Statistical energy analysis of dynamical systems : theory and applications PDF
Preview Statistical energy analysis of dynamical systems : theory and applications
STATISTCIALE NERYG ANAYLSISO FD YANMCIALS YSETMS: THEORYA NDA PPLCIAITOSN STTAITSICALE NEGRYA NALYSISO F DYNAMICALS YSETM:S THEROYA ND APPLCIATINOS Richadr H.L yon TheM ITP ress Cabmridge,M assachsuettsa,n dL ondoEnn,g alnd ACKNWLOEDGEMNT This book was writtena s a resulto f researchp erformed for the Aerospace Dynamics Branch, VehicleD ynamicsD ivision,W right-PattersoAni r Force Base,O hio. PUBLSIHER'NSO TE This format is intendedt o reduce the cost of publishingc ertainw orks in book form and to shortent he gap betweene ditorial preparatanido n final publication.T he time and expenseo f detailede ditinga nd compositioinn printh ave been avoidedb y photographintgh e text of the book directly from the author'st ypescript. Printedi n the United Staotfe sAm erica Copyright@ 1975 by the MassachusettIsn stituteo f Technology All rightsr eserved. No part of this bock may be reproducedi n any form or by any means,e lectronico r mechanicali,n cludingp hotocopyingr,e cording, or by any informations toragea nd retrievals ystem,w ithoutp ermission in writingf rom the publisher. Library of CongressC atalogingi n PublicationD ata Lyon, Richard H. StatisticalE nergyA nalysiso f Dynamical Systems. Bibliographyp:. Includesi ndex. 1. Structurald ynamics. 2. Vibration.I . Title. TA654.L9 624'.176 75-19074 ISBN0 -262-12071-2 ISBN0:- 262-62(1P7a5-p4e rback) CONTNETS PARTI . BASICTH EOYR Chpater TheD eevlompent oSft taistcialE nregyA nalyiss 3 1. 1.0 Introcdtuion3 1.1 A BreifH istiocarlS urvye 7 1.2T heG enerla Procdeureosf S EA 10 1.3 FuturDee evlopmnets ofS EA 14 1.4 Oragnziationo fP atr I1 6 Chapetr2 . EnergyD ecsriptino ofV ibratiSnygst esm 20 2.0 Introudctio2n0 2.1M odalR eosnatso r21 2.2M odalA nalysios fD istruitbedS ytsesm 34 2.3D ynamics of fIininteS ytsesm 43 2.4 Moda-lWaveD uailty 52 Chpater 3. EnregyS hraign byC opuledS ystesm 65 3.0I ntrodtuicon6 5 3.1 EnregyS hairng AmongR esontaosr 66 3.2E negry Exchnagei nM utliD-egre-eof-Frdeome Systesm 86 3.3 Recipriotcy ndaE nregyE xhcagnei nW aveB earinSgys tems 93 3.4 Some SampleA pplicatiosn ofS EA 103 Chapte4r. The Estimaitono fR esposnei nS tatsitical Enregy nAalysis 115 4.0 Inrtodutcino 115 4. 1M eanV alueE stimatse oDfy namicalRe sopnse 116 4.2C aclulatino of Varaincei nT epmoraMlea nS quarRees pones 128 4. 3C alcualtiono fC onifdenceC ofeficiesn t135 4.4C oherenEcfee fcst -PureT onea ndN arorw BandRe sposne1 40 AnnotateBdi lbiogrpahy forP artI 153 PATR II. ENIGNEEIRNGA PPLICATINOS Introdtuicno toP artI I 173 Sectni oI. TheU se ofS EA inP reliminayr Desing Chpatre 5. ResponseE stiamtoinD urig nPreliminary Desing 175 Chapter 6. Procdeureosf S tatsiticaEnle ryg Analyiss 178 6.0 Intordutcion1 78 6.1 Modleign the System 178 6.2 Evlauatitnhge P armaeters 179 6. 3 Solvnig for theE neryg Distribut1i8o1n 6.4 EvalutaingRe sposnef orE negryE stimate1 82 Chapter7 . Estiamtioonf D ynamcial Resposne 185 7.0 Intrdouctni 1o85 7.1 Represnetationso f theE nery gEstimate1 85 7.2 Covnersino fromE nregyt oO therV araibles 187 7.3 ResposneC oncenttirnoa Fatcosr 192 7.4 Varaince d aEnstimatiIontne vrasl 196 7.5 Using Varaincet oC alculea Stafeyt Factosr 199 Chpater8 . Estimaitng theE nregyo fV ibratio2n0 9 8.0 Inrtoudctoin 209 8.1 How the OvreallS ytsemi sD escriebd 209 8. 