Damping with Piezoelectric
Material
Mohammad Tawfik

Damping with Piezoelectric Materials
Mohammad Tawfik

WikiCourses#
ht...
Objectives
• General Introduction to smart materials
and structures
• Recognize the nature of piezoelectric
material
• Und...
Smart Structures

Damping with Piezoelectric Materials
Mohammad Tawfik

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Smart Structures: What?
• Controlled change in properties
– Change in mechanical properties
– Change in geometry
• Energy ...
Smart Structure: Why?
•
•
•
•
•

Vibration Damping
Shape Control
Noise Reduction
Vibration/Damage Sensing
Heat Sensing

Da...
Smart Structures: Classification

Wada, Fanson, and Crawly
Damping with Piezoelectric Materials
Mohammad Tawfik

WikiCours...
Piezoelectric Materials

Damping with Piezoelectric Materials
Mohammad Tawfik

WikiCourses#
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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

Damping with Piezoelectric Materials
Mohammad Tawfik

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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...
Other types of Piezo!

Damping with Piezoelectric Materials
Mohammad Tawfik

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1-3 Piezocomposites

T 3 =c

E

33

S 3 +e 33 E 3

D 3 =e 33 S 3 + ε

Damping with Piezoelectric Materials
Mohammad Tawfik...
Active Fiber Composites (AFC)
c

eff

11

=c

E

11

+

v p e2
31

(v

C

e

eff

31

p

ε 33+ v ε

=

p

v ε 33 +v ε
ε

D...
Applications of Piezoelectric
Materials in Vibration Control

Damping with Piezoelectric Materials
Mohammad Tawfik

WikiCo...
Collocated Sensor/Actuator

Damping with Piezoelectric Materials
Mohammad Tawfik

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Self-Sensing Actuator

Damping with Piezoelectric Materials
Mohammad Tawfik

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Hybrid Control

Damping with Piezoelectric Materials
Mohammad Tawfik

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Passive Damping / Shunted
Piezoelectric Patches

Damping with Piezoelectric Materials
Mohammad Tawfik

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Passively Shunted Networks

Resistive

Capacitive
Damping with Piezoelectric Materials
Mohammad Tawfik

Resonant

Switched...
Adaptive Structures

Passive Networks
Wada, Fanson, and Crawly
Damping with Piezoelectric Materials
Mohammad Tawfik

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How does it work?

Damping with Piezoelectric Materials
Mohammad Tawfik

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Shunted Piezoelectric Material
(Physical)

Damping with Piezoelectric Materials
Mohammad Tawfik

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Shunted Piezoelectric Material
(Physical)
•Mechanical energy is
converted to electrical
energy through
piezoelectric effec...
Shunted Piezoelectric Material
(Electric)

Damping with Piezoelectric Materials
Mohammad Tawfik

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Shunted Piezoelectric Material
(Energy)

Damping with Piezoelectric Materials
Mohammad Tawfik

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Mechanical Impedance /
Viscoelastic Analogy
Resistor Shunt
RES
Z 11 =1−

R-L Shunt

2
k 31

1+iρ3
2

δ
RSP
2
Z 11 =1−k 31 ...
Viscoelastic analogy
• The model of the shunted piezoelectric
patches, in many researches, is reduced
to an equivalent of ...
The Problem With Viscoelastic
Analogy!

Base structure

Damping with Piezoelectric Materials
Mohammad Tawfik

WikiCourses#...
Modeling of Piezoelectric
Structures

Damping with Piezoelectric Materials
Mohammad Tawfik

WikiCourses#
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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
• d31 is called the electromechanical
coupling factor (m/Volt)

Damping with Piezoelectric ...
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:
Damping with ...
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’:

(

2
S =s 11 1−k 31

)T 1

• ‘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 ...
Damping of Structural Vibration
with Piezoelectric Materials and
Passive Electrical Networks
N. W. HAGOOD AND A. VON
FLOTO...
The Constitutive Relations for
Piezoelectric Materials
• The constitutive relation is:
• Where:

Damping with Piezoelectri...
Constitutive relation (cont’d)
• The electric permiativity:
• Electromechanical
coupling:
• Mechanical
compliance:
Damping...
Electrical relation
• Into constitutive
relations:
• Where the
capacitance is:

Damping with Piezoelectric Materials
Moham...
The Electric admittance
• Introducing the electric admittance:

• Generally; with a shunt circuit:

Damping with Piezoelec...
The Electromechanical Model

Damping with Piezoelectric Materials
Mohammad Tawfik

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Remember
• The electric admittance is
the reciprocal of the electric
impedance.

