Aeronautical engineering - structural dynamics and aeroelasticity

Completed notes of the course

Complete course

Structural dynamics and aeroelasticity New paragraph/ new theorem Demonstration Definition Important Legend TYPICALSECTION•TYPICALSECTION:RIGIDWINGMOUNTEDOnanELASTICLOWAMPLITUDEPITCHOSCILLATIONSeareCONSIDERED°TOES/☐~:ITISTHETW'STINGOF☐NESECT/☐~OFANOBJECTWITHRESPECTTOANOTHER-SHEARCENTER:ITISAPOINTOFAZDSECTIONVVHEREIFYOUAPPLTATRANSVERSESHEARLOADITProductsZERORATEOFTWISTONTHESECT/On•ELASTICa✗1SiITISTHELOCUSOFSHEARCENTERSALONGTHESPANOFTHEWING•FLEXURALCENTER | TWISTCENTER)iitISTHEPointW#ERETHESHEAR(☐ADPAPPLIEDTOcantileverDOESNOTCAUSEtorsioniOFTHESECT/OnABC,butnotNECESSARILTinTHEWING /la) : Ò(b):O..FCLOT EA1zFLEXURALCENTERELASTICAXISa0•O=0ATTHEFinalSECT/OnWHERETHELOADISAPPLIED ② =0ATTHEFinalSECT/OnWHE.EETHELOADISAPPLIED•THEtorsioniisnull☐ultonTHESECTIONVVHERETHELOADISAPPLIED •THEFCOFEUERYSECT/ONISCLEARLYDEPENDENTonTHESECT/ONITSELF,butALSODEPENDEONTHEUTHERSECTIONSEA:ITISaproPERTYOFTHESINGLESECT/On { ,,,,,,,ayoga,aaaaa,o,,.ee,.ae«augure,,,aaaaa,,,,ma,,.am,gagnon,gg,.ae,,,o,con,,,,'NGTHEPC-rturb.ESForm•EQUILIBRIUMElevation:LE+Mac=Ka@9se/(la 0 + Clpp )+ qsccmpp =Ka @ • ( ka-qsecix/O=9s(ecc pp + ccmpp) •@=9SP(eccp + CCMP)trimSOLUTIONka-9Secca•Remarks:1)☐EC-NDSonTHEFLAPDEFLECTIOn2)THEDenominatoreISALWAYSTHEEFFECTIVESTIFFNESS: È =Ka-9Ka3)THEDERIVATIVESATTHEnumeratoreAREPHYSICALLYOPPOSITE:ROTATINGTHEFLAPSURFACEDOWNWARD1GETanINCREASEOFLIFTGENERATEDBYTHEFLAPandaCONSEQUENTNEGATIVEPITCHINGMOMENT:( Lp >0(MP ☐ 9dg9130907 1 / 1-✗)92 -1-19-1=o9,>☐ 9dLQdf913021-1±1-1+c,1 / 1-×)Qdf9130QDF91309,>☐ 9dL •9==2 In -×)91>☐ 9dL 29009☐f. QDO - 907±(9☐o9 ☐ f) = QDO -91>f+(qioq☐ f) ☐ 9dL211-×)21-9☐f±1-9Df+G|1-X)9☐fifTHESOLUTIONWITH"-"isNEGATIVEHASTOQDOQDOQDO°9☐=QDOBENEGLECTED211-×)9☐900•REMARKABLECASE:Unrestrainedflap |Kp =0)z1,5✗=0,5•01☐f=-'< P =o9☐=QDO1✗=o SfcfCap 1-✗✗=-10,59☐f900✗=0,59☐=29☐o• xp --o: | ✗=a.»☐nero.on.ca.namepressoreiss.a.e.cann.ae/--ea-c-☐INTEGRATIONBYParisonSwi: | " SO 'GJO' dy =_ SOGJO "-◦- § SOGJO'" dyf '8f8fg'0-,-Ll-|-•Swi=Swe:- 50050.0 - [ SOGJO'" dy = | SontdySO.GJO'+mt.=o O' ll)=o00101=0•GIVENTHEarbitrariNESSOFVIRTUAL☐ISPLACEMENTS:GJÒ""tMT=0STRONGFORM(DIFFERENTIAL) ✗L:SEMI-SpanMace-d•LOADFACTOR:N=LFi=WN=magnWe°remare:IHAVECHOOSENEA'~FRONTOFTHECGACmgn4EACG•STRONGForm:GJ0""tmt=☐•mi=9C/ 4) eCid(✗☐+0)+9c'(4)Cnac+ dmgn I•HP:ConstantPropertiesGJO"=GJO"•GJO"+9Cecco+qcec.la@+9C≥Cnac+ dmegn =o⊖"+9Clicca⊖=_9C ( ecco+c.Cnac)- dmSNGJGJGJ•ITISCONVENIENTTOUSEnon-DIMENSIONALVA R I A B L ES:• I =4Ls 0 "=d0 "^=1 DI 1dodi =d, do '= 0 LZd4didudu l dìdy **e10+9Clicca⊖=_9C ( ecco+c.Cnac)- dmSuL2GJGJGJ•@**t9C@Cla(2 ⊖ =-9cL≥(ecco+CCMAC)+- DMINÙlla:aerodinamicoFORCINGTERMGJ0JGJIn:INERTIALFORCINGTERM midaUn@**+ né@=latInHARMONICElevation• {0,01=00*11 )=0↑=4IFy=L,THEN I =1LHOMOGENEOUS Proponenti LET'SFin☐THEVA LU E SOFwe(I.EDYNAMICPRESTRE9)FORWHICHAnonTr i v i a lSOLUTIONEXISTSTHISTURNSOUTTOBEanC-IGENVALUEPROBLEM •FOREACHC-IGENVALUE µ ITISPOSSIBLETOFINDanASSOCIATEDEIGENFUNCTION Ò /Ù)≠☐THATSOLVESTHEElevationi Òly )= Asiatici )+BCOS HI )•LET'SSUBSTITUTETHEGENERICSOLUTIONINTOTHEEQUATION: vitasinlny )- più B costi -4)+12Asinlreyltribcostui )=0-_•_A Sintini )+ bcostuyl.lu?m2 )=o li = miaÒly )-1-0 ÒIo)=0:o-A+B=o•LET'SAPPLYTHEBC: Ò '/11=0: amicostui )- ribSink )-.