结构动力学第III篇-硕士知识资料P17知识资料石岩

PartIIIDistributed-­ParameterSystems第III篇分布参数体系Chapter17Partialdifferentialequationsofmotion34567891011121314151617181920212223242526（1）国内教材(点击阅读)刘晶波、杜修力--结构动力学俞载道--结构动力学基础于建华--高等结构动力学邱吉宝--计算结构动力学李明昭--桥梁结构动力分析

胡宗武--工程振动分析基础胡兆同--结构振动与稳定胡少伟--结构振动理论及其应用包世华--结构动力学彭俊生--结构动力学、抗震计算与SAP2000应用盛宏玉--结构动力学辅导与习题精解盛宏玉--结构动力学

（第二版）王光远--应用分析动力学（2）Clough教材资料(点击阅读)克拉夫--结构动力学(81版)结构动力学（第2版）中文版克拉夫--DynamicsofStructure（英文原版）结构动力学习题详解(Clough版)（3）Chopra教材资料(点击阅读)Chopra--结构动力学_理论及其在地震工程中的应用（第2版）(中文版)Chopra--DynamicsofStructures(4thEdition，英文原版)Chopra--(入门)结构动力学入门Chopra--(入门)DynamicsofStructures：APrimerChopra哈工大结构动力学讲义结构动力学学习资料下载链接：/post/109.html272024/2/25兰州理工大学土木工程学院韩建平3017-­1INTRODUCTIONThediscrete­-coordinatesystemsdescribedinPartTwoprovideaconvenientandpracticalapproachtothedynamicresponseanalysisofarbitrarystructures.However,thesolutionsobtainedcanonlyapproximatetheiractualdynamicbehaviorbecausethemotionsarerepresentedbyalimitednumberofdisplacementcoordinates.Theprecisionoftheresultscanbemadeasrefinedasdesiredbyincreasingthenumberofdegreesoffreedomconsideredintheanalyses.Inprinciple,however,aninfinitenumberofcoordinateswouldberequiredtoconvergetotheexactresultsforanyrealstructurehavingdistributedproperties;hencethisapproachtoobtaininganexactsolutionismanifestlyimpossible.Theformalmathematicalprocedureforconsideringthebehaviorofaninfinitenumberofconnectedpointsisbymeansofdifferentialequationsinwhichthepositioncoordinatesaretakenasindependentvariables.Inasmuchastimeisalsoanindependentvariableinadynamicresponseproblem,theformulationoftheequationsofmotioninthiswayleadstopartialdifferentialequations.Differentclassesofcontinuoussystemscanbeidentifiedinaccordancewiththenumberofindependentvariablesrequiredtodescribethedistributionoftheirphysicalproperties.Forexam­ple,thewave­-propagationformulasusedinseismologyandgeophysicsarederivedfromtheequationsofmotionexpressedforgeneralthree-­dimensionalsolids.Simi­larly,instudyingthedynamicbehaviorofthin­plateorthin­-shellstructures,specialequationsofmotionmustbederivedforthesetwo-­dimensionalsystems.Inthepresentdiscussion,however,attentionwillbelimitedtoone-­dimensionalstructures,thatis,beam­-androd­-typesystemswhichmayhavevariablemass,damping,andstiffnesspropertiesalongtheirelasticaxes.Thepartialdifferentialequationsofthesesystemsinvolveonlytwoindependentvariables:timeanddistancealongtheelasticaxisofeachcomponentmember.Itispossibletoderivetheequationsofmotionforrathercomplexone­-dimensionalstructures,includingassemblagesofmanymembersinthree­dimensionalspace.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.Chapter18Analysisofundampedfreevibration18­1BEAMFLEXURE:ELEMENTARYCASEFollowingthesamegeneralapproachemployedwithdiscrete­parametersys­tems,thefirststepinthedynamic­-responseanalysisofadistributed­-parametersystemistoevaluateitsundampedmodeshapesandfrequencies.Becauseofthemathematicalcomplicationsoftreatingsystemshavingvariableproperties,thefollowingdiscussionwillbelimitedtobeamshavinguniformpropertiesalongtheirlengthsandtoframesassembledfromsuchmembers.Thisisnotaseriouslimitation,however,becauseitismoreefficienttotreatanyvariable-­propertysystemsusingdiscrete­parametermodeling.（17-7）（18-1）（18-2）（18-3）First,letusconsidertheelementarycasepresentedinSection17-­2withandsetequaltoconstantsand,respectively.AsshownbyEq.(17­-7),thefree­-vibrationequationofmotionforthissystemisExampleE18­1.SimpleBeamConsideringtheuniformsimplebeamshowninFig.E18­1a,itsfourknownboundaryconditionsareFIGUREE18-1Simplebeam-vibrationanalysis:(a)basicpropertiesofsimplebeam;(b)firstthreevibrationmodes.第五章无限自由度体系的振动分析5.1运动方程的建立一.弯曲振动方程微段平衡方程挠曲微分方程消去内力,得加惯性力,得运动方程二.考虑轴力对弯曲的影响时的弯曲振动方程三.考虑剪切变形与惯性力矩对弯曲的影响时的弯曲振动方程1.考虑剪切变形时的几何方程杆轴转角截面转角2.惯性力矩的计算单位长度上的惯性力矩3.运动方程4.物理方程5.方程整理几何方程：物理方程：运动方程：对于等截面杆：对于等截面细长杆：四.考虑阻尼影响时的弯曲振动方程外阻尼力内阻尼力1.粘滞阻尼

