Closed loop finite element modeling of piezoelectric smart structures

13 Pages·2015·1.32 MB·English

ShockandVibration13(2006)1–12 1
IOSPress
Closed loop finite element modeling of
piezoelectric smart structures
a b a,∗ a
GuangMeng ,LinYe ,Xing-jianDong andKe-xiangWei
aStateKeyLab. ofVibration,Shock&Noise,ShanghaiJiaoTongUniversity,Shanghai,200030,People’sRepublic
ofChina
bCentreforAdvancedMaterialsTechnology,SchoolofAerospace,MechanicalandMechatronicEngineering
(J07),TheUniversityofSydney,Sydney,NSW2006,Australia
Received25July2004
Revised9November2004
Abstract.Theobjectiveofthispaperistodevelopageneraldesignandanalysisschemeforactivelycontrolledpiezoelectricsmart
structures. Theschemeinvolvesdynamicmodelingofasmartstructure,designingcontrollawsandclosed-loopsimulationina
finiteelementenvironment. Basedonthestructureresponsesdeterminedbyfiniteelementmethod,amodernsystemidentification
techniqueknownasObserver/KalmanfilterIdentification(OKID)techniqueisusedtodeterminethesystemMarkovparameters.
TheEigensystemRealizationAlgorithm(ERA)isthenemployedtodevelopanexplicitstatespacemodeloftheequivalentlinear
system for control law design. The Linear Quadratic Gaussian (LQG) control law design technique is employed to design a
controllaw. ByusingANSYSparametricdesignlanguage(APDL),thecontrollawisincorporatedintotheANSYSfiniteelement
model toperform closed loop simulations. Therefore, the control lawperformance can be evaluated inthecontext of afinite
elementenvironment. Finally,numericalexampleshavedemonstratedthevalidityandefficiencyoftheproposeddesignscheme.
Withoutanyfurthermodifications,thedesignschemecanbereadilyappliedtoothercomplexsmartstructures.
Keywords: Vibration control, eigensystem realization algorithm, system identification, finite element method, piezoelectric
materials
1. Introduction
Advancedapplications,suchaslarge-scalespacestructures,jetfightersandconceptautomobiles,havestructural
requirements including high strength, lightweight, high structural damping and low acoustic noise. Attempts at
achievingthese characteristicshaverecentlystimulatedextensiveresearchintosmartstructures, whichhavebeen
widelyappliedinthefieldofactivevibrationcontrolofstructures. Piezoelectricmaterials,suchasleadzirconate
titanate(PZT),havecoupledmechanicalandelectricalproperties. Bondingorembeddingpiezoelectricpatchesina
smartstructurecanactassensorstomonitororasactuatorstocontroltheresponseofthestructure[1].
Todesignpiezoelectricsmartstructuresforactivevibrationcontrol,bothstructuraldynamicsandcontroltheory
need to be considered. The finite element method, a widely accepted and powerful tool for analyzing complex
structures,iscapableofdealingwiththepiezoelectricsmartstructures. Specialfiniteelementshavebeendeveloped
toaccountforpiezoelectriceffect[2–4]. ANSYS,oneofthemostwidely-usedcommerciallyavailablefiniteelement
analysis (FEA) codes, has the ability to model piezoelectric materials. Unfortunately, finite element models are
always of high-orderand can not be used directly in controllerdesign whereas most controllaw design methods
require an explicit mathematical model of the system to be controlled. To overcome this difficulty, the mode
∗Correspondingauthor.Tel.:[phone];Fax:[phone];E-mail:[email protected].
ISSN1070-9622/06/$17.00©2006–IOSPressandtheauthors. Allrightsreserved
2 G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures
superpositiontechniqueiswidelyusedtotransformthecoupledfiniteelementequationsofmotioninthephysical
coordinates into a set of reduced uncoupled equations in the modal coordinates, and then the state space model
of the system is developed[5–7]. Commonly, the MatLAB Control System Toolbox is used for the control law
designandthenclosedloopsimulationis performedbasedonthestate spacemodel[6,8]. As canbeseen, in all
theseresearchworks[5–8],dynamicmodelingofthesmartstructuresanddesignofthecontrolsystemsareusually
carriedoutseparatelywithouttakingtheinteractionbetweenthestructureandthecontrolsystemintoaccount. Due
to theincreasinginterestin the designofcomplexpiezoelectricsmart structuresandthe needforfast andsimple
implementation of piezoelectric control systems, it is necessary to develop a general design scheme of actively
controlledpiezoelectricsmartstructures.