----☐1A0•= IcostivisiviB0-_--__°Tr i v i a lSOLUTION:A=B=0--01 lì =0•nonTr i v i a lSolution: dlt =0"¥ Cosmi =0 COSIvisivi =Ttmit,ME-_2☐IVERCENCEDYNAMICPRESSURE mi =9ce ciai G)22 mi=e2 :9cacca Ì =TI+MIT 9dm =TI+MIT0J2gj22EccolaLZ/µ =,+n,2•REMARZK:BEINGACONTINUOUSPROBLEMWEHAVE☐BTA'NEDanINFINITESETOFDYNAMICPRESSLEE.THE☐NERELEVANTISTHE(☐WESTiITISINTERESTINGTOCOMPARETHEDIVERGENEEDYNAMICPRESSUREOFTHECONTINUOUSMODELWITHTHE☐NECOMPUTEDUSINGTHE2""""""""°". /""=""2EccolaLZ2ITGJ=kaka= È GJ2EccedereCLCCla4L91>ts=KX=KX=kakaseccacheCla -I•THEASSOCIATEDtorsionaleDEFORMATIONMODES(EigerFunction)are: Òry =sinttMIT42L--FORCEDPROBLEMO"+ zio =la+In=IT(0,01=09*11 )=0•ConstantFORCINGTERMConstantparticularSolution:⑤+ NÉ@p=ItOp =t ri COMPLETEPROBLEM•COMPLETESolution: ②(Y)= Ò + Op =A sinoeiey )+Bcos(µ] )+ Itlez0Io)=oiBtet=onel•(C-T'SAPPLTTHEBC:0'/11=Oi amCosta )--Bsin Lu )=o-.----☐1A_QT-2•=~costeinsinuiB0-.----B=-llt→2 i Aucoin +QTA=-etTa r y nµ, µ S' µ =°µ,--•ANALYTICALSOLUTION: (Ù)=Qt- tamenSimon] )-cos(µ] )+1 ma --CASE1iVariationOFLIFTDistributionatConstantancheOFAttack•Hp:d=0NECLECTABLEinertialEFFECTS:Llm=0SYMMETRICAIRFOIL:(Mac=O-llt=lla+In=-9cL?(eccotCCMAC)0J •llt=_01cL>ECCO.GJ=-✗oConstantinTHISCASE→2GJqc@Cia E --_-•Cie=Cia@= Cisl -Xo)- tamensinoeny )- Cosby )+1=-[Lo- tamensinoeny )- Costley )+1----•Efficiencyratio:L=[LO+Cle=1+CLe= tan~sinh.ly )+cos Neuf )La[LoIS Cia9☐=^KO9☐=^KO1cos'1seccacos'1secca1-KO 5 tantiKyè2•remareiQDDEPENTSSOnIandè case1:1>0,è>o•KO [ [and>☐itispossibleto/☐entiF>aCriticalSWEEPanche50THATQD00☐a9☐☐)NEUERDIVERGE•WHENTHEWINGBENDSUPTHERE'SAREDUCTIONOFTHELocalLIFTWHICHISHIGHELTitanTHEINCREASEOFLIFTDUETOTWISTBENDINGHASaStars/liZincEFFECTcase2:1oKO Ò tante=^KO1KYè2cos'1secca1-KO 5 tanteKlfè2°QDLoversasIGrowsDivergenteOCCUREarlierandEarlierEXAMPLE:I=-30°QD~10..QD/1=0)e LIFTEFFECTIVENESS--- g- -è• Es -ll 1¥ = Ò✗0COSIA. ÈÙ =-Z---_--- è - ky + ilbtant-d-e-tandè2•=1 È ✗o. È .dei= èb-io-a-e'"^b-22--,-•(141=9Cciaa = È c- Cid × /Lo+ 01-41COSI-2-'lf)sind)= ÌCCia ✗ocosa= Èc-Cia ^cosa+ Ò -4- tann COSI✗TZ'= Y • Lrtyl = ÌECia ✗oCOSI•DEVELOPINGLI4)(SEESLIDE1301)WEGET:l=1=1=1LR1-ll1-91-9UDYDYDL1'+92 SU " EJRÉTT "dx, sui = | ,Bar:BEAMSUBJECTTOtorsionaleLOADUn=Oi3. | ma=-×>0in,ma[13iz0✗39M}O[12Xzil}=✗20/✗1)ein•Mi= | ✗2713-✗3712da= | ✗26 / ✗20")-✗3G / -✗30")da= / a 61 ✗ è +✗ 32)da0'=gj*0'aaGJ* E •mi=-IO Ò +9T ÷ dx,•EQUILIBRIUM:MttDMT+MiDX1-MT=0Mt+DMTI◦dmt+mt=0GJ@*'-IOÒ =-9Tdei VIRTUALWORKDUETOTORSIONALLOAD•Swi= |{ETEdu= | , SAIIna+ 5613713dad ✗1= | ,I-✗350')o / -✗30')+ 1×250 '/61×20 ') dadxn = Èlà .V= | ,so' |61×22 +✗ ÌIda0dai = | , solo >*0'duatEULERBERNOULLIBEAMRESUME1)axialLOAD:Ea"n"-miei=-91 Su'TEÀ ne'dx,swi=p ,"2)BENDINGLOAD: EJÌW "+MÙÙ=ma'+93Sui= SW " EJÉ W"dx, 1 ."c- JÉV "+nè=-ma'+92 Sui = | , SU " EJRÉV "dx,I3)To r s i o n a l eLOAD:GJ@*'-IOÒ =-9TSWI =SO'GJ#0'd✗1 I. LIMITOFTHEMODELsn sto snTHENORMALCOMPONENTMUSTHAVEACOMPANIONSTRESSONTHEFREEOUTSIDESURFACE,WHICHDOESN'TEXISTForHP.HENCE,THENORMALCOMPONENTSMMUSTb.EABSENT.ITMEANSTHATWar p/NC,WHICHISADMISSIBLEFORREALBEAMS,Can'TEXISTFORTHISMODEL•THESIMPLESTMODELTORECOVERanAV E R AG EEFFECTOFWarinGISTOCONSIDERTHATSECTIONSDON'TREMAINperpendicolareTOTHEBEAMAXRSTIMOSHENKOBEAMun=✗ 304 (✗1)Nz=ole}=w/✗1) FREEVIBRATIONSFORTHEBEAMBENDING•consideraTHEHOMOGENEOUSProblemforTHEuniformeBeau:C-JZW""""+MÙÙ=0•THISisaPDEINSPACEANDTIMETHATCOULDBERESOLVEDUSINGTHEAPPROACHOfSeparationOFVariables:W/×,t)= f- (×)9(t)•c- Jzf ""9+ nfòi =o•C-Jz f """=-È=è=constsincelhsisOrlyfunctionOFSPACEANDthisis☐NLYFUNCTIONOFTIMEitRESULTSTHATTHEOnu> mf 9VA L IDCONDITIONISTHATBOTHALEC-QUALTOCONSTANTÙÈ +ut9=o i f ""- nèf =0C-Jz•LET'Scall P "= MÙ sothati f- ""-P" f =OC-Jz"GENERALSOLUTION: f /✗|= fol "". | ✗4-p4)f-o=oTr i v i a lSolution: Fo =Oti =P NONTr i v i a lSolutioni✗4=4✗z=- P ✗3= jp 'a=- jp +ceP"_ jpx •THEGeneralSolutioncanBEWrittenas: f /×)=al×+ bè ×+de•sinhx=e"-e-×coshx=e'+e-×22•LET'scalla=B+ab=B-Ac=D+c d =D-C22222222•b+Ae×+B-Ae-P"=a EP "-e-P"+Be✗+e'×= Asinhpx + Bcoshpx 2222z2AEIRbeIR=( ein - IP ")+c☐+ j c eip "+D- j c e-IP '☐+e /eip "_ e-JP " ) =222222=D zcospx +C j(zjsinpx)=- Csinpx +☐ cospx 2ceIRdeIR- jpx +be-P"+< ejpx = g ,,,= aep ,,ga=a,,, g ,+ gaogygg ,+ ag.mg ,, gaoggg , PROBLEMOFACANTILEVER f- 10)=0MIL)=-C- Jzf "=oBc: f '/01=0SIc)=-C- Jzf '"=o5=dmdxt. f 'l×)= acoshpx + bsinhpx + ccospx - dsinpx 'IMPOSINGTHE4BEWEGET:--01o1'a'101ob=o-COSPL- sinplsinhplcoshpl C--COSPLsinplcoshplsinhpl .- d -H=°THEvonTr i v i a lSOLUTIONISGIVENBY: dlt/¥)=0,WHICHLEADSTO:cos plcostiGL =-1•LET'SSOLVEITGRAPH'CALLY: COSPL =_1 coshpl ,ei1iSOLUTIONS•WITHTHEKNOWNVA LU E SOF P l/SOALSOOF P)itisPOSSIBLETOIDENTITYEXACTLYTHEConstantS2,6,C,dANDSOALSOTHEEIGENSOLUTIONSANDTHEFREQUENCIES è :lui=EJz PÌ nn•ONCEHAVINGOBTAINEDWIITISPOSSIBLETOSOLVEÓI+ut01=0•9=qoe"◦(X2tU2)010èt =OnonTr i v i a lSolutions:✗=± ÌW •9It)= geiwt +ne- Int = gcoswti-jgsinwti-hcos.net - jhsinwt =(y-1h)cosut+ ljag - jh )simul= gsinwt +hcoswt8h PROPERTIES OFMODALFORMS Iwit •GenericBENDINGMODALSOLUTION:W/✗,t)=Wi/✗)eC-IGENFUNCTIONMODALFREQUENCY(MODALSHAPE)•(c-Jzw")"+ miei =ovi= jwiwi /×)e"it,vi=-wi≥wiix)@twit,w"=wi"(×)e"il•LET'STA K ETWODIFFETZENTMODALSOLUTIONSlui,WjFORWHOMi2(c-Jzwi")"-nwiwi=o |(c- Jzwj ")"-nwj≥wi=0eMULTIPLYTHEFIRSTElevationBY Wj ANDINTEGRATEOVERTHEDOMAIN:--zl | " wj . / c-Jzwi")"-maiwi. dia =0 wj/ Esami")"d.✗1- | , mwiwiwjdxn=o ti fg '0•SOLVETHEFIREINTEGRALBYParisTWICEi-" I wj/ Esami")"d.