full-strokestaticanddynamicanalysisofhigh-power piezoelectricactuators · 2012-08-16 · the linear...

Full-stroke Static and Dynamic Analysis of High-powerPiezoelectric Actuators

RADU POMIRLEANU AND VICTOR GIURGIUTIU*

Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA

ABSTRACT: A method for characterizing smart materials actuators in full-stroke static andlow frequency dynamic regimes is presented. At first, static and dynamic linear models of thepiezoelectric actuator response under electro-mechanical excitation are discussed. Then, athorough static and dynamic experimental characterization on a typical large-strokepiezoelectric actuator is performed. The measurements indicated a strong dependence of theactuator stiffness and piezoelectric properties on the electromechanical loading. Thecomparison of the model with the measured behavior is performed and the materialcoefficients are tuned to locally match the observed nonlinear behavior. The comparison alsoallows the identification of key parameters of the induced-strain actuator model. Theseparameters are necessary for design optimization towards maximum mechanical energyoutput and minimum electrical power input. The paper provides useful basic data for thedesign of actuation systems incorporating active materials actuators.

Key Words: PZT, piezoelectric, active materials, actuators, smart materials, induced strain,blocked force, full-stroke, high power, piezoelectric stack

INTRODUCTION

State of the Art in Piezoelectric Actuators Modeling and

Characterization

CONVENTIONAL modeling of piezoelectric materialresponse assumes the piezoelectric material as a

continuum, described by constitutive equations in termsof three material-properties tensors (the compliance,sijkl, the piezoelectric, dijk, and the dielectric permittivity,"ij) and four field quantities (strain, Sij, stress, Tkl,electric field, Ek, and electric displacement, Di):

Sij ¼ sEijklTkl þ dkijEk Di ¼ diklTkl þ "TikEk ð1Þ

Nonetheless, this electro-mechanical linear model canaccurately describe only a fraction of the full operatingrange of induced-strain actuators. Full-stroke operationis significantly nonlinear. Hence, the prediction of themaximum design force and the displacement as well asthe maximum available mechanical energy and therequired electrical energy of an active material actuatorcan be only approximately estimated using the linearconstitutive equations.

Krueger and Berlincourt (1961) characterized thenonlinear behavior of piezoelectric materials underhigh-stress conditions for naval transducer applicationsby relating the deviations from linear behavior to

macroscopic quantities such as dielectric permittivity,piezoelectric coefficient, loss factor, and couplingcoefficients. Although exclusively based on macroscopiceffects, this approach was useful in recognizing the basicdifferences in the responses of hard and soft PZTs.These early experiments showed that the tensorialmaterial properties varied significantly with the elec-tro-mechanical boundary conditions and that time-history effects are important. A strong dependence ofdielectric permittivity on stress and applied AC fieldamplitude was also observed (Krueger, 1967, 1968;Brennan and McGowan, 1997; Jordan et al., 2000).

Attempts to understand and explain the nonlinearpiezoelectric response considered the microstructurebehavior through domain switching under high electro-mechanical loads, where the domains inside a piezo-electric material crystal are defined as regions with thesame direction of the remnant polarization vector(Lynch, 1996). Carefully conducted experiments onvarious doped PZT ceramics have been pursued todetermine the polarization switching behavior under highelectric fields and compressive stresses (Zhang et al.,1997; Fan et al., 1999; Ching-Yu and Hagood, 2000; Linand Hagood, 2000; Yang et al., 2000).

Several modeling attempts have tried to bridge the gapbetween the micro-mechanics of the crystal structure andthe macroscopic properties of piezoceramics under highelectro-mechanical driving conditions (Chen et al., 1997;Fan et al., 1999; Hwang and McMeeking, 2000; Huberand Fleck, 2001). Such modeling efforts have beenconfined to the analysis of bulk piezoelectric material*Author to whom correspondence should be addressed. E-mail: [email protected]

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 13—May 2002 275

1045-389X/02/05 0275–15 $10.00/0 DOI: 10.1106/104538902029495� 2002 Sage Publications

+ [19.11.2002–2:59pm] [275–290] [Page No. 275] REVISED PROOFS I:/Sage/Jim/Jim13-5/JIM-29495.3d (JIM) Paper: JIM-29495 Keyword

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subjected to precisely known electro-mechanical excita-tion, and under negligible boundaries conditions influ-ence. However, in the case of stacked piezoelectricactuators consisting of hundreds of thin layers, theelectro-mechanical boundary conditions for every layerare insufficiently known and difficult to define. Conse-quently, the characterization of actual stack actuatorshas been, so far, phenomenological through measure-ment and data fitting rather than micromechanicalmodeling. Straub and Merkley (1995), Lee and Chopra(1999), Mitrovic et al. (1999, 2000), Straub et al. (1999),Pan et al. (2000) obtained essential design informationabout the mechanical stiffness, energy density, loadamplitude, temperature effects, and durability. Thesetests used various methods for external load application,such as dead weight (Lee and Chopra, 1999), MTSmachine (Mitrovic et al., 1999, 2000), variable impedancethrough a variable-stiffness spring (Straub et al., 1999),or constant stiffness spring mounted in series with thesmart material actuator (Pan et al., 2000).

