Aeronautical engineering - structural dynamics and aeroelasticity

Exercises of the course

Complete course

Structural dynamics and aeroelasticity exercises •coshx=e×+e-×2--◦coshx=0✗= j TtKIT2-_•ALTERNATIVEFORM:costi✗=COSIj ✗)=COSI- jx)costi(Ix)=cosi•Sinhx=e×-e-×2--•sinhx=0✗= j o+KIT-_•ALTERNATIVEFORM:sinhx=- j sin/jx )= j sin/- jx ) sinhljx )= j sin✗COSZX= 2✗=1+COSZX2•COSZX=COSI- Siri XCOSZX=1-251 IX sirt×=1-COSZX2 Slkx)=1SI×)K•Dirac'SDELTAPROPERTIES:00 fltlsltldt = [ 7101Sitidi= f- '01/Io Sltldt = fioi t .1+be-P"+< ejpx •FREEvibrations: f /×)=al×+de'P"= asinhpsx + bcoshpx + csinpxtdcospx SHAPEFUNCTIONSSELECTIONTORSIONKINEMATICBc:⊖/0)=0•014,ti=NOI4)90ft)POL>NOMIAL |NO=ay+C2=1NO=4010)=0HARMONIC•no/y)=Accosw4tAssinw4bb'naturalBC:②Io)=0Ac=☐,arbitrariLTAs=1•CinematicBc:GJ@"(b)=Ocostu=oun=IT2SUMMINCUP:no (4) =sinIT42bB.ENDINGcinematicoBC:Wto)=0W'(☐)=0•WIll,ti=NwltwPOL>NOMIALNw= 242 +C |wioi.coa-un =p w'10)=0 SWEPTWING•4=4-COSIè=eCOSIÈ =^ già =1 gv2COIL=9cos' 122•b= ÈCOSIè=ccosnh•BEINGL'FTTHESAMEINBOTHREFERENCES:L d4 = [dj •9CCina× dy = È c- CÌÀdit9CClaady =01 COSTI CCOSICiaè 4COSI•(la✗= CiaÓCOSI (la= ÉLCOSI✗= ÈCOSI/ TOTALCHANCEOFAOA)TORSIONy0costat= Ò COSIi0sind4EAINCREASEOFAOA0✗BENDING-2-141)zlyz)4I✗b=-W"sindè41-YEA'=dwW'COSI42dyDECREASEOFAOA4=CsinnhW'sind✗E•✗=Lt+db= ÒCOSI-w'/Ù/sind •To t a ll'FT:LT= | l/4)d4 = |L/4)d4THETo t a ll'FTMUSTBETHEsane'~•◦t"riferentesiTHEWINDISTHESAME'•consideriNGaninfinitesimalestrip: d4= dyCOSI" " =/ """"""" = {""""d""" " "="""""SECTIONALLIFTANDMOMENTSTILLWATTHEY-AXIS,BUTWrittenasFUNCTIONSOFLT•Mt =/m'4)COSIdi =m/y) dynty )= nlyl COSI4=MI=m/Ìf )COSItorsionaleMOMENT"""" | "ᵗeMImityEAMb=MI=m(Y)sindBENDINGMOMENTNEGLECTABLE✗ IMPULSERESPONSE-F=FoSlt)e-Swat e>0 | Fosinlwdt)mwde- Sunt • qlt )=COMPACTFORM:9(t)=Fosin/ndt)StepIt)mwdotw"-3 BZKJè +b" È =ow"-3.b>K è +b" Èjz=°J'LET'scall:È=K:W"-3ktWZ + Il g-z=°J3kt±sEJj v21.2 == 13 ± SIÈ22J -----v25+EI01•=☐-I-upJ+ZI.-02-•01=e / - ut -3+EI1+È02=o02=w>J-I= ut J-1èE(3+5)EJ-1=1+52JI2•213-s)EJ-1=1-52JI2--_-11• WE =(3ts)Iv1 =1+5 WE =(3-s)I12 =1-52J22J2-...