1、A Reproduced CopyOF/q,/f7CF)- 7-H-. Il as for r = 1qO_ rtains the values of c taken from a die.gram similar to Fig. 4.is employed.and in Fig. 5Table I con-Table !. Values ofo I o.o5r = 1.0 !.000 I 0.780 0.655 0.561 0.4850.8 0.800 i 0.690 0.600 0.523 0.4590.6 0.600 I 0.540 0.485 0.437 0.394.mtThis fo
2、rmula is constructed in a similsr manner to the formula _ L_for self-induced drag D_ wqbi: - into which it passes whenL_ = L_, b_ = b and G = O, whereby c equals 1.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- 7 -Table I. Values of _ (Cont.)r: !.0
3、0.80.60.2,5O. 420O.4CI0.5510.30.3700.3_50.3!50.350.5270.3150.3850.40.3900 “_20.2550,450.2580.,U52O. 2,Z10.50.?3000 _0.210The values ofand, therefore,imatioa betweenin the most innertent case, _here r = !b_ : b_ : b, arc represented (with good approx-O/b = 1/15 end _-b : 1/4) b7-“ (4)o, = I + 5.3 _,t
4、More exact is the approximztion formula_ I - o sol bdl = 1.05 + 3.7 3/-bwhich obtains between O/b = 1/15 and O/O = 1/2. The approxi-mate fornmla for r _l is less simply constructed. T;e fl._tb_ + b_calculate the value of _, corresponding to bm- 2 and,further, the auxi!isry quantities 0.8 x a_ (I - o
5、_) - 0.I = s0.56and = t_ + s - 0.22b_ -b_ _l -r_- 1 -I- rascure b_ + ua -fand, if (fcr the sp_ke of brevity) we_ _V_- “r ,“7,0 _._._ .:.iNumerical Examnle. Let a biplane h,._v_ an unDer-win_ s_nbI = 12 m (39.37 ft.) and _ ower-._ing span b2 = I0 m ($2.8 ft.)and let the gap O = 2 m (6.56 ft.) to calc
6、ulate the coefficientof mutual influence _ for the drag D_.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 -We first calculato -_:, for the mean s_an38.09 ft.) Then G/bm = 2/11 O.ISiCbm = (b_ + b_)/2 = II.0 m , = ._._ to equation (4), wc now obt_,i
7、n1 = 0.509_ = ! + 5.3 x 0.1918I_, for the sPke of compariscn, we make the cal_iation fromequation (5), we EetI - 0.6_ 0.i_!8 - 0.5091.035 + 3.7 x 0.$18Lastly_ by interpolation fror“ Fig. 3, we get J_ = 0.5il.The agreement i_, therefore, quite satisf:_.ctcry.Takin_ d = 0.511, we, obtain the auxiliar,
8、_, values:s = 0.8 x 0.511 x 0.489 - 0.I = 0.100t - 0.56 = !.4320.511 + O.lOO- 0.22r = bm/b_ = 10/12 = 0.85Z, = I - r = 0.0909l+rTherefore _/t - 0.0909 _ 0.0635._ !_“ n. + 0.06552 =_;hencc _ = 0.511 + 0.I -= 0.61! - 0.1195 = 0.6935Interpolation in Fitre 5 ires _ = 0.490.Provided by IHSNot for ResaleN
9、o reproduction or networking permitted without license from IHS-,-,-9-3. The Biplane.The induced drag of the upyer win=_, for tLe unstagscrGd bi-plane, isL.2 L I,2D: = D,I + D_2 r_q _ b_ b_ b_ Iand that of the lower wing isD2 -_ +D_2 =-rrq _ + L_ _ “_Where thcre is a oositive stagger, as is generall
10、y the case,the dra_ of +.he upper _ing is diminished by the upward air cur-rents oroduced b7 the bower wing; but, on the other hand, thedrag of the lowez wing is increased, to exactly the same extent,by the downward air c_Irrent produced by the upper wing, so thatthe total drag is the same as in the
11、 case oi“ 9.n unstaggered bi-olane and ,2L_ h -i.l_i_ + 3 _ L_ L2 + _ (7)D : D_ + D_ = _q b_ b_ b2 b_ _ /With the givcn values of the total lift, L, and of b_, b_,and _, the question naturally arises as to how the lift must bedistributed on the two wings so that the total dra=_ will be thessme as th
12、at of an unstaggered biplane.For this purpose, let L2 = Lx, or L I = L (I - x) and* The approximate formula (given in Technische Barichte, Volume I!,No. 2, p.275) for the induced drag, based on rather uncertain an-aloTies, does not satisfactorily stand the test by the more exactformula (7). Its a_re
13、ement with the measurements of Yunk s_emsto point to inaccuracies in these measurements, which were made inthe old wind tunnel.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- i0-let us seek a value of x for which the expression in brecketsin equatio
