Aeronautical engineering - Aeroserfvoelasticity of fixed and rotary wing aircraft

Completed notes of the course

Complete course

Aeroservoelasticity of fixed and rotary wing aircraft ILAF:IDENTICALLOCATEDACCELERATIONANDFORCE-IFIGENERATEAFORCESOTHATF&-UI'LLOBTAINADAMPINGEFFECTFORCEMUSTBEAPPLIEDWHEREIMEASURETHESPEEDSOTHATF,UWILLB.EINPHASENOINSTABILITIES•PROBLEM:THEDEFLECTIONOFTITEFLAPPRODUCESTHEDE512EDFORCE,B.UTALSOAMOMENTUU11-1CHMAKESTHINGSMORECOMPLICATED•GENERALLYSPEEDISOBTAINEDINTEGRATINGACCELERATIONCOMINGFROMACCELEROMETERSSENSORINTEGRATORHIGHPASSGA1NFILTERMOVABLEAEROELASTICITYSURFACEL"✗STRUCTURE"¥/h.0.P.i.i.fi ,ii.i.f) -KAERODYNAMICS-IFIGENERATEAFORCEPROPORTIONALTO- [ 'OBTAINADAMPINGEFFECT•REMARK:SINCEINOURMODELhISPOSITIVEDOWNWARDANDLPOSITIVEUPWARDIDON'TNEEDTOPUT-KPHYSICALPROBLEM:WECAN'TINJECTFORCEDIRECTLYINOURSYSTEM:WENEEDTOADDCONTROLSURFACESTHEOUTPUTOFOURCONTROLSYSTEMMUSTBEAROTATIONOFTHECONTROLSURFACE-HOWFARCANIINCREASETHEGAIN?THEMOREIINCREASETHEGAINTHEMOREENERGYISREQUIREDTOSTABILIZETHESYSTEM(I.E.DAMPOUTOSCILLATIONS)PROCEDURE:SETACERTAINSPEEDANDEVALUATETHEROOTLOCUSINCREASINGTHECA1NUNTILWEREACHTHELIMIT'it'5BEIT-62TOSIMULATETHERESPONSEOFTHESYSTEMTOACONTINUOUSINPUTINJECTASINUSOIDALGUSTSPEEDWITH f- =6HZ•IOBTAINAROTATIONOFTHEFLAPOF 14° 't'SEXCESSIVE:THERE'SNOROOMFORTITEPILOTTOCONTROLTITEAIRCRAFT•IFIFURTHERINCREASETHEGAINITGETSEVENWORSE >IFICHANGETHEFREQUENCYOFTHEDISTURBANCETO f- =2HZIOBTAINTHEOPPOSITEEFFECT:THEMOREIINCREASETHEGAINTHEMOREISTABILIZEWHY?ATf-=6HZIWA SCLOSETORESONANCE•WEHAVELEARNTTITATTHECONTROLSYSTEMISLIMITEDBY:THEREQUIREDPOWERTHEDEFLECTIONANGLEOFTHECONTROLSURFACE'MC-T11-0DSTODEVELOPACONTROLSYSTEM:1)WORKONSENSORSIMANAGETHEHINGEMOMENT2)WORKONACTUATORS:MANAGEP◦PROBLEMOFREALSITUATIONSREGARDINGOPTIMALCONTROL:FOLLOWALLTHEMODESATTHEREALTIMEISNOTT1-1ATEASYILAFCONTROLSYSTEMSAREMOSTLYUSED•IFIINCREASETHEDAMPINGWITHTHESEMETHODS,AtTHESAMETIME,IREDUCELOADSDUETOGUSTS-ANOTHERTECHNIQUECOULDBEFILTERINGINSTEADOFACONTROLSYSTEMPSEUDOINTEGRATOR^INTEGRATORI^PROBLEM:ifÑISCONSTANTIT'SINTEGRALISINFINITEs'THEREFORE,DIFFERENTPSEUDO-INTEGRATORSCANBEINTRODUCED"""£94,021^PSEUDO-INTEGRATOR:TS+1W ~RODUCTIONTOROFORS migbalenegetthe ·Fin .HUBPLANE(HP):PERPENDICULARTOTHEHUBMAST Meternate ·TIPPATHROTOR(TPP):FIXEDONTHEPLANIDENTIFIEDBYTHETIPSOFTHEBLADES Ea TA I