Brennan 1997 PiezoPower4ActiveControl

8/12/2019 Brennan 1997 PiezoPower4ActiveControl

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Piezoelectric Power onsumption forctive Vibration ontrol

ByMatthew Charles rennan

Mechanical Eng ineeringB .S . 5/95, The University of Vermont

A Thesis submitted to

The Faculty of

The School of Engineering and pp lied Scienceof The George Washing ton University in partial satisfaction

of the requirements for the degree of ster Science

July 25 997

Thesis directed by

Dr. Robert H. Tolsonrofessor of Engineering and pplied Science

This research was conducted at NS Langley Research Center

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BSTRACTA method for predicting the power consumption of piezoelectric actuators utilized foractive vibr tion control is presented. Analytical developments and experimental testsshow th t the maximum power required to control a structure using surf ce bondedpiezoele tri actuators can be estimated without modeling the dynamics between thepiezoele tri ctu tor and the host structure. The results demonstrate that for a perfectly-controlled structure the power consumption of the piezoelectric actuator is a function ofthe effe tive capacitance of the actuator and the voltage and frequency of the control lawoutput signal. Furthermore as control effectiveness decreases the power consumption ofthe piezoele tri actuator decreases. Fo r typical surface-bonded piezoelectric actuatorsthe power required for active vibration control can be conservatively determined within90 accuracy without modeling the structural dynamics of the actuator and hoststructure. Also a non line r behavior in the capacitance of piezoelectric actuators wasidentified. This research revealed a 0.44 increase in capacitance per volt or an 88increase in c p cit nce at the maximum operating voltage of the actuator. A method ispresented to ccount for the non-linearity. Also problems and associated solutions thatare en ountered with using piezoelectric actuators for control are discussed;


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KNOWLEDGEMENTSThis work was suppo rted by NS Grant NCC1208. I would like to extend specialthanks to Anna Mc owan for giving me drive and motiva tion in my research and for allthe ex tra support that went beyond her respons ibilities as my advisor to make my careeras mem orable enjoyable and successfu l as it has become. Thanks to Carol Weisman andJennifer Heeg for the ir he lp and suppor t throughou t my research. Also Thanks toeveryone in the eroelas ticity Branch at NASA Langley that made my time there moreen joyable. I also extend my app reciation to Rob ert Tolson for giving me this opportun ityto take part in the JIAFS program and to Tom Noll and Boyd Perry fo r allowing me toconduct my research at the eroelasticity Branch. Finally I want to thank Steve fo r be ingthere through the good and bad and helping me maintain som e sani ty ove r the past twoyears.


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T BLE OF CONTENTS

BSTR CT

CKNOWLEDGEMENTS iii

T BLE OF CONTENTS iv

V RI BLES ND NOMENCL TURE vi

LIST OF FIGURES ixCH PTER : TNTRODUCTION

1 Motivation Piezoelectric Actuators 2

1 2 ctiveControl with Piezoelectric Actuators 41 3 Power Amplifiers 71 4 Objective 91 5 Outline 1

CH PTER 2: LITER TURE REVIEW 122 Introduction to Literature Review 122 1 Electro Mechanical Impedance Model 122 2 Rayliegh Ritz pproximation 132 3 Summary 14

CH PTER 3: N LYTIC LDEVELOPMENT OF POWER 163 Introduction to Analysis 163 1 Solutions of Piezoelectric Constitutive Equations 19

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3. la Stress Strain Relationsh ip 203. lb Solutions For Curren t 21

3.2 Actua tor Strain 233.2a Structural Equation of Motion 253.2b Equation of Motion and Actua tor Strain 26

3.3 Struc tural Control 283.3a Perfect Control 29 .3b Imperfect Control 303.3c Boundar ies on Admittance 32

3.4 Piezoelectric Power Required fo r Active Control 35CHAPTER 4: RESULTS 38

4.0 In troduc tion to Resu lts 384.1 SingleDegree Of Freedom Test Structure 394.2 Non Linear Capac itance 414.3 Open Loop Excitation 454.4 Feed Forward Control constant am plitude excitation 474.5 Closed -Loop Con trol strain-feedb ack 53

CHAPTERS: CONCLUSION 57APPENDIX : Electrical Ana lys is 59APPENDIX 2: Non Linear Capacitance Plots 65APPENDIX 3: ACX Actuator Material Prope rties and Geometry 66REFERENCES 69


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VARIABLES AND NOMENCLATUREC Effective capacitancec Damping of host structure Electric displacement

Piezoelectric constantB Electric fieldFe External forceFa Piezoelectric actuator forceG g Generic variableh ctuator thicknessI Currentk Stiffness of host structureka Effective stiffness of piezoelectric actuatorL ctuator lengthm Mass of the host structureP PowerQ Charge Strain of piezoelectric actuator Complex variableT Stress acting on piezoelectric actuatort TimeV Voltagew ctuator width

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x-directio n, displacement in x-diIectionY E]ectrical adm ittance Piezoe lec tric elastic modu lus y-d irectionz z direction

Dielectric cons tant Phase angle

Natural frequency Radial frequency amping coefficient

Der iva tive of g with respect to tRe[G ] The real component of GIm[Gj Imaginary component of G The magnitude of GG* tU om plex conjugate of G4gt] Laplace Transform of gt

EOM Equation of motionS OF Single degreeo ffreedomMOF Multipledegree offreedom


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T Piezoelastic eroelastic Response Tailong InvestigationCRO T ctively Controlled Response Of uffet ffected Tails

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LIST OF FIGURES

Figure 1.1: illustration of Piezoelectric Effect. Elemen t excited in thickness 2with displacem ent in extensionFigure 1.2: Illustration of PZT Actuator Typesnote: or ien tat ion of axis 4change.Figure 1.3: Diagram of Moments Gene rated by Piezoelectric Actuator. 4Figure 4: General Diagram of Con troller mplifier Structure Setup. 5Figure 1.5: illus trat ion of Internal Structure of P RTI Wing 6Figure 1.6: PARTI Wing With Aerodynam ic Shells. 6Figure 1.7: Photograph of Internal Structure of P RTI Wing. 6Figure 1.8: F-18 Wind -Tunne l Test Model With PZT Actua tors Embedded In 7Vertical Ta il.Figu re 1.9: Vertica l Ta il Embedded With PZT Actuators. 7Figure 3.1: Diagram of Components in Piezoelectric Admittance 8Figure 3.2: Il lustration of Piezoelectric ac tua tor orientat ion 19Figure 3: Figure 3.3: Simplification of Cantilever Beam to SDOF Structure. 25Figure 3.4: Diagram of Actuator and External Force. Figure 3.5: Strain and Voltage Relationship for Active Control 33Figure 4.1: Picture Of Single-Degree-of-Freedom Test Structure 39Figure 4.2: Illustration Of Test Setup 40Figure 4.3: Capac itance vs. Excitation Voltage for the Actual, na lytical and 42the Empirical Model of CapacitanceFigure 4.4: Open-Loop Excitation Response of Admittance and Strain to 46Voltage Response. Both Plots Have BeenNormalized by MaxValues.Figure 4.5: Diagram of Open-Loop Control Law. 49


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Figure 4.6 A top and Boom The PSD of Strain for Test A and Test B 50With Piezoelectric Actuator On and Off.Figure 4.7: Relative dmittanceBetween Empirical model, Test A and Test 51

B.Figure 4.8: Percent Error Between the Empirical Model And Tests A and B. 52Figure 4.9: Diagram of losedLoop Control Law. 53Figure 4.10: Frequency Response of Calculated Maximum Admittance and 54Actual dmittanceFigure 4.11: Frequency Response of Estimated Admittance and Actual 55

dmittanceFigure 4.12: Plots of Actual Power and Estimated Power vs. Frequency. 56

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CHAPTER 1: INTRODUCTION1.0 Motivation

In recent years a new category of engineering optimization has evolved referred to ssmart structures Smart structures are step in engineering evolution from thetraditional static structures to more active kinetic structures. Smart structur estransgress from the optimization of static structures with material and structural eff iciencyto the passive or active control of the structures state displacement and rate andproperties stiffness and damping by utilizing kinetic ab ility through extensive actua tor,sensor and processor array s. This will al low smart structures to adapt to changingboundary conditions, expanding the capacity and efficiency of the structures under ava riety of conditions. 1

Smart structures have been researched n many aspects of engineering includingapplications in medical technology, civil structures and ae rospace The materials foundto be most succes sful are shape memory alloys and piezoelectric materials 1,2 . Shapememory alloys SMAsa re characterized by the abili ty to return to pre-stressed shapewith applied heat. SMAs typically have a bandwidth on the order of one hertz or lessand are suitable for shape control and passive vibration con trol For instance LockheedMartin is investigating the feasibility of using SMAs to act ively control the twist of naircraft wing to increase the flight envelope of the aircraft 3. The second group of smartmaterials are piezoelectric materials. Piezoelectric actuators induce strain or force whenexposed to an electric field With a bandwidth on the order of severa l kilohertz,piezoelectric actuators are more su itable for acoustics and vibration control .