14、n (7) is a minimum. Taking b2/b _ = r, a very simplecaJ.cuation givesr - (8)X - 1r+ _crIf this value of x is put in equation (7), we have, for theminimum value of the induced drag,Dmi n -L_“ I- _ _qb_ 2 z (r + ! _ 2_)r(9)In the special case when b_ = b_ and, therefore, r = I,Ithe formulas become sim
15、pler. As is easily seen, x = _, or,in other words, the lift i_ equally divided between the two wings*.We also haveLa 1 + _ (9a)L_ is the induced drag, D I, of a monoplane _-ith a_qb_ _spsn b_ _7hich gives the same lift as the biplane under consid-eration. The factor following this expression in equa
16、tions (9)and (ga) thus _ves the ratio D/D I = K In Figure 6 thecourse of K is plotted agsinst G/b_ and r = b_/b_.* These relations are not quite exact, since the influence of thecomponent of the disturbed flow, v, pawa!lel to V, has beenneglected for simplicity. With more precise computation, it ap-
17、pears that it is not the lift, but the circulations of the t_.owings whichmust be equal, in order to obtain the minimum drag.The lifts are then in the r_otio V + v to V - v. The effect ofthis correction on the mmgnitude of the drag, hc_ever, vanishesfor all practical purposec.*The quantity k, introd
18、uced by l_ur.k(TecbD_ische Ber1_s_te, Vol-ume II, No. 2, p.187) is equivalent to 1/j_Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- II -It i_ seen that all biplanes have less drag than the equi-r.lentmonoplane and that their minilu_m dr_ is obtaine
19、d when r = l,that is, when the uDper end loner wings Lsve the safe span. Itis also seen What, with the sr_me span, the drag decreases as thegap increases.The result must not, however, be _zisunderstood. It doesnot mean that the biplane is once and for all _aocrior to themonoplane. The analysis merel
20、y states (apart from _n_ fact thatit enables the drag to be calculated in each particular case).that among monoplanes and biplanes having the _ame and the same tot_.i load, the bini_ne, both of whose w.ngs _the given maximum span, is superior to other arrangements, itis only necessary to comps.re a
21、monoplane with the same load. asa given biplane an_ _.th a sFan I/J_-times greater than thatof the bipl_e, in order to be convinced that both ha_e the sametotal drag. In the same way, it is seen that a biplane with wingspans of 12 m (39.$7 ft.) and lO m (32.8 ft.) is a li_tl_ super-ior to one with t
22、wo wings of ll m (36.0o_ ft.) sT-an. Figure 6and the corresponding Table II, give it formation on all these11_t :e caieulaticn.relations with a very “_ _If the span of the lower wing is taken as smaller than thatof the upper wing, then the portion of the lift that must beassio_ned to the lower wing,
23、 in order to produce the mini:.mlm dra_,is s_.naller th_n in oroportion to the spans. If we adopt eoualloading on both wings (which would seem to be mort desir_-b_-e),Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- 12 -then the lower _,ingwill have
24、a smaller _15 + 1.20 - 0.980)L 2 0.865 x 15002D = _:nqb,_ - 5.14 x 52 x 144= 82.70.865kg (ZS2.S lb.)Table IiValues of K = D/D I for the Biplane.G/b,r -0._0.?0.80.9l,O0i.o001.0001.0001,0001.OCO0.050.9900.9820.9740.9500.890o.! I o.150 _Z!o l 0.9560.9320“0.8,o0.3270.9540.9260.8928470.7790.20_932O. 8970
25、.8550.8070.742G/b 1 0.25 _ 0.5 0.35 0.4 0.45 0.5r= 0.60.70.80.91.00.91!0.8710.8250.7750.710Li 0.9920.8490.8000.7440.6840.8750.SZO0.7780. 719O. 6620.8610.812O. 7580. 6990.6450.8480.7970.7400.6830.6290.8390.7830 _oO. 6710.615L_Values of x- - for 1:he BiolaneT + L,-,. ! ,_ _ 02_/bl 0 05 0.I 0.I .Ir= 0.
