ADAPTIVE STRUCTURES

Contents

� Constitutive Relations

� Beam, Plate and Shell Models

� Applications§Panel Flutter§Noise Attenuation

ADAPTIVE STRUCTURES

Material Functions

THERMALTEMPERATURE

MECHANICALSTRESS

ELECTRICAL FIELD

Electric Flux Density

Strain Entropy

ADAPTIVE STRUCTURES

Constitutive Relations

ð The constitutive relations are based on the assumption that the total strain in the actuator is the sum of the mechanical straininduced by the stress, the thermal strain due to temperature andthe controllable actuation strain due to the electric voltage.

?Td

?TaeT

T

ασεεσ

++=++=

SE

CE

ADAPTIVE STRUCTURES

Constitutive Relations

ð Re-writing the stress-strain equation:

ðIn a plane perpendicular to the piezo-polarization, it has isotropic properties, i.e. transversely isotropic material in the plane 1-2.ðFor orthotropic material, there is no temperature shear strain. However there is a shear strain induced due to the electrical fields E1and E2.

TEEE

dd

ddd

SS

SSSSSSSSSS

∆

+

+

=

000

0000000

000000

000000000000000000000000

3

2

1

3

2

1

15

15

33

31

31

12

31

23

3

2

1

66

55

44

333231

232221

131211

12

31

23

3

2

1

ααα

τττσσσ

γγγεεε

ADAPTIVE STRUCTURES

Constitutive Relations

ð For piezoceramics, the actuation strain is:

ðd33, d31 and d15 are called piezoelectric strain coefficients of a mechanical free piezo element.ðd31 characterizes strain in the 1 an 2 directions to an electrical field E3 in the 3 directionðd33 relates strain in the 3 direction due to field in the 3 directionðd15 characterizes 2-3 and 3-1 shear strains due a field E2 and E1, respectively.

=Λ3

2

1

15

15

33

31

31

0000000

000000

EEE

dd

ddd

ADAPTIVE STRUCTURES

Block Force Model

•If an electric field V is applied, then the maximum actuator strain (free strain) will be:

•The maximum block force (zero strain condition) is:

=Λ=

ctVd31maxε

VbEdF ccb 31=

ADAPTIVE STRUCTURES

Block Force Model

ð A piezo patch attached to the beam structure results in an axial force F in the beam due to potential V. The reactive force in the piezo element will be –F. Then the strain in the piezo becomes

ccccc

c

EtbF

tV

dll −=∆= 31ε

ADAPTIVE STRUCTURES

Block Force Model

ð Force-strain relation for constant field V:

ðThis plot can also be used to determine the properties of piezo materials experimentally.

ccc

c

tbF

E

Vt

d

1

max

max

max31

ε

ε

=

=

ADAPTIVE STRUCTURES

Pure Extensionð Two identical patches mounted on the surface of a beam, one on either side can produce pure extension

ðFor pure extension, same potential is applied to top and bottom actuators. The induced force is

ðFb is the block force for each piezo patch.ðIf piezo stiffness (beam stiffness), actuation force becomes zero though actuation strain equals free strain;ðIf the actuation strain becomes zero though actuation force equals block force

bbcc AEAE >>

bbbbbccccc

ccbb

bbb

ccbb

ccbb

c

tbEAEtbEAEAEAE

AEFAEAE

AEAEtVdF

==+

=+

=

;2231

bbcc AEAE <<

ADAPTIVE STRUCTURES

Pure Bending

ð For pure bending, an equal and opposite potential is applied to top and bottom actuators

ðThe induced bending is

ðMb is the block moment for each piezo patch.ðIf actuation moment becomes zeroðIf actuation strain becomes zero