Present Investigation

The present study took a design-oriented approach tomodel and characterize the static and dynamic behaviorof piezoelectric actuators. A high stroke/ high forcepiezoelectric actuator (PiezoSystems Jena PAHL120/20) was modeled using quasi-linear models.Systematic experiments were performed to characterizethe actuator behavior over the full range of external loadand excitation voltage. The linear piezoelectric modelwas found capable to provide a first-order approxima-tion for the actuator behavior only under no-loadconditions. The measurements performed under increas-ing external load showed increasing nonlinearity asso-ciated with the intrinsic material response under highmechanical stresses. For each load–voltage–frequencycondition, a quasi-linear model was tuned to theexperimental data, revealing consistent coefficientspatterns in terms of frequency, prestress level andapplied electric field. The experimental observationswere found to be consistent with existing microstructuretheories relating the macroscopic actuator behavior tothe polarization domains dynamics inside the activematerial ceramic. The actuator characterization resultsobtained in this study are presented in a form suitablefor direct application to induced-strain actuation design.

STATIC AND DYNAMIC MODELING OF

PIEZOELECTRIC STACK ACTUATORS

Piezoelectric Actuators Modeling

The PiezoSystems Jena piezoelectric actuator PAHL120/20 (Figure 1a) consists of an active material stack

prestressed to F0¼ 350 N by an internal spring inside asteel casing (Figure 1b). The mechanical prestress isneeded to prevent tensile stresses in the stack during staticand dynamic applications. The space between the steelcasing and the piezoelectric stack is filled with epoxy resinto protect the stack from shocks. The piezoelectric stacktransmits force through a steel push-rod. The piezo-electric material used is a soft PZT, with the manufac-turer provided characteristics shown in Table 1.

The linear piezoelectric equations (ANSI/IEEE Std.176, 1987) were used to describe the active materialbehavior by simplifying to the case where only the3-direction effects (i.e. in the direction of remnantpolarization) are retained, and the effects that takeplace in the plane perpendicular to the 3-direction(e.g. interaction with the resin layer, nonuniformdistribution of the electric field through the activematerial, etc.) are neglected. Consequently:

S3 ¼ sðEÞ33 T3 þ d33E3 D3 ¼ d33T3 þ " ðTÞ

3 E3 ð2Þ

Table 1. PiezoSystems Jena actuator properties.

Actuator Properties and Dimensions

Maximum displacement (mm) 120Voltage range (V) � 10 to 150Actuator internal stiffness (kN/mm) 30Maximum tensile force (N) 350Maximum blocked force (N) 3500Capacitance (for small strength electrical fields) (mF) 42Resonance frequency (kHz) 5Length (mm) 126Diameter (mm) 20Piezoelectric coefficient d33 (10�12

�m/V) 700Compliance sE

33 (10�12�m2/N) 20.8

Coupling factor �33 0.65Stack length L (mm) 118Stack area A (mm2) 64Layer thickness t (mm) 0.1

Figure 1. (a) Piezoelectric stack actuator PAHL 120/20; (b) idealizedactuator and external structure; (c) free-body diagram for thepushing rod.

276 R. POMIRLEANU AND V. GIURGIUTIU


STATIC MODELThe static model assumes that the application of

electrical and mechanical loads to the actuator consti-tutes an equilibrium process and, consequently, thestatic loads super-position can be applied. The loadingof the active material stack was considered applied in3 steps:

(a) The prestress force F0 is applied through thecompression of the internal spring kSP against thecasing. During this process, both the stack andthe internal spring are compressed;

(b) The force F ðbÞe is applied by compressing the external

spring ke against an external support. During thisprocess, the stack and the external spring arecompressed, while the internal spring expands;

(c) A positive voltage V is applied. The stack expands,remaining in compression.

The piezoelectric and stiffness properties of the stackare assumed to remain constant throughout this 3-stepprocess. After the first step, the stack displacement,strain, and stress are:

�L að Þ ¼L

kEð Þ

ST

Sað Þ

3 ¼1

LkEð Þ

STF0 Tað Þ

3 ¼F0

Að3Þ

The stack stiffness, kðEÞST , can be evaluated using the

closed-circuit material compliance value, sðEÞ33 , reported

by the manufacturer:

kðEÞST ¼

A

sEð Þ

33 Lð4Þ

During step (b), the application of the externalmechanical load Fe (Figure 1(c)) leads to:

Fbð Þ

ST ¼ F bð Þe þ F

bð Þ

SP ð5Þ

Consequently, the displacement increment from state (a)to state (b), the strain and the stress at the end of step(b) are:

�L bð Þ ¼F bð Þe

kEð Þ

ST þ kSP, S

bð Þ

3 ¼ð1þ ðkSP=k

Eð Þ

ST ÞÞF0 þFðbÞe

LðkEð Þ

ST þ kSPÞ,

Tbð Þ

3 ¼ð1þ ðkSP=k

Eð Þ

ST ÞÞF0 þFðbÞe

Að1þ ðkSP=kEð Þ

ST ÞÞ

ð6Þ

The application of the voltage in step (c) does notchange the free-body diagram (Figure 1c), but adds apositive stack displacement, i.e.:

�L cð Þ ¼Ad33

tsEð Þ

33

V

ke þ kSP þ kEð Þ

ST

ð7Þ

The corresponding strain and stress are:

Scð Þ

3 ¼ð1þ ðkSP=k

ðEÞST ÞÞF0 þF

ðbÞe

ðkST þ kSPÞLþd33

tkðEÞST

V

keþ kSP þ kðEÞST

Tcð Þ

3 ¼F0

AþF ðbÞe

A�

kðEÞST

kðEÞST þ kSP

�d33 �V

t � sðEÞ33

keþ kSP

kðEÞST þ keþ kSP

ð8Þ

Hence, the external force F ðcÞe , which can be directly

compared with the experimental data, results as:

