結(jié)構(gòu)動力學(xué)第III篇-碩士

PartIIIDistributed--ParameterSystems第III篇分布參數(shù)體系Chapter17Partialdifferentialequationsofmotion17--1INTRODUCTIONThediscrete--coordinatesystemsdescribedinPartTwoprovideaconvenientandpracticalapproachtothedynamicresponseanalysisofarbitrarystructures.However,thesolutionsobtainedcanonlyapproximatetheiractualdynamicbehaviorbecausethemotionsarerepresentedbyalimitednumberofdisplacementcoordinates.Theprecisionoftheresultscanbemadeasrefinedasdesiredbyincreasingthenumberofdegreesoffreedomconsideredintheanalyses.Inprinciple,however,aninfinitenumberofcoordinateswouldberequiredtoconvergetotheexactresultsforanyrealstructurehavingdistributedproperties;hencethisapproachtoobtaininganexactsolutionismanifestlyimpossible.Theformalmathematicalprocedureforconsideringthebehaviorofaninfinitenumberofconnectedpointsisbymeansofdifferentialequationsinwhichthepositioncoordinatesaretakenasindependentvariables.Inasmuchastimeisalsoanindependentvariableinadynamicresponseproblem,theformulationoftheequationsofmotioninthiswayleadstopartialdifferentialequations.Differentclassesofcontinuoussystemscanbeidentifiedinaccordancewiththenumberofindependentvariablesrequiredtodescribethedistributionoftheirphysicalproperties.Forexam-ple,thewave--propagationformulasusedinseismologyandgeophysicsarederivedfromtheequationsofmotionexpressedforgeneralthree--dimensionalsolids.Simi-larly,instudyingthedynamicbehaviorofthin-plateorthin--shellstructures,specialequationsofmotionmustbederivedforthesetwo--dimensionalsystems.Inthepresentdiscussion,however,attentionwillbelimitedtoone--dimensionalstructures,thatis,beam--androd--typesystemswhichmayhavevariablemass,damping,andstiffnesspropertiesalongtheirelasticaxes.Thepartialdifferentialequationsofthesesystemsinvolveonlytwoindependentvariables:timeanddistancealongtheelasticaxisofeachcomponentmember.Itispossibletoderivetheequationsofmotionforrathercomplexone--dimensionalstructures,includingassemblagesofmanymembersinthree-dimensionalspace.Moreover,theaxesoftheindividualmembersmightbearbitrarilycurvedinthree--dimensionalspace,andthephysicalpropertiesmightvaryasacomplicatedfunctionofpositionalongtheaxis.However,thesolutionsoftheequationsofmotionforsuchcomplexsystemsgenerallycanbeobtainedonlybynumericalmeans,andinmostcasesadiscrete--coordinateformulationispreferabletoacontinuous--coordinateformulation.Forthisreason,thepresenttreatmentwillbelimitedtosimplesystemsinvolvingmembershavingstraightelasticaxesandassemblagesofsuchmembers.Informulatingtheequationsofmotion,generalvariationsofthephysicalpropertiesalongeachaxiswillbepermitted,althoughinsubsequentsolutionsoftheseequations,thepropertiesofeachmemberwillbeassumedtobeconstant.Becauseoftheseseverelimitationsofthecaseswhichmaybeconsidered,thispresentationisintendedmainlytodemonstratethegeneralconceptsofthepartial--differential--equationformulationratherthantoprovideatoolforsignificantpracticalapplicationtocomplexsystems.Closedformsolutionsthroughthisformulationcan,however,beveryusefulwhentreatingsimpleuniformsystems.Chapter17PartialDifferentialEquationsofMotion17--2BeamFlexure:ElementaryCaseFIGURE17-1Basicbeamsubjectedtodynamicloading:(a)beampropertiesandcoordinates;(b)resultantforcesactingondifferentialelement.Afterdroppingthetwosecond--ordermomenttermsinvolvingtheinertiaandappliedloadings,onegetsThisisthepartialdifferentialequationofmotionfortheelementarycaseofbeamflexure.Thesolutionofthisequationmust,ofcourse,satisfytheprescribedboundaryconditionsatx=0andx=L.17--3BeamFlexure:IncludingAxial--ForceEffectsFIGURE17-2Beamwithstaticaxialloadinganddynamiclateralloading:(a)beamdeflectedduetoloadings;(b)resultantforcesactingondifferentialelement.17--4BeamFlexure:IncludingViscousDampingIntheprecedingformulationsofthepartialdifferentialequationsofmotionforbeam--typemembers,nodampingwasincluded.Nowdistributedviscousdampingoftwotypeswillbeincluded:(1)anexternaldampingforceperunitlengthasrepresentedbyc(x)inFig.8--3and(2)internalresistanceopposingthestrainvelocityasrepresentedbythesecondpartsofEqs.(8--8)and(8--9).17--6AXIALDEFORMATIONS:UNDAMPEDTheprecedingdiscussionsinSections17--2through17--5havebeenconcernedwithbeamflexure,inwhichcasethedynamicdisplacementsareinthedirectiontransversetotheelasticaxis.Whilethisbendingmechanismisthemostcommontypeofbehaviorencounteredinthedynamicanalysisofone--dimensionalmembers,someimportantcasesinvolveonlyaxialdisplacements,e.g.,apilesubjectedtohammerblowsduringthedrivingprocess.Theequationsofmotiongoverningsuchbehaviorcanbederivedbyaproceduresimilartothatusedindevelopingtheequationsofmotionforflexure.However,derivationissimplerfortheaxial--deformationcase,sinceequilibriumneedbeconsideredonlyinonedirectionratherthantwo.Inthisformulation,dampingisneglectedbecauseitusuallyhaslittleeffectonthebehaviorinaxialdeformation.FIGURE17-4Barsubjectedtodynamicaxialdeformations:(a)barpropertiesandcoordinates;(b)forcesactingondifferentialelement.Chapter18Analysisofundampedfreevibration18-1BEAMFLEXURE:ELEMENTARYCASEFollowingthesamegeneralapproachemployedwithdiscrete-parametersys-tems,thefirststepinthedynamic--responseanalysisofadistributed--parametersystemistoevaluateitsundampedmodeshapesandfrequencies.Becauseofthemathematicalcomplicationsoftreatingsystemshavingvariableproperties,thefollowingdiscussionwillbelimitedtobeamshavinguniformpropertiesalongtheirlengthsandtoframesassembledfromsuchmembers.Thisisnotaseriouslimitation,however,becauseitismoreefficienttotreatanyvariable--propertysystemsusingdiscrete-parametermodeling.（17-7）（18-1）（18-2）（18-3）First,letusconsidertheelementarycasepresentedinSection17--2withandsetequaltoconstantsand,respectively.AsshownbyEq.