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1、CHAPTER11KinematicsandDynamicsInthischapterweshallillustratehowextremelyusefulthetheoryoflinearalgebraingeneralandmatricesinparticulararefordescribingmotioninspace.Therearethreesections,onthekinematicsofrigidbodymotions,onthekinematicsoffluidflow,andonthedynamic
2、sofsmallvibrations.1.THEMOTIONOFRIGIDBODIESAnisometrywasdefinedinChapter7asamappingofaEuclideanspaceintoitselfthatpreservesdistances.Whentheisometryrelatesthepositionsofamechanicalsysteminthree-dimensionalrealspaceattwodifferenttimes,itiscalledarigidbodymotion.I
3、nthissectionweshallstudysuchmotions.Theorem10ofChapter7showsthatanisometryMthatpreservestheoriginislinearandsatisfiesM*M=1.(1)Asnotedinequation(33)ofthatchapter,thedeterminantofsuchanisometryisplusorminus1;itsvalueforallrigidbodymotionsis1.Theorem1(Euler).Anisom
4、etryMofthree-dimensionalrealEuclideanspacewithdeterminantplusIthatisnontrivial,thatisnotequaltoI,isarotation;ithasauniquelydefinedaxisofrotationandangleofrotation0.Proof.Pointsfontheaxisofrotationremainfixed,sotheysatisfyMf=f:(2)LinearAlgebraandItsApplications.S
5、econdEdition,byPeterD.LaxCopyright(.)2007JohnWiley&Sons,Inc.172KINEMATICSANDDYNAMICS173thatis,theyareeigenvectorsofMwitheigenvalue1.Weclaimthatanontrivialisometry,detM=1,hasexactlyoneeigenvalueequalto1.Toseethis,lookatthecharacteristicpolynomialofM,p(s)=det(sI-M
6、).SinceMisarealmatrix,p(s)hasrealcoefficients.Theleadingterminp(s)iss3,sop(s)tendsto+ocasstendsto+oo.Ontheotherhand,p(O)=det(-M)=-detM=-1.Sophasarootonthepositiveaxis;thatrootisaneigenvalueofM.SinceMisanisometry,thateigenvaluecanonlybeplus1.Furthermore,1isasimpl
7、eeigenvalue;forifasecondeigenvaluewereequalto1,then,sincetheproductofallthreeeigenvaluesequalsdetM=1,thethirdeigenvalueofMwouldalsobe1.SinceMisanormalmatrix,ithasafullsetofeigenvectors,allwitheigenvalue1;thatwouldmakeM=I,excludedasthetrivialcase.ToseethatMisarot
8、ationaroundtheaxisformedbythefixedvectors,werepresentMinanorthonormalbasisconsistingoffsatisfying(2),andtwoothervectors.Inthisbasisthecolumnvector(1,0,0)isaneigenvect
