結(jié)構(gòu)動(dòng)力學(xué)第I篇-碩士

結(jié)構(gòu)動(dòng)力學(xué)主講：韓建平教授、博導(dǎo)Email：jphan@,jphan1970@163.com電話2973781（辦）辦公室：本部綜合試驗(yàn)樓B-420室、8號(hào)樓211室TA:劉宗成結(jié)構(gòu)動(dòng)力學(xué)的主要內(nèi)容？結(jié)構(gòu)動(dòng)力學(xué)知識(shí)如何體現(xiàn)、反映在結(jié)構(gòu)分析與設(shè)計(jì)中？為什么要學(xué)習(xí)結(jié)構(gòu)動(dòng)力學(xué)？

結(jié)構(gòu)動(dòng)力學(xué)：研究結(jié)構(gòu)在動(dòng)力荷載作用下的動(dòng)力反應(yīng)。例如：地震荷載風(fēng)荷載沖擊荷載等2008“5.12”汶川地震，MW7.9、MS8.0

2010Haitiearthquake，MW7.9

2010Chileearthquake，MW8.8

2010青海玉樹(shù)地震，MW6.9、MS7.1

2010NewZealand

earthquake，MW7.1

2011NewZealand

earthquake，MW6.3

2011年日本東北部地震及海嘯，MW9.0

2013年7月22日，甘肅岷縣漳縣地震，MW5.9，MS6.6：

95人死亡，災(zāi)害損失300多億引言The2010Haitiearthquakewasacatastrophicmagnitude7.0Mwearthquake.ItsepicentrewasnearthetownofLéogane,approximately25

km(16miles)westofPort-au-Prince,Haiti'scapital.Theearthquakeoccurredat16:53localtime(21:53UTC)onTuesday,12January2010.By24January,atleast52aftershocksmeasuring4.5orgreaterhadbeenrecorded.AsofFeb.12,anestimatedthreemillionpeoplewereaffectedbythequake;theHaitianGovernmentreportsthatbetween217,000and230,000peoplehadbeenidentifiedasdead,anestimated300,000injured,andanestimated1,000,000homeless.

Thedeathtollisexpectedtorise.Theyalsoestimatedthat250,000residencesand30,000commercialbuildingshadcollapsedorwereseverelydamaged.——FromWikipediaThe2010ChileearthquakeoccurredoffthecoastoftheMauleRegionofChileonFebruary27,2010,at03:34localtime(06:34UTC),ratingamagnitudeof8.8onthemomentmagnitudescaleandlastingforaboutthreeminutes.Thecitiesexperiencingthestrongestshaking—VIII(Destructive)ontheMercalliintensityscale—wereTalcahuano,Arauco,Lota,Chiguayante,Ca?ete,andSanAntonio.TheearthquakewasfeltinthecapitalSantiagoatMercalliintensityscaleVII(VeryStrong).TremorswerefeltinmanyArgentinecities,includingBuenosAires,Córdoba,MendozaandLaRioja.TremorswerefeltasfarnorthasthecityofIcainsouthernPeru.Tsunamiwarningswereissuedin53countries,andatsunamiwasrecorded,withamplitudeofupto2.6m(8

ft6in)high,intheseaatValparaíso,Chile.PresidentMichelleBacheletdeclareda"stateofcatastrophe".Shealsoconfirmedthedeathsofatleast708people.Manymorehavebeenreportedmissing.——FromWikipediaThe2010Canterburyearthquake(alsoknownastheChristchurchearthquakeorDarfieldearthquake)wasa7.1magnitudeearthquake,whichstrucktheSouthIslandofNewZealandat4:35

amon4September2010localtime(16:353SeptemberUTC).Thequakecausedwidespreaddamageandseveralpoweroutages,particularlyinthecityofChristchurch,NewZealand'ssecondlargestcity.Tworesidentswereseriouslyinjured,onebyacollapsingchimneyandasecondbyflyingglass.Onepersondiedofaheartattacksufferedduringthequake,althoughthiscouldnotbedirectlylinkedtotheearthquake.Massfatalitieswereavoidedpartlyduetotherebeingfewhousesofunreinforcedconstruction,althoughthiswasalsoaidedbythequakeoccurringduringthenightwhenmostpeoplewereoffthestreet.Theearthquake'sepicentrewas40

kilometres(25mi)westofChristchurch,nearthetownofDarfield.Thehypocentrewasatashallowdepthof10

km.——FromWikipediaThe2011Christchurchearthquakewasa6.3-magnitudeearthquakethatstrucktheCanterburyregioninNewZealand'sSouthIslandat12:51

pmon22February2011localtime(23:5121FebruaryUTC),causingwidespreaddamageandmultiplefatalities.Theearthquakewascentred2

kilometres(1mi)westofthetownofLytteltonand10

kilometres(6mi)south-eastofChristchurch,NewZealand'ssecond-mostpopulouscity.Itfollowsnearlysixmonthsafterthe7.1magnitude2010Canterburyearthquake,whichhadcausedsignificantdamagetotheregionbutnofatalities.Atleast146peoplehavebeenconfirmeddeadandmorethan200peoplehavebeenreportedasmissingasof11:41amlocaltime,27February,makingtheearthquakeoneofthedeadliestnaturaldisastersinNewZealand'srecordedhistory.PrimeMinisterJohnKeystatedthat22February"maywellbeNewZealand'sdarkestday".Nationalsfrommorethan20countriesareamongthosemissing.——FromWikipediaChristChurchCathedral,afterthe2011earthquakecollapseditsspireEmergencyservicesworkersgathernearcollapsingbuildings.Photo:JohnKirk-Anderson

BeforeandafterphotosofthePyneGouldCorporationbuildinginChristchurch.Photo:ChelseaWinter/stuff.co.nz

RescuecrewslookforstaffinthedamagedPyneGouldGuinnessbuildingincentralChristchurch.Photo:SIMONBAKER/Reuters

