piezoelectric materials
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Piezoelectric Materials
Dr. Mohammad Tawfik
What is Piezoelectric Material?
• Piezoelectric Material is one that possesses the property of converting mechanical energy into electrical energy and vice versa.
Piezoelectric Materials
• Mechanical Stresses Electrical Potential Field : Sensor (Direct Effect)
• Electric Field Mechanical Strain : Actuator (Converse Effect)
Clark, Sounders, Gibbs, 1998
Conventional Setting
Conductive Pole
Piezoelectric Sensor
• When mechanical stresses are applied on the surface, electric charges are generated (sensor, direct effect).
• If those charges are collected on a conductor that is connected to a circuit, current is generated
Piezoelectric Actuator
• When electric potential (voltage) is applied to the surface of the piezoelectric material, mechanical strain is generated (actuator).
• If the piezoelectric material is bonded to a surface of a structure, it forces the structure to move with it.
Applications of Piezoelectric Materials in Vibration Control
Collocated Sensor/Actuator
Self-Sensing Actuator
Hybrid Control
Passive Damping / Shunted Piezoelectric Patches
Passively Shunted Networks
Resonant
Capacitive Switched
Resistive
Modeling of Piezoelectric Structures
Constitutive Relations
• The piezoelectric effect appears in the stress strain relations of the piezoelectric material in the form of an extra electric term
• Similarly, the mechanical effect appears in the electric relations EdD
Eds
33131
31111
Constitutive Relations
• ‘S’ (capital s) is the strain
• ‘T’ is the stress (N/m2)
• ‘E’ is the electric field (Volt/m)
• ‘s’ (small s) is the compliance; 1/stiffness (m2/N)
• ‘D’ is the electric displacement, charge per unit area (Coulomb/m2)
The Electromechanical Coupling
• Electric permittivity (Farade/m) or (Coulomb/mV)
• d31 is called the electromechanical coupling factor (m/Volt)
Manipulating the Equations
A
QD
As
IIdt
AD
1
• The electric displacement is
the charge per unit area:
• The rate of change of the
charge is the current:
• The electric field is the
electric potential per unit
length: t
VE
Using those relations:
• Using the relations:
• Introducing the capacitance:
• Or the electrical admittance:
Vt
sAsAdI
Vt
ds
33131
311111
CsVsAdI 131
YVsAdI 131
For open circuit (I=0)
• We get:
• Using that into the strain relation:
• Using the expression for the electric admittance:
131
Y
sAdV
1
2
311111
tY
Asds
1
1133
2
31111 1
s
ds
The electromechanical coupling factor
• Introducing the factor ‘k’:
• ‘k’ is called the electromechanical coupling factor (coefficient)
• ‘k’ presents the ratio between the mechanical energy and the electrical energy stored in the piezoelectric material.
• For the k13, the best conditions will give a value of 0.4
1
2
31111 1 ks
Different Conditions
• With open circuit conditions, the stiffness of the piezoelectric material appears to be higher (less compliance)
• While for short circuit conditions, the stiffness appears to be lower (more compliance)
11
2
31111 1 Dsks
Ess 11
Different Conditions
• Similar results could be obtained for the electric properties; electric properties are affected by the mechanical boundary conditions.
Zero-strain conditions (S=0)
• Using the relations:
• Introducing the capacitance:
• Or the electrical admittance:
Vt
ds 31
1110
Vs
d
t
AsI
1133
2
3133 1
VkYI 2
311
Other types of Piezo!
1-3 Piezocomposites
3333333
3333333
EeD
Eec
S
E
Active Fiber Composites (AFC)
3333
2
311111
SpC
p
Eeff
vv
evcc
3333
313331
SpC
eff
vv
ee
3333
333333
SpC
S
eff
vv
Actuation Action
• PZT and structure are assumed to be in perfect bonding
Axial Motion of Rods
• In this case, we will consider the case when the PZT and the structure are deforming axially only
Zero Voltage case
• If the structure is subject to axial force only, we get:
• And for the equilibrium:
sss
aaa
E
E
sssaaassaa EAEAAAF
xssaassaa EAEAAAF
Zero Voltage case
• From that, we may write the force strain relation to be:
ssaassaa
xEtEt
bF
EAEA
F
2
Zero Force case
• In this case, the strain of the of the PZT will be less than that induced by the electric field only!
• For equilibrium, F=0: sss
asapasaa
E
t
VdEEEE
31
031 sssaasaassaa EAt
VdEAEAAAF
ssaa
aa
sEAEA
t
VdEA
31
Homework #2
• Solve problems 1,2,&3 from textbook
• Due 27/11/2013 (11:59PM)
Beams with Piezoelectric Material
Review of Thin-Beam Theory
• The Euler-Benoulli beam theory assumes that the strain varies linearly through the thickness of the beam and inversely proportional to the radius of curvature.
2
2
dx
vdy
2
2
dx
vdEyE
Equilibrium
• The externally applied moment has to be in equilibrium with the internally generated moment.
• For homogeneous materials:
bydydx
vdEybydyM
h
h
h
h
2/
2/
2
22/
2/
2
22/
2/
2
2
2
dx
vdEIbdyy
dx
vdEM
h
h
Equilibrium
• Rearranging the terms:
2
2
dx
vd
EI
M
I
My
With piezoelectric materials
• Introducing change in the material property:
2/
2/
31
2/
2/
2/
2/
31
2/
2/
h
t a
aa
t
t
ss
t
h a
aa
h
h
s
s
s
s
ydyt
VdE
ydyEydyt
VdE
ydyb
M
With piezoelectric materials
• Expanding the integral
2/
2/
31
2/
2/
2
2
22/
2/
2
2
2
2/
2/
31
2/
2/
2
2
2
h
ta
a
h
t
a
t
t
s
t
ha
a
t
h
a
ss
s
s
ss
ydyt
VdEdyy
dx
vdEdyy
dx
vdE
ydyt
VdEdyy
dx
vdE
b
M
With piezoelectric materials
• Rearranging
2/
2/
31
2/
2/
31
2/
2/
2
2/
2/
2
2/
2/
2
2
2
h
ta
a
t
ha
a
h
t
a
t
t
s
t
h
a
s
s
s
s
s
s
ydyt
VdEydy
t
VdE
dyyEdyyEdyyEdx
vd
b
M
With piezoelectric materials
• Integrating
22
31
22
31
33333
2
2
88
224
1
s
a
as
a
a
sasssa
tht
VdEht
t
VdE
thEtEthEdx
vd
b
M
2231333
2
2
412
1s
a
asssa th
t
VdEtEthE
dx
vd
b
M
Remember:
• For homogeneous structures:
• Thus, in the absence of the voltage:
• OR:
12
333
sssaEquivalent
tEthEbEI
b
M
dx
vdEh
2
23
12
2231
2
2
4s
a
aEquivalent th
t
VbdE
dx
vdEIM
In the absence of load
• Thus, the structure will feel a moment:
2231
2
2
4s
Equivalenta
a thEIt
VbdE
dx
vd
2231
2
2
4s
Equivalenta
asssss th
EIt
VbdEIE
dx
vdIEM
Piezoelectric forces
• The above is equivalent of having a force applied by the piezoelectric material that is equal to:
2231
4s
Equivalentas
ass
s
sa th
EItt
VbdEIE
t
MF
Homework #3
• Solve problems 4,5,&6 from textbook
• Due 30/11/2013 (11:59PM)
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