✗1=. / c-Jzwi")" wj _☐- wj ' / Esservi")"dxei= ti fg 'gf ☐f'9fg'--L-I-LL=. / c-Jzwi")" wj _☐-.C-Jzwi" wj -o+ | ,EJ wj "wi" dufggf '--l_-LL•. / c-Jzwi")" wj _☐-C-Jzwi"wj'_o+ | ,EJ wj "Wi"dei- | " mwiwiwjdxi=0-0"" EJWJ Wi-muri≥wjwida =0• i PROPERTX1iORTHOCONALITYWITHRESPECTTOTHEMASSDISTRIBUTION^ | "Ejzwi" wj "du=cui≥ / " wjnwitaxio☐2^: lui?wj≥) [ mwiwjdei=02 | "Ejzwi" wj "dxe=wj≥ |wjn Witaxio☐ / ' mwiwjdei=o i # j OPROPERTX2:ORTHOCONALITYWITHRESPECTTOTHESTIFFNESSDISTRIBUTION1Ejzwi" wj "dxe=cui≥ [ wjnwi d.✗1 i 22^ic-Jwi"wj"dxn= lui + WI ) [ mwjwidei % '2Ejzwi" wj "du= wj ' f.wjn Widiet: L•BEINO mwiwjdei=o i ≠ j c-Jwi"wj"dxn=o i ≠ j ti 1 ." | """ "mi≥-wi j , " l' " "a.=.Omi2c-Jwi"dxn=Zw ?| " MWÌdei c-Jwi"≥ dei =lui≥ | "nwi≥d.✗1. I ☐☐mi mwiwjdei= miSij : µiiMODALMASSc-Jwi"wj"dxn= mini ≥ Sij| "EJwi"≥ dei •ra>LEIGHauotient:Wi≥= µ nwi≥taxi MODALDECOMPOSITIONa•CONS/DERTHECASEVVHERETHESOLUTIONISREPRESENTEDASSUPERIMPOSITIONOFallMODALFORMS:VV(✗,t)=e-=,Wi(X)Qi(t)L•LET'SUSETHEVWPFORTHEBENDINGproblem: | , SW "EJZW" du = § SÉ(Pg- MÙÙ )dxnSwi= Sue •APPLYINGTHEMODALFORM,FOREACH SHI WEHAVEi•• | L•••" | "ewi Sai j,C- Jzwj " aidxi +inwi Sai i.e nwjOÌIdei= /è inwi Soli pasditlo 0•TA K I N GADVARITAGEOFTHEARBITRARINESSOFTHEVIRTUALDISPLACEMENTS,LET'SCONSIDEREACH{"¥InullEXCEPTore:SOTHATWEcanAnal>ZE☐NESINGLEMODE•(L°LIlwiSaij,C- Jzwj " aidxi +y,wiSaii.e nwj9JwiSoliPasditt: -_'•"i." ÷; Iiii! • SoliÈ , [ c-Jzwi"wj" dei9J+ Soli in | " Wip ≥ dei ---O---e/Ei= j : Solimini ≥ ai + puioiiwip } dei =0- ti ---_2• SoliµÈi + miniqi - | " wip } dit =0ForEACHi-MODE-O-•THISEQUATIONSTATESTHATEUERYMODEBEHAVESLIKEAYDOFOscillatoreOFMASS MI ANDFREQUENCYWi•THEGENERAL'ZEDFORCEISONCEAGAINTHEWORKOFTHEEXTERNALDISTRIBUTEDFORCEPonTHEMODEWi•remare:IFPSISALUMPEDFORCE: Pz(✗,t)= SI ✗o-✗)F(t)Dirac'SDELTARITZ-GALERKINAPPROXIMATIONco•in/✗,-11=enilx)Yi/t)PROPERTIESOFTHESHAPEFUNCTIONSNi(✗)co 1) L'NEARINDIPENDENCE:ITEXISTALINEARCOMBINATIONOFCOEFFICIENTSSOTHATL'-1diNoi≠☐2)COMPLETENESS:DEFINRNGTHEErrorasEn=Un- Ùhn ,THENlieuEn=0n00 •LET'SAPPLYTHERITZ-GALEREinAPPDOXINATIONTOourMODEL,CONSIDERINGASSHAPEFUNCTIONSTHEPROPERORTHOGONALMODESL'LET'SSTARTFROMTHEWEAKFORMOFTHEVWPi | , SW "EJZW" d ✗1= | ," SÉ(Pg-nÙÙ)DX1co•W/×,t)=inNi/×)giIt),withni, Nj =o•• | L•••"• |; inNi Sai i.eC- Jznj " aidxi +inNi Sai i.e nnjOÌIdei= /è inNiSoli pasditlo °""""""""""""""""""""""""""""""""""""""""""""""""""°""SOTHATWEcanAnal>ZE☐NESINGLEMODE• IL °ni"saij,C- Jznj " oijdxi +NiSoliin nnj9JNiSoliPasdit1: ""= !: ---co-• Soli ÷,C-Jani"nj" dei9J+ nninjdxn9; = Soli | , Nipsdei • |; sai i. [ ------eii.i= j : Solimini ≥ ai + puioii - |; nip } dei =0---_2• SoliµÈi + miniai - /(Nip } dit =0ForEACHi-MODE------ii.i.- ÷ ; ; ;; """"""" i. =,÷. iii.ai K= Ip ===Oi-..._-e'._-._FINITEELEMENTMETHODco-Ii(✗,t)=i.i✗il×)Ti/t),WHE.aelocalfunctionSHAPESARECHOSENSOTHAT:Gilt)=Il/✗i,[)a) RITZMETHODiTHESHAPEFUNCT/0nsYi(t)AREDEFINEDGLOBALLYANDMUSTSATISFYTHEKINEMATICBOUNDARYCONDITIONSb)FEM:THESHAPEFUNCT/0nsYi(t)AREDEFINEDWITH/~THEELEMENT,inSUCHaWAYTHATTHEGENERAL'ZEDCOORDINATESCOINCIDEWITHTHENODALDiSPLACEMENTSMi{×)ANDROTATIONSÒI(×)OFTHEELEMENT COUPLEDFLEXtonandTORSIONWzzdeL0MacMÒDICOÒ..