DebrisoutsidetheChristchurchCatholicCathedral.Photo:DavidWethey/AP/NZPA

AcollapsedbuildinginChristchurch.Photo:DonScott/ThePress

SoilliquefactionThequakewasa"strike-slipeventwithobliquemotion"-mostlyhorizontalmovementwithsomeverticalmovement.However,theverticalaccelerationwasfargreaterthanthehorizontalacceleration.TheintensityfeltinChristchurchwasMMVIII.Thepeakgroundacceleration(PGA)intheChristchurchareaexceeded1.8g(i.e.1.8timestheaccelerationofgravity),withthehighestrecording2.2g,atHeathcoteValleyPrimarySchool,ashakingintensityequivalenttoMMX+.ThisisthehighestPGAeverrecordedinNewZealand;thehighestreadingduringtheSeptember2010eventwas1.26g,recordednearDarfield.ThePGAisalsooneoftheworsteverrecordedintheworld,andwasunusuallyhighfora6.3quake.Bycomparison,the7.0Mw2010HaitiearthquakehadanestimatedPGAof0.5g.Theaccelerationoccurredmainlyinaverticaldirection.Theforceofthequakewas"statisticallyunlikely"tooccurmorethanoncein1000years,accordingtooneseismicengineer,withaPGAgreaterthanmanymodernbuildingsweredesignedtowithstand.Bycomparison,the2010quake-inwhichdamagewaspredominatelytopre-1970sbuildings-exerted65%ofthedesignloadingonbuildings.——FromWikipediaAlthoughsmallerinmagnitudethanthe2010quake,theearthquakewasmoredamaginganddeadlyforanumberofreasons.TheepicentrewasclosertoChristchurch,andshallowerat

5

kilometres(3mi)underground,whereastheSeptemberquakewasmeasuredat10

kilometres(6mi)deep.TheFebruaryearthquakealsooccurredduringlunchtimeonaweekday,whentheCBDwasbusy,andmanybuildingswerealreadyweakenedfromthepreviousquakes.Liquefactionwas3–5timesgreaterthanthatofthe2010quake,with100,000-150,000tonnesofsandlikelytoberemovedfromaffectedsites.Theincreasedliquefactioncausedsignificantgroundmovementandunderminingmanyfoundationsanddestroyinginfrastructure.The2011earthquakeofthePacificcoastofTōh(huán)okuTōh(huán)oku-chihōTaiheiyōOkiJishin),alsoknownasthe2011Tohokuearthquake,ortheGreatEastJapanEarthquake,wasamagnitude9.0(Mw)underseamegathrustearthquakeoffthecoastofJapanthatoccurredat14:46JST(05:46UTC)onFriday,11March2011,withtheepicenterapproximately70kilometres(43

mi)eastoftheOshikaPeninsulaofTōh(huán)okuandthehypocenteratanunderwaterdepthofapproximately32

km(20

mi).ItwasthemostpowerfulknownearthquakeevertohavehitJapan,andoneofthefivemostpowerfulearthquakesintheworldsincemodernrecord-keepingbeganin1900.Theearthquaketriggeredpowerfultsunamiwavesthatreachedheightsofupto40.5metres(133

ft)inMiyakoinTōh(huán)oku'sIwatePrefecture,andwhich,intheSendaiarea,travelledupto10

km(6

mi)inland.TheearthquakemovedHonshu2.4

m(8

ft)eastandshiftedtheEarthonitsaxisbyestimatesofbetween10

cm(4

in)and25

cm(10

in).2011Tōh(huán)okuearthquakeandtsunami

MapoftheTōh(huán)okuearthquakeandaftershockson11–14March

AnaerialviewofdamageintheSendairegionwithblacksmokecomingfromtheNipponOilSendaioilrefinery

2011Tōh(huán)okuearthquakeandtsunami2011年日本東北部地震及海嘯仙臺(tái)一倒塌的鋼筋混凝土建筑（來(lái)源：?merAYDAN等）2011年日本東北地震及海嘯仙臺(tái)一嚴(yán)重受損的鋼筋混凝土建筑（來(lái)源：?merAYDAN等）The2018OaxacaearthquakeoccurredonFebruary16,2018at17:39localtime(23:39UTC)intheSierraMadredelSurinOaxacastateinSouthernMexico.Ithadamagnitudeof7.2onthemomentmagnitudescaleandamaximumfeltintensityofVII(verystrong)ontheMercalliintensityscale.Thehypocenterwaslocatedatadepthof24.6kmandapproximately37kmnortheastofPinotepadeDonLuis.Historically,severalsignificantearthquakeshaveoccurredalongthesoutherncoastofMexico.In1932,aM8.4thrustearthquakestruckintheregionofJalisco,severalhundredkilometerstothenorthwestoftheOaxacaevent.OnOctober9,1995,aM8.0earthquakestruckintheColima-Jaliscoregion,resultinginatleast49fatalitiesandleaving1,000peoplehomeless.ThedeadliestnearbyearthquakeoccurredonSeptember19,1985,intheMichoacanregion500

kmtothenorthwestoftheFebruary16thevent.ThisM8.0earthquakeresultedinatleast9,500fatalities,injuredabout30,000people,andleft100,000peoplehomeless.In2003,aM7.6earthquakeinColima,Mexico,resultedin29fatalities,destroyedmorethan2,000homesandleftmorethan10,000peoplehomeless.InMarch2012,aM7.4earthquake60

kmtothenorthwestoftheFebruary16,2018eventkilled2andinjured11intheOaxacaregion.ThehypocenteroftheM8.2earthquakeofftheshoreofChiapasinSeptember2017waslocated440

kmsouthwestofthisearthquake.TheChiapaseventcausedatleast78fatalitiesand250injuriesinOaxaca,andafurther16deathsinChiapas.Elevendayslater,aM7.1earthquakestruckclosertoMexicoCity,230

kmnortheastoftoday’searthquake,resultinginover300fatalitiesandsignificantdamageinMexicoCityandthesurroundingregion.2018OaxacaearthquakeOn25February2018,at17:44UTC,anearthquakeof7.5momentmagnitudestruckontheboundaryofHelaandSouthernHighlandsProvinces,14kilometres(10