F cð Þe ¼ F bð Þ

e � Ad33 � V

t � sðEÞ33

ke

ke þ kSP þ kðEÞST

ð9Þ

DYNAMIC LINEAR MODELThe dynamic model describes the response of the

piezoelectric actuator under in-phase electro-mechanicalloading. The wave equation can model the response ofan elastic medium under harmonic excitation. Thisapproach, proposed by Ikeda (1990) and Liang et al.(1994) in conjunction with thin PZT wafers modeling,was later expanded to piezoelectric and magnetostrictiveactuators by Giurgiutiu et al. (1994), Giurgiutiu andRogers (1996), and Ackerman et al. (1996). The one-dimensional wave equation is:

@2u

@�2¼ c2 @

2u

@x23

with c2 ¼1

� � s33ð10Þ

To account for mechanical losses, including materialdamping, losses in the bonding between electrodes andpiezo-material, and losses in the surrounding resin, theeffective complex compliance s�33 ¼ s33 � ð1 � i�Þ wasintroduced.

The excitation voltage is assumed to consist ofsuperposed DC and AC components, i.e.,

Vð�Þ ¼ V0 þ Vaei!� ð11Þ

where V0 and Va are the bias and the amplitude voltagesrespectively.

This model assumes that: (i) the stack can beassimilated to an uniform continuum; (ii) the excitationangular frequency, !, is much smaller than the ratiobetween wave speed, c, and the stack length, L; and (iii)the electric field distribution within the material isrelatively uniform, E3¼V/t. The boundary conditionsassociated with Equation (10) are:

uðx3 ¼ 0, �Þ ¼ 0, A � T3ðx3 ¼ L, �Þ ¼ FST ð�Þ ð12Þ

where FST(�) is the force exerted in the piezo-stack. FSTis made up of two components FST (�)¼FbþFa(�)

Full-stroke Static and Dynamic Analysis 277


where: (i) Fb is the bias force given by the internalspring prestress, F0, external preload, Fe, and the biasvoltage V0; (ii) Fa(�) is the oscillatory component thatcan be related to the velocity of the top of the stackthrough:

Fað�Þ ¼ �ZEXTð!Þ �@u

@x3

� �x3¼L, �

ð13Þ

where ZEXT(!) is the external structure impedance.Denote ¼!/c and assume a displacement solution of

the form:

uðx3, �Þ ¼ ðC1 sinðx3Þ þ C2 cosðx3ÞÞei! � � C3x3 ð14Þ

The first boundary condition at x3¼ 0 yieldsC2e

i!� ¼ 0, 8�: Hence, C2¼ 0. At x3 ¼ L,

S3ðx3 ¼ L, �Þ ¼@u3

@x3

� �x3¼L

¼ C1 cosð LÞei! � � C3

ð15Þ

Using Equations (2), (12), (14), (15) yields:

C1 ¼d�33 Vat

cosð LÞ þs�33

Ai!ZEXTð!Þ sinð LÞ

� ��1

,

C3 ¼ �s�33

AFb þ

d�33

tV0

� �ð16Þ

with d�33 ¼ d33ð1 � i�Þ, where � accounts for theimperfect piezoelectric energy conversion (Holland,1967). The bias force Fb in Equation (16) is expressedusing Equation (8) with V¼V0:

Fb ¼ F0 þ FðbÞe �

k�STk�ST þ kSP

�d�33 � V0

t � s�33

ke þ kSPk�ST þ ke þ kSP

where F ðbÞe is the static compressive force applied to the

actuator prior to the voltage being applied.

uðx3, tÞ ¼d�33Vae

i!�

t

� 1þs�33

A

i!ZEXTð!Þ

tanðLÞ

� ��1sinð xÞ

cosðLÞ

þx3s�33

AF0 þF

bð Þe

k�STk�ST þkSP

� �þd�33

tV0

k�STkeþkSPþk

�ST

� �ð17Þ

For the generic external load consisting of a mass-spring-damper system (Figure 1b), we can write:

kdð!Þ ¼ i!ZEXTð!Þ ¼ mð!2n � !2Þ þ i!cd

with !n ¼ke þ kSPm

� �1=2 ð18Þ

Hence, the actuator displacement, uST¼ u(x3¼L, �),becomes:

uST ð�Þ ¼d�33Va tanðLÞ ei! �

tð1 þ ðs�33=AÞðkd ð!Þ=Þ tanðLÞÞþ

F bð Þe

k�ST þ ksp

þL

td�33V0

k�STk�ST þ kSP þ ke

ð19Þ

Using Equation (19) in Equation (2), and recallingFST(�)¼A�T3(x3¼L, �), the force exerted in the piezo-stack is:

FST ð�Þ ¼ �FblockVaV0ei! � 1 þ

s�33

A

kdð!Þ

tanð LÞ

� ��1

þ F0 þ Fbð Þe

k�STk�ST þ kSP

þ Fblockke þ kSP

ke þ kSP þ k�ST

þVaV0ei! �

� �ð20Þ

where Fblock is the static blocked force corresponding tothe bias voltage, V0, i.e.,

Fblock ¼ �d�33

s�33

A

tV0 ð21Þ

The external force can be expressed as:

Feð�Þ ¼ FST ð�Þ � kSPuST ð�Þ � F0 ð22Þ

To model the electric current, the second part ofEquation (2) is written as:

D3 ¼ d�33

FST ð�Þ

Aþ"�33

tðV0 þ Vae

i! �Þ ð23Þ

where the electric field was replaced by E¼V/t, andelectric losses were modeled as "�33 ¼ "33ð1 � i�Þ. Theelectric displacement D is related to the free chargedistribution, �f, by the equation div D¼ �f (Maxwell,1891). For one-dimensional geometry, this equationreduces to Q¼D3A, where Q is the free chargeaccumulated on the electrodes. The instantaneous



current is the partial derivative of the free charge withrespect to time:

iST ð�Þ ¼dQ

d�¼L

ti!Vae

i! �

� d�33Fblock 1 þs�33

A

kdð!Þ

tanð LÞ

� ��1

�1

!þ"�33A

t

" #

ð24Þ

The actuator electrical impedance is given by:

ZST ð!Þ ¼Vae

i!�

iST ð�Þ¼

1

i!

t

L

� d�33Fblock 1þs�33

A

kdð!Þ

tanðLÞ

� ��1

�1

!þ"�33A

t

" #�1

ð25Þ

The instantaneous power is given by p(�)¼iST(�)� (�).The average active power per cycle can then becalculated as:

Pa ¼ f �

Zcycle

pð�Þ d� ¼ f �

Z 2�=!

0

ð�Þ � iST ð�Þ ð26Þ

where (�) and iST (�) are given by Equations (11) and(24), respectively.

EXPERIMENTAL CHARACTERIZATION OF

PIEZOELECTRIC STACK ACTUATORS

The purpose of the experiments was to evaluate theblocked force, the characteristic loops, and the mechan-ical and electrical envelopes for a piezoelectric stackactuator PAHL 120/20, in quasi-static and dynamicregimes. The characteristic loops considered here areforce–displacement, current–voltage, and voltage–displacement curves. The mechanical envelope for agiven frequency correlates the extreme values of voltage,displacement, and force. Likewise, the electrical envel-opes correlate force and voltage with the peak or theactive power, respectively.

Experimental Setup

The signal generated by a Hewlett Packard HP3312A function generator was amplified by a high-voltage 2-channel TREK 750/50 amplifier, and sent tothe PAHL 120/20 actuator (Figure 2). To increase thedynamic range, the two channels of the TREK 750/50amplifier were merged. The amplified signal wastapped to the HP5460B digital oscilloscope. A parallel

circuit consisting of a 1 k� resistor and a normallyopen switch was used for discharging the stack.The current going through the actuator was measuredusing the voltage drop across a calibrated 1� resistorplaced in series with the actuator. This signal waspassed through a low-pass filter and displayed onthe second channel of the HP5460B. The displacementof the actuator was measured with a Philtec D100optical displacement transducer and sent to aTektronix TDS210 digital oscilloscope (Figure 3a).The external load was applied using a LongYearcompression frame and a proving ring. The provingring was equipped with a dial gauge calibrated tomeasure force (accuracy 7.2 N) (Figure 3b) and fourstrain gages in full bridge configuration. The straingage signal was passed through a low-pass filter anddisplayed on the second channel of the TDS210. The 4waveforms captured on the digital oscilloscopes weredownloaded into a PC.

The measurements were taken in static and dynamicregimes. The maximum frequency attained duringthe dynamic regime was fC� 5 Hz. For a capacitiveload, the maximum achievable frequency is limited bythe maximum electric current capability of the poweramplifier (Figure 4). Given the 0.1 A maximum currentof the TREK 750/50 amplifier, a voltage duty-cycle of0–150 V, and the PAHL 120/20 piezoelectric actuatorcapacitance of 42 mF, yields a predicted maximumfrequency of 7.1 Hz. Experimental results showedcurrent saturation of the power supply to appear at aslow as 5 Hz, and thus the highest testing frequency usedin our experiment was fC¼ 5 Hz.

Figure 2. Experimental setup for dynamic testing of PAHL 120/20.



Measurements Procedures and Results

MEASUREMENTS OF THE BLOCKED FORCEThe blocked force is defined as the force develop-

ing in the actuator when pushing against a body ofinfinite stiffness, such that no external displacementis possible and all the induced strain is consumedinternally. Since the compression frame does nothave an infinite stiffness, special methods had to bedevised to create the condition of zero externaldisplacement.

Static Method 1 Apply a given voltage, thus expandingthe stack, and then compress the actuator until the netdisplacement is brought back to zero. The followingsteps were applied:

(a) Discharge the unloaded actuator, zero the signalgenerator and switch on the amplifier;

(b) Adjust the platen supporting the actuator (Figure3b) until its tip positively contacts the proving ring;

(c) Mark the initial position of the displacement signalwith the oscilloscope cursor;

(d) Increase the voltage up to the first voltage level: thestack expands and the proving ring compresses;

(e) Adjust the platen to compress the assembly until thedisplacement signal returns to the initial positionmarked by the cursor. When this step is completedthe stack has recovered its initial length, and theforce recorded by the proving ring is the blockedforce for the given voltage.

(f) Repeat from (d) for the other voltage levels.

Static Method 2 Compress the stack to a given force,and then adjust the voltage until the initial length of thestack is recovered. Record the final voltage and forcevalues. This method is done in the following steps:

(a) Discharge the unloaded stack, zero the signalgenerator and switch on the amplifier;

(b) Mark the initial position of the displacement signalusing the oscilloscope cursor;

(c) Introduce the first prestress level by compressing theactuator with the platen against the proving ring;

(d) Increase the voltage until the displacement signal onthe oscilloscope returns to the initial position andread the final voltage and the correspondingblocked force.