2 Altenartive Formo ft he EnregyE quatoins2 16 8.3 ParmaeterE vlauatiUosni ngt heE negry Equaitosn 217 8.4 UsefulA pproxiamtiso nandS ipmliifcaitosn 218 8.5 Summary 221 Chapt9er. The Menaign andU se ofS EA Parmaetesr 224 9.0 Inrtoudction2 24 9.1 Dissipaiton Paarmeters 224 9.2 PowerT rasnferP arameters 226 9 . 3 Modal Count of bSsyustems 229 9 . 4 InputP owerP rediction2 32 9.5 Conlcusiosn 235 Chaprt 1e0. Modelign the Ssytem 238 10.0 Introcdtuino 238 10.1 Definiiotno fS ubsytsems 238 10.2 Idenitfyniga ndE vlauatitnhge C ouplign Bewteen bSu systems 240 10. 3S ubsytsesm Withina Sectino oft he System2 42 10.4D iscussion 243 SeciotnI I. Evalautiono fS EA Paarmeters Chapter 11. Parmaeter Evlautaion- The EgninereingB aseo fS EA 248 11.0 Introdtuicon2 48 11.1W hatA ret heS EA Paarmetesr? 249 11.2H ow Exacyt Clan( Must)W e Know SEA Parameters? 250 113. How toU se TihsS ecotni 251 Chapter 12.T heD amipngP arameter 252 12.0 Intordutcino 252 12.1 Measuremneto fD ampign 253 12.2 DapmingV aluse forM aetrilas 255 12.3 Dampingo f Bulit-up Strutcuers 258 12.4 TheD ampign ofA dd-OnS ystems 260 Chapter 13. Evlauatgi tnhe ModeC ount2 78 13.0 Introdtuicon2 78 131. Expreiemntal Prcoeudres forD etremniingMo deC ount2 79 13.2 ModeC ouns t ofc ouAsticla Subsytsems2 80 13.3 Flat rSutcutrse 281 134. ModeC outn oCfu revdS trucrtesu 283 13.5 OneD-imnesionalS tructur2e88 Chapter 14. Evlauatgi Cnopulign Loss Factors 294 14.0 Introduct2i9o4n 14. 1 Copuling BetweenA coutsical Space2s95 14.2 Copulign Bewteen Structeusr andA coustical Space2s9 9 14.3 CouplingB etweenS tructulr Saubsytsems 303 Seciton II.I An Exampleo fR esposneE stimatiboyn H uw G.D avies Chapter 15. Virbatioonf a Reentyr Vehicel 317 15. 0 Introtdiunco 317 15. 1 Modeling hteV ehicel 317 15.2 ModalD esniyt 319 15. 3 CopulingL ossF acotr 323 15. 4 Prediciton oft heV ehicel SkiVni bratiLoevnel s 334 15.5 VibratioLne evls of the InstrumentS hlef 340 15.6 CofnidencLei imts 344 Refereenscf orP artI I 365 Glossaryo fS ybmosl 369 Index 374 AutohrI ndex3 87 PART I BASICT HEORY CHAPTER1 . THE DEVLEOPMTE NOFS TATITSIACL ENERGYA NALSYIS 1 .0 Introductoin Thisb ooiks a presentaotft ihoebn a sitch eoarnydp ro cedufroera pplicaotfia o bnr anocfhs tudoyfd ynamical systecmasl lSetda tistEincearlAg nya lyswihsi,cw hes hall refetroa s" SEA"T.h en amSeE Aw asc oiniendt hee arly 1960t'ose mphasciezret aaisnpc etso ft hinse wf ielodf studyS.t atisteimcpahla stihzaettsh es ystebmesi nsgt udied arep resumteodb ed rwanf rosmt atistpiocpaull athiaovnisn g knowdni stribuotfit ohnesid rya nmicpaalr ameteErnse.r gy denottehsep rimavrayr iaobfli en tereOstth.e dry nacmal i variasb sluecahs d isplatc,ep mreenssuertec,.a ,r ef ounfdr om thee nregyo fv ibratiTohen t.e rAmn alyissiu ss etdo e mpha sizteh aStE Ai sa framewoofrs ktu drya,t htehra an p ar ticultaerc hnique. Statisticla Vibratni Anoalysis. Statisticaapplr oacehs ind ynamIcala naylsIs havea lonqh istory. Inm echancis, we arem osfta miliar wiatphp ltihceatitorit ohnev ibrattihoant