---..-_---_-3)J001KK01Ci+ È =oa-oJ--02--KK--02-.(2_---v25+ BZK[ K•detl- v21 + E) =0: det =o-K-utJ+ È K.22•(_"=]+I|-È==oWGJZ -zigof =☐ up , /wz ,-z,y=o"^=°•'°'☐"☐☐E WE =ziaJ-----v25+EI01•=☐-I-U2J+I_-02-•01=e / - ut -3+EI1+È02=o02=w>J-I= ut J-1EI-1022EJ-1=1JE----ei2•un=oV1=wz>=ZIYZ=J--1.-1. 4)---..-_--_ a- -Jo01KKCi•+ b ≥=Ole-0J,-02-.KK--02-.(2_--11•CET'5WRITETHEMODALForm:4=--11_--_-..--1-1J0112J0• È = 4thµ=====-11_.0J-.-^^..02J_--_-..--1-1EI1100•~K= 4T Kµ======-11_.EI_.-^^..04E-------1-1(1C1-Cz• 4T F==-=-11_-CZ-_Ci+Cz----..-_-----2J02-e00ZuC1-Cz•t=olo-02J_-Zz-.04È_-Zz-_Ci+C2-z=2-osinnrt f- •Ci=Cz=CosinnrtÈ =ZOSZCOSSZT-- È =- Zoèsinnzt------__2J0000°- Iove Simart+2=0Start=Simart-02J..04E_.ZCO_--.-- 2522 00•2--0=-O4E-2J È _.ZCO_,__-O0--.- GÈ - 25h2 00•2--0=1= ✗Col - 25h21 =zco4È-25122J SE/25h2 -GE)-O -2512 ..ZCO. ¢512/25h2 -GE)-_-_ --------_.•e5)2J02-a0o2-eCi-[2+=oco-O2J_-Zz-.04E--Zz_.Cn+CZ,eC1=-Cz:--------_.•e2J02-e0o2-a-ZCZ+=...OLO-02J.-Zz-.04È_-Zz--O_ C-✗3^)EASOFMOTIONbamimassperun/TLENGHT2)MODESOFVibrationb010,bUnL1,Lz:LIFT^b2b3)DiUERGENCEDYNAMICPRESSURE1)G)FINDTHEMODESTHATCANDIVERGEb2jj,JÒZbFeli010,bUn^b2bFeli•SW=Labson+LabSoa-K/bon+boa)/bSon+b502)- JÒÌson- JÒISO ,=o22bLe-✗ È01-Kb>02- JÒÌ =o i b.la-✗ È02-Kb>01- Joi =o2---..-_---_-Jo01KK01Le2e+ b =b•.2-oJ.-02-.KK--02-.LZ.✗o=OASSUMPTION--•L=95.Cio+Ciao.•renane:È= È K---..-_---_-Jo01È- bqsclaE0102•t=au.OJ--02-.ÈÈ-la9Scia..02..0.2•renane:ka=bSCla2---..-_È---_-Jo01È-qkaÈ010•+=E-oJ-- Òz -.ÈE-QKA--02--O- È =Ks-QKA==2)MODESHAPES:det/-ut! + E) =o -_-ut]+ ÈE•det=o-I-WZJ+È.• / -mt>+E)=-E-=o WGJZ + Il _ ZJEWZ -è>=o v4 _z Èv2 + EZ -IZ=oJJZ•in"-2/E-91calè + IZ + 92KAZ-ZEQKA-IZ=oJJZ.u"-z/è-QKA) ut +QKA /9Ka-ZE) =oJJZ•=4/È-9KA/=-49ka /9Ka-ZE)= GÈ 2+49> KI -8 ÈOIKA - 492kt t8È9KA=g-2JZJZ=ZÈJ2/E-91cal±ZEZE-9KAzJJJ•un,z==È-QKA±IzJJJ-9KAJ-.- a- -ut]+ ÈE•=O-I-WZJ+È_-02-• ( -ut]+E-QKA)01 +EOz=oJZÈ-QKA+QKA-1=1IJè02 = ut ]-I+qka= è ]+QKA-1EKè•01=1,g-+9Ka-1=-1-JQKAEMODESHAPES----ei2•un=ZÈ-QKAV1= nè =-9KA 12 =JJ-1.