An investigate ion was made of the disturbed motion of a gas for the harmonic vibrations of a thin slightly cambered wing of finite span moving forward with supersonic velocity. This problem was considered by E. A. Krasilshchikova who applied the method of Fourier series and obtained a solution of the space problem for the condition that the Mach cones drawn through the leading edge of the wing intersect the wing or are tangent to it. In this paper, a different method of solution is given, which is free from the previously mentioned condition. In particular, the vibrations of a triangular wing lying within the Mach cone are considered

Falkovich, S. V.

Haskind, M. D.

UNT Digital Library

---, ,,NATIONAL ADVISORY COMMIITEEFOR AERONAUTICSTECHNICAL MEMORANDUM 1257VIBRATION OF A WING OF FINITE SPAN IN ASUPERSONIC FLOWBy M. D. Haskind and S. V. FalkovichTranslation‘ Kolebaniia Kryla Konechnogo Razmakha v Sverkhzvukovom Potoke.”Prikladnaya Matematika i Mekhanika, Vol. XI, 1947.WashingtonApril 1950-_-ETECH LIBRARY KAFB, NM :-l!lllllllllllllllllllllll!llllilli5=rl144h77I?JWIONALADVISORYCOIWZITEEFOR AERONAtJLLUa ““‘“”TECHNICALMEMORANDUM1257VDRATION OF A WING OF FINITESPAN IN ASUPERSONIC FLotwBy M. D. Haskindand S. V. FalkoviohAn investigateionwas made of the disturbedmotionof a gasfor the harmonicvibrationsof a thin slightlycamberedwing offinitespanmovingforwardwith supersonicvelocity. This problemwas consideredby E. A. =as ilshchikova(reference1) who appliedthe methodof Fourierseriesand obtaineda solutionof the spaceproblemfor the conditionthat the Mach conesdrawnthroughtheleadingedge of the wing intersectthe wing or are tangentto it.In this pqer, a differentmethodof solutionis given,whichis free from the previouslymentionedcondition. In particular,the vibrationsof a triangularwing lyingwithinthe Mach cone areconsidered.1. FUNDAMENTALEQUATIONSOF PROBLEMA thin,slightlycamberedwing,whose projectionon thex,y-planehas the form of an equilateraltrianglewith vertexangle 2yc 2a,where a is the Mach angle,is considered. Itis assumedthat the fundamentalmotionof the wing consistsof arectilineartranslationalmotionwith constantsupersonicvelocity u parallelto the x-axis. It is also assumedthatthe coordinateaxesmove with the s=e velocityand that thex-axisis taksn in the directionof the velocity u. On thefundamentalmotionof the wing,the additionalharmonicof itsvibrationwith freauency u is superposed,where the possi-bilityof theof deformationof the wing is not excluded. ThO equationsurfaceof the wing cam thenbe writtenin the formz(xsY}t)= fo(x~y)+ fl(x$y)coswt + f2(x,y)sinWt (1.1)*“KolebaniiaI@ylaKonechnogoRazmakhaT SverlihzvukovomPotoke.”PrikladnayaMatematikai Mekhanika,Vol. XI, 1947,pp. 371-376..—..2 NACA TM 1257It is aasumwithat the ratiosof the functions fk to the lin-ear dimensionsof the triangleand the derivativesafkibx ~hfkfiy are s~ll.Consideringthe nonverticalmotionof the gas,for thevelocitypotential@ (Xjy,zjt)the followinglinearizedequationmay be taken:where M = u/a is theMaoh num”erand a is the velocityof sound.The flow conditionthatmust be satisfiedon the surfaceofthe wingby the velocitypotentialQ(x,y,z,t) shallbe consideredas satisfiedon the pro~ectionof the wing in the plane z = O.The derivationof this equationshallbe basalon the work ofN. E. KochZn (reference2). A stationarysystemof coordi-nates xl? Y~$ - zkoonnectedwith the movingsystemofcoordinatesby the rela ions x . xl - utl, Y u Y~t z = Zl,and t = tl is used;when expressedin thesecoordinates,equa-tion (1.1)becomeszl = fO(X1-utl}yl)+ fl(xl-utl,yl) ms @t + f2(xl-utl,yl)sinUtFor the componentof the velceityof the gas particlesnormalto the wing,thereis obtainedBy introducingthe notationafo afl af2- u ~= Z(x$y) Ufz - u ~= Z1(XJY) -(l)fl- u ~= Z2(X,Y)the boundarycondition,whichmust be satisfiedby the velocitypotential @(x,y,z,t), is represented in the form()Lob= ~=o = ZO(xsY)+ Zl(xjY)cos~t + Z2(X,y)sin6Jt (1.3]A.On the surfaceof theMach cone, 4(%Y,z,t) = 0.