2.Reveiw.smart Part2

7/30/2019 2.Reveiw.smart Part2

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11

Control of Smart Structures 2. Review of Smart Materials and Structures (Part 2) 1

Department of Mechanical Engineering

Dr. G. Song, Associate Professor

2. Review of Smart Materials andStructures (part 2): Piezoceramics

=

6

5

4

3

2

1

333231

24

15

3

2

1

000ddd

00d000

0d0000

D

D

D


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Piezoelectricity

Piezoelectricity describes the phenomenon of generating an electric

charge in a material when subjecting it to mechanical stress (direct

effect), and conversely, generating a mechanical strain in response

to an applied electric field (converse effect).

PZT: It is an acronym for Lead Zirconate Titanate, which is a

commonly used piezoelectric ceramic material.


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Piezoelectric Ceramics

Piezoelectric elementary cell

(1) before poling

(2) after poling

Courtesy PI Polytec

The piezo effect exhibited by natural materials such as

quartz, tourmaline, Rochelle salt, etc. is very small.

Polycrystalline ferroelectric ceramic materials such as

BaTiO3 and Lead Zirconate Titanate (PZT) have beendeveloped with improved properties.

Ferroelectric ceramics become piezoelectric when poled.

PZT crystallites are centro-symmetric cubic (isotropic) before

poling and after poling exhibit tetragonal symmetry(anisotropic structure) below the Curie temperature (beyond

which the piezoelectricity is lost).


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Poling

Electric dipoles in Weiss domains

(1) unpoled ferroelectric ceramic, (2) during and

(3) after poling (piezoelectric ceramic)

Courtesy PI Polytec

Charge separation between the positive and negative ions is the reason for electric dipole behavior.

Groups of dipoles with parallel orientation are called Weiss domains.

The Weiss domains are randomly

oriented in the raw PZT material,

before the poling treatment has beenfinished.

During poling, an electric field (>

2000 V/mm) is applied to the

(heated) piezo ceramics.

With the field applied, the materialexpands along the axis of the field

and contracts perpendicular to that

axis.

The electric dipoles align and roughly

stay in alignment upon cooling. Thematerial now has a remnant

polarization (which can be degraded

by exceeding the mechanical,

thermal and electrical limits of the

material).


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Poling (cont)

When an electric voltage is applied to a poled piezoelectric

material, the Weiss domains increase their alignment proportional

to the voltage. The result is a change of the dimensions(expansion, contraction) of the PZT material.


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Direct Piezoelectric Effect

- Sensors

Poling axis

Electrodes

+

_V

Applied Force F

_

+

V

F


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Inverse Piezoelectric Effect

- Actuators

Poling axis

Electrodes

+

_V

Resulting Strain S

_

+

S


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More about Lead Zirconate Titanate (PZT)

Material PZTs offer the user several benefits and advantages over other motion

techniques

1. Repeatable nanometer and sub- nanometer sized steps at high frequency

can be achieved with PZTs because they derive their motion through solidstate crystal effects. There are no moving parts (no "stick-slip" effect).

2. PZTs can be designed to move heavy loads (several tons) or can be madeto move lighter loads at frequencies of several 10 kHz.

3. PZTs act as capacitive loads and require very little power in staticoperation, simplifying power supply needs.

4. PZTs require no maintenance because they are solid state and their motionis based on molecular effects within the ferroelectric crystals.

With high-reliability PZT materials a strain on the order of 1/1000 (0.1%)can be achieved; this means that a I00 mm long PZT actuator can

expand by 100 micrometers when the maximum allowable field is

applied.


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Curie Temperature

Above a certain temperature, called the Curie Point, a

piezoelectric material has a symmetric cubic crystal structure

and there is no net charge induced dipole

Below this temperature, the crystal structure becomes

tetragonal, the positive and negative charges no longer

coincide, producing a dipole


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Hysteresis (Open Loop PZTs)

Hysteresis curves of an open loop piezo

actuator for various peak voltages

Similar to electromagnetic devices, open loop piezo

actuators exhibit hysteresis (they are also referred to

as ferroelectric actuators). Hysteresis is based on

crystalline polarization effects and molecular friction.

