low frequency acoustic excitation and laser sensing of vibration as a tool for remote...

8/6/2019 Low Frequency Acoustic Excitation and Laser Sensing of Vibration as a Tool for Remote Characterization of Thin She

1/193

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


2/193


3/193

Low Frequency Acoustic Excitation and Laser

Sensing of Vibration as a Tool for Remote

Characterization of Thin Sheets

Etienne Mfoumou


4/193


5/193

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


6/193

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


7/193


8/193


9/193


10/193


11/193


12/193


13/193


14/193


15/193


16/193


17/193


18/193


19/193


20/193


21/193

m

mm


22/193


23/193


24/193


25/193


26/193


27/193

Kc

6.25m

c =2b

arccos

exp( K

2c

8a022b

)

c b


28/193

2a0

2 0

Kc


29/193

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,...


30/193

x

y

Oa

b

2mn = 2mn 1 +

d2

c4 2mn

1 + 2qm

Mcos2(m x0a )sin

2(n y0b )


31/193

M = hab 0 = 1 q = 2 (q > 1) 2mn

m

M

x0, y0


32/193

1 2

1 2

2 = 22

b2c21c

22

c21 + c22

c1 c2

c21 c22 c1 c c2 c (1 )


33/193

c =

T /(h) =E

d = 0

f20n =E n2

4 b2

f0n

f20n

f20n


34/193

y =mi=1

aiexp(t/i)

ai i

1 2


35/193

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 =


36/193

Kr2 =


37/193


38/193


39/193


40/193

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


41/193

strain

time

mm

mm

mm/s

mm

kS kS/s


42/193


43/193

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


44/193

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5


Normalizedstress

STRESS vs CRACK SIZE(LDPE)


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3


Normalizedstress

STRESS vs CRACK SIZE(Paper)



45/193

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%


46/193


47/193


48/193

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



Recovery2



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


49/193

K2s = 1.017 K2r = 0.983


50/193


51/193

m


52/193


53/193


54/193


55/193


56/193


57/193


58/193


59/193


60/193


61/193


62/193


63/193


64/193


65/193


66/193

th


67/193

m


68/193

m

m

g/cm3

c =Kc

a ( aw )

( aw

) =

sec( a2 w )

1 0.025

aw

2 + 0.06

aw

4


69/193

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


70/193

= 1, if

b

2

2

< y 0

d2A(1)

dx2 k21A(1) = 0, d

2A(2)

dx2+ k22A

(2) = 0


91/193

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


92/193

c21 c22 c1 c c2 c (1 )


93/193


94/193

a0


95/193

a0 = 3.5mm

a0 = 4.6mm a0 = 5.7mm


96/193


97/193

f /m


98/193


99/193


100/193


101/193


102/193

th


103/193


104/193


105/193

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) .


106/193

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


107/193

c2 < c1

11

= c1 c22c1

mm

mm mm

m

N

o

mm/min


108/193

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


109/193

3) m)

mm/min

mm/min


110/193


111/193

mn = c =

Fah

=ma

2+nb

2

mn =

m = 0, n = 2 = 2b

mn =2

b

=

E


112/193


113/193


114/193


115/193


116/193


117/193


118/193

m m


119/193


120/193


121/193


122/193


123/193

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)


124/193

=

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


125/193


126/193

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


127/193

V pp

kH z


128/193


129/193


130/193


131/193


132/193


133/193


134/193

GP a M P amm mm


135/193


136/193


137/193

N


138/193


139/193


140/193


141/193


142/193

th


143/193


144/193

E =


145/193

= 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


146/193

b

a h

y

z

F

y

z


147/193

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,...


148/193

(m = 0) f0n

f20n =E n2

4 b2

f20n

E

mm mm

o

N


149/193


150/193

mm/skH z


151/193

l(mm)

l(mm)

l(mm)

f2


152/193


153/193


154/193


155/193


156/193


157/193


158/193


159/193

m

m


160/193


161/193


162/193

o


163/193

Material Density (g/cm3) Length (cm) Width (cm) Thickness (m)

PPR 0.684 250 15 100

LDPE 0.91 250 15 27


164/193

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


165/193

mm

kS

mm/mm mm/mm


166/193


167/193

1.6 103

Data Min Max Std

Relax upper strain level ON-1 309.1 340.6 9.768



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


168/193

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


169/193

Data Min Max Std




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


170/193

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


171/193


172/193


173/193


174/193


175/193


176/193


177/193


178/193

mm

mm m


179/193


180/193

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


181/193

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 ,


182/193

Ed L

1 2

mm


183/193

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)


184/193

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


185/193

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


186/193

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 =


187/193

E

K

K

K =

K

2 8639

1 C0a21

T = 400 1s 162s 8639

1r 162r 3800


188/193


189/193


190/193


191/193


192/193


193/193

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

low frequency acoustic excitation and laser sensing of vibration as a tool for remote...

Documents