1、i TECXBICAL UOTXS NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS .-_- . _- - -. . - NO . 667 OFERETIONAL TRldTtiENT OF TZE NONUNIFORH-LIFT TKEORY iN AIRPLAXE DYBIMICS Sy Robert T. Jones Langley demorisl. Aeronautical Laboratory t . gashington October I.938 . ; -.- . Provided by IHSNot for ResaleNo repr
2、oduction or networking permitted without license from IHS-,-,-J I. . J NATIONAL AL)VISORY COMXITTES: FOR AEROBAUTICS TECHNICAL NOTE NO. 667 i OFERATIO2?AL TREATXEXT OF THE NONUNIBORM-LIFT T3XORY IN AIRPLAYE DYNAMICS By Robert T. Jones The method of operators is used in the application of . ._ nonuni
3、form-lift theory to problerils of airplane dynamics. The nethod is adapted to the determination of the lift under prescribed conditions of motion or to the detorafnation of tho motions with prescribed disturbing forces. - 11TRODUCTION : Problems in airplane dynamics are usually treated on the assump
4、tion that the air forces are instantly adjusted to each motion of the airplane. Since the developmsnt of recent theories for the nonuniform motion of airfoils, it - has become possible to consider more exact laws for the ad- justucnt of the lift. The nonuniform-lift theory has already been applied t
5、o certain dynamical problems, notably to the problen of flut- .- ter. These applications have, however, been confined either to approxinate solutions or to case-s in which the-type of - ;uotion is prescribed beforehand. The Eore usual problem, in which the resulting xnotion is unknown, roauircs the
6、so- lution of integral equations. .-. - *The present papsr shows ho:w ; solutions of these equations may be obtained fairly simply by operational nethods. SUPERPOSITION OF LIFTS In nearly every aerodynamic problen, the approximations that Llust be made to effect solutions are such as to lead to- .-
7、_ linear relations. Thus, in the case of the unsteady lift of a wing, Laplaces equation cotibined with the assumption of - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 X.A.C.A. Technical Wote 370. 667 . an undistorted wake leads to a linear rela
8、tion between the lift and the angle of attack. Such a relation means that the lift due to the ,sum of ttio variable motione is equal to the sum of the lifts for the two motions taken lndepend- ently. 1 .- In particular, if the lift following a sudden unit I jump of angle of attack 1-s known (see rof
9、eronco l), then the lift for any variable motion is easily obtainol by breaking the given motion down into a. succession of small. jumps or steps and adding the lifts incident to each one. A The case treated by Wagner thus becomes the key to the calculation of lift for any variable motion. IPagnerIs
10、 function (reference 1) gi.ving the lift after a sudden unit jump of angle of attack (two:-lmcnsiona.1 - case) may be denotod by c 1Js) l The superposition of lifts for any variablemotiin CL(S) 9 as previously exa plained, is accomplished by the integration of Duhnmalls integral L - - a I I 0 ci, (s
11、 - so cL(so -so . (1) ci (See reference 2.) OPERATIONAL SOLUTION OF IXTEGRAL ;IlQUATIONS It is evident that, in order to take account oLun- 1 steady air-flow Phenomena in the theory of airplane dynam- *I= I.- its (including,stability and related problems) the custom- ary instantaneous equations of.
12、mot.ion_mu.gt be reglaced.by 1: equations involving the integral (l), The equati,ons of_ -, ;gotion then become linear integral e.quations, Solutions of these equations may be convoniontly obtained by opera- tional mothods. Lot D reprosent tho.operator d/da and let 1 = l(s) ro-orosent the unit jump
13、fuction, thaeis, a function of having the value 1 at .- 0 ix- s 0 .%iYd having the -velue s 1: (s) Equation (6) becomes F(1/2) Jr X(s) - 2x(s) = (9. - 2) D2 1 (s) r(i/2) 2 w-L or D ,31” + C2ehzs where co = .2-l-r $ = 1 -0.330,rr c, = -0.670 l-r K, = -0.0455 .- A, Z -0.300 and where s refers to the h
14、alf-chord as unit, that is - - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 N.A.C.A. Technical Noto Ko. 667 vt c. S = - 42 In this form, the operational equivalent is readily found from the relation - oh* = JJ - t l(s) (See reforonce 2) (14) D-h
15、 whence D +D) = Co -I- C1 - D +c,- D - X1 D -Y A, (15) _.-.- The calculation of lift under a prescribed variation -r / of anglo of attack can be illustrated-by assuming that the airfoil is given a sinusoidal motion . . a(s) (or O(s) = R.P. or I.P. of eins (16) * - This variation is reduced to operat