The absolute displacement generated by an open loop

PZT depends on the applied electric field and the

piezo gain which is related to the remanent

polarization. Since the remanent polarization andtherefore the piezo gain is affected by the electric field

applied to the piezo, its deflection depends on whether

it was previously operated at a higher or a lower

voltage (and some other effects). Hysteresis is

typically on the order of 10 to 15 % of the commanded

motion.

Hysteresis can be eliminated by closed loop PZT

actuators.


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The process starts with mixing and ball milling of the raw materials. Next, the

mixture is heated to 75% of the sintering temperature to accelerate reaction of

the components. The polycrystalline, calcinated powder is ball milled again to

increase its reactivity. Granulation with the binder is next to improve

processing properties. After shaping and pressing the (green) ceramics is

heated to 750 C to burn out the binder.

The next phase is sintering at temperatures between 1250 C and 1350 C.

The ceramic block is cut, ground, polished, lapped, etc., to the desired shape

and tolerance. Electrodes are applied by sputtering or screen printing

processes.

The last step is the poling process which takes place in a heated oil bath at

electrical fields up to several kV/mm.

PZT Ceramics Manufacturing

Process


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Multi-layer PZT Manufacturing

ProcessMulti-layer PZT actuators require a different manufacturing process.

After milling a slurry is prepared. A foil casting process allows layer thickness

down to 20 m.

Next, the sheets are screen printed and laminated. A compacting process

increases density of the "green" ceramics and removes air trapped between the

layers.

The final steps are the binder burnout, sintering (co-firing) at temperatures

below 1100 C, end termination and poling.

All processes, especially the heating and sintering cycles must be controlled to

very tight tolerances. The smallest change affects quality and properties of thePZT material..


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Applications of Piezo Actuators

Micro engraving systems

Knife edge control in

extrusion toolsStimulation of vibrations

Piezo hammersHolography

Shock wave generation

Linear drivesMirror positioning

Shock wave generationMicro pumpsLaser tuning

Vibration cancellationAudiophysiological

stimulatiuonNeedle valve actuationAdaptive & active optics

Inspection systemsMicro dispensing

devicesWear correctionFast mirror scanners

MicrolithographyVibration

cancellationCell penetrationTool adjustment

Fiber optic alignment &

switching

Critical dimension

measurementDisk spin standsMicro manipulation

Out-of-roundness

grinding, drilling, tuningAuto focus systems

Wafer and mask

positioningPole tip recessionGene technologyStructural deformationScanning microscopy

Nano-metrologyMR head testingPatch-clamp drivesVibration cancellationImage stabilization

MicroelectronicsDisk DriveLife Science,Medicine, Biology

Precision Mechanics

and Mechanical

Engineering

Optics, Photonics andMeasuring Technology

14


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Applications: Mirror Positioning

Active secondary tip/tilt platform for IRTF

(Mauna Kea, Hawaii) with Hexapod 6-

Degrees-Of-Freedom alignment system.

Mirror diameter: 244 mm

Tip/tilt range: 250 rad

Resonant frequency: 490 Hz

Courtesy PI Polytec

C l f S S 2 R i f S M i l d S (P 2) 15


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Applications: A Micro Scanner Actuated by

PZTs

Source: IEEE

2002 MEMS

Conference

C t l f S t St t 2 R i f S t M t i l d St t (P t 2) 16


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C t l f S t St t 2 R i f S t M t i l d St t (P t 2) 17


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Control of Smart Str ct res 2 Re ie of Smart Materials and Str ct res (Part 2) 18


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Control of Smart Structures 2 Review of Smart Materials and Structures (Part 2) 19


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Control of Smart Structures 2. Review of Smart Materials and Structures (Part 2)





