结构动力学第II篇-硕士

PartⅡMulti­-Degree-­of-­FreedomSystems第Ⅱ篇多自由度体系Chapter9FormulationoftheMDOFEquationsofMotion第9章多自由度运动方程的建立ThediscussionpresentedinChapter8hasdemonstratedhowastructurecanberepresentedasaSDOFsystemthedynamicresponseofwhichcanbeevaluatedbythesolutionofasingledifferentialequationofmotion.Ifthephysicalpropertiesofthesystemaresuchthatitsmotioncanbedescribedbyasinglecoordinateandnoothermotionispossible,thenitactuallyisaSDOFsystemandthesolutionoftheequationprovidestheexactdynamicresponse.Ontheotherhand,ifthestructureactuallyhasmorethanonepossiblemodeofdisplacementanditisreducedmathematicallytoaSDOFapproximationbyassumingitsdeformedshape,thesolutionoftheequationofmotionisonlyanapproximationofthetruedynamicbehavior.9-­1SELECTIONOFTHEDEGREESOFFREEDOM

ThequalityoftheresultobtainedwithaSDOFapproximationdependsonmanyfactors,principallythespatialdistributionandtimevariationoftheloadingandthestiffnessandmasspropertiesofthestructure.Ifthephysicalpropertiesofthesystemconstrainittomovemosteasilywiththeassumedshape,andiftheloadingissuchastoexciteasignificantresponseinthisshape,theSDOFsolutionwillprobablybeagoodapproximation;otherwise,thetruebehaviormaybearlittleresemblancetothecomputedresponse.OneofthegreatestdisadvantagesoftheSDOFapproximationisthatitisdifficulttoassessthereliabilityoftheresultsobtainedfromit.

近似成SDOF体系的最大缺点之一是很难估计所得结果的可靠性。Ingeneral,thedynamicresponseofastructurecannotbedescribedadequatelybyaSDOFmodel;usuallytheresponseincludestimevariationsofthedisplacementshapeaswellasitsamplitude.Suchbehaviorcanbedescribedonlyintermsofmorethanonedisplacementcoordinate;thatis,themotionmustberepresentedbymorethanonedegreeoffreedom.AsnotedinChapter1,thedegreesoffreedominadiscrete-­parametersystemmaybetakenasthedisplacementamplitudesofcertainselectedpointsinthestructure,ortheymaybegeneralizedcoordinatesrepresentingtheamplitudesofaspecifiedsetofdisplacementpatterns.Inthepresentdiscussion,theformerapproachwillbeadopted;thisincludesboththefinite­-elementandthelumped­masstypeofidealization.Thegeneralized­-coordinateprocedurewillbediscussedinChapter16.Themotionofthisstructurewillbeassumedtobedefinedbythedisplacementsofasetofdiscretepointsonthebeam:v1(t),v2(t),:::,vi(t),:::,vN(t).Inprinciple,thesepointsmaybelocatedarbitrarilyonthestructure;inpractice,theyshouldbeassociatedwithspecificfeaturesofthephysicalpropertieswhichmaybesignificantandshouldbedistributedsoastoprovideagooddefinitionofthedeflectedshape.Thenumberofdegreesoffreedom(displacementcomponents)tobeconsideredislefttothediscretionoftheanalyst;greaternumbersprovidebetterapproximationsofthetruedynamicbehavior,butinmanycasesexcellentresultscanbeobtainedwithonlytwoorthreedegreesoffreedom.9-­2DYNAMIC­EQUILIBRIUMCONDITIONTheequationofmotionofthesystemofFig.9­1canbeformulatedbyexpressingtheequilibriumoftheeffectiveforcesassociatedwitheachofitsdegreesoffreedom.Ingeneralfourtypesofforceswillbeinvolvedatanypointi:theexternallyappliedloadpi(t)andtheforcesresultingfromthemotion,thatis,inertiafIi,dampingfDi,andelasticfSi.Thusforeachoftheseveraldegreesoffreedomthedynamicequilibriummaybeexpressedas(9-1)(9-2)Intheseexpressionsithasbeentacitlyassumedthatthestructuralbehaviorislinear,sothattheprincipleofsuperpositionapplies.Thecoefficientskijarecalledstiffnessinfluencecoefficients,definedasfollows:(9­4)Inmatrixform,thecompletesetofelastic­forcerelationshipsmaybewritten(9­6)……SubstitutingEqs.(9­-6),(9­-9),and(9­-12)intoEq.(9­-2)givesthecompletedynamicequilibriumofthestructure,consideringalldegreesoffreedom:(9-13)ThisequationistheMDOFequivalentofEq.(2-­3);eachtermoftheSDOFequationisrepresentedbyamatrixinEq.(9­-13),theorderofthematrixcorrespondingtothenumberofdegreesoffreedomusedindescribingthedisplacementsofthestructure.Thus,Eq.(9­-13)expressestheNequationsofmotionwhichservetodefinetheresponseoftheMDOFsystem.阻尼、质量dampinginfluencecoefficientsmassinfluencecoefficients(9-15)9­-3AXIAL-­FORCEEFFECTSItwasobservedinthediscussionofSDOFsystemsthataxialforcesoranyloadwhichmaytendtocausebucklingofastructuremayhaveasignificanteffectonthestiffnessofthestructure.SimilareffectsmaybeobservedinMDOFsystems;theforcecomponentactingparalleltotheoriginalaxisofthemembersleadstoadditionalloadcomponentswhichactinthedirection(andsense)ofthenodaldisplacementsandwhichwillbedenotedby