isr andionmt imoef a determinsiysstmti.ec W es haluls e thiasn alyisnip sa rtosfC hapt2e arns d3 . Iti su seftuol emphashiezrete h atth ei mportfaenatt uorfSe E Ai st he descripotfit ohnev ibratsiynsgt aesma memboefra statistical populaotrie onns embnoltew ,h ethoerrn ott het empobreal haviiosrr andom. Traditiaonnaally osfem se chansiycsatlve imb ratoifo n machiannedss t ructhuarvebese edni recattet dh el owefre w resonmaondteb se cautshee smeo detse ntdo h avteh eg reatest displacermeesnpstoei n nm aniyn stanacnedst ,h ef requencies ofe xeitawteirofena irlloyw .O fc oursteh,ev ibratoifo n wallasn df looarnsdt heihri gfhr equesnocuynr da diathiaovne beeonf i nterfeosrta lontgi m,e butm echanaincdasl t ructural enginehearvsbe e egne neraulnlayw aorfoe r u nconcewrintehd thiwso rk.T hea dveonftf airllayr gaen dl ightweaiegrhot spacset ruct,u arnedts heidrc tenntdh aigfhr equebnrcoya d banldo adhsas, m eanmtu cmho raet tenttiooh ni ghoerrd er modaaln alyfsoirsp urpoosfep sr edicsttirnugc tfuratailg ue, equipmfeanitl uarnedn oisper oduction. A characteroifhs itgihcoe rrd emro daen alyshiosw,e ver, isb asiucn certaiinnm toyd apla rameteTrhser. e sonance frequenacnidme osd seh apoefst hesmeo des gsrheoaswte nsi- Basci Theoyr 4 tiviy t tso malld etaliso fg emoetrayn dc ontsructino. In additio,n thec omptuerp rogmrsau sedt o evaatleut hem odes hpaes andf requeinecsa rek nwon to breat heri naccuratfeo rt he higheorr demro des, evenf orr atheri deailzde systmse. In lighto ft hesuen certaiien,st sa tatsiticamlo delo ft he modalp armaetersse esm quitnea trual anda pproipatre. If threei sc ausfeo rs tatsiticaalpp rocahesf rmo the naturoef t hed ynamicalp rolbem,t herei se qualm otiviaotn frmo thev iewpointo fa pplicati.o nMechnaicaaln ds trucutral deisgnres aroef tefna cewdi thm akinge nviornmentala nd responesset miates ata stagei na projectw heres tructaulr detaiils n otk nwon. Thesees tiamtesa rem adef or the qualifiticoano fe qiupmenta ndt hed eisgno f ilsaoiton, dampin,g osrtr ucturcaonlf iugratnisot op rotcete quimpent andp roetctt hei ntegyr iotft hes trucutre.H ighldye tailed analeyssr euqirnigs peicfikcn woledgoef s hap,e contsructi,o n loadinfgun ctinos, etc., aren ota pprroiptae. Simpler statsiticaalna lyticaelst iamtse ofr esopnset oe nvrionment thatp resveerp arametedre penden(cusec ha s dpamin,g avreage panel thicknes,s etc.) area pprporaitet ot hed esingers' needa tt hiss tage. Inspaitrionfso rt heS EA Approcah. Threei se xperience in adleingw ithd yanmicals ytsemsd esrcibebdy ramn dpoara mete.r sTwo notbalee xmaplest hath aves evreda s" touch stones" ine arlyd evleopmetnso fS EAa �e thet heoroyf r omo acuositc,s ansdta tsiticamle chiacn.s Rooma cuosticsd eals witht hee xcittaino ofs ytsemso fv eyr mayn degreeosff ree dom (hterem ayb eo vre am illoinm odeso fo sciltliano ofa goods izedr omo itnh e audiblfere quencryan ge)a ndt hei nter actinosb ewteens uchs ytsems( osundt ranssmsiiotnh rougah walli sa ne xapmle). Thea naylsesa rec arrieodu tu sing boht modala nd wamvodee l.s Thev eryl arqe numebro fd egrees off reeodm isa na davntaeg frmo as tatisitcalv iwepoitn itt endtso d iminitshhe f luctuatioinnp sr eidctin oof resopnes. We shalslh wo