Inthispaper,amodernsystemidentificationtechniqueisemployedtodevelopanexplicitstatespacemodelfor
controllaw designfromthe outputof a commercialfinite elementcodeANSYS. Systemidentificationtechnique
dealswiththerealizationofamodelfromtheexperimentresultsforthepurposeofcontrollawdesign. Thisrequires
theconstructionofaminimumordermodelfromthetestdatathatcharacterizesthedynamicsofthesystematthe
selectedcontrolandmeasurementpositions. Afiniteelementsimulationprovidestimehistoriesofthevariablesof
apiezoelectricsmartstructuresystem,butdoesnotgenerateanexplicitmathematicalmodelofthesystem. Finite
elementsimulationsareanalogoustoperformingexperimentalinvestigationswheretheonlydirectoutputsaretime
responses. Therefore,thestructuralresponsesdeterminedbyfiniteelementsimulationsareemployedastheinput
toanidentificationtechniqueknownasObserver/KalmanfilterIdentification(OKID)[9,10]approachthatidentifies
theMarkovparametersofanasymptoticallystableobserver. ThesystemMarkovparametersarethendetermined
recursivelyfromthe Markovparametersof the observersystem, fromwhich a state space realization is obtained
using the Eigensystem Realization Algorithm (ERA) [9,11]. The LQG design method is employed to design a
controllawbasedonthestatespacemodel. ANSYSparametricdesignlanguage(APDL)isthenusedtointegrate
thecontrollawintotheANSYSfiniteelementmodeltoperformclosedloopsimulations[12].
Insummary,undertheinteractionofstructureandcontroldisciplines,theobjectiveofthispaperistodevelopa
generalanalysisanddesignschemeofpiezoelectricsmartstructures. TheproposedschemecanmakeuseofANSYS
andimplementauserselectedcontrolalgorithm. ByusingAPDL,thecontrollawisincorporatedintotheANSYS
finiteelementmodelofthesmartstructureto performclosedloopsimulations. Theproposeddesignschemecan
furtherreducecost,aswellasshortendesigncycletimeforsmartstructures. Withoutanyfurthermodifications,it
canbereadilyappliedtolargeflexiblespacestructures.
2. Fromfiniteelementmodeltostatespacemodel
2.1. Finiteelementmodel
Modelingofpiezoelectricsmartstructuresbythefiniteelementmethodanditsimplementationforactivevibration
controlproblemshasbeenpresented[2–4,13,14]. Theglobalmatrixequationsgoverningasmartstructuresystem
canbewrittenas:
(cid:1) (cid:2)(cid:3) (cid:4) (cid:1) (cid:2) (cid:3) (cid:4) (cid:1) (cid:2) (cid:3) (cid:4) (cid:3) (cid:4)
M 0 u¨ C 0 u˙ K K u F
0uu 0 φ¨ + 0uu 0 φ˙ + KuTu Kuφ φ = Fu (1)
uφ φφ φ
where u denotes structural displacement, φ denotes electric potential, and a dot above a variable denotes a time
derivative;FudenotesthestructuralloadandFφdenotestheelectricload;MuuandKuuarethestructuralmassand
stiffness matrix, respectively, Kuφ is the piezoelectriccouplingmatrix and Kφφ is the dielectric stiffness matrix,
Cuu isthedampingmatrix,whichisusuallyassumedtobealinearcombinationofthestructuralmassmatrixMuu
andthestructuralmatrixKuu
[Cuu]=α[Muu]+β[Kuu] (2)
Theconstant α andβ are the dampingparameters[15]. It is notableto pointoutthat proportionaldampingis
assumedbutnotrequiredforsystemidentification.
Asmentionedbefore,thecommercialfiniteelementcodeANSYScanbeutilizedtomodelthepiezoelectricsmart
structurewithprescribedinputstoobtainasetofcorrespondingoutputtimehistories. Thenextstepistoemploya
systemidentificationtechnique,usingthetimehistoriesofoutputsandinputsoftheANSYSfiniteelementmodel,
todeterminean“equivalentlinearsystem”foruseasacontrollawdesignmodel.