--1. 3)TOFiredTHEDiverOENCEDinamicpressore:det(Ef)=o--È-qkaÈ2•det=o:(è_qka)-È>=o 92KÌ - ZÈOIKA =o.ÈE-aka-☐/☐=OTr i v i a lSOLUTION•9KA /9Ka-ZI)=o9,QD=ZIka 4) ------_.°- à -JÈ-Ta k aÈ010+=a0.OJ--02-.ÈÈ-Yo k a--02--O-☐IVERGENCECONCETLNSTHESTATICRESPONSEOFTHESYSTEM--_.È-9,>kaÈ01-=☐-ÈE-QDKA--02-•(E-QDKA)01+E02=o•arbitrario@1=1 02 = QD Ka-È=QDKa-1=ZIKA-1=1EÈKAE--1 Yd == µ DIVERGENCEAFFECTS☐nltTHEFIRSTMODE.1. c-✗G1)EllsOFMOTIONg,b2)MODESOFVibration2bb013)☐IVERGENCEDYNAMICPRESSURE^, { KG)FINDTHEMODESTHATCANDIVERGEJÒI1) ab2bsiib"Fee01^bFel- SW =- JÒ ,son- JÒ >Soa- 1 COSIpw ×)-Ac,sin /pw ×) ) •a"/✗)= PW(Ass .nu/pwxltAzcoshlPw ×)-A>sin /PW✗)-Ac, COSIpw ×))•LET'SAPDLYTHEBC: 1) wIo,t)=oiAztAc,=0 2)WIL,ti=0Ansinh/PWL )+azcosh/PWIl+A>sin /PWL )tAc, COSIPWL )=o3)EJW"/0,-1)=0A2-Ac,=0 4) EJW"/"¥t)=0Aeisinh/PWL )+Azcosh/PWL )-A>sin /PWL )-Ac, COSIpwLI=0az=AG=0 | a.s.nu,puoi,+a>sn ,puoi,=◦2^-3Sin /PWL )=0 pm L=KITaisinh/piu L)-A>sin /PWL )=oW=KIT2^=KIT2EJL P Lm•EIGENFUNCTIONS/MODESHAPES):2k(X)=AKSin↑KIT=1✗=AKSinKITXL p L 08042204EX1iFREEVIBRATIONSOFaCLAMPEDBarGJ,Jp✗0/x.tlLeSwi= |sotto >0' di 0 SÒI - MÒIdx - swe =/i 0a•- Swi + Sue =0:- |;sotto >0'dx+ | ' SÒI - Jpoldx =o0--l0a•. SÒOJO '_o+ /èsotto >0'/' di + | ' SÒI - Jpoldx =o0--l• | .so' /050"- JPÒ/dx-. SÒOJO '_o=0L GJO "- JPÒ =o |soiii.tio-o.ie ,-4=0o> O'ii.+i.otossina.nomen, sotto , -11650 '/0,t)=o010.tl=0•VA R I A B L ESEPARATION: Ò =a/×)b.If)O0•Cosa"b-JpaÈ =o65 aa "=b=- è JPb✗2a"+ JPWZ a=0GJ : 0O |b+ utb =o1a"+ sia =0.aix)=ae" /È +✗ 2) ae"=o✗=± ja =±j Jpw : - jinÌOW +are•a/✗I=A1E=Aisin/Onix)+AZCOS/✗WXI•a'/✗I=6W(ancos/Out)-Azsin/Dux))•LET'SAPDLYTHEBC: 1)010,t)=0:Az=02)GJO'IL,t)=0:ancos/OWL)=0OWL=IT+KIT=IT(210'di- | ,+ | ,-1.T◦o•_ [ solo > to' + PÒ '/di+ | ' SÒI - Jpoldx =o0--LL+,-I◦°•. SÒOJ/O' + PÓ ')-o+ | SO-0510 '+ PÒ '). dx + | ' SÒI - Jpoldx =o00--l• | ' sotto-30"+ PGJÒ "- JPÒIdx -. SÒOJ/O' + PÒ ')-o=o0 | "°"" " " ° """≤°--.- SOTIL , -1165.0 '/Liti + PÒ '/L.tl _=0 65.0 '/Liti + PÒ '/L.tl _=0.