2.滞变阻尼不计阻尼时计阻尼时习题：1.求剪切杆的运动方程。

2.求拉压杆的运动方程。一.运动方程及其解边界条件xyxyxy几何边界条件力边界条件混合边界条件初始条件已知函数5.2自由振动分析设方程的特解为代入方程，得方程(1)的通解为运动方程的特解为运动方程的通解由特解的线性组合确定设方程(2)的特解为代入方程(2)，得方程(2)的通解为或二.振型与频率振型方程xy频率方程振型18­-4BEAMFLEXURE:ORTHOGONALITYOFVIBRATIONMODESHAPESThevibrationmodeshapesderivedforbeamswithdistributedpropertieshaveorthogonalityrelationshipsequivalenttothosedefinedpreviouslyforthediscrete­parametersystems,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惯性力幅值作为静荷载所引起的静力位移曲线。由虚功互等定理振型对质量的正交性表达式物理意义为i振型上的惯性力在j振型上作的虚功为零。由变形体虚功定理振型对刚度的正交性表达式当体系中有质量块、弹簧等时的情况Clough:振型对刚度的正交性表达式5.3受迫振动一.振型分解法设方程的解为运动方程为代入方程,得设注意到方程两端乘以并积分----振型j的广义质量----振型j的广义荷载方程两端乘以并积分----振型j的广义质量----振型j的广义荷载令----j振型阻尼比内力计算若外力是集中力或集中力偶例：试求图示梁跨中点稳态振幅。已知：解：例：试求图示梁跨中点稳态振幅。已知：解：例：试求图示梁跨中点稳态振幅。解：二.初速度、初位移引起的振动设初位移、初速度已知，求位移反应。设方程的解为由和确定例：杆件落到支座时的速度为v0，不反弹，不计阻尼，求位移。解：例：杆件落到支座时的速度为v0，不反弹，不计阻尼，求位移。解：练习题：振型分解法求图示体系杆端转角的稳态幅值，不计阻尼。三.简谐荷载作用下的直接解法运动方程为设特解为若梁是等截面梁，且q(x)为常数令例：试求图示梁跨中点稳态振幅，不计阻尼。已知：解：例：试求图示梁跨中点稳态振幅，不计阻尼。已知：解：例：试求图示梁跨中点稳态振幅，