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Blekinge Institute o Technology
Doctoral Dissertation Series No. 2008:16
School o Engineering
low frequency acoustic excitation
and laser sensing of vibration as atool for remote characterization
of thin sheets
Etienne Moumou
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Low Frequency Acoustic Excitation and Laser
Sensing of Vibration as a Tool for Remote
Characterization of Thin Sheets
Etienne Mfoumou
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Low Frequency Acoustic Excitation and Laser
Sensing of Vibration as a Tool for Remote
Characterization of Thin Sheets
Etienne Mfoumou
Blekinge Institute of Technology Doctoral Dissertation Series
No 2008:16
ISSN 1653-2090ISBN 978-91-7295-155-6
Department of Mechanical Engineering
School of Engineering
Blekinge Institute of Technology
SWEDEN
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2008 Etienne Mfoumou
Department of Mechanical Engineering
School of Engineering
Publisher: Blekinge Institute of Technology
Printed by Printfabriken, Karlskrona, Sweden 2008
ISBN 978-91-7295-155-6
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m
mm
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Kc
6.25m
c =2b
arccos
exp( K
2c
8a022b
)
c b
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2a0
2 0
Kc
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2
t2 c2
2
x2+
2
y2
+ d2
2
x2+
2
y2
2 =
p (x,y,t)
h
,
c =
T /(h), d2 =Eh2
12 (1 2)
mn = c
ma
2+n
b
21 +
d2
2c2
ma
2+n
b
2,
m, n = 0, 1, 2, 3,...
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x
y
Oa
b
2mn = 2mn 1 +
d2
c4 2mn
1 + 2qm
Mcos2(m x0a )sin
2(n y0b )
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M = hab 0 = 1 q = 2 (q > 1) 2mn
m
M
x0, y0
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1 2
1 2
2 = 22
b2c21c
22
c21 + c22
c1 c2
c21 c22 c1 c c2 c (1 )
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c =
T /(h) =E
d = 0
f20n =E n2
4 b2
f0n
f20n
f20n
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y =mi=1
aiexp(t/i)
ai i
1 2
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y = K0 + a1exp(t/1) + a2exp(t/2)
a1 a2 1 2
K0 = 0
K0 = 6.4
1 2
2n
1 a1
1n C0 a21n C0 =5n=1 1n/a
21n/5 = 30.2
= a1nexp(t/(30.2 a21n)) + a2nexp(t/8639) .
K
f2 = KEs
4 L2 L = 0.25 = m3
f2
= K f20/0 f
20/0 =
1
4L2K Ks2 =
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Kr2 =
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150mm 15mm
0.3 0.35 0.4 0.45 0.5 0.550.5
1
1.5
2
2.5
3x 10
5
Strain (%)
Freq*Freq(Hz*Hz) y = 8.1e+005*x 1.5e+005
ExperimentCurve fit
f2
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strain
time
mm
mm
mm/s
mm
kS kS/s
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mm mm
mm mm
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
Normalized crack length
Normalizedstress
STRESS vs CRACK SIZE(Al foil)
ExperimentalLEFMStrip Yield Model
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0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
Normalized crack length
Normalizedstress
STRESS vs CRACK SIZE(LDPE)
ExperimentalLEFMStrip Yield Model
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
Normalized crack length
Normalizedstress
STRESS vs CRACK SIZE(Paper)
ExperimentalLEFMStrip Yield Model
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8
10
12
14
Normalized crack length d/a
Relativefrequencyshift(%)
Defect severity analysis of paperboard 550 x 30
Theory
Experiment
mm mm mm N
2.25%
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0 50 100 150 200 250 300 350 400 450150
200
250
300
350
Time (min)
Frequency(Hz)
Paperboard: Frequency change during successive conditionings ON and OFF
Relax upper strain level ON1
Relax lower strain level ON1
Recovery1
Relax upper strain level ON2
Relax lower strain level ON2
Recovery2
Relax upper strain level ON3
Relax lower strain level ON3
Recovery3
Cond. ON
1 Cond. ON
1 Cond. ON
1Cond.OFF1
Cond.OFF2
Cond.OFF3
1.6 103
0 500 1000 1500 2000 2500 3000 35000
