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1、SensorsandActuatorsA130131(2006)9198TheconstituentequationsofpiezoelectricmultilayerbendingactuatorsinclosedanalyticalformandexperimentalresultsRuedigerG.Ballasa,∗,H.F.Schlaaka,A.J.SchmidbaDarmstadtUniversityofTechnology,InstituteforElectromechanicalDesign,Merckstr.25,D-64283Darmstadt,GermanybFestoA
2、G&Co.KG,DepartmentTechnology&Testing,RuiterStr.82,D-73734Esslingen-Berkheim,GermanyReceived27May2005;receivedinrevisedform31October2005;accepted22November2005Availableonline24January2006AbstractInthispaper,wepresentforthefirsttimethederivationoftheconstituentequationsforanykindofclamped-freepiezoelec
3、tricmultilayerbendingactuatorsunderdifferentexcitationconditionsformulatedgenerallyforanypointovertheentirelengthoftheactuator.Theconstituentequationsarepresentedbya4×4matrixM,whichcombinestheextensiveparametersmechanicalmomentMattheendofthebender,forceFappliedperpendicularlytothetipofthebender,unif
4、ormpressureloadpappliedovertheentirelengthofthebenderandappliedelectricalvoltageUwiththeintensiveparametersangulardeflectionα,deflectionξ,volumedisplacementVandelectricchargeQ.Inordertoverifyapartofthederivedconstituentequationsforaclamped-freepiezoelectricmultilayerbendingactuatorthebendingcurvaturea
5、ndforce-deflectioncharacteristicsofarealizedbendingactuatoraredeterminedexperimentallyandcomparedwithanalyticalcalculations.©2005ElsevierB.V.Allrightsreserved.Keywords:Piezoelectricbendingactuator;Multilayer;Constituentequations;Closedformsolution;Energyconcept1.Introductionxovertheentirelengthoftheb
6、eamactuatoryet.Butjustthisaspectaddsimportantinsightsintopiezoelectricdesign.Thestaticconstituentequationsforacantileverstructurecon-Inthispapertheneutralaxis,theinternalpiezoelectrictainingtwolayers,oneconsistingofapiezoelectricandthemomentsandtheirinfluencesonthebendingbehaviourofthesecondofanelast
7、icmaterial,weredescribedbySmitsandneutralaxisaretakenintoconsideration.IncombinationwithChoi[1].WangandCross[2]presentedtheconstitutiveequa-thepiezoelectricconstitutiveequationsfortheelectricdisplace-