mi)southwestofKomo.核心設(shè)備為三維地震模擬振動(dòng)臺(tái),臺(tái)面面積20m×15m、厚度5.5m、重770t;X向和Y向加振器各5套,Z向加振器14套,加振系統(tǒng)為MTS驅(qū)動(dòng)設(shè)備、現(xiàn)代先進(jìn)液壓伺服控制,最大承載力1200t,最大水平和豎向加速度分別超過(guò)900cm/s2、1500cm/s2,最大水平和豎向速度分別為200cm/s、70cm/s,最大水平和豎向位移分別為±100cm、±70cm,最大水平和豎向轉(zhuǎn)矩分別為150MN·m、40MN·m,水平和豎向運(yùn)行頻率為0.1Hz～50Hz,能夠模擬阪神地震極震區(qū)的主震,并實(shí)施原型結(jié)構(gòu)的強(qiáng)震反應(yīng)、破壞、倒塌試驗(yàn)。E-DefenseHyogoEarthquakeengineeringResearchCenter,NationalResearchInstituteforEarthScienceandDisasterPrevention試驗(yàn)概況:6層RC住宅,2×3跨,跨度5m、層高2.5m,二樓和五樓配有原型家具,試驗(yàn)體總高16.2m、上臺(tái)重約1000t,輸入：JMAKobe波。FilmofTacomaNarrowsBridgeCollapseTheoriginalTacomaNarrowsBridgetwistedandvibratedviolentlyunder40-mile-per-hour(64

km/h)windsthedayofthecollapse.教材及參考文獻(xiàn)結(jié)構(gòu)動(dòng)力學(xué)(第2版修訂版)，R.克拉夫、J.彭津，王光遠(yuǎn)等譯校，

高等教育出版社，2006.11結(jié)構(gòu)動(dòng)力學(xué):理論及其在地震工程中的應(yīng)用(第2版)，AnilK.Chopra著，謝禮立、呂大剛等譯，高等教育出版社，2007.1DynamicsofStructures(3rdEdition)，RayW.CloughandJosephPenzien,Computers&Structures,Inc.,Berkeley,USADynamicsofStructures:TheoryandApplicationstoEarthquakeEngineering(3rdEdition),AnilK.Chopra結(jié)構(gòu)力學(xué)教程（Ⅱ），龍馭球、包世華等，高等教育出版社DynamicsofStructures(3rdEdition)學(xué)習(xí)要求Clarifythepurposesandrequirements.Takethenotes.Studyactively.Relatetoengineeringproblemsifpossible.…結(jié)構(gòu)動(dòng)力學(xué)第一篇單自由度體系第二篇多自由度體系第三篇具有分布參數(shù)的體系第四篇隨機(jī)振動(dòng)第五篇結(jié)構(gòu)地震反應(yīng)分析Theprimarypurposeofthisbookistopresentmethodsforanalyzingthestressesanddeflectionsdevelopedinanygiventypeofstructurewhenitissubjectedtoanarbitrarydynamicloading.Inonesense,thisobjectivemaybeconsideredtobeanextensionofstandardmethodsofstructuralanalysis,whichgenerallyareconcernedwithstaticloadingonly,topermitconsiderationofdynamicloadingaswell.Inthiscontext,thestatic-loadingconditionmaybelookeduponmerelyasaspecialformofdynamicloading.However,intheanalysisofalinearstructureitisconvenienttodistinguishbetweenthestaticandthedynamiccomponentsoftheappliedloading,toevaluatetheresponsetoeachtypeofloadingseparately,andthentosuperposethetworesponsecomponentstoobtaintheirtotaleffect.Whentreatedthusly,thestaticanddynamicmethodsofanalysisarefundamentallydifferentincharacter.Chapter1Overviewofstructuraldynamics雙語(yǔ)實(shí)踐1第一章結(jié)構(gòu)動(dòng)力學(xué)概述

結(jié)構(gòu)動(dòng)力學(xué)是結(jié)構(gòu)力學(xué)的一個(gè)分支，著重研究結(jié)構(gòu)對(duì)于動(dòng)荷載的響應(yīng)（如位移、應(yīng)力等的時(shí)間歷程），以便確定結(jié)構(gòu)的承載能力和動(dòng)力學(xué)特性，或?yàn)楦纳平Y(jié)構(gòu)的性能提供依據(jù)。動(dòng)荷載的特性結(jié)構(gòu)的動(dòng)力特性結(jié)構(gòu)響應(yīng)分析§1-1概述輸入（動(dòng)力荷載）結(jié)構(gòu)（系統(tǒng)）輸出（動(dòng)力反應(yīng)）第一類問(wèn)題：反應(yīng)分析（結(jié)構(gòu)動(dòng)力計(jì)算）第二類問(wèn)題：參數(shù)（或稱系統(tǒng)）識(shí)別輸入（動(dòng)力荷載）結(jié)構(gòu)（系統(tǒng)）輸出（動(dòng)力反應(yīng)）第三類問(wèn)題：荷載識(shí)別輸入（動(dòng)力荷載）結(jié)構(gòu)（系統(tǒng)）輸出（動(dòng)力反應(yīng)）第四類問(wèn)題：控制問(wèn)題輸入（動(dòng)力荷載）結(jié)構(gòu)（系統(tǒng)）輸出（動(dòng)力反應(yīng)）控制系統(tǒng)（裝置、能量）-----控制問(wèn)題輸入（動(dòng)力荷載）結(jié)構(gòu)（系統(tǒng)）輸出（動(dòng)力反應(yīng)）第一類問(wèn)題：反應(yīng)分析（結(jié)構(gòu)動(dòng)力計(jì)算）第二類問(wèn)題：參數(shù)（或稱系統(tǒng)）識(shí)別輸入（動(dòng)力荷載）結(jié)構(gòu)（系統(tǒng)）輸出（動(dòng)力反應(yīng)）第三類問(wèn)題：荷載識(shí)別。輸入（動(dòng)力荷載）結(jié)構(gòu)（系統(tǒng)）輸出（動(dòng)力反應(yīng)）-----正問(wèn)題-----反問(wèn)題-----反問(wèn)題剛度、約束桿件尺寸截面特性動(dòng)位移加速度速度動(dòng)應(yīng)力動(dòng)力系數(shù)隨時(shí)間變化質(zhì)量、剛度阻尼、約束頻率、振型輸入Input輸出Output結(jié)構(gòu)體系靜力響應(yīng)靜荷載位移內(nèi)力應(yīng)力大小方向作用點(diǎn)數(shù)值結(jié)構(gòu)體系動(dòng)力響應(yīng)輸入Input輸出Output動(dòng)荷載大小方向作用點(diǎn)時(shí)間變化時(shí)間函數(shù)結(jié)構(gòu)動(dòng)力體系§1-2動(dòng)荷載的定義和分類荷載：荷載三要素：荷載分類：作用在結(jié)構(gòu)上的主動(dòng)力大小、方向和作用點(diǎn)作用時(shí)間：作用位置：對(duì)結(jié)構(gòu)產(chǎn)生的動(dòng)力效應(yīng)：恒載活載固定荷載移動(dòng)荷載靜荷載動(dòng)荷載