- sotto , -1165.0 'Io,-11+ PÒ '/o.tl _=0010,ti=0 •VA R I A B L ESEPARATION: Ò =a/×)b.If)a/×)=a@+×1• | = zètstbix)= best°/✗'t)=a/✗Ibit)=Age>✗+se2-2itxtsti0=A2-Eg" yzz @''✗+st=• Ò =gzz@+✗+St•GJO"+ PGJÒ "- JPÒ =o /GJXZ + PGJXZS -jps2)ze"""=o•gj/1+ Ps)È -Jp52=0✗=±JPSCONDITION:s≠- è GJ/1+ ps )✗×_✗וa/×)=ane+azl=AiSinti(✗✗I+azcosh/✗×)•LET'SAPDLYTHEBC:1)010,ti=oa/0)bit)=0Az=0.- 2)65.0 '/Liti + PÒ '/L.tl .=oa'/ilbit)+ pa' ( lib /ti=0•✗aicosti/✗LI. Best t p >aicosti/✗L).SBC"=o•✗(1+ PS )cosh/✗L)=☐Tr i v i a lSOLUTION:✗=0IOnonTr i v i a l:XL= j ITtKIT= j IT(2K+ 1) 22--•✗= j 1Ti 1210'dx- / " SÒTPOJÒ ' dx - [ SÒJPÒdi + |;sotncdx =o00TT•- SÉ); NgOJNg^ Ldg9- 59T | , NgPOJNg 1 ldgÈ - SÌ [ NTJPNLdSÈ +e^EknnCnnMnn+ 59T | " n' -ne Slgl -LIS'tlldg 0Unit)iMnn È +Cnn È +Knn 9 =alt)-LET'SAPPIO✗'MATETHE1stEIGENMODE(n=0)^•Koo= | 29a"- v25 a+ qceceaale.int =o•a/y)=A@ JAY |""""""""""""""""""6J0'(e)+Koll)=01GJO'(0)-K010)=02--_-GJ.-ACTLsin/IL)+asAcos/IL).+K-Accos/IL)+Assiri/12L).=0• | ≠,»a0JAsthe-KAC=0•STATICAPPROX/NationTOCOMPUTEdivergenze:GJa"-utÌ a+9(ECLXa=o•0Ja"+qcec.laa=0 ( - 12265 +9Cecca/Al"""=o qts = ÈGJCECLX EXERCISEV00 TI 4eMacUllaJÒÌ.=dco[JÓÌ' 02 E'✗01ellaggg[..."01.02mìominio≥bWace=E01WCG1=-d@1WM=-t@12-92ewacz=@02+WM=CO2-tonwcoz=-d@z+un=-d@z-t@1✗....01ea9•_801K01-I502-501)K(02-01)tSWAGLa+SWACZLZ+SO≥Mac-501JÒÌ -502502-Succim Wich -SwcozmWÌOZ EXERCISEV00I4ealle=^t✗01eeeeec02b◦STATICPROBLEM•WAC1=E@1.«a02Lei=QSCLa01@WAC Z@"Act^01"Eaz t.ae •vvacz=@02-[01La=qsc.la02VWP--501Kon- /SO≥-501)K /02-01)+Swaenla+Svuaczlz=0•-501Kon- 1802 - 50111-3KKAOI+ K¥92 -Kkat q =o 92 -K3tt9+31snuoci.co2^-1sin/0L)=06L=o+MIT pur L=MIT-A1sin(0L)+1-3Sinn/6L)=0•ce=1 q MIT2L•an=ansin/✗✗)=Ansin /piu ✗I=Ansin P 1MIT2✗=Ansinnit× p LL •21=alsin1T [ ,22=A2sin21TXLb)VWPL-_•_ [ Sw"EIN"dx- | " Swniidx + | ,Sw- q 2vi+w'di=oMooWoo0----_-•W=NYIn =-2122-9h1- Quiz .--T- | ,ew"-EI NÉdxIn - / ' NÌnriwdx9¥ . nè nw' dxq.in =0•Sannwtriwdx9in -29 |; -'- Sao / j .