2

31

22

2

=

+=

+=

bccccc

ccbb

bbb

ccbb

ccbb

bc

ttbEIE

IEIEIEM

IEIEIEIE

ttVdM

bbcc IEIE >>

bbcc IEIE <<

ADAPTIVE STRUCTURES

Euler-Bernoulli Beam Model

ðBeam, adhesive and actuator form a continuous structureðBernoulli´s assumption: a plane section normal to the beam axis remains plane and normal to the beam axis after bendingðLinear distribution of strain in actuator and host structureðGenerally gives more accurate results than uniform strain model

( )( ) ( )

( ) ( ) netxx

net

zEzzz

zz

εσεε

κκεε

=Λ−=

=−= xx0 -w, ,

ADAPTIVE STRUCTURES

Bernoulli-Euler Beam Model

ð Axial force and bending moment expressions are:

where

ð F is the axial force in the beamð M is the bending moment in the beamð b(z) is the beam width

=

++

Λ

Λ

xxwEEEE

MMFF

,0

21

10 ε

( ) ( )

( ) ( )( ) widthbeam is

2h

2h-

2h

2h-

zb

zdzzzbM

dzzzbF

xx

xx

∫

∫

=

=

σ

σ

ADAPTIVE STRUCTURES

Euler–Bernoulli Beam Model

ð Axial force and bending moment due to induced stress:

ð If the placement of the actuators is symmetric, the coupling term will be zero; if not, this term will be non-zero: extension-bending coupling

( ) ( ) 210 ,2

2

,,jdzzzEzbEh

hj

j == ∫−

( ) ( ) ( ) ( ) ( ) ( )∫∫ Λ=Λ= ΛΛ2h

2h

-2h

2h

- , zdzzzEzbMdzzzEzbF

ADAPTIVE STRUCTURES

Uniform Strain and Euler-Bernoulli Beam Models

ðThe thickness ratio, T, determines if the strain variation across the piezo affects the analysis:

ðfor small T, the uniform strain model overpredicts strain (curvature)ðfor large T, the predicted induced bending curvatures are identical for both models

c

b

tt

T =

ADAPTIVE STRUCTURES

Plate with Induced Strain Actuation

ðInduced strain actuation is used to control the extension, bending and twisting of a plate

ðUsing tailored anisotropic plates with distributed piezoactuators, the control of specific static deformation can be augmented

ADAPTIVE STRUCTURES

Plate with Induced Strain Actuation

ð Assumptions to develop a consistent plate model:ð Actuators and substrates are integrated as plies of a laminated plateð A consistent deformation is assumed in the actuators and substratesð Generally, a thin classical laminated plate theory is adopted

ð For systems actuated in extension:ðAssume strains are constant across the thickness of actuators and plate

ð For systems actuated in pure bending:ð Assume strains vary linearly through the thickness

ADAPTIVE STRUCTURES

Plate with Induced Strain Actuation

ð Strain in the system:

ð Mid-plane strain:

ð Curvature:

{ }T

Txyyx x

vyu

yv

xu

∂∂+

∂∂

∂∂

∂∂== 0000 εεεε

κεε z+= 0

{ }T

Txyyx yx

wyw

xw

∂∂∂−

∂∂−

∂∂−==

2

2

2

2

2

2κκκκ

ADAPTIVE STRUCTURES

Plate with Induced Strain Actuation

ð Constitutive relation for any ply:

ð is the transformed reduced stiffness of the plateðThe second term represents an equivalent stress due to the actuation

ð Stress vector:

ðActuation strain vector

{ }Txyyx τσσσ =

( )Λ−= εσ Q

{ }Txyyx ΛΛΛ=Λ

Q

ADAPTIVE STRUCTURES

Plate with Induced Strain Actuation

ð Net forces and moments

=

xy

y

x

xy

y

x

z

y

x

z

y

x

DDDDDDDDD

BBBBBBBBB

BBBBBBBBB

AAAAAAAAA

MMMNNN

κκκγεε

0

0

0

62616

262211

161211

62616

262211

161211

62616

262211

161211

62616

262211

161211

ADAPTIVE STRUCTURES

Shells

t

Ta ng e ntia l stra in

HAxia l stra in

R

X

θ

o

Qx Nx

Qθ

NθN xθ

Nxθ

X

θ

o

Mx

MxθM xθ

Mθ

ADAPTIVE STRUCTURES

Shells

ð Strain-Displacement Relations

ε κ

ε κ

ε τ

θ θ

θ

x x

x

ux

wx

v wR R

wR

v

vx R

uR

wx R

vx

= ∂∂

= − ∂∂

= ∂∂θ

+ = − ∂∂θ

+ ∂∂θ

= ∂∂

+ ∂∂θ

= − ∂∂∂θ

+ ∂∂

; ;

; ;

;

2

2

2

2

2 2

2

1 1

1 2 2

ADAPTIVE STRUCTURES

Piezo Patch Contributions

ð Finite Patches

M M M H x H x H H

M M M H x H x H H

N N N H x H x H H S x S

N N N H x H x H H S x S

x x x

x x x

p pinner pouter

p pinner pouter

p pinner pouter

p pinner pouter

= + − −

= + − −

= + − −

= + − −

1 2 1 2

1 2 1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) $ ( )

( ) ( ) ( ) ( ) ( ) $ ( )

, ,

, ,

θ θ

θ θ

θ θ θ

θ θ θ

θ θ θ

θ θ θ

ADAPTIVE STRUCTURES

Concluding Remarks

ðAnalytical models for beam, plate and shell type elements have been presented.

ðThe weak form of the equations of motion are desirable since they circumvent the need to differentiate terms with patch force and moment terms.

ðThe analytical models provide a physical appreciation of the interaction between the structure and the actuating piezo patches

ADAPTIVE STRUCTURES

Finite Element Models

� Piezoelectric Finite ElementsF Solid, Plate and Beam Models

� Simple Plate Finite Element Model

� Actuation and Sensing ExamplesF Bimorph beamF Adaptive Composite Plate

ADAPTIVE STRUCTURES

Solid Elements

Allik and Hughes (1970)u,v,w, ϕ : linear16 dofStatic condensation of the electric dof

Gandhi and Hagood (1997)u,v,w, ϕ : linear16 dof + internal dofNonlinear constitutive relations

ADAPTIVE STRUCTURES

Solid Elements

Tzou and Tseng (1990)u,v,w,ϕ : linear + quadratic incompatible modes32 dofStatic condensation of the electric dof

Ha and Keilers (1992)u,v,w,ϕ : linear + quadratic incompatible modes32 dofEquivalent single layer modelStatic condensation of incompatible modes

ADAPTIVE STRUCTURES

Solid Elements

Chin and Varadan (1994)u,v,w,ϕ : linear32 dofLagrange method

Allik and Webman (1974)u,v,w,ϕ : quadratic80 dofSonar transducers

ADAPTIVE STRUCTURES

Shell Elements

Lammering (1991)u,v,w,βx, βy : linear28 dofShallow shell theoryUpper-lower nodal electric potential dof

Thirupati et al (1997)u,v,w,φ: quadratic32 dof3D degenerated shell theoryPiezo effect as initial strain problem

ADAPTIVE STRUCTURES

Shell Elements

Varadan et al (1993)u,w,φ: linear9 dofLagrange formulationMooney transducers

Tzou and Ye (1993)u,v,w,φ: in-plane quadratic, thickness linear48 dofLayerwise constant shear angle theoryLaminated piezo shell continuum

ADAPTIVE STRUCTURES

Plate Elements

Suleman and Venkayya (1995)u,v,w, θx, θy, θz : bilinearφ: linear24 dofMindlin plate element C0

1 dof per piezo patch/layer

Ray et al (1994)w: cubicφ: linear104 dofLinear potential in thickness1 dof per piezo patch/layer

