华科船海结构动力学大作业两端简支梁参数激励振动特性 英文版Word格式文档下载.docx
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华科船海结构动力学大作业两端简支梁参数激励振动特性 英文版Word格式文档下载.docx
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Faculty:
NavalArchitectureandOceanStructureDesign
Abstract:
1.Intheoreticalaspect,equilibriumequationofthegivenbeamstructureisdeducedanalyticallyviatheNewtonmethod,throughwhichnaturalvibrationfrequencyandEuler-CriticalLordcanbeobtained.Apartfromthese,inordertostudythedynamicalstabilitybehaviorofthesystem,thecontrolequationaswellasthekeyproblem“Mathieuequation”arealsoconcludedinthepassage.
2.WiththehelpoftheMatlabsoftware,theresponsecurvesofthebeamsubjecttodifferentcombinationsofaxial-forceparameteraredrawn,thusitenablesustotakefurtherresearchintothestableareaandunstableareabysolvingthenonlinearMathieuEquationviadifferencemethod.
3.ThepassagealsomakeprogressinapplyingnumericalmethodFEMinsolvingthedynamicalstabilityproblem,whichhasgreatersuperiorityoverdifferencemethodinstudyingtheessenceoftheparametervibrationandapplicationincomplexstructuresandload.FEMmethodisbasedonlargecommercialFEMsoftwareANSYS’stransicientmodule,eigen-bucklingmoduleanditsAPDLlanguage(ANSYSParameterDesignLanguage).ThecurvesofBucklingeigen-valuesofthebeamandtimetaregained.
4.Forsomespecialcaseswheretheexcitingloadsareharmonicload,thentheharmonicanalysiscanbeapplied.Byobservingtherelationbetweenmaximumresponsedisplacementandexcitingfrequency,“thedangerousfrequency”canbeobtained,whichisveryimportanttodesignerssincetheyneedtoensuretheexcitingfrequencyarefarfromthose“dangerousfrequency”.
Catalog
Abstract1
1>
PhysicalandMathematicalModels4
1.1PhysicalModel4
1.2EquilibriumEquation4
1.3FreeVibration6
1.4Stability6
1.5ControlEquation6
2>
TimeDomainSimulation7
2.1DifferenceMethod7
2.2ParameteroftheStructure8
2.3PreferenceSetting8
2.4SimulationviaMATLAB8
3>
ParameterResonanceandMathieuEquation11
3.1Instability11
3.2PropertiesoftheMathieuEquation12
3.3ComputingProcess14
3.4OptimizationoftheProcess15
3.5EvaluateoftheOptimization17
4>
FEMethods18
4.1IntroductiontoFEMethods18
4.2ValidationofFEMethods19
4.3Dynamicalbucklinganalysis21
4.4EvaluateofFEMethods23
4.5HarmonicAnalysis23
Reference26
Appendix27
PhysicalandMathematicalModels
AnEulerbeamsubjectedtoanaxialforce.Pleasegivethemovementequationanduseaproperdifferencemethodtosolvethisequation.Studythedynamicstabilityofthebeamrelatedtothefrequencyandamplitudeoftheforce.
Figure1theforcediagramoftheEulerBeam
1.1.PhysicalModel
Fetchoutanelementofthebeam,sketchtheforcediagramofit
Fig1.1Forceanalysisofthesmallelement
Fig1.1.1forcediagramoft
1.2.NewtonMethodtoDeducetheEquilibriumEquation
Thenwritethedynamicsequilibriumequation,whichisthemathematicalmodelofthissystem.
WhentheaxisforceNisnon-existence,derivetheforceequilibriumequationintheydirectionandmomentofforceequilibriumequation.
S(x,t)istheshearforce;
P(t)istheaxisforce;
M(x,t)istheforceofmoment;
isthedeflectionoftheelement;
isthelengthoftheelement;
Force:
<
1.2.1>
Momentofforce:
1.2.2>
Andotherrelationshipsbetweenthesequantitiesare:
1.2.3>
1.2.4>
Substituteallforceintheydirectionintotheeq1.2.1weobtain:
1.2.5>
AccordingtoTaylorExpansion:
1.2.6>
Substituteeq1.2.6intoeq1.2.5,thenbothsidesdivideby,weobtain:
1.2.7>
Accordingtoeq1.2.2:
UsingTaylorExpansion,thendividebyasabove:
1.2.8>
Combineeq1.2.4,eq1.2.7,eq1.2.8weobtain:
1.2.9>
AssumethequantitiesE(Young'
smodulus)andI(secondaxialmomentofarea)areconstant,andtheexcitationforceissimpleharmonicas.Theeq1.2.9canbeRewroteas:
1.2.10>
1.3.Freevibration
Somedynamicpropertiesoffreevibration(withoutexcitationforce)areintroducedbelow:
Thecontrolequationoffreevibrationis:
1.3.1>
canbesolvedbysegregationvariablemethod.Whilethebeamissimplesupported,thefirstordersolutionoffreevibrationis:
1.3.2>
Infreevibrationsystem,1storderangularfrequencyequalsto:
1.3.3>
1.4.Staticstability
ToresearchthedynamicstabilityofEulerbeam,weshouldintroducestaticstabilityofsimple-supportedbeamfirst.
AswelearnedfromMechanicsofmaterials,theEulerCriticalLordofthesimple-supportedbeamis:
1.4.1>
Whenconsiderthefirstordercondition,theminimumofEulerCriticalLordis:
1.4.2>
1.5.Controlequation
Thecontroleq1.2.10isanon-linearpartialdifferentialequationwhichisverycomplextosolve.Butasitsolvedinstep3,thefirstordermodalof
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