Oscillating Airfoil in Parallel Streams Separated by an Interface Citation Lo, Shih-Chun (1951) Oscillating Airfoil in Parallel Streams Separated by an Interface. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HR3R-5994. https://resolver.caltech.edu/CaltechETD:etd-10012002-100524 Abstract A new approach to tail buffeting is made by studying the problem of a thin airfoil performing a periodic oscillation of small amplitude in the presence of an interface across which the flow undergoes a constant change in density and velocity. A general solution to the problem is found. Lift and moment for some special cases are obtained in simple forms and are plotted in Figs. 3 and 4 for the two basic modes of oscillation: bending and torsion. A typical application to flutter analysis is made and it is found that tail flutter at low speeds is possible for the tail lying in the wake of the wing. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Aeronautics and Mathematics Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Tsien, Hsue Shen Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1951 Additional Information: Author changed the spelling of his name to Shijun Luo after his return to China in 1951. Record Number: CaltechETD:etd-10012002-100524 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10012002-100524 DOI: 10.7907/HR3R-5994 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3845 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 01 Oct 2002 Last Modified: 24 Apr 2023 22:14 Thesis Files PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

IN SSPARATE:D BY Thesis by Shih-Chun Lo In Partial Fulfillment-of the Requirements For the Degree of Docto.r of Fllilosophy California Institute of Technology Pasadena. California 1951. .;.;"'irst of al.1, the authcir wish-!s to ad::1'lowletlge ti.\e const'.:.nt help and encouragement which he received from the faculty and the staff of the Guggenheim 1ieronautica1 Laboratory. in particular Dr. Clark B. lv1illikan and Dr. Grne st .:.£:. s~chlei·. The inspiration for this attempt to investigate the fa.il buf. feting phenoinenon by studying the t!fiect of interface on oscillating airfoil was provided by Dr. Hsue .. Shen Tsien, l'-obert H. Gt)da<=~rd Professor of Jet Propulsion. Not only is the original cnnc1::ption ol the pa.per attributable to him, but the actual execution of tht:icalculation has been possible only with the aid of his constant :-i.dvice and encouragement. The author is i.foeply grateful to ~~~rofos~· sor Tsien for his brilliant and insFiring guidance. Tfi,BL:i!: O.f' CCNTE.;NT:S. {Cont'd) Page Appendix 10: Derivation 0£ E:q. (84) Table l: . Lift and Moment when h .. ,o Table Z: Lilt and Moment when h=o • l=I and ;(s;f ·Table 3: t\erodynamic Coefficients whett h:10 a.ncl ~ = .• 35 Table 4: .1:--unction M.sUU when , o<.l.-:.o:;1 s := o-s and and :j6 &._ .. o , o<L(.~ 87 , ~=o Jl=. o-3 Figure 1: "Ning Wake ;i..cting upon the Cscillating Tail Figure 2: Inb~rf;u~e with Thin .i"... irfoil Figure 3: Vector Diagram of the Lift and Moment on the Air£oil i.n Bending Oscillati.Ot1., as Functions 0£. It. f , i ';0 q 1. ··;;, and Jl.. Figure 4: V<Jctor Diagram of tlli;~ Lift <i1.mi ~'-<-QKn1::•1t on the l irfoil in "To1·rlional" Z)scill.atkin, as. Functions of n , .l ' :E. and Jl.. r;.3 Figure 5: ·Schematic Diagram of a .Fluttering Airfoil S·4 References 95 T A new a.pproach to tail buffeting is by studying the problem of a thin airfoil perfo:rming a periodic oscillation of arnplitude in the presence of an interface across 1;,.thich the flow undergoes a constant change in density and velocity. .A gen- eral solution to the problem is fol;lnd. Lift and moment for some special cases are obtained in simple forn-1s and arc }">lotted bi :b"'igm. J and torsion. 4 for the two basic of oscillation: ,~. . typical application to flutter analysis is made and it is fou,nd that tail flutter at low speeds is possible for the tail lying in. the wake of the wing. I. INTRODUCTION TaU buffeting, l. e. tall vibration under the aerodynamic action of the wake shed by the wing at Large angle of attack, i.$ a well ... ltnown phenomenon. Following the unusual accident of the J'u.nkera alrph.1.ne at Meopham, the problem took on a aerioua :aspect and led 11u;:ientiftc orgianiza.Uons tn: various couutries to u.n• dartake detailed investigationu . . (Refs. l-4}. i>ased on these and other itiVEtllltigl'ltfons, a l~Pge n.u1-nb.er of !limple dc,ivi.eeaJ ar• UQW available for the elimination of the tall buffeting. theories for predicting HowtJYlt~, ph~nomeuon arltll still la.eking. The main dU'fkulty in trcii\Ung the tail l:n,~t'f!i1Ung U.ea. in the fact that thfol actu~l na.tttre oft.he aerodynamic wake behind. the bodies has not yet been established. In Ref. 4, ,.i.\bdrasb.itov a.pproxiroated the effect ol the wake on the tail. by a harmonic disturbs.nee fo.rce and found that tb.e chara.ctera 0£ the ta:U vibra•.' ti.on 1.u·• fundamentally determined by (a) the ratio o:i the freq\lel\ey of the flow in the wake of the wing to the natural frequency of the tail surface. (b) the amplitude of the di$turbance forces produced. by the wa.k.e of the wing, (c) the magnitu.de of the speed of flight, and (d) the vertica.1 position of the tail surface relative to the wing* . In this paper. a11 ~n:tirely di.lferent approach is made. The aerodynamic wake shed by the wing is here approximated by a.n interf~ce across· which the 'f'low undergoes a constant change in density ~n4 v1t.lodty (Fias • l and Z) . The problem ir. then set to determine the possibilities o.£ the tail flutter in the presence of the interfacie at ts believed to be approximated by the 'ef!ect of mutual in.fluence betw~en the vQrti~lty 01:1 thtt interface and the vorticity generated by tll• osclllation of the a.i:rt'oll. It is found convenient to discuss the results in terms of . >t.h the two parameters: ."'£:" . • the vertical distance between the air.,.. loil and. the inter.face divided by the half chord, and R • the re- duced fl."equency (the product of the half chord and the vibr·ation freq,u.enc::y of the tail divided by the flying speed}. (a) When ~h" i. tbe influence of the interface is extremely small for all valuea of .fl. • Therefore the tail flutter has es.sen.ti.ally the tu.ttu.:re of the wing flutter (R.ef. 5 )', when. the vertical distance between the airfoil and the inter.face is greater than the half chord. · (b) When JL< l, the influence of the interface is small for au vat ... ues o! '!." . Hence in high speed range of flight (since JL < 1 means. *It is noted that he also made an important approximation in taking only the quad-steady values for the aerodynamic force and moment. i.e. neglecting the effect of the tail wake produced by the non-steady motion of the tail. .. 3 ... .If U > <:.::'. °"" ) •• the tail 1.lutter again has the nature o f the wing f lutter. (c) When ~h c = 0 and J'2. > 1, the in(luenc:e of the interface is so large a$ to render the tail flutter possible. Ul'llike the wing, the tail has the possibility of entering into flutter· at low speeds of flight, when it lies in the wake of the wing. -4- u. FOR~f.ULA TlON OF THE PROBLEM 'l'HE GENERAL PROC.EDIJRES To formu.l.ate the general problem the following a11sumptions at"• made: (a) the wake given o.«by the wing may be approximated. by an inter- face across whi¢h the flow u11det>goes a co1u1iant change in velocity and density; th• interface is flat• of zero thickneaa am.t eX:tend.1$ to lnflnlty tn all fitrectiona; (b) the tail surface i~ of infinit«t ~•pect retioi · (c). the oscillati:ng motion. is two dimenaional, i.e. every cros&"' section ta.keu. perpendicular to the &pan has identical motion· and remains in lte own plane during the motion; (d) the flow ia incompr.-usible and non-viecous; (e) the thickness o! the tail surface and the amplitude are small in compal'i&on with the ch9rd; {!)the oscillation is periodic; (&i) the tail a mean position p&rallel to the interface. Ustng the a1uumptiorul (a)-(c), th1e problem beeomem two dimensional. As shown in Fig. Z, the interface is located at y=h The airfoil is put on the x-axis with its leading edge at x=-1 its trailing edge at x =, . I • and The chord ia taken as a so that all d.i.s.tancea. in this. ana.lysl.a a.re mea9u.red. r•lativ~ to th• half chord. The interface divides the whole space in.to two regions. The region -S- whicb. the tail is situated, yt:::h i~ designated by u1P and the otliet• y ~h .is designated by region, 11 velocity and density of l 11 • _the u.ndintul."bed flow in region l arc denoted by rr, and fl . and the t:.orrespondin.g quantities in region a are denoted by rr:l. arid '" . The assumption (d) makes the flovvs potential in both r•egiona. i.e. the~e exist the potential functions, ~, and ~.4· satisfying the L~1.p1.ace differential equation for the flows detern-iining specified~ in regions l and 2. For £, and $.