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he cur ren t research is an investiga tion of the power requirem ents of piezoe lectricac tua tors during ac tive vibration control Whether piezoelectric actuators require nexcessive quantity of electrica l power s concern in assessing the feasibility ofpiezoelectric actuators utilized fo r active con trol. First, a brief introduction ofpiezoelectric materials will be discussed. Then the types of piezoe lec tric actua tors andhow they are applied will be introduced. Also, previous research tha t demons trates thecp ility of piezoelectric actua tors will be presented. Next, an overview of past effortsin iden tify ing piezoe lectric power characteristics will precede the analysis and anexperimental verifica tion will conc lude the paper.

1.1 ie zoelectric ctuatorsThe piezoelectric effect is the capacity of the materia l to change in dimension whenexposed to an electric field or gene rate a charge when exposed to a stress or strain. Theabi lity to coup le electrical energy with mechanica l energy is the un ique behavior tha tmke piezoelectric materials ideal for actuation and sensing Figure 1.1.

Electrical Energy Mechanical EnergyActuatorolt ge Strain

SensorPiezoelectcWafer

II_ AxDieleesric Dipoles \Conductive Elements

y APPLY VOLTAGEx igur : llustrtion of Piezoelectric Effect Element Excited in Thickness With Displacement inExtension


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PZT Stack Actuator PZT Surface-Bonded Laminar ctuator

PZT eenI

PZT

Y 2 :Ix I

Figure 1.2: fllustration of PZT Actuator Types note: orientation of axis change

1.2 Active Control with Piezoelectric ActuatorsUsing surface bonded laminar actuators, a net moment can be induced on a plate orbeam-like structure by adhering the piezoelectric actuato rs to the top and bottom face ofthe plate. By exciting the two actuators 180 degrees out of phase, a ne t moment isgenerated about the neutral axis of the beam or plate Figure 1.3 .

F Piezoelectric forces due to an applied voltage Resultant moment applied to structurenet

Figure 1.3: Diagram of Moments Generated By Piezoelectric Actuator A diagram of the general test setup during active vibration control is shown in Figure 1.4.

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Digital Controller StructurePowerPiezoelectric Actuator

Figure 14: General iagram Controller AmplifierStr ucture Setup

The benefits found in us ing piezoelectric ac tua tors fo r active control have been displayedin numerous ground tests and wind-tunnel demonstrations 4-16. The curren t reserchwas motiv ted by the Piezoe lec tric Aeroelastic Response Ta iloring Investigation PARTI4,5.. Dur ing this study, a four foot long semi-span wing model with 72 distriutedpiezoelectric actuators was used to demonstrate that piezoe lec tric ac tuation canef fectively control aeroelastic response. fliustrations and pho tog raphs of the PARTI wingare supp lied y NASA Langley Research Center Figures 1.5, 1.6 and 1.7 .


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igure 6: P RTI Wing With erodynamicSlreiIs

igure 5: Illustration Internal Structure of P RTI Wing

igure 7: Photograph of Internal Structure ofPkRT Wing

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A more recent program is the ACROBAT program 6 where piezoelectric actuators weresuccessfully used to reduce damaging vibra tion in the ve rtical stabilizers of an F-18 wind-tunnel test model Illustrations and photographs of ACROBAT test model are suppliedby NASA Langley Research Center F igures 1.8 and 1.9. Presently a full scale F- aircraft is being fitted with piezoelectric actuators to demonstrate that the actuators areeffective on full scale aircraft.

1.3 Power Amplifiers

The following analysis aiid verification will determine the power requirements of apiezoelectric actuator used for act ive vibration con trol Although this is the powerrequired by the actuator this is not necessa rily the wall power required to power theactuators Wall power refers to the amount of power required by the power amplifiers topower the aGtuators or thepower running through- the plug of the power amplifier Theana]ysis will show tha t the admittance of the actuator is primarily due to a capacitance

Figure 8: F18 WindTunnel TestModel With Figure 9: Vertical Tail Embedded With PZTPZT Actuators Embedded In Vertical Ta il Actuators


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oad. Capacitors require a reactive power, identified by the 90 degree phase lag betweenthe current and voltage. A description of reactive power is given in Appendix ifferent amplifiers power the reactive load in different ways, the result is a drastic

difference in the efficiency of power amplifiers. Efficiency of the amplifier refers to theamount of power generated relative to the amount of power required to generate a signal.

There are currently three types of power amplifiers available for piezoelectric actuation:linear amplifier, switching amplifier Piezo Systems, Inc. and switching capacitoramplifier Lucent Technologies, 28 . inear amplifiers are the more traditional amplifiersused in most piezoelectric applications. Due to the reactive load, of the piezoelectricactuators noted above, a new generation of power amplifiers have been developed,tailored to power the reactive load: switching amplifiers and switching capacitoramplifiers.

inear amplifiers are the least efficient because of the method used to amplify the reactivesignal. The linear amplifier charges the capacitor, or piezoelectric material, with theentire power output. On the discharge, all the stored charge in the capacitor is dumped toground. This is the mechanical equivalent of compressing a mass-spring system, takingthe mass off the spring to let the spring expand, and lifting the mass onto the spring again

only to repeat the process. The result is a tremendous loss in potential energy.

witching amplifiers and switching capacitor amplifiers are designed to store thepotential energy so that the energy can be used again to recharge the capac itor. By doingthis, the potential energy in the capacitor is maintained and the overall powerconsumption is drastically reduced. The switching amplifier and switching capacitor

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amplifier use different mechanisms to perform this task. Details of these mechanism areno t available at th is t ime because the amplifiers are a new technology and detailedinformation on the amplifiers is not ye t available. For the same reason the effectivenessof these amplifiers in the control of piezoelectric actuators is also unknown. inearamplifiers demonstrate good control behavior including large bandwidth low noiseconstant gain and negligible phase loss and are still the amplifiers of choice at this time.

In summary the equation of power developed in the Analysis Section applies to the linearamplifier. For the new generation of switching amplifiers and swithing capacitoramplifiers the reactive power requirements of the piezoelectric actuator would becomenegligible and the overall power loss would be du e to losses in the electrical system. Theelectrical system refers to the power amplifier and the piezoelectric actua tor. Thusknowing the type of power amplifier being used is crucial to determining the total powerconsumption of piezoelectric actuators.

4 ObjectiveOne critical issue raised by these studies is whether full scale airplanes and other largestructures will require excessive power to use piezoelectric actuators for active vibrationcontrol. Piezoelectric materials are complex to model as actuators due to thepiezoelectric effect that defines the electro mechanical coupling within the actuator. Thepiezoelectric power characteristics become a function of the material properties of thepiezoelectric actuator as well as the mechanical dynamics between the actuator and thehost structure. Modeling the dynamics between the piezoelectric actuator and the hoststructure has proven to be an extremely complex task to complete for structures as large


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or larger than the PARTI wing 17 ,18. Thus, piezoelectric power charac terization ofactuators control ling complex structures may be ex tremely complex to determ ine.

or simple plate and beam models with a few piezoelectric ac tuators, the actuator andhost dyn amics are reasonable to solve. Realistically, the host struc ture will be much morecomplex than a simple beam or plate and will require complex finite elemen t analysis todevelop an estimate of the actua tor and host structure dynamics.

The ob jec tive of the current research is to more close ly define the individual contribu tionsof the mater ial prope rties and mechanical dynamics on the power characteristics of thepiezoelectric actuators during the spec ific case of ac tive vibration control. The goal is todetermine the overall nece ssity of modeling the complex electro-mechan ica l re lationshipbetween the actuator and the hos t struc ture .

1.5 utlineThe following literature review will introduce the two founding techniques available forcha racterizing piezoelectric power consumption. Liang et. al. 19-22 and Hagood et. al.23 developed the techniques used by most researchers investiga ting piezoelectric powerconsumptionBoth techniques are shown to be highly accurate using two distinctlydifferent app roaches. The po sitive and negative attributes of each will be discussed. Athi rd techn ical reference is from Warkentin 24 , who used the techniques presen ted by agood et. al and applied them to closed-loop control.

All three researdh efforts requ ired an accura te model of the dynamics of the actuator andhost stru cture. The analytical section of this paper extends Warkentins model and proves


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that the influence of the ac tuator and ho st structural dynam ics are negligible inhr terizing piezoele tri power consumption fo r the specific case of active vibrationcontrol.

The experimeiital sec tion will cover in detail the verification of the ana lyt ical mode]presen ted A single degreeo f freedom cantilevered beam is used with a strain feedbackcontrol law. In add ition severa l re la ted topics will be discussed tht include non linearmaterial properties and power amplifiers for piezoe lectric actuator use.


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HAPTER 2: LITER TURE REVIEW2.0 Introduction to Literature ReviewAlthough piezoelectric actuators have been investigated extensively for active control,research on piezoelectric power consumption is limited to essentially two groups o fresearchers; Liang et. al. 19-22 and Hagood et. al. 23 . The pros and cons of eachtechnique will be discussed.