.Whentheseforcesareincluded,thedynamic­-equilibriumexpression,Eq.(9­-2),becomesChapter10EvaluationofStructural­PropertyMatrices第10章结构特性矩阵的计算10-­1ELASTICPROPERTIESFlexibility10-1弹性特性Beforediscussingtheelastic­stiffnessmatrixexpressedinEq.(9­5),itwillbeusefultodefinetheinverseflexibilityrelationship.Thedefinitionofaflexibilityinfluencecoefficientis=deflectionofcoordinateiduetounitloadappliedtocoordinatej

(10­1)ForthesimplebeamshowninFig.10­1,thephysicalsignificanceofsomeoftheflexibilityinfluencecoefficientsassociatedwithasetofvertical­displacementdegreesoffreedomisillustrated.Horizontalorrotationaldegreesoffreedommightalsohavebeenconsidered,inwhichcaseitwouldhavebeennecessarytousethecorrespondinghorizontalorrotationalunitloadsindefiningthecompletesetofinfluencecoefficients;however,itwillbeconvenienttorestrictthepresentdiscussiontotheverticalmotions.FIGURE10-1Definitionofflexibilityinfluencecoefficients.:结构的柔度矩阵(10-4)10-1弹性特性10­-1ELASTICPROPERTIESFIGURE10-2Definitionofstiffnessinfluencecoefficients.Betti定律柔度矩阵、刚度矩阵：对称Finite­-ElementStiffnessInprinciple,theflexibilityorstiffnesscoefficientsassociatedwithanyprescribedsetofnodaldisplacementscanbeobtainedbydirectapplicationoftheirdefinitions.Inpractice,however,thefinite­elementconcept,describedinChapter1,frequentlyprovidesthemostconvenientmeansforevaluatingtheelasticproperties.Bythisapproachthestructureisassumedtobedividedintoasystemofdiscreteelementswhichareinterconnectedonlyatafinitenumberofnodalpoints.Thepropertiesofthecompletestructurearethenfoundbyevaluatingthepropertiesoftheindividualfiniteelementsandsuperposingthemappropriately.FIGURE10-4Beamdeflectionsduetounitnodaldisplacementsatleftend.(10-22)Example9.6Formulatethefreevibrationequationsforthetwo-elementframeofFig.E9.6a.ForbothelementstheflexuralstiffnessisEI,andaxialdeformationsaretobeneglected.Theframeismasslesswithlumpedmassesatthetwonodesasshown.10­-2MASSPROPERTIES10-2质量特性Lumped­-MassMatrixThesimplestprocedurefordefiningthemasspropertiesofanystructureistoassumethattheentiremassisconcentratedatthepointsatwhichthetranslationaldisplacementsaredefined.Forasysteminwhichonlytranslationaldegreesoffreedomaredefined,thelumped­massmatrixhasadiagonalform;forthesystemofFig.10­6itwouldbeinwhichthereareasmanytermsastherearedegreesoffreedom.Theoff­diagonaltermsmijofthismatrixvanishbecauseanaccelerationofanymasspointproducesaninertialforceatthatpointonly.Theinertialforceatiduetoaunitaccelerationofpointiisobviouslyequaltothemassconcentratedatthatpoint;thusthemassinfluencecoefficientmii=miinalumped­masssystem.inwhichthereareasmanytermsastherearedegreesoffreedom.Theoff­-diagonaltermsmijofthismatrixvanishbecauseanaccelerationofanymasspointproducesaninertialforceatthatpointonly.Theinertialforceatiduetoaunitaccelerationofpointiisobviouslyequaltothemassconcentratedatthatpoint;thusthemassinfluencecoefficientmii=miinalumped-­masssystem.Consistent-­MassMatrixMakinguseofthefinite-­elementconcept,itispossibletoevaluatemassinfluencecoefficientsforeachelementofastructurebyaproceduresimilartotheanalysisofelementstiffnesscoefficients.FIGURE10-7Nodesubjectedtorealangularaccelerationandvirtualtranslation.Example9.2Auniformrigidbaroftotalmassmissupportedontwospringsk1andk2atthetwoendsandsubjectedtodynamicforcesshowninFig.E9.2a.Thebarisconstrainedsothatitcanmoveonlyverticallyintheplaneofthepaper;withthisconstraintthesystemhastwoDOFs.Example9.3FormulatetheequationsofmotionofthesystemofFig.E9.2awiththetwoDOFsdefinedatthecenterofmassOoftherigidbar:translationutandrotationuθ(Fig.E9.3a).FigureE9.3FigureE9.210-­6CHOICEOFPROPERTYFORMULATIONIntheprecedingdiscussion,twodifferentlevelsofapproximationhavebeenconsideredfortheevaluationofthemass,elastic­stiffness,geometric­stiffness,andexternal­loadproperties:(1)anelementaryapproachtakingaccountonlyofthetranslationaldegreesoffreedomofthestructureand(2)a“consistent”approach,whichaccountsfortherotationalaswellastranslationaldisplacements.Theelementaryapproachisconsiderablyeasiertoapply;notonlyaretheelementpropertiesdefinedmoresimplybutthenumberofcoordinatestobeconsideredintheanalysisismuchlessforagivenstructuralassemblage.Inprinciple,theconsistentapproachshouldleadtogreateraccuracyintheresults,butinpracticetheimprovementisoftenslight.Apparentlytherotationaldegreesoffreedomaremuchlesssignificantintheanalysisthanthetranslationalterms.Theprincipaladvantageoftheconsistentapproachisthatalltheenergycontributionstotheresponseofthestructureareevaluatedinaconsistentmanner,whichmakesitpossibletodrawcertainconclusionsregardingboundsonthevibrationfrequency;however,thisadvantageseldomoutweighstheadditionaleffortrequired.Theelementarylumped­massapproachpresentsaspecialproblemwhentheelastic­stiffnessmatrixhasbeenformulatedbythefinite­elementapproachorbyanyotherprocedurewhichincludestherotationaldegreesoffreedominthematrix.Iftheevaluationofalltheotherpropertieshasexcludedthesedegreesoffreedom,itisnecessarytoexcludethemalsofromthestiffnessmatrixbeforetheequationsofmotioncanbewritten.Theprocessofeliminatingtheseunwanteddegreesoffreedomfromthestiffnessmatrixiscalledstaticcondensation.partitionedformtranslationalelasticstiffness作业Chopra教材：9.5、9.6Clough教材：10-5~10-8多自由度体系自由振动多自由度结构体系运动方程的一般形式：柔度矩阵表示：刚度矩阵表示：无阻尼多自由度结构体系自由振动方程：无阻尼多自由度结构体系运动方程：质量矩阵刚度矩阵阻尼矩阵柔度矩阵位移向量等效荷载向量荷载位移向量无阻尼自由振动—振动频率分析略去阻尼矩阵和施加的荷载向量的影响：假定以上多自由度体系的振动是简谐振动：表示体系的形状，不随时间变化。无阻尼自由振动—振动频率分析设：上式的N个根，表述体系可能存在的N个振型的频率。可以证明，稳定的结构体系具有实的、对称的、正定的质量和刚度矩阵，频率方程所有的根都是实的和正的。称为频率方程或特征方程。将频率方程展开，可得到一个关于w2