“- .1.NACA TM 1257 3Thus the m?oblemis arrivtiat: To findwithinthe Mach conethe function ~(x,y,z,t) satisfyt~ equation(1.2)and condi-tion (1.3)and becomingzero on the Mach cone.When the steadyvibrationaloharacterof the motionof the gasis tskenintoaccount,equation(1.2)is transformedto a simplerform,setting(reference3)Q(X,Y,Z,*)=90(X,Y,Z) +91(X,7,2) Cos (@J +Ax) +‘Yz(x,y,z) sin @t(~ ==-)+Ax)By introducing the notationZ+x,y) + iz3(x,Y) = Z(%Y)P1(X,Y,Z)+ W3(X,Y,Z) =W%Y,Z)there is obtain~ for determiningthe functions~o(x,y,z) andq(x,y,z) afterstiplecaputationsfrom equatio~ (1.2)@ (1.3)the followingequations:a?po(1-M2)—axz(*O)\F)z.o = z&Y)t(1.4)(14f2)b?+b~+a~+k2~= O()a~o -i,AxT Z=o = iZ(x,y)e J~=-) (1.5)The functionqO(x,Y,z) determinedby equations(1.4)is thevelocitypotentialof the motionof the gsa for steadyforward[-i@.lt+Ax)motionof the wing and Re tix,y,z)e 1 is the velocitypotentialcorrespondingto the vibrationsof the wing.4 NACA TM 1257It is notedthatfor k = O, the system(1.5)goes over intothe system(1.4),so that it is sufficientto find the solutionofsystem(1.5).2. METHODOF SOLUTIONIn equations(1.5)Y“P2=22*and the curvilinearsystemof coordinatesare introduced(2.1)., —t=It is easilyseenthatx*+pV=== JwithintheMaoh cone #2-Y+$2-Z*2= O:the coordinates‘~, q,and ! uniquelydefhe the p&itlon of a ~oint.On theMaoh cone g=o, ana go. In the triangularregionof theprojectionof the wing, z = 0, IYI<@ 7 x,T =0aS((am hLG-z-cos y + sin 7In the new coord.inates,quation(1.1 (2.2)V’==7Cos y - sin yu5).assumesthe form(2.3).-——,.—.NACA TM 1257 5(2.4)By settingin equation(2.3) cp= x(~,q)*(~) afterseparationof the variablesthereresultsThe solutionsof equation(2.5)are expressedin Besselfunctions.Whenaocount is takenofMach cone (~ = O), theresults:$n(k~) =the finitenessof the function v on thefollowingsolutionof equation(2.5)(2.7)In orderto choosein equations(2.5)and (2.6)the constantthe arbitrsryfunction F(c) is expanded,analyticin the regioncontainingthe originof coordinates,into a seriesoffunctions $n(~):‘(t) = ~ ~$n(~)In orderto establishthe possibilityof this expansion,useis made of the formulaof Gegenbauer(reference4, p. 283):(2.8)n,..6 IVACATM 1257where &V(T) are the Gegenbauerpolynomialsdetemined by theequation .1S AnAnV(T)= 2v+n(v+n)2Tn+l -oBy setting v = $ in equationsZkt!m 1’2”In! o2 (2.9)(2.8)and (2.9),and by takingaccountof equation(2.7),the followingseriesis obtained:—where %(T) accordingto equation(2.9)will have the formWhen series(2.10)and the integralformula(2.10)(2.11]Or Cauchyare USti,(2.12)The coefficients~ can easilybe expressedin termsof thederivativesof the function F(t) for ~ . 0. Thus,by ’taki~intoaccountseries(2.10)and the C!auchyformulafor thederivatives,<In~ = 2n+l2~(n-2m) (2n-2m)!2n =() ‘0)m!(n-m)!(n-2m)!(2.13).-...NACA TM 1257 7For the steadymottonin equation(2.5) k = O, then$n(~) = t= and equations(2.12)bec~e the usualwer seriesIt is assumedthat in equation(2.4)the function f(f,c) isanalyticin ~; then accordingto equations(2.12),it can beexpandedin the functions$n(k~) intothe series(2.14)In this connection,the solutionof the problemis soughtinthe form(2.15)On the basis of equation(2.4)@ series(Z.14),the bo@~conditionforis obtainal:where a andthe func~ion %n(~,q), which satisfies-equation(2.6),axn()y q.o = f’n(~) (~<f<b) (2.16)b are determinedfrom equations(2.2). The problemthus reducesto the integrationof equa~ion(2.6)for the bo&darycondition(2.16). By means of the transformation% = ~n+l~*,equation(2.6)reducesto the equationof Darbouxfor thefunction xn*(~,q) and becausein this case n is an integer,the generalsolutionof equation(2.6)can be representedin theform (seereference5)(2.17)where ~(~,q) is an arbitr~y h~o~c function~ An is theLa@ace operatorof order n.83. OBTAININGTHE HARMONICFUNCTION ~NACA TM 1257By developingthe expression(2.17)and takinginto accountthat AUn(~,q) = O, thereis obtainedWhen condition(2.16)is satisfied,an ordinEwYdifferentialequationof Eulerfor (b~hv) ~=0 iS obtainelsc~.