whyt hisi ss oi nC hapte4r. Statsiticamle chnaicsd easl witht her adnomm otiono f sytsemsw iht eithera few orv eyr amnyd egreeosff reedmo. Howerve,i ti sr andmo motiono fa veyr speicalt yep,w hich we may·call "mxaimlalyd iosrder.e"dI nt his stateo f vibrat,i aonllm odes, whethetrh eyr eosnataet f reuqencies nera eacho theorr a ref ara par,t tendt oh avee quale nergy ofv ibratiaonndt oh ave incoheremnotti on. Wem usta dd the proivs,o "ignorinqgun atmu efefct.s"T he erngye of the modesi s, saidef rmo au nievrsaclo ntsan,t thes ytsem tem peraurte. Thes tateo fe qula modale nregyi ss pokeno fa s TheD evleopmento f ESA 5 "equipairttioonf e negry". InS EA,w es ometmies maket he equirptaitni aossmuptionf orm odest hat rensatoei nt he same freuqencbya nd, but not aflolmr o des. Statsiticamle chniac,s anidt sr elateds cinece, heat trasnfer, alsote achu st hatt hemral( arndomv ibrtaio)n energfyl wos frmo hottetro coolesry stmse, andt hatt her ate off lwo isp ropoirontalt ot empearture( modale nregy) dif Chanter3 . wseh wo thatt hisr esulatl soa pplise fp.rp.nce. In tod ynamical styemsse xictde bby roabda ndn osies ourecs. Noto nlyt hat, buts incew e alsosh wo iCn hpater3s and4 , thatn arrwo band sourceasr ee quivaletnotb roabda nds ources whens ytsema vreageasr et ake,n the resulctan beg enearlzie,d withp ropecra r,e tpou ret one.s Thes tudyo ft he statsiticamle chnaicso f electrical circuisthswo s thata reissotra tk nwon tempearturies equivaletnota thermarle esrvoir (ro temperatureb aht). The interactoifot nhe dynmaicals ytsem wiht thisr esevroiri s repreensteads a whiten oiseg eneartori nc icruitt heoyr. We aruamenta rounadn ds ay thata damper (mecnhiacal ('!i'ln turn the resistanecleme nt), inc onujnctiwoint h nao sieg enera,t or e reprseentsa thermalr esevroi,r andw e shoulndo tb es urprised whenc ertai"nth emral" resuldtesv eolpf romi tsa naliyss. TheA dvantageasn dL imitatioonsfS tatsiticaAln alyiss. Onea dvnatagoef s tatisticala naliyss ofs ytsemsm ay bes een frmo thep rcatciala specotsfr omo acoutsic.s If onter uly with. event hem .s.p ressruea ssociead t hH� 106 mnnp.� to deal wiht eac,h changiwnight time tahse fluteg ivse wayt ot he tympan,i wouldb ea hopeelssm ass ofi nfromatino toa ssmiilate. Whato ned ose intseadi st od escirbet hef ielbdy a few coherefneta tusr oeft hem odalp atter(ndr iectf ieladn da few earlyr efelctinos) andt hei nchoerenetne rgy (rveebrearnt fiel)d totalled intoa few freuqencbya nds. Thu,s intseda of 6 10 measuroefst hes oundf ield(wh ihc wouldb ei ncmopletien anyc aswei htoutt hec ohereen dcat)a,w e area blet od escbrei thes oundf ieldf eecftvielyb y1 02-0m easur.e s Thes tatsiticala naylsisa lsoa llowsf orm uchs impler decsriptioofnt he sytsem, whethero ned escribtehsef iedl by omdeso rw aves. Int hef omrerc as,e modald enisty, averagmeod ald amping,a ndc ertaiavne ragoefsm odali mpedance soucresa rer euqierd. Ina waved esrcipti,o snuhc to sound parmaeterass m eanf reep aht for vweas, surfacaned v olume absortpino, agned neragleo metrci confuirgatoin arer equried. Then umbero fi npupta rmaeteriss g enelrlay ibna lnacew ith then umb ero f amseurse (102-0)t ob e taken.
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