的n次代数方程。从频率方程可解得n个正实根；

开方得到各阶频率，记作：如果方程存在非零解，则系数行列式必为零，即：频率谱：w1<w2<…<wn，分别称作第1阶频率、第2阶频率…第n

阶频率。动力学问题转变为矩阵求特征值问题。频率向量：将频率方程展开，可得到关于w2

的n次代数方程。从频率方程可解得n个正实根；

开方得到各阶频率：频率方程：

自由振动方程：Nextstep?求v——

振型（Modeshapes）。无阻尼自由振动—振型分析则：上式中，振型的幅值不能确定。振动体系的形状可以按照任何一个坐标所表示的各点位移来确定。振型可理解为各自由度幅值的相对值！将运动方程写成与频率有关，因此对每一个振型都是不相同的假定位移向量的第一个元素是一个单位幅值无阻尼自由振动—振型分析展开：从而：无阻尼自由振动—振型分析即：故：以上为求解第n阶频率对应振型的方法。∴

第i个振型方程中的n个方程中只有n-1个是独立的！——无法得到j1i、

j2i、…、jni

的确定值，但可以确定各质点振幅之间的相对比值：

——

振型的幅值是任意的，但形状是惟一的。

∵j

称为振型矩阵；ji称为对应于第i阶频率wi的主振型，简称第i阶振型；为了描述振型的形状，进行规格化处理；振型规格化处理方式很多，原则：保持形状不变！最简单可取ji的第一个元素j1i=1；

振型方程：（4-15）按j1i=1进行振型规格化：得到按j1i=1规格化的振型：对于有n个自由度的体系，可以得到n个线性无关的主振型：规格化的主振型矩阵：无阻尼多自由度结构体系自由振动方程：第i阶振型的特解：用规格化振型表示成：这样的特解有n个！振型的物理意义定义:体系上所有质量按相同频率作自由振动时的振动形状称作体系的主振型。以任意初始条件开始，每个质量的运动不是简谐运动，运动的频率不能够定义，挠曲形状随时间变化。两自由度体系，存在两个特征挠曲形状，若将体系置于这些挠曲形状中的任何一个后释放，体系将以简谐运动形式振动，并维持初始挠曲形状，两个楼层同时达到最大位移，并同时通过平衡位置。无阻尼自由振动—振型分析将N个振型中的每一振型形式，用F表示N个振型所组成的方阵。以上矩阵为结构的振型矩阵，为一N*N方阵。无阻尼自由振动—振动分析的柔度法各项前乘，可得：即：注意：即使质量矩阵和柔度矩阵都是对称的，它们的乘积也是不对称的！求解结构特征值的另一种方法：解:例一.求图示体系的频率、振型.

