3. 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 strain
induced by the stress, the thermal strain due to temperature and
the controllable actuation strain due to the electric voltage.
?T
d
?T
a
e
T
T
α
σ
ε
ε
σ
+
+
=
+
+
=
S
E
C
E
4. 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 E1
and E2.
T
E
E
E
d
d
d
d
d
S
S
S
S
S
S
S
S
S
S
S
S
∆
+
+
=
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
2
1
3
2
1
15
15
33
31
31
12
31
23
3
2
1
66
55
44
33
32
31
23
22
21
13
12
11
12
31
23
3
2
1
α
α
α
τ
τ
τ
σ
σ
σ
γ
γ
γ
ε
ε
ε
5. 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 direction
ðd33 relates strain in the 3 direction due to field in the 3 direction
ðd15 characterizes 2-3 and 3-1 shear strains due a field E2 and E1,
respectively.
=
Λ
3
2
1
15
15
33
31
31
0
0
0
0
0
0
0
0
0
0
0
0
0
E
E
E
d
d
d
d
d
6. 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
=
c
t
V
d31
max
ε
V
b
E
d
F c
c
b 31
=
7. 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
c
c
c
c
c
c
E
t
b
F
t
V
d
l
l
−
=
∆
= 31
ε
8. 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.
c
c
c
c
t
b
F
E
V
t
d
1
max
max
max
31
ε
ε
=
=
9. ADAPTIVE STRUCTURES
Pure Extension
ð 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
b
b
c
c A
E
A
E >>
b
b
b
b
b
c
c
c
c
c
c
c
b
b
b
b
b
c
c
b
b
c
c
b
b
c
t
b
E
A
E
t
b
E
A
E
A
E
A
E
A
E
F
A
E
A
E
A
E
A
E
t
V
d
F
=
=
+
=
+
=
;
2
2
31
b
b
c
c A
E
A
E <<
10. 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 zero
ðIf actuation strain becomes zero
2
31
2
2
2
=
+
=
+
=
b
c
c
c
c
c
c
c
b
b
b
b
b
c
c
b
b
c
c
b
b
b
c
t
t
b
E
I
E
I
E
I
E
I
E
M
I
E
I
E
I
E
I
E
t
t
V
d
M
b
b
c
c I
E
I
E >>
b
b
c
c I
E
I
E <<
11. ADAPTIVE STRUCTURES
Euler-Bernoulli Beam Model
ðBeam, adhesive and actuator form a continuous
structure
ðBernoulli´s assumption: a plane section normal to the
beam axis remains plane and normal to the beam axis
after bending
ðLinear distribution of strain in actuator and host
structure
ðGenerally gives more accurate results than uniform
strain model
( )
( ) ( )
( ) ( ) net
xx
net
z
E
z
z
z
z
z
ε
σ
ε
ε
κ
κ
ε
ε
=
Λ
−
=
=
−
= xx
0 -w,
,
12. ADAPTIVE STRUCTURES
Bernoulli-Euler Beam Model
ð Axial force and bending moment expressions are:
where
ð F is the axial force in the beam
ð M is the bending moment in the beam
ð b(z) is the beam width
=
+
+
Λ
Λ
xx
w
E
E
E
E
M
M
F
F
,
0
2
1
1
0 ε
( ) ( )
( ) ( )
( ) width
beam
is
2
h
2
h
-
2
h
2
h
-
z
b
zdz
z
z
b
M
dz
z
z
b
F
xx
xx
∫
∫
=
=
σ
σ
13. 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
( ) ( ) 2
1
0
,
2
2
,
,
j
dz
z
z
E
z
b
E
h
h
j
j =
= ∫
−
( ) ( ) ( ) ( ) ( ) ( )
∫
∫ Λ
=
Λ
= Λ
Λ
2
h
2
h
-
2
h
2
h
-
, zdz
z
z
E
z
b
M
dz
z
z
E
z
b
F
14. 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
t
t
T =
15. 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 piezo
actuators, the control of specific static deformation can be
augmented
16. 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 plate
ð A consistent deformation is assumed in the
actuators and substrates
ð 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
17. ADAPTIVE STRUCTURES
Plate with Induced Strain Actuation
ð Strain in the system:
ð Mid-plane strain:
ð Curvature:
{ }
T
T
xy
y
x
x
v
y
u
y
v
x
u
∂
∂
+
∂
∂
∂
∂
∂
∂
=
= 0
0
0
0
ε
ε
ε
ε
κ
ε
ε z
+
= 0
{ }
T
T
xy
y
x
y
x
w
y
w
x
