, z

 b , t ) b ,

)

 input
F
_ signal
 input _ signal


F ( signal )
0.6
0.4
0.2
0
A

H
1
 
0 
' 2

 
' 2

 sinh

F ( signal )
' b


2

2

2

' 2 sinh sinh


' b
' b sinh

' b


-0.2
-0.4
-0.6
-0.8
-1
2.5
3 3.5
4 4.5
t x 10
-4 u r displacements of A
0 at r= PZT edge (blue), 5 (green), 10 (red) and 20cm (cyan)
~ r
 r , z ,
 

F ( signal ) H
1
 

 sinh
H
1
 
0

 sinh

' z


2 
2


2
'
2 sinh sinh


' b
' b sinh

' b

 2 
2


'
2
2 sinh

' b



' z


4
3
2
1 z
( r , z ,

)


H
0
H
1

'
2

'
 
'
2
 
0 

 cosh
'
2

 
'
2

' z


2
2


2

'
2 cosh cosh


' b
' b cosh

' z



 sinh

' b

 2 
2

2

'
2 sinh

' b


17
0
-1
2 2.5
3 3.5
4 4.5
5 u z displacements of A
0 t -4 x 10 at r= PZT edge (blue), 5 (green), 10 (red) and 20cm (cyan)

Numerical solutions and comparison
• Finite Element Method (
FEM )
A computer simulation which divides the plate into small elements and solves the energy relations between them.
0.8
0.8
0.6
analytical Ur at r
0
(5mm) analytical Ur at 11cm
FEM Ur at r
FEM U
FEM U
FEM U z
at r z z
0 z
(5mm)
0
at 11 cm
at 31 cm z y x
0.2
0.4
0
0.2
-0.2
-0.6
-0.2
-0.8
2.4
2.5
2.6
3
2.8
3.5
3 3.4
4.5
3.6
-4
-4
– The analytical solutions Were crucial in choosing the parameters for the FE
Models in order to receive the correct model
– The FEM enables solving even more complex cases (e.g. plate with a damage) z
18
Courtesy of Iddo Kressel of IAI ltd.
r

Analytical model
 Numerical model

19
• What is the analytical influence of cylindrical waves in Lamb wave detection by a FBG?
• The FBG signal is angular dependent (as opposed to PZT sensor)
• FBG parallel to the Plane wavefront- No
Signal in the tangential FBG
• Cylindrical wave- Signal (strain) Exists
 r

 u
 r r
1


 u r r
• The tangential strain is:
– Different in its shape than the radial strain
– It can not be neglected at close distance
– Decays faster
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
PZT wavefront wavefront
2.2
2.4
2.6
2.8
3 3.2
time x 10
-4 radial (blue) and tangential (green) strains at 21mm

Purpose- to measure an ultrasonic
Lamb wave via FBG sensor and validate analytical and numerical models
Signal
Processing
Basic measurement setup:
• Function Generator produces an input signal.
• The PZT transforms the electrical signal to an ultrasonic wave that propagates through the plate.
• The sound vibrations affect the FBG which is bonded to the plate.
• The FBG transforms the mechanical vibrations to an optical Bragg reflection shift.
• This shift is identified by the optical interrogation system.
Detector
Laser source
Function
Generator
+Amplifier
20 y

PZT Exciter
FBG sensor r
θ
 x x

Experimental Results
Tangential strain Radial strain
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
2.6
2.7
2.8
2.9
time
3 3.1
3.2
3.3
x 10
-4 measured (red) and analytic (blue) strains for tangential FBG at 7cm
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
2.4
2.6
2.8
3 3.2
3.4
3.6
3.8
ε r time x 10
-4 strains of measured (red) vs. analytic (blue) at 7cm, with analytic S
0 included
• These figures reinforce two claims:
– The tangential strain, though smaller and decaying faster than the radial strain, exists
21
– Experimental and analytical results match

y
Goal: to show the different angular dependence of FBGs for plane and cylindrical waves
• Plane wave: 
FBG
• Cylindrical wave:
  max cos
2
   xx cos
2

PZT
For θ=0 (PZT-FBG angle) the principal strains are in the x,y directions:
 xx
  r

 u
 r r

FBG
  xx cos
2
   yy sin
2
 
 u
 r r cos
2
  u r r
 yy
 

 u r r (
 
FBG
 max cos
2

)
1.2
1
PZT Exciter
FBG sensor sin
2

0.8
wavefront r
FBG

β x x
0.6
0.4
0.2
0
22
-0.2
0 15 30 45 60 75 90 105 120 135 150 165 180 angle
Analytical angular dependence for plane wave assumption (cos 2 (β), green line) and cylindrical assumption (blue line) at 21mm from the source