大小、方向和作用點(diǎn)不隨時(shí)間變化或變化很緩慢的荷載。靜荷載：動(dòng)荷載：

大小、方向或作用點(diǎn)隨時(shí)間變化很快的荷載。是否會(huì)使結(jié)構(gòu)產(chǎn)生顯著的加速度快慢標(biāo)準(zhǔn)：質(zhì)量運(yùn)動(dòng)加速度所引起的慣性力與其它外荷載相比是否可以忽略顯著標(biāo)準(zhǔn)：動(dòng)荷載的定義荷載在大小、方向或作用點(diǎn)方面隨時(shí)間變化，使得質(zhì)量運(yùn)動(dòng)加速度所引起的慣性力與荷載相比大到不可忽略時(shí)，則把這種荷載稱為動(dòng)荷載。問(wèn)題：你知道有哪些動(dòng)荷載？LoadinghistoriesTypicalexamplesFIGURE1-1Characteristicsandsourcesoftypicaldynamicloadings:(a)simpleharmonic;(b)complex;(c)impulsive;(d)long-duration.LoadinghistoriesTypicalexamples2024/2/26蘭州理工大學(xué)土木工程學(xué)院韓建平45Twobasicallydifferentapproachesareavailableforevaluatingstructuralresponsetodynamicloads:deterministicandnondeterministic.Thechoiceofmethodtobeusedinanygivencasedependsuponhowtheloadingisdefined.Ifthetimevariationofloadingisfullyknown,eventhoughitmaybehighlyoscillatoryorirregularincharacter,itwillbereferredtohereinasaprescribeddynamicloading;andtheanalysisoftheresponseofanyspecifiedstructuralsystemtoaprescribeddynamicloadingisdefinedasadeterministicanalysis.雙語(yǔ)實(shí)踐2Ingeneral,structuralresponsetoanydynamicloadingisexpressedbasicallyintermsofthedisplacementsofthestructure.Thus,adeterministicanalysisleadsdirectlytodisplacementtime--historiescorrespondingtotheprescribedloadinghistory;otherrelatedresponsequantities,suchasstresses,strains,internalforces,etc.,areusuallyobtainedasasecondaryphaseoftheanalysis.Ontheotherhand,anondeterministicanalysisprovidesonlystatisticalinformationaboutthedisplacementsresultingfromthestatisticallydefinedloading;correspondinginformationontherelatedresponsequantitiesarethengeneratedusingindependentnondeterministicanalysisprocedures.雙語(yǔ)實(shí)踐3動(dòng)荷載的分類：概念：動(dòng)荷載是時(shí)間的函數(shù)！分類：動(dòng)荷載確定性荷載非確定性荷載周期性荷載非周期性荷載prescribeddynamicloadingrandomdynamicloading突加荷載沖擊荷載確定性荷載：例如：簡(jiǎn)諧荷載荷載的變化是時(shí)間的確定性函數(shù)。例如：風(fēng)荷載地震作用平均風(fēng)脈動(dòng)風(fēng)非確定性荷載：荷載隨時(shí)間的變化是不確定的或不確知的，又稱為隨機(jī)荷載。結(jié)構(gòu)在確定性荷載作用下的響應(yīng)分析通常稱為結(jié)構(gòu)振動(dòng)分析。結(jié)構(gòu)在隨機(jī)荷載作用下的響應(yīng)分析，被稱為結(jié)構(gòu)的隨機(jī)振動(dòng)分析。本課程主要學(xué)習(xí)確定性荷載作用下的結(jié)構(gòu)振動(dòng)分析。與結(jié)構(gòu)靜力學(xué)相比，動(dòng)力學(xué)的復(fù)雜性表現(xiàn)在：§1-3動(dòng)力問(wèn)題的基本特性動(dòng)力問(wèn)題具有隨時(shí)間而變化的性質(zhì)；數(shù)學(xué)解答不是單一的數(shù)值，而是時(shí)間的函數(shù)；慣性力是結(jié)構(gòu)內(nèi)部彈性力所平衡的全部荷載的一個(gè)重要部分！引入慣性力后涉及到二階微分方程的求解；需考慮結(jié)構(gòu)本身的動(dòng)力特性：剛度分布、質(zhì)量分布、阻尼特性分布的影響；(a)(b)FIGURE1-2Basicdifferencebetweenstaticanddynamicloads:(a)staticloading;(b)dynamicloading.Astructural--dynamicproblemdiffersfromitsstatic--loadingcounterpartintwoimportantrespects.Thefirstdifferencetobenoted,bydefinition,isthetime--varyingnatureofthedynamicproblem.Becausebothloadingandresponsevarywithtime,itisevidentthatadynamicproblemdoesnothaveasinglesolution,asastaticproblemdoes;insteadtheanalystmustestablishasuccessionofsolutionscorrespondingtoalltimesofinterestintheresponsehistory.Thusadynamicanalysisisclearlymorecomplexandtime--consumingthanastaticanalysis.Thesecondandmorefundamentaldistinctionbetweenstaticanddynamicprob-lemsisillustratedinFig.1.-2.雙語(yǔ)實(shí)踐Inertialforceswhichresistaccelerationsofthestructureinthiswayarethemostimportantdistinguishingcharacteristicofastructural--dynamicsproblem.Ingeneral,iftheinertialforcesrepresentasignificantportionofthetotalloadequilibratedbytheinternalelasticforcesofthestructure,thenthedynamiccharacteroftheproblemmustbeaccountedforinitssolution.Ontheotherhand,ifthemotionsaresoslowthattheinertialforcesarenegligiblysmall,theanalysisofresponseforanydesiredinstantoftimemaybemadebystaticstructural--analysisprocedureseventhoughtheloadandresponsemaybetime--varying.雙語(yǔ)實(shí)踐§1-4離散化方法1.集中質(zhì)量法把結(jié)構(gòu)的分布質(zhì)量按一定的規(guī)則集中到結(jié)構(gòu)的某個(gè)或某些位置上，成為一系列離散的質(zhì)點(diǎn)或質(zhì)量塊。適用于大部分質(zhì)量集中在若干離散點(diǎn)上的結(jié)構(gòu)。例如：房屋結(jié)構(gòu)一般簡(jiǎn)化為層間剪切模型。Lumped-MassProcedure