2.1 Electro Mechanical Impedance ModelLiang et. al. 19-22 developed a piezoelectric electro-mechanical impedance model.This electro mechanical impedance model was derived from the equation of motion ofthe piezoelectric actuator which was modeled as a plate in longitudinal excitation. Theboundary conditions were prescribed by the piezoelectric constitutive equations andcoupled with the mechanical impedance of the host structure. Mechanical impedancerefers to the ratio of force input to structural velocity, or the ratio of input to response.Thus, the dynamics between the actuator and host structure are directly coupled. Thesolution includes a coupling between the electro-mechanics of the piezoelectric actuatorand the mechanics of the host structure through the impedance of the actuator and hoststructure. Liangs electro mechanical impedance model was then applied to determine thepower characteristics

Liangs model only investigated the case of open-loop excitat ion. Open-loop excitationre fe rs to the case when the piezoelectric actuator is used to excite motion in the structurewithout feedback The major benefits of Liarrgs mudel are the explicit solutions th-atallow the investigation of the electrical and mechanical energy exchange between actuator

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and host structure. Also by modeling the equation of motion of the actuator Liang wasable to define the region where the actuator behaves linearly and the reg ion where theactuato rs mechanical dynamics affect the contro l authority of the ac tuator. Liangs modelis an accu rate mode l of piezoelectric actua tors but it is difficul t with implement tocomplex structures . Also the solu tion is based on open loop exci tation Applyingclosedloop con tro l or active vibration control proves to be extrem ely complex.

2.2 Rayliegh Ritz Approximation

An alternative method is based on the deve lopm en t by Hagood et. al.. Their approachdetermined the equation of motion of the host structure excited by the piezoelectricac tuator. Using a Ray leigh Ritz approximation to couple the beam with the piezoelectricac tuator bonded to it the strain of the actuator can be determ ined. Incorporating thestra in of the actuator the piezoelectric constitutive equations are used to de termine thepower cha racteristics The difference between the two method s is that Liang solved fo rthe explicit equation of motion of the actuator bonded to a host structure where Hagooddetermined the equation of motion of the hos t struc ture exc ited by the piezoelectricactuator while igno ring the mass characteristics of the piezoelectric actuator. AlthoughHagoods model is not as accurate in modeling the piezoe lec tric ac tuator the modelsuff iciently couples the dynam ics between the ac tuator and the host structu re for mostcases. Also th is model can be readi ly appl ied to complex structures and closed loopcontrol .

A comprehensive study was presented by Warkent in of MIT. Warkentin presen ted adevelopment of piezoelectric power con sumpt ion for active vibra tion control using the


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n lytic l models developed by Hagood et. al.. Warkentins model directly addresses thech r cteriz tion of power for piezoelectric actuators during active vibration control.

2.3 umm ry

ubsequent research concerning the power consumption of piezoelectric actuators isprim rily founded on the two previous methods 14,15,16,24 . The conclusion of allprevious rese rch for both the excitat ion of a structure and the closed-loop control of astructure is that the electrical power of piezoelectric actuators is a function of themech nic l motion o f the structure and the electrical characteristics of the piezoelectricmaterial . Modeling the mechanical motion is the limiting factor in applying thesetechniques for active vibration control. As structures controlled by piezolectric actuatorsbecome more complex, modeling the mechanical motion of the actuator and hoststructure becomes extremely complex.

The current research takes Warkentins results a step further. An n lytic l model isdeveloped that shows the influence of the mechanical motion of the host structure arenegligible on the power characteristics of the piezoelectric actuator used for activevibr tion control . When comple te ly controlled, the structure is motionless, thus thepower requirements of the piezoelectric actuator are no longer a function of themech nic l motion of the structure. In th is ideal scenario, the power is only dependent ongeometry and material properties of the piezoelectric actuators and the voltage andfrequency of the control law signal. Furthermore, the current research finds that ascontrol effectiveness decreases, the power requirements of piezoelectric actuators

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CHAPTER 3: ANALYSIS

3.0 Introduction to AnalysisThe analysis is composed of the derivation of piezoelectric power consumption for thespecific case of active vibration control. Appendix defines important electricalquantities and tools required for developing the analysis. For an electrical system withfrequency content, the most straightforward way to characterize power requirements is todefine the electrical admittance of the circuit.

1 w 3.1wThe electrical admittance, Y w , defines the linear relationship between current, 1 w andvoltage, V w , in the frequency domain. Frequency is designated by w. dmittanceisrelated to power, P , through. the equation:

P oi Y w Vw 3.2The ana lysis will focus on the development of the admittance which can then be appliedto the equation of power . The admittance is determined in Section 3.1 f rom thepiezoelectric constitutive equations. Because piezoelectric actuators are electromechanical devices, the admittance of piezoelectric actuators is made up of an electricalcomponetit and an electro mechanical component as shown by this generic equation ofadmittance for piezoelectric materials.

Y c,. = electrical + electro-mechanical

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The electromechan ica l component couple s the motion of the ac tuator with the electricalcharacteristics. For an actua tor bond ed to structure the motion of the struc ture will becoupled with the electrical characteristics o f the ac tua tor s well. In fact the admittancecan be broken into the two physical app lications that piezoe lec tric mater ials are used foractuator and sensor:

Y oji actuator sensor

Here the electrical component is iden tified as the actuator and the electromechanicalcomponent is iden tified as the sensor . The solution for the electrica l or actuatorcomponent is developed direc tly from the cons titu tive equations of motion show n inSection 3.1. The electromech anical or sensor compo ne nt will be the predominant focusof the analysis. The electromechan ica l component will prove to be appl ication specific .For each application a model of the piezoelectric ac tuator and host dyn amics isnecessary, Figure 3.1 . For the current research the spec ific case of activ e vibrationcon tro l is investigated where the model of piezoe lec tric ac tuator and host structuredynamics predicts the respon se of the struc ture due to a rando m load and the response ofthe structu re due to the piezoelectric actuators be in g e xcited by specific con trol law.Due to the vast array of vibration co ntrol applications available the development of ageneric solution that is applicable to all ac tive vibration control system s is difficult toachieve.


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Electrical or ctuatorompon nt

Electro-Mechanicor SensorComponent

After thorough ana lysis and experim ental testing the sensor co mpo nent is fou nd to haveneg ligible in fluenc e on the overall admittance during active vibration con trol Thecon clusion of the analysis is that an accurate: pred iction of the adm itta nce and hencepower can be de termined without modeling the effects of the sensor com po nen t in thepiezoelectric admittance le ving just the actua tor com pon en t to solve for As will be seenin the analysis the ac tuator co mpo ne nt is only made up of material constants andgeom etry which drastica lly simplifies the solution of the piezoelectric adm ittance foractive vibration control The ana lys is begins in Section 3 with the dev elopment of the basic piezo electricequatio ns from the piezoelectric cons titutive equations including force strain and dmitt nce Se ction 3 2 de velops the equation of motio n of the ac tua tor bonded to asimple struc ture The equation of motion is then app lied in Section 33 to investigate thecase of active vibiEtion control Section 33 is broke n into sections The first sectioninvestigates the case of perfect control where struc tural motion is completely sup pressedThe second sectio n inve stigtes the more rea lis tic c ase of imperfect control where motionis not completely suppressed The th ird section investigates the est and worst casescena rios for structural control which are then app lied to the equation of admittance to

Response of Structure toExternal Force random

igure 3.1: iagram of Components in Piezoelectric dmittance

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7 or

determine when adm ittan ce is maximum for active cont rol. The analysis concludes withthe equation of power for piezoelectric ac tuators used for active vibratio n cont ro l3.1 Solutions of Piezoelectric Constitutive Equations:

The constitutive equations for a piezoelectric actuator 19 excited in the zdirection anddisplaced in the xdirection are o rien tat ion of actuator is illustrated in Figure 3.2:

=d

Conductiveelem en ts

3.3

3.4

y,2

material

x, I

z,3AL

igure 32: illustration of iezoelectric actuator orientation

As noted earlier th is paper is investigating surfacebonded lam inar actuators; thereforedisplacement in the i axis and y-axis are considered negligible and will not be consideredfor this derivation. Dielectric and piezoelectric losses are also considered negligible. Theelectric field , E, is defined as voltage divided by the piezoelectric material thickness: E

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z The strain in the actuator is designated by S or . The stress T acting on the

actuator can be determined from equation 3.3.T=Yd 3.5

3 designates the electric displ cement which is defined as the charge per unit area. Thech rge is determined by integrating the electric displacement over the surface area of thepiezoelectric actuator.

3.6Q= Dxdy

The current is determined by the time derivative of the charge.

3.7

Several rel tionships can be defined from these equations.

3.la Stress Strain Relationship

Prom equation 3.5 the relationship between stress and strain is developed. The blockingforce of the piezoelectric actuator is defined as the force output of the piezoelectricctu tor at zero strain.

b1ocking =TS =0 x re =dwh =wd 3.8The strdke of the actuator is determined by sett ing the stress to zero.

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ST d 3.9For a short-circuited piezoelectric ac tuator when bo th electrical po les are grounded or V 0) the stress strain relationship is defined by Hooks Law.

T=Y1 3.103.lb Solutions For Current

From equations 3.4, 3.6 and 3.7, the charge and the cu rrent are developed:

QjDxdywL[edJ 3.11i t==--[EdwL 3.12

The equat ion of cur ren t can be transformed into the complex domain using a Lap lacetransform allowing fo r easier interpretation . The Laplace transformof time derivativewith initial cond itions of zero is defined as:

Ed 3.134[-a_Gt j sGs

Where signifies the complex variab le. The solu tion for curren t becomes:Ys ) sfl s s[ss + s)]wL 314

Introducing the so lutIon of the stress, equation 3.5, the current can be rewritten.


