eptedforpublicationinJASA,2010.
Powerdissipationandtemperaturedistributioninpiezoelectricslabs
D.Thomasa),b),
D.D.Ebenezerb)andSivakumarM.Srinivasana)a)Dept.ofAppliedMechanics,IndianInstituteofTechnologyMadras,Chennai600036,Indiab)NavalPhysicalandOceanographicLaboratory,Thrikkakara,Kochi682021,India Abbreviatedtitle:Piezoelectricslabs Amethodispresentedtodeterminepowerdissipationinpiezoelectricslabswithinternallossesandtheresultingtemperaturedistribution.Thelengthoftheslabismuchgreaterthanthelateraldimensions.Lossesarerepresentedplexpiezoelectriccoefficients.Itisshownthatthespatiallynon-uniformpowerdissipationdensityintheslabcanbedeterminedbyconsideringeitherhysteresisloopsorPoynting’stheorem.Thetotalpowerdissipatedintheslabisobtainedbyintegratingthepowerdissipationdensityovertheslabandisshowntobeequaltothepowerinputtotheslabforspecialcasesofmechanicallyandelectricallyexcitedslabs.Theonedimensionalheatequationthatincludestheeffectofconductionandconvection,andtheboundaryconditionsarethenusedtodeterminethetemperaturedistribution.Whentheanalyticalexpressionforthepowerdissipationdensityissimple,directintegrationisused.ItisshownthatamodifiedFourierseriesapproachyieldsthesameresults.Forothercases,thetemperaturedistributionisdeterminedusingonlythelatterapproach.Numericalresultsarepresentedtoillustratetheeffectsofinternallosses,heatconductionandconvectioncoefficients,andboundaryconditionsonthetemperaturedistribution. PACSnumbers:43.38.Ar,43.38.Fx,43.20.Bi,43.40.At[AJZ]Page1of30
I.INTRODUCTION Piezoelectricmaterialsareusedintransducers,transformersandinvariousapplicationsbecauseoftheirquickresponseandhighcoupling.However,oneoftheirlimitationsarepredominatelyduetoselfheatingthatursbecauseofinternallosseswhileoperatingunderresonantconditions.1,2Hotspots,causedbynon-uniformheatgeneration,canquicklyleadtothermalrunawayinthesepiezoceramicmaterialsleadingtofailureofthedevices.Therefore,whiledesigningsuchdevices,itisnecessarytoanalyzefortemperatureriseduetoheatgenerationtoestimatethefactorofsafetyagainstsuchconditions.Variousauthorshaveattemptedtoestimatethisbyfirstfindingthepowerdissipationdensityoverthegeometryofthepiezoactuatorthatursunderthevariousmechanicalandelectricalexcitations.Thetemperatureriseordistributionundersteadystateandtransientconditionsareputedbycarryingoutaheattransferanalysisusingtheheatgeneratedduetothepowerdissipationasaheatsource. Ageneralexpressionforthepowerdissipationdensityduetodielectric,elasticandpiezoelectriclossesundermechanicalandelectricalexcitationwasfirstderivedbyHolland.3Thefrequencyandspatialdependencyofthepowerdissipationdensityresultsinanon-uniformtemperaturedistribution.Therefore,themaximumtemperatureisgenerallygreaterthantheaveragetemperatureestimated.However,inmostoftheliteraturerelatedtoheatgenerationanalysis,thepowerdissipationduetoadynamicelectricalexcitationisgenerallyassumeduniform.Thetotal/averagedensityofpowerdissipatedforpiezoelectricdeviceshavingsuchlossesisobtainedbyintegratingthepowerdissipationdensityusinganelectro-mechanicalimpedancemodel4anduseditforuniformheatgenerationcalculation.Inaddition,notalllossesareountedforputingthepowerdissipation.Forexample,heatinginlowfrequencysonarprojectors,5PMNsonarflex-tensionalprojectors6andinmultilayerpiezoelectricactuators7arestudiedbyassuminguniformpowerdissipationfromthepiezoelectricceramicstack.Inreferences6and7heatgenerationisassumedtobeduebydielectriclossesonly. Therearesomeworksreportedonthecalculationofnonuniformpowerdissipationconsideredintheanalysisofapplicationssuchaspiezoelectrictransformer.8Inthis,however,empiricalrelationshavebeenusedforheatgeneration.Finiteelementmodelshavealsobeenusedinsolvingfornon-uniformpowerdissipation.Forexample,highpoweractivetransducers Page2of30 aremodeledusingATILA,9piezoelectrictransformersaremodeledusingANSYS10andpiezoelectricponentsareanalyzedusinganin-housefiniteelementcode.11ItispossibletoplexmodulirelatedtoallthelossesintheATILAsoftwarewhiletheyareintroducedthroughRaleighdampingcoefficientsforanalysisinANSYS. Transientstatesolutionhasalsobeenstudiedfortemperaturerisewiththeassumptionofuniformandnonuniformpowerdissipation.12,13binationsofheattransfermodes(i.e.,conduction,convection,andradiation)andthermalboundaryconditionshavebeenusedintheabovetosolvethetemperaturedistributionproblem.However,simpleparametricstudiesanddesigniterationsmaybetime-consumingusingthesefiniteelementpackages.Therefore,thereisaneedforevolvingatasimpleanalyticaltoolforthecalculationofthetemperaturedistributioninapiezoelectricactuatorofasimplegeometry(suchasasimpleslab)tocarryoutthedesigniterationcalculationsandtheparametricstudiesthatwillprovideaquickanswertothepreliminaryquestionsaskedinadesignexercise. Thus,inthispaper,ananalyticalsolutionisproposedandformulatedforaonedimensionalpiezoelectricslabinvolvingallthemechanical,dielectricandpiezoelectriclosseswithrealisticconditions.Complexcoefficientsofthematerialpropertiesareconsideredtotakeintoountforthehystereticeffectsinthemechanicalandelectricalregimes.ClosedformsolutionsarederivedforpowerdissipationdensityandthetemperaturedistributioncalculationsarecarriedusingamodifiedFourierseriessolutionmethod. Theworkpresentedinthispaperanizedasfollows.Firstpowerdissipationofapiezoelectricslabissolvedforandthentemperatureestimationisobtainedbysolvingtheenergybalanceequation.Inthenextsection(SectionII),apiezoelectricslabwithinternallossesrepresentedplexcoefficientsisanalyzedfortimeaveragedpowerdissipationdensityandtotalpowerdissipated.Bothmechanicalandelectricalharmonicexcitationsareconsideredwithdifferentmechanicalboundaryconditions.Undermechanicalexcitation,piezoelectricbothshortcircuitedandopen-circuitedconditionsareanalyzed.Expressionsarederivedforthefrequencyandamplitudeatwhichmaximumpowerdissipationursforallcases.Twoapproachesareusedtoestimatethepowerdissipationdensityasaspatialdistributionandtotalpowerasafunctionoffrequency,bothyieldsthesameresult.InSectionIII,thetemperaturedistributionisestimatedbysolvingtheonedimensionalenergybalanceequationfortwothermalendconditionsusingamodifiedFourierseriesmethodingeneralandshownagreeingwithdirection Page3of30 integrationmethodforaspecificproblem.NumericalresultsarepresentedinSectionIVtoillustratespatialdistributionofpowerdissipationdensityandtemperatureinpiezoelectricslabsbeforeendingwithconcludingremarksinSectionV. II.POWERDISSIPATION Beforearrivingatthetemperaturedistributioninapiezoelectricactuatorofasimple geometrysuchasasimpleslab,itisnecessarytocalculatethepowerdissipationofa piezoelectricslab.Thus,apiezoelectricslabwithinternallossesrepresentedplex coefficientsisanalyzedinthefollowingfortimeaveragedpowerdissipationdensityandtotal powerdissipated.Powerdissipationdensityandthetotalpowerdissipatedintheslabare separatelyanalyzedanalyticallyformechanicalandelectricalexcitationsrespectively. ConsiderapiezoelectricslaboflengthLandbreadthBwhereL>>Basshownin Fig.1.ThefacesoftheslabwithareaLBareelectroded.Thedistancebetweentheelectrodesis handL>>h.Thedirections1,2,and3areshowninthefigure1,where3isthedirectionof polarization.NormalandshearstressesonthefaceswithareaLBarezero.Thedistancebetween thesefacesissmallanditis,therefore,assumedthatstressesthatarezeroonthesurfacearezero intheinterioralso.Similarly,stressesonthefaceswithareaLhandthecorrespondingstresses intheinteriorarezero. Theslabisharmonicallyexcitedmechanicallyorelectrically.Therefore,forexample,the normalstressinthe1orxdirectionisexpressedasτ(x,t)=Γ(x,ω)e+jωtwhereωisthe
1 1 angularfrequencyofexcitation,tdenotestime,τ1andΓ1arethetimeandfrequencydependent normalstresses,respectively,inthexor1direction.Γ1(x,ω)isassumedtoplexfor conveniencebutonlytherealpartofτ(x,t)hasphysicalsignificance.Lowercaseanduppercase lettersdenotetime-dependentandfrequency-dependentfunctions,respectively. Itisassumedthattheonlynon-zerostressisthenormalstressinthe1directionandthat theonlynon-zeroelectricalfieldisalongthe3direction.Therefore,thepiezoelectricequations ofstateare S1 ( x,ω ) = sEΓ111 ( x,ω ) + d31E3 (ω ) (1a) and D3 ( x,ω ) = d 31Γ
1 ( x,ω ) + ε T33 E3 (ω ) (1b) Page4of30 whereS1(x,ω)isponentofnormalstraininthexor1direction,andD3(x,ω)isthe componentofelectricdisplacementalongthezor3direction.E3(ω)istheelectricfieldinthe3 directionandisequalto−V/hwhereVisthepotentialdifferencebetweenandbottomelectrodes.ThestrainsS2(x,ω)andS3(x,ω),andthechargedensitiesD1(x,ω)andD2(x,ω), arenotzero. The piezoelectric material coefficients sE,11 ε T33 , and d31are real when there are no losses. Internallossesarerepresentedplexpiezoelectriccoefficients;andthecoefficientsand allfieldvariablesareexpressedasthesumofrealandimaginaryparts.Forexample, S=S'+jS"andsE=sE'+jsE"where'and"denoterealandimaginarypartsrespectively.
1 1
1 11 11 11 Theheatgeneratedinaslabthatisexcitedeithermechanicallyorelectricallyisof interest.Inthissectionexpressionsforthetime-averagedpowerdissipationdensityandthetotal heatarepresented.Thetotalheatisdeterminedbyintegratingthepowerdissipationdensityover thevolumeoftheslabandisshowntobeequaltotheinputpower.Anexpressionforthetime-averagedpowerdissipationdensity,Q(x,ω),intheslabis obtainedbyconsideringhysteresisloops.Integratingtodeterminetheareasundertheloopsand averagingovertimeyields Q(x,ω)=QE(x,ω)+QM(x,ω) []∫∫=1−d(x,t)de(t)+τ(x,t)ds(x,t) (2a)
T 3
3 1
1 whered3(x,t),e3(t),τ1(x,t),ands1(x,t)arethetime-dependentelectricdisplacement,electric field,longitudinalstress,andlongitudinalstrain,respectively.Theintegrationisdoneoveroneperiod,T=2π/ω,andeachintegralistheareaofahysteresisloop.Theterms,QEandQM, representthepowerdissipationintermsofelectricaldisplacementsandmechanicalstresses respectively.Itshouldbenotedthatthepiezoelectriccouplingisinbuiltinboththeexpressions. Then,usingaphasordiagramyields [∫]∫Q(x,ω)=−1d(x,t)de(t)=−1Td(x,t)de3(t)dt=(−ω/2)D(x,ω)E(ω)cos(ψ),(2b) ET33T03dt
3 3 whereψistheanglebetweenthejωE3(ω)andD3(x,ω)phasors. Page5of30 Forthespecialcaseofelectricalexcitation,itconvenienttoassumethatE3(ω)isrealanditthen followsthatψ=π/2−tan−1[Im(D3(x,ω))/Re(D3(x,ω))].Similarly, ∫∫
Q (x,ω)=
1 τ (x,t)ds (x,t)=
1 T τ (x,t)ds1(x,t)dt =(ω/2)Γ (x,ω)
S (x,ω)cos(ϕ) (2c) MT11T01dt
1 1 whereϕistheanglebetweenthestressandstrainratephasors. Inanalternativeapproach,Holland3usedthegeneralizedPoyntingvectortoobtainan expressionforthepowerdissipationdensityinapiezoelectricmedium.ForthespecialcasewhereΓ1(x,ω)andE3(ω)aretheonlyponentsofstressandelectricfield, respectively,theexpressionreducesto ()Q(x,ω) = (ω / 2)Im E3ε3T3*E3* + ΓsE*Γ*1111 + E3d3*1Γ1* + Γ1d3*1E3*
(3) where*plexconjugateandtheargumentsofthefunctionsaresuppressedfor convenience.Usingthenotationforrealandimaginarypartsintheaboveequationyields [()]Q(x,ω)=Q(x,ω)+Q(x,ω)+Q(x,ω)=−(ω/2)εT"E2+sE"Γ2+2d"E'Γ'+E"Γ".
(4)
1 2
3 333 111 3131 31 Itisshownlater,usingnumericalresultsforseveralcases,thatEqs.(2)and(4)areequivalent. Thetime-averagedtotalpowerdissipatedintheslabisexpressedas
L Π(ω)=Bh∫Q(x,ω)dx.
