AdvancedStructuralDynamicsProjectstabilityanalysistheverticalexcitationInstructor：Dr.LiWeiName：StudentID：1.1calculationmodelAnEularbeamsubjectedtoanaxialforce.Pleasebuildthedifferentialequationofmotionanduseaproperdifferencemethodtosolvethisdifferentialequation.Studythedynamicstabilityofthebeamrelatedtothefrequencyandamplitudeoftheforce.AsshownintheFig1.1.Fig1.11.2purposeandprocessarrangementa.learninghowtocreatemathematicalmodelofthecontinuoussystemandselectpropercalculationmethodtosolveit.b.learninghowtobuildbeamvibrationequationandsolveMathieuequation.c.usingFloquettheorytojudgevibrationsystem’sstabilityandanalyzetherelationshipamongthefrequencyandamplitudeoftheforceanddynamicresponse.Thisprojectwillintroducetheestablishmentofthemathematicalmodelofthecontinuoussysteminsection2,themovementequationandthenumericalsolutionofusingMATLABinsection3,ApplyingFloquenttheorytostudythedynamicstabilityofthebeamrelatedtothefrequencyandamplitudeoftheforceinsection4.Inthelastoftheproject,wegetsomeconclusionsinsection5.systemThegeometricmodelofthebeamandforce-simplifieddiagramisshowninFig.2.1.Weassumethatitsstiffness(EI)isconstantandthedeflectionofthebeamissmall,andtheboundaryconditionsissimplysupport.Nowthebeamsubjectedtoanaxialforce.Weassumetheforceisequalto.Pcost0yxFig.2.1Weselectthelengthofinanypositionofthebeam,thexfree-bodydiagramisshowninFig.2.2.FVFig.2.2Usingequationsofmovementequilibrium,thatistosay:F(m)a+（1）（2）yyM0GFromequation(1),wewillget:y2S(x,t)S(xx,t)Ax（3）t2Divideequation(3)byandtakethelimit:xSxy2A（4）x2Thensynthesizeequation(2),wecanget:M(xx,t)M(x,t)F[y(xx,t)y(x,t)]S(xx,t)x0（5）（6）Divideequation(5)byandtakethelimit:xMxyFSxCombineequation(4)with(6):Myy222FA0（7）（8）x2x2t2yAnd2M(x,t)EIx2Combineequation(7)with(8):2yyy222(EI)FA0（9）x2x2x2t2WeknowEIisaconstant,soyyy422F(t)m0（10）EIx4x2t2Inequation(10),misthemassofunitlength.Nowwewilluseassumed-modesmethod.Namednx,so:y(x,t)Tt)sinnlnnnxd2Tn242TFt)nTn0（11）ml4l2ld2Tn2Ft)PT02onn=1,2,......（12）dtFnnIntheequation(12)222nnP,Fl2mlnAnd,soFt)Fcostd2TnFPcostT0n=1,2,......（13）（14）2ondt2Fnnd2T(cos)tT0ndt2nIntheequation(12)F(n)2EIn()4AL2Equation(14)istheMathieuequation.itisdifficulttosolvetheanalyticalsolutiondirectly,thus,weusetheapproximatederivativenamelyanaverageaccelerationmethodtogetthenumericalsolutionfromthereference.3.Numericalsolution3.1usingMATLABtosolveequationWewillusetheNewmark-methodtosolveequation(14).We[1]ucanusetheinitialconditionutointegratethemoveequation:00u015（）Fig.3.1AsshowninFig.3.1t(uu)i（16）uui2i1ii1t2uu()（17）（18）uuuti4i1iiiii1cosu0wtuiiFromequation(16),(17)and(18),wewillget:icoswntuu（19）（20）i2()2uuutiiii2(t)u[coswnt]t2u（21）ii4[coswntti2WhenapplyingtheMATLAB,weneeddiscretetheprocessingtimet,gettimestep.Whensolvingthevibrationstabilityinterval,ttherearethreevariablestoparticipateinthediscussion,namely.So,c,wtakeaparticularwfirstanddiscusstheremainingtwoparameters.FromFloquenttheory，wecanuseparameterAtojudgestability.[2]d2ydt2dydtEquation（22）()t()0tyTaketwosetsofspecialsolution:y(0)y(0)0（20）（21）11y(0)y(0)12212Parameter[2]A[yT)()]T12Ifabs(A)islessthan1,thesystemisstability.Andifabs(A)isgreaterthan1,thesystemisinstability.Whenabs(A)isequalto1,thesystemiscriticalstate.WeuseMATLABCodestosolveequations.Weuseω=2MathieuEquationtojudgethevalidityofthecodes.FromFig.3.2andFig.3.3,wecanconsiderthecodesarecorrect.Inthesefollowfigures,ω=2,thehorizontalaxisisδ,verticalaxisisε.Fig.3.2.stabledomainin[3]and[2]Fig.3.3.stabledomaininMATLABsolutionComparedFig.3.2withFig.3.3,wecanseethatthestabilitydomainofnumericalsolutionsapplyingaverageaccelerationmethodareconsistentwiththestandardsolutions.itcanconcludedthatwhenthesystemhavesolutionwhosecycleisequaltoπor2π,n2(n)ThischapterdiscussestheaccuracyofthevibrationstabilitydeterminationwithFloquenttheory.Thenextchapterwilldiscussthenumericalsolutionandstabledomainandtwoparameters’influencesonthestabilityforthisquestion.4.ParametersinfluenceInthispart,weonlyconsidertwoparameters,namelythefrequencyandamplitudeoftheforce.4.1theinfluenceofthe’sfrequency4.1.1thestabilityofthesystemWhenwediscussthestabilityofthesystemrelatedtothefrequencyoftheforce,weshouldselectsomedifferentfrequencies,sowechooseω=1,2,4,6,8and10.UsingMATLABcodes,wecanobtainthefigsofthestability.WecanknowthestableregionisbiggerwiththeincreaseofthefrequencyinFig.4.1.ω=1ω=2ω=4ω=6ω=8Fig.4.1stabledomainwithdifferentω4.1.2theresponseofthesystemω=10Whenwediscusstheresponseofthesystem,thesystemshouldbestable.Sowechoose7，1，=2,4and6.InFig.4.2,thecycleoftheresponseincreaseandtherangeofthereactiveamplitudeissmallerwiththeincreaseofthefrequency.Fig.4.2responsesofthesystemwithdifferentω4.2theinfluenceoftheforce’samplitudeTheεisrelatedtotheforce’samplitudeP.Thecycleoftheresponsealittleincreaseandtherangeofthereactiveamplitudeisbiggerwiththeincreaseoftheforce’s,inFig.4.3.andFig4.4.Fig.4.3vibrationresponsecurvewithdifferentδTheredcurveisw=2,δ=12,ε=1;thebluecurveisw=2,δ=14,ε=1.Thisfigurestatethatthevibrationcycleissmallerandtheamplitudehavealittlechangewiththeincreaseoftheδ.Fig.4.3vibrationresponsecurvewithdifferentεTheredcurveisw=2,δ=12,ε=1;thebluecurveisw=2,2,ε=5.Thisfigurestatethattheamplitudeissmallerandthevibrationcyclehavealittlechangewiththeincreaseoftheε.(1)Withtheincreaseofthefrequency,thestableregionandthecycleoftheresponsearebigger,buttherangeofthereactiveamplitudeissmaller.(2)Withtheincreaseoftheforce’samplitude,thecycleoftheresponsealittleincreaseand