Piezoelectrics as Multifunctional Electromechanical Materials for Space Applications

Stewart Sherrit

Jet Propulsion Laboratory, California Institute of TechnologyPasadena, CA

3545 Lunchtime SeminarMonday, September 8, 2003

AcknowledgementThe NDEAA Lab team consisting of:• Dr. Yoseph Bar-Cohen, Senior Research Scientist and

Group Leader• Dr. Xiaoqi Bao• Dr. Zensheu Chang• Dr. Mike Lih

Outline

• Describe the various actuation and sensing mechanisms under development in the NDEAA Lab at JPL

• Review a variety of Electroactive Material Dielectric and StrainResponse

•Present the various groups of piezoelectric materials(Ceramic, Single Crystal, Polymer) and typical size of effects

• Review the Macroscopic description of Piezoelectricity and look a the limitations. (Nonlinear, Dispersion, Loss)

• Present the techniques used by transducer designers toamplify the limited strain (Cantilevers, Resonance, Stacking)

• Present the sensor examples (Quartz Crystal Microbalance QCM,Surface Acoustic Wave SAW Devices)

Anti-Ferroelectricity•Low fields - counteracting domains stable•Polarization curve opens up at large externalelectric field as domains switch

D vs E S vs E

Electroactive Response

Ferroelectricity•Presence of a spontaneous polarization •Polarization can be switched by anexternal electric field.

Electric Field E (MV/m)

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Stra

in S

(m/m

)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

EC

D vs E S vs E

Common applications include ferroelectric RAM, electro-optic devices and displays

Electroactive Response

Electrostriction Electrostriction is the name given to the quadratic relationship

between the strain S and the electric displacement D present in most materials. The most fundamental form of this relationship is

kjijkij DDQS =

.......+++= nmlkjijklmnlkjijkljiji EEEEEEEEED εεε

2kijkij ES λ≈

Applications Actuators, sonar projectors, ultrasonic transducers

Electroactive Response

D is electric displacement

E is field

Ds is the Saturation Electric Displacement

±(E1 – Ec) is the field at which the anti ferroelectric becomes stable

±(E1 + Ec) is the field at which the domains switch to parallel orientation.

k is saturation coefficient (k large switching occurs abruptly k = µ/KT)

normalized first time derivative of the electric field

normalized second derivative of the electric field wrt time

ε is a term to account for linear saturation

DDs2

tanh k E E1 R2⋅+ Ec R1⋅−( )⋅[ ]⋅Ds2

tanh k E E1 R2⋅− Ec R1⋅−( )⋅[ ]⋅+ ε E⋅+⎡⎢⎣

⎤⎥⎦

:=

dtdEdtdER

//1 =

22

22

//2dtEddtEdR =

Semi Empirical Modeling (Ising Spin model)Start with antiferroelectric

DDs2

tanh k E E1 R2⋅+ Ec R1⋅−( )⋅[ ]⋅Ds2

tanh k E E1 R2⋅− Ec R1⋅−( )⋅[ ]⋅+ ε E⋅+⎡⎢⎣

⎤⎥⎦

:=

2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4

0.2

0

0.2

0.40.4

0.4−

D

2 106×2− 106× E

E1 = 0 Ferroelectric

2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4

0.2

0

0.2

0.40.4

0.4−

D

2 106×2− 106× E

2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4

0.2

0

0.2

0.40.4

0.4−

D

2 106×2− 106× E

Ec small Electrostrictive

For all the above S = QD2

What about piezoelectrics?

All constants Antiferroelectric

2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4

0.2

0

0.2

0.40.4

0.4−

D

2 106×2− 106× E

S = Q(D + Dr)2

= (2QDr)D + QD2 + QDr2

= gD + nonlinear term + remanent strain

Piezoelectric coefficient in the poled materials is due to coupling of the electrostrictive coefficient with the remenant electric displacement.