(5)
0 Q(x,ω)andΠ(ω)fortheslabaredeterminedforanysetofboundaryconditionsbysubstituting theappropriateexpressionsforΓ1(x,ω)etcinEqs.(2)and
(4).Forspecialcases,Π(ω)isalso expressedasthepowerinputtotheslabbytheexcitation.
A.Mechanicalexcitation Considerthecaseofslabfixedatx=0andmechanicallyexcitedatx=Lbyaharmonic normalstressτ(L,t)=ΓejωtwhereΓisrealandindependentoffrequency.Theresponseof
1 0
0 theslabisthesameasthatofonehalfofaslaboflength2Lthatissymmetricallyexcitedat bothends.Theanalysisisdoneforshort-circuitandopen-circuitconditions. Themechanicalpowerinputtotheslabisequaltothetotalpowerdissipatedintheslab. Itisexpressedas ∫∫Π(ω)=Bhτ(L,t)du(L,t)=BhTτ(L,t)v(L,t)dt=0.5BhΓV(
L,ω)cos(ϑ) (6a) T1T010 Page6of30 whereu(L,t)andv(L,t)arethedisplacementandvelocityrespectivelyattheexcitedendand ϑ=tan−1[Im(V(
L,ω))/Re(V(
L,ω))]. (6b) istheanglebetweenthestressandvelocityphasors.Powerdissipationdensityandtotalpower dissipatedintheslabarederivedfortwoconditionsundermechanicalexcitationsinthispaper.
1.Short-circuitconditions Undershort-circuitconditions,thepotentialiszeroonboththeelectrodesandisassumed tobezeroeverywherewithintheslabbecausethedistancebetweentheelectrodesissmallwith respecttothelengthoftheslabandthevibrationalongthelengthisdominantandofinterest. Therefore,E3(ω)iszeroandtheEqs.ofstate(1)andotherequationsaresimplified.Solvingthe differentialequationsgoverningvibrationandusingtheboundaryconditionsatx=0andL yields U(x,ω)=sEΓsin(kx), (7a) 110kcos(kL) Γ(x,ω)=Γcos(kx), (7b)
1 0cos(kL) and S(x,ω)=sEΓcos(kx) (7c)
1 110cos(kL) where k =ω/cbE and cbE =(ρsE)−0.5.11 SubstitutingEqs.(7b)and(7c)ineitherEqs.(2)or(4)yieldsthetime-averagedpower dissipationdensity Q(x,ω)=−0.5ωsE"Γ2cos(2k'x)+cosh(2k"x).
(8) 110cos(2k'L)+cosh(2k"L) Thetime-averagedtotalpowerdissipatedintheslabisdeterminedbyusingEqs.(5)and
(8),andperformingtheintegrationanalyticallyasdoneinRef.14.Itisexpressedas Π(ω)=−Bh4ωks'1kE1""Γ02[k["csoins((22kk''LL))++cko'ssihn(h2(k2"kL")L])](9a) and,atlowfrequencies,reducesto Π(ω)≈−0.5LBhωs1E1"Γ02k'L,k"L<<
1 (9b) Page7of30 whereitisproportionaltothevolumeoftheslab.Alternatively,usingEq.(7a)todeterminethe velocity,substitutingtheresultinEq.(6)fortheinputpower,andfollowingthesameprocedureusedtodeterminethetotalheatgeneratedinanelasticrodwithinternallosses14yields Π(ω)=0.5BhΓ02(s1E1/ρ)0.5tan(kL)cos(ϑ). (10a) When2k'Lisapproximatelynπ,n=1,3,5,...,Π(ω)isclosetoalocalmaximumandis determinedusing ()()ϑ ≈ tan−1⎜⎛ − 2sE'11 ⎟⎞ + tan−1⎜⎛ ωρ0.5s1E1" ⎟⎞. ⎜⎝ sE"11 ⎟⎠ ⎜⎝ sE'11 0.5π−2ω(ρsE')0.511 ⎟⎠ (10b) Theangularfrequency,ωΠ,atwhichthetotalpowerdissipatedismaximumis determinedbydifferentiatingEq.(9a)withrespecttoωandequatingittozero.Then, anticipatingthat,atresonance,2k'Lisalittlelessthannπ,n=1,3,5,...,andusing k=ω(α'+jα")=ω(ρs1E1)0.5and2k"L≈0tosolvetheequationyields ω≈nπα'≈nπ(s1E1'/ρ)0.5. (11) Π2Lα'2+α"22LsE 11 ThepowerdissipatedatresonanceisthendeterminedbyusingEqs.(9a)and(11)andexpressed as ()Πmax ≈−BhΓ02 sE"11 (α '
2 +α"
2 ) 2nπα'α"
2 2BhΓ2sE'1.5 ≈− 011 nπ ρ 0.5 sE"11 (12) wherethenegativesignispresentbecausesE"isnegative.11
2.Open–circuitconditions Underopen-circuitconditions,thetotalcurrent,i(t),iszero,i.e., ∫i(t) = Bh
L ⎡⎢ dd3 (x,t) ⎥⎤dx =
0 . (13) 0⎣dt⎦ SolvingthegoverningdifferentialequationfordisplacementandusingtheconstitutiveEqs.
(1),theopen-circuitconditioninEq.(13),thefixedboundaryconditionsatx=0,andthespecifiedstressatx=Lyields Page8of30 U(x,ω)=Ls1E1Γsin(kx),
1 χ0sin(kL) (14a)
S (x,ω) = sE11 Γ ξ ,
1 χ
0 Γ1(x,ω)=Γ0(κ+ξ)/χ, D(x,ω)=Γd[ξ−1]/χ,
3 031 (14b)(14c)(14d) andE3(ω)=−Γ0s1E1κ/(d31χ). wherek321=d321/(s1E1ε3T3)istheelectromechanicalcouplingcoefficient,χ=κ+kLcot(kL),andξ=kLcos(kx)/sin(kL). (14d) κ = k321 /(1− k 231 ) , SubstitutingEqs.(14)inEq.(2)yieldsanexpressionforthepowerdissipationdensityin theform []Q(x,ω)=0.5ωχ−2ΓsEκ(ξ−1)cos(ψ)+sE(κ+ξ)ξcos(ϕ). (15) 011 11 SubstitutingEq.(12)inEq.(4)yieldsanequivalentexpression [()()]}Q(x,ω) = 0.5ωΓ(
L,ω) ⎨⎧⎩ χ −
2 ⎢⎣⎡ε3T3" s1E1κ / d31
2 + sE"11 κ +ξ
2 ⎤⎥⎦ + . (16) [][]2d"sEκ/(dχ)'(κ+ξ)/χ'+sEκ/(dχ)"(κ+ξ)/χ" 3111 31 11 31 ItisshownlaterusingnumericalresultsthatEqs.(15)and(16)obtainedusingthehysteresis loopandPoyntingtheorem,respectivelyareequivalent. Thetotalpowerdissipatedintheslabisdeterminedbynumericallyintegratingthepower dissipationdensityoverthevolumeoftheslab.Alternatively,becauseitisequaltothe mechanicalpowerinputtotheslab,itisexpressedusingEq.(6)as Π(ω) = LBhωΓ02 sE11 / χ cos(ϑ)/
2 . (17a) Atlowfrequencies,theexpressionreducesto [()]Π(ω)=0.5ωLBhΓ02(k3'21−1)s1"1E+k3"12s1'E1’ (17b) andthepowerisproportionaltothevolumeoftheslab.Theangularfrequencyatwhichtheinputpowerismaximum,ωΠ,isofinterest.Itisseen fromEq.(6)thatωΠisalsothefrequencyatwhichtherealpartof Page9of30 V(
L,ω)= jωL(s1E1'+js1E1"). (κ'+jκ")+(k'L+jk"L)cot(k'L+jk"L) (18a) ismaximumanditisdeterminedasfollows.Usingcot(kL)=sin(2k'L)−jsinh(2k"L),cosh(2k"L)−cos(2k'L) (18b) toseparateEq.(18a)intoitsrealandimaginarypartsyieldsanexpressionforV'(
L,ω).The numeratorofthisexpressionvariesnearlylinearlywithfrequencywhen2k'Lisintheneighborhoodofnπ.Thedenominatoristhesquareoftheabsolutevalueofplexfunctionandisneverzero.Thesignoftherealpartofthefunctionchangesinthisneighborhoodwhereastheimaginarypartisnearlyconstantanddoesnotchangesign.Therefore,V'(
L,ω)ismaximum approximatelyatthefrequencyatwhichtherealpartofplexfunctioniszero,i.e.,when Re[(κ'+jκ")[cosh(2k"L)−cos(2k'L)]+(k'L+jk"L)[sin(2k'L)−jsinh(2k"L)]]=
0. (19) SimplifyingEq.(19)byusing2k"L≈
0,α'>>α",andanticipatingthat,atresonance,2k'Lis approximatelynπ,n=1,3,5,...,yieldsanquadraticEq.inωthatissolvedtoobtain {[]}ωΠ≈nπ(1+2κ')+(1+2κ')2+4[4(nπ)−2−1]κ'(1+κ')0.5[4α'L(1+κ')]−
1. (20) ThepowerdissipatedatresonanceisthenobtainedbyusingEqs.(6)and(18).Usingallthe approximationusedtodetermineωΠyields Π≈LBhΓ02 ωπ sE'11 . (21) max 2κ"−0.5ωπ2ρL2s1E1" ItisseenfromEqs.(20)and(21)thatΠmaxisindependentofthelengthoftheslab.
3.Electricalexcitation Considernextthecasewheretheslabiselectricallyexcitedbyapplyingaharmonic voltage,v(t)=V0ejωt,acrosstheelectrodes.Inthiscase,itisconvenienttoconsideraslabof lengthLwithendsatx=±L/2.Thestressesattheendsarezero.Thefieldsofinterestare expressedas E3=−V0/h, (22a) S1(x,ω)=E3d31γ, (22b) Page10of30 Γ1(x,ω) = E3d31(γ −1)/sE11 ≈ 0.5E3d31k 2(0.25L2 − x2) / sE11 ,kL<<
1 (22c) and []D3(x,ω)=E3ε3T31+k321(γ−1) (22d) whereγ=cos(kx)/cos(kL/2).Theinputelectricaladmittance,Y(ω),oftheslabisexpressedas ∫()Y(ω) = G(ω)+ jΒ(ω) = −jωBh L/2
D (ω)dx =jω LBε
T ⎡⎢
1 − k
2 +k2 tan(kL/2)⎤⎥ (23) 3−L/2 h 33 ⎣ 3131(kL/2)⎦ whereGandΒaretheelectricalconductanceandsusceptance,respectively. ThepowerdissipationdensityisobtainedbysubstitutingEq.(22)inEq.(2)and expressedas [()]QE ( x,ω ) = 0.5ωE32 ε T33 k321γ(γ−1)cos(ϕ)− 1+k321(γ−1) cos(ψ). (24a) QM (x,ω) = 0.5ωE32 εT33 ⎣k321γ (γ −1) cos(ϕ)⎦ (24b) Alternatively,itisobtainedbyusingEqs.(20)and(4)andexpressedas Q(x,ω) = −0.5ω
E 23 ⎢⎣⎡ε T"33 + sE"11 (γ −1)d31 / sE11
2 + (2d3"1(γ −1)d31 / sE11 )' ⎤⎥⎦ . (24c) SubstitutingtheexpressionsforelectricfieldandstressatlowfrequenciesinEq.(22) intoEq.(4)yields Q(x,ω)=−0.5ωεT"E2,
1 333 (25a) Q(x,ω)≈−0.5ωE2sE"d/sE2k4(0.25L2−x2)
2,
2 3113111 (25b) and Q(x,ω)≈−0.5ω3d'd"E2ρ(0.25L2−x2).
3 31313 (25c) where Q2(x,ω)and Q3(x,ω)are maximum at x =
0. Then, it is seen by using tan(δε ) = εT"33 / εT'33 , tan(δs ) = sE"11 / sE'11 , and tan(δd ) = d3"
1 / d3'
1 that Q2(
0,ω)≈k2sin(δs)(kL)4<0,ω)≈k2sin(2δd)(kL)4<8
31
It,therefore,followsthat,atlowfrequencies,onlydielectriclossesaresignificant,
Q(x,ω)≈−ωε3T3"E32/2,
(26b)(27)
andΠ(ω)≈−LBhωε3T3"E32/2.(28)
Thetime-averagedtotalpowerdissipatedintheslab,atarbitraryfrequency,isobtained
byconsideringtheelectricpowerinputtotheslab.Itisexpressedas
Π(ω)=V02G(ω)/2
(29a)
wheretheinputelectricaladmittance
[()]G(ω)=−ωLB⎜⎛εT1−k2"+2[ζ'sinh(k"L)+ζ"sin(k'L)]⎟⎞
h⎜⎝33
31
cos(k'L)+cosh(k"L)⎟⎠
(29b)
is
obtained
by
rearranging
Eq.
