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Piezoelectric Materials
Dr. Mohammad Tawfik
What is Piezoelectric
Material?
• Piezoelectric Material is one that possesses
the property of converting mechanical energ...
Piezoelectric Materials
• Mechanical Stresses  Electrical Potential
Field : Sensor (Direct Effect)
• Electric Field  Mec...
Conventional Setting

Conductive Pole
Piezoelectric Sensor
• When mechanical stresses are applied on the
surface, electric charges are generated
(sensor, direct...
Piezoelectric Actuator
• When electric potential (voltage) is applied to
the surface of the piezoelectric material,
mechan...
Applications of Piezoelectric Materials in
Vibration Control
Collocated
Sensor/Actuator
Self-Sensing Actuator
Hybrid Control
Passive Damping / Shunted
Piezoelectric Patches
Passively Shunted Networks

Resistive

Capacitive

Resonant

Switched
Modeling of Piezoelectric
Structures
Constitutive Relations
• The piezoelectric effect
appears in the stress
strain relations of the
piezoelectric material in
...
Constitutive Relations
•
•
•
•

‘S’ (capital s) is the strain
‘T’ is the stress (N/m2)
‘E’ is the electric field (Volt/m)
...
The Electromechanical
Coupling
•

Electric permittivity (Farade/m) or

(Coulomb/mV)
• d31 is called the electromechanical...
Manipulating the
Equations
• The electric displacement is
the charge per unit area:
• The rate of change of the
charge is ...
Using those relations:
• Using the
relations:
• Introducing the
capacitance:
• Or the electrical
admittance:

d 31
 1  s...
For open circuit (I=0)
• We get:
• Using that into the
strain relation:
• Using the expression
for the electric
admittance...
The electromechanical
coupling factor
• Introducing the factor ‘k’:

1  s11 1  k  1
2
31

• ‘k’ is called the electr...
Different Conditions
• With open circuit conditions, the stiffness of
the piezoelectric material appears to be higher
(les...
Different Conditions
• Similar results could be obtained for the
electric properties; electric properties are
affected by ...
Zero-strain conditions
(S=0)
• Using the
relations:
• Introducing the
capacitance:
• Or the electrical
admittance:

d 31
0...
Other types of Piezo!
1-3 Piezocomposites

3  c

E

  e33 E 3

33 3

S

D3  e33 3  

33

E3
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 ...
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:
 a  Ea  a
 s  Es s

• And for the equil...
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
Zero Force case
• In this case, the strain of the of the PZT will be
less than that induced by the electric field
only!  ...
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
...
Equilibrium
• The externally applied moment has to be in
equilibrium with the internally generated
h/2
h/2
moment.
d 2v
M ...
Equilibrium
• Rearranging the terms:
M d 2v
 2
EI dx

My
 
I
With piezoelectric
materials
• Introducing change in the material property:
h/2

M
   ydy
b
h / 2
t s / 2


V
  ...
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

...
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...
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
...
Remember:
• For homogeneous structures: Eh3 d 2v

M

2
12 dx
b

• Thus, in the absence of the voltage:





Ea h  t s ...
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 momen...
Piezoelectric forces
• The above is equivalent of having a force
applied by the piezoelectric material that is
equal to:
M...
Homework #3
• Solve problems 4,5,&6 from textbook
• Due 30/11/2013 (11:59PM)
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Piezoelectric Materials

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

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Transcript of "Piezoelectric Materials"

  1. 1. Piezoelectric Materials Dr. Mohammad Tawfik
  2. 2. What is Piezoelectric Material? • Piezoelectric Material is one that possesses the property of converting mechanical energy into electrical energy and vice versa.
  3. 3. Piezoelectric Materials • Mechanical Stresses  Electrical Potential Field : Sensor (Direct Effect) • Electric Field  Mechanical Strain : Actuator (Converse Effect) Clark, Sounders, Gibbs, 1998
  4. 4. Conventional Setting Conductive Pole
  5. 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. 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. 7. Applications of Piezoelectric Materials in Vibration Control
  8. 8. Collocated Sensor/Actuator
  9. 9. Self-Sensing Actuator
  10. 10. Hybrid Control
  11. 11. Passive Damping / Shunted Piezoelectric Patches
  12. 12. Passively Shunted Networks Resistive Capacitive Resonant Switched
  13. 13. Modeling of Piezoelectric Structures
  14. 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. 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. 16. The Electromechanical Coupling • Electric permittivity (Farade/m) or (Coulomb/mV) • d31 is called the electromechanical coupling factor (m/Volt)
  17. 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. 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. 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. 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. 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. 22. Different Conditions • Similar results could be obtained for the electric properties; electric properties are affected by the mechanical boundary conditions.
  23. 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. 24. Other types of Piezo!
  25. 25. 1-3 Piezocomposites 3  c E   e33 E 3 33 3 S D3  e33 3   33 E3
  26. 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. 27. Actuation Action • PZT and structure are assumed to be in perfect bonding
  28. 28. Axial Motion of Rods • In this case, we will consider the case when the PZT and the structure are deforming axially only
  29. 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. 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. 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. 32. Homework #2 • Solve problems 1,2,&3 from textbook • Due 27/11/2013 (11:59PM)
  33. 33. Beams with Piezoelectric Material
  34. 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. 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. 36. Equilibrium • Rearranging the terms: M d 2v  2 EI dx My   I
  37. 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. 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. 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. 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. 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. 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. 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. 44. Homework #3 • Solve problems 4,5,&6 from textbook • Due 30/11/2013 (11:59PM)
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