~ , complete boundary conditions should be These conditions are (a) on the surface of the airfoil. the normal component of vc: lodty should be equal to that of the prescribed motion of the airfoil; (b) ~t infinity, i.e. poirits tar from the airfoil, the disturbance should (c:) at the interface~ the velocity vectors 011 the two sides and the st.a.tic pressures on the two sides are equal. lating lift and moment, only ~ b.oth if, v,d.ll be required. in- .l•"or calcu ... Ho·v1ever 1 since and .ffi.il.. enter into the conditions at the interface. they must be investigated sinmlta11eous ty. Using the assumption (e). tbe boundary conditions (a) and (c) can be applied at the u.ndisturbed or mean positions. The <=<a:::.u.·iu1-,- tions (e) ... (g) enable one to write for the boundary condition (a): -60) where v is the velocity component in the y-direction and _, 8 ..:: cos x as given bv .Eq ~ (7) below . .Becatuse the: Laplace differential equatiot~ is linear, the general solution can be obtained by superposing elementary solutions. The elementa1·y solution smtisfies the Laplact"; di!fe.1.·- ent.ial equation and part of the boundary conditions. The s·uperpo- sition ia then made in such· a manner th.at the rest of the boundary conditions are satisfied. The general procedures of the analysis are &$ follows: (a) Taking only .one term of Eq. (1). dition at the airfoil; eiA,-tco.sme, as the boundary con- </>om• the velocity potential for a. uniform flow without fnte:J:9faee is determined; (b) introducing the interface, the velocity potentials are so determined that tP11n and (A»t </>o,,,,+ <Pnn for the flow in region l and s6~11C for the flow in. region a satisfy the conditions at the interface with vanishing disturbance at infinity; (c) considering </>0 ,,,,-1-<Am as the elementary solution, the general solution is obtained by superposition (2) -7where Am is c;leteri:~'1ined in terms of Bri. by the boundary condition at the airfoil, Elq. (1) . (d) using l., , lift and moment acting on the oscillating airfoil arc calculated. i;n pa1·ticular for the two basic modes of oscillation: bending and torsion; (e) calculations for soine special cases are carded out. in detail and others are discussed; and (f) a typical application to the flutter analysis is made through a numerical example. It is noted that the airfoil has been set below tric interface (Le. '1 :>- o ) ~ Si.nee the aerodynamic force and moment bear a definite relation with the motion of tl1e airfoil normal to the intarface, the results obtained under the above condition can be easily inter- preted for the case of the airfoil lying above the inter!ac:e. III. (a.) DETERMINATION OF cf>o,,,_ ftJm is 4efined as the velocity poten~ia.l for the oscilla:ting motion of an air£oil described by v ( e, tJ = e'"'t: cos me (3) in a uniform 'now without interface . In accordance with the theory ol thin air:loilG in oscillating n'lotion (Rd. 6) th!llt whole a.yate.1.-i1 is represented by the two vortex sheets: (a) the bound vortex •he"'t lying along the chord of the airfoil• Le. . . l on the x•a.xis between X=-/ and. x= +I i and (b) the wake vort-ex sheet lying on the x•a:lds from x::+/ to x==+ro. assuming that the motion has been occu.rring so long tb~t transient phenomf;!n.a. have disappeared. The bound vortex sheet is considered to b•. made u,p of the two parts: (a.) the quasi-rdeady part which would be produced if the motion were steady•· or if the wake had no effect on the motion; and (b) the indu.eed pa.rt induced by the wake vortex. De'noting the distribution of the vortex strength. (or circulation) along the x-axis by r ·' the definition of circulation yields . r<X,t:J = Lim l.x [ .ax.-+o . x+4\x. J)\ U(X,o+,t) dx. + x 1:x+i!X l((X,o-)t) dx.] or, by carrying out the lirnit procedure , along where y=o denotes the velocity component in the x-direction U()(,o+it-) on the upper surface of the sheet y=o and u.txp-,t:) denotes the cor- responding quantity on the lower surface oi the sheet ,Y=O . ing om 0 • (4) and rll'n V.'rit- IY.i"t for the strength distributions of the quasi-steady vortex t .the induced vortex and the wake vortex respectively, the velocity potential, <J.,,,,,_ takes the form (5) according to the twlJ dimensional potentfo~ theory. ¥",,,. • The quasi-steady p::.trt of the bound vortex Calculation of is deterrnincd by the use of the conformal transformation i! .:::. ...L ( f; +- J_ ) ~ where ~=X+~y {6) ! and By this transformation, the airfoil in the z ... plane is transfo:rmeJ. into a unit circle in the ~ -plane with the re 1ation -10- The radial and tangential components of ti.le velocity ln the ~ -plane, 'f;i and ie . a1·e easily found tc be re lated to the u and v components of the velocity in the z-plane by t::. and (8) V Stn 9 'f ~ == - u sin 8 (9) By J;q. (8), 8q. (3) is transformed into the boundary condition on the unit circle, ~wt· 'f--"- = 1; · [sin (m+-l)B- Sin tm-tJB] The velocity potential in the (10) ~-plane evidently has the general form ( 11) where the p 's at·e undetermined cons.tanb. Using Eq. (10). aU constants except fo in J!:q. (11) are determined. For rn:::::. 0 I f1 =-I For In= I , tb~-f p, = f3 = ... = d For ,,,, I m-1 -.;Um-I) __1_ ! ~~, To determine - Inf.I - I - ;;;;i+1) (lZ) ofher p·d =o fo , the Kutta-Joukowsky condition is used. The con- cHtlon states that the velocity at the trailing edge 0£ the airfoil should· be finite. By Eqs. (8) a~d (9), the condition becomes that (IO). 'J4 satisfies the condition as seen frorn II= o at 1"= o The condition at fl =o yields (l3) .F'o:r. lrl=/ I B = .J,. (J ~ formula (14) the U:ie of which iti obtained . (4} a.mi (9) and the fact that . (11). Thus - Y.,"' ( ~t)::::; -~ e v iUJt /- C<>S {}sln8 = - ,;z,, e iW t;/i.1: . {x t-) =-oz.. e 1,wt Bm . f} = -a., e i.u.Jt F 1-x.:i flO I / ( 15) and ).- . t Uoml>v ) :: -de ilOb S111,mf) ' for fhe total circulation of the quasi-steady v~n·te:;:. by integrating ro,...., from x:::: -I to r:rn{i-)=-lrC:: ~w-t, x. = -1- I rorn . is obtained , and is Calculation of Yon . The induced part of the bound vortex may also evaluated by the .conformal transformation given by Eq. (6). The induced vor~ex dist:ributton on the airfoil due to a single free vortex with circulation r' located at x=:J and y:::o has been eva.i ... uated in Rel. 6 as olx) == · Putting r' . re ~-)( -1.... r'===- ril.#1 dj ~ /I+I ll+X 1-1 (17} and integrating the result frorr, X==I to X= 00 yield (18) The total circ:u\a.tion of the induced vortex, integration of Eq. (18) from X.=-1 r;m is obtained by to. X=.+-1 , and is '. (19) Calculation of o.;i.,n-i • .Assuming that the wake vortex is left in the fl.uid with invariable strength and position (i.e. the wake vortex is moving away from the airfoil at the velocity rr, and with fixed strength) its strength distribution, along the x-axis ~"' , can be ex- pressed as (ZO) where 7m. is undetermined constant. -13- To evaluate 'fm • the law of conservation of circu.latiou is used. Th.z: law states that tU.e total circulation o! the W;:..ole system is invariably zero. Denoting the ch-culation 1;1.bout tl1e airfoil by the increment of. r/'11 (::::: COWi + r,nt) rm, iii""' it must be equal and opposite to the Thua. there is obtained ( 21) Putting Sqs. {16), (19) and (lO) into .8q. (Zl) yields .3 I .::: f( t>( i..f'L) where -12- w U,, + /-(1 ( i J2.) ~~d K., ,and K, are the modified Bessel functions of the second kind and can be expres~ed by the Bessel functions of the first kind· as (23) *In this deduction the following integration formula given in Ref. 6 is used . ®-I) d~ = K lll\.) + K (.r.JL)--).... e-·fl.. j ....e -i.11..}( rr=/ J t I V Jt,. ( .22a) I -14Putting Zq. (Z2) iub Eq.(.20) gives (24) and for whi!re K,, and 11, are used as abbreviations for Koli..JL) an:d K, l,;AJ • Eq. (S) tog•th~r with Eqs ~ (lS), (Us) and (24) gives the complete dl!>- termination of ,Po..... -is .. IV. ELE:~vlENTARY. (b) DETERMINATION O.F' fan tfmt is defined as the velocity potential to be added to .