(23)
for
the
admittance
and
ζ
=
d321cbE
/
sE
11 . At low frequencies, it isseenbyusingEq.(27b)thatEq.(27a)reducestoEq.(27). Theconductanceand,therefore,thetotalpowerdissipatedaremaximuminthe neighborhoodofk'L=nπ.Usingapproximationsinthisneighborhoodandk"L≈0yields [()]G(ω)=−B⎜⎜⎛ωLεT1−k2 "+4[ζ'k"L+ζ"(nπ−k'L)]⎟⎞. ⎟ (30) h⎝ 33 31 (k"L)2+(nπ−k'L)2⎠ InEq.(30),thefirsttermandthenumeratorofthesecondtermarefirstdegreefunctionsof frequency.However,thedenominatorofthesecondtermisaseconddegreefunction.TheangularfrequencyatwhichGismaximum,ωs,isalsotheangularfrequencyatwhichthe maximumpowerisdissipated.Thisfrequencyisapproximatelythefrequencyatwhichthe denominatoriszeroandisexpressedasωπ=ωs≈L(αn'2π+αα'"2).(31) Finally,substitutingEq.(31)inEq.(29)todeterminethemaximumpowerdissipatedandusing sE"/sE'1111 <<
1 yields ()Π ≈−E2 Bh ⎜⎛nπ[εT(1−k2)]"+
8 d2'31 ⎟⎞ . max3ρs1E1'0.5⎜⎝3331nπs1E1"⎟⎠ (32) Page12of30 wherethefirsttermisoftenmuchsmallerthanthesecondbecausesE"ursinthedenominator11 ofthesecondterm.ItisofinteresttonotethatΠmaxdependsontherealandimaginarypartsofthecoefficientsandisindependentofthelengthoftheslab. III.TEMPERATUREDISTRIBUTION Havingarrivedatexpressionsforpowerdissipationfortheelectricalandmechanicalexcitations,wearenowreadytosolveforthetemperaturedistributionbasedonthethermodynamicbalancelaw.Differentthermalendconditionshavebeenconsideredhere. Considerthecasewhereheatconductionintheslabandconvectiontothesurroundingsaresignificantandaffectthetemperaturedistributionintheslabthatisofinterest.Asstatedearlier,itisassumedthatthecross-sectionissmallwithrespecttothelengthandthatthepowerdissipationdensityintheslabisconstantoverthecross-section.Itis,therefore,alsoassumedthatthetemperatureisconstantoverthecross-sectionandtheone-dimensionalheatequationisused. Thegoverningenergybalanceequation15isexpressedasΚd2θ(x,ω)−Μθ(x,ω)=−Q(x,ω),−d≤x≤d(33)dx2 whereθ(x,ω)isthedifferencebetweenthetemperatureintheslabandtheuniformambienttemperature,Κistheconductioncoefficient,andΜ=2hˆsl(B+h)/Bhandhˆslistheheatconvectioncoefficient,anddiscase-dependent.Theslabthatismechanicallyexcitedisfixedatx=0,andexcitedatx=
L.Itisalsothermallyinsulatedatx=
0.Therefore,thetemperatureisthesameasthatintheregion0≤x≤Lofaslaboflength2Lthatissymmetricallyexcitedatx=±
L.Inthiscase,d=
L.TheslabthatiselectricallyexcitedisoflengthL,haszerostressatbothends,andissymmetricallyexcitedoveritslength.Inthiscase,d=L/2.Inbothcases,dθ(
0,ω)/dx=0andthetemperaturedistributionissymmetricaboutx=
0 Twothermalboundaryconditionsareconsidered.Inthefirstcondition,thetemperature intheslabatx=disequaltotheambienttemperature,i.e. θ(d,ω)=
0. (34a) Inthesecondcondition,noheatislosttothesurroundingsatx=disassumed,i.e. dθ(d,ω)/dx=
0. (34b) Page13of30 TheenergybalanceequationandthethermalboundaryconditionsaresolvedinallcasesbyusingamodifiedFourierseriesmethod.Forsomecases,theyaresolvedbydirectintegrationalsoanditisshownthatthetwomethodsyieldthesameresultsthusvalidatingtheFourierseriesapproachthatismoregeneral.ThederivationofthetemperaturedistributionsolutionisdoneforageneralcaseofQ(x,ω)whichistheheatgeneratedatanylocation,x,forafrequencyof loading,ωatasteadystatecondition.Tohighlighttheimportanceofdeterminingthetemperaturedistributionforageneralcasesuchasthis,thesolutionforacaseinwhichheatgeneratedisuniformlydistributedovertheslab,i.e.,Q(x,ω)=Q0(ω)=Π(ω)/(LBh)isalso solvedforparedwiththetemperaturedistributionwhentheactualnon-uniformheat distribution. InthemodifiedFourierseriesmethod,thetemperaturedistributionintheslabthatis mechanicallyorelectricallyexcitedisexpressedintheform ∑∞ θ(x,ω)=Ax2+Bncos(knx) (35) n=
0 wherekn=nπ/d.Asnotedearlier,thetemperaturedistributionaboutx=0issymmetricand, therefore,thenon-symmetricsintermsarenotpresentinEq.(35).Itisnotedthatthefunctions,cos(knx),formaorthogonalpletesetintheregion x=[−d,d].Specifically,eachfunctioninthesetisorthogonaltoalltheothersbecause ∫∫⎧0m≠n dcos(kmx)cos(knx)dx=2dcos(kmx)cos(knx)dx=⎪⎨dm=n=1,2,...∞.(36) −d
0 ⎪⎩2dm=n=
0 Further,thesetpletebecausetheredoesnotexistanyfunctionf(x)thatissymmetric aboutx=0andorthogonaltoallthetermsoftheFourierseries,i.e.,nofunctionsatisfiesthe condition d ∫f(x)cos(knx)dx=0,n=0,1,
2,…∞. (37) −d However,differentiatingEq.(35)termbytermyields ∑d2θ(x,ω)=A−∞Bk2cos(kx) (38) dx2 nn n n=
1 Page14of30 anditisseenthatAinEq.(38)correspondstoB0inEq.(35)andthattheexpressionforthe secondderivativeofθ(x,ω)pleteonlywhenthetermAispresent.Atermthatis proportionaltoxisnotrequiredinEq.(35)becausethetemperaturedistributionissymmetricwithrespecttox=
0.TheseriesinEq.(35)iscalledamodifiedFourierseriesbecauseofthe additionalAx2term.Foranyparticularcase,substitutingtheexpressionforQ(x,ω)intoEq.(33),multiplying bothsidesbycos(kmx),m=0,1,2,...∞,integratingoverxfrom0tod,andusingthe orthogonalpropertyofthecosfunctionsinEq.(36)yields ()A2Κ−Μd2/3−B0Μ=−Q0(ω) (39a) ∫A(−1)m+14Μ/km2−Bm(km2Κ+Μ)=−Qm(ω)=−2d−10dQ(x,ω)cos(kmx)dxm=1,
2,…∞(39b) where ∫Q0(ω)=d−10dQ(x,ω)dx=Π(ω)/(LBh) (39c) istheaveragepowerdissipationdensityandQm(ω)=0,m=0,1,2,...∞forthecaseofuniform heatgeneration. SubstitutingtheformofthesolutioninEq.(35)intheboundaryconditionsinEqs.(34a) and(34b)yields ∑∞ AL2+B0+Bn(−1)n=0 n=
1 (40a) andA=
0, (40b) respectively.CombiningEqs.(39)and(40a)or(40b)andretainingtermsonlytermsuptoBN yieldsequationsthataresolvedbyusingmatrixmethodsorbyelimination.Forthespecialcaseofuniformheating,Q(x,ω)=Q0(ω).SolvingEq.(33)bydirect integrationandretainingonlythetermsthataresymmetricaboutx=0yields []θ(x,ω)=Fcosh(M/K)0.5x+(Q0/M) (41a) wherethethermalboundaryconditionsforthetwocasesinEqs.(34a)and(34b)aresatisfiedby
F =
M [−Q0 cosh(M/ K)0.5 d] (41b) Page15of30 and F=
0, (41c) respectively.InthemodifiedFourierseriesapproach,thetemperaturedistributionisdetermined byusingQm(ω)=0,m=0,1,2,...∞inEqs.(39)andtheappropriateboundaryconditioninEq. (40). FromthegeneralsolutionobtainedinEqs.(39)and(40),thetemperaturedistributionfor thespecificconditionsofmechanicalandelectricalexcitationcannowbedetermined.
A.MechanicalExcitation Consideragainthecaseofslabfixedatx=0andmechanicallyexcitedatx=Lbya harmonicnormalstressτ(L,t)=Γejωt.
1 0
1.Short-circuitconditions Thenon-uniformpowerdissipationdensityforthiscaseispresentedinEq.
(8).The temperaturedistribution,inthemodifiedFourierseriesapproach,isdeterminedbyusing Q0(ω)=0.5β(k'k")−1[k'sinh(2k"L)+k"sin(2k'L)], (42a) and where
Q (ω) = (−1)mβ ⎡k'sin(2k'L)⎢ + k"sin(2k"L)⎤⎥ m=0,1,2,...∞, m 2k'k" ⎣ (
4 k '
2 − k 2m ) (
4 k "
2 + k 2m ) ⎦ β= − ω Γ02 sE"11 cos(2k'L)+cosh(2k"L) (42b)(42c) inEq.(39)andtheappropriateboundaryconditioninEq.(40).Thetemperaturedistributionintheslabfornon-uniformheatgenerationisalso determinedbydirectlyintegratingEq.(33)aftersubstitutinginittheexpressionforQ(x,ω)in Eq.
(8).IntegratingandusingtheboundaryconditioninEq.(34)yields []θ(x,ω)=Fcosh(Μ/Κ)0.5x+β⎜⎛cos(2k'x)−cosh(2k"x)⎟⎞2⎝4k'2Κ+Μ4k"2Κ−Μ⎠ where
F = (Μ / Κ)0.5 [β sinh(Μ / Κ)0.5 L]⎜⎝⎛ k'sin(2k'L)4k'2Κ+Μ + k"sinh(2k"L)4k"2Κ−Μ ⎟⎞⎠ (43a)(43b) Page16of30 forthecaseofzerotemperaturedifferenceand
F = [−β 2cosh(Μ/ Κ)0.5 L]⎜⎝⎛ cos(2k'L)4k'2Κ+Μ − cosh(2k"L)4k"2Κ−Μ ⎟⎞⎠ , (43c) forthecaseofzeroheattransfer.Itisshowninthesectiononnumericalresultsthatthe temperaturedistributionsobtainedbyusingthemodifiedFourierseriesanddirectintegration methodsarefoundtobethesamefortheshort-circuitmechanicalexcitationcase.However,it maynotpossibletodirectlyintegrateforothercasessuchasopen-circuitandelectricalexcitation conditions.Forsuchcases,modifiedFourierseriesmethodisadoptedfordeterminingthe temperaturedistribution.
2.Open-circuitconditions Forthiscase,equivalentexpressionsforthenon-uniformpowerdissipationdensity(equivalentlytheheatgenerated),Q(x,ω),arepresentedinEqs.(15)and(16).Inthemodified Fourierseriesapproach,solvingEqs.(39)withtheappropriateboundaryconditionsinEq.
(4) usingtheaveragepowerdissipationdensity, Q0(ω) = ωΓ02 sE11 / χ cos(υ)/
2 (44) isobtainedfromEq.(17a)andQm(ω)obtainedbyusingnumericalmethods.Equation(15)or (16),yieldsthecoefficientsinEq.(33)thatarerequiredputethetemperaturedistribution.IfthepowerdissipationdensityisassumedtobeuniformthenQm(ω)=0form=0,1,2,...∞and thetemperaturedistributionismoreeasilydeterminedusingmodifiedFourierseries.
