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1、ADualNumberApproachforNumericalCalculationofVelocityandAccelerationintheSpherical4RMechanism1F.Pe~nu~nuri,2O.Mendoza,2CarlosA.Cruz-Villar,1R.Peon-Escalante,1C.Villanueva1FacultaddeIngeniera,UniversidadAutonomadeYucatan,A.P.150,Cordemex,Merida,Yucatan,Me
2、xico.2Cinvestav-IPN,DepartamentodeIngenieraElectrica,Av.IPN2508,A.P.14-740,07300,MexicoD.F.,Mexico.AbstractThispaperproposesamethodologytocalculateboththe rstandsecondderivativesofavectorfunctionofonevariableinasinglecomputationstep.Themethodisbasedontheapp
3、licationofthedualnumbers.IthasbeenimplementedinFortranlanguage,amodulewhichcontainsthedualversionofelementaryfunctionsaswellasmorecomplexfunctions,whicharecommoninthe eldofrotationalkinematics.Ourbasicnumericalentityhasthreeelements,then,foragivenvectorfunctionf
4、:R!Rm,itsdualversionwillhavetheformf~:R3!R3m.Asastudycase,theproposedmethodologyisusedtocalculatethevelocityandaccelerationofapointmovingonthecoupler-pointcurvegeneratedbyasphericalfour-barmechanism.Keywords:Dualnumbers,Numericalderivatives,Spherical4Rmechanism.
5、1.IntroductionThecalculationofvelocityandaccelerationishighlyneededinthe eldsofphysicsandengineering.Insomecases,thereisnopossibilitytoobtainthemanalytically.Inothercases,theanalyticalresultisquitecomplicatedtodealwith;inbothsituationsanumericaltreatmentisdesira
6、ble.Regardingthe eldofmechanisms,thecalculationofthe rstandsecondderivative,letsthedesignertoincludethevelocityandaccelerationinthesynthesisofbalancedmechanisms,dwellmechanisms,etc.,[1,2,3,4,5,6,7].Usually,theprocessofcalculatingaderivativeisnotdicult.However,f
7、orthecaseofasphericalmechanism,obtaining rstandsecondderivativesofthepositionvectorforthecouplerpointisnotsimple.Evenwhensuchderivativescanbeexplicitlyobtained,theresultingexpressionscouldbeofgreatcomplexityanduselessforpracticalpurposes.So,analternativesolution
8、istonumericallycalculatesuchderivatives.Nevertheless,traditionalmethodsforcalculatingnumericalderivatives( nite-di erences)aresubjecttobothtruncationandsubtractivecan