L·AZiMUTHANGE:4=et·IFLOWRATTO:=NORMALDISKVELOCITY2R j HEAD·ADVANCERATO:M= tangentianbis velocity·LAPANOLE:1P1TPPTPP·FLAANG: /4) = B0+iccOs4+es sInYAsymmetryofvelocityADVANCINGBLADED·higherrelativespeed,hencehigherLift·AxSPEEDA9po27048REREATINGBLADE2bue=oYLIFTUNBALANCEX·Lowerrelativespeed,hencelowerLiFT180·MimSPEEDAr270°400100THELIFTUNBALANCECAUSESAROLINGMOMENT·EREATINGBLADESTALL:RETREATINGBLADEHASSUCHalowspeedthatstalls.It'stheprimaryresponsibleforthehelicopter'sspeedlimitation 3LADE-ROTORCONNECTIONS-=LAPHINGE·THEASYMMERYOFVELOCITYCAUSESALIFUNBALANCE,HENCEABENDINGFORCEISTRANSMITEDTOTHEHELICOPTERbodycausingaRollingmoment·solution:MECHANICALCONTROLSYSTEM#LAPHINGEBETWEENROTORANDBLADESTHATPREVENTSTHEDEVELOPMENTOFBENDINGFORCESADVANCINGBLADE·THEADVANCINGBLADEFLAPSUPBECAUSEOFITSLIFTSURPLUS=CHORD&WW2auIRW:ROTATIONALrelativewind·LappinguptheAOAreduces,andsoalsoliviRWI:RESULTANTRELATIVEWINDRETREATINGBLADE·TheretreatingbladeflapsdownbecauseofitsliftdeficiaCHORDzaRW ·ensuretree ·FLAPPINGDOWNTHEAOAINCREASES,ANDSOALSOLIFT&WW(b)·GENERALLYCENTRiegalForceisEnoughtoPreventTHEBLADESFROMFLAPPINGTOALARGEDEGREEzFc·sincecentrievalforceisalwaystheSame,THEHIGHERISLIFTTHEHIGHER is (a)BzFaxBe4=AngledependsonTHELOADFactorofTHEMANeurer:n= La W LEADLA6HINGEIVnVtFC0R/Otis=2M✓tnI]Vt✓[CORIOLIS'CORIOLISFORCEPRODUCESOSCILLATINGBENDINGMOMENTSINPLANEWHICHCANLEADTOFATIGUEDAMAGESTHELEADLA6HINGEMAKESTHESEMOMENTSNWLLSOLVINGTHECONSEQUENTSTRUCTURALPROBLEMPITCHHINGE-ALLOWSBLADESTOHAVEDIFFERENTPITCHANGLESSWASHPLATE•COLLECTIVEALLOWSTOTRANSLATEUPWARDDOWNWARDBOTHTHEPLATESINCREASINGTHEPITCHANGLEOFALLTHEBLADESOFTHESAMEQUANTITY,OBTAININGACORRESPONDINGINCREASEDECREASEOFLIFTa•CYCLICALLOWSTOHAVEADIFFERENTIALWA RIA1-1ONOFTHEPITCHANGLE•ROLESOFAROTOR:1)THRUST2)PROPULSION3)CONTROLMOMENTS-HOWTOVA RYTHEFLAPANGLE:h1)ACTONTHESWASHPLATE → 2)INCREASETHRUST,HENCELIFT ·ONEOFTHEPROBLEMSOFTHEROTORISTHATBLADESPRODUCEAcouple,THEREFOREroNORUNLOADStheassociatedcounter-coupleontheFUSELAGE,WHICHSTARTSROATINGsonwriONi1)INTRODUCEAtallrotortoMANASEAwmovements2)introduceanothermainrotorwhichrotatesinoppositedirectionR.ORHUBDESIGN↑Eccentricity/e):distancefromthemasttoTHEHInGe-EE-SAW . .... &THEFLAPHINGEISATTHETOPANDCENTEROFTHEMUST:e=0Mb=0ATTHEROTORCENTER"See":FLAPPINGuPSEEIT·HINOE+2BLADES"saw":LAPPINGDownIDON'TSEEITANMORE-·SEE-SAWROTORSARESTFF-IN-PLANENoLEADLAGHINGE,THESTFNessISgivenbyTHEtorsionalFLEXIBILITYOFTHEMAST·UNDERSLUGroroconfiguration:BLADESAT