3.ElectricalExcitation Consider,next,thetemperaturedistributionwhentheslabiselectricallyexcitedbyanappliedvoltage.Thenon-uniformpowerdissipationdensityispresentedinEq.(24)anditisseenfromEqs.(39c)and(29)that [()]Q0 (ω) = −ωE32 ⎜⎜⎛ εT33 1− k321 "+δ'sinh(k"L)+δ"sin(k'L)⎟⎟⎞. (45) ⎝ cos(k'L)+cosh(k"L)⎠ InthemodifiedFourierseriesapproach,thetemperaturedistributionisdeterminedbyusingEqs. (39),(40),(45),(24),andnumericalmethodstoevaluate∫LQ(x,ω)cos(kmx)dx.Asintheother0 Page17of30 cases,thetemperaturedistributionwhenthepowerdissipationdensityisassumedtobeuniformisdeterminedbysettingQm(ω)=0,form=0,1,2,...∞.IV.NUMERICALRESULTSANDDISCUSSION Somenumericalresultsareshowntoillustratethenon-uniformheatgenerationandthenonuniformtemperaturedistributionofonedimensionalpiezoelectricslab.ApiezoelectricslabischosenwithlengthL=100mm,widthB=10mm,andthicknessh=5mm,andwithmaterialproperties,16ρ=7500Kg/m3,ε3T3=1300×8.85e−12(1−j0.004),d31=−123e−12(1−j0.005),ands1E1=12.3e−12(1−j0.005).Aharmonicstressof20kPaasmechanicalexcitationandanelectricalexcitationof10Vpeakisusedunlessstatedotherwise.Thesearenumbersthataretypicalforwhichtemperaturebuildupislikelyinlinearregion. Firstconsiderthepiezoelectricslabundermechanicalexcitation(short-circuitedcase)fixedatx=0andexcitedatx=
L.Thenon-uniformpowerdissipationdensityQ(x,ω),forfourdifferentfrequencies:atfirstandsecondresonancefrequenciesoftheslabinlongitudinalmodeandslightlybelowthesefrequencies,aredetermined.ThevariationofthepowerdissipationdensityisshowninFig.2forthesefrequencies.Significantvariationisobservedalongthelengthoftheslab.Thepeakvaluecorrespondstothefixedendwherethestressismaximum.Itisimportanttonoticethatevena0.5%deviationfromtheresonantfrequencycausesasmuchas75%reductionofmaximumheatgeneration.Thenumberofpeakscoincideswiththeresonancefrequencymode.ThetotalaveragepowerinputtotherodisconvertedintoheatputedusingEqs.(9)and(10)isshowninFig.3.Boththeequationsyieldthesameresultinallfrequencies.TheapproximatepeakvaluesΠ(ω)andthecorrespondingfrequencyfΠobtainedfromEq.(9)areincloseagreementwiththecalculatedvaluesusinganalyticalrelationsasinEqs.(11)and(12)asshowninTableI. Nextconsiderthepiezoelectricslabundermechanicalexcitationwithopen-circuitedconditionfixedatx=0andexcitedatx=
L.Thepowerdissipationdensitiesatandnear1stand2ndresonancefrequencyareplottedinFig.4.Bothapproaches-Eqs.(15)throughthehysteresisapproachand(16)throughthePoyntingtheoremgivethesameresultandoneparticularcase(1stresonance)isplottedtogetherforreference. Page18of30 TABLEI.Comparisonoffrequenciesandamplitudesatmaximumpowerdissipation Excitationtype Mechanical Electrical Assumptionsandboundaryconditions Short-circuited,oneendfixedandforcedatotherOpen-circuited,oneendfixedandforcedatotherBothendsstressfree AnalyticalfΠ(kHz),Πmax(watts) 8.231,2.57824.693,0.8598.611,2.51424.825,0.85716.462,3.30149.386,1.102 NumericalfΠ(kHz),Πmax(watts) 8.231,2.57824.693,0.8598.612,2.39324.825,0.85216.462,3.30142.767,1.101 TotalinputpowerdissipatediscalculatedbyEq.(17)andbyintegratingthepowerdissipation densityinEq.(15).BoththeequationsyieldthesameresultasshowninFig.5.Thefrequency andtheamplitudecorrespondingtothemaximumpowerdissipatedfromEq.(17)andthe analyticalexpressionaregiveninTableIwhichareincloseagreement.Fromtheresultsof mechanicalexcitationwecanseethattheopen-circuitconditionhasahigherresonance frequencyvaluethantheshortcircuitedslabdueaddedstiffnesseffect.However,maximum powerdissipatedwhichiscontrolledbytheimaginarypartofpliances1E1,isnearlythe sameforboththecases. Nextconsiderpiezoelectricslabwithelectricalexcitationwithendsatx=±L/2usedfor symmetry.ThepowerdissipationdensityfromEqs(24)isplottedinFig.6.BothhysteresisandPoyntingtheoremapproachyieldthesameresultandbothvaluesareplottedforthe2ndresonant frequency.TheinputpowerdissipatediscalculatedfromEq.(29)andnumericallyintegrating Eq.(24)isplottedinFig.7.Boththeapproachesyieldthesameresultsforallexcitation frequencies.Thefrequencyandtheamplitudecorrespondingtothemaximumpowerdissipated fromEqs.(31)and(32)paredwithnumericallycalculatedvaluesandpresentedinTable I,showgoodagreement. Theincreasein sE′′11 causesthefrequency fΠmaxtoshifttoahigher valueunlikethecaseinwhichthereisashifttoalowervaluewheredisplacementismaximum foradampedstructure.Thepeakamplitudeoftotalpowerdissipatedapproximatelydrops4 timesforlossof0.005to0.02asshowninFig.8.Itcanbeseenthat,therateofreductioninthe amplitudedecreaseswithincreaseinloss.Asimilareffectonthepeaktemperaturewithelastic losscanbeseeninFig.14. Page19of30 Temperaturedistributionsofthepiezoelectricslabfordifferentexcitationsandthermalboundaryconditionsarecalculated.Inalltemperaturecalculations,Κ=2.0W/moCandhˆsl=10W/m2oCareusedunlessstatedotherwise.5Figure9showsthetemperaturedistributionoverthepiezoelectricslabformechanicalexcitationwithshort-circuitedcondition,calculatedusingdifferentmethods:themodifiedFourierseriesandthedirectintegrationmethodsgivenbyEqs.(42)and(43)respectively.Boththeapproachesyieldthesameresult.Thetemperaturedistributionalsoshowsahighertemperatureatthefixedendandtapersdowntoambienttemperatureatx=Lsatisfyingθ(d,ω)=
0. OnlymodifiedFourierseriesisusedforcalculatingtemperaturedistributionofslabundermechanicalexcitationwithopen-circuitedconditionandtheresultsatandnearthefirsttworesonancefrequenciesareshowninFig.10.Itisnotedthatatsecondresonantfrequency,unliketheearlierresultforthefirstresonantfrequency,thetemperaturedistributiondoesnotfollowthesametrend,asthepowerdissipationdensitydistribution(Fig.4).Theotherboundaryconditiondθ(d,ω)/dx=0effectisplottedat1stresonancefrequencyandthetemperaturevariationisalmostthesameasθ(d,ω)=0exceptnearertotheexcitationend,x=
L. Theeffectofuniformandnon-uniformheatgenerationassumptionsinpredictingthepiezoelectricslabtemperaturedistributionunderelectricalexcitationisshowninFig.11forthetworesonancefrequencies.Anincreaseof6paredtouniformheatingcaseisnoticedatthefirstresonancefrequencybutthedifferenceisnotsignificantforthesecondresonancemode.Theeffectofthermalboundaryconditionsatx=Lparedhereandresultsareinagreementwiththeassumptions.Figure12showstheresultoftemperaturedistributionasafunctionofexcitationfrequencyfortwodifferentthermalconditionswhichbringsoutthattheendconditionsdoesnotaffectthemaximumtemperatureataparticularfrequency.Differentauthorshaveuseddifferentvaluesbinationsoftheheattransferconstantsintemperatureestimation.TheimplicationoftheselectionofheatconductivityandheattransfercoefficientvaluesareplottedinFig.13.Itcanbenoticedthattheresultsvarysignificantlywiththechoiceoftheheattransferconstants.Hence,properchoiceofheattransfercoefficientsbasedonstrongexperimentalresultsisabsolutelynecessary. Fromthenumericalresults,itisshownthattheincreaseintemperatureriseinapiezoelectricmaterialoractuatordependsontheexcitationvoltageandtheelasticlossfactor.Reductionin Page20of30 material elastic imaginary property sE"11 causes further increase in the heat generation and leads tohighertemperaturebuild-upandthermalrunaway.Moreover,estimationoftemperature distributioninthefirstfundamentalmodeusinguniformheatgenerationunderestimatesthepeak value.Therefore,non-uniformheatgenerationtogetherwithtemperaturedistributionshouldbe assumedlestonewouldmissthelocalhot-spotsorfailure.Thus,adetailedanalysis/parametric studyispossiblewiththesemethodstoselectpiezoelectricconfigurations,materiallosses,and resonancemodesoroperatingfrequenciesdependingupontheexcitationvoltageatdesignstage itself.
V.CONCLUSIONS Non-uniformpowerdissipationandtemperaturedistributioninonedimensionalpiezoelectric slabwithinternallossesundermechanicalandelectricalexcitationareinvestigated.Lossesare representedplexelectro-elasticcoefficients.Itisassumedthatplexmaterial propertiesareindependentoffrequencyandtemperature.Analyticalexpressionsarederivedfor powerdissipationdensityundershortandopencircuitconditionsundermechanicalexcitation. Theonlycoefficientthataffectstheresponseoftheshort-circuitedpiezoelectricslabiss1E1. Underopen-circuitconditions,theslabiseffectivelystifferandthefrequencyatwhich maximumpowerisdissipatedisalittlegreaterthanthatoftheshort-circuitedslab.However,in bothcases,themaximumtotalpowerdissipatedisnearlythesameandindependentofthelength oftheslab.Approximateexpressionsarepresentedforthefrequencyandtheamplitudeatwhich maximumpowerisdissipated.Numericalvaluesofthemaximumpowerdissipatedobtainedby plottingthepowerdissipatedversusfrequencyareingoodagreementwiththoseobtainedusing theapproximateexpressions.Themaximumpowerdissipatedisexactlyinverselyproportionalto sE"11 in some cases and approximately inversely proportional in others. The frequency at which maximum
powerdissipationursisonlyweaklydependentonlosses.Nearthefirstresonance, thenon-uniformheatgenerationresultsinasignificantlyhighermaximumtemperaturethan uniformtemperaturedistribution.Athigherresonances,thereismorethanonemaximuminthe powerdissipationdensityandtheeffectofnon-uniformheatgenerationislesssignificant.The modifiedFourierseriesmethodisfoundtobeagoodgeneralsolutionforenergybalance equationtoountdifferentheattransfermodesintheslabandthermalboundaryconditions. Page21of30 ACKNOWLEDGEMENTEncouragementandfacilitiesgivenbyDirector,NPOLtocarryoutthisworkare gratefullyacknowledged. REFERENCES1.D.A.Berlincourt,
D.R.Curran,andH.Jaffe,“Piezoelectricandicmaterialsandtheirapplicationintransducers,”PhysicalAcousticsVol.1A,editedbyW.P.Mason(Academic,NewYork,1964).2.J.Yang,“Piezoelectrictransformerstructuralmodeling–Areview,”IEEETrans.Ultrason.Ferroelectr.Freq.Control54
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K.Jung,“Analysisoftemperatureriseforpiezoelectrictransformerusingfinite-elementmethod,”IEEETrans.Ultrason.Ferroelectr.Freq.Control53,8,1449-1457(2006). Page22of30 11.RKeoschkerjan,
M.Harutyunyan,andH.Wurmus,“Analysisofself-heatingphenomenonofpiezoelectricponentsactuatedharmonically,”MicrosystemTechnologies9,75-80(2002). 12.M.S.Senousy,
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S.Priya,andS.Hirose,“Lossmechanismsandhighpowerpiezoelectrics,”JMater.Science41,217228(2006). 14.D.D.Ebenezer,
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I.Resonancemethods,”5thInternationalCongressonSoundandVibration,Adelaide,Australia,15–18December1997,pp.385–393. Page23of30 FIGURES321orx FIG.1.Piezoelectricslab,lengthexpander,lengthL,widthB,andthicknessh.Directionofpolarization3andelectrodeandbottomfaceofareaLB. FIG.2.PowerdissipationdensityQ(x,ω)inamechanicallyexcitedshort-circuitedpiezoelectricslab,fixedatx=0andexcitedatx=LthroughanalyticalrelationEq.8,.Dash-dottedline:8.19kHz;solidline:8.231kHz(1stresonance);dashedline:24.57kHz,dottedline:24.693(2ndresonance).AppliedharmonicstressΓ=20kN/m2
0 Page24of30 FIG.3.TotalpowerΠ(ω)dissipatedinamechanicallyexcitedshort-circuitedpiezoelectricslab,fixedatx=0andexcitedatx=L,withtwoapproaches.Solidline:AnalyticalsolutionintegrationofpowerdissipationdensitythroughhysteresisEq.(9a)andcircles:Inputpower,Eq.(10a). FIG.4.PowerdissipationdensityQ(x,ω)inamechanicallyexcitedpiezoelectricslabwithopen-circuitedcondition,fixedatx=0andexcitedatx=
L.Solidline:8.611kHz,24.825(1stand2ndresonance)–hysteresisapproach,Eq.(15);Circles:Poyntingtheorem,Eq.(16). Page25of30 FIG.5.TotalpowerΠ(ω)dissipatedinamechanicallyexcitedopencircuitpiezoelectricslab,fixedatx=0andexcitedatx=