ADAPTIVE STRUCTURES

Plate Elements

Yin and Shen (1997)u,v,w, βx, βy, φ: quadratic54 dofMindlin plate theory C0

Linear voltage but transverse field dof

ADAPTIVE STRUCTURES

Beam Elements

Shen (1994)U: linearW: cubic hermiteΒ: linear8 dofTimoshenko beam theory with Hu-Washizu Principle (Mixed)Offset nodes

ADAPTIVE STRUCTURES

Summary of Available Elements

Elements Shape and approximations

Solid 4-nodes linear tetrahedron8-nodes linear hexahedron20-nodes quadratichexahedron

availableavailableavailable

Shell 3-nodes linear axisymmetric flat triangle8-nodes quadratic axisymm. quadrangle4-nodes linear flat quadrangle8-nodes 3D-degenerated quadratic quad12-nodes 3D-degenerated quadratic prism

availableavailableavailablenot availableavailable

Plate 3-nodes linear triangle4-nodes linear quadrangle8-nodes quadrangle9-nodes quadrangle

not availableavailableavailableavailable

Beam 2-nodes linear element3-nodes quadratic element

availablenot available

ADAPTIVE STRUCTURES

Adaptive Composite Plate Model

•If an electric field V is applied, then maximum actuator strain (free strain) will be:

ADAPTIVE STRUCTURES

Kinetic, Potential and Electrical Energies

•The Hamiltonian for the system is

[ ] 02

1

=+Π−∫ dtWTt

t eδ

dVTSdVuuT cc

V

T

V

T∫∫ =Π=21

;21 &&ρ

pee

Ve dVTSW

T

p

∫=21

ADAPTIVE STRUCTURES

Stress-Strain Relations

ecc

ecTe

SST

SST

ec

e

−=+= ε

{ }{ }

pntsyz

tsxz

bxy

by

bx

mxy

my

mx

etsbm

EESSSSSSSS

SSSSS

−−==

...1

{ }{ }

pntsyz

tsxz

bxy

by

bx

mxy

my

mx

etsbm

DDTTTTTTTT

TTTTT

...1==

ADAPTIVE STRUCTURES

Stress-Strain Relations

T

SSSS

TT

T

m

e

ts

b

m

TT

e

c

∆

−

=

=

000

α

ε0ee0g00e0cce0cc

ADAPTIVE STRUCTURES

Strain-Displacement Relations

−−−

+

+

+=

=xy

yy

xx

yx

y

x

xy

y

x

bxy

by

bx

b

www

zww

ww

vuvu

SSS

S

,

,

,

,,

2,

2,

,,

,

,

2221

( )ξηηηξξ HC +++=41

N

{ }{ }

pne

iyxs

i

q

wvuq

φφ

θθ

...

;

1=

=

ADAPTIVE STRUCTURES

Strain-Displacement Relations

=

=e

s

e

s

e

s

qq

SS

Sb00b

el

si

si

si

si

si

si

si

si

si

si

si

si

si ni

Nx

N

Nx

Nx

Nz

xN

z

yN

z

xN

z

xN

yN

yN

xN

,,1 ;