1
Z = EL
Y
EL

• Also, you may have up to...
From Constitutive Relations
• The voltage may be written as:
• Into the strain equation

Damping with Piezoelectric Materi...
The Electromechanical Compliance
• The Electromechanical
Compliance
• Or
• Where
• Generally:

Damping with Piezoelectric ...
The Mi matrices

Damping with Piezoelectric Materials
Mohammad Tawfik

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Uniaxial Loading Cases
• The compliance:
• Introducing the
electromechanical
coupling coefficient:
• The compliance
become...
Uniaxial Loading Cases
• For open circuit, the
compliance becomes:
• A similar expressions for
capacitance in case of
zero...
Uniaxial Loading Cases
• Introducing the
mechanical impedance:
(Which is the reciprocal
of the compliance)
• We may write ...
The Complex Modulus
• Now, let’s reintroduce the complex
modulus of the viscoelastic material:

• Where:

Damping with Pie...
Resistive Shunt Example

• For the case or resistive shunting, the
resistance and the capacitance are in
parallel 1
1
1
1 ...
Resistive Shunt Example
• Recall:
• Using the previous
results:
• Simplifying:

Damping with Piezoelectric Materials
Moham...
Resistive Shunt Example
• Substituting s=iω and
• Introducing the non-dimensional
parameter ρ=RCω, we get:
Z RES =1−
jj

k...
Resistive Shunt Example
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
Damping with Piezoelectric Materials
Mohammad Tawfik

...
Homework #11
1. Derive the equations for the RL shunt
circuit.
2. Plot the frequency response of
piezoelectric bar with a ...
RL Shunt Example

• For the case or RL shunting, the resistance and
the inductance are in series and are in parallel
with ...
RL Shunt Example
• Recall:
• Using the previous
results:
• Simplifying:

Damping with Piezoelectric Materials
Mohammad Taw...
RL Shunt Example
• Ignoring (1-k2) in the denominator:
ω2
e
Z RSP =1−k 2
jj
ij

ω2
n
2

2

2

ωe

ωe

s
s
+ 2 RC ω n
+ 2
2...
RL Shunt Example
• Using the full
term:

(

( 1−k 2 )
ij
Z RSP =
jj

ω2
e

ω2
e

2

Z RSP =
jj
ωn

ω2
e

s
+ 2 RC
s+ 2
2
ω...
RL Shunt Example: Hagood results

Damping with Piezoelectric Materials
Mohammad Tawfik

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Comparing results

Damping with Piezoelectric Materials
Mohammad Tawfik

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Modulus of Elasticity
• Recall that:
• And:
• Where:

Damping with Piezoelectric Materials
Mohammad Tawfik

WikiCourses#
h...
Modulus of Elasticity
• Substituting:
• Getting the
stiffness:
• Simplifying:

Damping with Piezoelectric Materials
Mohamm...
Finite Element Model

Damping with Piezoelectric Materials
Mohammad Tawfik

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Recall
• Recall the constitutive
relations of
Piezoelectric materials:

S 1 =s11 T 1 +d 31 E 3
D3 =d 31 T 1 +¿ 33 E 3

1
T...
Where:
h 31=

d 31

(

2
s11 ∈33 1−k 31
s
¿33=¿ 33 1−k 2
31
D
2
s11=s 11 1−k 31

(
(

Damping with Piezoelectric Materials...
Potential Energy
• Writing the expression for the potential
energy of the shunted piezoelectric
material:
l
l
1
1
U = ∫ S ...
The interpolation functions