Dr

S s T d E

D E d Tp pq

Eq pm m

m mnT

n pm p

= +

= +ε

PiezoelectricityCoupled “Linear” Relationship between the strain S, electric displacement D and the electric field E and the stress T

Derived from thermodynamic potentials

•Superscripts designate boundary conditions•Subscripts designate anisotropy

•Notation: IEEE Standards on Piezoelectricity

Elastic Response – Strain S, Stress TS = sT s is compliance

T

S= ∆x/l

T = F/A

l

∆xslope = s

Dielectric Response – Field E, Electric Displacement DD = ε0E+P = εE ε is permittivity

E =V/l

D

D = Q/A

l

slope = ε

V

+ + + + +

- - - - -

Direct Piezoelectric Response – Electric Displacement D, Stress T

D = dT d is piezoelectric charge coefficient.

T

T = F/A

slope = d

Converse Piezoelectric Response – Field E, Strain SS = dE d is piezoelectric charge coeff.

E =V/l l

slope = d

V

+ + + + +

- - - - -

I

∫ ∂= tIA

D 1

S= ∆x/l ∆x

Coupling constant

lV

∆x

versaor vice Density Energy ElectricalInput

DensityEnergy MechanialOutput 2

22

EsTkε

==

Energy conversion per cycle

sdkε

22 ∝

( )G s T T g D T D DijklD

ij kl nij n ij mnT

m n112

122= − + + β

Sample under isothermal and adiabatic conditions and ignoring higher order effects the elastic Gibbs function may be described by

The linear equations of piezoelectricity for this potential are determined from the derivative of G1 and are

SGT

s T g D

EGD

D g T

ijij

ijklD

kl nij n

mm

mnT

n nij ij

= − = +

= = −

∂∂

∂∂

1

1 β

Thermodynamic Derivation

S s T g D

E D g Tp pq

Dq pm m

m mnT

m pm p

= +

= −β

S s T d E

D E d Tp pq

Eq pm m

m mnT

n pm p

= +

= +ε

T c S e E

D E e Sp pq

Eq pm m

m mnS

n pm p

= −

= +ε

T c S h D

E D h Sp pq

Dq pm m

m mnS

n pm p

= −

= −β

Other Forms of Linear Piezoelectricity

Reduced Matrix Form

ET

x

xd

xd

xs

DS

T

E

•=)33(

)36(

)63(

)66(

ε

T

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

•

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Τ33

Τ32

Τ31

Τ23

Τ22

Τ21

Τ13

Τ12

Τ11

3

2

1

6

5

4

3

2

1

363534333231

262524232221

161514131211

636261666564636261

535251565554535251

434241464544434241

333231363534333231

232221262524232221

131211161514131211

3

2

1

6

5

4

3

2

1

EEETTTTTT

dddddddddddddddddd

dddssssssdddssssssdddssssssdddssssssdddssssssdddssssss

DDDSSSSSS

EEEEEE

EEEEEE

EEEEEE

EEEEEE

EEEEEE

EEEEEE

εεεεεεεεε

PZT has C∞ symmetry.Therefore in ideal case only 10independent constants are required to describe the response.

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

•

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000000

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Τ33

Τ11

Τ11

3

2

1

6

5

4

3

2

1

331313

15

15

1211

1555

1555

33331313

13131112

13131211

3

2

1

6

5

4

3

2

1

0000000000000

000)(20000000000000000000

000000000000000

EEETTTTTT

dddd

dss

dsds

dsssdsssdsss

DDDSSSSSS

EE

E

E

EEE

EEE

EEE

εε

ε

Real or Actual Piezoelectrics

( ) ( )tTTEistTTEss ijiEmniiji

Emnr

Emn ,,,,,,,, ωω +=

( ) ( )tTTEidtTTEdd ijimniijimnrmn ,,,,,,,, ωω +=

( ) ( )tTTEitTTE ijiTmniiji

Tmnr

Τmn ,,,,,,,, ωεωεε +=

LossDispersion

Non-LinearitiesTemperature Dependence

AgingALL May be Significant

Linear Loss

where x and y can be the field E, Displacement D, stress T or strain S

x =x0sin(ω0 t)•b real then y and x are in phase.•b complex then y and x are out of phase and depending on the variables a energy loss is present .