<f~. to !ol"rn an elementary.velocity potential o~tiliOfying the boundary con- dition:s at tlle interface and at infinity for the flow in region. l, while . another potential . tPVn. is defined as the corresponding elementary velocity potential for the now in region l. at the interface t y='1 ft = Negle.;;ting u.1 !il.nd t./.:t. Th6 boundary conditions -.re given by at (~) at ( ;?6) in .:;omparison with rr; and. ~ and introducing the ve loeity potentials , Eq. ( 25) becomes at y= h (Z7) Bq. ( U) can. be replaced by (28) at because f 1 :::}'4 at y::::h and x=.-(X) for all values of t . Using the lh1earized Eulerian equations of motion in the x-direction -16- where the velocity potentials are introduced. To detetmine <Pm. and film. by ~Qt. (27) and (Z9). it is convenient to write them in the Fourier integral forms.. The appropriate expressiou for· ¢,,n and tfUlf satisfying the Laplace differential equation and vanishing at infinity are ~here ttm , bm. • .i<m. £or '/::>Ii (30) tor y 3 /, ( 31) and Jm a.i·e undetermined functions of .A • cff)m given by Eq. (5) can- also be written in the form (32) where, by the Fourier integ,,..-al formula. (J3) (34) -17Putting Eq. (5) into 'Sq. (33) yields A (J.) . =-.:Z.nA _i. [ € Lw'f /lJ?t 1' [ { + 'I ~"111 ~'I i:) ·] • J I Ytm (X, t) ..Sin .\J< aX + 1 1 (35) by the use of the integration formula· derived in Appendix l. Putting into l<:::q. (35), Eq. (lS) for h111 , 1'-:q. (18) for a,n. Eq. (24) for . r4n and yields, by the use of t_he integration formulas de- rived in Appendilt .? , (36) where are for 111.:;: ~I for J?t ::::; 4 b > J • • • 3 1 !i I 7, •• ' Mn. :i.s given in Appendix land calculated in f',ppenciix 3 and {37) Similarly, the following e>cpressions a1·e obtah"led for ~ from {f>J4) t - ;,...Q_ .vre dL Cc.5 A-A :sin:-' Ko+-K, A~-..n.~ +- ) I (J8} 1'iH B,,,C:t)-:::;~ [ {-).L«h11l J;s~..t)} for m::: 3, 5' 1 lf · ·· Puttins Eqa. (3o):..(3a) into Eqs. (a7) and. (a9) yields fou:r linear algebraic: equation& for the four unknowns am • b_.,,, • o<."" Solving them, the results for and /,,., a.,,, and b111 are (39) ·where f = J7.. (/+-.Ii) r= 1-1/l,-J.,,,,. 1J ;;:..fl_'-{1-)Li<.J (40) i = il.Ji}'k (1-"k) 52 = . ~ r:r; Zq. (30) together with E:qs. (39) (36) and (JS) completes the determination of </> 1111 • ~ " \ . "J'':"•'"""l"'"· ;, f L .i'.,.,. .. region I and is obtained by superposing ti.,~ cle ~ncnt.:.;.ry velocity ::o~· ~q. §, in this generv.l form satisfi..'.!:::. the i..aplaclf:: (l). Tl1e only condi"tion left is that at t1lr:: airfoil given h)' . infinity. (1). To satisfy this condition it is convenient to expand y ==o diffor- and -J "- ~ ... 1 into the Fourier series. .,~ <J y nt For <Poni , it is evi- dent that ;:::: For ¢1..,, , e :wt Co5 ml} («11) it may be written that (42) wher~ uy the 1f'ourie1· e'll:!ries form.ula Putting Sq. (30) into Z.q. (43) yield6 -!or n. .:::: 0, ,;z. 4, "· by USillg fhe_ integration formula ci~l'iVti.1d in iLppendix: ", Putting Eq. (39) into ~q. ( 44) arJ.d using Sqs. (36) tu1d-(38) •. the final results obtained after son1e sirnple algebra are (->'t Ca.n,o = ~-- [ jlNcVt,t (hJ + jS QrU-t,/ (Ii, aJ + if. ~n,o (h,JJ J+ (·;5) -2.2- for m? / for m ;;i.· / for m ~I for m ~I wher~ the following de.finite integrals (with their principal values, if improper) a.re defined. Nnm. '1tJ .= .J"° e - ,V,.J.. .J,. (J't) .Tm (.>.J cJ.>.. o for /n-m.J == 1, 3, 5", ·- · for /n-m/=~~.4;--. and h7 ~ / for n.-m=t,3,5",··· for n. == I, 3, !J, · · · -Z3for n = / 3, 5, · · · for n::::: D' i2.., 4 , · ·· 1 th• !ollowing fun~tions assocfat$d with thtit definite integrals ar~ daffo.ad. (46) and the other notationa are given by :::;q. (-40). -24- 1\dding Eqs. (41) and (42) and superposing the results give (47) by interchanging the repeated summations, provided that the result- ing aeries converge. Zquating the coefficients of cos 116 1.n Eqs. (l) and (47}, an infinite set of linear algebraic equations for the unknowns. A. 's are ob .. ta ~ned and arc 11us follow a P6 :::: Ila A'" - Bn Ao+ ,;z. ~ Com Am J_ h1=t> IXJ Al? + L Cnm l'>r=o . (48) di.. for n -= 1,oz.,·3, ... The general solution is then reduced to the determination of the A's from Eq. (48). In order that such an analysis has practical value, it is. required tha.t c,,ni for fixed n.. tends to zero so rapidly when h1. increases th.at approximate solutions can be found without too much labor and with enough accuracy. c"'""- is expressed in terms_· of a number of definite integrals as given by .Sq. (45). The evalua- tion of the definite integrals is then the next step of the analysis. -ZS- VL GENER.Al SOLUTION' (b) FORMULAS FOR THE DEFL~ITl-; INTEGRALS 1~ nwnber of formulas are obtain~d for the definite integrals defined in Eq. (46). Some integrals are evaluated under the two di£.fe:t."ent conditions: h.> o and h=n due to the difficulties in evaluating them in general forms. Evaluation of N"",,.<ftJ. For Nnm • the following formula ia ·estab- lished· in Append.ix 5 . where h >D n+m >-I and In addition, a recurrence formula is obtained for Nnm in Appendix 6 and is for n=O (50) for By the relation that J:-roiJ = t-P J. 11 0 J • it is evident that 1 m - f-J . Nn , -m = (-) n+m N_n , -rn (51) -26N11m= N,,m • the evaluation of Using J:!::qs. (50}, (5 l) and N nn • reduced t•.::i that of The first two integrals of N nm is Nr1n are easily obto.ined by the use ol. Eq. ( 49) and are ;, I 1v11 di) where K and £ =rr.J_r<. [(,R.,-b<J K (/-'· (.} - .(, [(._I.(.) J are the complete elliptic intagral& of the first and second kinds respectively. It is seen that Eq. (49) also holds for h=o when ln-rn /=M+J • The following formula is easily obtained N,rim (0) = 100J. n 0 (..\)Im (-l..) d.>... = -frj f' ca.s (11-mJIJ d~ = (-/ i<, 0 where Jn - m J = Evaluation of ils +- I (5 i!) C.i::>S f). . Onn/hJ . For 0,., 111 • the following formula is estab- lished in Appendix 5 • (53) -2.7- h >o where 011m. and can also be evaluated in terms of Nmn • Le . .:2. m Onm = N n, 111-1 + N n, m+-1 (54) which is obtained by the recurrence formula T J Jmc~) Jm-1 (.,..\) + 1?1+1 (.-\} = «..-m ~ (55) Also it satisfies the relation (56) Eq. (53) also holds for h= 0 and n+m >o and becomes In particular , Q n~ lo) = -'dl..17 [ 0 as required in this analysis. m. tor >1= for n..\; Jn and In-m I= o, .i, -6··· (5 7) -Z8- Evaluation of f?tm, Q1tni, integral En, ht . G11 and 1-Jl't whe.n h>o • The Pnm ca.n be written as The !irat definite integral of Eq. (58) can be expressed in terms of the Whittaker function. YV;;m a.a deduced in Appendix 1. By the formula for derived in Appendix 8 • Eq. (59) be- comes where Ei"l-:xJ is the exponential integral defined as for x 70 (60a) By the use o£ Zq. (60). the second definite integral of E.q. (58} is obtained where a.s given in .lite!. 7, pp. 1- l. Putting Eqtl. {60) and (61) into Eq. (58) yields. for n-m.::::f/,cf.,,4-,··· (6Z) By similar procedures, the following formulas are obtained -30I z s a> +TS= " -n-'m-.;u [Cnt-m+-.i!..s+I) {41,; s ! rt h+.Hf) rrm+.fH) rrnrrnl-SW) f-) (63) for n= o, d., 4-, ·-- (64) for n = I, 3, 5, (65) -31- for for (66) ft .:: 0 ) cl,/ 4- / ... (67) In Eq. (59). using the asymptotic expansion for the Whittaker function for Also, for ;LJ,-:; >:> I <.-hf,>> I gives -3l- Putting Eqa. (68) and (69) into Eq. (58) yield&, for ah&»/ Similarly, the following asymptotic expressions are obtained for (71) (7Z) (7 3) (74) (75) Evaluation of Pnm • 0..,m , £1'(. • F11 , Gri and J-1.. when 1,-:::o • The following .formula is established for Pnm in .:1.ppendix 9, By the use of Eq. (SS), it is easily obtained fr.om Eq. (76) that 1( =--I. .X.b n (6) Y.h1 f IJ (77} It should. be noticed :that the subscripts n and wt on the right side of Eqa. (76} and (77) are not interchangeable. Using the expansions (Ref. 8 • p. 22). the following formulas a.re obtained -34- (80) (81) where In and '>;:; are uaed as abbreviations for Ll i>) and Y,,t.J.) • The calculation 0£ £,.,, ~, &,, and Hn can be' airnpli£ied: by the u.ae. of Eq. (55) and the relation (Ref. 8, p. 77) The fil":E.it few -V,: defined ln Eq. (46) are obtained by the use of the above results and are v. 7r ~(, v (of)~- - e t .. e.' il, v (82) J!;va.lua.tiou oi R,, J S,, and Tn. when h= o • The infinite series R11 defined itl. L;q •. (46) becomes, when h=o (33) by the use of Eq. (52). Using the summation formulas derived in Appendix 10. Eq. (8J) becomes. R.v: l.ll;oJ = <~>n-1 [.;,, ~~ M-t.r..