--ma-①-KSMC' % eminioaWAC=in+E⊖Wcg=W•L=9CCla⊖+V6V00b•Sui= [ Sw" TEIW " dy + | , sotto > Òdyb•swe= [ Sul- niildy + | ,SwacLdy - /[ SOJÒdy b•- |? Sw" TEIW " dy - |?So' io > Òdy - | ,swniidy + |? lswi-esolqcc.ca0+vs dy - [ SOJÒdy =0V00•RITZ-GALERKIN:NW= LÌ ,NO=Y- sqèfbnw "-c- Inviidyqw - 590T /bnòtosnòdy90 - Shui)nwtnnwdy4in + STÌ9(bNÉ acca Noyo +vo dy + : Un000bb+ SOIÒq | NO'CeccaNoyo +vo dy - SOIÒ|NÉ >no dyÌÒ =oUn0obbb• sqii ' | nvinnwdy9in + | nw"" EINIIdy9W -9 | , nèoceano90 -9 | , nè coca dy vo'+ : un-☐-b..b-+ STÈ ' | , nè JNO dy40+ |NÒTGJNÒdy90 -9 |?notCeccanody90-9 | noecco dy vo i V0o-- -__-s--a__--Mww0ÙÙ'Kw w00kwo' 9W Fw•+-01=9V6••aUN-0MOO.-O__0ÉOO..0KOO._ 90 .-FO_EI _2•④vo= Hflcul W=b≥=bb Hf =b122+v2a+jwa_jwatjwnvoHfHfls )=VolsiVolsi=bnls)la+5)lo=bnsuo=bn-av6nls)ats◦ io /ti=-a.volti+bnlt)• 1È + I -9= È vo'"--◦-M-1k-M- 1e '-È-__90-=-=--o9=1009+onlt)--a-V0--0o-a--vo.. b .•LYA P O U N O VElevation:T'✗ È + TXÌat==t EWtf =•Mb=EIW"/01=EINW"lol9W EXERCISEVERYHARDACvoCGEEAElasticBEAMWITHNOMASS0e..QUASISTEADYAEROgj,LW/4)ELASTICWINGMassMPerun/TSPAN2-nbay✗ÉI,ÓJBENDINGPROBLEMNF |Ty =-Fne Sty =-1ììE•iMx=-FZE•i | Smx=-zL FZZdz=FL}kb=3 ÉI •1.W= | ,ÉI3.ÉI',kb=12EI4EI12(3☐ TORSIONM4Z✗•VWPTOFINDKtµMz=M1SMZ=1z"✗•z"✗•L.⊖=1µdzKt=GJ0J / ◦L---w--Fb-Kbo•=-Mt--0kt_. ⊖ .efb^^^^^.^"ja,MVÙPnnnnn.Mtmt✓ ? uuusaMiip⊖F, mbzcii ✓ Ì '✓✓✓0 nbzc •L= qcbCla0-vi2Un•Fb=L-miip-nbevi= qcbCla0-viu,-Mvi+C Ò -nb,cui22222•MI=Le- Mvip C- JAÒ = qcbeCla0-vi-Mvi+C Ù c22va22-Ja Ò ---µ__---_-_--w--M-nbc-MCG-1295cL✗Ovi☐12Sciakb022=,o,°,o,°=µ.. g ..."° Ò .--MCG-Ja- Mc2 8__0__-129secLa0..-O12Secca_-O2 31AUGUST2021e2. a) V6 i ✗8Un✗=V6- vi UaoUnUso { keqdes-vi0'dx.swi= |; L SÓI -IaÒ)dx + SÒIL)mt.Swe= / ◦ """ " "" "" iii.iii. ".vi14)bboa•MT=UnbV00 | -bLIFTCOUPLEINERTIACOUPLE•-SWI+Sue=0:- |;SÒTOJÒdx- |; SÒIAÒdy + SÒILI) ; qccca v00"hit)- ÒIL)y . ydy +UnbV00- SÒIL) / nw ÒIL )y) ydy =0 IlTZ-GALERKIN 0 =no90 | no=*+eno=×L @ (0)=0.