PDF-2CEE421L.MatrixStructuralAnalysis{DukeUniversity{Fall2013{H.P.GavinSol

Author : marina-yarberry | Published Date : 2015-11-02

fdg4827665517NotethatifK11K12K21K22thenthelinesareparallelThereisnouniquepointatwhichtheycrossandthereisnosolutionfdgK1fpgInotherwordsifK11K12K21K22thenthematrixKcannotbe

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2CEE421L.MatrixStructuralAnalysis{DukeUniversity{Fall2013{H.P.GavinSol: Transcript

fdg4827665517NotethatifK11K12K21K22thenthelinesareparallelThereisnouniquepointatwhichtheycrossandthereisnosolutionfdgK1fpgInotherwordsifK11K12K21K22thenthematrixKcannotbe. DualoflinearprogramininequalityformwedenetwoLPswiththesameparametersc2Rn,A2Rmn,b2RmanLPin`inequalityform'minimizecTxsubjecttoAxbanLPin`standardform'maximizebTzsubjecttoATz+c=0z0thisproblemiscal AddresscorrespondencetoMathiasFleck,DukeUniversity,LSRCBldg.,Box90999,Durham,NC27708,e-mail:mathias. 2!k+!k 2(7)Rayleighdampingcanbeextended.ItcanbeshownthatthedampingmatrixC=M+K+ MK1M+KM1K(8)isaclassicaldampingmatrix.AnextendedRayleighdampingmatrix,calledCaugheydamp-ing[1,2],canbecomputedfromC Linear-fractionalprogramminimizecTx+d gTx+hsubjecttoAxbgTx+h0ifneeded,weinterpreta=0asa=0=+1ifa0,a=0=1ifa0however,inmostapplications,AxbimpliesgTx+h0equivalentform(withaddedvariable)minimize Grantsponsor:LeverhulmeTrust*Correspondenceto:AnneStarling,DepartmentofBiologicalAnthropologyandAnatomy,DukeUniversity,08BiologicalSciencesBldg.,Box90383,Durham,NC27708-0383.E-mail:aps4@duke.eduReceiv Processing Draft,Fall2013.Submittedforinclusionin:Schwarz,Florian(ed.)UnderContract.ExperimentalPerspectivesonPresuppositions,editedvolumeforSpringer UniversityofArizona,DukeUniversity leeTEACHINGInstructor,MIT,UndergraduateseminarinDiscreteMathematics(Spring2014)Instructor,MIT,TopicsinMathematicswithapplicationsinnance(Fall2013,2014)Instructor,KMO(KoreanMathematicalOlympiad)sum Linearandanefunctionslinearfunction:afunctionf:Rn!Rislineariff(x+y)=f(x)+f(y)8x;y2Rn;;2Rproperty:fislinearifandonlyiff(x)=aTxforsomeaanefunction:afunctionf:Rn!Risaneiff(x+(1)y)=f(x)+(1) Subspacedenition:anonemptysubsetSofRnisasubspaceifx;y2S;;2R=)x+y2Sextendsrecursivelytolinearcombinationsofmorethantwovectors:x1;:::;xk2S;1;:::;k2R=)1x1++kxk2Sallsubspacescontaintheorigin LM(x)=M1 x L1!+M2x LThetotalpotentialenergyofabeamwiththeseforcesandmomentsis:U=1 2ZL0M2 EIdx+1 2ZL0V2 G(A=)dxByCastigliano'sTheorem,1=@U @M1=ZL0M(x)@M(x) @M1 EIdx+ZL0V(x)@V(x) @M1 G(A=)dx=0B@ZL0 k=(m+ma).Ifca=0,thesystemisun-dampedandr=fiszeroat!=q ka=ma.Theequationsofmotioninthiscasearemr(t)+kr(t)ca(_ra(t)_r(t))ka(ra(t)r(t))=fcos!t(7)mara(t)+ca(_ra(t)_r(t))+ka(ra(t)r(t))=0(8)Using DukeUniversity,UniversityofToronto,KLARU,BaycrestCentre,Toronto2003MassachusettsInstituteofTechnologyJournalofCognitiveNeuroscience15:2,pp.249 Ghee-CheePhuaMDisaffiliatedwiththeDivisionofPulmonary,Al-lergy,andCriticalCare,DepartmentofMedicine,DukeUniversity,Durham,NorthCarolina,andwiththeDepartmentofRespiratoryandCriticalCareMedicine,Singapo

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