000

000

000

0000

0000

000

0000

0000

L=

−∂

∂∂

∂∂

∂∂

∂−

∂∂−

∂∂

∂∂

∂∂

∂∂

∂∂

=b

ADAPTIVE STRUCTURES

System Matrices

elje

V

ejee

je

V

cjce

jc

V

cjcc

jV

Tjcc

njdV

dV

dV

dV

j

T

j

T

j

T

j

,,1,

,

,

,

L==

=

=

=

∫∫∫∫

forbebK

bebK

bcbK

NNM ρ

ADAPTIVE STRUCTURES

Geometric Stiffness

dANdANdANdAN xA

yxyyA

xxyyA

yyA

xg NNNNNNNNK TTTx

Tx ∫∫∫∫ +++=

ADAPTIVE STRUCTURES

Equations of Motion

=

+

+

+

+

∆∆

0000K

000K

KKK0

000K

000M

T Te

cg

e

c

e

c

eeec

cee

ccc

e

ccc

P

UU

UU

UU

UU

UU

484 76484 76

4 84 76484 76

&&&&

484 76

stiffness nonlinear stiffness thermal

stiffness piezostiffnesslinear inertia

ADAPTIVE STRUCTURES

Actuation and Sensing Mechanisms

U Uc e= −K Kcc ce1

U Ue ee ec c= −K K1

ADAPTIVE STRUCTURES

Bimorph Beam

ADAPTIVE STRUCTURES

Bimorph Beam

ADAPTIVE STRUCTURES

Composite Plate

ADAPTIVE STRUCTURES

Composite Plate

ADAPTIVE STRUCTURES

PANEL FLUTTER

V

25 cm

25 c

m

ADAPTIVE STRUCTURES

PANEL FLUTTER – BOUNDARY EFFECT

ADAPTIVE STRUCTURES

PANEL FLUTTER – IN PLANE LOADING

ADAPTIVE STRUCTURES

PANEL FLUTTER

=

+

+

+

+

+

∆∆

0000K

000K

000K

KKK0

000K

000G

000M

T Te

cg

e

ca

e

c

e

c

eeec

cee

ccc

e

c

e

ccc

PUU

UU

UU

UU

UU

UU

UU

484 76484 76484 76

4 84 76484 76

&&

484 76

&&&&

484 76

stiffness nonlinearstiffness aerostiffness thermal

stiffness piezostiffness lineardamping aeroinertia

ADAPTIVE STRUCTURES

AERODYNAMIC LOADS

−−

−−+

−−=

∞∞∞

∞

∞ttxa w

Mrw

VMM

wM

qp ,2,2

2

,2 12

1112

1

2

−+−= tt

axa w

aD

rw

aDg

waD

p ,3,40

,3 2 βλ

ωλ

( )( )1

2;