TAC H E DBELOWTHEHINGE(i.2.PLANEofROTION)WITHAconeancie Bo ·LAPPINGspeedisalwaysperpendiculartothevectorthatconnectsthepointofthebladetotheHinoEUNDERTHEHINGEABOVETHEHINGE--·THECORIOLISFORCEABOVETHEHingegoesintheoppositedirectionoftheoneunderTHEVeVFLAPPINGVILAPPINGVeINE·THEOVERALLCORIOLISFORCEISBALANCED,THEREFORELEADLACHINGEISNOTNECESSANANORE-Fcomions.VtVLFcomioLis =WILYARTICULATED Imm =>& ⑤ 5·INEEDmorethan2Blades,theneachbladewillneeditsownHINGEAsaconsequencethemultiplehingeswillhave20,thereforethereasmall,butnotnull,bendingmomentatTHECENTER·REMARK:FULLARTICULATEDROTORSNEEDEVERYHINGE·THECRITICALELEMENTSARETHEBEARINGSSINCEALLTHELOADSMUSTPASSTHROUGHTHESUBJECTTOANIALLOADS1INGELESS mining -mean ·CONFIGURATIONAimedatSIDLYINGTHEROTOR·INSTEADOFAFLAPHINGEAshortersectionisdesignedtotakeadvantageoftheflexibilityofthebladewhichactsasavirtualHINGEvirtualhingepointISESTIMATEDASTHEpointWHEREMb0THEmost·REMARK:WECANCREATEHYBRIDCONFIGURATIONSTHATAREHINLESSONLYALONGCERTAINDOFS·REMARK:BEARINGSASSOCIATEDWITHHInCESTHATHAVEBEENREMOVEDCANBEREMOVEDTOO1INGELESSELASTOMERIC·ELASTOMERICBEARINGSACTSASMECHANICALHINGESINWHICHTHERELATIVEROTATIONBETWEENTHEPARTSISgivenbythedeforrabilityoftheMATERIAL·THEY'REMADEOFLAYERSOFMETALSANDVULCANIZEDRUBBER·THEYCANALSODAMPOurSENTRIFUGALFORCES 321)VERYFLEXIBLEARMWHICHALLOWSOUTOFPLANEANDPITCHBENDING,1SUBSTITUTINGFLAPHINGEANDPITCHHINGE42)VERYSTRONGFORKAIMEDATDEALINGWITHAXIALLOADS3)LEAD-LA0HINGE4)ELASTOMERICBEARINGINTHEPITCHDIRECTIONBEARINGLESS &← 'MATERIALS/LIKECOMPOSITE)ARESOELASTICTHATALSOBEARINGSCANBEREMOVED-FIBERSCANBEPROPERLYTA I L O R E DASFUNCTIONOFTHEREQUIREDPROPERTIESCLASSIFICATIONOFROTORHEADS•STIFF-IN-PLANECONFIGURATIONSARESUBJECTTOHIGHERIN-PLANEMOMENTSSTRUCTURALPROB.LEMS CONTROLMOMENT•AWAYTOINCREASECONTROLLABILITYISTOTA K EADVANTAGEOFTHEMOMENTAROUNDTHE[G)WHICHISTHESUMOFTHEMOMENTATTHEROTORHUB+THEMOMENTOFTHETILTEDTHRUST I e •• •THEHIGHERISETHEHIGHERISTHECO~TROLLAB.tl/TYINASEE-SAWROTORTHEMOMENTATTHEHUBISNULL,TITEREFORETHEONLYWAYTOGAINCONTROLLABILITYISTOTILTTHRUST•AWAYTOENHANCETHEEFFECTOFTHEROTATIONOFTHETHRUSTVECTORISTOMAKETHEMASTVERYLONG/I.E.ARM)-HOWEVER,THETILTOFTHETHRUST,IFTHEMASTISVERYLONG,CANLEADTOTHEMASTBUMPINGFATALFAILURECONTROLLAPstl.IT/ONAM=0&LOADFACTOR12-CONSIDERAPARABOLICTRAJECTORY