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( )



Applications: A Micro PZT

Speaker

Source: IEEE

2002 MEMS

Conference



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( )





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( )




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3131Department of Mechanical Engineering




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Applications: Vibration Control using PZT Actuator in

3-1 Mode

Polingdirection

1, X

3, z 2, y



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Applications: Active Control of a 11-

Foot Composite I-Beam Using PZT (3-1 mode)

Base to Cantilever the 11-Foot Composite I-Beam

Piezo Patches as Sensors and Actuators

PC with Real-time Control System

Power Amplifier for Piezo Actuators

Poling

direction

1, X

3, z2, y



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Applications: Active Vibration Control

using PZT Stack Actuators (3-3 mode)

LPACT

52

14

BaseBay

35

27

Piezo Stack

Actuator

Force Sensor

Displacement to 90 m

Pushing Forces to 1000 N

Pulling Forces to 50 N

Sub-msec Response

Sub-nm Resolution

Preloaded PZT Stack Actuator (P-841 from PI Polytec)



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Applications: A Micro PZT Stack

Actuator

Source: IEEE2001 MEMS

Conference



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Ultrasonic Piezo Rotary Motor

by Piezo System, Inc

Applications: Ultrasonic Piezo Rotary

MotorHow it works: Voltage excitation,

tuned to the bending resonance of a

piezo/metal ring, produces elastic

bending oscillations which travel alongthe surface of the ring. This travelling

wave induces rotational motion in the

rotor pressed against it. When

excitation is stopped, the shaft is held

in place by friction force. By reversingelectrical excitation, the rotor turns in

the opposite direction.



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Definition of Directions

1

3

2

45

6

Poling axis

along 3 or Z

Because of the anisotropic nature of PZT

ceramics, effects are dependent on direction

(see figure).

To identify directions the axes, termed 1, 2,and 3, are introduced (analogous to X, Y, Z

of the classical right hand orthogonal axial

set).

The axes 4, 5 and 6 identify rotations

(shear).

The direction of polarization (3 axis) is

established during the poling process by a

strong electrical field applied between two

electrodes.

For actuator applications the piezo properties

along the poling axis are most essential

(largest deflection).

z

x

y



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Piezoceramic Coefficients

Primary coefficients:

The primary piezoceramic coefficients are the electromechanical coupling k

the voltage constants g

the strain constants d.

High electromechanical coupling is desired for all applications.

Actuators need high strain constants d.

Sensors need high voltage constants g.


St i ffi i t St i


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Strain coefficients or Strain

constants dijdij: Strain coefficients [m/Vl:

strain developed (m/m) per electric field applied (V/m) or (due to the sensor / actuator properties of

PZT material).

a measure of the strain produced by an applied electric field (The motor Effect )

- OR

Charge output coefficients IC/N]: charge density developed (C/m2) per given stress (N/m2)

a measure of the short circuit charge density to the applied stress (The generator Effect)

d31

Indicates the electrodes areperpendicular to the 3 axes

d33

d15

Indicates electrodes are perpendicular

to the 1 axes

Indicates that the piezoelectrically

induced strain or the applied stress is

shear around the 2 axes

d =Strain

Applied field

Typical units:

Indicates that the piezoelectrically induced

strain or the applied stress is in the 1 direction

Indicates the electrodes areperpendicular to the 3 axes


induced strain or the applied stress

is shear around the 3 direction

=Short ct charge/electrode A

Applied stress

Coulomb/m2

Newton/m2

The first subscript gives the direction of the excitation, the seconddescribes the direction of the system response.