26、6 I 007 I o0.8 _ 00.9 0!.o i o/oO. O6O0.1050.172O.3O30.500O, 1040.1640.246O- 359O. _000-340.2030.2850.3870. 500O- 1_=-70-2280.3100. 402500Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- 14-Table Ii (Colt.)Values of x = _,L for the _i_lane.G/b: 0.25
27、0.3 0.35 0.4 0.45 0.5r= 0.60.70.80.9!.0O. 1760.2490.327O. 412,0.500O. 19 _2-O. 2620.558O. Al90. 500O. 9020.272O. 34-70.4_5O. 5000.2110.2810.3550.4.990.5000.2180.288O. 361O. 431O. _000.2240.294O. 3640.4330.500In like manner, in the case of the biplane with two wingsof I! m (56.09 ft.) span (0 = 0.511
28、), _ = 0.755 andD - 0.755 !_00 _ = 86.0 !:_ (189.6 lb.)3.14 x 52 x 121Note: The frictional drag may be taken as 0.008 qS. In theabove example, for S = 40 m_ (430.6 ft _) this gives 16.6 kg(36.6 lb.) which, togethor with the induced drag, represents thetotal drag on the win T.4. The Triolane.The trip
29、!ane may be treated in the same way as the biplane.In order to avoid complicating the problem unnecessarily, let itbe _ssumed that all three wings have the same span and that thegap between the upper and middle wings is the same as betv:eenthe middle and lower wings. Under these conditions, from the
30、results obtained with the bio!ane it may be assumed at once thatthe upper and lower wings have the same lift for the minimmm drag(See footnote on p.!O). The lift of the middle wing, however,is different.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,
31、- 15-Patting L2 - L x, then L_ : L3 - L (I - x)/2,sum of the lifts must equal L.since theFor the coefficient of mu$_al influence, a we need to dis-Cti_uish between the adjacent wings, which are _ apart, and thetop and bottom nines whose distance apert is G. Let the corre-sponding coefficients be den
32、oted by o7 and %. The induced dragsof the individual wings are accordingly:1 (L_D_ - v.qb_ + _L_L2 + c=L_L3)1 (L_ + L(L, L2 L_L3)D_ - _qb _ . .D3 = _ (L_ = + O_ L_L 3 + o_ L_L3).if, in the above msmncr,terms of L and x, we haveL: L_ end L_ are expressed inL2 _ + o_ 2x (I + % 2 _ _D - 2r_qb_ + x2 (3
33、+ o_ - 4c_) ; (lo)This will be a minimum, whenX ! _- (;s - 8_ (I!)3 + _2 - 4_The values of x for different ratiosfrom the broken line in Fi E. 7. The value ofO/b can be seenx is always lessthan !/3 and the middle wine should, therefore, always have asmaller load than either the top or bottom wing, i
34、n order to ob-tain the minimum drag. The error due to the as_m-_ption of equsldistribution of the lift between the three wines (x = I/Z) is,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- i6 -however, smal., as shown by Figare 7 and Table Iii.Table
35、II!.Best Subdivision of Lift for a Triplane -L_Values of x-(L_ + L2 + L3 )o.o 1 o.os i o.! o.15 o._G/b I 1 ,- 1x = O. i90 0.202a) Binlaneb) Triplanewith x = I/Zc) Best triplaned) Best wing systemo i O lel _ o.177Values of _ = D/D i!.CO01.000!.0001.000O. 8900.8890.8850.865t“0.0270.8240.8130.7870.7790
36、.7740.7670.7287d20.7320.7240.678IG/b_ 0.25 0.5 ! C.55 0.4 0.45 0.5x= i L o._31 0.238 i0.212 I 0.222 IJ IValues of _ = D/D I,.70 0.684 0.662 o.c451a) Binlaneb) Triplanewith x = I/5c) Best triplaned) Best wing systsml0.244 0.2510.629! 0.6150.6950.6870.637 o.306371o6, lo ,to T1O. 656 0.650 0.607 O. 585