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I s = .c[ 833 d. s + s ]wL 3.15Defining the electric field in terms of voltage, E = V/h, and dividing the equation ofcurrent by the voltage, the admittance of the piezoelectric actuator is determined.

I s wLV s h V s 3.16

Equation 3.16 shows that the equation of admittance is made up of two parts. The firstpart is a group of constants that define the electrical, or actuator, component and isdefined by the material constants and the geometry of the piezoelectric actuator.

electrical actuator = s } wL 3.17

The second part of the equation of admittance designates the electro-mechanical couplingof the actuator. The electro-mechanical coupling designates the relationship between themechanical strain and the voltage applied to the actuator, or the sensor component of theadmittance.

S selectro-mechanical sensor = w d V s

By solving the equation of admittance for zero strain, Sj = 0, the effect of the actuatorcomponent on the admittance is found to act as the effective capacitance of thepiezoelectric material.

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O s s[8 dY] s{effective capacitance]3.19

The effective capacitance will be de fined as the capac itance C of the actuator . Theeffective capacitance is made up of dielectric piezoe lec tric and elastic properties of thepiezoelectric actuator. As a note: the capacitance quo ted by mos t manufacturers isdefined on ly by the dielectric prope rties of the piezoelectric actua tor. In this papercapacitance will refer to the effective capacitance C defined in equation 3.19. Theequa tion of admittance can be rew ritten as:

sjs 1 3.20Ys =s[C+wL.d sjEqua tion 3.20 shows the admittance is composed of the effec tive capacitance designatedby the constant C and the electro mechanical term or sensor componen t. The electromechanical term as defined in equation 3.18 is a function of the ratio of actuator strainto app lied vo ltage. The solution fo r the electro mechanical term is determined by theboundary cond itions acting on the ac tuator. The solution of the strain of a piezoelectricactuator bonded to a structure is now investigated .

3 Actuator Strain

The actuator strain is determ ined by the in teract ion of the actuator and host structure.The strain of the actuator is assumed to be the strain of the beam at the actuator bond;therefo re the strain of the ac tuator is determ ined from the equa tion of motion of theactuator and host structure. The equation ofmotion will include the dynam ics of the


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structure, the actuators bonded to the structure, the force input of the actuator and theexternal force acting on the structure. For simple beam or plate like structures, a ayleigh itzmodel can be applied to determine the equation of motion of the structurewith piezoelectric actuators. 23

As the structure becomes more complex, the solution of this relationship becomes moredifficult to determine. For instance, the motivation behind this research is to characterizethe power requirements of the PARTI wing which consists of 72 individual actuatorsacting on a complex, plate-like structure Figure 1.4 . A model of the coupled motion ofeach individual actuator acting on the wing during active control would be necessary tosolve the equation of motion. The model would include an estimate of the actuator forceacting on the wing, an estimate of the external force, which is both random and nonstationary, and an estimate of the response of the wing to these forces. Also, a closed-loop control scenario must be applied to the model to simulate active vibration control.The time and energy required for this solution would be extensive and questionablyaccurate.

To build a better understanding of the effects of the strain to voltage response of theactuator and host structure on the admittance, a single degree of freedom model isinvestigated. The simplified model allows for a more straightforward investigation thatcan later be applied to more complex structures. Thus, the beam with piezoelectricactuators is simplified as a SDOF mass, spring damper system. Figure 3.3

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ctuator Fa

Figure 3 3 : Simplification of antilever eam to SDOF Structure

First , the equation of motion of the SDOF structure in Figu re 3.3 will be deve loped.Theit the strain of actuator will be de termined froththe equation of motion befo reinvestigating the pecific case of activ e con tro l.

3.2a tructurl quation of Motion

For a single degree of freedom system exc ited by an externa l fo rce and a piezoelectricactuator the :equation ofmotion of the. structure is:

3.21

Where m and k represent the mass , damping, and stiffness of the host structureaccordingly. The actuator fo rce js designated by Fa and the ex ternal fo rce is designatedby The mass and damping contribut ions of the piezoelectric actuator are assumed tobe neg ligible The force of the ac tuator is de term ined by the cons titu tive equa tion ofstress equat ion 3.5:

F, =TxAreaw 3.22

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The constant, ka, designates the effective stiffness the actuator contributes to the host-structure. The equation of motion of the actuator-host structure Equa tion 3.21 can berewritten.

mx+ cx+k + ka = + Fe 3.27

The normalized equation of motion is:

3.28x+2coxco;x= +F/mThe solution for the equat ion of motion can be determined by taking the Laplacetransform of Equation 3.28 :

3.29ms + w + 2 s + F s

Using Equation 3.25 to defin e the strain of the actuator witli,the motion of the host.structure, the strain of the ac tua tor is determined

mL s3

Applying the definition of the actuator blocking force and dividing by the voltage, therat io of strain to voltage is de termined

.Fs 3.31 s + + 2 wd +


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s was expected from quation3.31 it can be concluded that the strain to voltage ratio isa function of the motion of the actuator host structure in response to the control force ofthe piezoelectric actuator and the external disturbance force.

Instead of solving for the explicit solution two cases of structural control areinvestigated perfect and imperfect control to gain a better understanding of the electromechanical interaction between the actuator and the host structure.

3.3 Structural Control

The solution would traditionally be determined by solving the strain to voltagerelationship developed in the previous section Equation 3.31 which is applied to theequation of admittance E4uatiorr3.20. Instead two cases of tructurai control areinvestigated perfect. contrGiand.imperfect control. Perfect structural control refers tothe ideal case where strain throughout the structure is completely suppressed.rnperfectcontrol refersto the.morerealisticcase.where str.aihthroughout the. structure is no tcompletely suppressed. Investigation of these two cases will reveal that an upper andlower bound on the admittance can be defined. The analysis wil l show that the maximumpower is consumed when the structure is perfectly controlled; and as control authority islost the power requirements will decrease. A minimum admittance is defined by thestructural strength of the piezoelectric actuator which corresponds to the worst casecontrol scenario.

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3.3a Perfect Control

As mentioned above, fo r a perfect lycontrolled structure, the strain th roughout the hoststructure is equa l to zero. As indicated by quation 3.25, the strain of the actuator willequa l zero for this case. The actuator and host structure strain are equal to zero when thefo rce generated by the piezoelectric actuator is equal and oppos ite to the external force asshown in quation 3.30:

7s s 3.32The maximum commanded force of the piezoelectric actuator is defined by the maximumblocking force:

maximum s s 3.33

The maximum voltage, s , can be defined by two lim itation s, the limitation of theactuator or the limitation of thecorrtroiler: lim ita tion of the actuator isthebreakdown vol tage of the piezoelectric material. The breakdown voltage is the voltagethat will destroy the piezoelectric material. The maximum voltage may also be definedby the lim itat ions of the controller, or the maximum voltage the controller can supply tothe ac tuator.

For per fec t control, the external force mus t be less than or equa l to the max imum actuatorblocking force:


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Ps wd1s 3.37Blocking Force,

Fe = Whd3lVmax

External Force,Fe = f

ig ur 34: IJiagrarn of-Actuator and xternal Force

The transform of the strain to:voltage Equation 3.31 can be simplified as:

sfss mL s +2w s+o .38

Applying this to the equation of admittance Equation 3.20 , the admittance fo rimperfec t control is:

f s wdY..Y s = C Vmsms2sj 3.39

The first important detail to not ice is the 180 degree phase shift between the added force ,fe and the excitation voltage which is designated by the negative sign before the force tovoltage rat io This relationship is logical because the excitation voltage is designated tooppose the force by the cont ro l Jaw defined in Equation 3.33 . Thus, the magnitude of the

Imperfec t Cont rol

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dded force, fe, actually reduces the total admittance of the actuator. Accordingly,assuming that the maximum capability of the piezoelectric actuator is used for control, ascontrol effectiveness decreases, admittance and thus power decreases.

3 3c Boundaries on Admittance

From the relationship defined above, an upper and lower bound for admittance can bedefined. For perfect and imperfect control, the admittance is defined as:

Perfect Control Imperfect Control ? s wds sC = s[C_ : S ,n s 12as .w j

Thusthe maximum-admittanceis defined for the perfect control scenario:

3.40

From the imperfect control section, the admittance will decrease as structural control is]ost or fe increases . From Equations 3.38 and 3.39, the admittance will decrease themost at the maximum strain to voltage ratio

3.41Y s = s[ _wli ]

The negative sign in Equation 3.41 corresponds with the phase shift defined in Section3.3b.