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Voltage coefficients or field output

coefficients

g31

Indicates the electrodes are

perpendicular to the 3 axes

g33

g15




induced strain or the applied stress is

shear around the 2 axes

g =Strain developed

Applied charge

density

Typical units:

Indicates that the piezoelectrically induced

strain or the applied stress is shear around the

1 axes




induced strain or the applied stress

is shear around the 3 axes

=Open ct electric field

Applied stress

V/m

Newton/m2

gij: Voltage coefficients or field output coefficients [Vm/N]: open circuit electric field developed (V/m) per applied mechanical stress (N/m2).

a measure of the electric field produced in the material by an applied mechanical

stress.

or (due to the sensor / actuator properties of PZT material) strain developed (m/m) per

applied charge density (C/m2).

a measure of the strain developed by an applied electric field.

The first subscript gives the direction of the excitation, thesecond describes the direction of the system response.



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Coupling coefficients

K31



Electromechanical coupling

Indicates the stress/strain is in the 1 directionKp

Indicates the electrodes areperpendicular to the 3 axes and

stress/strain is equal in all directions



Kr

Indicates that ceramic is disk shaped,

the electrodes are perpendicular to the 3

axes and the stress/ strain is radial


K2 =

Electrical energy converted tomechanical energy

Input electrical energy

Mechanical energy converted to

electrical energy

Input mechanical energy

kij : Coupling coefficients [no dimensions].

The coefficients are energy ratios describing the conversion from mechanical to electrical

energy or vice versa.

k2 is the ratio of energy stored (mechanical or electrical) to energy (mechanical or electrical)

applied. Quantifies how much energy remains in the material rather than being converted into

another form. Typical values range from 0.40 to 0.70.

The first subscript gives the direction of the excitation, the second

describes the direction of the system response.



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Elastic Compliance Constants Sij

Elastic compliance coefficient is ratio of strain in i-indirection due to stress in the j-

direction, provided there is no change of stress in other two directions.

Direct stresses and strains are denoted by indices from 1 to 3 and shears are denoted by

indices from 4 to 6.

For example, S12 denotes direct strain in direction-1 due to direct stress in direction-2 and

stresses in directions 1 and 3 are unchanged.

In a similar way, S55 denotes shear strain around the axis-2 due to shear stress around the

axis-2.



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If the electric field across the piezoelectric element is held constant, such as the case

with short circuiting the electrodes, the properties are denoted by superscript E.

If the electric charge density is held constant, such as the case with an open circuit at the

electrodes, it is denoted by superscript 'D.'

strainSij= stress

meter/meter per Newton/square meter, square meter/Newton

More about Elastic Compliance

Constants Sij



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Other coefficientsOther important parameters are:

Young's modulus Y (describing the elastic properties of the material)

Because mechanical stressing of the ceramic produces an electrical response which opposes the

resultant strain, the effective Young's Modulus with electrodes short circuited is lower than with the

electrodes open circuited. In addition, the stiffness is different in the 3 direction from that in the 1 or

2 direction. Therefore, in expressing such quantities both direction and electrical conditions must

be specified.

The relative dielectric coefficients (permittivity)

The dielectric constants

The relative dielectric constant is the ratio of the permittivity of the material, to

the permittivity of free space, in the unconstrained condition, i.e., well below the

mechanical resonance of the part.

c (describing the capacitance of the material)



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More Definitions

Definitions:

S = strain (constant if mechanically clamped)

T = stress (constant if not clamped)

E = electric field (constant for short circuit)

D = electrical displacement (constant for open circuit)



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PZT-5H characteristicsMorgan Matroc Inc

~ 5.5 kV/cmDepoling field (DC)

~ 20 kV/cmDielectric breakdown

~ 12 kV/cmPoling field

~ 11,000 psiStatic tensile strength

> 75,000 psiCompressive strength

3400K33

3130K11

193 deg. CCurie point

7500 kg/m3

-8.4510-12 m2/NS13E

-4.7810-12 m2/NS12E

43.510-12

m2

/NS44E

20.710-12 m2/NS33E

16.510-12 m2/NS11E

74110-12 m/Vd15

59310-12 m/Vd33

-27410-12 m/Vd31



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Major PZT manufacturers and PZT name

convention

Sensor Morgan-

Technology Matroc

BM300 - - - - -

BM400 5400 EC-64 PZT-4 Type-I

BM500 5500 EC-65 - PZT-5A Type-II

BM527 5600 EC-70 PZT-5J Type-VBM532 5700 EC-76 - PZT-5H Type-VI

BM740 PZT-7A

BM800 5800 EC-69 - PZT-8 Type-III

BM900 K81

BM920 - - K83 - -BM940 K85

Channel EDO Keramos Navy



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Different PZTs and Applications