例如：討論：動(dòng)力自由度Thenumberofdisplacementcomponentswhichmustbeconsideredinordertorepresenttheeffectsofallsignicantinertialforcesofastructuremaybetermedthenumberofdynamicdegreesoffreedomofthestructure.結(jié)構(gòu)的動(dòng)力自由度：描述體系在運(yùn)動(dòng)過(guò)程中任意時(shí)刻全部質(zhì)量的位置所需要的獨(dú)立幾何參數(shù)的數(shù)目。FIGURE1-3Lumped-massidealizationofasimplebeam.Forexample,ifthethreemassesinthesystemofFig.1--3arefullyconcentratedandareconstrainedsothatthecorrespondingmasspointstranslateonlyinaverticaldirection,thiswouldbecalledathree--degree--of--freedom(3DOF)system.Ontheotherhand,ifthesemassesarenotfullyconcentratedsothattheypossessfiniterotationalinertia,therotationaldisplacementsofthethreepointswillalsohavetobeconsidered,inwhichcasethesystemhas6DOF.Ifaxialdistortionsofthebeamaresignificant,translationdisplacementsparallelwiththebeamaxiswillalsoresultgivingthesystem9DOF.Moregenerally,ifthestructurecandeforminthree--dimensionalspace,eachmasswillhave6DOF;thenthesystemwillhave18DOF.However,ifthemassesarefullyconcentratedsothatnorotationalinertiaispresent,thethree--dimensionalsystemwillthenhave9DOF.Onthebasisoftheseconsiderations,itisclearthatasystemwithcontinuouslydistributedmass,asinFig.1--2b,hasaninfinitenumberofdegreesoffreedom.Thelumped--massidealizationdescribedaboveprovidesasimplemeansoflimitingthenumberofdegreesoffreedomthatmustbeconsideredinconductingadynamicanalysisofanarbitrarystructuralsystem.Thelumpingprocedureismosteffectiveintreatingsystemsinwhichalargeproportionofthetotalmassactuallyisconcentratedatafewdiscretepoints.Thenthemassofthestructurewhichsupportstheseconcentrationscanbeincludedinthelumps,allowingthestructureitselftobeconsideredweightless.雙語(yǔ)實(shí)踐適用于質(zhì)量分布比較均勻，形狀規(guī)則且邊界條件易于處理的結(jié)構(gòu)。例如：右圖簡(jiǎn)支梁的變形可以用三角函數(shù)的線性組合來(lái)表示。2.廣義坐標(biāo)法假定具有分布質(zhì)量的結(jié)構(gòu)在振動(dòng)時(shí)的位移曲線可用一系列規(guī)定的位移曲線的和來(lái)表示：GeneralizedDisplacements則組合系數(shù)Ak(t)稱為體系的廣義坐標(biāo)。定義假定具有分布質(zhì)量的結(jié)構(gòu)在振動(dòng)時(shí)的位移曲線為y(x,t)，可用一系列位移函數(shù)的線性組合來(lái)表示：廣義坐標(biāo)位移函數(shù)或形函數(shù)廣義坐標(biāo)表示相應(yīng)位移函數(shù)的幅值，是隨時(shí)間變化的函數(shù)。廣義坐標(biāo)確定后，可由給定的位移函數(shù)確定結(jié)構(gòu)振動(dòng)的位移曲線。以廣義坐標(biāo)作為自由度，將無(wú)限自由度體系轉(zhuǎn)化為有限個(gè)自由度。所采用的廣義坐標(biāo)數(shù)代表了所考慮的自由度數(shù)。3.有限單元法先把結(jié)構(gòu)劃分成適當(dāng)（任意）數(shù)量的單元；對(duì)每個(gè)單元施行廣義坐標(biāo)法，通常取單元的節(jié)點(diǎn)位移作為廣義坐標(biāo)；對(duì)每個(gè)廣義坐標(biāo)取相應(yīng)的位移函數(shù)（插值函數(shù)）；由此提供了一種有效的、標(biāo)準(zhǔn)化的、用一系列離散坐標(biāo)表示無(wú)限自由度的結(jié)構(gòu)體系。要點(diǎn)：——將有限元法的思想用于解決結(jié)構(gòu)的動(dòng)力計(jì)算問(wèn)題。對(duì)分布質(zhì)量的實(shí)際結(jié)構(gòu)，體系的自由度數(shù)為單元節(jié)點(diǎn)可發(fā)生的獨(dú)立位移未知量的總個(gè)數(shù)。綜合了集中質(zhì)量法和廣義坐標(biāo)法的某些特點(diǎn)，是最靈活有效的離散化方法，它提供了既方便又可靠的理想化模型，并特別適合于用電子計(jì)算機(jī)進(jìn)行分析，是目前最為流行的方法。已有不少專用的或通用的程序（如SAP、ANSYS、ABAQUS等）供結(jié)構(gòu)分析之用。包括靜力、動(dòng)力和穩(wěn)定分析。Athirdmethodofexpressingthedisplacementsofanygivenstructureintermsofafinitenumberofdiscretedisplacementcoordinates,whichcombinescertainfeaturesofboththelumped--massandthegeneralized--coordinateprocedures,hasnowbecomepopular.Thisapproach,whichisthebasisofthefinite--elementmethodofanalysisofstructuralcontinua,providesaconvenientandreliableidealizationofthesystemandisparticularlyeffectiveindigital--computeranalyses.