1

22

22

2

3

−−=

−=

∞

∞

∞M

Mg

MD

aqa

a

βµλλ

ADAPTIVE STRUCTURES

AERODYNAMIC LOADS

dAwtw

aD

rtw

aDg

xw

aDW

A

aa ∫

∂∂−

∂∂+

∂∂−= 34

03 2 β

λω

λ

eln,1,jfor L==

−=

∫∫

dA

dAaD

rg

xA

Tja

A

Ta

j

,

3 ,2

NNK

NNG

λβ

λ

ADAPTIVE STRUCTURES

RESULTS

Critical Aerodynamic Parameter8.36=criticalλ

Configuration#1 #2 #3 #4 #5

0 V 46.9 70.5 88.5 91.8 63.9400 V 66.7 93.5 92.5 99.2 76.5

criticalλ +42% +32% +5% +8% +20%Mass +17% +69% +86% +69% +52%

#1 #2 #3 #4 #5

ADAPTIVE STRUCTURES

ACTIVE CONTROL

1in_1

2Ou tport 2

C

C M a trix

M u x

M u x

x' = Ax+Bu y = Cx+Du

Pla te M odel

Sta te Nois eSou rce

Sys temVis u a liza tion

Ou tpu t Nois eSou rce

Ou tport 1

++

Su m

Dyna m ic M odel of Pla te w ith Piezoelectric Sens ors a nd Actu a tors

ADAPTIVE STRUCTURES

ACTIVE CONTROL

1st mode 2nd mode

ADAPTIVE STRUCTURES

ACTIVE CONTROL

ADAPTIVE STRUCTURES

ACTIVE CONTROLnωω

Mode OPEN LOOP

CLOSED LOOP

(rad/s) (rad/s) Damping Comments

57.30249 57.30438561 0.009520978 Structure

365.4731 70.25461695 0.927711305 Controller

1057.714 365.4493213 0.007791086 Structure

1133.858 567.3243875 0.52316159 Controller

1193.705 1056.67806 0.000688374 Structure

2178.078 1133.858129 0 Structure

ADAPTIVE STRUCTURES

NOISE SUPPRESSION

ADAPTIVE STRUCTURES

STRUCTURE BORNE NOISE

ADAPTIVE STRUCTURES

SPECTRUM OF CABIN NOISE

ADAPTIVE STRUCTURES

A composite shell element with electromechanical properties and with principal radii of curvature Rx and Ry has been formulated and implemented. This 8-noded isoparametric finite element has five degrees of freedom at each node, which includes three displacements and two rotations . To derive the equations of motion for the laminated composite shell, in an acoustic field with piezoelectricallycoupled electromechanical properties, we use the generalized form of Hamilton’s principle

[ ] 02

1

=−+Π−∫ dtWWTt

tpeδ

COMPOSITE SHELL

ADAPTIVE STRUCTURES

[ ] 02

1

=+Π−∫ dtWTt

tpppδ

01

2

2

22 =

∂∂−∇

tp

cp

•To derive the equations of motion for the acoustic cavity, we use the generalized form of Hamilton’s principle

boundary vibratingaat

boundary rigid aat 0

2

2

tw

np

np

a ∂∂−=

∂∂

=∂∂

ρ

With the following boundary conditions:

ACOUSTIC CAVITY MODEL

ADAPTIVE STRUCTURES

=

Θ−+

Θ

00

p

e

p

s

eees

pp

sess

e

p

s

ppTss F

UUU

U

UU

K0K0K0

KK

0000M00M

&&

&&

&&

.

and

matrix coupling acoustic-structural theis

matrix; stiffness"" acoustic theis 1

matrix; mass"" acoustic theis 1

2

∂

∂∂

∂∂

∂=

=Θ

=

=

∫∫

∫

zyx

dS

dV

dVc

pi

pi

pip

i

pS

Ts

pV

Tp

app

pV

Tp

app

NNNb

NN

bbK

NNM

ρ

ρ

EQUATIONS OF MOTION

ADAPTIVE STRUCTURES

( )( )( )

><= −−

−−

ozzd

ozzd

f zzePzzePzP

o

o

for for

( )

>−−

<−−

=θθ

θθθθ

θθθθθθ

θθ

θ

for

for

2

2

1

1

kj

o

okj

of

eP

ePP

Axial distribution:

Circumferential Distribution:

ASSUMED PRESSURE DISTRIBUTION

ADAPTIVE STRUCTURES

PropellerNoise

StiffeningForce

InteriorNoise

PiezoelectricActuator

Fuselage

PASSIVE ACTUATION MECHANISM

ADAPTIVE STRUCTURES

Acoustic Elements

Adaptive Composite Shellwith Piezo Layer

FINITE ELEMENT MESH

ADAPTIVE STRUCTURES

Propeller Plane

z = 3.5 m

z = 0

θ = 0

r=1.3

m

ACTUATOR CONFIGURATION

ADAPTIVE STRUCTURES

Propeller Plane

z = 3.5 m

z = 0

θ = 0

r=1.3

m

ACTUATOR CONFIGURATION

ADAPTIVE STRUCTURES

NOISE REDUCTION

0

20

40

60

80

100

120

140

ANGULAR POSITION (Deg)

NO

ISE

RE

DU

CT

ION

(dB

)

45 90

θ = 0

Frequency 90 HzActuation 400 VCase 2 - Line Pattern

Frequency 90 HzNo Actuation

Frequency 90 HzActuation 400 VCase 1 - Chess Pattern

Symmetric

360135 180 225 270 270

External Pressure Distribution

θ = 180

RESULTS

ADAPTIVE STRUCTURES

´ Analytical and finite element models with electromechanical properties have been presented.

´ Application of piezoelectric patches to control panel flutter has been demonstrated.

´ Internal noise reduction using a stiffened fuselage with piezo pacthes achieved considerable reduction in noise levels.

CONCLUSIONS