+-1 (./ZJ + f. ['(,,(.JV +t-L (.JL)J } (84) Using Eqr.;.. (76) and (77), the in.Linite series 5'.., and -,-;, defined in Eq. (46} take the following forms {85) -36VII. GENERAL EXPRESSIONS FOR LIFT .AND MOMEI'-!T The lift, L 1 a.nd the moment, M. are defined as the aero- dynamic forae a.nd moment acting on the oscillating airfoil. In the following analysis. Land M are taken at the mid-chord point, L is positive when it is in the positive y-direction and M is positive when it causes di vlng motion. 1 ike ~' in .Eq. ( .?) , l. and M can be ex- presaed aa co L::! .?:. A,.,., Lht h"t--o (87) tV /V1= z_ ff1"'0 AmMrx where Ln. and Mn.. are the lift and moment· produced in the flow rep- resented by the elementary potential, cfo,,,.+- f1H<t A1 , and the constants are those of Eq. (Z) and are determined by Eq. (48). In the flow represented by tfun , the static pressure varies contlnuoualy, because the Bernoulli's equation gives where both ~WI i>t: and 1:tR are continuous as seen from Eq. ( 30 ) . ""''"' . The forces acting on the upper and lower surface of the thin airfoil are equal and opposite and therefore no lift and moment is produced. As defined in. Section. III, the velocity potential Po,.., repre .. sent& the oscillating n1otion of a thin airfoil in a uniform flow with- _37 .. out interface. The lift and moment produced in such a flow have been calculated in Ref. 6 t IU, by the momentum conaidera.tion of the vortex system reproaenting tho motion. Uaing the notations of the present paper, they are (88) where the first terms are the quasi-steady values. the second . I terms are the contributions o! the apparent mass and th.e last terms are the direct contributions of the wake vorticity. Putting into. Eq. ( 88). Eq. (15) for J;,wi • Eq. ( .14) !or r .u... and Eq. (16) !or rem• , carrying out the &imple integrations and summing the results according to Eq .. (87}. the final results are . ij!,ff, J A + k, A ,;JL A .L =-7l';, Ui e ;wt [ ti.JL-t Kb+K, Xo+K, . ' - T 'l0 J .:vK.c. A -(·'-R+ _& ) A -A + i.JZ. A M : : ; tr/,a,ir, e~""t[ lto+K, o 4 Ko+K, ' + 3 l(, where K 0 and J (89} K, are the abbreviations for . Ko<ifl.) and K, ( .:...R.) • It is seen from Eq". (89) that only the first four constants, A 0 , A,. -38- AOL and A~ are t'equired to be solved from Sq. (48) for the determination of the lift and the moment. In the following analysis, the two basic modes o£ oscillation: bending and torsion, are conaid~red in d.eta.H. The bending oac:Ula- tion is defined as the translatory motion normal to the flight direc- tion and the torsio11al oscillation is defined as the rotational motion a.bout the mld-chord polnt. The boundary condition for the former may be· expressed as (90) where Be> has the dimension of velocity. The boundary condition for the latter may be expressed as vto, tJ = e~wt: [ ~ 8, + B, cos I) · where J (91) B. has tb.e dimension of angular velocity, the second term represent& the upwash due to the angular ve loc::ity, and the first term represents the upwash due to the angle of attack at rotated positions.. Since the !irst term of Eq. ( 91) can be included in Eq. (90), it is convenient £or presenting the results on lift and moment to use only the second term of Eq, (91) as the boundary condition for the torsional oacilla.tion. The oscUla.tlon represented by the aecond term of Eq. (91) will be denoted aa the "torsional 11 oscillation in order to differentiate lt from the real torsional oacillation which -39 is rep1·esented by tbe complete expression of I.:.q. (91). The lift and moment for the real torsional oscillation are then the sums 0£ those for the 1 'torsionaln oscillation and those £or the bending oscillation with 8,, == '-~ B, In accordance with Ref. 6 , tb.e lilt and moment will be presented in the non-dimensional forms 1 and /VJ fv1o where, for the bending oscillation Q L 013 -- ,.,..,, 7X, Voe ~LOt (9Z} Moa = ·rre,u:; B .x... 0 e i.wt: and for the 11 torsiona1 1 : oscillation (93) Lo 8 ~ IVlofJ and l oT in Eqs. {92) and (93) are the respective quasi- steady values in a uniform flow without interface .. The c:orresponding value for MoT is zero. The value of Mor given in Eq. (93) is arbitrary but with the dimension of moment. In the preceding analysis only the case, h>:-o i.e. the tail lying below the interface is considered. It can now be seen that the lift and moment for the case of the tail lying above the interface -40are the same as those for the case of the tail lying below the interface with the same vertical distance. h , provided that the density and velocity of the undisturbed flow in which the tail lies are and p, -a; and those of the other flow are f-<- and U~ for both cases. Based on the playaical fact that the force and moment depend only on the motion o£·the airfoil normal to the interface, Eq. (89) should give the force and moment acting on the airfoil lying above the interiace if !orce • moment and motion are now positive Ln the re- versed directions. However, a change in the Bign of motion means a change in the aigna of lift and mon"lent. because in Eq. (89). L ,M ,,..., e'twt Therefore Eq. {89) ean be used without chanfie for the case of the airfoil lying above the interface. Before calculating the lift and moment in detail, the ranges tor the various parameters under consideration are Hated . i. (= u, CGJ .I r= f,A) h = 0 where the case, h =o and 00 ~k7t is excluded. bec:auae its lift and moment can be similarly interpreted from those obtained for the case, h:::o and l>'t. ?-O. .f Ls sometimes taken as unity for conven- ience in the following analysis, which is consistent with the assumption o! inco;.npressaibiU.ty. -41VIII. LIE'T AND lv:Ol\.'IZNT WH.SN '&cfr. >I. Due to the difficulties of evaluating the definite integrals in general forms. only special cases are carried out in detail. These spel';;i.al ca.JSes, be11d.Cles having advantages in mathemat- ical manipulation, are a_lso important from. the physical point af view. The first case studied is tations of the present paper. take~ as 2). ¥»I or h >>I in the no- (Slnce c, the chord, hae. been Using the formulas of Section VI, ·the asymptotic results £01.• the. case; ft >>I , .are easily obtained.. i\ s seen from the equations listed below, they actually yield good approximationa even when Ir= il. • which is ve?:y close to the value usad in conventional airplane designa. The investigation of thi& case therefore has p:c:actica! value. rt i5 found convenient:to discuss the case, It 7'>/ , by considering tbe different v~luea of J2. separately . (a) When st is so small that h.JL<:..< I. 'I'he asymptotic expansions of N,,,w/lr) and O,,,m(h) for h>>I are obtained by the use of Sqs. (49} and (53). The approximate expressions of P,,"'t/,,SJ, Qnmfh,~J.£,,(h,S) Eqs. (62)-(67) ~ Using the above results, the associated £unctions of ;:'.;q. (46) and then the coefficients, C>im lated. of .<l:q. (45) are calcu- For the bending ogcillation., putting 8,=8~::::.·.-=o in Eq. (4.8) and then solving for A0 , A,, A.z. and AJ , :"Cq. (89) gives L and lvi. -42Dividing L and M by L.~ cmd Moe of Eq. (9Z). the .final results are (94) =Ir i-Jl.. l••rJl.+ 1rti.IJ,<i-"{-ei.~'.J?.""'1'1JL-.L~"'· L+O(.fl. /,JL~..11..~1.Jt.. ..L J (M.) Mo 6 Cl ( 1+/i_«-J.,__ ({ 8 1 +./-It.,_, Jr:l. ., ' h k" · .:i..,. (48). similar procedures yield L and M. Dividing them by Lt:>/ and Mor of Eq. {93) give a (95) (b} When .Jl Ls of the order of unity. and H..,(/i,S; pansions for For this ease• the asymptotic are given by Eqa. (70)-(75). Using the same ex- N,,,,.p) and {2,J/•J and following the final results are similar procedures t -43- (96) Eq. ( 96) are also valid when J2. >)I . However, for this case simpler cxpre::;::; iun1;; can be obtained as bl! luw. (c)\Nhen Jl >> I . Introducing the .:.syrnptotic e;rp::.nsions of l'\o( c'JU, (40} for various notutions, the fin.al results are _ iJL _f_ _j_ _ _L 1-Jk_ iJL+I + O( _L _j_ _L ) ( !::_) LoB-- 01.- +CJ. 8JL lb 1+/fl h".IL"-, h~JLJ /.r4 (~ 7) (L ) .L i, I 1-)i _L 1-07""' ~ - 8'.Jl -lb l+.f{ t,;,, ( M) Z.fL +...L+_i_ + 7fH.'J U T := 1- d., 8JL + 0 ( .L 5/~-!iFJi... .JL;;.,,) _L _L ) h:i.A) J,+ 1--l-l _L + O(.-L 7+71;_ 4"- .JLZ/ .-L h'4-) h-'-JI../ It is seen that the thin airfoil oscillating in a uniforrn flow Ir= cO without interface is a special case corresponding to either or .f=-:k=I . Th~ expression of the lift "-nd mcment have been obtained in Raf. 6 p. 385 as follows ( '-?/:":) '_. which have been evaluated and plotted in f~ef. 6, ;; . 