Dynamic Method The following steps were used:

(a) Discharge the unloaded stack and bring the stack incontact with the proving ring;

(b) Set the first voltage duty cycle on the signalgenerator and turn on the amplifier – the voltage,displacement, and force will harmonically cyclebetween zero and the maximum value; place theoscilloscope cursor on the minimum value of thedisplacement waveform;

(c) Compress the assembly by adjusting the platen untilthe maximum value of the displacement waveformcoincides with the oscilloscope cursor;

(d) Record the blocked force value as the maximum ofthe force waveform;

(e) Repeat for other voltage duty cycles.

The results of blocked force measurements for staticand dynamic excitation are shown in Figure 5.Superposed on Figure 5 is also the theoretical linepredicted by Equation (21). As seen from Figure 5, the

Figure 3. (a) Schematic for the experimental setup for the piezo-electric PAHL 120/20 actuator; (b) compression frame.

Figure 4. Power supply frequency restrictions for the operation ofinductive and capacitive loads.



linear theory over-predicts the quasi-static blocked forcemeasurements, but seems to be a good approximation tothe dynamic blocked force measurements.

Regarding the static data, the two testing methodsgive similar results for voltages up to 70% of themaximum value. At higher voltages, the results of thetwo methods start to diverge, with 14% maximumdifference being observed at 150 V.

STATIC AND DYNAMIC ACTUATORCHARACTERIZATION

To completely describe the piezoelectric actuatorbehavior under quasi-static and dynamic conditions,the actuator displacement under combined electro-mechanical load (force and voltage) had to be recorded.Due to the spring-like character of the externalload, distinction had to be made between the measure-ment procedures for the points situated aboveand respectively below the external stiffness line ofFigure 6.For points situated above and on the external stiffness

line, the following steps were taken:

(a) Discharge the stack and bring the push-rod incontact with the proving ring;

(b) Superpose the first oscilloscope cursor on thedisplacement signal and record the initial position v1;

(c) Compress the whole assembly up to the first pre-stress level;

(d) Apply the first voltage level, V: the actuator expandsand the proving ring further compresses;

(e) Move the second oscilloscope cursor until it super-poses on the displacement signal on the oscilloscopedisplay, and record v2. The displacement of theactuator is given by v2� v1;

(f) Read the compressive force F;(g) Repeat from step (d) for other voltage levels, up to

the maximum voltage;

(h) Repeat from step (c) for the other prestress levels.

Data points evaluated using this procedure lie on curvesparallel to the external stiffness curve, above it. Thisprocedure cannot be used for the determination of thepoints situated below the external stiffness line, becausethis would require placing the stack in tension.

For points below the external stiffness line, thefollowing procedure was used:

(a) Discharge the stack;(b) Apply an initial voltage to the unloaded actuator;(c) Bring the top of the actuator in contact with the

bottom of proving ring;(d) Apply a higher voltage, V, and record the force, F,

and the displacement, u;(e) Repeat from step (d) for the other voltage levels up

to the maximum voltage;(f) Repeat from step (a) with higher initial voltages.

The data points will also lie on curves parallel tothe external stiffness curve, but situated below it(Figure 6).

The procedure for dynamic measurements resemblesthe procedure used for static measurements, withmodifications to account for the dynamic effects. Theexcitation voltage was of the form given in Equation(11), with Va ¼ V0.

For the points situated above and on the externalstiffness curve, the following procedure was used foreach tested frequency:

(a) Discharge the stack and set the desired prestresslevel by compressing the actuator with the platenagainst the proving ring;

(b) Set the voltage duty-cycle on the function generatorand turn on the amplifier;

(c) Download the waveforms into the PC;(d) Repeat from step (b) for the next voltage duty-cycle,

up to the maximum allowable voltage;(e) Repeat from step (a) for the next prestress level, up

to the prestress corresponding to the blocked forcefor maximum allowable voltage.

Figure 5. Blocked force variation with voltage: theory andexperiment.

Figure 6. Force–displacement–voltage correlation.



Dynamic data was complemented with no-loadmeasurements of the piezoelectric actuator for eachfrequency.

DATA PROCESSING AND COMPARISON WITH

THEORETICAL PREDICTIONS

Static Results

Figure 7 indicates that the piezoelectric linear staticmodel shows good correlation with experimental resultsfor zero load. However, the static model is a poorpredictor for the static behavior of the piezostack underhigh force condition, which shows significant non-linearity. The experimentally observed deviations fromthe linear model were addressed through coefficienttuning, using a bivariate cubic regression of the actuatordisplacement (Figure 8):

u ¼ C0F3e þ C1F

2e V þ C2FeV

2 þ C3V3 þ C4F

2e

þ C5FeV þ C6V2 þ C7Fe þ C8V

ð27Þ

Besides the intrinsic behavior of the piezoelectricmaterial under electro-mechanical loading, the overall

actuator displacement also contains the influence of theprestress spring, electrodes, and resin. The nonuniformelectric-field distribution inside the active material layeralso adds nonlinearity to the actuator behavior.Nevertheless, an approximate description of the overallactuator characteristics under electro-mechanical load-ing can be obtained, if we neglect the effect of elect-rodes, resin and electric field nonuniformity. The forcein the active material stack, FST, is simply the sum ofthe measured force and the force exerted by theprestress spring, kSP, i.e.,

FST ¼ Fe þ F0 � kSP � u ð28Þ

where Fe is the external force, and u is the actuatordisplacement. Substitution of Equation (28) into theregression Equation (27) provides the full description ofthe overall actuator behavior, in terms of the force in theactive material, FST, voltage and actuator displacement.Thus, a numeric solution of the form u¼ u (FST,V)could be determined.