it :# °ATA=0EQUILIBRIUMCONDITIONTHRUSTCAN'TBEROTATEDSINCETHEWEIGHTBALANCESTHECENTRIFUGALFORCEHILL•SINCESEE-SAWROTORSCONTROLLABILITYDEPENDSONLYONTHRUSTTILT,AT17=0MANEUVERSTHESEAIRCRAFTARENOTCONTROLLABLECONTROLLABILITY-LOADFACTORI•@SETTLINGTIMECONTROLLABILITY FLAPDYNAMICSvpKZBEQUIVALENT1HINGEKloofj1)→• jBeIfi' Ii •It=a-r-if^ turn ±)- Eun ±-2In1-RELATIVEACC.ACC.OFTHEREF.FRAMECORIOLISACC.K'•if=I=→sinPI '+ NRCOSPI'HP:I=consti.' L .◦I=1- I ≈t I 'ti•±=It-e){E'zr=It-e)FE'•REMARK:Ian/b-^≤)=b-/a-◦≤I-≤/f.◦b-I--•canturn±I=≈/yo±)-±two µ)=/→sinPI '+ scoopI'/It sinp -1- SEE '=-sit_ 11 - strip/I '- sinpcosp≤'.=--=-sit.[ OS2PI '- sinpcosp E'.=-sit/I '-PE')put •if^I=0SINCEµ=const--i' j 'k'•-2if^It=det→sinP0 scoop =2hsinpIt-e)§I '≈2h }It-e)PI '00It-e){ -_ -NOWWECANWRITEALLTHEFORCESACTINGONTHEBLADETOWRITETHEMOMENTUMBALANCEK•KµSIMULATESTHESTIFFNESSOFTHEBLADEINTHEVIRTUALFLAPHINGEOFAEROFORCES:dFZAHINGELESSROTORd'CENTRIFUGALFORCES:DFC=MJLZTdtKp•IFTHEROTORISASEE-SAWTHERE'SAREALHINGE,HENCEK=0me,,a,,- i1B--det-2p-290=2er /P1054-9sin4)k -rsin4&COS40-·Thefaxcavationsecone: " + r = 1) macomious/reldr · " + re- e) wrasposp-asmpl/reldr" + re- e) wrtrendsposp-asmp ·REMEMBERINGTHAT: == (esr-eide,speared · " + 5 = =r)* + esp)/pcosp-asin4l 2· " + 5 = x 1+ e/Posp-psin4) flaresponsewithmotorHuorchingandrocinainvacuo • z b ' Born y @ %FEÉus ⑦ É'EEE:Sp ÷ _ I•THESTEADY-STATESOLUTIONISPERIODICWITHTHESAMEFREQUENCYOFTHEFORCINGTE2M: @9 _fTEoTEE- TA I L"¥P= Pac cosy Pes sin4PROJECTIONOFTHEFLAPPINGMOTIONWITHTHEAZIMUTHANGLEONTHEFIXEDREFERENCEFRAMEF-HEAD•P"+ VIP =2IteSPIpcosy-9s/ny)-Paccosy-pass.my+V }/Pac cosy Pes sin 41 =2IteSPIpcosy-9s.in4) I=pI=p-.-_-P1C1UI -1)_2IteSPPcosy+pas/v } -1)+2IteSP9sorry=0I=pI=p--__2IteSp=ppic/v } -1)_2IteSPP=O Pac =I=pa/v3 _e) I •2IteSp=ppas/v } -1)+201=0ygIteSPpas =9=pa/v3 _e)•THISISACLASSICGYROSCOPICEFFECT90°DEGREESPHASEDELAY:THEMOTIONISORTHOGONALTOTHEAPPLIEDROTATIONINRESPONSETOAROLLRATETHEROTORDISKTILTSLONGITUDINALLYOFP1CINRESPONSETOAPITCHRATETHEROTORDISKTILTSLATERALLYOFP1SPacPesis Is P9KBKB MomentTr a n s m i t t e dbyasinglebladetoTHESHAFT·YPOTHESIS:2=0·BENDINGMOMENTOFTHEITBLADE:Mbi= kBBj =kBSinYinz- kBBcOS4jue ·zoningmoment:Li=-Mbisin4=-kBSinY ↑:im i·Icingmoment:Mi=-MbicoSY=- kBB COS4iv·Mi=- kB/BuccosYBessinlcOS4 =- kB1ccoy-ksinpcosp=-Ekbx)1 + cos24)-ssinzY!