G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures 3
2.2. IdentificationofthesystemMarkovparametersbyOKIDapproach
TheObserver/KalmanfilterIdentification(OKID)[9,10]methodwasdevelopedtocomputetheMarkovparameters
ofalinearsystem,whicharethesameasitspulseresponsesamples. Themethodisformulatedentirelyinthetime
domain,andiscapableofhandlinggeneralresponsedata. OneofthekeystotheOKIDalgorithmistheintroduction
ofanobserverintotheidentificationprocess. ThefirststepoftheprocessisthecalculationoftheobserverMarkov
parameters. Then the system Markov parameters are determined recursivelyfrom the Markov parameters of the
observersystem. Inthefollowingwesummarizeitinbrief.
Firstly,considerageneraldiscretemultivariablelinearsystemexpressedinthestatespaceformat
x(i+1)=Ax(i)+Bu(i)
(3)
y(i)=Cx(i)+Du(i)
wherexisann×1statevector,uanm×1inputorcontrolvector,yaq×1outputvector. MatricesA,B,CandD
arerespectivelythestatematrix,inputmatrix,outputmatrix,anddirectinfluencematrix. Theintegeriisthesample
indicator. Theinput-outputdescriptionoftheabovesystemwithzeroinitialconditionscanbeobtainedfromEq.(3)
as
(cid:5)i−1
y(i)= Yτu(i−τ −1)+Du(i) (4)
τ=0
wheretheparameters
Y =CAτB
τ
together with D, are known as the Markov parameters of the system, which are also the system pulse response
samples. ToreducethenumberofMarkovparametersneededtoadequatelymodelasystem,anobserverisintroduced
intotheOKIDtechnique. If(A,C)isanobservablepair,thenthereexistsanobserveroftheform
(cid:1)x(i+1)=A(cid:1)x(i)+Bu(i)−M[y(i)−(cid:1)y(i)]
=(A+MC)(cid:1)x(i)+(B+MD)u(i)−My(i) (5)
yˆ(i)=C(cid:1)x(i)+Du(i)
ThematrixM canbeinterpretedasanobservergain. ConsiderthespecialcasewhereM isadeadbeatobserver
gainsuchthatalleigenvaluesofA+MCarezero,theestimatedstatexˆconvergestothetruestatex(i)afteratmost
nstepswherenistheorderofthesystem. Equation(5)thenbecomes
x(i+1)=(A+MC)x(i)+(B+MD)u(i)−My(i)
(6)
y(i)=Cx(i)+Du(i)
Theinput-outputdescriptionofthesysteminEq.(6)is
n(cid:5)−1
y(i)= Y¯τ[u(i−τ −1)y(i−τ −1)]T +Du(i) i(cid:1)n (7)
τ=0
where
Y¯ =[C(A+MC)τ(B+MD)−C(A+MC)τM]=[Y¯(1) Y¯(2)]
τ τ τ
Y¯τ andD aretheMarkovparametersofanobserversystem. Aparticularfeatureofthedeadbeatobserveristhat
theMarkovparameters Y¯τ willbecomeidenticallyzeroafterafinitenumberoftimesteps. Thestandardrecursive
least-squarestechniqueisusedtosolveEq.(7)andthentheobserverMarkovparametersarecomputed. Thereisan
algebraicrelationshipbetweentheMarkovparametersoftheobserversystemandthoseoftheactualsystem
τ(cid:5)−1
Yτ =CAτB =Y¯τ(1)+ Y¯i(2)Yτ−i−1+Y¯τ(2)D (8)
i=0
A complete description for the computation of the Markov parameters is given in reference [9,16]. Once the
Markovparametersof the observersystem are identified, the actual system Markovparameterscan be recovered
accordingtoEq.(8). TheactualsystemMarkovparametersarethenusedtoobtainastatespacemodelofthesystem
byarealizationprocedurecalledtheEigensystemRealizationAlgorithm(ERA)[9,11].
4 G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures
2.3. MinimumrealizationbytheEigensystemRealizationAlgorithm
TheEigensystemRealizationAlgorithm(ERA)isasystemrealizationmethodthathasbeenstudiedextensively[9,
11]. Itdeterminesa state space model(Ar,Br,Cr,Dr) of a structurefromthe systemMarkovparameters. The
algorithmbeginsbyformingthel×lblockHankelmatrixdenotedbyH(l,τ)
⎡ ⎤
Y Y ··· Y
τ τ+1 τ+l−1
⎢ Y Y ··· Y ⎥
⎢ τ+1 τ+2 τ+l ⎥
H(l,τ)=⎢ . . . ⎥ (9)
⎣ . . . ⎦
. . .