已知:m1m211.61810.618例二.求图示体系的频率、振型解令1111第一振型第二振型1111第一振型第二振型对称体系的振型分成两组:一组为对称振型一组为反对称振型1111第一振型第二振型对称系的振型分成两组:一组为对称振型一组为反对称振型按对称振型振动=1l/3按反对称振型振动11第二振型对称系的振型分成两组:一组为对称振型一组为反对称振型按对称振型振动=1l/3按反对称振型振动对称系的振型分成两组:一组为对称振型一组为反对称振型按对称振型振动=1l/3按反对称振型振动=1l/9例三.求图示体系的频率、振型解:令例三.求图示体系的频率、振型解:令例三.求图示体系的频率、振型解:令无阻尼自由振动—轴向力的影响频率方程：自由振动情况：此时，只须将组合刚度中的矩阵代替弹性刚度矩阵，分析方法如前所述，对于任何给定的轴向荷载都可以计算其几何刚度以及组合刚度。体系在轴向压力作用下，减小了结构的有效刚度，振动频率亦因此降低。。无阻尼自由振动—轴向力的影响引入基准荷载作用下的几何刚度：屈曲荷载：若振动频率为零，则：实际上，只有第一阶屈曲荷载以及形状才是有意义的。无阻尼自由振动—轴向力的影响动力平衡方程：简谐振动的屈曲：若结构受外力作用定义：无阻尼自由振动—轴向力的影响若允许荷载向量的幅值趋近于零：零轴向荷载条件引起不受力结构按自振频率振动屈曲。It’sinterestingtonotethatazero-axial-loadconditioncauses“buckling”attheunstressednatural-vibrationfrequencyaccordingtothisdefinition.If：两个n维向量A1和A2存在如下关系：称向量A1和A2正交。If：存在一个方阵B，使得：称向量A1和A2加权正交。称向量A1和A2对矩阵B正交。B称为权矩阵。无阻尼自由振动—正交条件无阻尼自由振动—正交条件由教材图11-1，m处惯性力在n处产生的挠度，等于n处惯性力在m处产生的挠度：惯性力等于弹性力：则：又因为：无阻尼自由振动—正交条件故：当时，在二个振型频率不相同情况下，上述正交条件成立！同理：当时，正交条件仅对二个振型频率不相同情况适用，而对具有相同频率的两个振型，不适用。振型正交性i振型i振型上的惯性力j振型i振型上的惯性力在j振型上作的虚功j振型上的惯性力在i振型上作的虚功由虚功互等定理i振型上的惯性力在j振型上作的虚功j振型上的惯性力在i振型上作的虚功由虚功互等定理振型对质量的正交性的物理意义i振型上的惯性力在j振型上作的虚功等于0振型对刚度的正交性:振型对质量的正交性的物理意义i振型上的惯性力在j振型上作的虚功等于0振型对刚度的正交性:振型对刚度的正交性的物理意义i振型上的弹性力在j振型上作的虚功等于0振型正交性的应用1.检验求解出的振型的正确性。例:试验证振型的正确性2.对耦联运动微分方程组作解耦运算等等.例:已知图示体系的第一振型,

试求第二振型.解:例:已知图示体系在动荷载作用下的振幅为解:试从其中去掉第一振型分量.无阻尼自由振动—振型的规格化特征值问题的解得到的振型幅值是任意的，任何幅值都满足基本频率方程，只有振型的形状是唯一的。一个自由度的幅值取1，并以这个指定值为基准确定其他位移，这叫做关于特定坐标的振型的规格化。另一种规格化方法是，取最大的一个振幅为1，而不取特定的坐标值。最常用的规格化方法，是调整每个振型振幅，使满足无阻尼自由振动—振型的规格化由于，只需将振型除以即可。此外，由于此方法规格化的振型叫做对应于质量矩阵正交规格化振型。作业Chopra教材：10.6、10.8Clough教材：11-1、11-6第12章动力反应分析——叠加法§12-1正规坐标IntheprecedingdiscussionofanarbitraryN­DOFlinearsystem,thedisplacedpositionwasdefinedbytheNcomponentsinthevectorv.However,forthepurposeofdynamic­responseanalysis,itisoftenadvantageoustoexpressthispositionintermsofthefree­vibrationmodeshapes.TheseshapesconstituteNindependentdisplacementpatterns,theamplitudesofwhichmayserveasgeneralizedcoordinatestoexpressanysetofdisplacements.ThemodeshapesthusservethesamepurposeasthetrigonometricfunctionsinaFourierseries,andtheyareusedforthesamereasons;because:(1)theypossessorthogonalitypropertiesand(2)theyareefficientinthesensethattheyusuallycandescribeallNdisplacementswithsufficientaccuracyemployingonlyafewshapes.NormalCoordinates若特征值互异（），由线性代数可知，不同特征值对应的特征向量是线性无关的，因此n个振型是线性无关的。任意n个线性无关的向量都可以构成n维空间的一组基底，任何一个n维向量都可以用这组基底的线性组合表示。FIGURE12-1Representingdeflectionsassumofmodalcomponents.Inthisequation,itisapparentthattheN×Nmode­shapematrixservestotransformthegeneralizedcoordinatevectorY