[ ])t +(ωy=|b| x δ00 sin

y = bx

• Material Loss - First order correction • Rayleigh’s law Y= (a0 +αXm)X ± α/2(Xm

2 - X2) where the response increases linearly with maximum field/stress and the remanence increases with the square of the field/stress

S,T,D,E

-1.0 -0.5 0.0 0.5 1.0

S,T,D

,E

-0.9

-0.4

0.0

0.4

0.9

Rayleigh Behavior (1st order)

DispersionCausal (response follows input)•Passivity (Can only dissipate) Example PVDF-TRFE

( ) ( )" ' 333333 ωεωεε SSS i+=

( ) ( )" ' ωω ttt ikkk +=

tk

( ) ( )" ' 333333 ωω DDD iccc +=

Non Linearities

• First order corrections – Rayleigh•Higher order expand the elastic Gibbsenergy in field and stress

• Definition of how material constants are measured1. Average response (∆Y/∆X)2. Differential response (dY/dX)3. Biased small signal differential response(∂Y/∂X at X0)

D, S

EEBIAS

Under a large low frequency field domainstructure changes irreversibly

Under a small high frequency field domain walls vibrate reversibly

Piezoelectric Properties - Ceramics

TABLE 5.2-1: Piezoelectric coefficients and permittivity at three temperatures for three typical commercial PZT materials [Based on: Hooker, 1998]. Performance Standard set out by Navy eg, Navy I , Navy II, etc

Property PZT 4 PZT 5A PZT 5H*

-150oC

25oC 250o

C-

150oC25oC 250o

C-

150o

C

25oC 180o

C

d33 (pC/N) 210 225 360 190 350 440 260 585 900

d31 (pC/N) -70 -85 -130 -100 -190 -270 -115 -265 -500

ε (10-9 F/m)

7.1 9.7 27 5.3 10 27 7.5 28 124

*maximum in dielectric constant is at 180oC

Piezoelectric Properties - Polymers

d31 = 28 pC/N, d32 =4 pC/N, d33 =-35 pC/N

PVDF

Relative Permittivity =10-15Mechanical Q = 10Coupling up to 0.15

d31 = 10.7 pC/N, d32 =10.1 pC/N, d33 =-33.5 pC/N

Relative Permittivity =5-10Mechanical Q = 10Coupling up to 0.25

PVDF/TrFE 75/25 Copolymer

Table 2: The material constant for the length extensional resonance mode, at roomtemperature, sample 2.

Material constant sD

33 (m2/V) Re × 10-11 1.73

Im × 10-13 -1.76

sE

33 (m2/V) Re × 10-10

Im × 10-12

1.30 -2.17

d33 (C/N)

Re × 10-9

Im × 10-11

1.97 -3.32

εT

33 (F/m) Re × 10-8

Im × 10-10

3.42 -5.48

k33

Re Im x 10-4

0.93 -5.06

Res

ista

nce

(Ohm

s)

1000.0

10000.0

100000.0

1000000.0

10000000.0

100000000.0

Frequency (kHz)

0 500 1000 1500 2000 2500

Rea

ctan

ce (O

hms)

-2.0e+7

-1.0e+7

0.0e+0

1.0e+7

2.0e+7

Le mode PZN-PT 8%

E l e c t r i c F ie l d ( M V / m )

- 2 . 0 - 1 . 0 0 . 0 1 . 0 2 . 0

Elec

tric

Dis

plac

emen

t (C

/m2 )

- 0 . 4- 0 . 3- 0 . 2- 0 . 1

0 . 00 . 10 . 20 . 30 . 4

E l e c t r i c F i e l d ( M V / m )

- 2 . 0 - 1 . 0 0 . 0 1 . 0 2 . 0

Stra

in (µ

m/m

)

- 1 0 0 0

0

1 0 0 0

2 0 0 0

3 0 0 0

4 0 0 0

E le c t r i c F i e l d ( M V / m )

0 . 0 1 . 0 2 . 0

Elec

tric

Dis

plac

emen

t (C

/m2 )

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

0 . 2 5

E l e c t r i c F i e l d ( M V / m )

0 . 0 1 . 0 2 . 0

Stra

in (µ

m/m

)