@BtEAacnùCGVWPL•Sui= |SWIÌEJZSW " dx + | ' sotto>po ' dx o0BENDINGTo r s/0NL.swe= | , SwtPzdi + | Sotnxdx ',LOADSoPz=L-nii+ MÒD ◦mx=Mac+Le- /ICGÒ + MÒddl + niid =Mac+Le_(Ico +md'/Ò +niidIO•NECLECTINGAEDNODTNAMICFORCES:◦ Pz =-miei+ MÒD Oa•mx=- IOO + nivd •RESUMInoVWP:oa | " SÌ "E Jzsw " dx + [ SÒTGJPO ' di = |; sèI -miei+ mòd)dx + | ' SÒI - IOO + niidldx 00RITZ-GALERKINnw__--• wlx.tl =i.eNwiIx)qui/t)=nlw9¥ ----nw• 0/x.tl =i.eNoi1×190il -1)=ne9,0 00.11T'l'T' nènrlwdx9W + 59 |; nwE>a NYdi9¥ + STÈ | " no GJP Iodx90+ 59 |; --Ot•°+ Sopot (y t••..- SYÈ |; NwndIodi9.0NOIONIdi90 - SIÒ [ NÉndrlwdx9¥ =o----l'tIlL• sé - fi "- fi .-NwE Janyudi9¥ + | , nènrlwdi9WNÉmdIodi9.0 +in_.---T,T|T••LToe'+ 599 ' [ NO GJP Iodx9,0 + | " IOIOIOdx9g - | , IOndNYdi9¥ =o-0- ( """"""""""""""""""""""""""""""--,_..-T-i -1 : nènewdxEw +-Nwmdno di9.0 -1 |; NÉ EJ≥ NÉdi9¥ =o ti ....-----MwvvMWOkww---- NÉIOIOdxÈ '- [ NÉndrlwdi9in ++ NÌTGJPNÉdx9,0 =o ti . ti ----MowMOOKOO---..--___--MwwMuroKw w9¥==910E==◦+=a@-MOWMOO--9-0--KOO--9-0--E-===•REMark:MWO=MÒW==•REMARK:ONLYINERTIAISCOUPLINGBENDINGandtoastcon ----•Mww=Inni [ WW=nlrwiluw?=i.i.--------i....2• 100 =noi [ 00=→iwoi'_.-i.-- MODALANALYSISEIGENVA LU E SANDEIGENVECTOR• 1È +K9=0=-°GeneralSolution:= Yi e]"itC-IGENMODEEIGENFUNCTION/MODALSHAPE)mYi=-Ky i•- [WÌ + ETi =0 Eli =cui≥Miei=---=-t--'_.--2• E = 1I WHERE0= -91 ,..., In -,1=lui=='_.--STATESPACEAPPROACH•a :-.?? °GENERALSolution:=e>£• la -✗ E)I =o ≤il =✗ I =--'_.--•☐c-Finivo:=_ & ,..., In -,1=✗[ = I ='_.--•BEING1Diagonali= ¥ 1=tA∅STARTINGFROMASTATESPACEFORM,WEcanAPPLYTHISTr a n s f o r m a t i o nto=====MAKETHESTARTINGMatrix/A)Diagonal()=MODALSHAPESNORMALITA Z'0N°MODALSHAPESAREDEFINE☐unLESSFORaConstant.SEVERALNormalIZATIONSAREUSEDi1)unitariMaximum☐ISPLACEMENT:Max{Di}=1----=i.i.T2)unitariMODALMass: lui =1⑦iMi=1M=Mi=1===='_.-i..-- E'GENMODESTODECOWLE• 1È +K9=0=-'GeneralSolution:= Qi e]"it•- qui + ETi =0 Eli =cui≥mai--=-'LET'SPRE-MULTI(YTHISEXPRESSIONBY jt i-2'wi YJ T?Ii = YITKYi--=-22TwjIit?Ti = IiIi @SINCE MIT?Li 'SSCALAR,itisElevateTO Iit[j.THESANEFOR YI t-. Eli e21: lui ?- wj ≥) ipjtmYi=0-=-ORTHOCONAL'TYWITHRESPECTTOMassMatrix: MITLi =0IFi≠ j -e21: lui + wj ?)YjtmYi=2 YjtLi -=--i#jORTHOCONALITYWITHRESPECTTOMassMatrix: lfjt KYi=0/Ei≠ j -=-MODALTRANSFORMATION• 1È +K9=F=----◦MODALTRANSFORMATION:=EWHERE∅=_ & ,..., In -=LagrangianiCOORDINATESNATURALCOORDINATE5• I.È + [? =F 0T M È +' ≤? =TFTMÈ+TKz= TF -==-=======-----e.....•TMTK=mi=mini≥===)===- i. i--•FOREACHMODE: miÈ+mi≥2-= Yi "F-- TRANSFORMATIONTOTHEFIRSTORDERSTATE-SPACEFORMe MÌ +K9=0==-DEFINE90= ÌI 9;D+ E =0---.---.- E 090 E9.= ÈÈ - ÈÌ O IÌD K0QD-_-_-=_.