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To determine the lower bound on the adm ittance, the maximum strain to voltage ratiomust be determ ined The maximum strain is de termined by the ultimate strength of thepiezoelectric material If the strain in the piezoelectric actuator exceeds the ultimatestrength of the material, the actuator will fail arid pow er will no longer flow through theactuator. The vol tage is determ ined by the contro l law. However, assuming the capacityof the piezoelectric actuators is exhausted and is used, then the maximum strain tovol tage ratio is de fined by:

i s MAX

f silur s

u x s

increasing

34

Figure 5: Strain and Voltage Relationship for ctive ontrol

pp lying this relationship back into the equation of admittance fo r imperfect con trolEquation 3.39 :

s = SJC ailures ]

n xs

The lower boundary of the admittance is now defined. The mos t impo rtant detail-tonotice is that both the upper bound and the lower bound of the admittance are de fined

3.43


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independent of the dynamics of the host structure. Thus, the admittance of a piezoelectricctu tor can be estimated within a region for active vibration control. or example, usingan ACX QP4OW PZT actuator Appendix 3), the percent difference between the best andworst case for the admittance is determined.

S s 3.44failurc \ IW dpercent difference

x 100 10

Even at the worst case scenario for vibr tion control, there is only a 10 reduction inadmittance. Thus, the boundaries for the admittance are defined between the equation forperfect control, Equation 3.35, and the minimum admittance, Equation 3.43. Thus, formost control situations, a conservative estimate of the admittance.can be m de by.neglecting the influence of the additional external force, fe Note , the above estim te ofadmittance assumes two things; the actuator is excited at the maximum voltage and thatthe actuators is used for vibration control. This estimate of error is not valid unless theseconditions are met. The est imate of admittance is now defined and can be applied to theequation of power mentioned earlier.

s cC 3.45eslirnaie \

Where this estimate is always greater than the actual admittance with greater than 90accuracy for active vibration control.

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3.4 Piezoelectric Power Required for Active Control

This sec tion will present the equations for predicting the power requ ired of piezoelectricactuators du ring active vibration control. Add itional cons ide rations associated withpiezoelectric power are also discussed. onse rvat ive calculations of the powerconsumption of piezoelectric actuators can be determ ined withou t direct con sidera tion ofthe dynamics of the actuator and host structure. The results of Sec tion 3.3 indicate thatthe admittance can be determ ined with greater than 90 accuracy by neglecting thesedynamics. Transforming Equation 3.45 from the complex domain to the frequencydomain, the admittance becomes:

wL 3.46Yco) jisd=VjcUCImIy J1 ph ase angle, p tan 2

Transform ing the admittan ce back to the time domain and solving fo r the current:

dV 3.47 It=C-

For a voltage signal with sinusoidal motion, Vft=Vsinut, the power becomes:

Pt =--oCVsin2at 3.48

quation 3.48 is a conservative estimate of the piezoe lec tric pow er con sumption foractive control given the capacitance of the actuator and the voltage and frequency of the


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ontrol law output signal Th e capacitance of the actuator is determined from Equation

3 19 For multiple piezoelectric actuators connected in parallel the total capacitance isthe sum of the capacitance of each piezoelectric actuator:

l w 9

C = Effective Capacitance ofActuator

n = Number of Actuators eing UsedThe ability to predict the maximum power consumption of piezoelectric actuators can beextremely useful in the design of structures that will use piezoelectric actuators for activecontrol For example: once an estimate of the number mdtype of piezoelectric actuatorsare established the magnitude of the maximum power required to control vibration on a

large airplane wing can be calculated as follows:

1rnxQ = 3 50

Where o is the maximum frequency of the control law output signal Generally thisfrequency is defined by the frequency of the highest mode of interest for control purposesV is the maximum voltage that can be appliedto the actuators This is determined byeither the maximum voltage the piezoelectric actuators can withstand or th e maximumvoltage the controller can generate

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The benefits of this analysis can be seen in Equation 3 48 and 3 50 where there is nodependency on the structural dynamics between the actuator and host structure Thecomputational effort in characterizing the power requirements of a piezoelectric actuatorduring active control are drastically reduced from previous models of piezoelectric powerMaking the application and development of piezoelectric technology more accessible

An additional consideration in calculating pwer is that traditional power amplifiers are not designed for a capacitance load like that of piezoelectric actuators or this reason

the total power consumption is based on the combined magnitude of the imaginary powerand real power or the apparent power A detailed description of imaginary and realpower is given in ppendix As noted in the introduction a new generation of poweramplifiers are being developed that are capable of storing the imaginary power or thepower load of a piezoelectric actuator The only power loss would be due to losses withinthe amplifier actuator and circuitry Thus application of the new generation of poweramplifiers may proveto be a drastic reductiorrinpiezoelectricpower consumptionLinear amplifiers are the most dependable and widely used amplifier at this t ime

Note: the above developments assume the capacitance of piezoelectric actuators isconstant with respect to voltage Although this assumption is commonly made theexperimental tests conducted as a part of the current research show that capacitanceactually increases with voltage Thus to accurately predict the power consumption usingquation 3 48 or 3 50 the measured capacitance of the piezoelectric actuators must beased urther descriptitn of this pherromna is uss in the x pnmlVerification chapter that follows

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CH PTER 4: EXPERIMENT L VERIFIC TION

4.0 Introduction to Experimental Verification

This chapter focuses on the experimental verification of the previous analyticaldevelopment. The conclusion of the analysis is that for active vibration controlpiezoelectric admittance can be determined without modeling the actuator and hoststructures dynamics. The admittance can then be applied to the equation of power todetermine the power characteristics of the piezoelectric actuators utilized for activevibration control.

During the experimental verification of the analysis a non linearity in the materialproperties oftbe piezoelectric actuator was discovered. The non linearity will bediscussed before the results of the experimental verification are presented because anunderstanding of the non linearity is vital to interpreting the experimental results.

The verification of the analytical results will be presented for three cases: open loopexcitation feed forward control and closed loop control cases. During open loopexcitation the piezoelectric actuators are being used to excite motion in the structurewithout feedback. During feed forward and closed loop control an external disturbanceis applied by a force shaker and the piezoelectric actuators actively inhibit the motion inthe structure using feed forward or feedback control laws.

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4.1 Single Degree Of Freedom Test Structure

For the experimental tests, a singledegree-of-freedom model was used Figure 4.1 . Themodel is a 12 inch long cantilevered beam made of an aluminum honeycomb coresandwiched between graphite epoxy face sheets. An aluminum mass was fixed to the tipof the beam to isolate the first bending mode simu]ating a single degree of freedomSDOF system. The aluminum mass reduced the natural frequency of the first bendingmode of the beam to a frequency that made data acquisition and analysis of the data moreconvenient The piezoe]ectric actuators were adhered at the root of the beam because theroot of the beam is the region of highest curvature in the first bending mode.Piezoelectric actuators are essentially strain actuators and have been shown to have themost control effectiveness in the region of highest curvature. Two ACX type QP 40WPZT actuators Appendix 3 were used. The actuators were adhered to the beam in the

same orientation as illustrated in the introductory section in Figure 1.3.

Ahiminum MassACX QP4OW 1 AccelerometerPZT Actuator

Load CellStrain Gage

Electro MagneticCantilever Beam Force Shaker

Figure 4.1: Picture Of Single Degree of FreedornTest Structure

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Figure 4.2: Illustration Of Test Setup

During closed loop testing, the external force was introduced by applying an l tromagnetic force shaker that was orientated to supply a force at the center of gravity of thealuminum mass. Force output of the shaker was measured through a load cell at the tip ofthe stinger connecting the shaker to the structure. A strain gage was adhered adjacent tothe piezoeectric actuators at the root of the beam to measure strain in the beam. Anaccelerometer was positioned at the center of gravity of the aluminum mass to measuretip acceleration. An ammeter was used to measure the current in the piezoelectricactuator circuit Figure 4.2

Digital ontroller Control Law ImpleineniatData AcquisitionD1A and AID Conversion Voltage Piezoelectric ActuatorAcceleromcier

AluminumBlock

p Excitation Signal: Linear Sine Sweep 2O 4OHz

Load CellGage

Electro-MagneticForce Shaker

Data acquisition and control was performed using an SG I workstation. The signal used tocommand the piezoelectric actuator was generated at 2 kHz and then passed through alow pass filter with a cut off frequency of 1000 Hz. Filtering the excitation signal wasnecessary due to the response of the piezoelectric actuator to a digitally generated

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excitation signal with a sampling rate below the bandwidth of the piezoelectric actuator.Digitally generated signals resemble a staircase in struc ture, or a step input at each timestep. Although this does not typically influence the con trol effectiveness of the ac tuator ,the current signal becomes extremely no isy and difficu lt to measu re if the excitationsignal is not filtered. The response of the actuator to the step input of the exci tationsignal is a spike in the current that gives the current a sawtooth appearance. Current isone of the essential quantities that defines admittance; the refore an accurate measure ofthe current was important in these experiments. This problem was solved by conditioningthe digital signal with a lowpass fil ter . The result was a smooth excitation signal thatmore closely resembles an ana log signal. Using a linea r power amplifier the filteredsignal was amplified.with a gaiirof 7 Further discussion:of the amplifier typesand characteristics will be introduced in Section 4.6. Data acquisition is performed at asampling rate of 250 Hz yielding a Nyquist frequency of 25 Hz. An anti aliasing filterwas implernented.w .ith a cutoff frequency of 100 Hz.