- BM300 BM400 BM500 BM527 BM532 BM740 BM800 BM900 BM920 BM940

Sonar Projectors - X - - - -- X - - -

Hydrophones X X X X X - X - X -

Depth Sounders - X X - - X X - - -

Communication - X X X X X X - - -

Sonobuoys X - X X X - - - X -

Linear Arrays X X X X X - - - X -

Level, Flow - X X X X X X - - -

Flaw Detection (NDE) X X X - X X X X X X

Thickness - X X - - X X X X X

Accelerometers - X X X X X X X - X

Actuators - X X - X - X - - -

NDT - - - - - X - X X X

Transducers X X X - X X X X X -

Sterilizers - X - - - - X - - -

Cleaners - X - - - - X - - -

Degreasers - X - - - - X - - -

Welders - - - - - - - - - -

Alarms - X - - - - X - - -

- BM300 BM400 BM500 BM527 BM532 BM740 BM800 BM900 BM920 BM940

Underwater

Measurement

Medical

Industrial



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Electric behavior of an unstressed medium under the influence of an electric

field Eis

D = E

where is the permittivity.

The mechanical behavior (forE= 0) is

S = s T

where s is the compliance.

Basic Relationships



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TdED

dETsS

'T

E

+=

+=

S = Strain (6x1)

T= Stress (6x1)

E= Electric field (3x1)

SE = Compliance (zero field) (6x6)

d= Piezoelectric coefficient (6x3)

D = Electric displacement (3x1)

T= Dielectric constant (zero stress) (3x3)

Input Voltage

Micro-strain

0

Constitutive Equation



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More about PZT Properties

PZT materials exhibit most of the characteristics of ceramics, namely a high

elastic modulus, brittleness and low tensile strength.

The material itself is mechanically isotropic, and by virtue of the poling

process, is assumed transversely isotropic in the plane normal to the poling

direction as far as piezoelectric properties are concerned.

This means that for PZT materials,

s11 = s22, s13 = s23, s44 = s55

d31 = d32 and d15 = d24 .



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Constitutive Equation

Poling

direction

1, X

3, z2, y

1 11 12 13 31

2 12 22 13 31

3 13 13 33 33

23 44 15

31 44 15

12 66

1 15 11

2 15 11

3 31 31 33 33

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0

S S S d

S S S d

S S S d

S d

S dS

D d

D d

D d d d

=

1

2

3

23

31

12

1

2

3

E

EdD

Eds '

+=

+=



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Piezo Sensor Equation

=

=

12

31

23

3

2

1

6

5

4

3

2

1

The stress vector is written as

=

6

5

4

3

2

1

333231

24

15

3

2

1

000ddd00d000

0d0000

DD

D

This equation summarizes the principle of operation of piezoelectric

sensors. A stress field causes an electric displacement to be generated as a

result of the direct piezoelectric effect.

Note that shear stress in the 1-2 plane, 6 is not capable of generating any

electric response.


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Piezo Actuator Equation

11 12 131 1 31

12 11 132 2 31

13 13 333 3 33

4423 23 15

4431 31 15

6612 12

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

S S S d

S S S d

S S S d

S d

S d

S

= +

1

2

1

3

2

3

0

0

0

E

E TE

+

thermal coefficients of expansion, 1/K

temperature change, K

3 2

1

In a plane perpendicular to the piezo-polarization, it has isotropic

properties, i.e., transversely isotropic material in the plane 1-2. For

orthotropic material, there is no temperature shear strain. However, there

is a shear strain induced due to electrical field El and E2.