- 5 0 00

5 0 01 0 0 01 5 0 02 0 0 02 5 0 03 0 0 03 5 0 04 0 0 0

High field studies PZN-PT 4.5%Bipolar Excitation Unipolar Excitation

Piezoelectric Properties – Single Crystals

Piezoelectric Properties – Single Crystals

AT Cut Quartz

SεDc

kh

5.46x10-11(1 - 0.024i)2.96x1010(1 + 0.00012i)

0.072(1 - 0.0088i)1.67x109(1 + 0.021 i)

Density = 2650 kg/m3

Q’s up to 100,000106 in vacuum

Temperature stability extremely high about RTWhich is the reason they are used timing

Limitation of Piezoelectrics

Consider Motorola 3203 HD PZTwhich has a fairly large d33 constant

S = d33E T = e33E

For E = 105 V/md33 = 6x10-10 m/V e33 = 25 C/m2

S = 6x10-5T = 2.5 MPa

Q. What do you do now?A. Design around this. → Resonators or

other Strain Amplification techniques

Stacks and Bimorphs

3333 ndd eff =L

AnCT33

2ε= 2

2313tVLdy =∆

2

231

23

tVLdy =∆

Uni-morphs

Example – Clemson’s RAINBOW- NASA’s THUNDER

Flextensional Devices

Cantilevers

Resonator Example Thickness Extensional

T c S h D3 33 3 33 3= −D

E h S D3 33 3 33 3= − + β s

∂∂ ρ

∂∂

23

233

23

32

ut

ux

D

=c

Wave Equation

Linear EquationsOf Piezoelectricity

Also S = du/dx and

B.C. T3 = 0 at x3 = 0T3 = 0 at x3 = t

vc

D

D

= 33

ρ

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

D

D

D

DD

xt

tx

vv

vvc

DhS 3333

33

33

sinsin

1coscos ω

ω

ωω

( ) 30

33333330

3 dxdxt

st

∫∫ −=−= DShEV ε

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−−=

D

DD

Ds

t

t

l

v

vc

vDhDV 3

ω

ω

ωsin

2cos2

33

233

333ε

ID

D= = −Ad

d ti A3

3ω

Ai

ttD

DDs

ω

ωω ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

=vc

vh

Z2

tan233

233

33ε

Strain from Wave Equationand Boundary Conditions

Calculate Voltage

Calculate Current

Finally Determinethe complex ImpedanceVERY IMPORTANT

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

p

p

f

fk

Z

4

4tan

1

2

33ω

ω

ω

t

SAitε

SSD

pDSt l 3333

3333

2332 1 24

εβ

β===

vc

hk f

Simple Measurement Method to determine the ElasticDielectric and Piezoelectric Constants and loss by measuring the complex impedance as a function of frequency

Impedance Analyzer

Sample

Common Modes of Sensing

Select crystal properties, axis to monitor ∆f due tospecific environmental variable, Temperature, Pressure, Field, Viscosity or Mass change due to thermal or electrochemical deposition.

Select interface layer axis to monitor ∆f due tospecific reactant, adsorbent in enviroment

Integrate heater/cooler for thermal adsorption-desorption curves.

Quartz shear resonator examples

Network RepresentationSolution to the wave equationwith open boundary conditions

Current/Voltage←Transformer→Force/Velocity

-C0

C0

1

N

ZTZT

ZS

tASε

=0Ch0CN =

DD AA cv ρρ ==0Z

Dcρωω

== DvΓ

( )2/tan0 tiZ T ΓZ=

( )tiZ S Γcsc0Z−=

Mason’s Model

Acoustic Layer NetworksSolution to the wave equation with open boundary conditions

Force = VoltageVelocity = Current

Layer ParametersL thicknessA areav velocityρ density

Frequency (kHz)

0 500 1000 1500 2000

log|

Vel

ocity

| (1

m/s

)

-6

-5

-4

-3

-2

-1

0

1

2

Rigidly Fixed Free

= F0 / mω

= F0 ωL/c33A

Combination of Piezoelectric and 1 layer

Acoustic Load Regimes

Depending on the acoustic length in the layer compared to the acoustic length of the piezoelectric. Three distinct loadingregimes can load the piezoelectric differently. These include:

• Mass loading•Elastic loading•Radiation loading

Mass Loading

ωωρωρ imLAiLAil

llll

lllL =⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

vv

vvZ tan

Elastic Loading

⎟⎟⎠

⎞⎜⎜⎝

⎛=

llllL

LAiv

vZ ωρ tan

Radiation Loading

( ) lllllll

lllL AiAiLAi vvv

vZ ρρωρ =−=⎟⎟⎠

⎞⎜⎜⎝

⎛= tan

The constants (including loss) of the piezoelectriccan be determine using curve fitting techniques

prior to the addition of the layer

Known prior to deposition by analysis of free resonator

⎟⎟⎠

⎞⎜⎜⎝

⎛=

llllL

LAiv

vZ ωρ tan

( )TSA

SATTL ZZZ

ZZZZZ

−−−−

=22

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

C

CA

ZZ

NZZ

1

2

Since ZC, ZT, ZS, N can be determined fromthe free spectra prior to deposition or attachment of the layer then we can determine the acousticload directly for the layer using

Where Z is the impedance spectra of the compositeresonator

Transform applied to mass loading - QCM

ZL

m* =ZL/iω

( ) ( )( )iff

iffiffmf

ss

ssssS

11

221102

* )(∆+

∆+−∆+=m

Complex version of Sauerbrey equation

m* theory= (8.48-1.02x10-5i) mg

m* transform = (8.48-1.03x10-5i) mg

Error Propagation in Transform

Add 0.1 % Error to the composite resonator Impedance

Error does not propagateinto fs

Error Propagation in Transform – Cont.

Add Stray capacitance to the composite resonator Impedance

Again slope is

constant about

fs

Transform applied to water load

SεDc

kh

AT Cut Quartz Resonator in air

(F/m)(N/m2)

(#)(V/m)

ρ (kg/m3)

5.45933x10-11(1 - 0.02388i)2.94472x1010(1 + 0.00012i)

0.0717744(1 - 0.00884i)1.67x109(1 + 0.02084 i)

2650t (m) 0.000333

A (m2) 4.195x10-5

m0 (kg) 3.701x10-5

fsq (Hz) (quartz) 4.99477x106 + 110.9i

fs w(Hz) (quartz + water) 4.99390x106 + 990.8i

∆m (kg) (Sauerbrey eq.) 6.45x10-9-6.52x10-9i

∆m (kg) (m*(Re(fSq )) Figure 13.

6.44x10-9-6.03x10-9i

( ) ( ) ωηωρρρωρ *2/1tan imiAAiAiLAi llllllll

lllL ===−=⎟⎟⎠

⎞⎜⎜⎝

⎛= vv

vvZ

Water Load is special case of Radiation Loading

m*= 6.44x10-9- 6.03x10-9i kg with ω= 3.138x107 rads/s, A= 4.195x10-5 m2, ρ=1000 kg/m3

From previous table

η=1.38(1+0.066i) x10-3 Ns/m2

Book Value of η=1.12 x10-3 Ns/m2 at 60 oF

complex part of η determined above is a measure of the experimental error (no correction for parasitic electrical component), surface roughness anddeviations from pure Newtonian fluid

Curing/evaporation studies

Combination of Piezoelectric and 1 layer

Zp

Also present is a parasiticelectrical impedance that shiftsthe anti-resonance frequency fa

Free SAW Resonance Equation

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛++⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

000

20

2

00

20 2sin

2tan4

2tan

2sin

2tan

22

ωπω

ωπω

ωπω

πω

ωωπω

ωπω

πω

ω NNKci

NciNKc

Y ss

s

K = coupling factorω0 = resonance frequency ∝ Rayleigh velocitycs = capacitance of IDT pairN = Number of electrode Pairs

SAW Resonator with reflectors (Simplest model BVD circuit)

L1

R1

C1

C0

L1, C1, R1 = Motional componentsC0 = Clamped capacitance

Saw Example

SAW Example (55.2 MHz)

Q = 22000

SAW Example (62.2 MHz)

Q = 27000

SAW Example (49.6 MHz) : Poor Quality resonance

Q = 1000

SAW Stability (55.2 MHz)