= --1kB~1+ epletcoszu + asin. 2 -(58 -1)-·2=- /BucosPassing/sin4 =- kecospsin-ksi=-.snay + pase-cosaa= 21+ e- ---kB- Sin24- q/1-cos24). 2 -(58 -1)1 + eS -- 1 + eS1 + eS ·Mi=- -(58 -1)-p+ Pc0s24 +9Sin24-= -(58 -1)--m1r8 -1)- Posz + 9sinz4. 1 + eS -- 1 + eS1 + eS ·Li=- -(58 -1)- PS1n24-9 + 9cos24.=mir-e)*-(58 -1)- Psin24 + qcos24. ·It,applyarollrate pi obtainasteadyprichingmoment)duetoTHEpoPHASEshiftgerweentheappliedrotationandthemotionoftheDiskANDAHOBBLEATTWO-PER-RENinBOTHP. - C HANDDOLMOMENTS·i,applyaprichrate9,obtainasteadyrokingmoment)duetothe90PhaseshiftgerweentheappliedrotationandthemotionoftheDiskANDAHOBBLEATTWO-PER-RENinBOTHP. - C HANDDOLMOMENTSICONCLUSION,ROTORSAREGOODSYSTEMStogenerateamomentbut,ATTHESAMETIME,theintroductionofaRollingPrchingRatePRODUCesHIGHFREQUENCYVIBRATIONSAT2REV •SUMMINGUPTHECOMPONENTSOFALLTHENBLADESTHEOSCILLATORYCOMPONENTATTWOREVISCANCELEDOUT,LEAVINGONLYSTEADYVA LU E S:NN2 P1C&=i=,Li=-NKPPas M=i=,Mi=-NKP2THESTEADYMOMENTSTHATDISPLACETHEFLIGHTPATHAREPROPORTIONALTOTHETILTOFTHEROTORDISKFLAPMOTIONINAIR.✗=⊖+∅=0+atanUn=01UnUtUty ☐ _ t 0•Wt=ItUt•Fz=LCO50+☐sin∅=L④0:STABLEATAC O R I~oCO:NOTPHYSICALxo=-Mo2 f(x,nd =-2x0=2wo-xMo>0:rusTABLE(efreneml·STABLEEQUILIBRIUMUNSTABLEELVILIBRIWMGavinrudEVOLVEUNTILSTABILITY·-x FCO ESnoGaviziumcombirion,soalways,3.FURCAION point HENCETHESYSTEMDIVERGESTOWARDS-N con,aun EVOLVETOA↑LW T E R ↑RANSCRITICAL·CONSIDERTHEVECTORFIELD:i= f(x,u) =rx-x2·T H EFIXEDPOINTOFTHESYSTEMARE:Moxo- x8 =0xo=0,Xo=MoFOREVERYMOTHEREARE2FIXEDpoints·STABILITY:2 f(x,nd =no-zx0THEPOINTXo=0,Mo=0isabifurcationpoint-x~oCO:STABLEAT T R AC T O Rx0=02 f(x,nd =no-xMos0:rnSTABLE(aerekemlto0:STABLEATA c r o R·STABLEEQUILIBRIUMUNSTABLEELVILIBRIWMaucquic EQUILIBRIUM· PITCHFORK·CONSIDERTHEVECTORFIELD:X:f(x,u) =rx-x3·T H EFIXEDPOINTOFTHESYSTEMARE:Moxo- x8 =oxo=0,Xo=Ino~o>0:3FIXEDpoints IMOC0:1FIXEDPOINT2·siagiiry: 2f(x0,Mb =no-3x0THEPOINTXo=0,Mo=0isabifurcationpoint-x~oCO:STABLEATAC O R Ix0=02 f(x,nd =no-xMos0:rnSTABLE(aerekeml~oCO:NOTPHYSICALxo=Mo2 f(x,no) =-zwo-xMO>0:STABLEATA c r o R~oCO:NOTPHYSICALxo=-Mo2 f(x,no) =-zwo-xMO>0:STABLEATA c r o