Piezoelectric Shunt Damping

Damping with Piezoelectric
Material
Mohammad Tawfik

Damping with Piezoelectric Materials
Mohammad Tawfik

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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
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Smart Structures

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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…

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Smart Structure: Why?
•
•
•
•
•

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

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Smart Structures: Classification

Wada, Fanson, and Crawly
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Piezoelectric Materials

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

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

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Conventional Setting

Conductive Pole

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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

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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.

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Other types of Piezo!

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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

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S

33

E3

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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

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

)

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Applications of Piezoelectric
Materials in Vibration Control

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Collocated Sensor/Actuator

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Self-Sensing Actuator

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Hybrid Control

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Passive Damping / Shunted
Piezoelectric Patches

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Passively Shunted Networks

Resistive

Capacitive
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Resonant

Switched
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Adaptive Structures

Passive Networks
Wada, Fanson, and Crawly
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How does it work?

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Shunted Piezoelectric Material
(Physical)

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(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
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(Electric)

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(Energy)

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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 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

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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!

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The Problem With Viscoelastic
Analogy!

Base structure

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Modeling of Piezoelectric
Structures

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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
Mohammad Tawfik

S=s 11 T +d 31 E
D=d 31 T 1 +¿33 E
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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)
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The Electromechanical Coupling
• d31 is called the electromechanical
coupling factor (m/Volt)

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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:
Mohammad Tawfik

Q
D=
A
1
I
D= ∫ Idt=
A
As

V
E=
t

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Using those relations:
• Using the
relations:
• Introducing the
capacitance:
• Or the electrical
admittance:
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
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For open circuit (I=0)
• We get:
• Using that into the
strain relation:
• Using the
expression for the
electric admittance:
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

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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
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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

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

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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
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The Constitutive Relations for
• The constitutive relation is:
• Where:

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

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The Electric admittance
• Introducing the electric admittance:

• Generally; with a shunt circuit:

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The Electromechanical Model

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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
three circuits:

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From Constitutive Relations
• The voltage may be written as:
• Into the strain equation

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The Electromechanical Compliance
• The Electromechanical
Compliance
• Or
• Where
• Generally:

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The Mi matrices

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Uniaxial Loading Cases
• The compliance:
• Introducing the
electromechanical
coupling coefficient:
• The compliance
becomes:
• For open circuit
conditions
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• For open circuit, the
compliance becomes:
• A similar expressions for
capacitance in case of
zero stress is:

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• Introducing the
mechanical impedance:
(Which is the reciprocal
of the compliance)
• We may write the nondimensional mechanical
impedance:

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The Complex Modulus
• Now, let’s reintroduce the complex
modulus of the viscoelastic material:

• Where:

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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

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SU

=Cs+

R

=

R

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• Recall:
• Using the previous
results:
• Simplifying:

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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

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• 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+ (

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

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)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
Mohammad Tawfik

E Current
Eta Current
E von Flotto
Eta von Flotto

1

10
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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)

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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
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RL Shunt Example
• Recall:
• Using the previous
results:
• Simplifying:

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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 )
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)

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

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2

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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
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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 )
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RL Shunt Example: Hagood results

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Comparing results

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

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Modulus of Elasticity
• Substituting:
• Getting the
stiffness:
• Simplifying:

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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

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Finite Element Model

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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 1 = D S 1 −h31 D3
• Rearranging the terms:
s
11

1
E 3 =−h 31 S 1 + S D 3
¿ 33
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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

(
(

Mohammad Tawfik

)

)
)

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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
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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
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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
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Kinetic Energy
l

1
2
T = ∫ ρ u Adx
˙
20
l

me = ρA ∫ { N } ⌊ N ⌋ dx
0

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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

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l

c D= AbR∫ { N } ⌊ N ⌋ dx
0

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Element Equation

[

me
0

0
mD

]{ } [ ]{ } [
ue
¨
0
+
¨
De 0

0
cD

Mohammad Tawfik

ue
ke
˙
+
˙
De
k De

k eD
kD

]{ } { }
ue
f
=
0
De

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Piezoelectric Shunt Damping

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