雙語(yǔ)實(shí)踐高層建筑結(jié)構(gòu)的有限元模型——法門(mén)寺合十舍利塔效果圖振動(dòng)臺(tái)實(shí)驗(yàn)?zāi)Ｐ虯BAQUS有限元模型大型橋梁結(jié)構(gòu)的有限元模型第2次課結(jié)束（2學(xué)時(shí)）§1-5運(yùn)動(dòng)方程的建立在結(jié)構(gòu)動(dòng)力分析中，描述體系質(zhì)量運(yùn)動(dòng)規(guī)律的數(shù)學(xué)方程，稱為體系的運(yùn)動(dòng)微分方程，簡(jiǎn)稱運(yùn)動(dòng)方程。定義運(yùn)動(dòng)方程的解揭示了體系在各自由度方向的位移隨時(shí)間變化的規(guī)律。建立運(yùn)動(dòng)方程是求解結(jié)構(gòu)振動(dòng)問(wèn)題的重要基礎(chǔ)。常用方法：直接平衡法、虛功法、變分法。建立體系運(yùn)動(dòng)方程的方法直接平衡法，又稱動(dòng)靜法、慣性力法，將動(dòng)力學(xué)問(wèn)題轉(zhuǎn)化為任一時(shí)刻的靜力學(xué)問(wèn)題：根據(jù)達(dá)朗貝爾原理（d’Alembert’sprinciple），把慣性力作為附加的虛擬力，并考慮阻尼力、彈性力和作用在結(jié)構(gòu)上的外荷載，使體系處于動(dòng)力平衡條件，按照靜力學(xué)中建立平衡方程的思路，直接寫(xiě)出運(yùn)動(dòng)方程。虛功法:根據(jù)虛功原理，即作用在體系上的全部力在虛位移上所做的虛功總和為零的條件，導(dǎo)出以廣義坐標(biāo)表示的運(yùn)動(dòng)方程。變分法:通過(guò)對(duì)表示能量關(guān)系的泛函的變分建立方程。根據(jù)理論力學(xué)中的哈密頓原理或其等價(jià)形式的拉格朗日方程導(dǎo)出以廣義坐標(biāo)表示的運(yùn)動(dòng)方程。Hamilton原理：在任何時(shí)間區(qū)間t1到t2內(nèi)，動(dòng)能和勢(shì)能的變分加上所考慮的非保守力所作的功的變分必須等于零。優(yōu)點(diǎn)：只和純粹的標(biāo)量——能量有關(guān)?！w系的總動(dòng)能；——體系的位能，包括應(yīng)變能及任何保守外力的勢(shì)能；——作用于體系上的非保守力（包括阻尼力及任意外荷）所作的功；——在指定時(shí)間區(qū)間內(nèi)所取的變分。拉格朗日方程：——拉格朗日函數(shù)，動(dòng)能與勢(shì)能之差：或拉格朗日方程可由Hamilton原理導(dǎo)出！單自由度體系模型質(zhì)量塊m，用來(lái)表示結(jié)構(gòu)的質(zhì)量和慣性特性自由度只有一個(gè)：水平位移y(t)無(wú)重彈簧，剛度為k，提供結(jié)構(gòu)的彈性恢復(fù)力無(wú)重阻尼器，阻尼系數(shù)c，表示結(jié)構(gòu)的能量耗散，提供結(jié)構(gòu)的阻尼力隨時(shí)間變化的荷載F(t)第二章運(yùn)動(dòng)方程的建立承受動(dòng)力荷載的任何線性結(jié)構(gòu)體系的主要物理特性是體系的質(zhì)量、彈性特性（剛度或柔度）、能量耗散機(jī)理或阻尼、以及外部擾力或荷載單自由度體系運(yùn)動(dòng)方程的建立（直接平衡法）建立計(jì)算模型取質(zhì)點(diǎn)為隔離體畫(huà)平衡力系建立平衡方程直接平衡法，又稱動(dòng)靜法，將動(dòng)力學(xué)問(wèn)題轉(zhuǎn)化為任一時(shí)刻的靜力學(xué)問(wèn)題：根據(jù)達(dá)朗貝爾原理，把慣性力作為附加的虛擬力，并考慮阻尼力、彈性力和作用在結(jié)構(gòu)上的外荷載，使體系處于動(dòng)力平衡條件，按照靜力學(xué)中建立平衡方程的思路，直接寫(xiě)出運(yùn)動(dòng)方程。直接平衡法根據(jù)所用平衡方程的不同，直接平衡法又分為剛度法和柔度法。平衡方程：根據(jù)D’Alembert原理：等于彈簧剛度與位移的乘積：阻尼力等于阻尼系數(shù)與速度的乘積：由此得到體系的運(yùn)動(dòng)方程：（2-3）慣性力：彈性力：阻尼力：剛度法:取每一運(yùn)動(dòng)質(zhì)量為隔離體，通過(guò)分析所受的全部外力，建立質(zhì)量各自由度的瞬時(shí)力平衡方程，得到體系的運(yùn)動(dòng)方程。平衡方程：剛度法步驟：1.在質(zhì)量上沿位移正向加慣性力；2.求發(fā)生位移y所需之力；3.令該力等于體系外力和慣性力。試用剛度法建立圖示剛架的運(yùn)動(dòng)方程[解]1）確定自由度數(shù):橫梁剛性，柱子無(wú)軸向變形。2）確定自由度的位移參數(shù)。3）質(zhì)量受力分析：取剛梁為隔離體，確定所受的所有外力！4）列動(dòng)平衡方程：1個(gè)自由度。其中各力的大?。何灰品ǎ褐右欢水a(chǎn)生單位平移時(shí)的桿端剪力等效粘滯阻尼力：柱端發(fā)生平移y

時(shí)產(chǎn)生的梁-柱間剪力：由此得到體系的運(yùn)動(dòng)方程：慣性力：彈性力Fs=Fs1+Fs2：由此得到體系的運(yùn)動(dòng)方程：比較：；k為（等效）剛度系數(shù)。令：運(yùn)動(dòng)方程與(2-3)的形式是一樣的！(2-3)柔度法以結(jié)構(gòu)整體為研究對(duì)象，通過(guò)分析所受的全部外力，利用結(jié)構(gòu)靜力分析中計(jì)算位移的方法，根據(jù)位移協(xié)調(diào)條件建立體系的運(yùn)動(dòng)方程。柔度法步驟：1.在質(zhì)量上沿位移正向加慣性力；2.求外力和慣性力引起的位移；3.令該位移等于體系位移。[例]試用柔度法建立圖示簡(jiǎn)支梁的運(yùn)動(dòng)方程[解]1）確定自由度數(shù):集中質(zhì)量，僅豎向位移：2）確定自由度的位移參數(shù)：質(zhì)量m

的位移：3）體系受力分析：取梁整體為隔離體，確定所受的所有外力！1個(gè)自由度。4）列位移方程：改寫(xiě)成：Dp為動(dòng)荷載q(t)

引起的質(zhì)量沿y方向的位移：其中：d為自由度方向加單位力所引起的位移，即柔度：慣性力：阻尼力：由此得到體系的運(yùn)動(dòng)方程：位移方程：比較：含義：等效動(dòng)荷載直接作用在質(zhì)量自由度上產(chǎn)生的動(dòng)位移與實(shí)際動(dòng)荷載產(chǎn)生的位移相等！令：FE(t)