-•IFMèisaPOSITIVEDEFINITEMatrix,THEN: È =Ù-^ È9= ȧ ==-COINC/DENTEIGENVALUES•COINCIDENTEIGENVALUESTYPICALLYAPPEARWHENTHEREareSYMMETRIESINTHESTRUCTUREALGEBRA/EMULTIPLICITY:NUMBEROFTIMESanEIGENVALWE'5REPEATED t GEOMETRIEMULTIPLICITY"DIMENSIONOFTHEEIGENSPACEEVASSOCIATEDWITHanEIGENVALUE✗i•'FALGEBRA/EMULTIPLICITY=GEOMETRIEMULTIPLICITYTHEEIGENUALUESARELinearINDEPENDENTANDTHEMATRICESAREDIAGONAL12A@LERIGIDBODYMODES•isTRANSLATIONS+3ROTATIONS:6RIGIDMODESAREC-✗PECTED°RIGIDBODYMODESCONDITION:lui=☐' IÌ +K9=0=- jwit •GENERALSOLUTION:=E•_cui≥+ Ti =0 pi =cui≥MHiKYi=0MiTKIi =Ono☐c-FORMATIONFORRIGIDMODES--=-=--=◦KIS6TIMESSINGULARFoaaFREEFLYINGAIRCRZAFT=°""""i"""""""""""""""""°"°"""iSwi= Sw " EJÌW " da | anno,a.+/¨ qi =.meanstm.tt#....eana.noe..ae-☐•cui✗c☐✗CK-non>-cui☐==1--cuiK-nè__✗S--Fo_.✗S-(K-nè)>+/cui).awk-nè-.Fo_•AMPLITUDE:✗=✗ il +✗s?=/CUI> FÙ +(K-nè)"Fo>=Fo-- .lk -nè)>+lcw)'-=./K-nè)>+ ICWÌ -=(K-nè)>+lcwl'--•§=CLICLOADINGEXHIBITATYPEOFINTERNALDAMPINGCAUSINGENERGYLOSSESHYSTERESISVzV•✓'SCOUSDanINGiPROPORTIONALITYBETWEENDAMOINGandFREQUENCYiWc=TIC✗☐Len@☐=(Un § "=ClunK2Kh2•It>STETZETIC☐AMPING:noPROPORTIONALITYBETWEENDAMOINGANDFREQUENCY:WC=IT"¥✗ono☐EPENDENCYconUnGENERICDAMPINGCOEFFICIENTFORHYSTERETICDAMPINGh•@☐=1Wc=1 ITL+02 =h21TWK21T1K✗ è K2 g ""•=ED="¥22Kmi+ci+kx=FMÌt2§nunÌ+Kx=F | ,= zg n,=,gn,= zgnw ,nhh=h•☐amanoForce:Fc=cÌ= 25m unÌ=nunÈ=hun Ì =hI 2Kun≥unh'Tr a n s f o r m'NGINLAPLACEDomain:Fc=hS✗(s)unh•Tr a n s f o r m e r sinFourierDomain:S= JunFc=hj un✗= jh ✗Un "WEHAVEOBTAINEDACOMPLEXDAMPINGFORCE.ITISPHYSICALLYMEAN/NGLESS,psutITALLONSTOHAVEALINEARRELATIONBETWEEN☐ANPINGFORCEANDDISPLACEMENTWITHOUTANYDEPENDENCETOFREQUENCY.MODAL☐AMPING• 1È + cit +E? =o=_--°MODALTRANSFORMATION:9=EWHERE∅=_ & ,..., In --=LagrangianiCOORDINATESNATURALCOORDINATE5• È?È + QÈ + È.EE?=0•THENormal☐AMPINGISDEF/NEDASACASEWHE@ETHEMODALDAMPINGMATRIXISDIAGONAL--__--i.i.i.TtKTM=miC= zgi~iw.ee =mini≥===I===I===-e'._,e'._,e'.-c.= 25 nunRAYLEIGHDAMPING- ≤ =× ? + PESi =1×+ puoi 2lui•THEConstant5✗,APHASESHIFT,THEREFOREWEWON'TOBTAINTHEMAXIMUMCHANCEOFLIFTATTHEMAXIMUMCHANCEOFAOACONDITIONMAXPHASESHIFTATK=0,20- "WHEN✗INCREASESTHELIFTISLOWERTitanTHESTATICVA LU E•VVHEN✗DECREAS.ESTHELIFTISIt'CHERTitanTHESTATICVA LU EeiDominantLcL=Clk)La+LNCiLCiDOMINANTLNCeiDominantLciDOMINANTLNC.CICALAaPPTLOACH•THISISAMOREGENERALAPPROACHWHICHcanBEUSEDTODETERMINETHESANERESULTS☐b.TA I N E DUSINGTHETHEODORSENFUNCTION.WEONLYNEEDTOSETTHEPITCHANDPLUNGEMOVEMENTSASDOFS-THISMETHODCanALSOBEUSEDFORFLATPLATESTHATCHANCESHAPE,ASMORPHINGPROFILESandMOVATSLESURFACES°REMARK:THISMETHODIsinFREQUENCYDOMAINFORZDFLOWS(asTHETHEODORSENFUNCTIONMETHOD)WAG N E RAPPROACH•ITCOMPUTESTHERESPONSEOFTHEAIRFOILTOASTEPCHANGE'~THEANGIEOFAttackusingTHEINICIALFUNCTIONKNOWNASWAG N E RFUNCTION°REMARK:THISMETHODISINTIMEDOMAINFORZDFLOWS MoriròSMETHOD-BASEDONTHEGREENTHEOREM•☐'veroENCETHEOREMi | ,_°b-= | b-°IdsOv'THEGENERICVECTORb-canBEMADESPECIFICTOOURPROBLEM:b-=Y-G• | ,_◦ Il _coldv=Y_6°Ids1 .•WEcanalsowrite: | ,-◦(Y-G)dv= | , Y ,0-GtG°- fdv-◦ | , Y .