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Free StrainsFor piezoceramic materials:

Actuation Strain

=

3

2

1

15

15

33

31

31

E

EE

000

00d

0d0

d00

d00

d00

These are called free strains

=

0

Ed

Ed

EdEd

Ed

115

215

331

331

331

6

5

4

3

2

1

where d33, d3l, and dl5 are piezoelectric strain

coefficients.



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PZT Actuator: Longitudinal (d33 mode)

Motor

d33 mode

d31 mode

Poling

direction

1, X

3, z2, y

PZT Actuator: Transverse (d31 mode) Motor



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PZT Actuator: Double-layer

Extension Mode

PZT Actuator: Double-layer Bending Mode



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PZT Actuator: Multi-layer



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Lateral (31) Mode Piezoelectric

Sensor

l

t

w

Structure

1

2

3

P

V1

+

Top & Bottom Surfaces

Electroplated

Principle: Bond or embed a Piezo wafer to a surface

and monitor electric potential on electrodes. Piezo ispoled perpendicular to plated surfaces

Piezoelectric sensors produce a charge when strainedlaterally.



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Axial Force Sensor

tp1

2

3

Polarity

V

F

Principle: Sense dynamic (AC) load in the poling direction.Sensor is placed in the series load path. If a parallel load path

exists, make sensor as possible to attract a majority of the load.



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PZT Shear Mode Sensor

p p

Strain

P

P

V1

+

V2

+

Strain Direction

Electroplated

Top & Bottom

Part of Structure

Being Monitored

Bridge

3

2

1

Principle: Two PZTs poled

transverse to their electrodes are

bonded to a surface and connected

with a bridge. Longitudinal strain

induces shear in the PZTs and a

Voltage through the d15 coefficient.

Kistler Instrument A.G. of Switzerland has a patent on the basic idea.

A Similar approach (shear mode) is employed in most accelerometers and load

cells since there is much lower pyroelectric sensitivity and hence better

coherence at low frequency.



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A Specific Example

Voltage applied in the 3-direction and the Deflection Out is a 3D effect

For an element of thickness t, length L and width w, an applied

voltage Vproduces the following change in shape:

Vdtt

LVdL

t

wVdw 33

3131 ,, ===


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Voltage-Force Relationship

ELV = AFT=

TgE 33=

C

FdF

A

LgV 3333 ==

25 N generates about 100 volts, about the force generated from

squeezing your hand

and

C - Capacitance



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Displacement-Voltage Relation

===L

LLVdEdS 3333

VdL 33=

16kV or E = 800 V/m yields L = 10 micro meters.

All the above is static analysis



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Finite Element Modeling of Piezoceramic

Smart Structures

Euler-Bernoulli Model

No axial load

Shear deformation can be neglected

Uniform along its length

Predicts the correct curvature of actuator and base over the entire

contact area

Considers the actuator as a layer Treating bimorphs, the correct neutral axis is used which is the

neutral axis of the layered system



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tp

z

x

h

tp

wp

1

W1W(z) W2

y

x

2

yz

Poling

direction 1, X

3, z2, y

The Piezo Element Rigidly Bonded to a

Structure

wp

h


Z (W)


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X (u)

Z (W)

We have , where u is the displacement in x direction.

From the deformed configuration shown in the upper figure, we can get

Then,

dx

dux =

dx

dW

dx

dWz

dxdWzu =

2

2

dx

Wdz

dx

dux ==



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In terms of nodal displacements, W1, W2, 1 and 2 the bendingdisplacement is given by

2

32

2

32

1

32

1

32

23

2231)(

+

+

+

+

+

+

=

hh

x

h

xW

h

x

h

x

hh

x

h

x

h

xW

h

x

h

xxW

(1)

Let q1 = W1, q2 = 1, q3 = W2, q4 = 2, then

)()()(4

1

tqxxW ii

i=

=

(2)

(3)



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where

hh

x

h

xx

h

x

h

xx

hh

x

h

x

h

xx

h

x

h

xx

+

=

+

=

+

=

+

=

32

4

32

3

32

2

32

1

)(

23)(

2)(

231)(

(4)



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The governing equations of motion for smart structures are

based upon the electromechanical coupling effect. The

piezoceramic material shown in the figure satisfy the following

equation:

=

1

3

1131

313

1

3

T

E

Sd

d

S

DE

T

(5)

Where D3 represents the electric displacement, charge per

unit area, E3 represents the applied field intensity, S1represents the strain, T1 is the stress, is the permitivity of

the piezoelectric material, d31 is the piezoelectric charge

coefficient and is the elastic constant for the

piezoelectric material.

T

3

p

E ES /111

=

Poling

direction 1, X

3, z2, y


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+

=

h tT

p

p

dxdzS

E

T

DW

0 1

3

1

3

10

01

2

1(9)

The strain S1 can be written as

2

2

1x

WzS x

== (10)

Substituting Eq. (6) into Eq. (9), we get

+

+

+=

=

ht

xpxpp

T

p

h t

xpp

pp

TT

x

pp

p

p

dxdzEEEdEEdW

dxdzE

EEd

EdEdEWU

0

2

331

2

3

2

313

0

3

31

31

2

3133

]2)[(2

1

2

1

(11)



83/98



By using the expression in Eq. (10), we get

+

+=

h t

ppp

T

pp

p

dxdz

x

WzE

x

WzEEdEEdWU

0

2

2

22

2

2

331

2

3

2

313 ]2)[(

2

1

(12)

Substituting W(x) from Eq. (3), we get

qkqbeqeU pTT

p2

1

2

1 2 = (13)



84/98



where

( )

dxdx

d

dx

dttEtWk

dxdx

dtWEdb

Ete

Edt

hW

j

h

ip

ppppijp

h

ip

ppi

p

p

T

p

p

2

2

02

22

2

0

2

2

31

3

2

313

3][

2

++=

+=

=

=

(14)

By substituting i from Eq. (4), we get

+=

+=

==

2

2

0

314

312

31

p

pp

p

pp

t

WEdb

tWEdb

bb

(15)



85/98



Stiffness Matrices:

Define

++=

3

2

2 p

pppp

ttEtWk

23424

223214

313212

44333

2

2

22

33

2

2

11

6,

2

6,

6

12,

6

4,

12

44

3

12123

h

kk

h

kk

h

kk

h

kk

h

kk

h

kk

h

kk

h

kk

h

k

h

ttEtWk

hk

httEtWk

p

pppp

p

pppp

==

==

==

==

=

++=

=

++=

(16)



86/98



By including the beam element for the elastic energy, we can

rewrite Eq. (13) as

kqqBeqeU TT 2

1

2

12

=

where

T

pb bbbbBkkk ],,,[; 4321=+=

(17)

(18)

where

kb = stiffness matrix for the structure

kp = stiffness matrix for the piezoceramic member



87/98



The kinetic energy of piezoceramic can be written as

dxwTh

pp

2

02

1

= (19)

where p is mass per unit length for the piezoceramics. Thekinetic energy for the piezoceramic element can be written as

= qMqT pT

p2

1

where

[ ] =h

jipijpdxxxm

0

)()( (20)



88/98



The total kinetic energy is then given by

= qMqTT

2

1

where M = Mb + Mp

Mb = mass matrix for beamMp = mass matrix for piezoceramic

The Lagrangian function, L , is given by

L = T U

kqqBeqeqMq TTp

T

2

1

2

1

2

1 2 +=

The Lagrangian equation is

0=

kk

q

L

q

L

dt

d

(21)

(22)

(23)



89/98



Substituting Eq. (22) into Eq. (23) and using q as generalized

coordinates, we get

aBeqkqm =+

][][where

][][][

][][][

pb

pb

kkk

mmm

+=

+=

and ea is applied voltage.

