What are piezoelectric materials? How to use them? How to perform analysis on structures with piezoelectric components?

1. Piezoelectric Materials
Dr. Mohammad Tawfik

2. What is Piezoelectric
Material?
• Piezoelectric Material is one that possesses
the property of converting mechanical energy
into electrical energy and vice versa.

3. Piezoelectric Materials
• Mechanical Stresses  Electrical Potential
Field : Sensor (Direct Effect)
• Electric Field  Mechanical Strain :
Actuator (Converse Effect)
Clark, Sounders, Gibbs, 1998

4. Conventional Setting
Conductive Pole

5. 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

6. 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.

7. Applications of Piezoelectric Materials in
Vibration Control

8. Collocated
Sensor/Actuator

9. Self-Sensing Actuator

10. Hybrid Control

11. Passive Damping / Shunted
Piezoelectric Patches

12. Passively Shunted Networks
Resistive
Capacitive
Resonant
Switched

13. Modeling of Piezoelectric
Structures

14. 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
  s11 1  d 31 E
D  d 31 1  33 E

15. 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)

16. The Electromechanical
Coupling
•
Electric permittivity (Farade/m) or
(Coulomb/mV)
• d31 is called the electromechanical coupling
factor (m/Volt)

17. Manipulating the
Equations
• 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:
Q
D
A
1
I
D   Idt 
A
As
V
E
t

18. Using those relations:
• Using the
relations:
• Introducing the
capacitance:
• Or the electrical
admittance:
d 31
 1  s11 1  V
t
A 33 s
I  Ad 31s 1 
V
t
I  Ad 31s 1  CsV
I  Ad 31s 1  YV

19. For open circuit (I=0)
• We get:
• Using that into the
strain relation:
• Using the expression
for the electric
admittance:
Ad 31s
V 
1
Y
2
31
Asd
1  s11 1 
1
tY
2

d 31 
 1
 1  s11 1 
  s 
33 11 


20. The electromechanical
coupling factor
• Introducing the factor ‘k’:
1  s11 1  k  1
2
31
• ‘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

21. Different Conditions
• With open circuit conditions, the stiffness of
the piezoelectric material appears to be higher
(less compliance)
1  s11 1  k  1  s  1
2
31
D
• While for short circuit conditions, the stiffness
appears to be lower (more compliance)
  s11  s 
E

22. Different Conditions
• Similar results could be obtained for the
electric properties; electric properties are
affected by the mechanical boundary
conditions.

23. Zero-strain conditions
(S=0)
• Using the
relations:
• Introducing the
capacitance:
• Or the electrical
admittance:
d 31
0  s11 1 
V
t
2

As 33
d 31 
1 
V
I 
t  33 s11 


I  Y 1 k V
2
31

24. Other types of Piezo!

25. 1-3 Piezocomposites
3  c
E
  e33 E 3
33 3
S
D3  e33 3  
33
E3

26. Active Fiber Composites (AFC)
c eff 11  c E11 
e
eff
31

v 
C
2
v p e31
 v p S 33
33
 33e31
v C  33  v p S 33
 33 S 33
 eff 33  C
v  33  v p S 33




27. Actuation Action
• PZT and structure are assumed to be in
perfect bonding

28. Axial Motion of Rods
• In this case, we will consider the case when
the PZT and the structure are deforming
axially only

29. Zero Voltage case
• If the structure is subject to axial force only,
we get:
 a  Ea  a
 s  Es s
• And for the equilibrium:
F  Aa a  As s  Aa Ea a  As Es s
F  Aa a  As s   Aa Ea  As Es  x

30. Zero Voltage case
• From that, we may write the force strain
relation to be:
F
F b
x 

Aa Ea  As Es 2ta Ea  t s Es

31. Zero Force case
• In this case, the strain of the of the PZT will be
less than that induced by the electric field
only!   E   E   E   E d V
a
a s
a
p
a s
a
31
t
 s  Es  s
• For equilibrium, F=0:
V
F  Aa a  As s  Aa Ea s  Aa Ea d31  As Es s  0
t
V
Aa Ea d 31
t
s 
 Aa Ea  As Es 

32. Homework #2
• Solve problems 1,2,&3 from textbook
• Due 27/11/2013 (11:59PM)

33. Beams with Piezoelectric
Material

34. 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.
d 2v
  y 2
dx
d 2v
  E   Ey 2
dx

35. Equilibrium
• The externally applied moment has to be in
equilibrium with the internally generated
h/2
h/2
moment.
d 2v
M    bydy 
h / 2
 Ey dx
2
h/2
• For homogeneous materials:
2
h/2
d v
d 2v
2
M  E 2  y bdy  EI 2
dx h / 2
dx
bydy

36. Equilibrium
• Rearranging the terms:
M d 2v
 2
EI dx
My
 
I

37. With piezoelectric
materials
• Introducing change in the material property:
h/2
M
   ydy
b
h / 2
t s / 2

V
   Ea   a  d 31  ydy   Es s ydy

ta 


h / 2
t s / 2
ts / 2

V
  Ea   a  d 31  ydy

ta 


ts / 2
h/2

38. With piezoelectric
materials
• Expanding the integral
2
M
d v
 Ea 2
b
dx
ts / 2
V
/ 2y dy  Ea d31 ta
h
ts / 2
 ydy
2
2
ts / 2
h / 2
2
h/2
d v
d v
V
2
2
 Es 2  y dy  Ea 2  y dy  Ea d 31
dx t s / 2
dx t s / 2
ta
h/2
 ydy
ts / 2

39. With piezoelectric
materials
• Rearranging
ts / 2
h/2
 ts / 2 2

M d v
 2 Ea  y dy  Es  y 2 dy  Ea  y 2 dy 

b dx   h / 2
t s / 2
ts / 2


2
V
 Ea d 31
ta
t s / 2
V
/ 2ydy  Ea d31 ta
h
h/2
 ydy
ts / 2

40. With piezoelectric
materials
• Integrating
 





M
1 d 2v
3
3
3

Ea h 3  t s  2 E s t s  Ea h 3  t s
b 24 dx 2
V 2
V 2
2
2
 Ea d 31
t s  h  Ea d 31
h  ts
8t a
8t a

 




Ea d 31V 2
M
1 d 2v
3
3
2
3

Ea h  t s  E s t s 
h  ts
2
b 12 dx
4ta


41. Remember:
• For homogeneous structures: Eh3 d 2v
M

2
12 dx
b
• Thus, in the absence of the voltage:


Ea h  t s  Es t s
EI Equivalent  b
12
• OR:
M  EI Equivalent
3
3
3

d 2 v Ea bd31V 2
2

h  ts
2
dx
4t a


42. In the absence of load
2

Ea bd31V
d v
2
2

h  ts
2
dx
4ta EI Equivalent

• Thus, the structure will feel a moment:

Es I s Eabd31V 2
d 2v
2
M s  Es I s 2  
h  ts
dx
4ta EI Equivalent


43. Piezoelectric forces
• The above is equivalent of having a force
applied by the piezoelectric material that is
equal to:
Ms
Es I s Ea bd 31V 2
2
Fa 
ts

4t s t a EI Equivalent
h
 ts


44. Homework #3
• Solve problems 4,5,&6 from textbook
• Due 30/11/2013 (11:59PM)