Viscoelastic Damping and Smart Structures Using Piezoelectric Materials

Viscoelastic Damping
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Vibration Damping
using
Smart Materials

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Recommended Reference
Serinivasan, A. V. and McFarland, D.
Michael, “ Smart Structures, Analysis
and Design,” Cambridge University Press,
UK, 2001.

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Damping with
Piezoelectric
Material

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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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Piezoelectric Materials
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
3333333
3333333
ESeD
EeScT
S
E



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Active Fiber Composites (AFC)
 3333
2
31
1111 SpC
p
Eeff
vv
ev
cc
 

3333
3133
31 SpC
eff
vv
e
e




 3333
3333
33 SpC
S
eff
vv 





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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
Resonant
Capacitive Switched
Resistive

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Adaptive Structures
Wada, Fanson, and Crawly
Passive Networks

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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
3
2
31
11
1
1
i
k
Z RES


222
2
2
3111 1




r
kZ RSP
Resistor Shunt
R-L Shunt
)parametertuningfrequencyricpiezoelectshuntedresonant(
)frequencyldimensiona-noncomplex(
)parametern tuningdissipatio(
n
e
n
n
s
RCr










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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 ETdD
EdTsS
33131
3111



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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
A
Q
D 
As
I
Idt
A
D  
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
V
E 

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Using those relations:
Using the
relations:
Introducing the
capacitance:
Or the electrical
admittance:
V
t
sA
sTAdI
V
t
d
TsS
33
131
31
11



CsVsTAdI  131
YVsTAdI  131

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For open circuit (I=0)
We get:
Using that into the
strain relation:
Using the expression
for the electric
admittance:
1
31
T
Y
sAd
V 
1
2
31
11 T
tY
Asd
TsS 
1
1133
2
31
11 1 T
s
d
sS 








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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
3111 1 TksS 

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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
3111 1 TsTksS D

TsTsS E
 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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Classical Models

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Objectives
•Recognize the nature of viscoelastic
material
•Understand the damping models of
viscoelastic material
•Dynamics of structures with viscoelastic
material

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What is Viscoelastic Material?
•Materials that Exhibit, both, viscous and
elastic characteristics.
•The material may be modeled in many
different ways. Classical models include:
–Mawxell Model
–Kalvin-Voight Model

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Maxwell Model
•The Maxwell model
describes the material
as a viscous damper in
series with an elastic
stiffness.
•When stress is applied,
it is uniform through the
element.
•The strain may be
written as:
𝜀 = 𝜀 𝑠 + 𝜀 𝑑

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Maxwell Model
Maxwell Model Video

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Stress-Strain Relation
•The stress is equal in
both elements and is
given by the relation:
•From which we may
write:
•Or:
𝜎 = 𝐸𝑠 𝜀 𝑠 = 𝐶 𝑑 𝜀 𝑑
𝜀 𝑠 =
𝜎
𝐸𝑠
, 𝜀 𝑑 =
𝜎
𝐶 𝑑
݀‫ݐ‬
𝜀 =
𝜎
𝐸𝑠
+
𝜎
𝐶 𝑑
݀‫ݐ‬ ∧ 𝜀 =
𝜎
𝐸𝑠
+
𝜎
𝐶 𝑑

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Three Main Characteristics
•Creep
Strain changing with time for the same stress
•Relaxation
Stress changing with time for constant strain
•Storage and Loss Moduli
Effective modulus of elasticity in response to
frequency excitation

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Maxwell Model Characteristics
•Creep:
–For constant stress, we get:
–Which gives:
•Which indicates that the strain will
grow to an unbound value as
time increases!
𝜀 =
𝜎
𝐸𝑠
⏟
‫݋ݎ݁ݖ‬
+
𝜎
𝐶 𝑑
𝜀 =
𝜎
𝐶 𝑑
𝑡