(17--7),thefree--vibrationequationofmotionforthissystemisExampleE18-1.SimpleBeamConsideringtheuniformsimplebeamshowninFig.E18-1a,itsfourknownboundaryconditionsareFIGUREE18-1Simplebeam-vibrationanalysis:(a)basicpropertiesofsimplebeam;(b)firstthreevibrationmodes.第五章無限自由度體系的振動分析5.1運動方程的建立一.彎曲振動方程微段平衡方程撓曲微分方程消去內(nèi)力,得加慣性力,得運動方程二.考慮軸力對彎曲的影響時的彎曲振動方程三.考慮剪切變形與慣性力矩對彎曲的影響時的彎曲振動方程1.考慮剪切變形時的幾何方程桿軸轉(zhuǎn)角截面轉(zhuǎn)角2.慣性力矩的計算單位長度上的慣性力矩3.運動方程4.物理方程5.方程整理幾何方程：物理方程：運動方程：對于等截面桿：對于等截面細(xì)長桿：四.考慮阻尼影響時的彎曲振動方程外阻尼力內(nèi)阻尼力1.粘滯阻尼

2.滯變阻尼不計阻尼時計阻尼時習(xí)題：1.求剪切桿的運動方程。

2.求拉壓桿的運動方程。一.運動方程及其解邊界條件xyxyxy幾何邊界條件力邊界條件混合邊界條件初始條件已知函數(shù)5.2自由振動分析設(shè)方程的特解為代入方程，得方程(1)的通解為運動方程的特解為運動方程的通解由特解的線性組合確定設(shè)方程(2)的特解為代入方程(2)，得方程(2)的通解為或二.振型與頻率振型方程xy頻率方程振型18--4BEAMFLEXURE:ORTHOGONALITYOFVIBRATIONMODESHAPESThevibrationmodeshapesderivedforbeamswithdistributedpropertieshaveorthogonalityrelationshipsequivalenttothosedefinedpreviouslyforthediscrete-parametersystems,whichcanbedemonstratedinessentiallythesame—byapplicationofBetti'slaw.ConsiderthebeamshowninFig.18--1.Forthisdiscussion,thebeammayhavearbitrarilyvaryingstiffnessandmassalongitslength,anditcouldhavearbitrarysupportconditions,althoughonlysimplesupportsareshown.Twodifferentvibrationmodes,mandn,areshownforthebeam.Ineachmode,thedisplacedshapeandtheinertialforcesproducingthedisplacementsareindicated.Betti'slawappliedtothesetwodeflectionpatternsmeansthattheworkdonebytheinertialforcesofmodenactingonthedeflectionofmodemisequaltotheworkoftheforcesofmodemactingonthedisplacementofmoden;thatis,（18-31）（18-34）Thefirsttwotermsinthisequationrepresenttheworkdonebytheboundaryverticalsectionforcesofmodenactingontheenddisplacementsofmodemandtheworkdonebytheendmomentsofmodenonthecorrespondingrotationsofmodem.Forthestandardclamped--,hinged--,orfree--endconditions,thesetermswillvanish.However,theycontributetotheorthogonalityrelationshipifthebeamhaselasticsupportsorifithasalumpedmassatitsend;thereforetheymustberetainedintheexpressionwhenconsideringsuchcases.（18-35）（18-40）三.振型的正交性振型可看作是慣性力幅值作為靜荷載所引起的靜力位移曲線。由虛功互等定理振型對質(zhì)量的正交性表達(dá)式物理意義為i振型上的慣性力在j振型上作的虛功為零。由變形體虛功定理振型對剛度的正交性表達(dá)式當(dāng)體系中有質(zhì)量塊、彈簧等時的情況Clough:振型對剛度的正交性表達(dá)式5.3受迫振動一.振型分解法設(shè)方程的解為運動方程為代入方程,得設(shè)注意到方程兩端乘以并積分----振型j的廣義質(zhì)量----振型j的廣義荷載方程兩端乘以并積分----振型j的廣義質(zhì)量----振型j的廣義荷載令----j振型阻尼比內(nèi)力計算若外力是集中力或集中力偶例：試求圖示梁跨中點穩(wěn)態(tài)振幅。已知：解：例：試求圖示梁跨中點穩(wěn)態(tài)振幅。已知：解：例：試求圖示梁跨中點穩(wěn)態(tài)振幅。解：二.初速度、初位移引起的振動設(shè)初位移、初速度已知，求位移反應(yīng)。設(shè)方程的解為由和確定例：桿件落到支座時的速度為v0，不反彈，不計阻尼，求位移。解：例：桿件落到支座時的速度為v0，不反彈，不計阻尼，求位移。解：練習(xí)題：振型分解法求圖示體系桿端轉(zhuǎn)角的穩(wěn)態(tài)幅值，不計阻尼。三.簡諧荷載作用下的直接解法運動方程為設(shè)特解為若梁是等截面梁，且q(x)為常數(shù)令例：試求圖示梁跨中點穩(wěn)態(tài)振幅，不計阻尼。已知：解：例：試求圖示梁跨中點穩(wěn)態(tài)振幅，不計阻尼。已知：解：例：試求圖示梁跨中點穩(wěn)態(tài)振幅，不計阻尼。已知：解：作業(yè)Clough：17-118-4練習(xí)題：試求桿端彎矩穩(wěn)態(tài)幅值，不計阻尼。5.4自振頻率的近似解法