Engin

eerin

g & In

forma

tion T

echn

ology Modeling and Characterization of

High-Force/High-Stroke Piezoelectric Actuator

Radu PomirleanuVictor Giurgiutiu

University of South CarolinaMechanical Engineering Department

9th International Workshop onAeroelasticity of Rotorcraft Systems

October 22-24, 2001University of Michigan, Ann Arbor

#2

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Acknowledgements

The financial support of the Army Research Office through the Grant No. DAAD 19-00-1-0017 with Dr. Gary Anderson as Technical Monitor is gratefully acknowledged.

#3

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Outline

State-of-the-art in actuator characterizationModeling and experimental results• Quasi-static and dynamic modeling• Experimental set-up• Comparison of modeled and empiric data

Design tools• Mechanical and electrical envelopes• Design guidelines

Conclusions

#4

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

State-of-the-art in actuators characterization

Typical manufacturer dataPhysik Instrumente, Kinetic Ceramics, PiezoSystems Jena, Etrema Inc., etc.

Response under variable pre-stress and voltage cycleLee et al. (1999), Mitrovic et al. (1999), Pan et al. (2000), Straub et al. (1999), Mitrovic (2000)

Lee et al. (1999), Pan et al. (2000)

GoalProvide characterization of actuator force, displacement, energy and power under various load conditions

#5

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Quasi-static piezoelectric actuator modeling

m

ke cd

kSP

x3

x1

Model development:Apply the loads in 3 steps:

a) Apply the internal pre-stress due to the internal spring, F0

b) Apply the external force F(b)

c) Apply the voltage V

Equations:( ) ( ) 33

( ) ( )33

c b ee e E E

e SP ST

d V kF F At s k k k

⋅= −

⋅ + +

a) linear constitutive equations

b) equilibrium condition

Hypotheses:

How it works as an actuator…33 33

3 33 3 33 3

EE s dS s T d E u L F L V

A t= + ⇒ = +

( )

( )

(1)0( )

( )333 ( )

1 SPeE

c ST EST E

ST SP e SP ST

k F Fk d Vu L kk k t k k k

+ +

= ++ + +

After applying a positive voltage, V V

#6

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Dynamic piezoelectric actuator modeling (I)

- linear constitutive equations- the piezoelectric stack is modeled as a continuous medium - the material losses are introduced through complex coefficients:

s33(E)* = s33

(E) (1-iη); ε33(E)* = ε33

(E) (1-iδ); d33*=d33 (1-iλ)

Hypotheses:

Model development:(0, ) 0u τ =( ) ( )2 2

3 322 2

3

, ,u x u xc

xτ τ

τ∂ ∂

=∂ ∂ ( ) ( )3 , STA T L Fτ τ⋅ =

Equation: Boundary conditions:

3Q D A=div D = ρfreedQidτ

=

( ) ( )3

* 03 33*

333

,i

aE

x L

V V eA uF x L dx ts

ωτ

τ=

+∂ = = − ∂

( ) ( ) ( )( )3 1 3 2 3 3 3, sin cos iu x C x C x e C xωττ γ γ= + −( ) 0i

aV V V e ωττ = +With , assume

Electrical modeling:

#7

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Dynamic piezoelectric actuator modeling (II)

Displacement, force and electric current equations:

( )( )

** 3333 ( )*

33

1 11 tan

iST a block E

EXT

AdQ Li i V e d Fd t ts i Z L

A

ωτ ετ ωτ ω γ

γ

= = − +

+

( )( )

( )

( )

( )

( )

( )

* 1 *33 *

33 0* **33

tan

1 tan

i Ea e ST

ST E EEST sp ST sp eEXT

d V L e F kLu d Vtk k k k ks i Zt L

A

ωτγ

ωγ γγ

= + + + + ++

( ) ( )( )

( )( )

( ) ( )

*0 33

*

0 * *0

1

1 tan

iaST block E

EXT

Eb iST e SP a

e blockE EST SP e SP ST

VF F eV s i Z L

A

k k k VF F F eVk k k k k

ωτ

ωτ

τω γγ

= − +

+

+ + + + + + +

#8

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Testing of PAHL 120/20 actuator Function generator HP 33120

TREK 750/50 Ch1 Ch2

Provingring

Displacement transducer Philtec D100

Active materialactuator

PC

Discharger

Tektronix TDS210

Low Pass Ch1 Ch2 Filter

HP 5460B

Strain gages(full bridge)

10Ω 10Ω 1Ω1kΩ

Calibrated resistor for electric current measurements

Proving ring

Force dial gaugeSpacer (cylindrical)Stack actuatorPlatenCompressionmechanismColumns