R·STABLEEQUILIBRIUMUNSTABLEELVILIBRIWM··REMARK:THEPITCHFORKISONLYTHEOREICAL,THEREALWORLDISNEVERperfectSMMERICALTHECOMPRESSIONOFABARBEHAVESACCORDINGTOTHEPKdeformation.Inreality,thedeformationdependsonTHEimperfectionsofTHEMAERIALx0x>0 ·Let'snowStudyTHEREVERSEDPITCHFORK·CONSIDERTHEVECTORFifLD:x= f(x,u) =-mx+x·T H EFIXEDPOINTOFTHESYSTEMARE:-Moxo+ x8 =oxo=0,Xo=Ino I ~o>0:3FIXEDpointsMOC0:1FIXEDPOINT2·Stagiiry: yf(x0,0 =-uo+3x0THEPOINTXo=0,Mo=0isabifurcationpoint-xto0:rusTABLE(aerekeml~oCO:NOTPHYSICALxo--no 0f(x,Mo =zwo-xMo>0:rusTABLE(aepeem)X·THEREVERSEDPITCHFORKISMUCHMOREDANGEROUS:IF,moveoursideTHEPARABOLAiLLNEVERREACHSTABILITYAGAINr HOPFx=mx1-xz-x/x+x2)·CONSIDERTHETWO-DIMENSIONALVECTORFIELD: Ixz=x1+rxz- xz)x1 +x2)·THEPOINTX10=0,X20=0ISAFIXEDPOINTOFTHESYSTEM----u-x+x2- 2x -1-2xx2--1·Stagiviry: 81(X,Md =-3X-1-2x1x2u- x - x2 - 2x2- -e->---x-1·eigenvalues:det-W u + x2 - zux +1=0--u-x- x-zux +(1+ u) =0mc0:STABLEFIXEDpointx1,2=2 =2j =x= j=4 - x(1 + ) = zjU0:UNSTABLEFIXEDpoint·OUNDERSTANDIFTHEREAREanOTHERinvariantsetsitmaybeusefultotransformthesystemofervationintopolarFormINRODULimTHECOMPLEXVA R I A B L Ez=x1+jxz·i=x+jxz =mx-xz-x/x+x2)+ jx +rx- xz/x + x2) =az+ jz -z z2j = g(n - 52) ·Expressingz= gei1; + jgei = (n + j - g2)geibII =I·LOOKINGATTHEERVATIONISITPOSSIBLETOSEETHATTHEREARETWOSOLUTIONSFORWHOMTHEAMPLIuDEIisnorchangins:90=0,g0 =MO1)So=0z0=0x10+ jx20 =0x10=0,X20=0:THISFIXEDPOINTHASALREADYBEENFOUNDBEFORE2)Igo =MoTHISISARorATIONwithconstantSPEED,HENCEAperiodicorbit 60 =1·:AssLuncoauaeEvonv·THEHOPEbifurcationimpliesthetransformationofafixedpointINTOAPERIODICORBIT·whentotheperiodicorbitdoesntexist(notphysicall ·PROCEDURE:1)LINEARIZETHESSEABOUTAnarmcondition2)studytheevolutionorthesystemuntilIFINDACOUPLEOFCOMPLEXCONUSAeDCROSSINGTHEIMAXIS3)IDENTIFYTHEHOPFBIFURATIONPOINTS4)IDENTIFYTHELOSAmplitudeTRIRPROBLEM5)investigateTHESTABILITYOFTHELCOSINEEDAnewTHEORtostudyperiodicSTABILIT·EMARK:T'SconsideraSTABLELCD.Evenifit'sSTABLE,INCREASINGExcessively o/ i.e.isamolutubalcanleadtoveryhighloadswhichRESULTTOBEWNSAFEREMARKABOUTnOnLINEARSTABILIT·Intherealworldanon-linearsystemwillalwaystendtofallintotheexistingstablesolutionByNATURE(ATRACTORS).THISMAKESTHEINSTABILITIESlessperceivableFROMARATHEMATICALpointofviewunstablesolutionsareobjectofinterest,sincearesimplynotreachableIrepellers)·flutter:imnotchangingfromastargletoanunstablecondition.WhatchancesisatractorofthesolutionInowspeed:classicstaticgravitiairmhighspeed:10 ( 1,1CVA R T I N GLINEARSYSTEMSx(t)= 1(t)y(z) ·CONSIDETHEFOLLOWINGAUTONOnOUSLINEAR1StoRDEesisterofafavATONS:1I(0)=10·IT'SPOSSIBLETOPROVETHATTHESYSTEMADMITSANWIRUECONTNUOUSIDIFFERENTIAL SouONI ·Thesenteensolutionofatradethewaterswouldsa.