constitutive relations beam, plate and shell models ...yyaman/avt086/suleman/afzal_suleman...c v e...
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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
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