Analysis of Active Constrained Layer Dampingin composite sandwiched plates

Charles K. KimNASA-Goddard Space Flight Center, Code 542

Greenbelt, MD 20771

Chul H. ParkWilcoxon Labs

Wilcoxon Research Inc.Gaithersburg, MD 20878

Amr BazMechanical Engineering Department

University of MarylandCollege Park, MD 20742

FEMCI WorkshopMay 18, 2000

Abstract

This paper describes how a new class of composite sandwich platesis implemented to control vibration of the bending and torsional modes usingbuilt-in Active Constrained Layer Damping (ACLD). The paper also gives aninsight into the fundamentals governing the dynamics and active/passivecontrol of smart composite sandwiched plate structures to use as the potentialbasic building block of structures where effective vibration damping isessential to their successful operation.

To improve predictions of the dynamics and controlled vibration of thecomposite sandwiched plate/ACLD, a powerful assumed displacement field forthe finite element modeling is introduced. This assumed displacement fielddiffers from the classical laminated theory and offers a definite advantage infinite element modeling as it gives a displacement distribution along the wholethickness of the laminates and requires fewer degrees of freedoms torepresent the kinematical relationships for viscoelastic layers withpiezoelectric layers in ACLD. Also, the predictions of the finite element modelusing this assumed displacement field have been validated by comparing ofmodal frequencies and damping loss factors with experiment and are found tobe in close agreement.

Outline of Presentation

z Introduction

z Active Constrained Layer Damping (ACLD)

z Theoretical Development (Finite Element Modeling)

z Experimental Performance

z Comparison between Theory & Experiments

z Summary

Passive and active layer damping(a)-Passive: unconstrained(b)-Passive: constrained (c)-Active: constrained

Viscoelastic material

Variation of storage modulus, and loss factor, with temperature[Nashif et al. 1985]

Variation of storage modulus, and loss factor, with frequency [Nashif et al. 1985]

s

44

33

2210)( fafafafaafE ++++= 4

43

32

210)( fbfbfbfbbf ++++=η

Interaction processes between theelectrical, mechanical, and thermal system

[Ikeda Takuro; Fundamental of Piezoelectricity,1990]

Active constrained layer damping

• Combines active & passive damping control• Enhances energy dissipation characteristics

Operating principle of sandwichedplate/ACLD system.

Theoretical Development(Formulation)

z Displacement Field

z Displacement-Strain

z Stress-Strain

z Sensor & Actuator Equation

z Equation of Motion

Schematic drawing of six-layer cantileverplate/ACLD system

Composite sandwiched plate coordinate

Displacement field

),,()(),,(),,,(0

1

0

1 tyxztyxutzyxuNkj

jjN jN∑

−=

=± ±

+= φψ

),,()(),,(),,,(0

2

0

2 tyxztyxvtzyxuNkj

jjN jN∑

−=

=± ±

+= φψ

),,(),,(3 tyxwtyxu =

where the functions of the thicknessψ i z( ) and the in-planecoordinate φ 1i

( , , )x y t and φ 2i( , , )x y t are defined as follows:

ψ i z( ) = − −z z i , z z zi i− − −≥ ≥ 1

= z , z z z0 1≤ ≤ or z z z− ≤ ≤1 0

= −z zi , z z zi i≤ ≤ +1

xtyxw

atyxii ∂∂

φ),,(

),,( 11 −=

ytyxw

atyxii ∂∂

φ),,(

),,( 22 −=

where the al i's denotes adjustment coefficients which will be

determined later. Physically, awxi1

∂∂

and awyi2

∂∂

define the slopes of

the i th layer due to bending in the x and y directions, respectively

Displacement-Strain Relations

ε ε∂∂

∂∂

p

x x

p

x x

p

i

j

p

j

ii j j i

ux

ux

= = +12

( ) ( , , , )i j = 1 2 3 .