90 Hz over 18 hrs = 1.7 ppm

4:30 pm

10:30 am

Example of SAW Sensor - Dust

Clean SAW Resonator

SAW Resonator withLimestone Dust

Passive Saw Interrogating SensorsTable 1: SAW Properties of Common Materials (Yili Wu et al. “Principals of Surface Acoustic Waves and its application in Electronic Technology” Defense Industrial Press, PRC, 1983, (In Chinese)

Material SAW velocity v (m/s)

Coupling K2(%)

ε ( pF/m) αT(ppm/oC)

Quartz ST-X 3158 0.16 55 ≈0

LiNbO3 (YZ) 3485 4.5 460 91

LiNbO3 (128o) 3921 5.7 - 57

ZnO 2715 1 - 40

Bi12GeO20(100) (011)

1681 1.5 400 130

Correction Techniques for cross sensitivities

σσ

γγ

ββ

σγ∆

+∆

+∆

+∆

+∆

=∆

≈∆−

=∆ kk

TTk

cck

mmk

ff

vv

Tcm

v= velocityβ= propagation factorm = massc = elastic stiffnessT = Temperatureγ = Surface tensionσ = film stress

The k factors above are material parameters so chose 5 sensors with different materials, cuts, or frequencies to get cross sensitivity matrix ( 1-5 designate specific sensor) (k normalized by dividing through by initial condition) (Hietala et al. IEEE Trans UFFC 48, pp. 262-267)

[ ] [ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

⇒∆=∆⇒

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

−

5

4

3

2

11

55555

44444

33333

22222

11111

55555

44444

33333

22222

11111

5

4

3

2

1

fffff

kkkkkkkkkkkkkkkkkkkkkkkkk

Tcm

xkfTcm

kkkkkkkkkkkkkkkkkkkkkkkkk

fffff

Tcm

Tcm

Tcm

Tcm

Tcm

i

Tcm

Tcm

Tcm

Tcm

Tcm

σγ

σγ

σγ

σγ

σγ

σγ

σγ

σγ

σγ

σγ

σγ

σγ

jjii Xkf ,=∆

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆

∆∆∆

nnnnnn

ijiii

nj

nj

nj

n X

XXX

kkkkkkkk

kkkkkkkkkkkkkkk

f

fff

M

K

MM3

2

1

321

321

33333231

22232221

11131211

3

2

1

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆

∆∆∆

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡−

nnnnnn

ijiii

nj

nj

nj

n f

fff

kkkkkkkk

kkkkkkkkkkkkkkk

X

XXX

M

K

MM3

2

11

321

321

33333231

22232221

11131211

3

2

1

⇒

General Solution

Example

Passive Saw Interrogating Sensors

Transmitter/Receiver

Antennae

SAW

Reflector

eg. Bao et al. IEEE Ultrasonics 1987, pp. 583-585

Phase difference in reflected pulse is a measure of the strain due to thermal expansion

USDC Technology

Ultrasonic Actuator(Horn/Stack/Backing)

Free Mass

Drill StemRock

Device is drivenat ultrasonic frequenciesbut producessonic and ultrasonic Impacts.

Various Applications of the USDC TechnologyURATUltrasonic Rock Abrasion Tool

Smart USDC with Integrated Sensors

2 cm.

Folded Horns

Various Applications of the USDC Technology

Deep Drill

Rock Crusher

Powdering Tool

Surface Acoustic Wave (SAW) Motors

128o Y-cut LiNiO3

V up to 10 m/s

Force proportional to µN

Signal On

IDT excite a surface wave on piezoelectricsubstrate causes a mass to surf towards

the source

Piezoelectric Motors (Early Versions)

Ultrasonic Motors

Stator surface

Friction layer

Rotor ω

v =velocity of traveling wave in stator

F

τ

Elliptical motion of

contact point

-

V0cosωt V0sinωt

High Torque DensityLow RPMSelf Braking

Piezoelectric Pump

Current Specifications4-5 cc/min1100 PaNo Moving PartsPeristaltic

For More Information see

Devices -http://ndeaa.jpl.nasa.gov/

Materials –http://www.rmc.ca/academic/physics/ferroelectrics/Publications_e.html