4.2 NonLinear Capacitance

As noted previously experimental tests revealed a nonlinear characteristic in thecapacitance of the piezoelectric actuator. Brief mention of this phenomenon was made byWarkentin 24 and a more detailed investigation was recent ly presented by Sherrit 25 .Experimental tests revealed that the capacitance of the piezoelectric actuators increaseswith voltage as indicated from Figure 4.3. Capacitance was measured by clamping thehost structure to inhibit all motion Recalling from the equation of admittance when thestrain is zero the admittance is a function of frequency and capacitance alone. The


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p it ne of the piezoele tri actuator is measured from the slope of the admittanceversus frequen y plot.

0.9U

0.8ct5 0.72

Experimentalo 0.6Analytical

0 17 34 _Empcal__8Voltage V

Figure 4.3: Capacitance vs. Excitation Voltage for The Actual, Analydcal and the Empirical Modelof Capacitance.

An empirical model of p it n e as a function of voltage was developed to model thevarying p it n e behavior. This was done by performing a first order least squaresapproximation on the experimentally determined values of capacitance. The resultsfound a 0.44 increase in capacitance per volt. Resulting man empirical model ofcapacitance:

ac 4.1Ce Ipiricd = C + --V coO 661iFarads

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0.00440

Piezoelectric actuators are commonly operated in the 100 to 200 volt range . In this vo ltrange approximately a 40-90 increase in capacitance can be expected over themanufacturers quoted va ]ue. The manufacturer s quoted value of capacitance is shown inFigure 4.3 as the analytica l va lue , 0.66 pFarads.

The empfrical solution above is a linear relationship Equation 4.1 , but the effect of thenonconstant capacitance makes the admittance nonlinear because the admittance is nowa function of voltage as well:

Yw Jco[{Co +Va - dIt is common for electrica l engineers to refer to nonconstant capacitance as non -lin ear.Not because of the relatior iship..be tweewcapac itance:and voltage, but forthe no1ineari.ty.introduced in the admittance as shown above. Because the non-linearity was introducedand accounted for in the frequency domain the ana lyt ical solution of admittance can notbe transformed in to the tim e domain. Thus, the analytical equations Of power can not bedetrmined in the time domain. This does not pose a problem in the characterization ofadmittance or power. There is no evidence that suggest that admittance may change withtime; therefore a sufficiently accurate mode l of admittance and power can becharacterized in the frequency domain . However quantify ing the ene rgy consumption ofa piezoelectric actuator require s the power as a function of time. Thus , the only way to


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etermine energy consumption for a test is to experimentally measure the voltage andcurrent and determine the power as a function of time directly.

The admittance is now a function of the voltage or control signal that is applied to thepiezoelectric actuator the frequency of the control law signal as well as the materialproperties of the piezoelectric material. It is important to realize that the admittancemust be determined for individual control signals as the control signals may changebetween tests, control laws or boundary conditions

During this research efforts were made to investigate the origin of the non linearbehavior of the capacitance. Measurement techniques and meter calibration werechecked and verified to be accurate. Investigations were made exploring thermal effectsThe high voltages exposed to piezoelectric actuators were thought to increase thetemperature within the material which may result in thechange incapacitance.Experimental test measured a change in temperature of one degree Celsius between 17and 85 volts. Thus thermal effects were considered negligible.

Sherrits 25 presentation identified that the dielectric piezoelectric and elastic constantsare influenced by the voltage applied to the material and the s tre ss acting on the material.The consequence is that the material constants will change under different bbundaryconditions and under different excitation signals. Some plots that illustrate the change incapacitance under different boundary conditions are found in Appendix 2. The resultsverify that the capacitance will change with both the boundary conditions and voltage. Atthis point too many unknowns still exist to completely characterize how the material

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properties of piezoelectric ac tuators are affected by the applied voltage or the stress act ingon the ac tuator.

The author believes that due to the inherent nature of piezoelectric materials changes inmaterial properties should be expected with changes in excita tion . As the materialchanges dimension with excitation the molecular or ion ic structure of the dipoles withinthe material also shift rotate or change shape . The change in position or shape of thedipoles may change the material properties of the piezoelectric material. This is only ahypothesis with the conclusion that more research needs to be conducted to trulyunderstand the material properties of piezoelectric materials

43 OpenLoop Excitation

This section investigates the admittance during open loop excitation. During open loopexcitation the piezoelectric actuators are utilized to exc ite motion in the structure withoutfeedback The purpose of inyestigatingopenloop excitation is to iLlu stratethe effects ofthe phase angle between strain and vo ltage

The strain response due to voltage input transfer funct ion for a laminar piezoelectricactuator fixed to a single degree of freedom struc ture resembles an ideal second ordersystem. Below the natura l frequency the phase is zero degrees . Above the naturalfrequency the phase is 180 degrees This means that the ratio of stra in to voltage ispositive below the natural frequency and negative above the natural frequency. Insertingthis strain to voltage relationship for open loop excitat ion into the equation of admittance:


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0.9

0.8a 0.7x

0.3

0.20 1

0

aW j

IYQ I=[C_w I II v jCO D 4 2b

The admittance is found to increase due to the strain to voltage response for frequenciesbelow the natural frequency and decrease for frequencies greater than the naturalfrequency. However, the increase and decrease in admittance will only cause minorfluctuations about the mean admittance generated by the effective capacitance C

II

= capacitaiice

Admittance, V Strain to Voltage Response [SN]

I II II I

I

a

20 22 23 24 25 27 28 29 30 31 33 34 35 36 38Frequency Hz

Figure 4 4 : Open-Loop Excitation Response of Admittance and Strain to Voltage Response

i t 4.4 a fth fru6n tns iJ f tri t vOltage arid theadmittance of the piezoelectric actuator during open loop excitatin. As predicted by

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xc it tionSignal

Figure 5: Diagram of Feed-Forward ontrol Law

The excitation signal was a sine sweep between 20 and 40 Hz to ensure the firs t mode ofthe structure is excited 3 2 Hz). The increase in natura l frequency between the open loopexcitation test 29 Hz) and the closed-loop control test 32 Hz) was due to the presence ofthe electro-magnetic force shaker which adds st iffness and damping to the host structure.

Open-loop excitation will refer to the case where the ex ternal force shaker is excitingmotion in the structure, feed-forward control will refer to the case the piezoelectricactuators::are used to.inhibitthemotion generatedby the ex ternaLforce. In Test A, feedforward control demonstrated a 53.5 attenuation in strain from the Open-loop excitationcase Fi gure 4.6A). Feed-forward control in Test B demonstrated a 41.5 at tenuation. instrain from open-loop exci tation Figure 4.6B).

Both test successfully demonstrated vibration control. The admittance between Test Aand Test B can now be exam ined.. During feed-forward control the strain of the structurein Test B was 600 greater than the stra in in Test A. The analysis predicted that anincrease in strain will cau se a dec rease in admittance. Thus, the admittance in Test B isexpected to be less than the admittance in Test A. The admittance of the piezoelectric

8 phase shift

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Figure 4.6 A top and B bottom : The PS D of Strain for Test A and Test B With PiezoelectricActuator On and Off.

actuators during feed forward control for Tests A and B are compared with the empiricalmodel of admittance in Figure 4.7. As expected the admittance for both tests are lessthan the empirical model. As expected by the analysis the greater structural strain in TestB resulted in a greater reduction in admittance as compared to Test A.

Test A Open LoopTest A Closed-Loop

0.035 : 3

0.025002

4Co0.01

0.0050 57333537 40

Frequency, Hz.

Ce4

0.90.8 TestS, Open-Loop0 7 Test B Closed Loop

25 335374Frequency, Hz

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0 25

0 23th 0 21

0 19E0 170 15

0 130.110 09

Frequency Hz

Figure 4 7 Relative Admittance Between Empirical model Test A and Test B.

The analysis also predicted that the increase in strain would be insignificant to the overalladmittance Figure 4.8 compares the perceTit effor between the empirical model ofadmittance and the experimental admittance. Again, the error in Test B is greater becauseof the higher strain energy in the structure. Although the error is significantly greater inTest B 14 ) than. Test A 4 ), it should be noted that the strain in the structure is 600greater in Test B. Thus a 600 increase in stfucturai notion only increased the error inthe admittance by 10 , which verifies that the mechanical dynamics, or strain, has arelatively insignificant influence on the admittance during vibration control.

20 22 25 27 30 32 35 37 40

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14

12108

IIu

4

20

Figure 4.8: Percent Error :BetweentheEmpirical Modei d Tests A and B.

Furthermore the excitation signal utilizedduring both t es ts was only 37 of thepiezoelectric actuators maximum allowable voltage Recalling from Equation 3.48 if themaximum voltage was supplied to the actuator, the error in the admittance could havebeen reduced by factor of 3.

The important details highlighted by these tsts are 1 the actual admjttance of thepiezoelectric actuators never exceed the empirical model of admittance that ignores themechanical motion of the actuator and host structure and 2 a significant quantity of straininduces a small decrease in the admittance of the piezoelectric actuator. Thus, this testsupports the theory developed in the analytical section that a onserv tive model ofadmittance can be determined without modeling the mechanical dynamics of the actuator

Frequency Hz

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and host structure. The next experiment verifies this theory for the case of closed loopcontrol or active vibration control using a more realistic strain feedback control law.