16、ional form (see equation (14): D E(D) = - D - in (17) czn(s) = p D D - .-+ 0 - in (See equations (12) and (15),) The resulting operator may be-evaluated by the Heavl-.- _ ti side expansion theorem: f(D) f(O) c f(h) As !qEj- l =k(o)+ GFq-ye where the AS are the roots of F(D) = 0. (19) . .- l ., Provi
17、ded by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. .* N,A,.C.I. Technical Note No, 667 7 CL. (s) = - v inelns + co+cl-=- + c, in l.,inst . A: . .-* n in- h, in-h, _I _ .i + Cl Al ha _L- *hls + c, - h 1 - in A, -in QhC (20) The terms involving ehs disappear
18、 in time and hence may be disregarded ina continuous oscillation. Tha terms I- G 1 - z. -:;-zL-z yield approximate expressions of the l:ift functions fOT .L- tno oscillating airfoil fntroduced by Thoodorsen (refer- ence 3): n2 2 2rrF(n) n = co -I- c, -+ca -s Ala + n2 2 h, fn a (2% 2rrG(n) = - C, h1n
19、 - ca h2n + Al2 + n2 haa +n VI - -.- As pointed out by Garrick (reference 4), Thoodrosenls function for sinusoidal motion -_ - - : . . C(in) = F(n) + iG(n) (equation (21) -_ may bo regarded as the operational equivalent of Wagnors curve, i.e., 2rrC(D)l = c, (D)l = c1 (5) (23) 1 . t , This fact may b
20、e verified by referring to equation (15). This relation is especially intoresting because i-t shows a - connection between the Fourier and the operational analyses. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 N.A.C.A. Technical Note No 667 Thus
21、, if the response of a linear sy-stem to a cant-inuous sinusoidal excitation is known, I Rn(s) = f(in) oins (24) _ = Then the function f furnishes immediately the operational equivalent of the unit-response so that for any variable excitation Z(s), R(s) = f(D) ,2(E) = f(D) z(D) 1 (25) In general, th
22、e motion of the airfoil or airplane will not be prescribed beforehand but must be determined from dynamical equations, This type of problem can be il- lustrated simply by considering the disturbed vertical mo- tion of the airplane without pitching. The dynamical equation in this ca,se is . . dw m- r
23、esisting force = impressed force, Z (26) at . . . where w is the vertical velocity of the airplane and m is the mass including the virtual additional-mass of the wing. Since mdw 2m V2 da = - dt x s /i/2 c/2 x - - s P/2 c c/2 ds Xaking ths,substltution 2 9 - =-Q s P/2 c (27) - (28). (29) and writing
24、the equation in coefficient form, aDa +-c, (I) a = c1 (s) 1 0 (30) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N.A.C.A. Technfcal.Noto No. 657 9 where cLO is the lift coefficient of the given disturb- ing force. The operational solution is - .- .
25、- a(s) = 1. Cl (d (31) aD -I- cl,(D) o Again, as in the case.of the lift, the solution for.the elementary jump is the key to solutions for variable con- ditions. 1 a, (4 = 1 oD + F,;D) (32) Replacing “I (D) by (15) and simplffyrng: 3. (D - a,(s) = - A,) (D - ha) 1, = - f(D) 1 (33) aD3 + bD2 -I- CD -
26、I- d F(D) . which is in standard form for evaluation by the expansion theroem (19). Finally, a(s) = Cl (0 0 s S 1 a,(s) + a,(s - so 0 ) 3 I(so) ds, (34) 0 The extension of this treatment to problems involv- -.- _ .- ing a number of degrees of freedom will be evident, - -.v Langley Memorial Aeronauti
27、cal Laboratory,. Xational Advisory Committ.ee for -haronautics, Langley-Field, Va., September 12, 1938. . i Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 N.A.C.A. Technical Note.No. 667 REFERENCES 1 .: 1. Wagner, Herbert: Uber die Entstehung des
28、 dynamischen Auftriebes-von fragfltigeln. Z.f.a.M.IL., Bd. 5, 1 Heft 1, Feb. 1925, S. 17-35. 2, Carson, J. R.: Electric Circuit Theory and Operational ._ _ Calculus. McGraw-Hill Book Co., Inc., 1926, .- 3. Theodorscn, Theodore: General Theory of Aerodynamic s Instability and the Mechanism of Flutter
29、. T.R. No, - - 496, N.A.C.A., 1934, 4. Garrick, I. E.: On Some Reciprocal Relations in the Theory of Nonstationary Flows. T.R. No. 629, W.A.C.A., -_ 1938. . - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N.A.C.A. Technical Note No. 667 Fig. 1 8 - 150 -4 25 - d V - 0 L - -. ._ Pigure l.- Moving azcs. a = w/V ; 8 = vt c/2 I . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