50
100
150
200
250
300
Resonance
frequency
Strain
Stress
Time (s)
Cycle: 2 3 4 5 6
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K2s = 1.017 K2r = 0.983
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m
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th
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m
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m
m
g/cm3
c =Kc
a ( aw )
( aw
) =
sec( a2 w )
1 0.025
aw
2 + 0.06
aw
4
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c Kc
c =2 b
asec
exp
K2c
8 a 2 2b
b
2
t2 c2
2
y2=
p (y, t)
h
m d2b2
dt2
= p S = p 2w
c =
T
hT
2
t2 c2
2
y2 = m
h 2 w d2 b2
dt2
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= 1, if
b
2
2
< y 0
d2A(1)
dx2 k21A(1) = 0, d
2A(2)
dx2+ k22A
(2) = 0
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21 =2
b
2
2
c
2
1
, 22 =2
c
2
2
2
b
2
A(1) = 1 cosh (1x) + 1 sinh(1x)
A(2) = 2 cos(2x) + 2 sin(2x)
1, 2, 1 2
dA(1)
dx |x=0= 0 dA
(2)
dx |x=a= 0
12
A(1) = 1 cosh (1x)
A(2) = 2cos (2a x)
cos (2a)
A(1)(x = a/2) = A(2)(x = a/2) 1 cosh
1a2
= 2
cos
2 a2
cos(2a)
dA(1)
dx|x=a/2 =
dA(2)
dx|x=a/2 11 sinh
1
a
2
= 22
sin
2a2
cos (2a)
1 tanh
1a2
= 2 tan
2
a2
1a2
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c21 c22 c1 c c2 c (1 )
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a0
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a0 = 3.5mm
a0 = 4.6mm a0 = 5.7mm
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f /m
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th
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2
t2 c2
2
x2+
2
y2
= 0
c = T /(h)T
h
=
m,n=0
mn =
m,n=0
Amn cosmx
a sinn
y
b sin(mnt + mn) .
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Amn mn mn
mn = c
ma
2+n
b
2, m, n = 0, 1, 2, 3,...
a, b
mn (x, y = 0) = 0, mn (x, y = b) = 0
ddx (x = 0, y) = 0, ddx (x = a, y) = 0.
y =
0, y = b x = 0, x = a
2(1,2)
t2 c21,2
2(1,2)
x2+
2(1,2)
y2
= 0
1 2
m1
2 1
2 < < 1
(1,2) = A(1,2) (x)sin
y
b
cos(t)
= 1
1 c1 c22c1
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c2 < c1
11
= c1 c22c1
mm
mm mm
m
N
o
mm/min
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V pp
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8
10
12
14
Normalized crack length d/a
Relativefrequ
encyshift(%)
Defect severity analysis of paperboard 550 x 30
Theory
Experiment
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3) m)
mm/min
mm/min
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mn = c =
Fah
=ma
2+nb
2
mn =
m = 0, n = 2 = 2b
mn =2
b
=
E
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m m
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2
t2 c2
2
x2+
2
y2+ d2
2
x2+
2
y22
=p (x,y,t)
h
z = 0
T /(h)
d2 = Eh2
12(12)
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=
m,n=0
mn =
m,n=0
Amn cos
m xa
sin
n yb
sin(mnt + mn) .
Amn, mn mn
mn = c
ma2
+
nb2
1 + d
2
2c2
ma2
+
nb2
m = 0, 1, 2, 3,..., n = 1, 2, 3,...
mn (x, y = 0) = 0, mn (x, y = b) = 0,
ddx
(x = 0, y) = 0,ddx
(x = a, y) = 0.
y =
0, y = b x = 0, x = a
=
2
24
Eh3
(1 2)T a2
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T /(h) =
E
mn = 2fmn =
E
ma
2+n
b
2,
m = 0, 1, 2, 3,..., n = 1, 2, 3...
f20n =E n2
4
b2
mm mm
m mmo
mmmm mm mm mm mm
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V pp
kH z
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GP a M P amm mm
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N
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th
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E =
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= i = iEi, i = 1, 2, 3,...
i Ei i
FL = i
Fi
Ai
L
L =
i FiiAi
L =
i itit
EL =
iEitit
EL Ei
S =
1N
Ni=1
(Ei Eav)2
N Eav
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b
a h
y
z
F
y
z
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2
t2
c22x2
+ 2
y2
+ d2
2
x2+
2
y22
= p (x,y,t)h
(x, y)
c
p
c d
c = T /(h) d2 = Eh212(1
2
)
E T
mn
mn = cm
a2
+n
b2
1 +
d2
2c2m
a2
+n
b2
,
m, n = 0, 1, 2, 3,...
m n
mn = c
ma
2+n
b
2, m, n = 0, 1, 2, 3,...