( ) ()
( ) (
) () ⌊
( )
( )
( ) ()

x
x
u ( x )= 1− u1 +
u 2 = ⌊ N ( x ) ⌋ {u e }
l
l

x
x
e
...
The Stiffness Matrices
l

1
k e = D ∫ { N x }⌊ N x ⌋ Adx
s 11 0
l

k eD =−h31∫ { N x } ⌊ N ⌋ Adx
l

0

1
k D= S ∫ { N } ⌊ ...
Kinetic Energy
l

1
2
T = ∫ ρ u Adx
˙
20
l

me = ρA ∫ { N } ⌊ N ⌋ dx
0

Damping with Piezoelectric Materials
Mohammad Tawf...
External Work
l

l

l

˙
¨
˙
W =∫ VDbdx=∫ ( L I + RI ) Dbdx=∫ ( L Q+R Q ) Dbdx
0

0

0

l

¨
˙
¿∫ A ( L D+ R D ) Dbdx
0

l...
Element Equation

[

me
0

0
mD

]{ } [ ]{ } [
ue
¨
0
+
¨
De 0

0
cD

Damping with Piezoelectric Materials
Mohammad Tawfik...
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Piezoelectric Shunt Damping

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Passive damping using shunt circuits
Finite Element Modeling

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Piezoelectric Shunt Damping

  1. 1. Damping with Piezoelectric Material Mohammad Tawfik Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  2. 2. Objectives • General Introduction to smart materials and structures • Recognize the nature of piezoelectric material • Understand the use of passive shunt circuits • Dynamics of structures with shunt piezoelectric materials Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  3. 3. Smart Structures Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  4. 4. Smart Structures: What? • Controlled change in properties – Change in mechanical properties – Change in geometry • Energy Converters! – Mechanical Electrical (Piezoelectric) – Heat  Mechanical (SMA) – Mechanical  Heat (Viscoelastic) – Etc… Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  5. 5. Smart Structure: Why? • • • • • Vibration Damping Shape Control Noise Reduction Vibration/Damage Sensing Heat Sensing Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  6. 6. Smart Structures: Classification Wada, Fanson, and Crawly Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  7. 7. Piezoelectric Materials Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  8. 8. What is Piezoelectric Material? • Piezoelectric Material is one that possesses the property of converting mechanical energy into electrical energy and vice versa. Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  9. 9. Piezoelectric Materials • Mechanical Stresses  Electrical Potential Field : Sensor (Direct Effect) • Electric Field  Mechanical Strain : Actuator (Converse Effect) Clark, Sounders, Gibbs, 1998 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  10. 10. Conventional Setting Conductive Pole Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  11. 11. 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 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  12. 12. 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. Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  13. 13. Other types of Piezo! Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  14. 14. 1-3 Piezocomposites T 3 =c E 33 S 3 +e 33 E 3 D 3 =e 33 S 3 + ε Damping with Piezoelectric Materials Mohammad Tawfik S 33 E3 WikiCourses# http://WikiCourses.WikiSpaces.com
  15. 15. Active Fiber Composites (AFC) c eff 11 =c E 11 + v p e2 31 (v C e eff 31 p ε 33+ v ε = p v ε 33 +v ε ε Damping with Piezoelectric Materials Mohammad Tawfik 33 = 33 ) ε 33 e 31 C eff S ε 33 ε (v C S S 33 33 p ε 33 + v ε S 33 ) WikiCourses# http://WikiCourses.WikiSpaces.com
  16. 16. Applications of Piezoelectric Materials in Vibration Control Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  17. 17. Collocated Sensor/Actuator Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  18. 18. Self-Sensing Actuator Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  19. 19. Hybrid Control Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  20. 20. Passive Damping / Shunted Piezoelectric Patches Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  21. 21. Passively Shunted Networks Resistive Capacitive Damping with Piezoelectric Materials Mohammad Tawfik Resonant Switched WikiCourses# http://WikiCourses.WikiSpaces.com