- SOIÒ / < NÒTOJNÒdx90- Sotto | ' NÒIanodiÈÒ + SOIÒNOILI/9cacav00"niti- NOILIÌOY. ydy UnbV0000b- SOIÒNOILI| .,MWNOILIOÌÒY/ydy =0-Arts'Tr a a lNESSOFVIRTUALDISPLACEMENTS:- | " NÒOJNÒdi90 - / " NÒIanodiÈÒ +NO/LI / b1 42dy Voohit)-Nolli/b9casaNOIL)42dyÓO +00-b"°b-b"°-NOILI | bnwNolli42dyOÌÒ -b-b-_◦ NÒIanodi+ |MWNÓILI42dyOIÒ + | ] 9cci.snNÒIL)42dg ' gia + ti V00--b---MC-b-1 42voohlt)+' | ' NÒOJNÒdi90 =' | -"¥"°'"uaob-0---KFe MOIÒ + còo + kqo =Fvoohltl2. b) STEPRESPONSE•qwIt)=Fuoco(1- l'§""cos/wdt)) withun=Kea,§=LSIT È +.Kt+ gu2b2ls.IT .✗= [LSSUITCIKIVG 28F--..KMC D= • 0g =[2+WZ• 0g = Hf/a)2W =d.dHflwl =d1[+ In e- jw [+jw'^Hf"Hp(s= jw )=Volsivols)= Hfls)nls)=din/s)nls)TTS•(e+S)Volsi=dnls)svols)= dn (s)-Ivo(s)•anti-Tr a n s f o r m/NG:N°6(t)=-TV0(t)+dn(t)FILTERDYNAMICElevation- à ---- à --M-1c-M-1K-FCIKI-- È =1ooa+.O0 d .nlt)-ÙG_-O0-I_.V6_a±=•LYA P O L N O VEQ:ASSYSTEM+WHITENOISE:AT≥+T≥ At + BWBT =0FIND TEX =======GUARANTEEDONOFORClk)=1•Mb=KTX=4- à ---•y=.OKT0.✗C=-V6_◦[2= Co2 2A}COSIPWX )+1 pm )> ÀC ,sin1 pm ✗)◦APPLYTHEBC:AztAc,=0AttA-3=0 ( a. annusa,+a.,,µ,_a .sn/pu,.aa..s,pu,= ..---K-A1Sinh/pull +azcosh/PWL)+ A3s.in/pwlItA4C0slpuLl--EIlpw3).A1C0ShlpwLI+Azsinh/pull -A}COSIpwlltaqsinlpwll.io SOLVINGTHISSYSTEM1'LLFINDanEQUATIONi & (n)=0FINDle 3.b)- qui --MODESHAPES--w/✗,t)= NY91 =_nunNWZ--9WZ-'PISTONTHEORY:P-Poo=-012vi+W'MooWooVWP•Swi= | "Sw"Iw"dxO-.swe= |; SÉ 1-nii)dx- SÌ /L,t)KW/L,t)- | " sè ' q 2vi+w'diMooWooo--USINGBERNOULLIiSaaLLt--ilT•- 59W/ Nw EINIIdxqui- SÉW) , NÉnnydxqiv-SIw.n-wtlllknwlll.ly -291 SIÈ/ " NÉriwdxòw +------_-Maov00☐☐-2 qsqtwfjnwtnvidxq.in =o--maL--_-L-ilT NÌmuydi9in + gaao |; nènwdx9in + | ewEINIIdx+ NÌILIKNW/LI+29nè nw' di9¥ =o :| ........ma / '---__-o0-MCKSKa====3.C)• In = [ ◦e"•(+21 ++ ≤ +K)90e"ᵗ=odet/II. ++ E + E) =o=•THISEllDEPENDSOn01,THEREFOREICANSETUPanITERATIVEMETHODi1.TiXACERTAINRANGEOFQ2.FOREACHQVERITYIFIGETASOLUTIONSOTHATRE(X)=0FlutterOCCURSUVHENRe(t)TURNSFROMNEGATIVETOPOSITIVE3.IFIIDENTITYMOREFlutterCONDITIONS,THEONEASSOCIATEDTOTHELOVUESTQHASaPH>51calMEANING 9FEBRUARY2022aB Lo e..W=LOniiL=9scusaV6-È=ascesaV6- LÒ V00WooWooV00 Li °• -2501