3<1G. is reduced to the asyn:ptotic ~}q.i:::nsion of ~:q. (91-;). :::etti~1rr ::.:etting ft --->' oo -45and hJL_,,.. 0 , Eqs. (94) anci. (95) are reduced to e'.:pansion of Sq. (98) for Jt-;... o . Noting this fact. the lift and moment for h » / can be evaluated by the perturbation method using the relations of. Eq. (92) as bases .?..nci the terms contaihing '1 i.n :Sqs. (94+--(97) as P'~rturb21.- tions, and superposing the latter to the forn."ler. :.:iuch a method is good if the perturbations are small relative to the basic values. For h::::: ~ , 0 <,f .::.ro and i=o and co 1 the perturbations are found. less than 10°/o of the corresponding values to which they are superposed when o~ J2..-< .04Sl. ~. 5 by the use of ~:qs. (94) and (95) and whe11 by the use of Eq. (97). Believing that the approxima- tions are good for practical usaQ:e, the results for h ~ , o<- .I..:.. co and -k = o and co are plotted in Figs. 3 and 4 together with those for Ii= co cited above . The significance of the results is that the winr, wake has very little effect upon the tail oscillation when the tail is loc::..te.d away from the wake with a vertical distance equal to or greater than its chord. In other words. the tail flutter under this co~1- ditfon possesses the nature cf the wing flutter. Sinct.? the flutter speeds for wing are normally high re luti ve to the flight speeds, the tail buffeting which usually appears at low speeds ca.n ce. avoidad by putting the fa.il s.ufficientl}' a\·,iay fro::ri the path ol: the wing wake. It rnay be.n:ientiorn~cl that putting Jl::::o in Eqs. (94) and -46(95) yields U1e 1 Tatios of the quasi-stcac.ly values of the lift and moment in z.. flow with interface at y~ h interface.> exeept tM) lMo T to those in a flow without of .:::q. (<)5) which states that the quasi- steady moment r~mc..ins zero in a flow with interface. the above ratios /:ire less than one when and grea~er than one when .I k-<-> J , (or ~./i.oreovet, ., :-. ..-.T•· lX. .4« ..ii,. (_b) • o:.· h= o in the the oscillating tail lying in the ·v.;<;,.ke of the 'tK notations of the present paper. The mather.:1atical calculf.;tions are facilitated by the closed forr.ns obtain::cl for t~1e definite int~grals at h =O However. the difficulties b ite series S,..Ul,o,$) and s~m~-ciing the:: 7':iul.,o)) give:1 by fine the investigation to the~ two special ca.se~,-;: infiri.- .( (a) Sl. <.<.I and (or -k=o ) . In the forn1er th·~ series arc S?lmr.ncC.. app:i.·o:.dmately. fa the latter the su:n~.,ations <a~e avoided aml ex<:.:.ct expres.siom:; are obta.irled fo;.· lift <.J.ncl. ~n-:.m:1ent. tigo.tion is of i·:'l.terest. This in.ves- c&uze it woulo. reveal the essential featu:::es of the; interc:icticm, between the interface 2.nd th::: tail os-· tigation of thi:; t&H flutt<."!r 2.t all -48(a) Jl <<I . Nnw/o) ancl and Rn (JL,t>) for Sl <-<-I and (84). Onnf 0 ) are given by L:qs. (52) und (57). • are obtained by the use of .Sqs. (76 )-(f;l) ~""allowing the procedures stated in the last section, the final results for lift and mornent are {99) l' .ht • ls see:1 t'11nt' ·-, ..,..,q, (O~) ~'J ' ve i·y close- to tbe c or res ponding ex.pan.is sion of ~:q_. (913) and ii.; reduced to it by putting 1:::: k:=; I . rt ::nay then be said that the functional ci.ependeuce of the lift anci mornent actin:;r upon tlle oscillating tail ;..i.pon the :re:duced f1·equency is not much influenc0d by the we.ke when the flying speed is so high that R <. I • In this high speed range 1 the tail flutter sti.11 has the na- ture of the wing flutter even though the tail lies in the wake. Set- ting Jl.=o , Eq. (99) yields the ratios of the quasi-steatly lift and rnoment at h =O to those at h =t>O. {b) k= o *. Putting "k==o in Zq. (45), the terms containing S.,.rsz..,of) and Tn (Sl..,o,$) disappear and C111n:;= o for In ~4. and n.9' ltt. • Due to this fact, the ·. 's can be solved from Sq. (48) in the exact forms. Using Eqs. (76)-(81) and (84) for the various integrals and functions , the final results for -lift an.cl mc~11ent are *On c.ccount of the lineari:wation used i.n deriving c:q. (27). it is necessary to interpret this as ~.<::<Tis while the condition, U..,;i_, v~ (the perturba.tion velocity components in region 2) « U,z. is still satisfied, -SO- (lOC) where ~ ,;, -<.. C.== ( - - -..n..)::>...e ...rt. and ~ - iJL-1T l Vl~+~ J;., ) -7tJ,( · .L +(, ) -7f vJ ( I o I ..(.. 0 T ~ .fl.., 1 ·) --1.,. . .fl. and 'ti are used es abbreviations for l;(Jl) and 'r:,C.ll..) By the use 0£ Eq. (100), the lift o.nci moment arc calculated £or o ~fl -6 ol.4- • The results are tabulated in Tu.ble land are plotted out in .Figs. 3 and 4. It is seen that when JZ.< I the variations cf lift and moment with . Jl as shown in case (a) above. entirely different. are very si:nilar to these of Zq. (9S) 'Nhen R >I , the: vari:ritions becor.:1e It is therefor~ interesting to investi!5ate the asymptotic expansions of .Sq. (100) which are obtained as follows: l ~JZ.. I ~,· I-~ 6>i r,;;:-;; ( 13 - e e,os 1J) + ( - ) = -4 - -b { 1 - e c...osI J + ./q vlr.11.. . Lo 8 t' +J..Jl. [J.J§.. 8 ( 3 1!Ji. .1.. .:u,· J (L lb +y.;r- a+3rr) e casu-+ e ("as tJ- + () :Ji.-h) 11 <i, (101) : -l tr:+ d-J9 €°:<.. _J_ 4 &~· ...L ;;zA -'l. J <i +a-371 + ( 8 - 3-irJ e cos&_,..Jl. I 4 e <'oS 5 +- o J2w where By comparing the values calculated by Eq. (101) with those calculated by Eq. (100), it ~s founci that, neglecting the tern'1s of the order Jt and the higher orders the expansions for ( ~~ c..nci f ~)T of Eq. (101} yield vcr}' good approximations (error less than 5°/ 0 ) when .51.. >3 • By compr..:.ring the coefficient? of thi? v~u:- ious expansions of i!:q. (101), it is expected that the expansions for (£,,Jg and ( tJr are accurate when J2. "> /00 . l1 s JL---? co • the L. ( L)r and the ree::.1 parts of limiting values of and 1 are periodic in J2. ( ~) with the pericci, TC • Fra:n Figa. 3 and 4, it is seen that the variations o! the lift <'l.U'l niomcnt against Jt are peculiar when Jl. > 1 • Hence, when the -52flying speed is so low that J1. >/ ~ the Util lying in thi:: wake of the wing inay be subject to se 1£-cxcited. vibr~tion, i.e. flutter. Such a possibility will later be verified in the nurnerice.l example. Befo1·c doing this. the cases excluded from this and the last section are investigated. -5 3- The c~sce rcma.ining to be studied o.rc (u) e>o f ~ h >o and ~-t. ~o and (b) h =o, l>-k. >o and J2>/. The exp1.·essions for lift and moment have not been obtained due to mathewatical difficulties. However by considering the continuity character of the various functions with respect to h and k, and using the results for the boundary cases obtained in the last two sections, it is pos.siblc to r:.1ake some qualitative remarks about the lift and moment variations (a) I :::::- h with .J2 >o . ,/ and 00 ~"'.k ~ O • The difficulty lies in finding the ascending power series in h for the definite integrals of the following type (Eq. 46) f'n111. ch,$) hi e'f"idently continuous in h. first derivative exists there. at h ?O , becat.u:H.: its It is also continuous in h for A>,,o. This in seen by considerir~g th~ difference (10 2) ..~.rherG '(.. is <:. small positive quantity • . (102) c&n be \Vritten as -54- (103) .1.. where T is taken equal to t-~ e:.- o ( 1 e • .:1s · • the £irst term of Eq. (103) approaches ~ei·o, !or -.:t.t/I J-:)> o ; and the second term approaches zero for which approaches zero when Hence and is greater than S R.m(h, ') L/m T-ro i.e.. t--+ o is continuous iu h at h=o and therefore R.m (h,$)::::: Pnm (o,6) = - : J; !SJ 'fm UJ h-4() as given by :::q. (76). ()ther definite integrals of Zei.. (46) have the same prOj_:lerty. Therefore the lift z.nci moment vary contin- uous ly from those· at from ·Z to 0. k::::..i.. to those at /, == o , when h ciecreo.se s . By the use of the findings of the last bvo sections, the following :;nay be saici about the variations of the lift and mornent with 52. (i) dhcn J1 £../ , they are similar to those for h== CO • -55(ii) ",•,:hen Jl '7 / and h is close to one• they are aguin similar to those for h : : : . a? • (iii) ·vlhen Jl >/ and h is close to zero, they are different from tho:::e for it=o::J and becorne sirnilar to tho5e for A~ o . ( b) h ::::. o , I > i1t > o and Sl. > I The calculation requires the summation of the series of the following type (:Z:q. 46) or, by the use of Eq. (77) where /V/s{JL) of J;,CJZ..) is defined ir1 E:c.:. (37) and is expressible in terms and Y.,(Jt....) as sho·,;;,·n in .Appendix 3 ~ and The e>:act .summation has not b·~en obtained. mation for .IL»/ The asymptotic sum- is complicated. by fac fact that th.er~ e:dst thl'ee different asymptotic eicp<=.msions for J;,l.Jl..) a.ncl both h anclJl are forge according as equal to or greater than one. 