4.5 Closed-Loop Control strain-feedback

During closed loop control the piezoelectric actuators are used to inhibit motion in thestructure using a feedback control law as illustrated in Figure 4.9. Th e addition of thecontrol law introduces an active vibration control scheme where the control law reacts tothe structural displacement attempting to actively minimize structural displacement.

ExcitationSignal

The digital controller is used to measure the strain in the structure at the root of the beamdirectly adjacent to the piezoelectric actuator. The strain is fed to the control law whichgenerates a signal 180 degrees out of phase with the strain at an amplitude assigned by theuser.

Since experimental tests showed that capacitance increases with voltage, all values forcapacitance were calculated using the empirical model of capacitance, Equation 4.1. Asmentioned in the previous section the strain in the structure will reduce the admittance of

External

Figure 4.9: Diagram of closed oop control law

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he actuator. Thus, the empirical model of admittance should always be greater than orequal to the experimental admittance measured during the test.

Two estimates of piezoelectric admittance during active vibration control are presentedusing Equation 4 1 Both estimates of admittance are plotted against the experimentaladmittance. The first estimate of admittance as seen in Figure 4.10 assumes the controllaw output voltage is unknown; therefore, the maximum voltage of the actuator is used todetermine the capacitance. From Figure 4.10 the maximum admittance is clearly greaterthan the experimental admittance The second estimate of admittance, as seen in figure4 11 uses the power spectral density of the control law output voltage to calculate thecapacitance. This yields an estimate Of the admittance for this specific control outputvoltage. From .Figure 4.11 the estimated admittance is greater than or equal to theexperimental admittance.

0.25

o Experimenta 0.09007 0.05 H

20 24Frequency Hz

32 36

Figure 4 10: Frequency Response of Calculated Maximum Admittance and ExperimentalAdmittance. 54


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0.21

0.190.170.15

0.13iC 0.110.090.070.05

20Frequency Hz

Figire 4;Lh FiquencyResponse of.Esthnated Admittance.and :Experiinenta[Athnittnnce.

This is consistent with what was stated in Section 3.4. The control law used for thisexample. displayed a 20 reduction in structural vibration or 80 of the strain energystill exists. For this control law, the difference between the experimental and theestimated admittance was never greater.than 3 . Thus, the structural motion has anegligible effect on the total admittance of the piezoelectric actuator for active vibration.control. The admittance is greater than the estimate at the ends of the frequency rangedue to an abrupt loss of frequency content or a problem referred to as Gibbsphenomenon.

With an estimate of admittance, the power required by the piezoelectric actuator foractive vibration control can easily be calculated. Using Equation 3.2, the power isdetermined in the frequency domain Figure 4.12)

24 28 32 36

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43 53

252

1.5

0.50

20Frequency Hz

Figure 412 Plots of ctual Power and EstimatedPower vsFrequency

As expected the. estimate of power is greater than or equa l to theactual power acrQss theentire bandwidth Also the error is insignificant l ess than 3 ) across the bandwidth

24 28 32 36 40


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CHAPTERS: CONCLUSIONAnalytical models of piezoelectric power consumption during active vibration controlwere developed. These developments showed that piezoelectric power consumption isessentially a function of the material and geometric properties of the piezoelectricactuators. The structural dynamics of the hos t structure and actuator have a minimaleffect on the power consumed by the actuators and in fact will reduce the powerconsumed by piezoelectric actuators Thus maximum power is consumed when thevibration of the host structure is completely controlled.

A single degree of freedom experimental..model-was used to verify that a conservativeestimate of power consumption can be made by ignoring structural dynamic effectsVerification of the analysis was performed using two cases of closed loop vibrationcontrol. Both test cases verified theresuJ.Ls. Also it was verified :thattheeffect ofincreased strain reduces admittance The experimental results showed that even withlarge quantities of strain in the structure and the actuator the admittance was reduced aninsignificant amount and the analytical model of admittance was always greater than theadmittance measured during the experiment. The experimental verification also revealeda non linearity in the capacitance of the piezoelectric materiaL The non linearity wasaccounted for with an empirical model. Without the empiricaimodel the error in theestimate of admittance and power could b.c 40 to 90 within the actuator operatingrange. Note these conclusions are only valid for vibration control using piezoelectricactuators do appJy to Qpn-lQQp ition

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The n ys s also showed that the l load of the piezoeecic ctu to r ispredomin ntly a capacitor. Becau se capacitors create a reac tive electrical load the typeof power amplifier used will greatly influence the total power consumption of thepiezoelectric ac tuator . In additio n to examining piezoelectric po wer consumption thisresea rch revealed a no n linear behavior in the material properties of piezoelectricmaterials. uture work includes a characterization of piezoelectric material proper tieswith a concentrtion on defining the material non linearity validating the powerco nsumption results using multiple deg ree of freedoth exper imental mode ls and aninve stig tion of piezoelectric actuator se lf sensing capabilities for feedback control laws.


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APPENDIX-i: Electrical Analysis:

A short introduction to the tools required for e]ectrical analysis will be presented. Anunderstanding of the fundamentals behind electrical engineering is required to understandthe characterization of electrical power consumption.

A i ia Definitions Reference 27Table Ai i Important Electrical QuantitiesQuan tity Symbol Definition Unit Abb reviation alternateEne rgy w ab ility to do joule J N*mworkPower p energy/unit watt W J/stime harge q quantity o coulomb C A*selec tric ityurren t i rate of flow ampere A Cisof charge Voltage. v energyunit volt V W/Acha rge

Electric Field E force/unit voltlm eter V/rn N /CchargeResistance R vo ltage/unit ohms I/s iemensof cu rrentapacitance C cu rrent/time farads Frate of change

of vol tage

Power measures the rate at which energy is transferred. Power is defined by the time rateof change of energy and is measured in units of watts.

dw


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harge is the quantity of electricity harge is defined as conservative because it canneither be created or destroyed and is measured in units of coulombs

The current through an area is defined by the electric charge passing through the area perunit time and is measured in units of amperes

dq A l 2

Voltage is a measure of electricfl potential difference The energy transfer capability of aflow of electric charge is determined by the voltage Voltage is defined by the quantity ofenergy per unit charge and is measured in volts

dw A l 3dq

Rsistance a measure of the conductance of aEmateriai Itisdefinedby the ratioofvoltage to current and is measure in units of ohms

v A l 4

Conductivity is the inverse of resistance;

Capacitance defines the ability of a material to store a charge The current in a capacitoris found to be proportional to the time rate of change of the applied voltage

dv A l 5i dt

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Material capacitance is defined by the permittivity, e, of the material and its geometryarea A-l.6C=s

thicknessThese definitions make up a small part of electrical engineering but will be sufficient inthe analysis of the power requirements of piezoelectric actuators. A brief introduction toelectrical analysis is introduced an will then be applied to piezoelectric actuators.

A-1.lb Electrical Analysis Reference 27

Consider the linear system:

v t Linear System i t

Where the voltage and current are related through the convolution integral. This systemcan be transferred into the frequency domain

V in r System J

Where the relationship can be expressed linearly using the frequency response fhnction,

I a =Y o V o A-l.7

In this case, the frequency response function, Y co , is defihed as the admittance of thecircuit; Because-of-the-reiative easeof-appiyin-g alinear systenriirthe frequency domainadmittance is commonly used in defining the characteristics of electrical circuits. The


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Inits of admittance are 1/ohms or siemens. Admittance generally de fines the conductanceof a circuit with frequency content .

Impedance is also commonly used to define the linear relationship between current andvoltage.

VoJ =Z aI a A.-l.8

The impedance, Zco, of a circuit is the inverse of the admittance with units of ohm s.Impedance de fines the resistance of a circuit with frequency content . oth admittanceand impedance are often complex functions, for this reason, d mitt nce and impedanceare defined in term s of magnitude and phase.

magn irnde: IY 0 {Re[Y J +Im[Y j A-1.9Im[Ywj A-1.1Ophase: q t Re[Y

Yoi IYwIe A1.11Where the Re[g c ] and Im{gw j notation sign ify the real and imaginary components ofthe function, go, respectively. The- imaginary number 1)1/2 is designated by thevan able Electrical power can be characterized in terms of the admittance: First , power is definedin terms of the voltage and current through reJjor sbip:


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dw dwdq A-1.12pThe power in the frequency domain is de fined as:

PQ A-I .13

Applying the definition of admittance in Equation A-1.7, the power characteristics canbe defined in terms of the admittan ce and the vol tage

P co Y w Vo A-I.14A-1.lc Discussion of Power Reference 27

A few solutions can be developed from the previous definition of power, Equation A i4. By breaking down the exponen tial function into the sine and co sine parts, thepower can beexpressedbyits real andimaginary components:..

PIco =1Yc iV .cosa + sin qw A- 1.15Re[P ] iYa1V2 o cosq a A-i. 16

Im[P wj Y0 1V2 .a sino A-I .17

The real component of power is common ly referred to as the dissipa tive power.Dissipative power consumption is gene ral ly due to a resistive load in the circuit. hisenergy loss is dissipated in the form of heat, thus, the power loss is referred to asdissipative pow er The imaginary component is commonly referred to as reactivepower. Theoretically, the reactive component of power doe s not consume power.