Y Y ···Y
τ+l−1 τ+l τ+2l−2
TheorderofthesystemisdeterminedfromthesingularvaluedecompositionofH(l,0)
H(l,0)=UΣVT (10)
wherethematrixU andV areunitarymatrices. Assume
Σ=diag[σ1σ2···σnσn+1···σN] (11)
with
σ (cid:1)σ (cid:1)···(cid:1)σ (cid:1)σ (cid:1)···σ
1 2 n n+1 N
Chooseanumberδasazerothresholdsuchthat
σ σ
n >δ n+1 (cid:2)δ (12)
σ σ
1 1
thenthematrixH(l,0)is consideredto haverankn. Take Σn as an×n diagonalmatrixofthefirst n singular
valuesin Σ. UnandVncorrespondtothesingularvaluesin Σn. Equation(10)thenbecomes
H(l,0)=UnΣnVnT (13)
Definingaq×lqmatrixETandanm×lmmatrixET madeupofidentityandnullmatricesoftheform
q m
EqT =[Iq 0q×(l−1)q], EmT =[Im 0m×(l−1)m] (14)
adiscrete-timeminimalorderrealizationofthesystemcanbewrittenas
Ar =Σ−n1/2UnTH(l,1)VnΣ−n1/2 (15a)
Br =Σ1n/2VnTEm (15b)
Cr =EqTUnΣ1n/2 (15c)
ThedirectinfluencematrixDr canbeidentifiedbysolvingEq.(7). Theabovestatespacemodelofthesystem
dynamicsisusedtodesignaLQGbasedrobustcontroller.
G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures 5
3. Controlalgorithmandclosedloopsimulation
3.1. LQGdesign
Inpractice,themodelingofthesmartstructureisseldomperfectandthesystemparameterslikedensity,elastic
stiffness anddampingusually varyduringthe courseof operationof the structuredue to corrosion, damage, etc.
Moreover,thesystemmightbesubjectedtovaryingdynamicloadswhichmayexcitemultiplemodesofvibration[7].
A LQG based robust controller which is less sensitive to system parameter variations, and also works efficiently
whenmultiplemodesofvibrationarepresentintheresponse,isanexcellentchoiceforcontrollingthevibrationsof
suchsystems. Inthissection,therobustcontrollerdesignusingtheLQGisdiscussed. Consideringtheprocessand
measurementnoisew(i)andv(i),thestatespaceequationforthepiezoelectricsmartstructurecanbewrittenas
x(i+1)=A x(i)+B u(i)+Gw(i)
r r
(16)
y(i)=C x(i)+D u(i)+v(i)
r r
IntheLQGalgorithm,theprocessnoisew(i)andthemeasurementnoisev(i)arebothassumedtobestationary,
zero-mean,Gaussianwhite,andtohavecovariancematricessatisfying
E{w(i)wT(j)}=V1δi,j E{v(i)vT(j)}=V2δi,j (17)
whereE{·}denotestheexpectedvalue,δi,j denotestheKroneckerdelta. Inaddition,w(i) andv(i) areassumed
tobeuncorrelated. Thecontrollerconsistsofastatefeedbackmoduleandastateestimationmodule. Theformer
correspondstothefollowingcontrollaw
u(i)=−Kx(i) (18)
inwhichK denotesthefeedbackgainmatrix. Thefeedbackgainmatrixisobtainedsoastominimizethequadratic
costfunctionoftheform
(cid:12) (cid:13)
(cid:5)∞
J =E [xT(i)Qx(i)+uT(i)Ru(i)] (19)
i=1
whereQ is asymmetricpositivesemi-definitematrixandR is asymmetricpositive-definitematrix. Theoptimal
feedbackgainmatrixthatminimizestheabovecostfunctionisgivenby
K =(R+BrTP1Br)−1BrTP1Ar (20)
wherethepositivedefinitematrixP1 isthesolutiontothefollowingalgebraicRiccatiequation
ATrP1Ar −P1−ATrP1Br(R+BrTP1Br)−1BrTP1Ar+Q=0 (21)
In Eq. (18) it is assumed that all of the states are available for feedback. However, in practice only the system
outputsareavailableforfeedback. Toestimatethestatesofthesystemfromsensoroutputs,astateestimatorbased
ontheKalmanfilterisdesigned
(cid:1)x(i+1)=Ar(cid:1)x(i)+Bru(i)+L[y(i)−Crxˆ(i)−Dru(i)] (22)
wherexˆ(i)denotestheestimatedstateandtheobservergainmatrixisobtainedform
L=ArP2CrT(CrP2CrT+V2)−1 (23)
inwhichthepositiveP2 satisfiesthefollowingalgebraicRiccatiequation
ATrP2Ar +GV1GT−ArP2CrT(CrP2CrT+V2)−1CrP2ATr −P2 =0 (24)
NotethatthestatevectorinEq.(18)shouldbereplacedbytheestimatedstatexˆ(i). OneofthemeritsoftheLQG
techniqueliesintheseparationprinciple,whichstatesthatthedesignofthefeedbackandtheestimatorcanbecarried
outseparately. AblockdiagramoftheLQGcontrollerisshowninFig.1.