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•Relaxation:
–For constant strain, we get:
–Which gives:
•Which means that the stress will
decrease as time grows for the
same strain
0 =
𝜎
𝐸𝑠
+
𝜎
𝐶 𝑑
𝜎 = 𝜎0 𝑒−‫ܧݐ‬𝑠 𝐶 𝑑

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•Storage and Loss Factors:
–For harmonic stress:
–Which drives the strain harmonically:
–Giving:
𝑗߱ߝ 𝑜 =
𝑗𝜔
𝐸𝑠
+
1
𝐶 𝑑
𝜎𝑜
𝜎 = 𝜎0 𝑒 𝑗𝜔𝑡
𝜀 = 𝜀0 𝑒 𝑗𝜔𝑡
𝜎𝑜 =
𝐸𝑠 𝐶 𝑑 𝑗𝜔
𝐸𝑠 + 𝑗𝜔𝐶 𝑑
𝜀 𝑜

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𝜎𝑜 =
𝐶 𝑑2 𝐸𝑠 𝜔2 + 𝐸 𝑠2 𝐶 𝑑 𝑗𝜔
𝐸 𝑠2 + 𝜔2 𝐶 𝑑2
𝜀 𝑜
𝜎𝑜 =
𝐶 𝑑2 𝐸𝑠 𝜔2
𝐸 𝑠2 + 𝜔2 𝐶 𝑑2
+ 𝑗
𝐸 𝑠2 𝐶 𝑑 𝜔
𝐸 𝑠2 + 𝜔2 𝐶 𝑑2
𝜀 𝑜
𝜎𝑜 = 𝐸′ 1 + 𝑗𝜂 𝜀 𝑜

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Storage and Loss Moduli
•The stress strain relation of the
viscoelastic material appears to
contain a complex modulus of
elasticity!
•The real part is called the storage
modulus
•The imaginary part is called the
loss modulus
•And their ratio is called the loss
factor
𝜎 𝑜 = 𝐸′ 1 + 𝑗𝜂 𝜀 𝑜

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Frequency Dependent Behavior

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Notes on the Maxwell Model
•Under static loading, the stiffness, storage
modulus is zero, and the loss factor is
infinity!
•For very high frequencies, the loss factor
becomes zero!

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Kalvin-Voigt Model
•The Kalvin-Voigt model
describes the material as a
viscous damper in parallel
with an elastic stiffness.
•When stress is applied, it is
distributed through the
elements.
•The stress strain relation
may be written as:
𝜎 = 𝜎𝑠 + 𝜎 𝑑
𝜎 = 𝐸𝑠 𝜀 𝑠 + 𝐶 𝑑 𝜀 𝑑

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Kalvin-Voigt Model Characteristics
•Creep:
–For constant stress, we get:
•Which indicates that the strain will grow to
a constant value as time increases!
𝜀 =
𝜎
𝐸𝑠
1 − 𝑒−𝐸𝑠 𝑡 𝐶 𝑑

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•Relaxation:
–For constant strain, we get:
•Which means that the stress will
stay constant as time grows for
the same strain!
𝜎 = 𝐸𝑠 𝜀0

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Creep Relaxation Summary

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•Storage and Loss Factors:
–For harmonic stress:
–Which drives the strain harmonically:
–Giving:
𝜎 = 𝐸𝑠 + 𝑗𝜔𝐶 𝑑 𝜀 𝑜
𝜎 = 𝜎0 𝑒 𝑗𝜔𝑡
𝜀 = 𝜀0 𝑒 𝑗𝜔𝑡

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Notes on the Kalvin-Voigt Model
•Under all loading, storage modulus is equal
to the stiffness of the spring, and the loss
factor is zero.
•For very high frequencies, the loss factor
becomes grows unbound!

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Assignment
•Study the creep, relaxation, and frequency
response characteristics of the Zener model
shown in the following sketch

Viscoelastic Damping and Smart Structures Using Piezoelectric Materials

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