L.Solidline:inputpowerEq.17aandcircles:integrationofpowerdissipationdensitythroughhysteresisQ(x,ω),Eq.(16),overvolumebySimpson'srule FIG.6PowerdissipationdensityQ(x,ω)inanelectricallyexcitedpiezoelectricslabat10Vpeak,stressfreeatx=0andL.Solidline:16.38kHz,dashedline:16.462kHz(1stresonance),dasheddottedline:49.139kHz,anddottedline:49.386kHz(2ndresonance)Eqs.(24a)and(24b);Circles:PoyntingtheoremapproachEq.(24c). Page26of30 FIG.7.TotalpowerΠ(ω)dissipatedinanelectricallyexcitedpiezoelectricslabat10Vpeak,stressfreeatx=0andL.Circles:integrationofQ(x,ω),Eq.(24)overvolumebySimpson'sruleandsolidline:inputpowerEq.(29a). FIG.8.Effectofelasticlossfactoratinfundamentalresonancefrequencyofpiezoelectricslabelectricallyexcitedat10Vpeakwithstressfreeatx=0andL. Page27of30 FIG.9.Temperaturedistributionofpiezoelectricslabundermechanicalexcitationundershortcircuitconditionwithθ(d,ω)=0,at8.231kHz(1stresonance).Dottedline:solutionthroughdirectintegration,stars:modifiedFourierseriessolution,Q(x,ω)estimatednumerically,andcircles:modifiedFourierseriessolution,usedanalyticalQ(x,ω)asinEq.8.AppliedharmonicstressΓ=20kN/m2
0 FIG.10.Temperaturedistributionofpiezoelectricslab,undermechanicalexcitationwithopencircuitedconditionwithθ(d,ω)=
0.Solidline:1stresonance8.611kHz,circles:at2ndresonance24.825kHz,andstars:1stresonancewithdθ(d,ω)/dx=
0. Page28of30 FIG.11.Temperaturedistributionofpiezoelectricslabonehalfsection,underelectricalexcitationatfirstresonance16.462kHzwithθ(d,ω)=0,solidline:non-uniformheatgeneration,starline:uniformheatgeneration,anddasheddottedline:non-uniformheatgenerationwithdθ(d,ω)/dx=
0.Atsecondreson
D.Thomasa),b),
D.D.Ebenezerb)andSivakumarM.Srinivasana)a)Dept.ofAppliedMechanics,IndianInstituteofTechnologyMadras,Chennai600036,Indiab)NavalPhysicalandOceanographicLaboratory,Thrikkakara,Kochi682021,India Abbreviatedtitle:Piezoelectricslabs Amethodispresentedtodeterminepowerdissipationinpiezoelectricslabswithinternallossesandtheresultingtemperaturedistribution.Thelengthoftheslabismuchgreaterthanthelateraldimensions.Lossesarerepresentedplexpiezoelectriccoefficients.Itisshownthatthespatiallynon-uniformpowerdissipationdensityintheslabcanbedeterminedbyconsideringeitherhysteresisloopsorPoynting’stheorem.Thetotalpowerdissipatedintheslabisobtainedbyintegratingthepowerdissipationdensityovertheslabandisshowntobeequaltothepowerinputtotheslabforspecialcasesofmechanicallyandelectricallyexcitedslabs.Theonedimensionalheatequationthatincludestheeffectofconductionandconvection,andtheboundaryconditionsarethenusedtodeterminethetemperaturedistribution.Whentheanalyticalexpressionforthepowerdissipationdensityissimple,directintegrationisused.ItisshownthatamodifiedFourierseriesapproachyieldsthesameresults.Forothercases,thetemperaturedistributionisdeterminedusingonlythelatterapproach.Numericalresultsarepresentedtoillustratetheeffectsofinternallosses,heatconductionandconvectioncoefficients,andboundaryconditionsonthetemperaturedistribution. PACSnumbers:43.38.Ar,43.38.Fx,43.20.Bi,43.40.At[AJZ]Page1of30
I.INTRODUCTION Piezoelectricmaterialsareusedintransducers,transformersandinvariousapplicationsbecauseoftheirquickresponseandhighcoupling.However,oneoftheirlimitationsarepredominatelyduetoselfheatingthatursbecauseofinternallosseswhileoperatingunderresonantconditions.1,2Hotspots,causedbynon-uniformheatgeneration,canquicklyleadtothermalrunawayinthesepiezoceramicmaterialsleadingtofailureofthedevices.Therefore,whiledesigningsuchdevices,itisnecessarytoanalyzefortemperatureriseduetoheatgenerationtoestimatethefactorofsafetyagainstsuchconditions.Variousauthorshaveattemptedtoestimatethisbyfirstfindingthepowerdissipationdensityoverthegeometryofthepiezoactuatorthatursunderthevariousmechanicalandelectricalexcitations.Thetemperatureriseordistributionundersteadystateandtransientconditionsareputedbycarryingoutaheattransferanalysisusingtheheatgeneratedduetothepowerdissipationasaheatsource. Ageneralexpressionforthepowerdissipationdensityduetodielectric,elasticandpiezoelectriclossesundermechanicalandelectricalexcitationwasfirstderivedbyHolland.3Thefrequencyandspatialdependencyofthepowerdissipationdensityresultsinanon-uniformtemperaturedistribution.Therefore,themaximumtemperatureisgenerallygreaterthantheaveragetemperatureestimated.However,inmostoftheliteraturerelatedtoheatgenerationanalysis,thepowerdissipationduetoadynamicelectricalexcitationisgenerallyassumeduniform.Thetotal/averagedensityofpowerdissipatedforpiezoelectricdeviceshavingsuchlossesisobtainedbyintegratingthepowerdissipationdensityusinganelectro-mechanicalimpedancemodel4anduseditforuniformheatgenerationcalculation.Inaddition,notalllossesareountedforputingthepowerdissipation.Forexample,heatinginlowfrequencysonarprojectors,5PMNsonarflex-tensionalprojectors6andinmultilayerpiezoelectricactuators7arestudiedbyassuminguniformpowerdissipationfromthepiezoelectricceramicstack.Inreferences6and7heatgenerationisassumedtobeduebydielectriclossesonly. Therearesomeworksreportedonthecalculationofnonuniformpowerdissipationconsideredintheanalysisofapplicationssuchaspiezoelectrictransformer.8Inthis,however,empiricalrelationshavebeenusedforheatgeneration.Finiteelementmodelshavealsobeenusedinsolvingfornon-uniformpowerdissipation.Forexample,highpoweractivetransducers Page2of30 aremodeledusingATILA,9piezoelectrictransformersaremodeledusingANSYS10andpiezoelectricponentsareanalyzedusinganin-housefiniteelementcode.11ItispossibletoplexmodulirelatedtoallthelossesintheATILAsoftwarewhiletheyareintroducedthroughRaleighdampingcoefficientsforanalysisinANSYS. Transientstatesolutionhasalsobeenstudiedfortemperaturerisewiththeassumptionofuniformandnonuniformpowerdissipation.12,13binationsofheattransfermodes(i.e.,conduction,convection,andradiation)andthermalboundaryconditionshavebeenusedintheabovetosolvethetemperaturedistributionproblem.However,simpleparametricstudiesanddesigniterationsmaybetime-consumingusingthesefiniteelementpackages.Therefore,thereisaneedforevolvingatasimpleanalyticaltoolforthecalculationofthetemperaturedistributioninapiezoelectricactuatorofasimplegeometry(suchasasimpleslab)tocarryoutthedesigniterationcalculationsandtheparametricstudiesthatwillprovideaquickanswertothepreliminaryquestionsaskedinadesignexercise. Thus,inthispaper,ananalyticalsolutionisproposedandformulatedforaonedimensionalpiezoelectricslabinvolvingallthemechanical,dielectricandpiezoelectriclosseswithrealisticconditions.Complexcoefficientsofthematerialpropertiesareconsideredtotakeintoountforthehystereticeffectsinthemechanicalandelectricalregimes.ClosedformsolutionsarederivedforpowerdissipationdensityandthetemperaturedistributioncalculationsarecarriedusingamodifiedFourierseriessolutionmethod. Theworkpresentedinthispaperanizedasfollows.Firstpowerdissipationofapiezoelectricslabissolvedforandthentemperatureestimationisobtainedbysolvingtheenergybalanceequation.Inthenextsection(SectionII),apiezoelectricslabwithinternallossesrepresentedplexcoefficientsisanalyzedfortimeaveragedpowerdissipationdensityandtotalpowerdissipated.Bothmechanicalandelectricalharmonicexcitationsareconsideredwithdifferentmechanicalboundaryconditions.Undermechanicalexcitation,piezoelectricbothshortcircuitedandopen-circuitedconditionsareanalyzed.Expressionsarederivedforthefrequencyandamplitudeatwhichmaximumpowerdissipationursforallcases.Twoapproachesareusedtoestimatethepowerdissipationdensityasaspatialdistributionandtotalpowerasafunctionoffrequency,bothyieldsthesameresult.InSectionIII,thetemperaturedistributionisestimatedbysolvingtheonedimensionalenergybalanceequationfortwothermalendconditionsusingamodifiedFourierseriesmethodingeneralandshownagreeingwithdirection Page3of30 integrationmethodforaspecificproblem.NumericalresultsarepresentedinSectionIVtoillustratespatialdistributionofpowerdissipationdensityandtemperatureinpiezoelectricslabsbeforeendingwithconcludingremarksinSectionV. II.POWERDISSIPATION Beforearrivingatthetemperaturedistributioninapiezoelectricactuatorofasimple geometrysuchasasimpleslab,itisnecessarytocalculatethepowerdissipationofa piezoelectricslab.Thus,apiezoelectricslabwithinternallossesrepresentedplex coefficientsisanalyzedinthefollowingfortimeaveragedpowerdissipationdensityandtotal powerdissipated.Powerdissipationdensityandthetotalpowerdissipatedintheslabare separatelyanalyzedanalyticallyformechanicalandelectricalexcitationsrespectively. ConsiderapiezoelectricslaboflengthLandbreadthBwhereL>>Basshownin Fig.1.ThefacesoftheslabwithareaLBareelectroded.Thedistancebetweentheelectrodesis handL>>h.Thedirections1,2,and3areshowninthefigure1,where3isthedirectionof polarization.NormalandshearstressesonthefaceswithareaLBarezero.Thedistancebetween thesefacesissmallanditis,therefore,assumedthatstressesthatarezeroonthesurfacearezero intheinterioralso.Similarly,stressesonthefaceswithareaLhandthecorrespondingstresses intheinteriorarezero. Theslabisharmonicallyexcitedmechanicallyorelectrically.Therefore,forexample,the normalstressinthe1orxdirectionisexpressedasτ(x,t)=Γ(x,ω)e+jωtwhereωisthe
1 1 angularfrequencyofexcitation,tdenotestime,τ1andΓ1arethetimeandfrequencydependent normalstresses,respectively,inthexor1direction.Γ1(x,ω)isassumedtoplexfor conveniencebutonlytherealpartofτ(x,t)hasphysicalsignificance.Lowercaseanduppercase lettersdenotetime-dependentandfrequency-dependentfunctions,respectively. Itisassumedthattheonlynon-zerostressisthenormalstressinthe1directionandthat theonlynon-zeroelectricalfieldisalongthe3direction.Therefore,thepiezoelectricequations ofstateare S1 ( x,ω ) = sEΓ111 ( x,ω ) + d31E3 (ω ) (1a) and D3 ( x,ω ) = d 31Γ
1 ( x,ω ) + ε T33 E3 (ω ) (1b) Page4of30 whereS1(x,ω)isponentofnormalstraininthexor1direction,andD3(x,ω)isthe componentofelectricdisplacementalongthezor3direction.E3(ω)istheelectricfieldinthe3 directionandisequalto−V/hwhereVisthepotentialdifferencebetweenandbottomelectrodes.ThestrainsS2(x,ω)andS3(x,ω),andthechargedensitiesD1(x,ω)andD2(x,ω), arenotzero. The piezoelectric material coefficients sE,11 ε T33 , and d31are real when there are no losses. Internallossesarerepresentedplexpiezoelectriccoefficients;andthecoefficientsand allfieldvariablesareexpressedasthesumofrealandimaginaryparts.Forexample, S=S'+jS"andsE=sE'+jsE"where'and"denoterealandimaginarypartsrespectively.
1 1
1 11 11 11 Theheatgeneratedinaslabthatisexcitedeithermechanicallyorelectricallyisof interest.Inthissectionexpressionsforthetime-averagedpowerdissipationdensityandthetotal heatarepresented.Thetotalheatisdeterminedbyintegratingthepowerdissipationdensityover thevolumeoftheslabandisshowntobeequaltotheinputpower.Anexpressionforthetime-averagedpowerdissipationdensity,Q(x,ω),intheslabis obtainedbyconsideringhysteresisloops.Integratingtodeterminetheareasundertheloopsand averagingovertimeyields Q(x,ω)=QE(x,ω)+QM(x,ω) []∫∫=1−d(x,t)de(t)+τ(x,t)ds(x,t) (2a)
T 3
3 1
1 whered3(x,t),e3(t),τ1(x,t),ands1(x,t)arethetime-dependentelectricdisplacement,electric field,longitudinalstress,andlongitudinalstrain,respectively.Theintegrationisdoneoveroneperiod,T=2π/ω,andeachintegralistheareaofahysteresisloop.Theterms,QEandQM, representthepowerdissipationintermsofelectricaldisplacementsandmechanicalstresses respectively.Itshouldbenotedthatthepiezoelectriccouplingisinbuiltinboththeexpressions. Then,usingaphasordiagramyields [∫]∫Q(x,ω)=−1d(x,t)de(t)=−1Td(x,t)de3(t)dt=(−ω/2)D(x,ω)E(ω)cos(ψ),(2b) ET33T03dt
3 3 whereψistheanglebetweenthejωE3(ω)andD3(x,ω)phasors. Page5of30 Forthespecialcaseofelectricalexcitation,itconvenienttoassumethatE3(ω)isrealanditthen followsthatψ=π/2−tan−1[Im(D3(x,ω))/Re(D3(x,ω))].Similarly, ∫∫
Q (x,ω)=
1 τ (x,t)ds (x,t)=
1 T τ (x,t)ds1(x,t)dt =(ω/2)Γ (x,ω)
S (x,ω)cos(ϕ) (2c) MT11T01dt
1 1 whereϕistheanglebetweenthestressandstrainratephasors. Inanalternativeapproach,Holland3usedthegeneralizedPoyntingvectortoobtainan expressionforthepowerdissipationdensityinapiezoelectricmedium.ForthespecialcasewhereΓ1(x,ω)andE3(ω)aretheonlyponentsofstressandelectricfield, respectively,theexpressionreducesto ()Q(x,ω) = (ω / 2)Im E3ε3T3*E3* + ΓsE*Γ*1111 + E3d3*1Γ1* + Γ1d3*1E3*
(3) where*plexconjugateandtheargumentsofthefunctionsaresuppressedfor convenience.Usingthenotationforrealandimaginarypartsintheaboveequationyields [()]Q(x,ω)=Q(x,ω)+Q(x,ω)+Q(x,ω)=−(ω/2)εT"E2+sE"Γ2+2d"E'Γ'+E"Γ".