ACTUATOR STATIC STIFFNESSFor nonlinear behavior, the actuator stiffness can be

defined as the local derivative of the force–displacementcurve (tangent stiffness) or as the ratio of the local forceand displacement values (secant stiffness). To calculatethe tangent stiffness, we used the inverse of the tangentcompliance, i.e. k ¼ ð@u=@FeÞ

�1: The results presentedin Figure 9 indicate that the tangent stiffness ofthe piezoelectric actuator increased significantly withthe compressive mechanical load. The variation of theactuator stiffness with voltage is, however, almostnegligible.

COMPLIANCE AND PIEZOELECTRICCOEFFICIENTS

If we assume that the piezoelectric materialbehavior is sufficiently well described by the lawS33¼ s33(V,F) �Fþ d33(V,F) �V, the displacement deri-vatives with respect to force and voltage can bewritten as:

A

L0i

@u

@FST

� �¼ s33 þ F

@s33

@FSTþA

tV

@d33

@FST,

t

L0i

@u

@V

� �¼t

AFST

@s33

@Vþ V

@d33

@Vþ d33

ð29Þ

By neglecting the cross-coupling terms, (ANSI/IEEEStd.176, 1987; Mitrovic et al., 1999) we obtain afirst-order approximation of the compliance and

Figure 8. Curve-fitting of experimental data using a bivariateregression.

Figure 7. Comparison of the PiezoSystems Jena actuator staticlinear model prediction with experimental data.



piezoelectric coefficients for the active material of thetested actuator:

s33 �A

L0i

@u

@FST

� �; d33 �

t

L0i

@u

@V

� �ð30Þ

The resulting s33 and d33 curves are presented inFigure 10. The compliance coefficient, s33, is seen todecrease with increasing stress (Figure 10a) and not tovary significantly with the applied electric field. Thepiezoelectric coefficient d33 displays a maximum value atapproximately half the maximum electric field. Itsdependence on stress is not negligible (Figure 10b).

Dynamic Results

DATA PROCESSINGFor each frequency–prestress–voltage case, four

waveforms (displacement, force, current, voltage) wererecorded and further processed in several stages usingMatlabTM software (Figure 11). The first stage of dataprocessing consisted of:

(a) Adjust the current and force related signals toaccount for the phase and amplitude changeintroduced by the analog low-pass filter, using thefilter calibration;

(b) Filter out the spikes from the electric current, forceand displacement related waveforms using a cas-caded algorithm for outliers elimination;

(c) Apply a high-order Butterworth digital filter to allfour waveforms (Figure 12);

(d) Use the calibration data for the proving ring, straingages full-bridge, and the 1� resistor to convertelectrical waveforms into force, electric current, anddisplacement waveforms.

The next stage of data processing consisted ofbuilding the characteristics loops of synchronizedwaveforms, paired as follows: displacement versusforce, current versus voltage, and displacement versusvoltage (Figure 13d–f). These characteristic loops for

each frequency–prestress–voltage range were used fortuning the dynamic model of the actuator.

Finally, the mechanical and electrical envelopes weresynthesized using the extreme values of the characteristicloops (Figures 16–18). These envelopes give a generalview of actuator capabilities and point to its usefulnessfor specific applications.

Figure 9. Actuator stiffness: (a) variation with external force; (b) variation with voltage.

Figure 10. Effective piezoelectric material coefficients evaluatedfrom the derivatives of the displacement with respect to force andvoltage: (a) compliance, s33; (b) piezoelectric coefficient, d33.

Figure 11. Data processing logic flow.



TUNING AND IMPROVEMENT OF THEDYNAMIC MODEL

The dynamic model parameters and their observedcorrelation with the actuator behavior are listed inTable 2. The values of these parameters were determinedby tuning the theoretical model prediction to the

experimental data. During this process, special attentionwas given to the parameters "33 and �, which representthe dielectric permittivity and dielectric loss, respec-tively. Unlike the other model parameters that could beassumed constant throughout a loading–unloadingcycle, "33 and � had to be allowed to vary duringthe cycle. While the dynamic model predictedsinusoidal electric current waveforms, the experimentalresults showed significant deviations from the harmonicwaveform on the electro-mechanical unloadingpath (Figure 13c). During unloading, the measuredcurrent lags behind the linear-model prediction, andthe instantaneous minimum current is smaller inabsolute value than the maximum current. In orderto account for these phenomena, the followingnonlinear laws of "33 and � variation inside the cyclewere used:

"33 ¼ "0½1 � 0:5ðk" � 1ÞfeðtÞðHðtendÞ �HðtstartÞÞ ð31aÞ

� ¼ �0½1 þ 0:5ðk� � 1ÞfeðtÞðHðtendÞ �HðtstartÞÞ ð31bÞ

where "0 and �0 are the values during the loading half-cycle, and k� and k" are numerical values that accountfor the phase shift and amplitude reduction of theelectric current respectively. H(t) is the Heavisidefunction with tstart and tend being the start and the endof the electro-mechanical unloading part of the cycle.The function 0.5 feðtÞðHðtendÞ �HðtstartÞÞ is a Hanningwindow, which ensures smooth transition between theloading and unloading portions of the cycle (Figure 14).The improved model correlates better with the observedactuator behavior, as shown in Figure 15.

Thus, for each combination of frequency, voltage,and prestress, a set of 8 effective parameters weredetermined: d33, �, s33, �, "0, �0, k�, k" (Table 3).