定義為體系的等效動(dòng)荷載或等效干擾力。其通用表達(dá)式（2-3）已經(jīng)知道柔度d和剛度k

之間的關(guān)系為：結(jié)論：任何一個(gè)單自由度體系的運(yùn)動(dòng)方程都可以抽象成為一質(zhì)量、彈簧、阻尼器體系的運(yùn)動(dòng)方程，一般形式為：比較：剛架：（2-3）基本質(zhì)量彈簧體系：表達(dá)式成為：簡(jiǎn)支梁：[例]：應(yīng)用D’Alembert原理推導(dǎo)圖示質(zhì)量-彈簧-阻尼體系的運(yùn)動(dòng)方程，假定運(yùn)動(dòng)方向是豎向的，不計(jì)空氣阻力。重力的影響Thus,theweightforcecanbeeliminatedfromthedifferentialequationofmotionofanSDOFsystemifthedisplacementofthemassismeasuredrelativetoitsstaticequilibriumposition.[例]：應(yīng)用D’Alembert原理推導(dǎo)圖示體系的運(yùn)動(dòng)方程，按相對(duì)于基礎(chǔ)的位移列出運(yùn)動(dòng)方程表達(dá)式。地基運(yùn)動(dòng)的影響Thus,basemotionhastheeffectofaddingareversedinertialforcetotheotherappliedforces.Sincerelativemotionmaybemoreimportantthanabsolutemotion,andsincethebaseacceleration,,ismucheasiertomeasurethanthebasevelocityorbasedisplacement,therearemanyapplicationswhereEq.6ismoreusefulthanEq.列運(yùn)動(dòng)方程例題例3.mEIlEIl1例4.mEIl/2EIl/21層間側(cè)移剛度mEIlEIl1

對(duì)于帶剛性橫梁的剛架(剪切型剛架),當(dāng)兩層之間發(fā)生相對(duì)單位水平位移時(shí),兩層之間的所有柱子中的剪力之和稱作該層的層間側(cè)移剛度.EIllEIEIEI三、列運(yùn)動(dòng)方程例題列運(yùn)動(dòng)方程時(shí)可不考慮重力影響例5.mEIl/2l/2W---P(t)引起的動(dòng)位移---重力引起的位移質(zhì)點(diǎn)的總位移為加速度為列運(yùn)動(dòng)方程例題例6.m1EIl/3l/3l/3m2=簡(jiǎn)記為位移向量柔度矩陣荷載向量質(zhì)量矩陣加速度向量例7.m1m2=剛度矩陣?yán)?.m1m2=+例7.m1m2例8建立圖示體系的運(yùn)動(dòng)方程m2mlllkAy(t)2y(t)3y(t)作業(yè)Chopra教材1.1-1.5、1.12-1.16第3次課結(jié)束（2學(xué)時(shí)）第三章

自由振動(dòng)反應(yīng)表征結(jié)構(gòu)動(dòng)力響應(yīng)特性的一些固有量稱為結(jié)構(gòu)的動(dòng)力特性，又稱自振特性。定義結(jié)構(gòu)的振動(dòng)反應(yīng)結(jié)構(gòu)的動(dòng)力特性與結(jié)構(gòu)的質(zhì)量、剛度、阻尼及其分布有關(guān)。定義結(jié)構(gòu)受外部干擾后發(fā)生振動(dòng)，而在干擾消失后繼續(xù)振動(dòng)，這種振動(dòng)稱為結(jié)構(gòu)的自由振動(dòng)。如果結(jié)構(gòu)在振動(dòng)過(guò)程中不斷地受到外部干擾力作用，這種振動(dòng)稱為結(jié)構(gòu)的強(qiáng)迫振動(dòng)，又稱受迫振動(dòng)。結(jié)構(gòu)的自由振動(dòng)與受迫振動(dòng)固有頻率質(zhì)點(diǎn)在運(yùn)動(dòng)過(guò)程中完成一個(gè)完整的循環(huán)所需要的時(shí)間稱為周期，單位時(shí)間內(nèi)完成的循環(huán)次數(shù)稱為頻率。結(jié)構(gòu)在自由振動(dòng)時(shí)的頻率稱為結(jié)構(gòu)的自振頻率或固有頻率。對(duì)大部分工程結(jié)構(gòu)，結(jié)構(gòu)的自振頻率的個(gè)數(shù)與結(jié)構(gòu)的動(dòng)力自由度數(shù)相等。結(jié)構(gòu)的自振頻率與結(jié)構(gòu)的質(zhì)量和剛度有關(guān)。阻尼結(jié)構(gòu)在振動(dòng)過(guò)程中的能量耗散作用稱為阻尼。結(jié)構(gòu)的自由振動(dòng)會(huì)因?yàn)樽枘嶙饔枚S時(shí)間衰減并最終停止。由于阻尼而使振動(dòng)衰減的結(jié)構(gòu)系統(tǒng)稱為有阻尼系統(tǒng)。阻尼原因復(fù)雜：內(nèi)摩擦、連接摩擦、周圍介質(zhì)阻力等。等效粘滯阻尼：以阻尼器表示結(jié)構(gòu)阻尼作用：c

為阻尼系數(shù)，為質(zhì)量的速度?！?-1運(yùn)動(dòng)方程的解最簡(jiǎn)單的由剛體、彈簧和阻尼器組成的單自由度體系.

已經(jīng)得到單自由度體系的運(yùn)動(dòng)方程：（3-1）這個(gè)運(yùn)動(dòng)方程也適用于可轉(zhuǎn)換為單自由度體系的任何復(fù)雜結(jié)構(gòu)體系的廣義坐標(biāo)反應(yīng)。運(yùn)動(dòng)方程：等效動(dòng)荷載為零的情況下的振動(dòng)稱為自由振動(dòng)。定義自由振動(dòng)產(chǎn)生的原因：初始時(shí)刻的干擾！

初始位移；初始速度；初始位移+初始速度結(jié)構(gòu)受外部干擾后發(fā)生振動(dòng)，而在干擾消失后繼續(xù)振動(dòng)，這種振動(dòng)稱為結(jié)構(gòu)的自由振動(dòng)。去掉外荷載p(t)=0!上式稱為（二階線性常系數(shù)）齊次方程；