w
∂
∂
∂
−
∂
∂
−
∂
∂
−
=
=
2
2
2
2
2
2
κ
κ
κ
κ
18. ADAPTIVE STRUCTURES
Plate with Induced Strain Actuation
ð Constitutive relation for any ply:
ð is the transformed reduced stiffness of the
plate
ðThe second term represents an equivalent stress
due to the actuation
ð Stress vector:
ðActuation strain vector
{ }
T
xy
y
x τ
σ
σ
σ =
( )
Λ
−
= ε
σ Q
{ }
T
xy
y
x Λ
Λ
Λ
=
Λ
Q
19. ADAPTIVE STRUCTURES
Plate with Induced Strain Actuation
ð Net forces and moments
=
xy
y
x
xy
y
x
z
y
x
z
y
x
D
D
D
D
D
D
D
D
D
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
A
A
A
A
A
A
A
A
A
M
M
M
N
N
N
κ
κ
κ
γ
ε
ε
0
0
0
6
26
16
26
22
11
16
12
11
6
26
16
26
22
11
16
12
11
6
26
16
26
22
11
16
12
11
6
26
16
26
22
11
16
12
11
21. ADAPTIVE STRUCTURES
Shells
ð Strain-Displacement Relations
ε κ
ε κ
ε τ
θ θ
θ
x x
x
u
x
w
x
v w
R R
w
R
v
v
x R
u
R
w
x R
v
x
=
∂
∂
= −
∂
∂
=
∂
∂
θ
+ = −
∂
∂
θ
+
∂
∂
θ
=
∂
∂
+
∂
∂
θ
= −
∂
∂∂
θ
+
∂
∂
; ;
; ;
;
2
2
2
2
2 2
2
1 1
1 2 2
22. 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
23. 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
24. ADAPTIVE STRUCTURES
Finite Element Models
ΠPiezoelectric Finite Elements
F Solid, Plate and Beam Models
• Simple Plate Finite Element Model
Ž Actuation and Sensing Examples
F Bimorph beam
F Adaptive Composite Plate
25. ADAPTIVE STRUCTURES
Solid Elements
Allik and Hughes (1970)
u,v,w, ϕ : linear
16 dof
Static condensation of the electric dof
Gandhi and Hagood (1997)
u,v,w, ϕ : linear
16 dof + internal dof
Nonlinear constitutive relations
26. ADAPTIVE STRUCTURES
Solid Elements
Tzou and Tseng (1990)
u,v,w,ϕ : linear + quadratic incompatible modes
32 dof
Static condensation of the electric dof
Ha and Keilers (1992)
u,v,w,ϕ : linear + quadratic incompatible modes
32 dof
Equivalent single layer model
Static condensation of incompatible modes
27. ADAPTIVE STRUCTURES
Solid Elements
Chin and Varadan (1994)
u,v,w,ϕ : linear
32 dof
Lagrange method
Allik and Webman (1974)
u,v,w,ϕ : quadratic
80 dof
Sonar transducers
28. ADAPTIVE STRUCTURES
Shell Elements
Lammering (1991)
u,v,w,β
x, β
y : linear
28 dof
Shallow shell theory
Upper-lower nodal electric potential dof
Thirupati et al (1997)
u,v,w,φ
: quadratic
32 dof
3D degenerated shell theory
Piezo effect as initial strain problem
29. ADAPTIVE STRUCTURES
Shell Elements
Varadan et al (1993)
u,w,φ: linear
9 dof
Lagrange formulation
Mooney transducers
Tzou and Ye (1993)
u,v,w,φ: in-plane quadratic, thickness linear
48 dof
Layerwise constant shear angle theory
Laminated piezo shell continuum
30. ADAPTIVE STRUCTURES
Plate Elements
Suleman and Venkayya (1995)
u,v,w, θx,θy,θz : bilinear
φ
: linear
24 dof
Mindlin plate element C0
1 dof per piezo patch/layer
Ray et al (1994)
w: cubic
φ
: linear
104 dof
Linear potential in thickness
1 dof per piezo patch/layer
31. ADAPTIVE STRUCTURES
Plate Elements
Yin and Shen (1997)
u,v,w, β
x,β
y, φ: quadratic
54 dof
Mindlin plate theory C0
Linear voltage but transverse field dof
32. ADAPTIVE STRUCTURES
Beam Elements
Shen (1994)
U: linear
W: cubic hermite
Β: linear
8 dof
Timoshenko beam theory with Hu-Washizu
Principle (Mixed)
Offset nodes
33. ADAPTIVE STRUCTURES
Summary of Available Elements
Elements Shape and approximations
Solid 4-nodes linear tetrahedron
8-nodes linear hexahedron