DISCUSSION

The blocked force for a given voltage is an importantparameter in describing a piezoelectric actuator, becauseit helps to establish an estimate of the availableactuating energy under static loading (Figure 16). Thetwo experimental methods devised for blocked forceestimation showed consistency for voltages up to 70%of the maximum voltage. Above this value, the twomethods started to depart from each other, with asmuch as 14% difference being observed at 150 V. Theseresults indicate that the hypothesis of linear super-position holds only up to 70% of the operational range.Above 70%, ferroelastic and ferroelectric switchingtime-history effects seems to become important, sincethe same inputs applied in a different order lead todifferent electro-mechanical equilibrium states. The

Figure 12. Signal processing steps simultaneously shown onthe same chart (5 Hz, 1150 N prestress, 0–100 V): (a) force;(b) displacement; (c) electric current.



explanation for this behavior resides in the different netnumber of ferroelastic domain switching that occurswhen the order of loading is changed. For dynamicloading, these ferroelectric–elastic switching effects seemto reach a state of dynamic equilibrium, which yields analmost linear dependence between voltage and blockedforce. In addition, trends of increased blocked force withfrequency were also observed, indicating that the time-history effects are not negligible even for the limitedfrequency range tested.

For dynamic application, effective values of modelparameters (Table 3) were obtained in a cycle-averagesense. Since most of the parameters display a consistentpattern of variation when frequency, voltage range, andprestress level are varied, interpolation of the data inTable 3 between the tested cases seems appropriate forthe tested actuator.

Data presented in Table 3 suggests that the cycle-equivalent value of the d33 coefficient increases withvoltage. This can be explained through the increasednumber of domain ferroelastic switching when thevoltage is cycled to a higher value. The cycle-equivalentvalues of d33 are consistent with the corresponding staticdata (Figure 10b), when the bias voltage is varied.

The electrical envelopes for the maximum peak powershow a consistent dependence on frequency, force, andvoltage cycle (Figure 17), qualifying the peak power as auseful indicator when evaluating a piezoelectric actuatorfor a specific application. Hence, besides the expectedproportionality with the applied voltage and frequency,the peak power is seen also to increase with force. Thisincrease (up to 10%), may be due to an increaseddielectric permittivity along a direction that is no longer

Figure 13. Comparison of experimental data with the model prediction for 5 Hz, 1150 prestress and 0–100 V: (a) displacement waveform;(b) force waveform; (c) electric current waveform; (d) displacement–force loop; (e) electric current–voltage loop; (f) voltage– displacement loop.

Figure 14. Hanning window acting on the unloading part of thecycle.



aligned with the dipole axis of the PZT crystal, due toinherent crystal anisotropy (Du et al., 1997; Uchino,2000). As the stress increases, more ferroelastic switchesoccurs, leading to more domains having their electricdipoles misaligned from the x3 axis, resulting in anincreased permittivity at the polycrystallite level. Thebehavior of the cycle-equivalent electrical permittivity,"33, correlates well with this explanation.

The average active power per cycle consumed by theactuator (Figure 18) appears to show a trend towardsincreasing with the voltage and decreasing with

Figure 15. Improved model predictions for: (a) electric current;(b) instantaneous power.

Table 2. Correlation between tuning coefficients andexperimental curves.

MaterialParameter Influence on the Characteristic Curves

d33 Proportional to the amplitude of thedisplacement waveform and the secantslope of the force–displacement characteristicloop.

Influences amplitudes of the force anddisplacement waveforms

s33 The ratio of thestack stiffness kST to theexternal dynamicstiffness kd influences theslope of the force–displacement loop.

"33 Related to the electric current amplitude onthe loading path.

� Proportional with the hysteresis in thedisplacement–voltage characteristic loop.

� No significant influence.� Global tilt of the current–voltage

characteristic loop.

Figure 16. Mechanical envelope for the piezoelectric actuatorPiezoSystems Jena PAHL 120/20 under dynamic testing at 5 Hz.

Figure 17. Electrical envelope for the PiezoSystems Jena PAHL120/20: peak power at 5 Hz.

Figure 18. Electrical envelope for the PiezoSystems Jena PAHL120/20: active power at 5 Hz.



prestress level. The increase with voltage is morepronounced than the theory predicts with a constantdielectric loss coefficient, �. These characteristics couldbe explained through domain switching: for a givenvoltage cycle, increasing the prestress is equivalent to

impeding more domains to perform ferroelastic switch-ing, while with a given prestress level, an increasedvoltage cycle will help more domains to perform 90�

switching. Since ferroelastic switching is associated withenergy loss, as shown by Kessler and Balke (2001), the

Table 3. Piezoelectric model parameters – variation with frequency, voltage and prestress level.

Voltage Range � d0�10�12 "0�10�8 s33�10�12 �33 �0 k� �0�k� k"

1 Hz, External load not coupled0–60 V 0.03 650 7.1 20.8* 0.29* 0.13 2.0 0.26 1.000–100V 0.08 730 7.7 20.8* 0.33* 0.10 4.5 0.45 1.060–150V 0.07 750 7.8 20.8* 0.35* 0.05 12.0 0.60 1.10

1 Hz, � 1100 N Prestress0–60 V 0.08 620 7.2 15 0.36 0.13 1.5 0.20 1.000–100 V 0.03 665 7.6 15 0.39 0.15 2.6 0.39 1.010–150 V 0.04 710 8.0 17 0.37 0.05 12.0 0.60 1.09