(4)
1 2
3 333 111 3131 31 Itisshownlater,usingnumericalresultsforseveralcases,thatEqs.(2)and(4)areequivalent. Thetime-averagedtotalpowerdissipatedintheslabisexpressedas
L Π(ω)=Bh∫Q(x,ω)dx.
(5)
0 Q(x,ω)andΠ(ω)fortheslabaredeterminedforanysetofboundaryconditionsbysubstituting theappropriateexpressionsforΓ1(x,ω)etcinEqs.(2)and
(4).Forspecialcases,Π(ω)isalso expressedasthepowerinputtotheslabbytheexcitation.
A.Mechanicalexcitation Considerthecaseofslabfixedatx=0andmechanicallyexcitedatx=Lbyaharmonic normalstressτ(L,t)=ΓejωtwhereΓisrealandindependentoffrequency.Theresponseof
1 0
0 theslabisthesameasthatofonehalfofaslaboflength2Lthatissymmetricallyexcitedat bothends.Theanalysisisdoneforshort-circuitandopen-circuitconditions. Themechanicalpowerinputtotheslabisequaltothetotalpowerdissipatedintheslab. Itisexpressedas ∫∫Π(ω)=Bhτ(L,t)du(L,t)=BhTτ(L,t)v(L,t)dt=0.5BhΓV(
L,ω)cos(ϑ) (6a) T1T010 Page6of30 whereu(L,t)andv(L,t)arethedisplacementandvelocityrespectivelyattheexcitedendand ϑ=tan−1[Im(V(
L,ω))/Re(V(
L,ω))]. (6b) istheanglebetweenthestressandvelocityphasors.Powerdissipationdensityandtotalpower dissipatedintheslabarederivedfortwoconditionsundermechanicalexcitationsinthispaper.
1.Short-circuitconditions Undershort-circuitconditions,thepotentialiszeroonboththeelectrodesandisassumed tobezeroeverywherewithintheslabbecausethedistancebetweentheelectrodesissmallwith respecttothelengthoftheslabandthevibrationalongthelengthisdominantandofinterest. Therefore,E3(ω)iszeroandtheEqs.ofstate(1)andotherequationsaresimplified.Solvingthe differentialequationsgoverningvibrationandusingtheboundaryconditionsatx=0andL yields U(x,ω)=sEΓsin(kx), (7a) 110kcos(kL) Γ(x,ω)=Γcos(kx), (7b)
1 0cos(kL) and S(x,ω)=sEΓcos(kx) (7c)
1 110cos(kL) where k =ω/cbE and cbE =(ρsE)−0.5.11 SubstitutingEqs.(7b)and(7c)ineitherEqs.(2)or(4)yieldsthetime-averagedpower dissipationdensity Q(x,ω)=−0.5ωsE"Γ2cos(2k'x)+cosh(2k"x).
(8) 110cos(2k'L)+cosh(2k"L) Thetime-averagedtotalpowerdissipatedintheslabisdeterminedbyusingEqs.(5)and
(8),andperformingtheintegrationanalyticallyasdoneinRef.14.Itisexpressedas Π(ω)=−Bh4ωks'1kE1""Γ02[k["csoins((22kk''LL))++cko'ssihn(h2(k2"kL")L])](9a) and,atlowfrequencies,reducesto Π(ω)≈−0.5LBhωs1E1"Γ02k'L,k"L<<
1 (9b) Page7of30 whereitisproportionaltothevolumeoftheslab.Alternatively,usingEq.(7a)todeterminethe velocity,substitutingtheresultinEq.(6)fortheinputpower,andfollowingthesameprocedureusedtodeterminethetotalheatgeneratedinanelasticrodwithinternallosses14yields Π(ω)=0.5BhΓ02(s1E1/ρ)0.5tan(kL)cos(ϑ). (10a) When2k'Lisapproximatelynπ,n=1,3,5,...,Π(ω)isclosetoalocalmaximumandis determinedusing ()()ϑ ≈ tan−1⎜⎛ − 2sE'11 ⎟⎞ + tan−1⎜⎛ ωρ0.5s1E1" ⎟⎞. ⎜⎝ sE"11 ⎟⎠ ⎜⎝ sE'11 0.5π−2ω(ρsE')0.511 ⎟⎠ (10b) Theangularfrequency,ωΠ,atwhichthetotalpowerdissipatedismaximumis determinedbydifferentiatingEq.(9a)withrespecttoωandequatingittozero.Then, anticipatingthat,atresonance,2k'Lisalittlelessthannπ,n=1,3,5,...,andusing k=ω(α'+jα")=ω(ρs1E1)0.5and2k"L≈0tosolvetheequationyields ω≈nπα'≈nπ(s1E1'/ρ)0.5. (11) Π2Lα'2+α"22LsE 11 ThepowerdissipatedatresonanceisthendeterminedbyusingEqs.(9a)and(11)andexpressed as ()Πmax ≈−BhΓ02 sE"11 (α '
2 +α"
2 ) 2nπα'α"
2 2BhΓ2sE'1.5 ≈− 011 nπ ρ 0.5 sE"11 (12) wherethenegativesignispresentbecausesE"isnegative.11
2.Open–circuitconditions Underopen-circuitconditions,thetotalcurrent,i(t),iszero,i.e., ∫i(t) = Bh
L ⎡⎢ dd3 (x,t) ⎥⎤dx =
0 . (13) 0⎣dt⎦ SolvingthegoverningdifferentialequationfordisplacementandusingtheconstitutiveEqs.
(1),theopen-circuitconditioninEq.(13),thefixedboundaryconditionsatx=0,andthespecifiedstressatx=Lyields Page8of30 U(x,ω)=Ls1E1Γsin(kx),
1 χ0sin(kL) (14a)
S (x,ω) = sE11 Γ ξ ,
1 χ
0 Γ1(x,ω)=Γ0(κ+ξ)/χ, D(x,ω)=Γd[ξ−1]/χ,
3 031 (14b)(14c)(14d) andE3(ω)=−Γ0s1E1κ/(d31χ). wherek321=d321/(s1E1ε3T3)istheelectromechanicalcouplingcoefficient,χ=κ+kLcot(kL),andξ=kLcos(kx)/sin(kL). (14d) κ = k321 /(1− k 231 ) , SubstitutingEqs.(14)inEq.(2)yieldsanexpressionforthepowerdissipationdensityin theform []Q(x,ω)=0.5ωχ−2ΓsEκ(ξ−1)cos(ψ)+sE(κ+ξ)ξcos(ϕ). (15) 011 11 SubstitutingEq.(12)inEq.(4)yieldsanequivalentexpression [()()]}Q(x,ω) = 0.5ωΓ(
L,ω) ⎨⎧⎩ χ −
2 ⎢⎣⎡ε3T3" s1E1κ / d31
2 + sE"11 κ +ξ
2 ⎤⎥⎦ + . (16) [][]2d"sEκ/(dχ)'(κ+ξ)/χ'+sEκ/(dχ)"(κ+ξ)/χ" 3111 31 11 31 ItisshownlaterusingnumericalresultsthatEqs.(15)and(16)obtainedusingthehysteresis loopandPoyntingtheorem,respectivelyareequivalent. Thetotalpowerdissipatedintheslabisdeterminedbynumericallyintegratingthepower dissipationdensityoverthevolumeoftheslab.Alternatively,becauseitisequaltothe mechanicalpowerinputtotheslab,itisexpressedusingEq.(6)as Π(ω) = LBhωΓ02 sE11 / χ cos(ϑ)/
2 . (17a) Atlowfrequencies,theexpressionreducesto [()]Π(ω)=0.5ωLBhΓ02(k3'21−1)s1"1E+k3"12s1'E1’ (17b) andthepowerisproportionaltothevolumeoftheslab.Theangularfrequencyatwhichtheinputpowerismaximum,ωΠ,isofinterest.Itisseen fromEq.(6)thatωΠisalsothefrequencyatwhichtherealpartof Page9of30 V(
L,ω)= jωL(s1E1'+js1E1"). (κ'+jκ")+(k'L+jk"L)cot(k'L+jk"L) (18a) ismaximumanditisdeterminedasfollows.Usingcot(kL)=sin(2k'L)−jsinh(2k"L),cosh(2k"L)−cos(2k'L) (18b) toseparateEq.(18a)intoitsrealandimaginarypartsyieldsanexpressionforV'(
L,ω).The numeratorofthisexpressionvariesnearlylinearlywithfrequencywhen2k'Lisintheneighborhoodofnπ.Thedenominatoristhesquareoftheabsolutevalueofplexfunctionandisneverzero.Thesignoftherealpartofthefunctionchangesinthisneighborhoodwhereastheimaginarypartisnearlyconstantanddoesnotchangesign.Therefore,V'(
L,ω)ismaximum approximatelyatthefrequencyatwhichtherealpartofplexfunctioniszero,i.e.,when Re[(κ'+jκ")[cosh(2k"L)−cos(2k'L)]+(k'L+jk"L)[sin(2k'L)−jsinh(2k"L)]]=
0. (19) SimplifyingEq.(19)byusing2k"L≈
0,α'>>α",andanticipatingthat,atresonance,2k'Lis approximatelynπ,n=1,3,5,...,yieldsanquadraticEq.inωthatissolvedtoobtain {[]}ωΠ≈nπ(1+2κ')+(1+2κ')2+4[4(nπ)−2−1]κ'(1+κ')0.5[4α'L(1+κ')]−
1. (20) ThepowerdissipatedatresonanceisthenobtainedbyusingEqs.(6)and(18).Usingallthe approximationusedtodetermineωΠyields Π≈LBhΓ02 ωπ sE'11 . (21) max 2κ"−0.5ωπ2ρL2s1E1" ItisseenfromEqs.(20)and(21)thatΠmaxisindependentofthelengthoftheslab.
3.Electricalexcitation Considernextthecasewheretheslabiselectricallyexcitedbyapplyingaharmonic voltage,v(t)=V0ejωt,acrosstheelectrodes.Inthiscase,itisconvenienttoconsideraslabof lengthLwithendsatx=±L/2.Thestressesattheendsarezero.Thefieldsofinterestare expressedas E3=−V0/h, (22a) S1(x,ω)=E3d31γ, (22b) Page10of30 Γ1(x,ω) = E3d31(γ −1)/sE11 ≈ 0.5E3d31k 2(0.25L2 − x2) / sE11 ,kL<<
1 (22c) and []D3(x,ω)=E3ε3T31+k321(γ−1) (22d) whereγ=cos(kx)/cos(kL/2).Theinputelectricaladmittance,Y(ω),oftheslabisexpressedas ∫()Y(ω) = G(ω)+ jΒ(ω) = −jωBh L/2
D (ω)dx =jω LBε
T ⎡⎢
1 − k
2 +k2 tan(kL/2)⎤⎥ (23) 3−L/2 h 33 ⎣ 3131(kL/2)⎦ whereGandΒaretheelectricalconductanceandsusceptance,respectively. ThepowerdissipationdensityisobtainedbysubstitutingEq.(22)inEq.(2)and expressedas [()]QE ( x,ω ) = 0.5ωE32 ε T33 k321γ(γ−1)cos(ϕ)− 1+k321(γ−1) cos(ψ). (24a) QM (x,ω) = 0.5ωE32 εT33 ⎣k321γ (γ −1) cos(ϕ)⎦ (24b) Alternatively,itisobtainedbyusingEqs.(20)and(4)andexpressedas Q(x,ω) = −0.5ω
E 23 ⎢⎣⎡ε T"33 + sE"11 (γ −1)d31 / sE11
2 + (2d3"1(γ −1)d31 / sE11 )' ⎤⎥⎦ . (24c) SubstitutingtheexpressionsforelectricfieldandstressatlowfrequenciesinEq.(22) intoEq.(4)yields Q(x,ω)=−0.5ωεT"E2,
1 333 (25a) Q(x,ω)≈−0.5ωE2sE"d/sE2k4(0.25L2−x2)
2,
2 3113111 (25b) and Q(x,ω)≈−0.5ω3d'd"E2ρ(0.25L2−x2).