齊次方程的求解：可設(shè)齊次方程解的形式為：（3-3）

其特征方程為：或：代入（3-2）可得：（3-4）（3-2）稱為（二階線性常系數(shù)）齊次方程(Homogeneousequation)；式中ω2=k/m，ω是體系振動(dòng)的圓頻率。根據(jù)阻尼系數(shù)c值的不同，解出的特征參數(shù)s值將具有不同的特性。（3-2）§3-2無(wú)阻尼自由振動(dòng)Ifc=0：

特征方程：自由振動(dòng)方程：（3-7）

引入Euler方程：

代入(3-2)得：（3-9）A和B是由初始條件決定的常數(shù)。得無(wú)阻尼自由振動(dòng)的位移反應(yīng)：（3-10）（3-2）設(shè)t=0時(shí)：代入：代入：?jiǎn)巫杂啥葻o(wú)阻尼體系運(yùn)動(dòng)方程的解：（3-11）或?qū)懗桑海?-14）位移反應(yīng)：（3-10）三角關(guān)系：對(duì)比(3-11)，顯然有：(3-13)成為：即：（3-14）（3-11）（3-14）物理意義：（3-11）（3-14）物理意義：（3-11）定義對(duì)于無(wú)阻尼體系，運(yùn)動(dòng)完全是反復(fù)進(jìn)行的。運(yùn)動(dòng)的最大位移稱為振幅。運(yùn)動(dòng)完成一個(gè)完整循環(huán)所需時(shí)間稱為自振周期，由于對(duì)應(yīng)每個(gè)角增量

2p

便發(fā)生一個(gè)完整循環(huán)，自振周期就是：

單位時(shí)間內(nèi)的循環(huán)次數(shù)稱為自振頻率：

運(yùn)動(dòng)的角速度稱為自振圓頻率：§3-3阻尼自由振動(dòng)對(duì)于有阻尼的單自由度體系

特征方程：自由振動(dòng)方程：∵

則：隨著根號(hào)中值的符號(hào)的不同，這個(gè)表達(dá)式可以描述臨界阻尼、低阻尼和超阻尼三種體系的運(yùn)動(dòng)型式。本課程只講臨界阻尼和低阻尼兩種情況。（3-2）1.臨界阻尼當(dāng)根式中的值為零時(shí)，對(duì)應(yīng)的阻尼值稱為臨界阻尼，記作cc。顯然，應(yīng)有cc/2m=w，即：

特征方程的根：

這時(shí)，對(duì)應(yīng)的s

值為：自由振動(dòng)方程：臨界阻尼自由振動(dòng)方程的解為：（3-19）（3-20）（3-2）由初始條件：得到臨界阻尼體系反應(yīng)的最終形式：

臨界阻尼位移解：臨界阻尼體系反應(yīng)不是簡(jiǎn)諧振動(dòng)，體系的位移反應(yīng)從開(kāi)始時(shí)的，依照指數(shù)規(guī)律衰減，回復(fù)到零點(diǎn)。臨界阻尼的物理意義是：在自由振動(dòng)反應(yīng)中不出現(xiàn)震蕩所需要的最小阻尼值。速度（3-20）2.低阻尼

特征方程：自由振動(dòng)方程：如果體系的阻尼比臨界阻尼小，則顯然有

，這時(shí)，特征方程根式中的值必然為負(fù)值，則s值成為：引入符號(hào)：（3-2）如果體系的阻尼比臨界阻尼小，則顯然有

，這時(shí)，特征方程根式中的值必然為負(fù)值，則s值成為：其中

表示體系阻尼與臨界阻尼的比值，稱為阻尼比，則：成為：低阻尼自由振動(dòng)方程：的解為：

引入Euler方程：引入符號(hào)：其中wd

稱為有阻尼振動(dòng)頻率。則（3-25）則利用初始條件：得到低阻尼體系動(dòng)力反應(yīng)的最終形式：

（3-25）寫(xiě)成矢量表達(dá)式：運(yùn)動(dòng)的振幅（矢量的模）和初相位分別為：（3-27）低阻尼體系動(dòng)力反應(yīng)：

物理意義：低阻尼體系的自由振動(dòng)具有不變的圓頻率ωd

，并圍繞中心位置振蕩，而其振幅則隨時(shí)間呈指數(shù)e-xωt

衰減。如果反應(yīng)的時(shí)間足夠長(zhǎng)，最終會(huì)衰減到零。2.超阻尼體系

特征方程：自由振動(dòng)方程：如果體系的阻尼比臨界阻尼大，則顯然有c/2m>ω

，這時(shí)，特征方程根式中的值為正值，則s值成為：（3-2）（3-38）超阻尼體系反應(yīng)不是震蕩的，體系的位移反應(yīng)從開(kāi)始時(shí)的，依照雙曲函數(shù)規(guī)律衰減，回復(fù)到零點(diǎn)。返回速度較臨界阻尼時(shí)更快。確定體系阻尼比的一種方法體系的阻尼比可以通過(guò)測(cè)試體系運(yùn)動(dòng)的衰減規(guī)律得到：阻尼體系動(dòng)力反應(yīng)：體系從任一時(shí)刻經(jīng)幾個(gè)周期后的振幅比為：取對(duì)數(shù)后：（3-35）阻尼很小時(shí)：體系阻尼的測(cè)試：2）計(jì)算阻尼比：確定結(jié)構(gòu)體系阻尼的其它方法。1）實(shí)測(cè)體系經(jīng)過(guò)n個(gè)周期后的位移幅值比：3）計(jì)算阻尼系數(shù)：（3-36）模態(tài)參數(shù)識(shí)別計(jì)算圖示剛架的阻尼系數(shù)已知：

柱子無(wú)重、剛性梁；

F=90kN使大梁產(chǎn)生5mm的初位移；擺動(dòng)1周后的位移4mm；周期為1.4s.[解]確定梁的有效質(zhì)量:計(jì)算阻尼系數(shù)：阻尼特性:確定體系的自振頻率:六周以后振幅：作業(yè)Clough教材：P26:2-1、2-2、2-3Chopra教材2.1、2.4、2.7、2.132-1COMPONENTSOFTHEBASICDYNAMICSYSTEMTheessentialphysicalpropertiesofanylinearlyelasticstructuralormechanicalsystemsubjectedtoanexternalsourceofexcitationordynamicloadingareitsmass,elasticproperties(flexibilityorstiffness),andenergy--lossmechanismordamping.InthesimplestmodelofaSDOFsystem,eachofthesepro