'f:rCJl..) \vhen ..ft Yi"" is less th2n, neE.rly J:.;esides, the usefulness of the asyE'.lptatic expansions is in doubt, because at i=o some of thern yield good approximation only when 12.?loo as given by -56Zq. (101). The numerical su~.-nmatior. . for specific values of .1 , k and SL c:.::.n however be easily done. .1, . 4, l. 5 &.nci 3 \~·it'..1 an en· or less than tions are made for Jl = The results arc tul:t... latcd hi. I'i:o.blc 2:. and shown ir. Figs. 3 10°/0 • and 4 by broken lines. Though the points a;:e too fe•N to deter- mine the curves precise l~', they definitely show that the variations of the lift and. rr.on.:.ent lar to those foi~ i=o with J2. than to those for the lift and mo.-nent are conth1uous in for k-= k=/ ±n.rc :no re s imi- \vhen ..n. >I 5inc€i: k as seen from ({5) I it may be maid that- the vari<..-.tions of the lift ~~1d ::-:1.omen.t with .52.. fr:.>r ±. ;::.;k, >o h::>.n) the character p~rtairiing tc• those £0:. t ere s t.1ng t·o con 1nue t•lns . ln . ve s t. .... f 01· ./ - 4§._ ·1g:~ ~ion ,q_It i s 1n t• • l or for i:::::: /- t where £: lnine t!1e value of is a positive= s1-:iD.ll :-:Juantity b cleter- i corresponding to the transitic:n in tl1e ''a.r- iations of the lift and morr1ent. question is that: at However, the tL01;·e irr1port:i~1t Do the peculiar va.ri2.tions o! thr: lift and mo~1H::::-1t i= o really cause the t2-i l flutter z.t lo·.;· of f1 ? -57- XL TYPIC/.l i\PPLICATICI'\ TO .ET UTT.:!:;:;. ! ND DlSCUSSICN OF TH:r.:. RE~~ULTS To illustrate the essential features of the tdl flutter under the influence of the interface, a flutter analysis is l1ere carried out using typical cas::: h= ~=O • one. the lift and 'moment obtained for the The flutter an"!-lyi:;;is is a t\vo-dimensional P, tail of unit span with locked.elevator is considereC.. ~;uch a tail has two degrees of freedom, namely bending and torsion. The procedures of the analysis used below follow mainly those given in Ref. 9. Calculation of the Aerodynamic Coefficients. A representative section of the tail with all re lated notations is shown in Figure 5. E denotes the elastic center and G the center of gravity. The ti· ans latory moti011 of E and the rotationary ;.lotion about E are expressed as \ y = y e Zwt: and (10~.i.) e= e respectively. The aerodynamic force F and moment T acting c.t S are expressible in terms of the lift and moment defined in Section VII and is obtained that Y and e of Eq. (104). J.:-ccording to Fig. 5, it -58- F = - n/ w.<.-( A Y +. c. B 1f ) _e ~wi: (105) T = wz 1 c. w ~ Cc Y+ c. D 8 ) e ,·w-ewhere and A, .B , C and 75 are kno ..~m a$ the aerodynamic <.~oefficients and are given by (l06) Using Table l and taking ~ coefficients are evaluated for = . 35 in Eq. (i06). the aerociynamic o :S.JZ..<«..4- and are tabulated i.n Table 3 ~ Calculation of the Flutter Coeffi.cie11ts. Denoti.ng the :ne.ss of the tail by rn , the mass moment of inerti2,· by J , the elastic constant for tr an& lation at E by ca and the elastic constant fo:.:rotation about E by CT and assuming vanishing damping, the -~9- y+ s ii> + cs y - F < rri. (lC 7) +• •• .T e + rn s y + Cr B :::: -r Putting .Sqs. (104) and (105) into 1 (l:Yt) rields e=o !r1? +8 c -y- -r p' 1J ;:::::o (IOS) I \Vhere If', 13' ~ c 1 and o' are k.r:Q•~,tn 8..5 the flutter CQefficicntc ai1d are given by 8 1 8 = -Jn.5 + rn'c (W9} _, D where L.Q6 an.a Wr a::e the natural fre<;ucnc.ies of the bendi:':.C vi- bration z.nd the torsional vibration of the tail in v1.. cuum anC: ~.~·e. . . rr1v"r. t.. ......... ,..... •"') f!J.s ,,.,., ;'.nd /¥ /II respectively. -! n=C=O ·-601 1 8 =- !5 =0 (b} In pure torsion, (110} _,_, - , -1 -1-.1 AD - BC = o (c) in coupled bern:Hng~torsion, -) - r'or a specific structure, A, 8 _, C and D or U, and W are functions of 52.. and w ) as give<• by l.::1:,,,b. (109) ar~::. (106). (£or Jt "":;,,;;, Th~ task is to determine: the values of U, and w \;1bich satisfy There r;.1ny e~::ist a nu.n.1ber of solutions the criterion of .L:::q. (110}. for one criterion. J Those with U, lyini; it: the spec:::--: i·anGe of flight or nearest to the range are of interest and faese valu(;e; of rI, are called us the flutter speeds. In th~ generd \">'iag flutter ar.al- ysis, only (c) of Eq. (110) has solutions and the Hutter speeci.s ar\! higher than the £lyir1g speed. The numerical e:rnm;:ilc under conskle.ratio~1 Numerical :C:xample. is described as follows. The tail.: c = 100 in. • a = 35 in. , s = 5 in. Span = l in. , m = .009 lb-secz;hi.., S = 5 .62. lb-in.-sec. 2 w 5 = 40 rad./ sec. , w 1 = 50 rr;;.c..{/ sec. :Vamping constr.nts = 0 ·:.'he flovv: h=oJ V<- u, O ::.nd -<--/. 4f, =f,,_:::: l./47XIO -1 ! 6 -5ec./tJ1. /;.ccorciing to the two dimensional winr;, fli.;.tter andysis, only the coupleC. bending-torsion is possible in <:.. uniforr,.., flo·"'' v.·ith no in- terface and. the £1utte1· speed for the tail under this condition is equal ta l80 m.p.h. by the use of Graph 1-f·.(n) in .flef. 5. The v«lue is rather low relative t:::> the flying speed of modern <:i.irpknes. -61This is mainly due to the vanishing clarnpina a.sEun1ed z.bove. For the discussion of the present example, it \•?ill h'! assum~?d that the flying speed is about 100 m. p .h. sc that no flutter occurs for the tail in the absence of interface. To calculate the flutter speeds of the tail in the presence of the interface. 2:q. (no) with I::.q. (109) and Table 3 are used. The results are as fellows: (a) No flutter occurs in pure bending. (b) N,, flutter occurs in pure. torsion. (c) Flutter. occurs in coupled bending-torsion. And it occurs at speeds below as wel 1 as above the flying speed. The lowest i:peec. above the flying speed which may be called the upper flutter. speec: is equal to 160 m.p.h. and the highest speed below the flying speed which may be called the lower flutter speed lies between zero and 6 m.p.h. No flutter speed lies between 6 m.p.h. and 160 m. p. h: Tb.0 levier flutter speed has not beer. cletermit'l.ed eJ:actly because it rcc;.uires the lift ancl :no=.1ent of Eq. (100) to be calcu.lated fo1· .n. >;i...4 (Table l). lf Hutter occurred at .Jl=~4-, the flutter speed would be about 6 r.n.l;.h. The ei:istence of the flutter speeds greater than zero is shown by the use of the aaymp- totic e:.:pansions for the lift anc< moment. The criterion (c) of ::::q. (110) becomes -.o:;;ro8 + ,;i.1~0 sinti.-.ll.- .U6t3!J- :rln~JLt-;r/;:-.JI.., [ 4,3b3 -/.!li?..5 s;n ceJZ..+ +.14-W Cos tl.JL + , o5'J 79 ( si"n..z.<....1z. - Si'ri&-JL cos d...Jl..) J+- Q ti_)= o ( l 11) -6Zby the u::>e of i:;q:;;. (lOl), (106) s.nc.~ (lu\:) 2:.:-.t..i <.!.SS\.ix.-...ing that J2. < w .. Sq. (Hl} Ci.as real solutions. I'\eglecdng -;;h~~ te.nns of the o::.·c1er m and the higher 01•ders in I!::q. (111) yie l-::is the sok.tim1s Jl::::: rnr+.13/ where: n and n TT + /,44-0 (112) is an integer. Eq. (112) gives good approximation \vhen. 4 n-> 10 • It is in.tere.:>tinf to note that corresponding to .E:q. (112), the anguL~r f're1~uency is equal to 37.7 rad./sec. 'Vhich ties very close to w 13 • FroG1 Cc:_. (ID!:!), is obtained (113) which represents the ratio of the displace:::.1ent of th~~ trailing edge due to torsion to that due to bendine. u~;i:>.s: :_~(~- (112) • .>.__ is equa1 to . flt..:5, -.:.:hich indicates tha.t bendinr.; is _t>l'e1~.C~Td.n2.nt during th•..; flutter at low speeds. For the £lutte ::.· ;:;t lGO 1:-;. p. h. , the angtc.lc.r frequency is 50 raci./sec. and ,..\ is 3. 44 v-.-:<ich inaic.:-=.t12s t!:u:;t torsior-. is predcrr1inant at the upper flutter c;_::r::('.!Lt. cusseci.. Taking 4 n.:::.10 to . 0034:2 m. p. h. 'This \-;ill later be C:is- in E~. (112) yields a flutter speed eq_u.al Taking the te r:n; of -/.;;_ into co:i.s iderai.:ion in Eq_. (111) gives flutter speeds of the order .3 m.l).h. o.nd A. re- maining less than one. of Eq_. (llO) yi>.?lcis Finally setting 12..::::co the criterion (c) -63- = 37 .9 end sa .o raci../scc. w '.vhich a:z.·e the natu1·al £r"1quenciec 0£ the bcr•.:iing vibr.