PartIIIDistributed--ParameterSystems第III篇分布參數(shù)體系Chapter17Partialdifferentialequationsofmotion34567891011121314151617181920212223242526（1）國內(nèi)教材(點擊閱讀)劉晶波、杜修力--結(jié)構(gòu)動力學(xué)俞載道--結(jié)構(gòu)動力學(xué)基礎(chǔ)于建華--高等結(jié)構(gòu)動力學(xué)邱吉寶--計算結(jié)構(gòu)動力學(xué)李明昭--橋梁結(jié)構(gòu)動力分析

胡宗武--工程振動分析基礎(chǔ)胡兆同--結(jié)構(gòu)振動與穩(wěn)定胡少偉--結(jié)構(gòu)振動理論及其應(yīng)用包世華--結(jié)構(gòu)動力學(xué)彭俊生--結(jié)構(gòu)動力學(xué)、抗震計算與SAP2000應(yīng)用盛宏玉--結(jié)構(gòu)動力學(xué)輔導(dǎo)與習題精解盛宏玉--結(jié)構(gòu)動力學(xué)

（第二版）王光遠--應(yīng)用分析動力學(xué)（2）Clough教材資料(點擊閱讀)克拉夫--結(jié)構(gòu)動力學(xué)(81版)結(jié)構(gòu)動力學(xué)（第2版）中文版克拉夫--DynamicsofStructure（英文原版）結(jié)構(gòu)動力學(xué)習題詳解(Clough版)（3）Chopra教材資料(點擊閱讀)Chopra--結(jié)構(gòu)動力學(xué)_理論及其在地震工程中的應(yīng)用（第2版）(中文版)Chopra--DynamicsofStructures(4thEdition，英文原版)Chopra--(入門)結(jié)構(gòu)動力學(xué)入門Chopra--(入門)DynamicsofStructures：APrimerChopra哈工大結(jié)構(gòu)動力學(xué)講義結(jié)構(gòu)動力學(xué)學(xué)習資料下載鏈接：/post/109.html272024/2/25蘭州理工大學(xué)土木工程學(xué)院韓建平3017--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式當體系中有質(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，