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(m = 0) f0n
f20n =E n2
4 b2
f20n
E
mm mm
o
N
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mm/skH z
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l(mm)
l(mm)
l(mm)
f2
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m
m
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o
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Material Density (g/cm3) Length (cm) Width (cm) Thickness (m)
PPR 0.684 250 15 100
LDPE 0.91 250 15 27
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T e n s i le m a c h i n e
L o a d c e l l
P n e u m a t ic g r ip
S a m p l e
L o u d s p e a k e r
L a s e r Do p . V i b .
F u n c . G e n .
Oscil loscope
P C c o n tr o l M T S
P C c o n tr o l D A Q
T e n s i le m a c h i n e
L o a d c e l l
P n e u m a t ic g r ip
S a m p l e
L o u d s p e a k e r
L a s e r Do p . V i b .
F u n c . G e n .
Oscil loscope
P C c o n tr o l M T S
P C c o n tr o l D A Q
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mm
kS
mm/mm mm/mm
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1.6 103
Data Min Max Std
Relax upper strain level ON-1 309.1 340.6 9.768
Relax upper strain level ON-2 306.8 312.8 1.897
Relax upper strain level ON-3 305.3 316.6 3.337
Relax lower strain level ON-1 169.5 187.5 5.987
Relax lower strain level ON-2 169.5 174 1.388Relax lower strain level ON-3 167.3 171.8 1.611
Recovery 1 150.8 176.3 2.58
Recovery 2 154.5 180 2.243
Recovery 3 153 176.3 2.594Min=Minimum Max=Maximum Std=standard deviation
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0 10 20 30 40 50 60 70305
310
315
320
325
330
335
340
345
Time (min)
Frequency(Hz)
Paperboard: Frequency change at upper strain level during conditioning ON
Conditioning ON1Conditioning ON2Conditioning ON3
1.6 103
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Data Min Max Std
Relax upper strain level ON-1 257.3 270.8 3.941
Relax upper strain level ON-2 257.1 264.8 2.383
Relax upper strain level ON-3 255.1 263.3 2.359
Relax lower strain level ON-1 142.5 163.5 6.796
Relax lower strain level ON-2 133.5 141 2.625
Relax lower strain level ON-3 132 138.8 1.883
Recovery 1 120.8 150.8 3.089
Recovery 2 114.8 144 3.331
Recovery 3 114.8 140.3 2.628
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Comparison TenCate et. al. Mfoumou et. al.
Samples Regular solid materials (rocks,
metals, concrete, ...)
Thin sheets having no bending
stiffness (paperboard, LDPE, ...)
Conditioning Harmonic acoustic wave
incremented through the
fundamental longitudinal
resonance frequency
Harmonic mechanical loading and
unloading (not related to the
resonance frequency)
Conditioning
frequency range
High (1 to 10kHz) Low (0.0025Hz)
Conditioning
duration
About 1000 seconds About 4000 seconds
Offset pre-stressed
for conditioning
Not required Required in order to give bending
stiffness to the sample
Method ofinvestigation
Resonance method(longitudinal)
Resonance method(bending)
Resonance
frequency range
Several kilohertz Below 500Hz
Source PZT (contact method) Loudspeaker (non-contact
method)
Receiver (sensing) Accelerometer cemented on
the sample (contact method)
Laser beam (non-contact method)
Featuring
observation
Drop in Youngs modulus and
increase in material damping
Drop in Youngs modulus.