°_6+_6º- Pdv= | Y_6°Idsin)su•WEcanREVUEITETHEPROCEDUREWITH:b-=G- µ • | ,o_°- Y t- Y °_6dv= | o_ Y °Ids12)sua11)-12):y2g-G 29dv = | (Y -G-G- f) ◦adsy2g-G 29dv = | 9so-004ds| . | .sesaDvDv•GISTHEFUNCTIONSOTHATi % =S/ ✗-XO)1IF✗0ISINTERNALTOTHEVOLUME YSI ✗-✗◦ Idv = eixoiyixoiw.tn eixo) =/ oii.✗◦isexreana.- | ,4'odv= | ,12IF✗☐ISONTHEBOUNDARY-El✗ 0141×0)= | 9so-074ds+ { G 29dv sesasue'Io.IoE/Io)=0E/Io)=e RaoIn•LET'SSPLITTHEVOLUMEin3i11BODY:lb-RwIb2)WA K E:RWR3)INFINITE:100INCOMPRESSIBLEFLOWS 24 =0°EQUATIONTOSOLVEi•BCONTHEINFINITEilineY = Y go=0R00•BCONTHEWA K E:THEWA K EISALIGNEDWITHTHEFLOW:NEW☐2W=OI✗ W=OJaw•E'✗ 0191×0)= | 9so_oalds+9so_oalds+9so-oalds+G 29dv sasesasesase | , tu . tu .SrbSTEADYCASE-THEREISNOTAJUMPOFPRESSUREACROSSTHEWA K E,HENCEiP=- ] UaoI✗+24=☐wOxatW•CPw= Pw =-2Un2I+a9=☐uè◦×seti guai W2•STEADYFLOW:CPW=-2Un7I+a4=0 ✗ w/y)=constuè◦×setW° 4L =UINSTEADYINCOMPRESSIBLEFLOWSTHEJUMPOFPOTENTIALACROSSTHEWA K EISNULLALONG✗,HENCEITISONLYFUNCTIONOFTHESpar4ÈM.KUTIACONDITION: fw = ( n+ { in ✗ te(y)w=↑(Y)pn DISCRETIZATION-PANELMETHOD1)DIVIDETHEAITZFOILinN=Nx✗NyPANELS2)☐c-FINETHETYPEOFAPPROX/NATIONFORTHEPOTENTIAL Y OVEREACHPANEL•0ᵗʰorder: Lf ConstantonEACHPANEL•1st☐zder: Y LinearonEACHPANEL3)DEFINEHOWANDVVHEREIMPOSETHEBCOnEACHPANEL:ForTHE0ᵗʰORDERTHEBC15GENERALEAPPLIEDATTHECENTEROFEACHPANEL•BEINGTHEJUMPOFPOTENTIALACROSSTHEWA K ECONSTANT(✗ te(Y)=↑(Y))ALONG✗WECANDISCRETI2ETHEWA K Einstrips50THATTHEVA LU EOF 4 acrossTHEVVAKESDEPENDSONL>On4I42+2- 2) =×+le- nè )/42-12-2)≥o71[1,TZ>02✗+ p >I42+2- 2) =✗2+le- nè )/42+2-2)≤ 2) IF1- p ==0Te,72EE71•THEBODYISHOVINGATSPEEDV00,UVHILETHEINFORMATIONISPROPAGATINGATSPEED200P.•V>200VI>200ITHEBODYISALVVA>5OUTSIDETHEPERTURBATIONITHASJUSTGENERATED.-IF[1,TZ>0EACHPOINTP15r-CACHED.BYTWOPERTURBATI☐NSGENERATEDATDIFFERENTINSTANTSinTIMEFROM☐IFFERENTPOSITIONSTHEPOINTISWITHINTHEMACHCONE-IF[1,72E[THEPOINTISOWTSIDETHEMACHCONE,HercECAN'TBEREACHEDBYANYPERTURBATI0nsSUPERSONICFLOWPANEL/ZATION § °""≤">1-PISTONTHEORY FORMULATI0nsOFTHEAEROELASTICSTABILITYPropsLEM•STRUCTURAL☐ISPLACEMENTiNLS=N9-=_0a0•THESTRUCTURALMODELWIllBEWrittenas: I +C9+KG= 9FA ,WHERE 9FA ARETHEUNSTEADYAERODYNAMICFORCES=-=_'LET'SUNDERSTANDHOW4Fa 15COMPOSEDTHROUGHTHEVWP:T• Swa = 9|ShaoRaCPIIAIds -As•WEMEEDTODEFINEanINTERFACEOPERATORTOTRANSFORMTHEINFORMATIONSCOMINGFROMTHESTRUCTURALMODELINTOINFORMATIONSFORTHEAerodynamicsMODEL:Ma=¥2,5 -Ed•SWA=9fa , 51stHto na colla ) ds =9{9T | NTHT .laCPIIAIds = 959T i | n'- Htonadscplxai )=-==-==CONSTANTONAsAsiEACHPANELinOURDISCRETEMODEL" Swa =01 59T D= § ,WHERE☐'SAGEOMETRIEMATRIXWHICHCONTAINSTHEINTERFACETRANSFERMatrix=•DEVELOPINGTHEEXPRESSIONOFTHECPACCORDINGTOTHEQuasi-STEADYTHEORYWEcanwrite: Ia /K)=Han(K,M)9=-Han(K,M):FREQUENCYRESPONSEMATRIXOFTHEAERODYNAMICFORCESITDEPENDSONTHEREDUCEDFREQUENCYANDONMACH=•HOWEVER,OURSTRUCTURALMODELSTARTSFROMTHETIMEDOMAIN:-tt-1-•ACCORDINGTOTHEDEFINITIONOFCONVOLUTION:Fa/t)= 7 . Ian /u)?lui).=han/t)Flt)= | ,han/Te)/t-E)DI =/ teamIt-I)/E)de-=-=OFOURIERANTI-TRANSFORMOFM f- -HENCE,inTIMEDomain: I ÓI+C È +KG=9han/t-E)/E)DI-=-=-=0'NONTHATWEHAVETHECOMPLETEMODELINTIMEDOMAINLET'SMOUETOFREQUENCYDOMAIN:-_-52M+SCtK-9ham/ P,M). 