6 G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures
State estimator System dynamics
w

B G
r
xˆ(i) x(i)
_ L z-1 -K Br z-1

A A C
r r r

C v
r

Fig.1.Ablockdiagramofthediscrete-timeLQGcontroller.
F
e
v v
ANSYS model of
a s
smart structure
LQG controller
Fig.2.Blockdiagramoftheclosed-loopsimulation.
TheselectionofQ andRisvitalinthecontroldesignprocess. QandRarethefreeparametersofdesignand
stipulatetherelativeimportanceofthecontrolresultandthecontroleffort. AlargeQputshigherdemandoncontrol
result,andalargeRputsmorelimitoncontroleffort[17]. TheoptimalvaluesforQandRaretypicallyobtained
bytrial-and-errormethods. Inthispaper,QiscomputedfromCr with
Q=CrTCr (25)
WeightingmatrixRcanbesetasρIwithρasascalardesignparameter. Therefore,thechallengingtaskofchoosing
QandRreducestochoosingoneparameterρ. Consequently,thecontrolperformancesuchassettlingtimeandthe
maximumvalueoftheactuatorvoltagecanbetunedthroughchangingthevalueofρ. Thisprocedureisusedinthe
illustrativeexamplepresentednext.
3.2. Closedloopsimulationinfiniteelementenvironment
Inthissection,byusingANSYSparametricdesignlanguage(APDL),theANSYSfiniteelementmodelofasmart
structureismodifiedtoacceptcontrollawsandperformclosed-loopsimulations. Therefore,thecontrollawcanbe
evaluatedin thecontextoffiniteelementenvironment. Theblockdiagramoftheanalysisis showninFig.2. Fe
isthevibrationgeneratingforce. Theinstantaneousvalueofthevibrationgeneratingforceisdefinedateachtime
steps. vs isthesensorsignalatasensorlocation. Forinstance,foracantileverplate,verticalvelocitycanserveas
sensorsignal. vaistheactuatorvoltageanditisdeterminedbytheLQGcontrollerusingFeandvs.
Inthisstudy,toconstructanANSYSfiniteelementmodelforapiezoelectricsmartstructure,SOLID45elements
are used for the metal part and SOLID5 elements are used for the piezoelectric part of the structure. A macro
hasbeenwrittentoperformthetransientanalysis, inwhichthecontrollawis accepted. Inthetransientanalysis,
theinstantaneousvalueofthesensorsignaliscalculatedatatimestepbyusingtheclassicalNewmarkintegration
method. Thenthe feedbackvoltage, whichwill be used as the inputto the piezoelectricactuatorduringthe next
timestep,isdeterminedbytheLQGcontroller. Thisprocesswillcontinueforthepredefinedtimedurationofthe
computedresponse. Throughtheresultsoftransientanalysis,thecontrollawcanbeevaluatedinthefiniteelement
G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures 7
Fig.3.Theconfigurationofanactivelycontrolledcantileverplate.
simulation. Ifthecontrollawperformanceisnotadequate,thecontrollawcanberedesignedandevaluatedagain
untilthedesiredperformanceisobtained.