3 31313 (25c) where Q2(x,ω)and Q3(x,ω)are maximum at x =
0. Then, it is seen by using tan(δε ) = εT"33 / εT'33 , tan(δs ) = sE"11 / sE'11 , and tan(δd ) = d3"
1 / d3'
1 that Q2(
0,ω)≈k2sin(δs)(kL)4<
11 . At low frequencies, it isseenbyusingEq.(27b)thatEq.(27a)reducestoEq.(27). Theconductanceand,therefore,thetotalpowerdissipatedaremaximuminthe neighborhoodofk'L=nπ.Usingapproximationsinthisneighborhoodandk"L≈0yields [()]G(ω)=−B⎜⎜⎛ωLεT1−k2 "+4[ζ'k"L+ζ"(nπ−k'L)]⎟⎞. ⎟ (30) h⎝ 33 31 (k"L)2+(nπ−k'L)2⎠ InEq.(30),thefirsttermandthenumeratorofthesecondtermarefirstdegreefunctionsof frequency.However,thedenominatorofthesecondtermisaseconddegreefunction.TheangularfrequencyatwhichGismaximum,ωs,isalsotheangularfrequencyatwhichthe maximumpowerisdissipated.Thisfrequencyisapproximatelythefrequencyatwhichthe denominatoriszeroandisexpressedasωπ=ωs≈L(αn'2π+αα'"2).(31) Finally,substitutingEq.(31)inEq.(29)todeterminethemaximumpowerdissipatedandusing sE"/sE'1111 <<
1 yields ()Π ≈−E2 Bh ⎜⎛nπ[εT(1−k2)]"+
8 d2'31 ⎟⎞ . max3ρs1E1'0.5⎜⎝3331nπs1E1"⎟⎠ (32) Page12of30 wherethefirsttermisoftenmuchsmallerthanthesecondbecausesE"ursinthedenominator11 ofthesecondterm.ItisofinteresttonotethatΠmaxdependsontherealandimaginarypartsofthecoefficientsandisindependentofthelengthoftheslab. III.TEMPERATUREDISTRIBUTION Havingarrivedatexpressionsforpowerdissipationfortheelectricalandmechanicalexcitations,wearenowreadytosolveforthetemperaturedistributionbasedonthethermodynamicbalancelaw.Differentthermalendconditionshavebeenconsideredhere. Considerthecasewhereheatconductionintheslabandconvectiontothesurroundingsaresignificantandaffectthetemperaturedistributionintheslabthatisofinterest.Asstatedearlier,itisassumedthatthecross-sectionissmallwithrespecttothelengthandthatthepowerdissipationdensityintheslabisconstantoverthecross-section.Itis,therefore,alsoassumedthatthetemperatureisconstantoverthecross-sectionandtheone-dimensionalheatequationisused. Thegoverningenergybalanceequation15isexpressedasΚd2θ(x,ω)−Μθ(x,ω)=−Q(x,ω),−d≤x≤d(33)dx2 whereθ(x,ω)isthedifferencebetweenthetemperatureintheslabandtheuniformambienttemperature,Κistheconductioncoefficient,andΜ=2hˆsl(B+h)/Bhandhˆslistheheatconvectioncoefficient,anddiscase-dependent.Theslabthatismechanicallyexcitedisfixedatx=0,andexcitedatx=
L.Itisalsothermallyinsulatedatx=
0.Therefore,thetemperatureisthesameasthatintheregion0≤x≤Lofaslaboflength2Lthatissymmetricallyexcitedatx=±
L.Inthiscase,d=
L.TheslabthatiselectricallyexcitedisoflengthL,haszerostressatbothends,andissymmetricallyexcitedoveritslength.Inthiscase,d=L/2.Inbothcases,dθ(
0,ω)/dx=0andthetemperaturedistributionissymmetricaboutx=
0 Twothermalboundaryconditionsareconsidered.Inthefirstcondition,thetemperature intheslabatx=disequaltotheambienttemperature,i.e. θ(d,ω)=
0. (34a) Inthesecondcondition,noheatislosttothesurroundingsatx=disassumed,i.e. dθ(d,ω)/dx=
0. (34b) Page13of30 TheenergybalanceequationandthethermalboundaryconditionsaresolvedinallcasesbyusingamodifiedFourierseriesmethod.Forsomecases,theyaresolvedbydirectintegrationalsoanditisshownthatthetwomethodsyieldthesameresultsthusvalidatingtheFourierseriesapproachthatismoregeneral.ThederivationofthetemperaturedistributionsolutionisdoneforageneralcaseofQ(x,ω)whichistheheatgeneratedatanylocation,x,forafrequencyof loading,ωatasteadystatecondition.Tohighlighttheimportanceofdeterminingthetemperaturedistributionforageneralcasesuchasthis,thesolutionforacaseinwhichheatgeneratedisuniformlydistributedovertheslab,i.e.,Q(x,ω)=Q0(ω)=Π(ω)/(LBh)isalso solvedforparedwiththetemperaturedistributionwhentheactualnon-uniformheat distribution. InthemodifiedFourierseriesmethod,thetemperaturedistributionintheslabthatis mechanicallyorelectricallyexcitedisexpressedintheform ∑∞ θ(x,ω)=Ax2+Bncos(knx) (35) n=
0 wherekn=nπ/d.Asnotedearlier,thetemperaturedistributionaboutx=0issymmetricand, therefore,thenon-symmetricsintermsarenotpresentinEq.(35).Itisnotedthatthefunctions,cos(knx),formaorthogonalpletesetintheregion x=[−d,d].Specifically,eachfunctioninthesetisorthogonaltoalltheothersbecause ∫∫⎧0m≠n dcos(kmx)cos(knx)dx=2dcos(kmx)cos(knx)dx=⎪⎨dm=n=1,2,...∞.(36) −d
0 ⎪⎩2dm=n=
0 Further,thesetpletebecausetheredoesnotexistanyfunctionf(x)thatissymmetric aboutx=0andorthogonaltoallthetermsoftheFourierseries,i.e.,nofunctionsatisfiesthe condition d ∫f(x)cos(knx)dx=0,n=0,1,
2,…∞. (37) −d However,differentiatingEq.(35)termbytermyields ∑d2θ(x,ω)=A−∞Bk2cos(kx) (38) dx2 nn n n=
1 Page14of30 anditisseenthatAinEq.(38)correspondstoB0inEq.(35)andthattheexpressionforthe secondderivativeofθ(x,ω)pleteonlywhenthetermAispresent.Atermthatis proportionaltoxisnotrequiredinEq.(35)becausethetemperaturedistributionissymmetricwithrespecttox=
0.TheseriesinEq.(35)iscalledamodifiedFourierseriesbecauseofthe additionalAx2term.Foranyparticularcase,substitutingtheexpressionforQ(x,ω)intoEq.(33),multiplying bothsidesbycos(kmx),m=0,1,2,...∞,integratingoverxfrom0tod,andusingthe orthogonalpropertyofthecosfunctionsinEq.(36)yields ()A2Κ−Μd2/3−B0Μ=−Q0(ω) (39a) ∫A(−1)m+14Μ/km2−Bm(km2Κ+Μ)=−Qm(ω)=−2d−10dQ(x,ω)cos(kmx)dxm=1,
2,…∞(39b) where ∫Q0(ω)=d−10dQ(x,ω)dx=Π(ω)/(LBh) (39c) istheaveragepowerdissipationdensityandQm(ω)=0,m=0,1,2,...∞forthecaseofuniform heatgeneration. SubstitutingtheformofthesolutioninEq.(35)intheboundaryconditionsinEqs.(34a) and(34b)yields ∑∞ AL2+B0+Bn(−1)n=0 n=
1 (40a) andA=
0, (40b) respectively.CombiningEqs.(39)and(40a)or(40b)andretainingtermsonlytermsuptoBN yieldsequationsthataresolvedbyusingmatrixmethodsorbyelimination.Forthespecialcaseofuniformheating,Q(x,ω)=Q0(ω).SolvingEq.(33)bydirect integrationandretainingonlythetermsthataresymmetricaboutx=0yields []θ(x,ω)=Fcosh(M/K)0.5x+(Q0/M) (41a) wherethethermalboundaryconditionsforthetwocasesinEqs.(34a)and(34b)aresatisfiedby
F =
M [−Q0 cosh(M/ K)0.5 d] (41b) Page15of30 and F=
0, (41c) respectively.InthemodifiedFourierseriesapproach,thetemperaturedistributionisdetermined byusingQm(ω)=0,m=0,1,2,...∞inEqs.(39)andtheappropriateboundaryconditioninEq. (40). FromthegeneralsolutionobtainedinEqs.(39)and(40),thetemperaturedistributionfor thespecificconditionsofmechanicalandelectricalexcitationcannowbedetermined.
A.MechanicalExcitation Consideragainthecaseofslabfixedatx=0andmechanicallyexcitedatx=Lbya harmonicnormalstressτ(L,t)=Γejωt.
1 0
1.Short-circuitconditions Thenon-uniformpowerdissipationdensityforthiscaseispresentedinEq.
(8).The temperaturedistribution,inthemodifiedFourierseriesapproach,isdeterminedbyusing Q0(ω)=0.5β(k'k")−1[k'sinh(2k"L)+k"sin(2k'L)], (42a) and where
Q (ω) = (−1)mβ ⎡k'sin(2k'L)⎢ + k"sin(2k"L)⎤⎥ m=0,1,2,...∞, m 2k'k" ⎣ (
4 k '
2 − k 2m ) (
4 k "
2 + k 2m ) ⎦ β= − ω Γ02 sE"11 cos(2k'L)+cosh(2k"L) (42b)(42c) inEq.(39)andtheappropriateboundaryconditioninEq.(40).Thetemperaturedistributionintheslabfornon-uniformheatgenerationisalso determinedbydirectlyintegratingEq.(33)aftersubstitutinginittheexpressionforQ(x,ω)in Eq.
(8).IntegratingandusingtheboundaryconditioninEq.(34)yields []θ(x,ω)=Fcosh(Μ/Κ)0.5x+β⎜⎛cos(2k'x)−cosh(2k"x)⎟⎞2⎝4k'2Κ+Μ4k"2Κ−Μ⎠ where
F = (Μ / Κ)0.5 [β sinh(Μ / Κ)0.5 L]⎜⎝⎛ k'sin(2k'L)4k'2Κ+Μ + k"sinh(2k"L)4k"2Κ−Μ ⎟⎞⎠ (43a)(43b) Page16of30 forthecaseofzerotemperaturedifferenceand
F = [−β 2cosh(Μ/ Κ)0.5 L]⎜⎝⎛ cos(2k'L)4k'2Κ+Μ − cosh(2k"L)4k"2Κ−Μ ⎟⎞⎠ , (43c) forthecaseofzeroheattransfer.Itisshowninthesectiononnumericalresultsthatthe temperaturedistributionsobtainedbyusingthemodifiedFourierseriesanddirectintegration methodsarefoundtobethesamefortheshort-circuitmechanicalexcitationcase.However,it maynotpossibletodirectlyintegrateforothercasessuchasopen-circuitandelectricalexcitation conditions.Forsuchcases,modifiedFourierseriesmethodisadoptedfordeterminingthe temperaturedistribution.
2.Open-circuitconditions Forthiscase,equivalentexpressionsforthenon-uniformpowerdissipationdensity(equivalentlytheheatgenerated),Q(x,ω),arepresentedinEqs.(15)and(16).Inthemodified Fourierseriesapproach,solvingEqs.(39)withtheappropriateboundaryconditionsinEq.
(4) usingtheaveragepowerdissipationdensity, Q0(ω) = ωΓ02 sE11 / χ cos(υ)/
2 (44) isobtainedfromEq.(17a)andQm(ω)obtainedbyusingnumericalmethods.Equation(15)or (16),yieldsthecoefficientsinEq.(33)thatarerequiredputethetemperaturedistribution.IfthepowerdissipationdensityisassumedtobeuniformthenQm(ω)=0form=0,1,2,...∞and thetemperaturedistributionismoreeasilydeterminedusingmodifiedFourierseries.