((t)= -z(s,as -XΔTHISISFALSEFORNON-LINEARSYSTEMS,UNLESS:2 E(t) =1t2)E(t)1(s(ds = 1(s(d5 i(t)1 o 1. ·HENCE,THESOLUTIONCANBEGENERALWRITTENAS:((t)= 1(t,to)X0 ·Ta n s i t i o nMA+21x (t,t0): MAPSTHEINITIALCONbITIONSTOTHEcuezfNSTATE.i= 0x0 = 1c =10108(t,t0) = 1(t(f(t,to) ·p20PC2+1fs: 18(t0,t0)= |t-to)=2)forconstantcoefficientsystems: eIt, toe 3)(t,t) = d(t,t1)d(t1,z)a)!(t,t) -1= 1(z,t) 5)dd !(t,z1 =- d(t,t)E(z)6)det(g(t,z1) = etr/zsslas THETRANSITIONMATRIXISNEVERSINCULAR,HENCEIT'SALWAYSINVERTIBLETHERE'SNOANALOTICEXPRESSIONFORTHETRANSITIONMATRIX.ICANBEFOUNDNUMERICALLYBYINCORATINGTHESYSTEMMTIMESWITHASETOFMLINEARINDEPENDENTINITIALCONDITIONS ↑(R10x(CLINEA25T5TEMLET'SCONSIDERTHESOCIALCASEOFTHEUNTINGLINEAnsistenwHere: (t +i)= 1(t), Tso·THESOLUTIONOFTHESISTERINTHISCASEISACAIN:1(t)= 1(t,to)X0 ·THetensition17+21x1S81-p(210Dic: f(t +T, t o+T)= (t,to) ·10005207: 7A r 2 , x . 1 ( t ) = 1(t +T, t )ISTHETRANSITIONMATRIXOVERONEP(218D020PCR+15:--11...-1)31-p210x14+0. 8(z + T, t ) = 1(z,t - T) t-[Ltx+ +2)!(z+ T, z ) d ( z , t ) = 1(t,t)!(t,t - +)3)1(z)8(z,t) = d( =+ T, z ) d ( x , t ) = 1(t,t)!(t,t -1)= 1(t,t)b(z + T, z ) = d(t,t)1(t) z1 a(f(x) = f(z,t)f(t(v(z,t) -1SIMILARMATRICESSHARETHESAMEEICENVALUESFLOQUETTHEORYFLOQUET-LADOUNONTRANSFORMATION·THEIDEAISTOFINDreANSFORMATIONMATRIX&ABLETOtaNSFORMAEndthePERIODICsistentoALINERTIMENUARANTSISTENIFTHETIMENARANSASTERISSTABLEweknowHOWTOfunLunTESTABILITY: E), THENalSOTHEORIGINALSISTERWILLBESTABLENORDERroDOSO,THETRANSFORMATIONMATRIXTUSTBETIMEINDEPENDENT·Lets+85tofixaT-p(210D1c+exus=0nax0 1(t +T)= f(t) oftheStatevectoeinoatetoOBTAINANEWSISTERWITHACONSTANTCOEFFICIENTMATRIX =#(t) = f(t1 -11(t)·X(t)= f(t)f(t) ↓ x = 1+1x2x = (1f - p)5= 18 -1 11-1 - 1)x =11= 1=1E: nus30+1=invxz1x+· E = f(t) - A(t)f(t) - p(t) - 18(t)·11 = 1f - p2 = 18 - 11 ·is80sSiB2)T0veliFrThatTHesouvot0 ( = 1f - 811s.f(t) = 1(t,t)f(z)2E(z -t)·p(z+1)= 1(z + T, t ) f ( z ) e - E8(z) = f(z(f(t)e - ET THISSYSTEMADUITSWFIN+EsaurIONS (8,E)f(z)= (z +i)1(z)ET .