Vibration damping using smart materials

Viscoelastic Damping

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Vibration Damping

using

Smart Materials




Recommended Reference

Serinivasan, A. V. and McFarland, D.

Michael, “ Smart Structures, Analysis

and Design,” Cambridge University Press,

UK, 2001.




Damping with

Piezoelectric

Material




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




Smart Structures




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…




Smart Structure: Why?

Vibration Damping

Shape Control

Noise Reduction

Vibration/Damage Sensing

Heat Sensing




Smart Structures: Classification

Wada, Fanson, and Crawly




Piezoelectric

Materials




What is Piezoelectric Material?

Piezoelectric Material is one that possesses

the property of converting mechanical

energy into electrical energy and vice versa.




Piezoelectric Materials

Mechanical Stresses Electrical

Potential Field : Sensor (Direct Effect)

Electric Field Mechanical Strain :

Actuator (Converse Effect)

Clark, Sounders, Gibbs, 1998




Conventional Setting

Conductive Pole




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




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.




Other types of

Piezo!




1-3 Piezocomposites

3333333

3333333

ESeD

EeScT

S

E




Active Fiber Composites (AFC)

3333

2

311111

SpC

p

Eeff

vv

evcc

3333

313331

SpC

eff

vv

ee

3333

333333

SpC

S

eff

vv




Applications of Piezoelectric

Materials in Vibration Control




Collocated Sensor/Actuator




Self-Sensing Actuator




Hybrid Control




Passive Damping / Shunted

Piezoelectric Patches




Passively Shunted Networks

Resonant

Capacitive Switched

Resistive




Adaptive Structures

Wada, Fanson, and Crawly

Passive Networks




How does it work?




Shunted Piezoelectric Material

(Physical)





(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





(Electric)





(Energy)




Mechanical Impedance /

Viscoelastic Analogy

3

2

3111

11

i

kZ RES

222

22

3111 1

rkZ RSP

Resistor Shunt

R-L Shunt

)parameter tuningfrequency ricpiezoelect shuntedresonant (

)frequency ldimensiona-noncomplex (

)parameter n tuningdissipatio(

n

e

n

n

s

RCr




Modeling of

Piezoelectric

Structures




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




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)




The Electromechanical Coupling

d31 is called the electromechanical coupling

factor (m/Volt)




Manipulating the Equations

A

QD

As

IIdt

AD

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

VE




Using those relations:

Using the

relations:

Introducing the

capacitance:

Or the electrical

admittance:

Vt

sAsTAdI

Vt

dTsS

33131

3111

CsVsTAdI 131

YVsTAdI 131




For open circuit (I=0)

We get:

Using that into the strain relation:

Using the expression for the electric admittance:

131 T

Y

sAdV

1

2

3111 T

tY

AsdTsS

1

1133

2

3111 1 T

s

dsS




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




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




Different Conditions

Similar results could be obtained for the

electric properties; electric properties are

affected by the mechanical boundary

conditions.




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





Classical Models




Objectives

•Recognize the nature of viscoelastic

material

•Understand the damping models of

viscoelastic material

•Dynamics of structures with viscoelastic

material




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




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 = 𝜀𝑠 + 𝜀𝑑




Maxwell Model

Maxwell Model Video

WK 10/Clips/Simple Physical Maxwell Model of Viscoelasticity.mp4




Stress-Strain Relation

•The stress is equal in

both elements and is

given by the relation:

•From which we may

write:

•Or:

𝜎 = 𝐸𝑠𝜀𝑠 = 𝐶𝑑𝜀 𝑑

𝜀𝑠 =𝜎

𝐸𝑠, 𝜀𝑑 =

𝜎

𝐶𝑑 ݐ݀

𝜀 =𝜎

𝐸𝑠+

𝜎

𝐶𝑑ݐ݀ ∧ 𝜀 =

𝜎

𝐸𝑠+

𝜎

𝐶𝑑




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




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

𝐶𝑑

𝜀 =𝜎

𝐶𝑑𝑡





•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

𝐶𝑑

𝜎 = 𝜎0𝑒𝑠ܧݐ− 𝐶𝑑





•Storage and Loss Factors:

–For harmonic stress:

–Which drives the strain harmonically:

–Giving:

𝑗߱ߝ𝑜 =𝑗𝜔

𝐸𝑠+

1

𝐶𝑑𝜎𝑜

𝜎 = 𝜎0𝑒𝑗𝜔𝑡

𝜀 = 𝜀0𝑒𝑗𝜔𝑡

𝜎𝑜 =𝐸𝑠𝐶𝑑𝑗𝜔

𝐸𝑠 + 𝑗𝜔𝐶𝑑𝜀𝑜





𝜎𝑜 =𝐶𝑑2𝐸𝑠𝜔

2 + 𝐸𝑠2𝐶𝑑𝑗𝜔

𝐸𝑠2 +𝜔2𝐶𝑑2𝜀𝑜

𝜎𝑜 =𝐶𝑑2𝐸𝑠𝜔

2

𝐸𝑠2 +𝜔2𝐶𝑑2+ 𝑗

𝐸𝑠2𝐶𝑑𝜔

𝐸𝑠2 + 𝜔2𝐶𝑑2𝜀𝑜

𝜎𝑜 = 𝐸′ 1 + 𝑗𝜂 𝜀𝑜




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

𝜎𝑜 = 𝐸′ 1 + 𝑗𝜂 𝜀𝑜




Frequency Dependent Behavior




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!




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 + 𝜎𝑑

𝜎 = 𝐸𝑠𝜀𝑠 + 𝐶𝑑𝜀 𝑑




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

𝐸𝑠1 − 𝑒−𝐸𝑠𝑡 𝐶𝑑





•Relaxation:

–For constant strain, we get:

•Which means that the stress will

stay constant as time grows for

the same strain!

𝜎 = 𝐸𝑠𝜀0




Creep Relaxation Summary





•Storage and Loss Factors:

–For harmonic stress:

–Which drives the strain harmonically:

–Giving:

𝜎 = 𝐸𝑠 + 𝑗𝜔𝐶𝑑 𝜀𝑜

𝜎 = 𝜎0𝑒𝑗𝜔𝑡

𝜀 = 𝜀0𝑒𝑗𝜔𝑡




Frequency Dependent Behavior




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!




Assignment

•Study the creep, relaxation, and frequency

response characteristics of the Zener model

shown in the following sketch

Vibration damping using smart materials

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