tothegeometriccoordinatevectorv.ThegeneralizedcomponentsinvectorY

arecalledthenormalcoordinatesofthestructure.TheproceduredescribedabovecanbeusedtoobtainanindependentSDOFequationforeachmodeofvibrationoftheundampedstructure.ThustheuseofthenormalcoordinatesservestotransformtheequationsofmotionfromasetofNsimultaneousdifferentialequations,whicharecoupledbytheoff­-diagonaltermsinthemassandstiffnessmatrices,toasetofNindependentnormal-­coordinateequations.Thedynamicresponsethereforecanbeobtainedbysolvingseparatelyfortheresponseofeachnormal(modal)coordinateandthensuperposingthesebyEq.(12­-3)toobtaintheresponseintheoriginalgeometriccoordinates.Thisprocedureiscalledthemode­-superpositionmethod,ormorepreciselythemodedisplacementsuperpositionmethod.§12-2非耦合的运动方程：无阻尼12­2UncoupledEquationsofMotion:Undamped振型的正交性解耦振型叠加法§12-3非耦合的运动方程：粘滞阻尼12­3UncoupledEquationsofMotion:ViscousDamping有阻尼运动方程的解耦！！Aswasnotedearlier,itgenerallyismoreconvenientandphysicallyreasonabletodefinethedampingofaMDOFsystemusingthedampingratioforeachmodeinthiswayratherthantoevaluatethecoefficientsofthedampingmatrixcbecausethemodaldampingratiosncanbedeterminedexperimentallyorestimatedwithadequateprecisioninmanycases.§12-4用振型位移叠加法进行反应分析12­-4ResponseAnalysisbyModeDisplacementSuperposition振型位移叠加法的要点！！会计算！！ViscousDampingComplex­-StiffnessDamping（12-16）（12-17）whichsuperposestheseparatemodaldisplacementcontributions;hence,thecom­monlyreferredtonamemodesuperpositionmethod.Itshouldbenotedthatformosttypesofloadingsthedisplacementcontributionsgenerallyaregreatestforthelowermodesandtendtodecreaseforthehighermodes.Consequently,itusuallyisnotnecessarytoincludeallthehighermodesofvibrationinthesuperpositionprocess;theseriescanbetruncatedwhentheresponsehasbeenobtainedtoanydesireddegreeofaccuracy.Moreover,itshouldbekeptinmindthatthemathematicalidealizationofanycomplexstructuralsystemalsotendstobelessreliableinpredictingthehighermodesofvibration;forthisreason,too,itiswelltolimitthenumberofmodesconsideredinadynamic-­responseanalysis.(12-29)Thedisplacementtime­-historiesinvectorv(t)maybeconsideredtobethebasicmeasureofastructure'soverallresponsetodynamicloading.Ingeneral,otherresponseparameterssuchasstressesorforcesdevelopedinvariousstructuralcomponentscanbeevaluateddirectlyfromthedisplacements.Forexample,theelasticforcesfSwhichresistthedeformationofthestructurearegivendirectlybyBecauseeachmodalcontributionismultipliedbythesquareofthemodalfrequencyinEq.(12­-33),itisevidentthatthehighermodesareofgreatersignificanceindefiningtheforcesinthestructurethantheyareinthedisplacements.Consequently,itwillbenecessarytoincludemoremodalcomponentstodefinetheforcestoanydesireddegreeofaccuracythantodefinethedisplacements.12­-5ConstructionofProportionalViscousDampingMatrices§12-5比例粘滞阻尼矩阵的建立RayleighDampingAswasstatedabove,generallythereisnoneedtoexpressthedampingofatypicalviscouslydampedMDOFsystembymeansofthedampingmatrixbecauseitisrepresentedmoreconvenientlyintermsofthemodaldampingratios().However,inatleasttwodynamicanalysissituationstheresponseisnotobtainedbysuperpositionoftheuncoupledmodalresponses,sothedampingcannotbeexpressedbythedampingratios—insteadanexplicitdampingmatrixisneeded.Thesetwosituationsare:

nonlinearresponses,forwhichthemodeshapesarenotfixedbutarechangingwithchangesofstiffness,and(2)analysisofalinearsystemhavingnonproportionaldamping.Inbothofthesecircumstances,themosteffectivewaytodeterminetherequireddampingmatrixistofirstevaluateoneormoreproportionaldampingmatrices.Inperforminganonlinearanalysis,itisappropriatetodefinetheproportionaldampingmatrixfortheinitialelasticstateofthesystem(beforenonlineardeformationshaveoccurred)andtoassumethatthisdampingpropertyremainsconstantduringtheresponseeventhoughthestiffnessmaybechangingandcausinghystereticenergylossesinadditiontotheviscousdampinglosses.Incaseswherethedampingisconsideredtobenonproportional,anappropriatedampingmatrixcanbeconstructedbyassemblingasetofsuitablyderivedproportionaldampingmatrices,asexplainedlaterinthissection.Thusforthesetwosituations,itisnecessarytobeabletoderiveappropriateproportionaldampingmatrices.12­5ConstructionofProportionalViscousDampingMatrices§12-5比例粘滞阻尼矩阵的建立12­5ConstructionofProportionalViscousDampingMatrices§12-5比例粘滞阻尼矩阵的建立RayleighDampingFIGURE12-2Relationshipbetweendampingratioandfrequency(forRayleighdamping).12­5ConstructionofProportionalViscousDampingMatrices§12-5比例粘滞阻尼矩阵的建立ExtendedRayleighDampingAlternativeFormulationConstructionofNonproportionalDampingMatricesTheproportionaldampingmatricesdescribedintheprecedingparagraphsaresuitableformodelingthebehaviorofmoststructuralsystems,inwhichthedampingmechanismisdistributedratheruniformlythroughoutthestructure.However,forstructuresmadeupofmorethanasingletypeofmaterial,wherethedifferentmaterialsprovidedrasticallydifferingenergy­lossmechanismsinvariouspartsofthestructure,thedistributionofdampingforceswillnotbesimilartothedistributionoftheinertialandelasticforces;inotherwords,theresultingdampingwillbenonproportional.12­6ResponseAnalysisUsingCoupledEquationsofMotion§12-6采用耦合运动方程的反应分析ModesuperpositionisaveryeffectivemeansofevaluatingthedynamicresponseofstructureshavingmanydegreesoffreedombecausetheresponseanalysisisperformedonlyforaseriesofSDOFsystems.However,thecomputationalcostinthistypeofcalculationistransferredfromtheMDOFdynamicanalysistothesolutionoftheNdegreeoffreedomundampedeigenproblemfollowedbythemodalcoordinatetransformation,whichmustbedonebeforetheindividualmodalresponsescanbeevaluated.Certainlytheeigenproblemsolutionrepresentsthemajorpartofthecostofatypicalmodesuperpositionanalysis,butalsoitmustberecalledthattheequationsofmotionwillbeuncoupledbytheresultingundampedmodeshapesonlyifthedampingisrepresentedbyaproportionaldampingmatrix.12­6ResponseAnalysisUsingCoupledEquationsofMotion§12-6采用耦合运动方程的反应分析Forthesereasonsitisusefultoexaminethepossibilityofavoidingthemodalcoordinatetransformationbycarryingoutthedynamicresponseanalysisdirectlyintheoriginalgeometriccoordinateequationsofmotion.Oneapproachtothesolutionofthissetofcoupledequationsthatoftenmaybeworthconsiderationisthestep-­by­-stepprocedure,asisdescribedinChapter15.However,forlinearsystemstowhichsuperpositionisapplicable,amoreconvenientsolutionmaybeobtainedbyFouriertransform(frequency­domain)procedures,aswellas—atleastinprinciple—byapplyingconvolutionintegral(time­domain)methods;theseMDOFproceduresareanalogoustothecorrespondingmethodsdescribedpreviouslyforSDOFsystems.Abriefconceptualdescriptionofthesetechniquesfollows;however,theconvolutionintegralapproachisnotgenerallysuitableforpracticaluse,anditisnotdiscussedfurtherafterthisbriefdescription.作业Clough教材：12-4~12-7第13章振动分析的矩阵迭代法§13­1PreliminarycommentsChapter13VibrationanalysisbymatrixiterationItisevidentfromtheprecedingdiscussionthatthemodedisplacementsuperpositionmethodprovidesanefficientmeansofevaluatingthedynamicresponseofmoststructures—thoseforwhichtheundampedmodeshapesservetouncoupletheequationsofmotion.Theresponseanalysisfortheindividualmodalequationsrequiresverylittlecomputationaleffort,andinmostcasesonlyarelativelysmallnumberofthelowestmodesofvibrationneedbeincludedinthesuperposition.Inthisregard,itisimportanttorealizethatthephysicalpropertiesofthestructureandthecharacteristicsofthedynamicloadinggenerallyareknownonlyapproximately;hencethestructuralidealizationandthesolutionprocedureshouldbeformulatedtoprovideonlyacorrespondinglevelofaccuracy.Nevertheless,themathematicalmodelsdevelopedtosolvepracticalproblemsinstructuraldynamicsrangefromverysimplifiedsystemshavingonlyafewdegreesoffreedomtohighlysophisticatedfiniteelementmodelsincludinghundredsoreventhousandsofdegreesoffreedominwhichasmanyas50to100modesmaycontributesignificantlytotheresponse.Todealeffectivelywiththesepracticalproblems,muchmoreefficientmeansofvibrationanalysisareneededthanthedeterminantalsolutionproceduredescribedearlier,andthischapterdescribesthematrixiterationapproachwhichisthebasisofmanyofthevibrationor“eigenproblem”solutiontechniquesthatareusedinpractice.Thebasicconceptisexplainedfirstwithreferencetothesimplestapplication,theevaluationofthefundamental(orfirst­)modeshapeandfrequencyofanN­-degree-­of­freedomsystem.Thisisfollowedbyaproofofthefactthattheiterationwillconvergetothefirst-­modeproperties;theessentialconceptoftheproofisthenusedasameansforevaluatingthehighermodesofvibration,onemodeatatimeinsequence.Becausethisprocedureinvolvesincreasingcomputationalcostsasmoremodesarecalculated,analternativemethodthatemploys“shifting”oftheeigenvalues(frequencies)isdescribed.Alsoincludedisabriefdiscussionofelasticbuckling,notingthatboththevibrationsandbucklingarerepresentedbyequivalenteigenproblemequations.4.2迭代法

对于给定的方阵,满足上式的向量和数值称作的特征向量和特征值，合称为特征对.有限自由度体系求频率、振型，属于矩阵特征值问题。---标准特征值问题---广义特征值问题柔度法建立的振型方程令---动力矩阵---标准特征值问题刚度法建立的振型方程---广义特征值问题一.迭代法求基频和基本振型1.作法假设振型,计算,若是真的振型,则下式成立即与成比例.柔度法建立的振型方程令---动力矩阵---标准特征值问题若不成比例,不是振型.迭代式为这时将归一化,得;再将其作为新的假设振型继续计算.一直算到