  22. 22. Adaptive Structures Passive Networks Wada, Fanson, and Crawly Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  23. 23. How does it work? Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  24. 24. Shunted Piezoelectric Material (Physical) Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  25. 25. Shunted Piezoelectric Material (Physical) •Mechanical energy is converted to electrical energy through piezoelectric effect •Electric charge is driven by potential difference through the circuit •Energy is dissipated in the resistance Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  26. 26. Shunted Piezoelectric Material (Electric) Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  27. 27. Shunted Piezoelectric Material (Energy) Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  28. 28. Mechanical Impedance / Viscoelastic Analogy Resistor Shunt RES Z 11 =1− R-L Shunt 2 k 31 1+iρ3 2 δ RSP 2 Z 11 =1−k 31 2 2 γ +δ rγ+ δ 2 r =RC ωn (dissipation tuning parameter ) s γ= ( complex non-dimensional frequency ) ωn ω δ= e (resonant shunted piezoelectric frequency tuning parameter ) ωn Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  29. 29. Viscoelastic analogy • The model of the shunted piezoelectric patches, in many researches, is reduced to an equivalent of a viscoelastic patch. • But Piezoelectric patches are elements that respond to the total strain rather than the local strain! Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  30. 30. The Problem With Viscoelastic Analogy! Base structure Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  31. 31. Modeling of Piezoelectric Structures Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  32. 32. 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 Damping with Piezoelectric Materials Mohammad Tawfik S=s 11 T +d 31 E D=d 31 T 1 +¿33 E WikiCourses# http://WikiCourses.WikiSpaces.com
  33. 33. 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/m) • ¿ Electric permittivity (Farade/m) Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  34. 34. The Electromechanical Coupling • d31 is called the electromechanical coupling factor (m/Volt) Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  35. 35. 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: Damping with Piezoelectric Materials Mohammad Tawfik Q D= A 1 I D= ∫ Idt= A As V E= t WikiCourses# http://WikiCourses.WikiSpaces.com
  36. 36. Using those relations: • Using the relations: • Introducing the capacitance: • Or the electrical admittance: Damping with Piezoelectric Materials Mohammad Tawfik S =s 11 T + d 31 t I = Ad 31 sT 1 + V A ∈33 s t V I = Ad 31 sT 1 +CsV I = Ad 31 sT 1 +YV WikiCourses# http://WikiCourses.WikiSpaces.com
  37. 37. For open circuit (I=0) • We get: • Using that into the strain relation: • Using the expression for the electric admittance: Damping with Piezoelectric Materials Mohammad Tawfik V =− Ad 31 s S =s 11 T − ( S =s 11 1− Y T1 2 Asd 31 tY d2 31 ¿33 s 11 T1 ) T1 WikiCourses# http://WikiCourses.WikiSpaces.com
  38. 38. The electromechanical coupling factor • Introducing the factor ‘k’: ( 2 S =s 11 1−k 31 )T 1 • ‘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 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  39. 39. Different Conditions • With open circuit conditions, the stiffness of the piezoelectric material appears to be higher (less compliance) ( 2 S =s 11 1−k 31 ) T 1 == s D T1 • While for short circuit conditions, the stiffness appears to be lower (more compliance) S =s T =s E T 11 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  40. 40. Different Conditions • Similar results could be obtained for the electric properties; electric properties are affected by the mechanical boundary conditions. Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  41. 41. Damping of Structural Vibration with Piezoelectric Materials and Passive Electrical Networks N. W. HAGOOD AND A. VON FLOTOW Journal of Sound and Vibration (1991) 146(2), 243-268 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  42. 42. The Constitutive Relations for Piezoelectric Materials • The constitutive relation is: • Where: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  43. 43. Constitutive relation (cont’d) • The electric permiativity: • Electromechanical coupling: • Mechanical compliance: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  44. 44. Electrical relation • Into constitutive relations: • Where the capacitance is: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  45. 45. The Electric admittance • Introducing the electric admittance: • Generally; with a shunt circuit: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  46. 46. The Electromechanical Model Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  47. 47. Remember • The electric admittance is the reciprocal of the electric impedance. 1 Z = EL Y EL • Also, you may have up to three circuits: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  48. 48. From Constitutive Relations • The voltage may be written as: • Into the strain equation Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  49. 49. The Electromechanical Compliance • The Electromechanical Compliance • Or • Where • Generally: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  50. 50. The Mi matrices Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  51. 51. Uniaxial Loading Cases • The compliance: • Introducing the electromechanical coupling coefficient: • The compliance becomes: • For open circuit conditions Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  52. 52. Uniaxial Loading Cases • For open circuit, the compliance becomes: • A similar expressions for capacitance in case of zero stress is: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  53. 53. Uniaxial Loading Cases • Introducing the mechanical impedance: (Which is the reciprocal of the compliance) • We may write the nondimensional mechanical impedance: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  54. 54. The Complex Modulus • Now, let’s reintroduce the complex modulus of the viscoelastic material: • Where: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  55. 55. Resistive Shunt Example • For the case or resistive shunting, the resistance and the capacitance are in parallel 1 1 1 1 RCs+1 Z EL = Z D + Z Damping with Piezoelectric Materials Mohammad Tawfik SU =Cs+ R = R WikiCourses# http://WikiCourses.WikiSpaces.com
  56. 56. Resistive Shunt Example • Recall: • Using the previous results: • Simplifying: Damping with Piezoelectric Materials Mohammad Tawfik ME 2 1−k ij RES Z jj =Z jj = 1−k 2 ij ( RCs 1+ RCs ) 2 RES Z jj =1− k ij 1+( 1−k 2 ) RCs ij WikiCourses# http://WikiCourses.WikiSpaces.com
  57. 57. Resistive Shunt Example • Substituting s=iω and • Introducing the non-dimensional parameter ρ=RCω, we get: Z RES =1− jj k2 ij 1+( • Finally: 2 1−k ij ) ρi ( RES Z jj = =1− 1− 2 k ij 1+( 2 2 1−k ij ) ρ ρ )( 2 k ij 1+ ( Damping with Piezoelectric Materials Mohammad Tawfik + 2 2 1−k ij ) 2 2 2 2 k ij ( 1−k ij ) ρi 2 2 1−k ij 1+ ( 1+ ) 2 k ij 1+ ( ρ2 ρ 1−k 2 ij )ρ 2 i WikiCourses# http://WikiCourses.WikiSpaces.com )
  58. 58. Resistive Shunt Example 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 Damping with Piezoelectric Materials Mohammad Tawfik E Current Eta Current E von Flotto Eta von Flotto 1 10 WikiCourses# http://WikiCourses.WikiSpaces.com