Upper beam

Strain gages

Proving ring

Digital O-scopes

Displacement transducer

Trek voltage amplifier

HP Function generator

Piezo actuator

Strain gages

PC System

PiezoSystems Jena PAHL 120/20 piezoelectric actuator

Compression frame

Experimental set-up

Experimentalset-upschematic

#9

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

PAHL 120/20: Blocked forceDefinition Blocked force = the force generated by the actuator when the

displacement is completely denied

Testing methodsMethod 1 (quasi-static): apply first the voltage (actuator expands) and then a compressive force until the initial length is recovered

Method 2 (quasi-static): apply first a force (actuator compresses), and then a voltage until the initial length is recovered

Method 3 (dynamic): apply a biased harmonic voltage and then increase the compressive force until the maximum displacement corresponds to the undeformed actuator

Theory33

33block E

d AF Vs t

= −

Results

0 20 40 60 80 100 120 140 160

1

2

3

4

Voltage [V]

Blo

cked

forc

e [N

]

Method 1 Method 2 Method 3 @1HzTheory

#10

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

PAHL 120/20: Quasi-static test results

Actuator displacement [µm] -100 -50 50 100 0 125

0

1000

2000

3000

Com

pres

sive

forc

e [N

] Test 150 V

Model 150 VTest 105 V

Model 105 VTest 60 V

Model 60 VTest 20V

Model 20V

40 0 40 80 120

2000

3000

1000

0

Actuator displacement [µm]

Com

pres

sive

forc

e [N

]

150V 80V

3 2 2 30 1 2 3

2 24 5 6 7 8

( , )u F V C F C F V C FV C V

C F C FV C V C F C V

= + + +

+ + + + +

Bi-cubic regression

Linear model

#11

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

PAHL 120/20: Quasi-static s33 and d33

0 10 20 30 40 50 60 0

10

20

30

40

50

Stress (MPa)

s 33 (×1

0-12 V

/m)

20 V80 V

150 V

Manufacturer reported compliance

33ST

A usL F

∂≈

∂Compliance coefficient:

0 20 40 60 80 100 120 140 160500

550

600

650

700

750

Voltage (V)

d 33 (×1

0-12 m

/V)

5.4 MPa15.6 MPa31.3 MPa

Manufacturer reported d33

Piezoelectric coefficient: 33t udL V∂

≈∂

33 0ST E SPF T A F F k u= = + − ⋅

Force in the stack:

0 1000 Force [N]

Bulk

stif

fnes

s [N

/ µm

]

2000 30000

40

60

80

100

20 V80 V

150 V

Manufacturer reported stiffness

Bulk actuator stiffness:1

BukF

−∂ ≈ ∂

#12

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

PAHL 120/20: Dynamic test results (I)

Voltage range

Load case Pre-stress (N)

Load:

0 570 1150 1700 2300 2900

No external load

0 – 20 V

0 – 40 V 0 – 60 V 0 – 80 V

0 – 100 V 0 – 120 V 0 – 140 V 0 – 150 V

Frequency=1Hz Frequency=2Hz

Frequency=4Hz Frequency=5Hz

ke =6.5⋅106 , ωn=450Hz, ζ=0.05

Test matrix for dynamic measurements

Number of processed files:(4 frequencies)x(8 voltage cycles)x[(6 loaded cases)x(4 signals)+(1 no load case)x(3 signals)] = 864 Freq.

Elec

tric

curr

ent

Power supply maximum current

fC ˜ 5Hz

#13

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

PAHL 120/20: Dynamic test results (II)Data processing flow

Clean the raw data

Filtered data

Digital O-scope data for a specificied frequency/load/voltage

Synchronize waveforms Characteristic

loops

Electrical and Mechanical Envelopes

Model tuning

0 20 40 60-1600

-1400

-1200

-1000

Forc

e [N

]

Relative displacement [µm]

ExperimentModel

-0.1 -0.05 0 0.05 0.10

50

100

150

Vol

tage

[V]

Current [A]

ExperimentModel

0 20 40 60 80 100 1200

20

40

60

80

Rel

ativ

e di

spal

cem

emt [µ m

]

Voltage [V]

ExperimentModel

#14

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

PAHL 120/20: Mechanical envelopeFrequency = 1Hz

Displacement [µm]

Max

imum

forc

e pe

r cyc

le[N

]

Model prediction for 0- 150 V

Model prediction for 0- 20 V

#15

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

PAHL 120/20: Electrical EnvelopesFrequency = 1Hz

Maximum force per cycle [N]

Max

imum

pea

k po

wer

[W]

Model :0-150V

Act

ive

pow

er p

er c

ycle

[W]

Maximum force per cycle [N]

Model 0-150 V

Peak power

Average active power per cycle

#16

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Dynamic model improvement

( ) ( ) ( ) ( )( )0 1 0.5 1 e end startk f t H t H tεε ε = − − −

ε33(E)* = ε33

(E) (1-iδ)

Initial model:

0 0.05 0.1 0.15-5

0

5

Time [s]

Pow

er [W

]

ExperimentInitial modelImproved model:

( ) ( ) ( ) ( )( )0 1 0.5 1 e end startk f t H t H tδδ δ = + − −

0 0.05 0.1 0.15

0.5

1

0

Time [s]