Eq. (24) represents the equation for the actuator.

(24)



90/98



Taking q as a generalized coordinate, the equation for

sensor voltage output in terms of q is

qBe Ts

= (25)

Eq. (25) represents the voltage output from a piezoceramicsensor. Up to now, we have considered only an element.

The equation for the global form is determined by combining

the equations.



91/98



Example

Let us consider a system as shown in the following figure.

1 2 3 4

Actuator Sensor

11,W

22,W 33 ,W 44 ,W 55 ,W

The system consists of four elements. Elements 1 and 4 arebeam elements. Element 2 is a piezoceramic beam element and

is used as an actuator. Element 3 is a piezoceramic/beam

element and is used as a sensor.



92/98



Let us assume that

=

=

1

44

1

43

1

42

1

41

1

34

1

33

1

32

1

31

1

24

1

23

1

22

1

21

1

14

1

13

1

12

1

11

1

1

44

1

43

1

42

1

41

1

34

1

33

1

32

1

31

1

24

1

23

1

22

1

21

1

14

1

13

1

12

1

11

1

][

][

mmmm

mmmm

mmmm

mmmm

m

kkkk

kkkk

kkkk

kkkk

k

Beam element 1



93/98



=

=

aaaa

aaaa

aaaa

aaaa

a

aaaa

aaaa

aaaa

aaaa

a

mmmm

mmmm

mmmm

mmmm

m

kkkk

kkkk

kkkk

kkkk

k

44434241

34333231

24232221

14131211

44434241

34333231

24232221

14131211

][

][

Actuator element ( beam + piezoceramic)



94/98



=

=

ssss

ssss

ssss

ssss

s

assss

ssss

ssss

ssss

s

mmmm

mmmm

mmmm

mmmm

m

kkkk

kkkk

kkkk

kkkk

k

44434241

34333231

24232221

14131211

44434241

34333231

24232221

14131211

][

][

Sensor element ( beam + piezoceramic)



95/98



=

=

444

443

442

441

4

34

4

33

4

32

4

31

4

24

4

23

4

22

4

21

4

14

4

13

4

12

4

11

4

4

44

4

43

4

42

4

41

4

34

4

33

4

32

4

31

4

24

4

23

4

22

4

21

4

14

4

13

4

12

4

11

4

][

][

mmmm

mmmm

mmmmmmmm

m

kkkk

kkkk

kkkk

kkkk

k

Beam element 4



96/98



++

++

++++

++

++

5

5

4

4

3

3

2

2

4

44

4

43

4

42

4

41

4

34

4

33

4

32

4

31

4

24

4

23

4

2244

4

21434241

4

14

4

13

4

1234

4

11333231

242322

2

4422434241

1413123411333231

242322

1

4421

1

43

141312

1

3422

1

33

W

W

W

W

mmmm

mmmm

mmmmmmmm

mmmmmmmm

mmmmmmmmmmmmmmmm

mmmmmm

mmmmmm

ssss

ssss

ssssaaa

sssasaaa

aaaa

aaaa



97/98



=

++

++

++++

++

++

+

0

0

0

02 4

3

2

1

5

5

4

4

3

3

2

2

4

44

4

43

4

42

4

41

4

34

4

33

4

32

4

31

4

24

4

23

4

2244

4

21434241

4

14

4

13

4

1234

4

11333231

242322

2

4422434241

1413123411333231

242322

1

4421

1

43

141312

1

3422

1

33

a

a

a

a

a

ssss

ssss

ssssaaa

sssasaaa

aaaa

aaaa

bb

b

b

e

W

W

W

W

kkkk

kkkk

kkkkkkkk

kkkkkkkk

kkkkkkkkkkkkkkkk

kkkkkk

kkkkkk

(26)



98/98

and

=

4

4

3

3

4321][

1

W

W

bbbbe sssss

(27)

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2.Reveiw.smart Part2