37、 C. 5650.601 0.572 0.545 f 0.521 0.500Two additional curves have been plotted in Fig. 7 for the pur-pose of comparison, one for a biplane, the other for a _ving systemlike Figure 8, that is, for a biplane closed laterally by panels_nd so arranged that the upper Dortion is subject to outward ores-sur
38、e and the lower portion to inward pressure. The induced dragof this _ng system has been evaluated accordin_ to a method whichI/I. -JProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- 17-cannot here be gone into in detal.* This _“inT system has the leo_
39、tinduced drag of al! wing systems cf like span and the same totalh_ht (sum of the individuel gs-ps). If we orocecd from a triplaneto a multiplane, v.hile maintaining the over-all dimensions s_nd in-serting further supporting surfaces, the induced drag continues todecree_se, the closer the :Tings are
40、 placed to_ather, provided Zherequired subiivision of the load is theoretically maintained, inwhich the extreme top and bottom wings carry more load then thent_r._ed_a_e :?ings. If _= consider the !imitin_ case oz an infi-nite number of v;ings vithin the outside dimensions of b and G,_ _e the same i
41、nduced dragwe obtain, in the case of the ._ultl_.la_ ,as for the win_ system of Fig_re 8.For the calculation of this d_g, _ea-_ - am indeb _ _ to _essrs.Grammel and Pohhausen. The results may be taken frcm Fire,re 7and Table III. Approximatelyi_ 0 ,_D (_2)= 1.04 + 2._i !bNumerical Exem,_le.- It is d
42、esired to obtain the coefficientof induced drag for a tri_iane _ith a span of I0 m (32.81 ft)actly at the center of the total height. The mutual-influencecoefficients, for G/b = 0.125 and for G/b = 0.25, are found tobe o_ = 0.606 and _2 = 0.421. Hence (from equation (!I)I ,x- 0.21 and (from equation
43、 (I0) _ = _ _the e:preszion in* ! h_ve hrief!y indicated the line of er_ament in my lecture be-fore the Hamburg meeting of the ?issenschaftliche Oesellschaft furLuftfahrt, and it will be included in the printed report of thelecture.and a total height of 2.5 m (8.2 ft.) ,hen the middle “lint is ex-Pr
44、ovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-/- 18 -1brackets) = C _ iustead, we take .x = 3.007. If, - we get _ = r_.593The biplane which is derived frcm thc triplane by removing thc mid-dle wing, while keepin_ the sane distance between the two ou_
45、exwino_s, gives K = 0_710. The three values thus differ slightly.The biplane, who_e _ap is equal to the sum of the two gaps of thetriolsne and has G = 1.25 m (4.1 ft.) will, on thc contrarF, havea nT_ch greater drag. In this case_ _ = 0.803. The wing sFstem ofFi_ure 8 gives K = 0.637.Translated byN9
46、_ional Advisory Con:_rizteefor Aeronautics.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Fig,l.tJ I!| :i in I _ J t r _ _ _Elliptical and _ectangulazdi_tzibutiou of lift.!/Fig.2. Distuzbance of flowaround _ wing.J_Provided by IHSNot for ResaleNo re
47、production or networking permitted without license from IHS-,-,-1.00o80.60,40,20-0.2-0.4-0,6-0.8-_.0Figs,3 ),lProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I i _ I i0.9 _ l -!L.5H “ ! r _ _J.:_! x=_/3_ t :._ t 0.5 o .l .a .3 ._ .5G/DFig.7. Efficiency K of various wing systems andcoefficient of distribution of wing loadsx, for the triplane.3 %.2 H1)lane,_ith x=l/3ibFig.8. Best _ving system.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
