Identification of local material damping in vibrating plates using full-field measurements

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A. Giraudeau, F. Pierron - 2009 MPSVA conference

Identification of local material damping in

vibrating plates using full-field measurements

Alain Giraudeau, Fabrice PIERRON

LMPF research groupArts et Métiers ParisTech

Châlons en Champagne - France

7th International Conference on Modern Practice in Stress and Vibration Analysis, 8-10th September 2009, Cambridge

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Motivation / Objectives

Material damping Difficult to measure Problem with connections/joints Most of the standard techniques rely on beams Difficult (but important) to get both complex stiffnesses

on same testN.W. Tschoegl, W.G. Knauss, and I. Emri. Poisson's ratio in linear viscoelasticity - a critical review. Mechanics of Time-Dependent Materials, 6:3-51, 2002.

Objectives Simultaneous identification of both complex

stiffnesses (isotropy) on single test Method to cancel the effect of connections Explore the use of full-field measurements

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Identification of material properties :Stiffness

Damping

Thin plate specimens

Dynamic loading

Attachements : Parasistic effects Unknown energy loss

Equilibrium of forces No energy consideration

F(t)

Principle

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Dynamic equilibrium :

Point M :

iij,ij f

dVudVufdVu ii

V

i

V

ijij

V

,

dVudVufdSuTdV ii

V

i

V

ii

V

ijij

V

*:

Observed area

Internal f. Boundary f. Volumic f. Inertial f.

Principle of Virtual Work (P V W)

:,* yxui- Continuous

- Differentiable

ui* ui*ui*

Principle F(t)

M

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Considered case

Rectangular thin plate

Sine driving movement

Inertial excitation

Full field measurements

Data processing using Virtual Fields Method (V.F.M.)

Clamped on small area

Principle

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Principle of Virtual Work

u*

Thin plates Sine vibrating response

*=(u*)

xyD

xxDConstitutive law

,, xyxx DD

uFg , uuh ,,

Unknown

Thin plates <=> k

dVudSuTdV ii

V

ii

V

ijij

V

*:

kkDDf xyxx ,,,

SelectedMeasured

Virtual Fields Method

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dVudSuTdV ii

V

ii

V

ijij

V

*:

uuhuFgkkDDf xyxx ,,,,,,

along V. F. Selection :

= 0

= 0

uuhkkDDf xyxx ,,,,,

0*

0*u


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P V W uuhkkDDf xyxx ,,,,,

UD

DHG

xy

xx

Measured (w(x,y), k(x,y))

Selected

k*(x,y), w*(x,y)

(w(x,y),w*(x,y))

2

1

22

1)1(

U

U

D

D

HG

HG

xy

xx

Two selected virtuals fields : VF1, VF2

Direct resolution of an inverse problem


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Deflectometry

CCDO

M

P

Q

Deflection fieldsCurvature fields

Slope fields

In phase

/2 lag

d

. d

Experimental arrangement

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Image Processing

Grid ImagesAt rest Deformed

Phasesxx

y y

-

Slopes

x

-y

Spatial phase evaluation


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Experimental set-up


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Rea

lIm

agin

ary

Slope / X

0.41 mrad / col.

Slope / Y

0.17 mrad / col.

Deflection

34 m / col.

80Hz (out of resonance)Images (examples)

Results

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Piecewise virtual fields (Toussaint et al.,

IJMS, 2006)

w*(x,y) = 0

*(x,y) = 0

Virtual fields selection

Results

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Reference valuesCoupons: polycarbonate beams

h = 4mm, l = 10mm, L= 107-114mm

Clamped-free conditions

Free vibrations, first bending mode

E

(GPa)

( = tan() )

Means 4.900.055

C. variation

2.4% 5.2%

DMTA

0.056

Results

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Results

80 Hz 100 Hz reference (Beams, DMTA)(non res.) (res.)

)ˆ1(h

D12E

3xx

xx

xy

D

Dˆ

0,00

0,02

0,04

0,06

0,08

0,10

tan(delta)

)ERe(

)EIm()tan(

0,00

0,02

0,04

0,06

0,08

0,10

Poisson rat. loss factor

)ˆRe(

)ˆIm(

0

1

2

3

4

5

6

Young mod.

GPa

)ERe(E

0,0

0,1

0,2

0,3

0,4

0,5

Poisson rat.

)ˆRe(

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Boundary condition effects

Rubber washers

2nd accelerometer

= 34°

tan()=0.672nd accel.

Driving mvt. 100 Hz

Results

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0

1

2

3

4

5

6

Young mod.0,0

0,1

0,2

0,3

0,4

0,5

Poisson rat.

GPa

0,00

0,02

0,04

0,06

0,08

0,10

tan(delta)

0,000

0,020

0,040

0,060

0,080

0,100


Initial clamping With rubber washers

Boundary condition effects (2) 100 Hz

tan()=0.67

Results

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Additional damping effect

Viscoelastic patch at back surface

w*(x,y) = 0

*(x,y) = 0

Results

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0

1

2

3

4

5

6

Young mod.0,0

0,1

0,2

0,3

0,4

0,5

Poisson rat.

GPa

0,00

0,02

0,04

0,06

0,08

0,10

tan(delta)

0,00

0,02

0,04

0,06

0,08

0,10


Initial clamping With rubber washers

Additional damping effect (2) 100 Hz

Results

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Conclusion

Simultaneous identification of stiffness and damping

Material damping

Resonant or non resonant response Information on localization phenomena

Low sensitivity to boundary conditions

Could be applied to in-plane vibrations

Plate of any shape Part of a component

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Conclusion

Soundboard of a guitar

Local damping

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Conclusion

Improvements Time synchronization: high speed camera Better fixture (sweep over large frequency range)

Future work Apply to composite plates (anisotropy) Test on more complex structures

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Identification of local material damping in vibrating plates using full-field measurements

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