20-nodes quadratichexahedron
available
available
available
Shell 3-nodes linear axisymmetric flat triangle
8-nodes quadratic axisymm. quadrangle
4-nodes linear flat quadrangle
8-nodes 3D-degenerated quadratic quad
12-nodes 3D-degenerated quadratic prism
available
available
available
not available
available
Plate 3-nodes linear triangle
4-nodes linear quadrangle
8-nodes quadrangle
9-nodes quadrangle
not available
available
available
available
Beam 2-nodes linear element
3-nodes quadratic element
available
not available
35. ADAPTIVE STRUCTURES
Kinetic, Potential and Electrical Energies
•The Hamiltonian for the system is
[ ] 0
2
1
=
+
Π
−
∫ dt
W
T
t
t
e
δ
dV
T
S
dV
u
u
T c
c
V
T
V
T
∫
∫ =
Π
=
2
1
;
2
1 &
&
ρ
p
e
e
V
e dV
T
S
W
T
p
∫
=
2
1
39. ADAPTIVE STRUCTURES
Strain-Displacement Relations
=
= e
s
e
s
e
s
q
q
S
S
S
b
0
0
b
el
s
i
s
i
s
i
s
i
s
i
s
i
s
i
s
i
s
i
s
i
s
i
s
i
s
i n
i
N
x
N
N
x
N
x
N
z
x
N
z
y
N
z
x
N
z
x
N
y
N
y
N
x
N
,
,
1
;
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
L
=
−
∂
∂
∂
∂
∂
∂
∂
∂
−
∂
∂
−
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
b
42. ADAPTIVE STRUCTURES
Equations of Motion
=
+
+
+
+
∆
∆
0
0
0
0
K
0
0
0
K
K
K
K
0
0
0
0
K
0
0
0
M
T T
e
c
g
e
c
e
c
ee
ec
ce
e
c
cc
e
c
cc
P
U
U
U
U
U
U
U
U
U
U
48
4 7
6
48
4 7
6
4 8
4 7
6
48
4 7
6
&
&
&
&
48
4 7
6
stiffness
nonlinear
stiffness
thermal
stiffness
piezo
stiffness
linear
inertia
55. ADAPTIVE STRUCTURES
ACTIVE CONTROL
1
i
n
_1
2
Ou t
por
t2
C
C M a t
r
i
x
M u x
M u x
x
' = A
x
+Bu
y = Cx
+Du
Pl
a t
e M odel
St
a t
e Noi
s e
Sou r
ce
Sys t
em
Vi
s u a l
i
za t
i
on
Ou t
pu tNoi
s e
Sou r
ce
Ou t
po
+
+
Su m
Dyn
a m i
c M odelofPl
a t
e w i
t
h Pi
ezoel
ect
r
i
c Sen
s or
s a n
d A
ct
u a t
or
s
62. 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 piezoelectrically
coupled electromechanical properties, we use the
generalized form of Hamilton’
s principle
[ ] 0
2
1
=
−
+
Π
−
∫ dt
W
W
T
t
t
p
e
δ
COMPOSITE SHELL
63. ADAPTIVE STRUCTURES
[ ] 0
2
1
=
+
Π
−
∫ dt
W
T
t
t
p
p
p
δ
0
1
2
2
2
2
=
∂
∂
−
∇
t
p
c
p
•
To derive the equations of motion for the acoustic cavity, we use the
generalized form of Hamilton’
s principle
boundary
vibrating
a
at
boundary
rigid
a
at
0
2
2
t
w
n
p
n
p
a
∂
∂
−
=
∂
∂
=
∂
∂
ρ
With the following boundary conditions:
ACOUSTIC CAVITY MODEL
65. ADAPTIVE STRUCTURES
( )
( )
( )
>
<
= −
−
−
−
o
z
z
d
o
z
z
d
f
z
z
e
P
z
z
e
P
z
P
o
o
for
for
( )
>
−
−
<
−
−
=
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
for
for
2
2
1
1
k
j
o
o
k
j
o
f
e
P
e
P
P
Axial distribution:
Circumferential Distribution:
ASSUMED PRESSURE DISTRIBUTION
70. ADAPTIVE STRUCTURES
NOISE REDUCTION
0
20
40
60
80
100
120
140
ANGULAR POSITION (Deg)
NOISE
REDUCTION
(dB)
45 90
θ = 0
Frequency 90 Hz
Actuation 400 V
Case 2 - Line Pattern
Frequency 90 Hz
No Actuation
Frequency 90 Hz
Actuation 400 V
Case 1 - Chess Pattern
Symmetric
360
135 180 225 270 270
External Pressure
Distribution
θ = 180
RESULTS
71. ADAPTIVE STRUCTURES
´ Analytical and finite element models with
electromechanical properties have been presented.
´ Application of piezoelectric patches to control pane
flutter has been demonstrated.
´ Internal noise reduction using a stiffened fuselage
with piezo pacthes achieved considerable reduction in
noise levels.
CONCLUSIONS