3.ElectricalExcitation Consider,next,thetemperaturedistributionwhentheslabiselectricallyexcitedbyanappliedvoltage.Thenon-uniformpowerdissipationdensityispresentedinEq.(24)anditisseenfromEqs.(39c)and(29)that [()]Q0 (ω) = −ωE32 ⎜⎜⎛ εT33 1− k321 "+δ'sinh(k"L)+δ"sin(k'L)⎟⎟⎞. (45) ⎝ cos(k'L)+cosh(k"L)⎠ InthemodifiedFourierseriesapproach,thetemperaturedistributionisdeterminedbyusingEqs. (39),(40),(45),(24),andnumericalmethodstoevaluate∫LQ(x,ω)cos(kmx)dx.Asintheother0 Page17of30 cases,thetemperaturedistributionwhenthepowerdissipationdensityisassumedtobeuniformisdeterminedbysettingQm(ω)=0,form=0,1,2,...∞.IV.NUMERICALRESULTSANDDISCUSSION Somenumericalresultsareshowntoillustratethenon-uniformheatgenerationandthenonuniformtemperaturedistributionofonedimensionalpiezoelectricslab.ApiezoelectricslabischosenwithlengthL=100mm,widthB=10mm,andthicknessh=5mm,andwithmaterialproperties,16ρ=7500Kg/m3,ε3T3=1300×8.85e−12(1−j0.004),d31=−123e−12(1−j0.005),ands1E1=12.3e−12(1−j0.005).Aharmonicstressof20kPaasmechanicalexcitationandanelectricalexcitationof10Vpeakisusedunlessstatedotherwise.Thesearenumbersthataretypicalforwhichtemperaturebuildupislikelyinlinearregion. Firstconsiderthepiezoelectricslabundermechanicalexcitation(short-circuitedcase)fixedatx=0andexcitedatx=
L.Thenon-uniformpowerdissipationdensityQ(x,ω),forfourdifferentfrequencies:atfirstandsecondresonancefrequenciesoftheslabinlongitudinalmodeandslightlybelowthesefrequencies,aredetermined.ThevariationofthepowerdissipationdensityisshowninFig.2forthesefrequencies.Significantvariationisobservedalongthelengthoftheslab.Thepeakvaluecorrespondstothefixedendwherethestressismaximum.Itisimportanttonoticethatevena0.5%deviationfromtheresonantfrequencycausesasmuchas75%reductionofmaximumheatgeneration.Thenumberofpeakscoincideswiththeresonancefrequencymode.ThetotalaveragepowerinputtotherodisconvertedintoheatputedusingEqs.(9)and(10)isshowninFig.3.Boththeequationsyieldthesameresultinallfrequencies.TheapproximatepeakvaluesΠ(ω)andthecorrespondingfrequencyfΠobtainedfromEq.(9)areincloseagreementwiththecalculatedvaluesusinganalyticalrelationsasinEqs.(11)and(12)asshowninTableI. Nextconsiderthepiezoelectricslabundermechanicalexcitationwithopen-circuitedconditionfixedatx=0andexcitedatx=
L.Thepowerdissipationdensitiesatandnear1stand2ndresonancefrequencyareplottedinFig.4.Bothapproaches-Eqs.(15)throughthehysteresisapproachand(16)throughthePoyntingtheoremgivethesameresultandoneparticularcase(1stresonance)isplottedtogetherforreference. Page18of30 TABLEI.Comparisonoffrequenciesandamplitudesatmaximumpowerdissipation Excitationtype Mechanical Electrical Assumptionsandboundaryconditions Short-circuited,oneendfixedandforcedatotherOpen-circuited,oneendfixedandforcedatotherBothendsstressfree AnalyticalfΠ(kHz),Πmax(watts) 8.231,2.57824.693,0.8598.611,2.51424.825,0.85716.462,3.30149.386,1.102 NumericalfΠ(kHz),Πmax(watts) 8.231,2.57824.693,0.8598.612,2.39324.825,0.85216.462,3.30142.767,1.101 TotalinputpowerdissipatediscalculatedbyEq.(17)andbyintegratingthepowerdissipation densityinEq.(15).BoththeequationsyieldthesameresultasshowninFig.5.Thefrequency andtheamplitudecorrespondingtothemaximumpowerdissipatedfromEq.(17)andthe analyticalexpressionaregiveninTableIwhichareincloseagreement.Fromtheresultsof mechanicalexcitationwecanseethattheopen-circuitconditionhasahigherresonance frequencyvaluethantheshortcircuitedslabdueaddedstiffnesseffect.However,maximum powerdissipatedwhichiscontrolledbytheimaginarypartofpliances1E1,isnearlythe sameforboththecases. Nextconsiderpiezoelectricslabwithelectricalexcitationwithendsatx=±L/2usedfor symmetry.ThepowerdissipationdensityfromEqs(24)isplottedinFig.6.BothhysteresisandPoyntingtheoremapproachyieldthesameresultandbothvaluesareplottedforthe2ndresonant frequency.TheinputpowerdissipatediscalculatedfromEq.(29)andnumericallyintegrating Eq.(24)isplottedinFig.7.Boththeapproachesyieldthesameresultsforallexcitation frequencies.Thefrequencyandtheamplitudecorrespondingtothemaximumpowerdissipated fromEqs.(31)and(32)paredwithnumericallycalculatedvaluesandpresentedinTable I,showgoodagreement. Theincreasein sE′′11 causesthefrequency fΠmaxtoshifttoahigher valueunlikethecaseinwhichthereisashifttoalowervaluewheredisplacementismaximum foradampedstructure.Thepeakamplitudeoftotalpowerdissipatedapproximatelydrops4 timesforlossof0.005to0.02asshowninFig.8.Itcanbeseenthat,therateofreductioninthe amplitudedecreaseswithincreaseinloss.Asimilareffectonthepeaktemperaturewithelastic losscanbeseeninFig.14. Page19of30 Temperaturedistributionsofthepiezoelectricslabfordifferentexcitationsandthermalboundaryconditionsarecalculated.Inalltemperaturecalculations,Κ=2.0W/moCandhˆsl=10W/m2oCareusedunlessstatedotherwise.5Figure9showsthetemperaturedistributionoverthepiezoelectricslabformechanicalexcitationwithshort-circuitedcondition,calculatedusingdifferentmethods:themodifiedFourierseriesandthedirectintegrationmethodsgivenbyEqs.(42)and(43)respectively.Boththeapproachesyieldthesameresult.Thetemperaturedistributionalsoshowsahighertemperatureatthefixedendandtapersdowntoambienttemperatureatx=Lsatisfyingθ(d,ω)=
0. OnlymodifiedFourierseriesisusedforcalculatingtemperaturedistributionofslabundermechanicalexcitationwithopen-circuitedconditionandtheresultsatandnearthefirsttworesonancefrequenciesareshowninFig.10.Itisnotedthatatsecondresonantfrequency,unliketheearlierresultforthefirstresonantfrequency,thetemperaturedistributiondoesnotfollowthesametrend,asthepowerdissipationdensitydistribution(Fig.4).Theotherboundaryconditiondθ(d,ω)/dx=0effectisplottedat1stresonancefrequencyandthetemperaturevariationisalmostthesameasθ(d,ω)=0exceptnearertotheexcitationend,x=
L. Theeffectofuniformandnon-uniformheatgenerationassumptionsinpredictingthepiezoelectricslabtemperaturedistributionunderelectricalexcitationisshowninFig.11forthetworesonancefrequencies.Anincreaseof6paredtouniformheatingcaseisnoticedatthefirstresonancefrequencybutthedifferenceisnotsignificantforthesecondresonancemode.Theeffectofthermalboundaryconditionsatx=Lparedhereandresultsareinagreementwiththeassumptions.Figure12showstheresultoftemperaturedistributionasafunctionofexcitationfrequencyfortwodifferentthermalconditionswhichbringsoutthattheendconditionsdoesnotaffectthemaximumtemperatureataparticularfrequency.Differentauthorshaveuseddifferentvaluesbinationsoftheheattransferconstantsintemperatureestimation.TheimplicationoftheselectionofheatconductivityandheattransfercoefficientvaluesareplottedinFig.13.Itcanbenoticedthattheresultsvarysignificantlywiththechoiceoftheheattransferconstants.Hence,properchoiceofheattransfercoefficientsbasedonstrongexperimentalresultsisabsolutelynecessary. Fromthenumericalresults,itisshownthattheincreaseintemperatureriseinapiezoelectricmaterialoractuatordependsontheexcitationvoltageandtheelasticlossfactor.Reductionin Page20of30 material elastic imaginary property sE"11 causes further increase in the heat generation and leads tohighertemperaturebuild-upandthermalrunaway.Moreover,estimationoftemperature distributioninthefirstfundamentalmodeusinguniformheatgenerationunderestimatesthepeak value.Therefore,non-uniformheatgenerationtogetherwithtemperaturedistributionshouldbe assumedlestonewouldmissthelocalhot-spotsorfailure.Thus,adetailedanalysis/parametric studyispossiblewiththesemethodstoselectpiezoelectricconfigurations,materiallosses,and resonancemodesoroperatingfrequenciesdependingupontheexcitationvoltageatdesignstage itself.
V.CONCLUSIONS Non-uniformpowerdissipationandtemperaturedistributioninonedimensionalpiezoelectric slabwithinternallossesundermechanicalandelectricalexcitationareinvestigated.Lossesare representedplexelectro-elasticcoefficients.Itisassumedthatplexmaterial propertiesareindependentoffrequencyandtemperature.Analyticalexpressionsarederivedfor powerdissipationdensityundershortandopencircuitconditionsundermechanicalexcitation. Theonlycoefficientthataffectstheresponseoftheshort-circuitedpiezoelectricslabiss1E1. Underopen-circuitconditions,theslabiseffectivelystifferandthefrequencyatwhich maximumpowerisdissipatedisalittlegreaterthanthatoftheshort-circuitedslab.However,in bothcases,themaximumtotalpowerdissipatedisnearlythesameandindependentofthelength oftheslab.Approximateexpressionsarepresentedforthefrequencyandtheamplitudeatwhich maximumpowerisdissipated.Numericalvaluesofthemaximumpowerdissipatedobtainedby plottingthepowerdissipatedversusfrequencyareingoodagreementwiththoseobtainedusing theapproximateexpressions.Themaximumpowerdissipatedisexactlyinverselyproportionalto sE"11 in some cases and approximately inversely proportional in others. The frequency at which maximum
powerdissipationursisonlyweaklydependentonlosses.Nearthefirstresonance, thenon-uniformheatgenerationresultsinasignificantlyhighermaximumtemperaturethan uniformtemperaturedistribution.Athigherresonances,thereismorethanonemaximuminthe powerdissipationdensityandtheeffectofnon-uniformheatgenerationislesssignificant.The modifiedFourierseriesmethodisfoundtobeagoodgeneralsolutionforenergybalance equationtoountdifferentheattransfermodesintheslabandthermalboundaryconditions. Page21of30 ACKNOWLEDGEMENTEncouragementandfacilitiesgivenbyDirector,NPOLtocarryoutthisworkare gratefullyacknowledged. REFERENCES1.D.A.Berlincourt,
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K.Jung,“Analysisoftemperatureriseforpiezoelectrictransformerusingfinite-elementmethod,”IEEETrans.Ultrason.Ferroelectr.Freq.Control53,8,1449-1457(2006). Page22of30 11.RKeoschkerjan,
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S.Priya,andS.Hirose,“Lossmechanismsandhighpowerpiezoelectrics,”JMater.Science41,217228(2006). 14.D.D.Ebenezer,
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J.SoundVib.32,892-902(2007). 15.J.P.Hollman,Heattransfer,McGrawHill1997,8thed.,pp.55.16.B.K.MukherjeeandS.Sherrit,“Characterisationofpiezoelectricandelectrostrictive materialsforacoustictransducers:
I.Resonancemethods,”5thInternationalCongressonSoundandVibration,Adelaide,Australia,15–18December1997,pp.385–393. Page23of30 FIGURES321orx FIG.1.Piezoelectricslab,lengthexpander,lengthL,widthB,andthicknessh.Directionofpolarization3andelectrodeandbottomfaceofareaLB. FIG.2.PowerdissipationdensityQ(x,ω)inamechanicallyexcitedshort-circuitedpiezoelectricslab,fixedatx=0andexcitedatx=LthroughanalyticalrelationEq.8,.Dash-dottedline:8.19kHz;solidline:8.231kHz(1stresonance);dashedline:24.57kHz,dottedline:24.693(2ndresonance).AppliedharmonicstressΓ=20kN/m2
0 Page24of30 FIG.3.TotalpowerΠ(ω)dissipatedinamechanicallyexcitedshort-circuitedpiezoelectricslab,fixedatx=0andexcitedatx=L,withtwoapproaches.Solidline:AnalyticalsolutionintegrationofpowerdissipationdensitythroughhysteresisEq.(9a)andcircles:Inputpower,Eq.(10a). FIG.4.PowerdissipationdensityQ(x,ω)inamechanicallyexcitedpiezoelectricslabwithopen-circuitedcondition,fixedatx=0andexcitedatx=
L.Solidline:8.611kHz,24.825(1stand2ndresonance)–hysteresisapproach,Eq.(15);Circles:Poyntingtheorem,Eq.(16). Page25of30 FIG.5.TotalpowerΠ(ω)dissipatedinamechanicallyexcitedopencircuitpiezoelectricslab,fixedatx=0andexcitedatx=
L.Solidline:inputpowerEq.17aandcircles:integrationofpowerdissipationdensitythroughhysteresisQ(x,ω),Eq.(16),overvolumebySimpson'srule FIG.6PowerdissipationdensityQ(x,ω)inanelectricallyexcitedpiezoelectricslabat10Vpeak,stressfreeatx=0andL.Solidline:16.38kHz,dashedline:16.462kHz(1stresonance),dasheddottedline:49.139kHz,anddottedline:49.386kHz(2ndresonance)Eqs.(24a)and(24b);Circles:PoyntingtheoremapproachEq.(24c). Page26of30 FIG.7.TotalpowerΠ(ω)dissipatedinanelectricallyexcitedpiezoelectricslabat10Vpeak,stressfreeatx=0andL.Circles:integrationofQ(x,ω),Eq.(24)overvolumebySimpson'sruleandsolidline:inputpowerEq.(29a). FIG.8.Effectofelasticlossfactoratinfundamentalresonancefrequencyofpiezoelectricslabelectricallyexcitedat10Vpeakwithstressfreeatx=0andL. Page27of30 FIG.9.Temperaturedistributionofpiezoelectricslabundermechanicalexcitationundershortcircuitconditionwithθ(d,ω)=0,at8.231kHz(1stresonance).Dottedline:solutionthroughdirectintegration,stars:modifiedFourierseriessolution,Q(x,ω)estimatednumerically,andcircles:modifiedFourierseriessolution,usedanalyticalQ(x,ω)asinEq.8.AppliedharmonicstressΓ=20kN/m2
0 FIG.10.Temperaturedistributionofpiezoelectricslab,undermechanicalexcitationwithopencircuitedconditionwithθ(d,ω)=
0.Solidline:1stresonance8.611kHz,circles:at2ndresonance24.825kHz,andstars:1stresonancewithdθ(d,ω)/dx=
0. Page28of30 FIG.11.Temperaturedistributionofpiezoelectricslabonehalfsection,underelectricalexcitationatfirstresonance16.462kHzwithθ(d,ω)=0,solidline:non-uniformheatgeneration,starline:uniformheatgeneration,anddasheddottedline:non-uniformheatgenerationwithdθ(d,ω)/dx=
0.Atsecondreson
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