εεγ

ε

ε

γ

ψ1

2

6

1

0

2

0

6

0 1

1

2

6

31 3

32 3

0

=

+

+

=

=

∑ ii

i k

i

d x yd x y

ΚΚΚ

Ε ΛΕ Λ

( , )( , )

γγ

∂∂∂∂

5

4

1

2

31 3

31 3

1

1

=

−

−

+

∫

∫k

l

vem

w

vem

awx

awy

d x y dx

hd x y dy

h

k

k

( )

( )

( , )

( , )

Ε Λ

Ε Λ

Stress-Strain Relation

σσσ

εεγ

1

2

6

11 12 16

12 22 26

16 26 66

1

2

6

=

k k k

Q Q QQ Q QQ Q Q

σσ

γγ

5

4

55 45

45 44

5

4

=

k k k

Q QQ Q

.

Sensor & Actuator Equation

∫

=A

T

dAdd

tq

6

2

1032

031

0)(

σσσ

VC

q tC

dd dAs

T

A

= =

∫1 1

0

310

320

1

2

6

( )σσσ

V K i K VC p d S= − +( )ω

Equation of Motion & Eigenvalue Analysis

[ ] [ ]{ } { } { }GGGGG

G FFKt

M ~2

2

+=∆+

∆

∂∂

[ ] [ ]{ }{ } 02* =Φ− GG MK ω

where ω *2 are complex eigenvalues.

The n th eigenvalue is written as follows:

)1(22*nnn iηωω +=

Experimental setup for compositesandwiched plate/ACLD

Response to random excitationsfor bending control

0 5 10 15 20 25 30 35

Frequency (Hz)

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Am

plit

ud

e (m

m)

PCLDACLD, Gain AACLD, Gain BACLD, Gain C

Response to random excitationsfor torsion control

0 5 10 15 20 25 30 35

Frequency (Hz)

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3A

mp

litu

de

(mm

) PCLDACLD, Gain DACLD, Gain EACLD, Gain F

Comparison between theory &experiments for first bending mode

with different control gains

4.00 4.20 4.40 4.60 4.80 5.00

Theoretical Natural Frequency (Hz)

4.00

4.20

4.40

4.60

4.80

5.00

Exp

erim

enta

l Nat

ural

Fre

quen

cy (H

z) PCLDACLD, Gain AACLD, Gain BACLD, Gain C

(a)

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Theoretical Loss Factor

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Exp

erim

enta

l Los

s Fa

ctor

PCLDACLD, Gain AACLD, Gain BACLD, Gain C

(b)

Comparison between theory &experiments for second bending mode

with different control gains

30.00 31.00 32.00 33.00 34.00 35.00

Theoretical Natural Frequency (Hz)

30.00

31.00

32.00

33.00

34.00

35.00

Exp

erim

enta

l Nat

ural

Fre

quen

cy (H

z) PCLDACLD, Gain AACLD, Gain BACLD, Gain C

(a)

0.00 0.02 0.04 0.06 0.08 0.10

Theoretical Loss Factor

0.00

0.02

0.04

0.06

0.08

0.10

Exp

erim

enta

l Los

s Fa

ctor

PCLDACLD, Gain AACLD, Gain BACLD, Gain C

(b)

Comparison between theory &experiments for torsional mode

with different control gains

15.00 20.00 25.00

Theoretical Natural Frequency (Hz)

15.00

20.00

25.00

Exp

erim

enta

l Nat

ural

Fre

quen

cy (H

z) PCLDACLD, Gain DACLD, GainEACLD, GainF

(a)

0.00 0.05 0.10 0.15 0.20

Theoretical Loss Factor

0.00

0.05

0.10

0.15

0.20

Exp

erim

enta

l Los

s Fa

ctor

PCLDACLD, Gain DACLD, Gain EACLD, Gain F

(b)

Summary

z Introduced a a new class of composite sandwiched plates

z Developed the theoretical analysis of a composite plate/ACLD

z Theoretical Analysis was validated experimentally

z The ACLD is found to be effective in controlling the vibration of the plates