  59. 59. Homework #11 1. Derive the equations for the RL shunt circuit. 2. Plot the frequency response of piezoelectric bar with a shunt circuit (R & RL) Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  60. 60. RL Shunt Example • For the case or RL shunting, the resistance and the inductance are in series and are in parallel with the capacitance 2 1 1 1 1 LCs + RCs+1 = D + SU =Cs+ = EL Ls+ R Ls+ R Z Z Z LCs 2 + RCs Z EL = LCs 2 + RCs+1 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  61. 61. RL Shunt Example • Recall: • Using the previous results: • Simplifying: Damping with Piezoelectric Materials Mohammad Tawfik 2 ME 1−k ij RSP Z jj =Z jj = 2 1−k ij Z RSP = jj ( LCs 2 + RCs LCs 2 + RCs+1 1−k 2 ) ( LCs 2 + RCs+1 ) ( ij 2 2 1+( 1−k ij ) ( LCs + RCs ) WikiCourses# http://WikiCourses.WikiSpaces.com )
  62. 62. RL Shunt Example • Ignoring (1-k2) in the denominator: ω2 e Z RSP =1−k 2 jj ij ω2 n 2 2 2 ωe ωe s s + 2 RC ω n + 2 2 ωn ω n ωn ωn Z RSP =1−k 2 2 jj ij δ 2 2 γ +δ rγ+δ Damping with Piezoelectric Materials Mohammad Tawfik 2 WikiCourses# http://WikiCourses.WikiSpaces.com
  63. 63. RL Shunt Example • Using the full term: ( ( 1−k 2 ) ij Z RSP = jj ω2 e ω2 e 2 Z RSP = jj ωn ω2 e s + 2 RC s+ 2 2 ωn ωn ω n ωn ( 2 + ( 1−k ij ) ω2 n ( ( 1−k 2 ) ij 2 ωn s 2 ωe + 2 RC s 2 ωn ω n ωn ) ) Z RSP = jj Damping with Piezoelectric Materials Mohammad Tawfik ωn s2 + RC s +1 2 ωn ωe ( 1+( 1−k 2 ) ij ωn s2 + RC s 2 ωn ωe ) ) 1−k 2 ) ( γ 2 + δ 2 rγ + δ 2 ) ( ij 2 2 2 2 δ + (1−k ij ) ( γ + δ rγ ) WikiCourses# http://WikiCourses.WikiSpaces.com
  64. 64. RL Shunt Example: Hagood results Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  65. 65. Comparing results Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  66. 66. Modulus of Elasticity • Recall that: • And: • Where: Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  67. 67. Modulus of Elasticity • Substituting: • Getting the stiffness: • Simplifying: Damping with Piezoelectric Materials Mohammad Tawfik SU s jj = SU E jj = SU Z jj Lj s Aj SU E jj = = D s jj SU Z jj Lj s D ME Z jj Z jj L j s ME Z jj Aj Aj ME = Z jj s E (1−k 2 ) jj ij WikiCourses# http://WikiCourses.WikiSpaces.com
  68. 68. Finite Element Model Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  69. 69. Recall • Recall the constitutive relations of Piezoelectric materials: S 1 =s11 T 1 +d 31 E 3 D3 =d 31 T 1 +¿ 33 E 3 1 T 1 = D S 1 −h31 D3 • Rearranging the terms: s 11 1 E 3 =−h 31 S 1 + S D 3 ¿ 33 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  70. 70. Where: h 31= d 31 ( 2 s11 ∈33 1−k 31 s ¿33=¿ 33 1−k 2 31 D 2 s11=s 11 1−k 31 ( ( Damping with Piezoelectric Materials Mohammad Tawfik ) ) ) WikiCourses# http://WikiCourses.WikiSpaces.com
  71. 71. Potential Energy • Writing the expression for the potential energy of the shunted piezoelectric material: l l 1 1 U = ∫ S 1 T 1 Adx+ ∫ D3 E 3 Adx 20 20 l ( S1 ) l ( ) D3 1 1 U = ∫ S 1 D −h 31 D 3 Adx+ ∫ D3 −h31 D 3 + S Adx 20 2 0 s11 ¿ 33 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  72. 72. The interpolation functions ( ) () ( ) ( ) () ⌊ ( ) ( ) ( ) () x x u ( x )= 1− u1 + u 2 = ⌊ N ( x ) ⌋ {u e } l l x x e d x = 1− d 1 + d 2= N ( x ) ⌋ {d } l l du x 1 1 S1 x = = − u 1+ u2 = ⌊ N x ( x ) ⌋ {ue } dx l l Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  73. 73. The Stiffness Matrices l 1 k e = D ∫ { N x }⌊ N x ⌋ Adx s 11 0 l k eD =−h31∫ { N x } ⌊ N ⌋ Adx l 0 1 k D= S ∫ { N } ⌊ N ⌋ Adx ¿33 0 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  74. 74. Kinetic Energy l 1 2 T = ∫ ρ u Adx ˙ 20 l me = ρA ∫ { N } ⌊ N ⌋ dx 0 Damping with Piezoelectric Materials Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
  75. 75. External Work l l l ˙ ¨ ˙ W =∫ VDbdx=∫ ( L I + RI ) Dbdx=∫ ( L Q+R Q ) Dbdx 0 0 0 l ¨ ˙ ¿∫ A ( L D+ R D ) Dbdx 0 l m D= AbL∫ { N } ⌊ N ⌋ dx 0 Damping with Piezoelectric Materials Mohammad Tawfik l c D= AbR∫ { N } ⌊ N ⌋ dx 0 WikiCourses# http://WikiCourses.WikiSpaces.com
  76. 76. Element Equation [ me 0 0 mD ]{ } [ ]{ } [ ue ¨ 0 + ¨ De 0 0 cD Damping with Piezoelectric Materials Mohammad Tawfik ue ke ˙ + ˙ De k De k eD kD ]{ } { } ue f = 0 De WikiCourses# http://WikiCourses.WikiSpaces.com
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