4. Illustrativeexample
Inthis section, a cantileveredplate is considered. Thestructureconsists of a hostaluminumplate (700mm×
150mm×1.2mm),APC-850piezoelectricpatches(70mm×15mm×0.5mm)andvelocitysensors. Theadhesive
layers are neglected. The piezoelectric patches work as actuators are perfectly bonded on the upper and lower
surfacesoftheplatestructureatthesamelocation. Itisassumedthatthesamemagnitudebuttheoppositedirection
ofelectricfieldisappliedtotheupperandlowerpiezoelectricpatchessoas tocreateapurebendingmomentfor
vibrationsuppression. Velocity sensors are bondedon top of the actuators. The configurationof the structureis
shown in Fig. 3. As can be seen in Fig. 3, there are three actuator pairs marked with actuator 1, actuator 2 and
actuator3separately. Also,therearethreevelocitysensorsmarkedwithsensor1,sensor2andsensor3separately.
ItshouldbepointedoutthatinANSYSfiniteelementmodeltheelasticandinertialpropertiesofthesensorsarenot
takenintoaccount. Thetransversevelocitiesatsensorlocationsarecalculatedateachtimestepandthenserveas
sensorsignals. ThematerialpropertiesofAPC850andaluminumarelistedbelow.
APC850:
E11 =E22 =63.0GPa,ν12=0.35,ρ=7600kg/m3,
G12 =G23=24.6GPa,G13=30.6GPa,
d31=-1.75×10−10m/V,d33=4.0×10−10m/V
Aluminum:
E =69.0GPa,G=27.0GPa,ρ=2700kg/m3
ThefiniteelementmodelforthecantileveredplatewithpiezoelectricpitchesisshowninFig.4. Atotalof17248
three-dimensionalsolidelementsareusedtomeshthestructure. Amongthoseelements,1008SOLID5elementsare
utilizedtomodelpiezoelectricpatches,and16240SOLID45elementsareemployedtomodelhoststructure. The
timestepischosenas0.0025.
Whitenoiseisselectedastheinputsignalsinceitsatisfiestheconditionofpersistentexcitation,asrequiredbya
reliableidentification[18]. FiniteelementsimulationusingANSYSisperformedtoobtainasetofcorresponding
outputtimehistories. Theinput-outputtimehistoryisdiscretizedatasamplingrateof400Hzforsystemidentifi-
cation. Theaforementionedsystemidentificationtechniqueisthenemployed,usingthetimehistoriesofinputsand
outputs,tosetupastatespacemodelforuseasacontrollawdesignmodel. Assumeanorderforthesystemtobe
identified,denotedbyn. Inthisstudy,onlythefirst6modesaretakenintoaccount,thus n ischosenas12. The
identifiedmodelisoforder12andisgivenintheAppendix.
8 G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures
Fig.4.Finiteelementmodelofanactivelycontrolledcantileverplate.
0 ERA
ERA
0 ANSYS ANSYS
-15
B) B)
e(d -20 e(d -30
d d
u u
gnit -40 gnit -45
a a
M M
-60
-60
-75
-80
1 10 100 1000 1 10 100 1000
Frequency (Rad/Sec) Frequency (Rad/Sec)
(a) From actuator 1 to sensor 3 (b) From actuator 3 to sensor 2
Fig.5.Frequencyresponsefunctions.
10 Control off
8 Control on
m) 6
m
4
(
nt 2
e
m
e 0
c
a -2
pl
s
Di -4
-6
-8
-10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Time (Sec)
Fig.6.Displacementresponsetoimpulseexcitation.
Frequencyresponsefunctionscanbeobtainedfromthestatespacemodel. Forbrevity,onlytwoofthemareshown
inFig.5. TheresultsfromtheANSYSfiniteelementmodelarealsoshowninFig.5forthepurposeofcomparison.
Excellentagreementisobservedbetweenthetworesultsexceptforthefrequencyrangeabove300rad/sec. Formost
structuresystemsunderpracticalloadingconditions,onlythefirstfewmodesplayanimportantpartinthevibration
response of the system. Therefore, it can be concluded that the state space model developed by identification
techniquecharacterizesthedynamicsofthesystemandcanbeusedforcontrollawdesign.
FortheLQGcontroller,thefreeparametersρ,V1andV2 selectedasρ=0.001,V1 =10−3InandV2 =10−6Iq.