<.d;ion c.nd the torsional vibration of the tail in the still air \vith the cicnsity given w.bove. Discussion of the Results. The most important results of the cxarnple are summarized and ditcussed ::;..s fol~ov,;s. (a) There elt:ist speeci. a lower flutter speed ::.:is well as an upper flutter The uppei· flutter speed is very close to the flutter speecl obtained by the wing flutter analysis, i.e. neglecting the ·effect of the interface. The t~xistence of the lower flutter spee<l is entirely new to the usual flutter a1~alysis and is caused by the presence of the interface. The occu1·rence of the tail lH.11'fetiug m::ar sto.ll.inr; speeds is qualitatively confirmed.. (b} At the upper flutter speed• the predominant J:'i..'10de of vibration. is torsion and at the lovl speeds of flutter, the preC..omina.nt moc.e is bending. n~.e The former is found in the \Ving flutter ana1ysis. iatter is at;ait1 new tc. the unual flut'..:er <:.nalysis and confirrns tho finding of the :i~nglish inve stigata:r s that the failure of the JU l3 irose £:-orn the flexural str,.:!sses en the tail (:.'':.e£. l). :J3:;:.sed en the discuasiot1s of the lift and mcm<ent in Section X 1 it is belkveci. thnt these i·esults are also valid when the flow is describeci. by h << I and a~ i. < I while the othc r data re ma in unc h<:rnged.. this ex.ample, it is seen that the theory c.nd. the observed facts ln -64agree in the essential features and therefore a continuation of the investigation along the line of approach of the :;:n:esent thesis ni.ay be profitable. :c.he integration form~.l;:is used in L~q. (3.5) aJ~~ for .-1 y.;::;r- o anu l..:Cerivation . (115) 'i'he integr&l of ::.::c:. (114) is by p<:.rtial int0gration for Similarly, .L::q. (115) is derived. .A.y > o by Corm ulc: 490 i:l Ref. 10 INTEGR.A TION :ff'OR)/UL.\S USED IN -~C. (36) The integration forrr.ulas used in 1~:q, (%) are 1 r;::x J_, l!+x - rr J,t>.> sin.A><. dx. (116) J 'jt=X >d x _ - eris.,..:.::: l+X -1 (117) P1-'<, J I • • \ 1 Sin m{} $1>1"")( a.( = [ o , -1 (I . J S/n m& Cos>.J<. dx. -J wi1ere x =cos& (-)---zrn"l('~()..)/J...., r o , 'L(_-)7 >h7[ J:,(>.) /.>... form= 2,4,6,.. for m = l , 3 , 5 , ... (llE.} f\,.:r /11 = Z, t, 6 , . ~ for /11. = l , 3 , 5 •... (119) 1 or Eq. (7). { l.2;0} (121} ( 12.2) where i\ 1; n is ·given by Eq. (3 7) ilnd ce..lculatcd in the ne~t a~)j.:Kn1dix. i1erhratinns. Eqs. (116)-(119) are imrneclic:.tdy obt::.ined by puttine -67X= c.o0 & and then using the e)i:punsi<..m. for!riulas (Ref. S, p. Zl). (:>:> Yf. L, H ..Lt.>..) cos ou1B cos (.A. coso) :::::- J:(.>..) +c2 (124) n""/ s/n (J.... Cos()) ::::: (125} The integral of Eq, (120) is :::. € J _z,.n_[ C.OSA jive-t,J2Xs:nJ.x +. $/nJ. j<oe -Z.12.x Cos...\~ dx 0 _;. .h... :::=. e >. c..os-A. -1- L.. ..rz. .s ln .;>.,, ""<-<-_ .11.:.i... 0 / by setting )=X+J by Formulas 414 and 415 of Ref. 10. Similarly, Eq. (121) is derived. The integral of Sq. (llZ) is = F -iJ~;H·l d< J';1-X ~d.x. . ~ e ~-1 s _, i+x 3-x , provideC.. that the resr.ilting inter:;ral ..::xists by puttir~g X = cos() provided the re:,;:ulting series exist. rt is then required to evaluate the integral n is an integer. Evidently, where by setting for j :>I by the residue theorr.;:;:. are used to derive Eq. (123). = jcv e-Wl-j /f! J LrJ' ·eosJ.x dX.-{ ;-nj' ~-/ ~ -I.//-)("-' I co ::: -<l-7i 2 (-f J..?.S (,J.._} /VI i .J7..) .S=o d.. -I Co5v\.x ,/f-.X.-{., cf_x.__- ~ X J - t.n... -~-..L[K+v-~J d.... (;) f\1 i,J'L and (37) for s = 1, 2, 3, .. -70·· APPENDIX 3 CA LC UL.I\ TION OiT /VI s (J(.) General Formula for Mst.nJ , s = 1, 2:, 3. . . The integral for Ms(f!.) given by Eq. (37) is J e -2} ( J/X] Ms(.Fl-) = 2::::: c..'Jl. where 1 "" i n-:::.o [-{l _ f _:__~"'--- ~-cJ JS-f'l (j :._ 1/t 17(f r{5-rt-H) I , by the binomial theorern Using the integration formula obtained from Eq. (29) on p. 50 of Ref. 11, (126a) where f arq. ~!.::.. "1f, i.:=Fo and RI< ~+1) > o , yields immediately The right h<4nd side of i::q. (126} can be put in tern1.s of kv (z.) the use of the formula {Ref. 8, p. 79) ( 127) -71- J,, [.fL) and 3.nd '(,,(.fl.) K n+-;f; ( ~JL) by the formula (Ref. 9, pp. I 3- ·rn) '.:vhere n : : 0, l, 2 .... can be put in terms of sirnµlc t::ansc;:nr.:.ental functions by the use of the asymptotic e:i::pansior;. of (l~S') •,vhich obviously termh1ates for Y==n+-± and thus yields e:;;:act vc:.h1e!::. /v75 (J?..J Evaluation <.>f , s= o-S . . Using Eqs. (12.6)-(129), Ms (s::: t- 5 l !v'l. are cbtc.ine<l as fr.Hows .to;;ether vJith ' -·-JL.+ -7' ( V + f..,Jn) , r +/I.,JT (J,/ - ·v /1) 10 o Mo (JL) =< _ J::._ e ;L.J1... . I 4 I ..n.)==(-!:..--:z.-.)e M( I .JL ..,fl; 7r (Y. '')' --;--,+(..J, 0<..JL -L'.il L, 3 .:1.A-i::i.1> -~fl. ·r !v13 (.JL) =;t. _ ./\... - - .....,i..; -~ + - r + )e +- ~Jl... ( v'J" + ,,.J ~) .J1.: -72- 3q. (130) are evaluated for J2. = 0, .1, .2, .4, 1. 2 :1.ncl 3 and the results are given in Table 4. Asymptotic E~cpansion of M.sU'-) for J7.. >JI . By the uf.e of i:q. ~ {126) , it is obtained that, putting = ~·J2.. (131) Introducing Sc... (129) into :!i:c... (131} yields, for J2 >> J (132) E::pansion 0£ Ms(.nJ £0-:::- .r >.>I . The integral of {() -i.J1.J O-lr--1 - - sd >i M/Jt) == J e . I - J.. ~ [1 e ttl -(S·i)X e :x - la:>e -(..St-I))( -l.TLCCJJl.-X_) e c1.x.. e - ;fLcoshx is analytic in :;.: , when Ix l 6 ] x -3- M (133} . Noting M itive large value and. has the exp:.lnston, when a.no when'~ is pcsitive and =Co..sh K 0 It is the11 required to consider' that ~ by setting -i....IZ..O>shxd. 0 is /0,,-(Jt) \Vhere J:-1: is (;. po5.Ix/ :S M ~73- the Wat son's Lemma (Ref. 12:, p. 218) gives Puttbg Eq. (134) into Eq. (13 3) yields (135) \~here :S>>I. f rrCOS 11/}. CO.J (.\ COJ (Jj C1 Dl ::::: 1 0 .II [ (-0102..,,. I,. rJ..) for n =G , 2, 4 , . . forn =l,3,5, ... (t 36) forn:::0,2,4 .. forn=l,3,5 .. (13?) which are easily derived by the use of oSqs. (124) <:.nd (125). -75- :'he integral of Eq_. ( ~9) is by i:q_. ( !) on p .. 150 cf Ref. 8 , by Eq. (3) on p. 3f.6 of Ref. 3 R/{n+m) >-t V.'here a ncl /(/(h)>o • The integral of Zq. (53) is by E:q. (7) on p. 386 of Ref. 8 where I -~ J<,-- , If.I (n+m) > D and R/(/,J>o T!1e function of .E:q. (50) is defined by Consider in.g r -t L when () n ""rn""' o when 11 and mare otherwise. and, also J;. f.'\) Using the recurrence 1·elation of and equating the two e:x:.pressions for I N n, ·rt+ IV tt+l,ITI;;:;:;. f\J 1 Yn-i,171-4-h N yield the recur-rence form- · J<i1. <J1hen ft.,,.rr?= '1,m -tNn,nr-1 +lo otherwise o {138) ?utting n=ni {50). -78J.,PF1!:NDIX 7 DI!:RIV,\TICN' OlT EC1. {59) The integral of Eq. (59) is by Eq. (l} on p. 147 of Ref. 8, (139) pi·ovided the l:et>ulting .series converge. rt is now requh-ed to evaluate the integral ,\J sing the integral representation of the .di.ittaker function, given on p. 310 of Ref. 13 yie k1fj where /cugh!/<.rr and .!fir-&) >-I ?utti1:1g; Eq. (140) into Eq. (139) gives irnmediatclr Sq. (59). Wnrn Putting ula of ~l1.. (..l() and in the ex pans ion fol."m- given en p. lt6 0£ H. eC 14 yielos -"t t: -I + (.-x) 2- ra-,P-J (-x) ,,..,._ J ')'-:>:.() r = i:ulcr 's constant = .5 77 2 and ~q. i = 0 , l •l . . . (60a) gives wh~re x :> 0 by puttin.3 by putting VV_...t. 0 (X) .;{.., ·by r "" f'~ (140) I ( 1·11) -80by ::~e;. (141) Thus , feil· x>o z (Kt:) y,,.o Putting Eq. (142) into Eq. (141) gives th•.::; expansion formula fo1· J?V:.-1-1 t- t- ~ (:-) w _-t-4-J f.. <'<. J .:l.. where -¢f.f l( ~ (X)= - - r{-f:::.H) [-e Ei:. t =0 • l, 2, ... x .x i. -~ (-.><) + e and X :> o . -t (-X) 't ) ~ 2 r<t-~) {-:x) J ~"" o £-, ~ -t~ 1- The inte f;:ra l of. :::;c,_. (76) is t Using th0 contour inte gra ti on of the Hanke 11 s type ( 13 . 5 3, p. 4 2.G P~ef. 8), the following integral is consid.ereG. •Nb.ere ~ is real and positive and the contou1· C is shown in y (.3) ------+--·-+---.J:...i-.--. . .-R o H +f\ x -~ the figm.·e where By the residue theorem, .I= 0 Denoting the integrand of I by . Q (i!.) , L can be written as I::: J Qf2Jd:C +I (){z.)dt +I Qr:t)t/i: -+- f Q{e) d:c. + p r R Q(x)dX. (/) (~) (j) l4) J-R. v-.rhere the paths (1), ( Z), (?.>) and. {4) are sho·.v:'l in the figure and P denotes the principal v~lue 0£ the integ1·21. when I~ I~ ro where the first terr:.1 approaches zero by the Jordan's L~mma 06.222, p. llS, Ref. 13) and the second term approaches ze1·0 by the results of j 6. 22 on p. 113 of Ref. 13. C: J Qf?:)d 2- --;)> 0 :::::: - 7r~ n-nt+-1 d3. I where M is a constant, when Jzj_,.o providecl that n + :i .> ni { Residue of Qt'!.) at i!: =; } (3) J,;lS) [ Jn(..~J +; 'r;,,_c;.) J = . rr,: d,, j Q L:r.) di: (.4-) ::::: ~n < L Residue of Q(?:) at r""-~ } Int~) f-:Tmt{,) -ri. /m(t)] = - !Ji d, provided ,..,_,.., = 0, Z, 4 •... -83by the use of the re la Hons that J"(-X.)