Dynamics of a Cantilever Beam Attached to a Moving

VOL.10,NO.2,MARCH-APRIL1987J.GUIDANCE139DynamicsofaCantileverBeamAttachedtoaMovingBaseT.R.Kane*andR.R.RyanfStanfordUniversity,Stanford,CaliforniaandA.K.Banerjeerj:LockheedMissilesandSpaceCompany,Sunnyvale,CaliforniaThebehaviorofacantileverbeambuiltintoarigidbodythatisperformingaspecifiedmotionofrotationandtranslationisstudiedwithtwoobjectivesinmind.First,becausethesubjectisofinterestinconnectionwithspacecraftantennae,helicopterrotorblades,robotarms,andothersystemsthatperformcomplexmotions,wecreateanalgorithmthatcanbeusedtopredictthebehaviorofthebeamwhenthebaseundergoesgeneralthree-dimensionalmotions.EffectssuchascentrifugalstiffeningandvibrationsinducedbyCoriolisforcesareaccommodatedautomatically,ratherthanwiththeaidofadhocprovisions.Thesecondobjectiveistodrawattentiontofundamentalflawsincertainmultibodycomputerprogramscurrentlyunderdevelopmentoralreadyinuse.Tothisend,weconstructasecondsimulationalgorithm,onethatembodiestheprocedureapparentlyemployedintheprogramsinquestion,andthencomparesimulationresultsproducedbycomputerprogramsbasedonthetwoalgorithms.Conflictsbetweenthetwoapproachesthatthuscometolightarediscussedindetail.I.IntroductionSTUDYofthebehaviorofflexiblebodiesattachedtomovingsupportshasbeenvigorouslypursuedforoverfiftyyearsinconnectionwithanumberofdiversedisciplines,suchasmachinedesign,robotics,aircraftdynamics,andspacecraftdynamics.Inparticular,beamsattachedtomovingbaseshavereceivedattentioninhundredsoftechnicalpapersdealingvariouslywithelasticlinkages,rotatingmachinery,roboticmanipulatorarms,aircraftpropellors,helicopterrotorblades,andflexiblesatellites.Indeed,theexistingliteratureissovoluminousastoprecludecomprehensivereviewwithintheconfinesofatechnicalpaper.(A1974review,byModi,1oftheliteratureonrigidbodieswithflexibleappendagescontainedmorethantwohundredreferences.)Accordingly,weshallmentiononlyrepresentativepublicationsthatpermitustodirectattentiontoissuesofprimaryconcern.Intheareaofmachinedesign,thepublishedanalyses25dealmainlywithplanarbendingofrotatinglinksorwithpuretorsionofshaftedsystems.Forthemostpart,thegoaloftheseanalysesistodetermineforcesorstressesofinterestincon-nectionwithmachinecomponentsizing.Inrobotics,69dis-placements,suchasthosegivingrisetobendingofamanipu-latorarm,areatthecenterofattention,andvariousbeameffectsarefrequentlyneglectedinanattempttoincreaseon-linecomputationalspeed.Analysesperformedinconnec-tionwithaircraftdynamics1014addressquestionsconcerningtapered,twisted,androtatingbeams;andinthefieldofspacecraftdynamics,1521theeffectofvehicleelasticityonattitudemotionsisofparticularinterest.Whatismissingfromthisvastliteratureisacomprehensivetheoryfordealingwithsmallvibrationsofageneralbeamattachedtoabasethatisundergoinganarbitrary,prescribedmotion(translationandrotation).Byageneralbeamwemeanonethathasanarbitrarycrosssectionandmaterialpropertiesthatvaryfrompointtopointalongthebeam;andacompre-ReceivedAug.30,1985;revisionreceivedApril8,1986.Copyright©AmericanInstituteofAeronauticsandAstronautics,Inc.,1987.Allrightsreserved.*ProfessorofAppliedMechanics.fResearchAssistant.MemberAIAA.:j:StaffEngineer.hensivetheoryisonethataccommodatesallofthefollowing:extension,bendingintwoprincipalplanes,torsion(uniformornonuniform),sheardisplacements,andwarping.Intheexistingliterature,oneoftenfindsthataparticulareffectisintroducedinanadhocfashion.Forexample,therearemanyanalysesthatmakespecialprovisionsforcentrifugalstiffening,Coriolisforces,etc.Mostoftheseanalysesarebasedononeoftwomethodologies.Thefirst22involvesstartingwithequationsapplicabletoabeamwithaninertiallyfixedsupport,andthenaddingtermsintendedtoreflectaneffectofinterest,suchastheCoriolisaccelerationassociatedwitharotationalmotionofthebasethatsupportsthebeam.Inthesecondapproach,21'23whichisbasedonaRayleigh-Ritztechnique,oneworkswithtrialfunctionsorassumedmodes,asthesefunctionsareoftencalled,someofwhicharedirectlyassociatedwiththegyroscopiceffectsorcentrifugaleffectspresentinabeamperformingasimplespinningmotionataconstantspinrate.Thesetwoapproachesshareabasicshort-coming:theydonotaccount,inageneralway,forarbitrarybasemotion.Whilemostpreviousanalysesdealeitherwithspinningbeamsorbeamsattachedtoabaseundergoingrectilineartranslation,someattemptstoconstructmorecomprehensivetheorieshavebeenundertaken.Forexample,Benedetti,24formulatingequationsofmotionforauniformEuler-Bernoullibeamwithasymmetriccrosssection,letthebasesupportingthebeammovewitharbitraryangularandtranslationalaccel-erations;however,termsassociatedwithstiffeningduetorotationwereneglectedbecausetheanalysiswasintendedforthestudyofthebehaviorofflexibleprojectiles,inwhichconnectionitisappropriatetodropthesetermssincethedominantangularvelocitycomponentisparalleltotheaxisoftheprojectile.Thefactthatthisworkisrestrictedtobeamswithsymmetriccrosssectionsprecludesitsuseforthepurposeofanalyzingmotionsofopensectionbeams,suchasSTEM(self-storingtubularextensiblemodule)booms,whicharecommonlyusedonsatellites.Laskinetal.25alsoattemptedtddevelopageneraltheoryforafree-free,symmetriccrosssectionRay