(-2)m u =Eafn(g)(2n-m)! ~m dm aun ,_, (a$~<b)m!(n-m)t d~m aq rI=O n!(3.2)The generalintegralof an Euler equation,as is known,hae the form$3wAi2~’’2m-1+un(g)‘a<‘<b) ‘303).where ~(~) is a particularintegralfor equation(3.2)and ~’is an arbitraryconstant.The complexvariable T=~+iq is intrcxiucedand thefunctionof thisvariable Wn(T) = Un + iVn is considered.BYmakinguse of the Cauchy-Riemannequations,the boundaryconditionsthat’mustbe satisfiedby the functionVn(~,O)‘vn(E) ‘~n(~)Wn(7) are obtained:(3.4)where Cm are new constantsof integrationand ~< ~<b....*,i..NACA TM 1257Further,the Maoh cone9from equations(2.15)d (3.1),it followsthat on.x~{~n(k~)liln 5 (-2) m (2n-m)\+O,c-o ~o En -O m!(n-m)%When the propertiesof the function $n(kt) and equa-tions (2.1)M tfien intoaocount,fT(oJ%o) =E n:(x+y)n“~(0,~)*OBecauseon the surfaceof theMaoh cone,the function~becomeszero,it is necessarythat Un(0)~)= O. Moreover,onaocountof the symmetry%(W = o (O<~<a, ~>b) (3.5)Condition(3.5)permitsthe anal.@icalcontinuationof thefunotion Wn(T) intothe lefthalf-plane.Finally,the function9 and therefore,accomiingto equa-tion (3.1),the entirefunction ~(m)(~jq) must be finitefor q = Oand ~ = a, and ~ = b. This conditionwill be satisfiedif thefinitenessof the function Wn(m)(T) is requiredfor m = O, l,...jn.The satisfyingof the lastconditionleadsto equationsdeterminingthe constants ~ in equations(3.4). Thus the followingboundsx’y-conditionproblemof the theoryof functionsis arrivedat: Todeterminethe analyticalfunotion Wn(d(T), regularin the upperha~-plane, finiteat the points T =&a and T =&b, and SdiS-fyingthe conditionson the real axis (q=o):Iill Wn(m)= Vn(m) (~) (+ac ~c+b)Re Wn(m) = O (a>1~1,]@b)Zulwnb)= Vn+qf) (-begC -a)(+(-g) = (-lp Vnkqg))-,●10 NACATM 1257 .By applyingthe formulaof Keldishand Sedov (references6 and 7), ~Equations(3.6)givefor Wn(m) finit valuesat the pointsT = *a. The conditionof finitenessOf Wn?m)(~) at 7=*b leads—to the systemof n+l equationsb1~SW Vn(m)(S)dSs -a b -S.0 (m=O,l,e..,n) (3.7)where ~ = O for m even and v = 1 forBy mmputin$ vn(m)(~) frum equationsthe resultin equation(3.7),the systemofbroughtto the formL~=o c2B2m=D2 (2=0,1,..,n)m odd.(3.4)and substitutingequation(3.7)‘canbe.—where%, 2U+1= 2(2-v)BZ,2V= 22(22-1)...(22V)V).Ja~2(2-v)~J2(2-V)=o 4/(x2-l)mJ Sv@)(S) daDz=-(O=:)U=a {(sz-a2)(~s~(3.8)~2(2-v)—J2(2 -P)b.—{0 for 2 even1 for 2 odd)NACA !Bd1257The integrals J2tcan beellipticintegralsofI he firstrecurrencerelation11expressedin termsof completeand secondkindmakinguse of the ‘(22+3)k2J21A u2(2+l)(l+@)J2(l+1)- (22+1)J2ZJO = Kr.1(3.9)J2= K’+()kt 2E,T(k’-A/”)where K* and Et are completeellipticintegralsof the firstand secondkind of the auxiliarymodulus k?. All the coefficientsBlm of the system(3.8)’canthusbe computed.Translatedby S. Reiss,NationalA&visoryCommitteefor Aeronautics.1.2.3.4.5.6.7.REFERENCESKrasilshchikova,E. A.: DisturbedMotionofVibrationof a Wing Movingwith SupersonicMat. iMekh., T. XI, 1947.the Air DuringtheVelocLty. Prik.Kochin,N. E.: The SteadyVibrationsof awing of CircularYlanForm. Prik.Mat. i Mel&.,T. VI, 1942.Haskind,M. D.: The Vibrationsof a Wing in the SubsonicFlowof a Gas. Prik.Mat. i Mel&.,T. XI, 1947.Watson,G.: A Treatiseon the Theoryof BesselFunctions.CambridgeUniv.Press,1922.Darboux,G: Leconssur la theorieg&&ale des surface.Vol. II. Paris,1915.Keldish,M. V., and Sedov,L. I.: EffectiveSolutionof CertatinBoundaryProblemsfor HarmonicFunctions. DAN SSSR,T. XVI,No. 1, 1937. g_+-Sedov,L. I.: Theoryof the PlaneMotfon of an IdealFluid. 1939.- NAcA-Langley -4-12-50 -1000

Kolebaniia Kryla Konechnogo Razmakha v Sverkhzvukovom Potoke

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