0.5⋅fe(t) (H(tstart)-H(tend)V(t)/Vmax

0 0.05 0.1 0.15-5

0

5

Time [s]Po

wer

[W]

ExperimentImproved model

#17

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Actuation design with PAHL 120/20 (I)

Displacement amplification

G,ηm, ϕa

F`e, u`e Fe, ue

PAHL 120/20

me ke

ζ

Ze

ZEXT

' 'e e m e eF u F uη =

External load: me=10kg, ωne =150Hz; ζ=0.05

Actuation parameters: 5Hz frequency

Displacement amplification parameters: G=7, ηm = 0.8, ϕa = 00

Required minimum displacement: 0.5 mm

Power supply ratings: 120V, maximum current imax or maximum power pmax.

( ) ( )' aie eu Gu e ϕτ τ=

Problem formulation

Given:

#18

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Actuation design with PAHL 120/20 (II)

2ai

EXT em

GZ Z e ϕ

η=

( )2

2 2 2 aid EXT e n n

m

Gk i Z m i e ϕω ω ω ζ ωωη

= = − +

Reduce the problem to a known case: F`e, u`e

PAHL 120/20

ZEXT

Re(kd) = 9.6⋅N/µm

Displacement [µm]

Max

imum

forc

e pe

r cyc

le[N

]

force –displacement for 120 V External

stiffness

Operating point

The maximum displacement :75 µm x G = 0.525 mm

The peak electric current:

max 2 0.103a aL Ai CV f Vt tεω π ≅ = =

A

The peak power (electrical envelope)

max 1.75p ≅ W

#19

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Optimal quasi-static energy transfer (I)Problem formulation

Displacement amplification mechanism

(DAM)

F`e, u`e Fe, ue

r0

uISA

ki

kδ

ISA stack

δ Mδ

Given:

' 'e

e em

e

F uF u

η ⋅=

⋅

Definitions:

'e

e

uGu

=

: DAM work efficiency

: DAM gain

20

i

d

krk r

=

Required angular deflection, δ: +/- 3 deg

Aerodynamic stiffness, kδ: 47 Nm/rad

Hinge radius, r0: 5 mm

Actuator free induced displacement uISA: +/- 60 µm.

: stiffness ratio

0

ISA

ruδη = : kinematic gain

2

2 2'0.5

oute

i ISA i ISA

E kEk u k u

δδ= = : energy transfer coefficient

#20

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Optimal quasi-static energy transfer (II)Solution

Required gain:

2

1 1 4

2m

m

r

Gr

ηη

ηη

+ −=

Critical actuator stiffness:

4i i cr

m ISA

kk kuδ

η≥ =

' 2eE rη=

Energy transfer coefficient:

150 200

6

250 300 350 4005

7

8

Actuator stiffness, ki

Req

uire

d ga

in, G

ηm = 0.8

150 0 200 250 300 350 400

0.3

0.2

0.1

Actuator stiffness ki (kN/mm)

Ener

gy tr

ansf

er

coef

ficie

nt, E

e’ ηm = 0.8

Optimal

Optimal

#21

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Optimal quasi-static energy transfer (III)

( )( )

' p

epq q

ER

G= x = 4rη2/ηm

( )

( )22

1 1

p qp qp p q

q m qxR

xη η

+− +−=

+ −

( )( )242

opt

p p qx

p q+

=+

100 ⋅

R11

Actuator internal stiffness (kN/mm)

ηm = 0.6ηm = 0.7ηm = 0.8ηm = 0.9ηm = 1.0

150 200 250 300 400 450 1.0

1.5

2.0

2.5

3.0

3.5

4.0p=1 q=1

• less complexity involved in the DAM design

• smaller and lighter DAM

• lighter actuator for a particular application

Optimization criteria:Smaller required gain

Greater energytransfer coefficient

#22

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Optimal quasi-static energy transfer (IV)

Actuator selection:

Option 1: PiezoSystems Jena PAHL 120/20: ki = 30 – 80 N/µm

Option 2: Kinetic Ceramics D125120: ki = 205 N/µm

Option 3: Physik Instrumente P247-70: ki =400 N/µm

The critical stiffness ki cr =154 N/µm

For equal optimal criteria weights, p=q=1, xopt = 8/9Choose ηm = 0.8

ki_opt = 200 N/µm

Option 1

Option 2

#23

Engin

eerin

g & In

forma

tion T

echn

ology

© PRO, 2001

USC

Conclusions

Review of the state-of-the-art for high-force/high stroke smart material actuators characterizationModeling and characterization of a high-force/high-stroke piezoelectric actuator. Model improvement was addressed, based collected experimental dataEffective design tools were proposed based on mechanical and electrical envelopes. Examples were given for the particular case of the PAHL 120/20 actuatorDesign guidelines for dynamic actuation systems incorporating piezoelectric actuators were produces