1 Hz, � 2840 Prestress0–60 V 0.10 600 7.5 13 0.37 0.14 1.6 0.22 1.000–100 V 0.07 650 8.1 13 0.40 0.11 3.3 0.36 1.050–150 V 0.04 690 8.2 13 0.45 0.03 15.0 0.45 1.05

2 Hz, External load not coupled0–60 V 0.30 630 7.3 20.8* 0.26* 0.10 1.8 0.18 1.070–100 V 0.10 710 7.7 20.8* 0.32* 0.10 3.3 0.33 1.040–150 V 0.07 720 7.6 20.8* 0.33* 0.05 12.0 0.60 1.10


2 Hz, � 2840 Prestress0–60 V 0.03 590 7.5 13 0.36 0.12 1.4 0.17 1.000–100 V 0.03 630 7.9 13 0.38 0.12 3.3 0.4 1.070–150 V 0.08 690 8.1 13 0.45 0.03 15.0 0.45 1.07

4 Hz, External load not coupled0–60 V 0.03 570 6.8 20.8* 0.23* 0.27 1.5 0.41 1.030–100 V 0.08 630 7.4 20.8* 0.26* 0.14 3.5 0.49 1.060–150 V 0.07 650 7.5 20.8* 0.27* 0.05 12.0 0.60 1.13


4 Hz, � 2840 Prestress0–60 V 0.03 550 7.2 13 0.27 0.10 2.0 0.20 1.010–100 V 0.02 610 7.6 13 0.38 0.12 3.3 0.40 1.060–150 V 0.03 660 7.8 13 0.42 0.06 8.0 0.48 1.08

5 Hz, External load not coupled0–60 V 0.12 620 6.6 22* 0.27* 0.28 1.5 0.42 1.030–100 V 0.17 700 7.0 22* 0.32* 0.30 2.0 0.60 1.100–150 V 0.12 720 6.8 22* 0.35* 0.12 5.5 0.66 1.12


5 Hz, � 2840 Prestress0–60 V 0.12 570 7.0 16 0.29 0.27 1.6 0.43 1.030–100 V 0.17 610 7.6 13 0.38 0.12 3.2 0.38 1.050–150 V 0.17 660 7.3 13 0.46 0.02 25 0.5 1.09



implications to the active power consumption areimmediate. These observations correlate with themodel coefficients mostly through the variation of the�0�k� product (Table 3), that represents the peakdielectric loss on the unloading cycle. This quantityrepresents a better modeling parameter than k�, since itcaptures both the above mentioned measured trendsseen in the active power envelopes, and its variation iscontained within a narrower range. For example, for the0–150 V duty cycle, the numerical value of k� is found tovary between 6 and 35, depending on the frequency andprestress level, but the variation of the �0�k� product liesonly in the range 0.4–0.6.

CONCLUSIONS

The present work has presented an engineeringapproach to multidisciplinary characterization of apiezoelectric actuator, using modeling and electro-mechanical measurements.

The thorough characterization of a commerciallyavailable piezoelectric actuator (PiezoSystems JenaPAHL 120/20) yielded correlations between outputforce, displacement, electrical current, and voltage overa wide range of voltages, prestress levels, and frequen-cies. For high static electro-mechanical loading, it wasnoticed from the blocked force measurements that theelectro-mechanical equilibrium depends on the order ofload application, and thus, the loads superpositionprinciple is valid only up to 70% of the maximumvoltage. The microstructural ferroelastic effects leadingto this behavior also induce a strong nonlinearity in thestatic mechanical envelope. However, for dynamicloading, this nonlinearity appears to be negligible.

The quasi-linear dynamic model employed in thisanalysis permitted the evaluation of cycle-averagedparameters influencing the active material behavior,giving a quantitative insight into optimization opportu-nities arising from varying the prestress and frequency.Comparison of experimental data and model predictionsallowed model improvement. The consideration ofhigher dielectric losses on the unloading path is animportant novel feature of the present work. A goodmetric for these losses was found to be the peakdielectric loss on the unloading path. This parameterwas found to increase with voltage cycle and decreasewith the level of applied prestress, correlating well withthe active power required by the piezoelectric actuator.The microstructure theory of polarization domainswitching correlates qualitatively with the observedbehavior of the piezoelectric stack.

The measurements of the peak and active powerconsumed by actuator yielded important characteriza-tion charts (electrical envelopes), showing consistentdependence of these quantities on prestress level andvoltage duty-cycle. Along with the mechanical envelopes,

they provide a complete data set for the design ofactuating systems incorporating piezoelectric actuators.

Although the numerical values and the actuatorbehavior trends noticed during this study are particularto the actuator under consideration and its piezoelectricmaterial, the characterization approach can be extendedto analysis of other actuators of this type.

NOMENCLATURE

A¼ cross-sectional area of the stackL¼ length of the stackt¼ active layer thicknessS3 ¼ contracted notation for the strain in the polariza-

tion direction, S33

T3 ¼ contracted notation for the stress T33

D3 ¼ electric displacement in the polarization directionE3 ¼ electric field in the polarization directionFST¼ force exerted in the piezoelectric stackV¼ voltaged33 ¼ contracted notation for the d333 piezoelectric

coefficients33 ¼ contracted notation for the s3333 compliance

coefficient"33 ¼ permittivity coefficientkST¼ stack stiffnesskSP¼ internal spring stiffnesske¼ external stiffnessF0 ¼ prestress forcec¼wave speed� ¼ time�¼ density�¼ elastic loss�¼ dielectric loss ¼ ratio !/c�¼ piezoelectric energy conversion lossiST¼ electric current

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