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Engin
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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
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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.
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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
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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
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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
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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:
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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
= − +
+
+ + + + + + +
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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
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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
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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
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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
−∂ ≈ ∂
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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
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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
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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
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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
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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
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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:
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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
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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
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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
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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
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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
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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
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