>::t-tJn {>(.J ::c.nc:l ,,, U:> nt U) ~ H (-j(J =l-J [1-t,.,..1xJ-Vcx) ,,., . Using the above results yields imme!diately ~'.:q. (76). DERIVi~T!ON C·F ZC. (84) i.Cq. (83) can be written as :Z. Mil.Sl-'l.J- 2.., Ma.~/ Jl.) J =I ri [ Mol.n.HJ.., CD 5-::/ .1 where It is req_uirt:d to sum for .:J -3 f , and (145) -85- by .similar :::;tepis above by Eq. (127) by Sq. {lZ6a) 'i.'hus, ~ Meuf-A) = ± [K/i.AJ -/ c:~ ki (~.11) J ,:/,%-./ Putting Eqs.. (145) and (146} into .Sq. (144) anci using Sq.s. {128) an~~ (129) give immediately Eq. (84). -86- .n. l(f)r 0 .50000 0 .soooo 0 .50000 .l .2 .403 .345 .305 .3016 . 2661 .19386 .12747 . l l 954 -.086 -· .076 A04 .341 .400 - .OZ.39 .0214 .2:74 ... 106 - .137 -.0883 .315 -.0693 .3016 .2165 . 2961 . 01290 . 2661 -.0908 -.0336 .3730 . 2368 .0589 .lCJ3b6 -.002.ll • G633 .4·6103 .51984 .17973 .18093 . 3203 .60785 .21720 • 75359 .23566 .29932 .12747 .1195 '( . l 6•Ji:26. .... 03900 - .105 z .16426 .03049 -.03172 -.06675 -.04.392 .02221 . 217 . 2270~ . l "/403 .~ .5 1.0 1.5 2·.o 2.5 3.0 3.5 «LO 4.5 s.o 6.0 7.0 s.o 9.0 10. 11 12 13 14 15 16 li lg 19 zz .22704 .94158 . l 7403 l .1156 . i0005 l. 2255 .10667 1.3538 . 20177 l .6653 .09225 1.985~ .06450 2.1128. . l 7 816 2 . 6 281 .09474 Z.7416 .032720 2.8787 . 15 341 3 . l 234 .10251 3 .4921 .0104-03 3.6495 .12661 3 .8585 . 11 11 4. 2.36 2 -.003047 4.4239 . 09798 4.598~' . l 2085 ~.9748 z , l; .346 - .125 .164 . 302 - .113 . ! 72 . 12~ . l 32 e l .~ J. 9 zo .185 .31 . 20062 .18090 .U.62.8 . 29500 .27860 .17111 .271'..iZ .Ol.0408 -.LQ042 - . 05763 .07038 .12661 -.14177 . 11211 -.013792 • 2032 .13383 . 2G7?3 -. .31408 .12085 ~.00815 .18&02. . 068-i:9 5.3442 S.7090 5. 966S" . 2666 7 . 198 J. s . 038560 . l 70 . 046991 - . %5 ~·'.; • L,.r 7'f c(i ""i .56 .80878 .2937.:~ -.OBS~D r-.GujO<l'? -.OTUH3 -.1s112·· , ! .1'7161 l.2/c:J2 .z,:,79s i. 2947 .338"~8 l .4.25 ·::. . l 7975 lo665[ .3&26 . 30620 .29674 s 7 .185 96 5. 1956 (100) .OS5 - .1422 - .12146 .31911 - .058439 .27372. .07509 - .009381 • 20503 .06267 .1000s - .02451 .22141 -.05874 .10667 -.14562: .00203 . 20177 -.08414 . 32890 .22850 .07537 . 092.25 -.01433 . 20623 - . 04826 . 06450 -.1372 . 32839 - . 0 zo 25 l . l 7 816 -.10734: .osoz1 .09.:.i14 -.00839 . 25207 .19371 -.032977 .032720 -.12113 . 32025 - .040781 .15341 - .12665 .30785' .06343 .10251 - .008G82 -.00815 .12'.670 ... 005 75 0 0 '7.111 .3L;,.:17 ...' .:H(lt} Z,Z•.",3.:~ .212;;; 2..~:"":::·:.. -· . 0 25 c9 2 . 1. (l •.. 05tJ:2SO - .15·575 - .0409% .3119 .233'.:l'17 l -.03306 • j.IJ 1s2 2. -87- ,r J!. B 1. .re z.: LI~T AND l'/iOlv:ZNT .NHEN h=o, ./::::.1 .1 .A.ND L Jl Rlf)8 I {L)a RC~)a I(tf;)s Rr t)r llr)T 0 .63077 .503 .384 . 260 .<..25 .63077 0 - .108 .505 .0149 • 300 I , .454 .265 I. 25 .603 0 - .137 - .l0l .60000 .477 .386 0 - .141 .033 -.03 .210 .l 1.5 3.0 .386 -.132 - .067 - . Ot; 1 -It,- .1. - "' /{(;:,,~ IfM> MoT a .376 0 .153 .170 .2.% .'ii08 .349 .139 .223 -88- 'f /~. B L.~C: 3: f-,:.::RODYN ..\1/.1C COXL.k'/t:i.CIF:l':'T:~; ViHEN h=-0 1 O<j <CO • k=o _4,!\:).) ~ =.35 *~ *2 .n.. AR Ax 0 "" (Jf) .5 - .106 1 .....n B..c. cf( Cz Q:> cZI <t> .5882 .0851 .4.294 .05850 .Z69Z .05500 .061532 BR Q!I l. 206 - - .45126 - . tJ.75 79 1. l89 .53.GZ .Zl79 .2587 .o 1233 .12747 - .027537 • 0956 32. - . Cl 2.22.Z . l 0951 -.018834 .12411 - . 0 29511 . 1135 2. -.049124 . 077346 ... 064'738 .04002.4- .067077 .035482- .052598 .057648- .061206 - . 4-9l43 - . 46951. .02306 - .07Zl27 .040252 .01434 .... 06 2008 .. 031817 12 13 -.52562 -.4i9848 - .4 7979 - .48052 .03564 -.071694 .0172 -.073608 .005457- .066694 .023602- .066300 .040535 .02954 .022173 11 - .49887 .01464 .... 01'4020 15 -.48660 16 - .4B231 - . .\:9838 -.49154 -.4';5404 - .4·9748 - .49482 -.<:8584 . oZ l 38g-. 069596 .01583 .... 06 7422. .0-'3675 .Ol.6775 .021403 .01320 - .073962. .019934 .0103 -.068340 .01206 - • 0736<;3 -.0 3 775 -.072950 .016863 .o l727 l l .s -.4330 ·•, rtf)w/3 2.0 ~ .'::6103 Z.5 - .41587 - .4052~ 3.0 3.5 4.0 4.5 5.0 6.0 /' . 0 ELO 9.0 10 11 17 18 19 ~o 21 22 23 2.4 1 •l<cf. -.43062 - .4 7079 - .19582 - .490ZO - .-'.. 9643 - .•tt97 24 . (106) .16626 .12143 . ll l34 .11117 .104:26 .086030 .06502.S .051837 .057044 . 02.7923 -.o 3 3386-.071592 . 013519 .011470 .0 26 ") ") -.069193 .o 13533 .....:;. .01101 -.073331 - . 0 3 48 -.073907 .015199 .010167 . 0687 2.5 .07192 .070868 .067843 .066031 .067263 • 07 2585 .071225 .0697S3 .073107 .079856 . 0 71123 .073342 • 0 73648 .o 7 i?56 7 .07346 .074148 .072686 .073540 .074432 .073l04 .073560 .134 .0683 .04013 .025812 .021841 .022851 . 02414.::1 .022861 Dr DR Gt:) - GI? .165 - .1565 .05436 -.06..rs .03223 •. 043 s< .02794 -.036.?,ll .028771 ~ .03009S .0294{ 7 - . 0236 i~· ' .028672 -.01&32.Z .026870 - . 0 l 1i:96 2. .o u;s11 . 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"'1 ~~ ~~I ~ [:i1 ~T ~ <:i.:... ~ ::J 0 I-< µ.. -32 - 16.0 J.O 2.5 /25 l----\-----------0--4-1-t-0-----~------3. o IL ) ?...b l'z. ZZ-z ( r;,,8: c =i , ot.. ~-< <XI, -«, =o (J .£.} ' 2 h . : : oo (or P.~ = ~ : . 1) ('-oa·c -a: 4.5 I L), ,-l.h v ( z. !OQ i 9' C ;:! r1 "'1 z.o I I 4.0 0 I ~ '!Z, £:, "< -;;-t:.. c;Q I";:;:" :: a:J ! I I I I I JS \. /.S \ 3.0 I\ 1.S \ 1.5 \ \ \ \ \ \ .25 0 . .75 2S I. 0 F/GUR£ 3: I/EC TOR LI/AGRAM OF ThE LIFT AN.O MOMENT ON THE AIRFOIL IN 8£/vlfJ/NG OSCll..l.ATION1 AS F(JNCTIONS OF h, f, 1$., -¥-JL -!JJ- 12 .f.O 125 /0 ,o<fc:.ex> , '01 (11 J1- 9.0 ll I 9.S Mc 1 1 .,f =o 1()0 I(:M) Mor I I 4.0 I 2.f1 1 zrl (~),. .· c =2. . o<T <oo ii:: 0 (M <!h P. l[ - ) :-:co{or~Mo r c 1 - :q l : I ) tv1 ) ·---2 .2 h P. .ll:! (/'10 f • c ' O< ~(~ p ir. :: <X) I I I J.S 7.0 J.O s.o Jl 2.0 ..fO 1.5 17GUR£ 4: VECTOR OIAGl?AM OF Th'E LIFT ANO MOMENT ON THE AIRFOIL !N TOR.S/ONAL.' OSC/iLATION. AS roNCTi'OAl.5' OF h, .t, ~,ii-JL. 11 -94- ---·-- Equil~~~'.'.!-!:'osi~~--·T · Y --- T~-~~ ,---.;;;;;,-.· - r-.., - -t,----- ~!~- C - - FIGURE 5: SCHEl\/'.ATIC DLA.GRAlv: OF A FLUTTERING . "".IRFCIJ REFE;RENC.~S 1. Technical Report by the ,,<\c:ddents Investigation Subcominittee .on the Accident to the Aeroplane ' ZK at Meopha.m. Kent, • &: lvL No. 1360. British ~\.R.C. 1 (1931). on 2ld July, 1930. . Duncan, W. J .•.8llis • D. L., and Scruton, C.: First Report on the General investigation of Tail .t)uffeting. R. & M. No. 145 7, British A. :l. C. , (1932:). · Duncan., Vl . J. , Ellis• D. L . , an.cl Smyth, E.: Second Report on the General Iuvestigat\on of Tail Buffeting. R. & M. No. 1541, British '.: .R .C .• (1933). 2. l:Henk., H., Hertel. H., and Thalan, K.: The German Investigation of the .6ccid~nt at "h.1eopham (England). T .M. No. 669. N.A.C . .A., {l93Z). Biec:hfeler, C.: Tests. for the Elimination of Tail 1:nutter. T .?v1. No. 710~ N .JLC .A.~ (1933). 3. White, J. A., and Hood, M. J.: Wing-Fuselage Interferen<:e, Tail Buife'tin.g, and Air Flow about the Tail of a I ow-Wing Monoplane. Rep. No. 482. N.,A.C.IL. (1934). 4. Abqrashitoz, G.,; Tail .Buffeting. T. M. No. \041, N. A. C .A. 1943. S. Theodorsen, T., and Garrick, LE.: Mechanism of Flutter. A Theoretical and Experimental. Investigation ()f the Flutter Problem. Rep. No. 685, N.A.C. \, (1940). for 6. v. Karman, Th. and Sears, W.R.: Airfoil Theory NonUniform Motion• Journal of the Aeronautical Sciences, Vol. · 5; pp. 379-390. (1938). 7. Jahnke, E. and Emde• F.: Tables of !''unctions with Formulae and Curves. Fou.rth edition, Dover l:"ublications. New York (19"45). 8. Watson, G. N.~ A Tr~atise on the Theory of .Bessel Functions. Second .Sdition, Cambridge University Press. London (1944) .. 9 . .c,;~ykle;.;t:r.!.J, N. "'.. 259. ~tb.·;;;t ·~diticm, Vibra.tL:m :\nalJsi3, ~'.,1.:!dio:i ·v I.-6; J?). 2·b·:;:.. :cGraw-HiU .Boo~~ C:om_?tn1.y, ;:'i~·:; Yor:.r. (19,1-s~). /:..Short .J:'.~ble of integrals, Thh-d revised editiu:.:i. Ginn and Company. Boston (1929}. J.J. :·1erc(;, .ti. 0.: 11. , ,ii•• , ;~,.iathews , G. B. and l\,~ac:~.obert, T . .i.\li.• : A Treatise on Bessel Functions and Their 'P:?lications to Ph;,rsics. Second edition, Lv:acl\,:illan and Cor.::lpany, Ltd .• I_ondon (i931). 12:. Cops on, Z. T.: ..1=-.n Introduction to foe Theory of Functions of a Co~11ple::c Variable. First edition. Ox.ford University Press, London (1935). l 3. 'JVhittaker, ~"C. T . .and W<1tson, G. N.: A Course of ModF>l"t'l. Analysis, .:F'ourth edition reprinted, Cambridge University l?r..::ss, London (1950). 14. Magnus. N. and Oberhettinger t F.: Fornveln und S!itze fur die Speziellen Funktionen der ;;,:athematischen rhysik. :~'..\.·1eb:o P.uflage, Springer-Verlag Chg., Berlin (1948).