9 0-=====•P'STHEnonDIMENSIONALLAPLACEVariable:P=Sb=Tb+ Jwb = Tb + j KV00UnUsoV00REDUCEDFREQUENCY-_•LET'SSOLVETHISSYSTEM dlt _52MtSCtK-9Ham(p,M)- 9 0THE"S"THATSOIUETHISPROBLEM-=====ARETHEEIGENVA LU E SOFTHEPROBLEMTHEN,DEPENDINGonTHESlonOFTHEREALPARTOFTHEEIGENVALUESWEcan☐C-TERMINETHESTABILITYOFTHEAEROELASTICSYSTEM LLUASI-STEADYAPPROXIMATION•PROBLEM:WEDON'TKNOWTHEGENERALAERODYNAMICTRANSFERMATRIXHan(p,M),SINCEWEONLYKNOWITSimmaginari=partHAM(K,M).WEWEEDTOUNDERSTANDHOWWEcanDEALWITHTHISLIMITEDKNOWLEDGEOFTHETRANSFERMATRIX="SOLUTION:IT'SPOSSIBLETOEXPANDHam(K,M)ASATAY L O RSERIESCENTEREDinS=Oi=-Han|5,MI= Hans ,+2 # ams+^Ò# a" è t...352055=05=0•Fa15)= Hans ,+° # a"s+^Ò# a" è (SI◦S5=020s5=0HanIK,M)=•BACK-Tr a n s f o r m'v6INTIMEDomain:--,--z--2ham'0)Fait)=Han10) L '"91s)+2ham'" L '_591s).+1= L' ^_ 5291s ).=-_-=_-=ngg2752-=Han/ 019 /E)+°#an'"Ó,it)+1 ÒHAM '°) È/[|==rgs2752Ka•CHAINRULEi7ham/01=7 #auto) 3P ==bcaIS If Isun=CA27ham/01=a7ham/0) op7ham/0) ap = Titanio ) op ====2DP===Ma È752IS If as rap as If as rap ≥Is=voiMA-Fatti=ka9It)+b ≤ aq:( ti+1 ÈmaÈ/t)-=-un2uè=-Oa 19 + CÌ +Kq= 9Fai•SINCETHEINITIALSTRUCTURALMODELWA S--=-=--- 1È + CÌ +Kq= 9 Ka9It)+b ≤ aq:( ti+1 ÈmaÈ/t)-=-=-=-un2uè=-----_-__ È ma È + ≤ - 9bEa =◦• I-9^- È + E-9 KA 9 2 VI =Un-=_------ COMPUTATIONOFFLUTIERSPEED•MZ=µ≥ut=U2= eeieasIo✗ ! CZ=Ort6P•90=1Son? »M≥=290P=2902xp6h2h-D : Maxh'MACHISOLINES:THEREISaMMAXCURVE15...ATLOWALTITUDESTHEENVELOPEISLIMITEDBY 9 ,SINCEIT'SPROPORT'ONALTOTHELOADSeMaxALTITUDE"MaxVEASFLIGHTENVELOPEFLUITERCERTIFICATION:IHAVETODEMONSTRATETOHAVENOFLWTIERINTHEENVELOPEINCREASEDBYTHE15..15..VEAS È 90MaxeCONSTANTALTITUDELINES ÈÈuna"M☐ITZECTSOLUTIONFORFLUTTER"THEFLUITERINSTABILITYStartWHENTHERC(X)=T=0•5=o+ ju • p = rb + jk V00-_•TOFINDTHEEIGENVALUES: dlt -52MtSCtK-9ham/p,M),=☐====-TF=0,2•atflutter: dlt _-WFMt j WFCtK-QFHan/KF,MF). 9 =0====-IFIF'✗MF,THENTHISisanonL'NEAREQUATIONTHATWILLCIVEASRESULTLUFANDKF •remare:inREALITYITISMOREConventENTTOCOMPUTEallTHEDESIREDC-IGENVALUESForaRANGEOFEASVELOCITIESANDEVACUATETHEIRREALPART.FlutterisDETECTEDASTHELOWERSPEEDATWHICHTHECORRESPONDINGEIGENVALUECROSSESTHE0-AxisFROMNEGATIVETOPOSITIVEP-KMETHODW-..B'lKNOWTHEVA LU EOF HIM inBandIWA N TTOUSEITTOCOMPUTETHEVA LU EOFTHETRANSFERFUNCTION/~AAPPROXIMATEHaminAUSINGATAY L O RSERIESOFHAMCENTEREDinBT==BEFOREiDIDTHESAME,BUTCENTER.noTHESERIESin5=0(non1CENTERitin5=JW). p =Tb+ jk Un•Han/P)=Han/IKI +°#am/p_jk)+1 Ò#am/p- JKÌrap 2 rapa •OPTIONS:thO-OEDERAPPROXIMATION 1st -OlderAPPROXIMATION|HOSTCOMMON:NASTRAN)ALGHORITM•FORAGIVENSETOFPARZAMETERSiV00,9,h,M,...1)COMPUTEAPRELIMINARYGUESSFOR☐NEOFTHEC-IGENUALUES:✗ I| TA K EStructuralFREQUENCIESATZEROSPEED)2)mi=In/✗ilki=uuibUN-_3)SOLVETHEEQUATION: dlt - IM t✗ctK-9Ham(p,M).=0USINGHam(p)APPROXIMATEDWITHTHEP-KMETHOD=====FINDTHEEIGENVALUES4)'DENTIFYTHEEICENVALUE✗IWHICHISClosedTO✗ È5)ifti- ii°