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eactive power is analogous to a mass spring system in motion The power will continueto cycle until dissipated by a resistive element or a damper in the mass spring systemRealistically the power requirements of an electrical system are characterized by theapparent power or the total magnitude of the power

A l.18

Typically amplifiers that supply the power to a ircuit are not capable of storing the

reactive power The reactive power is required to be replenished after every cycle Forthis reason the power requirements of a circuit are defined by the apparent power Morecomplex amplifiers are being produced that store the reactive component of the powertherefore the total power consumption would be characterized by the dissipative poweralone Because of the variability between amplifier types knowing the type of amplifieris vital to determining the power requirements of the circuit

64


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APPENDIX2: Non Linear Capacitance Plots Highlighting Effects ChangingBoundary Conditions.

Capacitance vs. Voltage of ACX QP W Actuator Excited in Bending440E04

420E 0494OOE 04

3BOE 04

360E 043 40E 04

320E043 E 4

5

ZZZZZZZZZZZZZZ10 15

Voltage20 25

Capacitance vsVoltage ACX QPOW PZT Actuator Excited inExtension and Contraction

E

C

t8 1598 205 2564 2862Voltage

6


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PPENDIX3 X ctuator Material Properties and GeometryContact information:

Active Control Experts Inc 215 First StreetCambridge MA 021241 1227

Tel: 617577 0700

Fax: 617 577 656 email: [email protected]: www.acx.com

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Model QP4OW Specifications

Application type: stra in actuator onlyDevice size in: 4.00 x 1.50 x 0.03Device weight oz : 0.51Active elements: 2 stacks of 2 piezosPiezo wafer size in: 1.81 x 1.31 x 0.010Device capacitance: iiF : 0.40FIiIT scale voltage range V:

Functional Diagrarn.

Device poled with positive voltag e applied to pins 2 and 3.Bonded Configuration

Full scale strain, extension p : 280

F

il

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QuickPack ActuatorPiezoelectric Properties

easurd at less than Hz .

Dielectric Loss FactotT -Curie Point3 Direct Charge Coeffioicnt1 Transverse Charge Coefficient1 Truisversc VoltageCoefficient3 Direct Voltage CoefficientK Planar Piectromechanical. Coupling Cocfficient

K1 Transverse ElcetromeehaniciilPree Dielectric ConstantMeastired Along Poling Axis

PerinlUivity of free spaeeK irect Elect ITlechanicalCoupling Coefficient

Direct Youngs Modulus

Property Symbol Vnit,g ValueDielectric Constant I KHz 1800Die.lecthc Losx Factor I KHz tan6 1.8Curie Point T 350Density kg/ni 7700Coercive Fielda E. KV/cni 14.9

KCoupling Coefficients K 0.70

1 0.30K 0.404 Cool. 13 xlO 5

. . NewtonPaezoelectnc Charge CoefficientsDisplacement Cuelficiern er -179meters. c 10voltPiczoelcctric Voltage Coefficient .g3 volt meters 24.2Vciltagc Coellicicnt Newton -11.0

yE Newton l 6.9 meter 55

K E 1wPlate Capacitance l =

Deftnitionzp Mass DensIty of CeramicI Length inw -Width mt Thkk m

Cocrdvc Field

L

ElasticModulus 68


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REFERENCES Rogers C Liang C. Some Interesting Mechanics Issues for SmartMaterial Systems

Proceed ings of the International Congress on Experimen tal Mechanics AnnualLas Vegas Nevada Vol. p. 608614.

2. Weisshaar T. A. Aeroservoe lastic Control Concepts with Active Materials School ofAeronau tics and Astronautics Purdue University West Lafayette Indiana.

3. Kudva I. N. Appa K. Martin C. A. Jardine A. Overview ofRecen t Progress onthe DARPA/WL SmartMaterials and Structures Wing Prograin SPIEs AnnualSymposium on Smart Structures and Materials San Diego March 1997.

4. McGowan AM.R. HeegJ. Lake R.C.: Results ofWindTunnel Testing From thePiezoelectric Aeroelastic Response Investigation Proceed ings of the 37th AIAAStructural Dynamics and Materials Conference Salt Lake City UT April 1996.

5. Heeg J. McGowan AM.R. Crawley E. Lin C.: The Piezoelectric AeroelasticResponse Tailoring Inve stigation Analysis and OpenLoop Testing; CEASInternational Forum on Aeroelastic ity and Structural Dynam ics Minchester UK June1995.

6. Moses Robert W.: Vertical Tail Buffeting Allevia tion Using Piezoelectric Actuatorsand Rudder Results of the Actively ControlledResponse Of iff t Affected TailsAROBAT Program HighAngle of Atta ck Technology Conference September1719 1996.

7. Pinker ton J.L. McGow anAM.R. Moses R.W. Scott R.C. Heeg I.: ControlledAeroelastic Response andAiifoil Shaping Using Adaptive Materials and Integrated


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Systems Proce edings of the 37th ALkA Struc tural Dynamics and M aterialsConference Sa lt Lake City UT April 1996.

8. Crawley E. F. de Luis 3. Use ofPiezoelectric Actuators as Elements ofIntelligentStructures AIA A Journal Vol. 25 NO. 10 1987.

9. Crawley E. F. Anderson E. H. DetailedModels ofPiezoceramic Actuation ofBeams Journal of Inte lligen t Systems and Structures Vol. 1-January 1990 pp. 4-25 .

10. W arken tin D. 3. Crawey E. F. Senturia S. D. The Feas ibility ofEmbeddedElectronicsor Intelligent Structures Journal of intelligent System s and StructuresVol. 3.-July 1992 pp. 462-482.

11. Dosch 3. Inman D. J. Garcia E. A Self-Sensing PizoelectricActuatororCoiiocatedControl Journal of Intelligent Systems and Structures Vol. 3 Jan uary1992 pp. l66185

12. Pan 3. Hansem H. Snyder S. D. A study of the..Response of a Simply SupportedBeam to Excitation by a PiezoelectricActuator Journal of In telligent Sys tems andStructures Vol. 3-January 1992 pp. 3-16.

13. Song 0 Lirescue L. Rogers C. A. Applicat ion ofAdapt ive Technology to StaticAeroelastic ontrdl ofWing Structures AIAA Journa l Vol. 30 No. 12 December192 pp. 2882-2889

14. Nam C. Kim J-S. Robus t Controller Design Ofa Wing with Piezoelectric Materialso r Flutter Suppression The 10th VPI SU Symposium on Struc tural Dynamics andControl May 8-10 1995 Biacksburg VA.


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15. Akella P. Chen X. Hughes D. Wen 3. T. Mod eling and Con trol ofSmartStructures with Bonded Piezoelectric Sensors andActuators A Passiv ity ApproachSPIEs 1994 North American Conference on Smart Structures and Materials Feb.1994 Orlando FL pp. 1081 19.

16. Layton B. An Analysis ofFlutter Supp ression Using Adaptive Materials IncludingPower onsumption AIA A 1995.

17. Freed B. BabuskaV. init Element Modeling of Composite PiezoelectricStructures With MSC/NASTRL4N SPIEs nnu al Sym po sium on Smart Struc turesand Materials San Diego CA March 1997.

18. Zaglave r H.W. La st B. Mod eling Techn iques in the Design ofSmart Structures forActive Vibration Control SPIEs Annual Symposium on Smart Structures andMaterials San Diego CA March 1997.

19. Liang C. Sun S. Rogers C.A.: Dynam ic Output Characteristics OfPiezoelectricActuators SP IEs 1993 North American Conferen ce on Smart Struc tures andMaterials Albuq uerque 1 4 Fe bruary 1993.

20. Liang C. Fanpirig S. Rogers C. A. Determination of the OptimalActuatorLocations and Configurations Based On Actuator Power Factors Proceed ing sFour th In ternationa l Con ference on Adap tive Structures Cologne Germany Nov. 24 1993.

21. Liang C. Sun F.P. Rogers C.A.: investigation of the Energy Transfer and PowerConsumptio n ofAdap tive Structures Proceedings of the 31st Conference on Decisionand Control Tucson Arizona December 1992.


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22. Zhou S. Liang C. Rogers C.A.: Coup led ElectroMechanical Impedance Mode lingto Predict Power Requ iremen t and Energy Efficiency ofPiezoelectric ActuatorsIntegrated with PlateLike Structures AIAA Paper No. 941762 Proceedings of theAIAAJASME Adaptive Structures Forum SC April 1994.

23 . Hagood N. W Chung W. H. von Flotow A. Mode ling OfPiezoelectric ActuatorDynamics ForActive Structural Con trol Journal of Intelligent Mater ial Systems andStructures Vol. 1 No. 3 pp 327-354.

24. Warkentin David J. Craw ley Edward F. PowerAmplfication for PiezoelectricActuators in Controlled Structures MIT Space Engineering Research CenterCambridge Massachusetts SERC 4-95 May 1995.

25 Sherrit S. Wiederick H. D. Mukherjee B. K. Field Dependency of onzplexPiezoetectricDielectric and Elastic Components ofMotarola PZT 32.03 HDCeramic SPIEs Annual Symposium on Smart Structures and Materials SanDiego CA March 1997.

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