Asmentionedpreviously,theseparametersaredeterminedbytrailanderrormethodstomosteffectivelycontrolthe
G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures 9
100
60 150
Actuator Voltage (V) -55000 Actuator Voltage (V) --244200000 Actuator Voltage (V) 1-0550000
-100
-60
-100
-150 -80
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Time (Sec) Time (Sec) Time (Sec)
(a) Actuator 1 (b) Actuator 2 (c) Actuator 3
Fig.7.Timehistoryofactuatorvoltageforcase1.
8
Control off
6 Control on
)
m
m 4
(
nt
e 2
m
e
c 0
a
pl
s -2
Di
-4
-6
0 1 2 3 4 5
Time (Sec)
Fig.8.Displacementresponsetorandomexcitation.
structure. Atthesametime,theseparametersarelimitedforthesakeofthebreakdownvoltageofthepiezoelectric
materials. Themaximumvoltageperthicknessofthepiezoelectricmaterialistakenas300Vmm −1. Thefeedback
gainmatrixK isgivenintheAppendix. OncetheLQGcontrollerdesigniscompleted,APDLisusedtointegrate
controllaw intothe ANSYS finite elementmodelto performclosed loopsimulations. Finally, thecontrollaw is
evaluatedinthefinite elementsimulation. Ifthe controllawperformanceis notadequate,the controllawcanbe
redesignedandevaluatedagainuntilthedesiredperformanceisobtained.
The external load is assumed to be applied at the free end of the plate and the dynamic responses at the free
end,i.e.thetipdisplacements,aregiventoevaluatethecontrollaw. Forcase1,atransverseimpulsiveloadingof
0.1Nisappliedatthefreeendoftheplatefor0.1s. Transientresponsesoftheplatetipdisplacementareshown
in Fig.6. Results whenthe controlis offarealso shownforcomparison. Actuatorvoltagesareshownin Fig.7.
Ascanbeseen,thecontrollersuccessfullydampsthevibrationoftheplate. Theactuatorvoltagesarelowerthan
the breakdownvoltage of piezoelectricmaterials. For case 2, bandedlimited white noise in the frequencyrange
0–200rad/sec is appliedat the free end ofthe plate. The plate tip displacementandactuatorvoltages are shown
inFigs8and9respectively. Thecontrollerisalsoeffectiveincontrollingtherandomvibration. Still,theactuator
voltagesarelowerthanthebreakdownvoltageofpiezoelectricmaterials.
5. Conclusions
Undertheinteractionofstructureandcontroldisciplines,ageneralschemeofanalyzinganddesigningpiezoelectric
smartstructuresissuccessfullydevelopedinthispaper. Theschemeinvolvesusingamodernsystemidentification
10 G.Mengetal./Closedloopfiniteelementmodelingofpiezoelectricsmartstructures
100 80
8 0 60 150
Actuator Voltage (V) --24642 000000 Actuator Voltage (V) -2240000 Actuator Voltage (V) 1-0550000
-40
-60 -100
-80 -60
-150
-10 0 -80
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
Time (Sec) Time (Sec) Time (Sec)
(a) Actuator 1 (b) Actuator 2 (c) Actuator 3
Fig.9.Timehistoryofactuatorvoltageforcase2.
techniqueto developanequivalentlinearmodelfromtheoutputsofthecommercialfiniteelementcodeANSYS.
Standardcontrollawdesigntechniquesarethenusedtodesigncontrollaws. ByusingANSYSparametricdesign
language(APDL),theresultingcontrollawsarethenincorporatedintotransientanalysisforevaluation. Resultsof
anumericalstudyapplyingthismethodologytovibrationcontrolofacantileveredplatewithpiezoelectricactuators
arepresented.
Numerical results indicate that the equivalent linear models developed by employing a system identification
technique can represent the input-outputrelationship of a finite simulation very well. The proposedscheme can
integrateauserselectedcontrolalgorithmintotheANSYSfiniteelementmodeltoperformclosedloopsimulation.
Thecontrollaw can beevaluatedin the contextof a finite elementenvironment. The techniquepresentedin this
papermaybealsoappliedtootherdifferentvibrationcontrolproblems,basedonpassiveorhybridtechniques.
Acknowledgements
Support from the National Natural Science Foundation of China (NSFC) under Grant No. 50390063 and the
foundation of National Key Lab. of Defense Technology under Grant No. 51463040403JW0301 is gratefully
acknowledged. Theauthorsalsogratefullyacknowledgethehelpfulsuggestionofthereviewers.
References
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