After stress
removal
The material properties
recover towards their original
values
The material properties recover
towards their original values
Process of
conditioning and
recovery
Assymetric Assymetric
Overall A retarded effect ressembling
creep appears, which cannot be
explained with equilibrium
elasticity theory
A similar effect appears here,
though much faster, which can
also not be explained with
equilibrium elasticity theory
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mm
mm m
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mm mm/s
mm
mm mm
Time (s)
Elongation (mm)
1.0
0.6
200 s
200 s
200 s
200 s
Loading level
Reverse loading level
Cycle 1 Cycle 2
kS kS/s
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y =m
i=1
aiexp(t/i)
ai i
1 2
y = K0 + a1exp(t/1) + a2exp(t/2)
a1 a2 1 2
K0 = 0
K0 = 6.4
exp[(t/s)p
f1
f21 =Ed
4 L2 ,
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Ed L
1 2
mm
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0 500 1000 1500 2000 2500 3000 35000
50
100
150
200
250
300
Resonance
frequency
Strain
Stress
Time (s)
Cycle: 2 3 4 5 6
0 20 40 60 80 100 120 140 160 180 20018.8
19
19.2
19.4
19.6
19.8
20
20.2
20.4
20.6
Time (s)
Stress(MPa)
Relaxation experimental data together with curve fitting for cycles 2 to 6
data cycle 2
curve fit 2
data cycle 3
curve fit 3
data cycle 4
curve fit 4
data cycle 5
curve fit 5
data cycle 6
curve fit 6 = 0.7221*exp(t/14.6349)+19.22*exp(t/8818.3)
= 0.7261*exp(t/16.2364)+19.26*exp(t/8396.3)
= 0.75*exp(t/17.6491)+19.31*exp(t/8733.6)
= 0.761*exp(t/19.1975)+19.38*exp(t/8190)
= 0.9527*exp(t/28.0426)+19.47*exp(t/9058)
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1 2 3 4 510
11
12
13
14
15
16
17
18
19
20
Cycle number
a2
20a1
b2
300b1
Cycle number1 2 3 4 5
3000
4000
5000
6000
7000
8000
9000
10000
1
2 2n
n = 2 1
a1 1n C0 a21nC0 =
5n=1 1n/a
21n/5 = 30.2
= a1nexp(t/(30.2 a21n)) + a2nexp(t/8639) .
a1 a2
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0 20 40 60 80 100 120 140 160 180 200
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
Time (s)
Stress(MPa)
Reverse relaxation experimental data together with curve fitting for cycles 2 to 6
data cycle 2
curve fit 1
data cycle 3
curve fit 2
data cycle 4
curve fit 3
data cycle 5
curve fit 4
data cycle 6
curve fit 5
= 6.40.7419*exp(t/16.8322)+5.791exp(t/3766.5)
=6.4 0.7435*exp(t/19.2160)+5.64*exp(t/3935.5)
= 6.40.7161*exp(t/20.0884)+5.585*exp(t/4478.3)
= 6.40.6992*exp(t/15.1930)+5.433*exp(t/3222.7)
= 6.40.6564*exp(t/16.3479)+5.151*exp(t/3204.1)
1 2 3 4 51000
1500
2000
2500
3000
3500
4000
4500
5000b
1
b2
7a1
b2
200b1
Cycle number1 2 3 4 5
4
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
Cycle number
a2
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K
f2 = KEs
4 L2
L = 0.25 = m3
f2 = K
f20/0
f20/0 =1
4L2K Ks2 =
Kr2 =
0 20 40 60 80 100 120 140 160 180 2000.930.94
0.95
0.960.97
0.98
0.99
1.00
1.01
1.02
Time (s)
Cycle 2, relaxation
f /f2
01.017 /
02
0 20 40 60 80 100 120 140 160 180 2001.00
1.05
1.1
1.15
1.2
1.25
Time (s)
Cycle 2, reverse relaxation
f /f
2
0/1.0170
2
K2s = 1.017 K2r = 0.983
Ks2 =
Ks3 = Ks4 = Ks5 = Ks6 =
Kr2 = Kr3 = Kr4 =
Kr5 = Kr6 =
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E
K
K
K =
K
2 8639
1 C0a21
T = 400 1s 162s 8639
1r 162r 3800
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There is a need to monitor the existence and e-
ects o damage in structural materials. Bulk com-
ponents provide a much publicized example, but
the need exists in a variety o other structures,such as layered materials used in ood packaging
industries. While several techniques and models
have been proposed or characterization and con-
dition monitoring o bulk materials, less attention
has been devoted to thin flms having no bending
rigidity. This study is thereore devoted to the de-
velopment o a new method or remote acoustic
non-destructive testing and characterization o
thin flms used in ood packaging materials or si-milar structures.
A method or assessin the stren th in the resen
the resonance requency and the materials elastic
property or single layers as well as or laminates,
which yields a new modality or sheet materials
remote characterization.
Further, the method has allowed demonstrating
that thin sheets having no bending stiness exhi-
bit a slow non-equilibrium dynamics when slightly
loaded within their elastic region and monitored
at constant strain. We ound that the resonance
requency shits downward in response to a con-
ditioning strain and to the number o cycles. This
is an indication o a long-time slow dynamics re-laxation, similar to that observed on bulk materi-
als o many types. Dierences and similarities in
the set as ell as eat res obser ed are oin
abstract