Cylindrical wave (cont.):
When ignoring the tangential effect, the error could be large. For example, at β=75 degrees:
0.2
0
-0.2
-0.4
1
0.8
0.6
0.4
-0.6
2.5
2.6
2.7
2.8
time
2.9
3 3.1
3.2
x 10
-4
75 degrees comparison of general analytic strain (blue) measured signal (green) and analytic without tangential strain (red) y
PZT Exciter
FBG sensor r
Conclusion : The tangential strain affects the angular dependence and cannot be ignored at small distances from the source.
23
 x

Rosette Calculations
• Rosettes are used for damage location in NDT
• Prior work uses only plane wave rosettes

Our work intends to:

Enable accurate location of damages in a close range
 Present different calculation for each wave (planar/ cylindrical)
What is a rosette?
24

Rosette Calculations
• For a Plane wave:
– Only 2 FBGs are required!
– Signals are in-phase.
– Max. values can be used
-0.2
-0.4
-0.6
-0.8
0.4
0.2
0
1
0.8
0.6
25

FBG
  max cos
2

3.5
4 4.5
time
5 5.5
x 10
-4
Signals of 2 FBGs oriented at different angles in the plane wave case

Rosette Calculations
• For a cylindrical wave:
– 3 FBGs are required!
– Signals are not necessarily in phase and differ from each other!!!
– Signal values should be taken at a specific time!
1
0.8
0.6
y
C

C
FB
G
B
26
FBG A



C
A
0 °
B
45 °
90 °

B

A
B
A x
0.4
0.2
0
-0.2
-0.4
-0.6
2.4
2.6
2.8
time
3 3.2
3.4
x 10
-4
Measured strains for angles: 0 (blue), 45 (green) and 90 (red) degrees

Rosette Calculations
• Cylindrical wave (cont.)
Angle to the source:
– Analytically (θ is 0).
tan( 2

)

2

45

0



90

90
 
0 angle strain 0 deg.
strain 45 deg.
strain 90 deg.
80
60
40
20
0
-20
-40
-60
-80
27
2.6
2.65
2.7
2.75
2.8
2.85
2.9
2.95
3 time x 10
-4
Estimated angle to the source (bold blue line), using analytical strain solutions (also added for time reference) y r
 x
PZT Exciter
FBG sensor y
C



C
A
0 °
B
45 °
90 °
B

C
FB
G
B
FBG A

B

A
A x

Rosette Calculations
• Cylindrical wave (cont.)
Angle to the source:
– Angle from measured signal was not as expected!!!
– Applying a 1mm shift to one of the FBGs in the analytical calculation shows a similar effect
☺ tan( 2

)

2

45

0



90

90
 
0
40
30
20
10
0
-10
-20
-30 angle strain 0 deg.
strain 45 deg.
strain 90 deg.
-40
2.6
2.65
2.7
time
2.75
2.8
Estimated angle to the source (bold blue line), using measured strain signals (also added for time
28 reference) x 10
-4
40
30
20
10 angle strain 0 deg.
strain 45 deg.
strain 90 deg.
0
-10
-20
-30
-40
2.65
2.7
2.75
2.8
time x 10
-4
The effect of 8*10 -7 [sec] time shift (~1 mm) of one of the analytical strain solutions on the angle estimation capability

Rosette Calculations
• Conclusions and Implications
– Realistically, the estimated angle will never be constant
– Improved analysis method for cylindrical rosettes:
• Perform analysis for each time step
• Choose the angle for which the denominator is maximal
– In plane wave rosettes this problem does not exist since it is possible to assume signals are in phase
Golden Rule: For long distance use plane wave rosette, For short distance- cylindrical wave rosette
29 tan( 2

)

2

45

0



90

90
 
0
0
-10
-20
-30
40
30
20
10 angle strain 0 deg.
strain 45 deg.
strain 90 deg.
-40
2.6
2.65
2.7
time
2.75
2.8
x 10
-4
Estimated angle to the source (bold blue line), using measured strain signals (also added for time reference)

30
Exact analytical solutions for a pulse of plane and cylindrical Lamb waves was calculated.
Parameters for a Finite Element Model were determined.
The angular dependence of FBGs at close range to the transducer, where the wave is cylindrical, was analyzed and measured.
Three FBG rosette calculations were performed and the effect of the tangential strain on the angle finding was analyzed.
The effect of co-location error was demonstrated.

Applying FBGs to NDT system for damage detection
Real time monitoring
High accuracy at all distances
31
Anisotropy
Composite plates, which are common in the industry, are usually anisotropic
Ability to embed optical fibers
Phase and group velocities are angle dependent
0.20
120
0.15
150
0.10
0.05
180
0.05
0.10
0.15
0.20
210
240
90
60
S
0
Slowness Curve
300
30
330
0
270

• Prof. Moshe Tur
• Lab colleagues, and especially:
– Yakov Botsev
– Dr. Nahum Gorbatov
• Iddo Kressel
• Shoham
32