Piezoelectric Materials

Dr. Mohammad Tawfik

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.

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

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 EdD

Eds

33131

31111

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)

The Electromechanical Coupling

• Electric permittivity (Farade/m) or (Coulomb/mV)

• 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

sAsAdI

Vt

ds

33131

311111

CsVsAdI 131

YVsAdI 131

For open circuit (I=0)

• We get:

• Using that into the strain relation:

• Using the expression for the electric admittance:

131

Y

sAdV

1

2

311111

tY

Asds

1

1133

2

31111 1

s

ds

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

31111 1 ks

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

31111 1 Dsks

Ess 11

Different Conditions

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

Zero-strain conditions (S=0)

• Using the relations:

• Introducing the capacitance:

• Or the electrical admittance:

Vt

ds 31

1110

Vs

d

t

AsI

1133

2

3133 1

VkYI 2

311

Other types of Piezo!

1-3 Piezocomposites

3333333

3333333

EeD

Eec

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

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:

• And for the equilibrium:

sss

aaa

E

E

sssaaassaa EAEAAAF

xssaassaa EAEAAAF

Zero Voltage case

• From that, we may write the force strain relation to be:

ssaassaa

xEtEt

bF

EAEA

F

2

Zero Force case

• In this case, the strain of the of the PZT will be less than that induced by the electric field only!

• For equilibrium, F=0: sss

asapasaa

E

t

VdEEEE

31

031 sssaasaassaa EAt

VdEAEAAAF

ssaa

aa

sEAEA

t

VdEA

31

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 of the beam and inversely proportional to the radius of curvature.

2

2

dx

vdy

2

2

dx

vdEyE

Equilibrium

• The externally applied moment has to be in equilibrium with the internally generated moment.

• For homogeneous materials:

bydydx

vdEybydyM

h

h

h

h

2/

2/

2

22/

2/

2

22/

2/

2

2

2

dx

vdEIbdyy

dx

vdEM

h

h

Equilibrium

• Rearranging the terms:

2

2

dx

vd

EI

M

I

My

With piezoelectric materials

• Introducing change in the material property:

2/

2/

31

2/

2/

2/

2/

31

2/

2/

h

t a

aa

t

t

ss

t

h a

aa

h

h

s

s

s

s

ydyt

VdE

ydyEydyt

VdE

ydyb

M

With piezoelectric materials

• Expanding the integral

2/

2/

31

2/

2/

2

2

22/

2/

2

2

2

2/

2/

31

2/

2/

2

2

2

h

ta

a

h

t

a

t

t

s

t

ha

a

t

h

a

ss

s

s

ss

ydyt

VdEdyy

dx

vdEdyy

dx

vdE

ydyt

VdEdyy

dx

vdE

b

M

With piezoelectric materials

• Rearranging

2/

2/

31

2/

2/

31

2/

2/

2

2/

2/

2

2/

2/

2

2

2

h

ta

a

t

ha

a

h

t

a

t

t

s

t

h

a

s

s

s

s

s

s

ydyt

VdEydy

t

VdE

dyyEdyyEdyyEdx

vd

b

M

With piezoelectric materials

• Integrating

22

31

22

31

33333

2

2

88

224

1

s

a

as

a

a

sasssa

tht

VdEht

t

VdE

thEtEthEdx

vd

b

M

2231333

2

2

412

1s

a

asssa th

t

VdEtEthE

dx

vd

b

M

Remember:

• For homogeneous structures:

• Thus, in the absence of the voltage:

• OR:

12

333

sssaEquivalent

tEthEbEI

b

M

dx

vdEh

2

23

12

2231

2

2

4s

a

aEquivalent th

t

VbdE

dx

vdEIM

In the absence of load

• Thus, the structure will feel a moment:

2231

2

2

4s

Equivalenta

a thEIt

VbdE

dx

vd

2231

2

2

4s

Equivalenta

asssss th

EIt

VbdEIE

dx

vdIEM

Piezoelectric forces

• The above is equivalent of having a force applied by the piezoelectric material that is equal to:

2231

4s

Equivalentas

ass

s

sa th

EItt

VbdEIE

t

MF

Homework #3

• Solve problems 4,5,&6 from textbook

• Due 30/11/2013 (11:59PM)