(ET'S+Ax- f(t) = Iy(z) =eFROMTHEnOrODRONSMATRIXItispossibletoefcOvER , RESPONSIBLEFORTHESTABILITYOFTHESISTER·Hance,THEPERIODICTRANSFORMATIONMATECANBEDEFINEDAs: f(t) = d(t,t(eE(z -t)(05.N.(t)=E·ForaperiodicundersoftenwithECA)= E(T +t)therefusitionmatelyHasTHEFORM: d(t,t0) = 1(t(eE(t -50)·WITH (t) +HA1+ASTHE1rpar100F Δ aNb f(t0) = IET·y(t0) =2 TE =(M·i 1ISDION17B,i.e.y = 111 - 1, THer. E = yX((1(t -1·NOTETHATINSENERAL A , CC, unlessallnetLfldtuvaluesof1ARePOSITIVETHElOARITHMOFacomplexfunction (1) HASANBRANCHESTHATDIFFERB0Multiplesof CI MORCOrGe,EACHBRANCHDIFFERSOFACHANCEOFPHASESTABILINOFPERIODICSYSTEMS·FoGuEtUUITIOUfRS gi: GIGENVALUESOFTHEMONODIOMSMArRix]fi =eT0i·FloeverexponentSi:dervalues of ·Si= I(((i) + jsi + i 1 Ilogs ogi.1 ....relsor canopicsenseseeisescoronasteronenovercomonessensessome,onsomeEXPONENTSSATSEY Re/8i)*↓·THeF12StSTEPTOASSESSPERIODICSTABILITYISTOCOMPUTETHEMONODEORTAT Z I X :IPOSEUN.T(ITALCONDITIONSANDINTECRATEAFTER1PERIOD1)N.T1A)CONDITIONFORDISPLACEMENTS:·1x=E--T5(0)=-10.·suurtiontoan1cF02 DIsanenens:y(t) =x(t)=i(t)EVALUATEAFTERIPERIOD:((t+T·cnresmatestressaresenop:x(t)=xt1= )i(t) dt EVALUATEAFTERIPERIOD:C(t+T1)INITIALCONDITIONFORSPEED:x=E--T·15(0)=.01.·suurriortoan 1(702S0tD:y(t) =x(t)=s(t)EVALUATEAFTERIPERIOD:S(t+T·Inresmatestressaresenop:x(t)=s(t) =)s(t)dt EVALUATEAFTERIPERIOD:S(t+T-c(t+T)s(t+T)- x(t +1)- 2(t +T)s(t+i)-- x(t)- ·M(t)==~:(t++1j(t+1)-- y(t + T).c(t +T)j(t+ T). . 4(t) -SocuONTOTHE1StCSOcuaONTOTHE2ndCX(t+T)x(t)·T H EPRECISIONOFTHEMETHODDEPENDSONTHEINECRATIONMETHODTOCOMPUTEISTEMMSCANOBTAINUNPRECISEEICENVALUES:WRONGSTABILITYCONSIDERATIONS -x(t)-1- x(t +T)- ((t +T)-==- 4(t).. Δ-> -y(t + T).c(t +1)-x(t)X(t+T)·INDECD,IFi-x(t)-⑤- x(t +T)- S(t +T)-==- 4(t).. 1-> -y(t + T)..j(t +1)-x(t)X(t+T)·LETSFINDTHEGIONVALUESOFTHEMONODROMSMATRIX:-c(t+T)-x5(t+T)-·det=0 (c(t +1)-x)(S(t+T)- x) -s(t+1)i(t+1)=0:(t++1j(t+T)-x-~·c(t++(j(t++1+ x- -x j(t +i)+c(t+11 - s(t+1)j(t++1=0. x -x j(t +1)+c(t++ 1 + c(t ++15(t+11-s(t+1)i(t++1=0x-tr(1(x + det(y) =0·p20pCr+125. 1)det(z) = eftr/zs)as =1=x1x2x+x2=102) tr(y) =j(t+i)+c(t+1)=x1+x21x1+x2=j(t+i)+c(t+T).POSSIBLESCENAR105:1)tr>2.SinceXe= 1, oneofthe2iocurturesMrs.Becheaterthanunstablesolution2)tr=2+i=xz=11ALIMITCYCLESOLUTIONARISES·x1=x2=1:THESOLUTIONISPERIODICOFPERIODx1=x2=-1:THESOLUTIONISPERIODICOFPERIOD2T3)tr