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1、ieeetransactionsonultrasonics,ferroelectrics,andfrequencycontrol,vol.46,no.6,november19991343ConstitutiveEquationsofSymmetricalTripleLayerPiezoelectricBendersQing-MingWangandL.EricCross,Fellow,IEEEAbstract—Piezoelectrictriplelayerbenders,withastructureoftwopiezoelect
2、rictopandbottomlayerssand-wichedbyanon-piezoelectricelasticcentrallayer,areoneofthemostcommonlyusedpiezoelectricdevices.Inthispa-per,wepresentthederivationoftheconstitutiveequationsofasymmetricaltriplelayerpiezoelectricbenderunderdif-ferentexcitationconditions.Thecon
3、stitutiveequationsarepresentedbya44matrixwithanexternalmomentM,anexternaltipforceF,auniformloadp,andanappliedelectricvoltageVastheextensiveparameters,withthegeneratedtipangulardeflection(slope),tipdeflection,volumedisplacement,andelectricchargeQastheinternalparamete
4、rs.Furtheranalysisontheelectromechanicalbe-haviorofthetriplelayerpiezoelectricbendercanbemadeontheconstitutiveequations.I.Introductionnrecentyears,solidstatedevicesutilizingpiezoelectricFig.1.Structuresofcantilever-mountedpiezoelectricunimorph,bi-Ieffectshavedrawnmuch
5、attentionbothscientificallymorph,andtriplelayermorphbenders:(a)unimorph,(b)bimorphinandtechnologicallybecausepiezoelectricactuators,sen-seriesconnection,(c)bimorphinparallelconnection,and(d)triplelayerbender.sors,andtransducershavebeenwidelyusedinmanyelec-tromechanica
6、lapplicationsincludingactiveandpassivevi-brationdamping,ultrasonicmotors,ultrasonicbiomedi-tions,andtheelectricfieldisappliedacrosseachindivid-calimaging,loudspeakers,accelerometers,resonators,mi-uallayerwithoppositepolarity.Inbothcases,oneplatecropositioner,acoustics
7、ensor,etc.[1],[2].Manynewde-expandsunderelectricfield,andtheothercontracts.Asvicesandapplicationsareexploredintensively,andmanyaresult,bendingdeformationoccursinthewholestruc-newtechnologicaldevelopmentsbasedonthesedevicesture.Similarly,intheunimorphbender[Fig.1(a)],t
8、heareemerging[3]–[9].Mostrecently,weareinvolvedinconstraintofthenon-piezoelectriclayeronthetransversearesearchprogramtodevelopanewt