Wing Mass Formula for Subsonic Aircraft

J. AIRCRAFT, VOL. 29, NO. 4: ENGINEERING NOTES 725

lytical Methods Inc., Washington, DC, 1982. Inboard Midboard wing Outboard wing 7 wing ^ Lan, C. E., and S. C. Mehrotra, "An Improved Woodward's "fuselage joint Panel Method for Calculating Leading-Edge and Side-Edge Suction Force t Subsonisa Supersonid an c c Speeds," NAS 3205R AC , 1979. 8Hardy . PS . . Fiddesd C.B ,an , , "Predictio f Vorteno x Lifn o t Non-Planar Wings by the Leading-Edge Suction Analogy," Aero- nautical Journal, Vol. 92, No. 914, 1988, 154-164.

Wing Mass Formula for Subsonic Aircraft

Serge . UdinV iWillia d *an m J. Andersont University of Michigan, Arbor,Ann Michigan 48109

Fig. 1 Wing geometry. Description OST wing mass formulae1-2 are based on statistical in- The relative masses of structure counteracting the bending M formation. As a consequence, these formulae usually momen shead e an t ar r have good accuracy onlrestrictea n winf yo o t gdse data A .

theoretical wing mass derivation based on a simplified concept Plower , Pup model of a wing is presented in Ref. 3. This method takes 4pT into consideratio detail e aircrafe th th l f nal so t wing, including lactic complicated geometry and concentrated loads, such as the engines. Formulae to estimate mass due to swept moment, mQ = 0.1 mM (3) twiso t mas e t du smoment sheao t masd pree e ar an r, - sdu sented below. The mass of other wing components, which where p, powed uppedensitiee an r ar r lowef so uppe d ran r panel total about 30% of the wing mass, can be estimated by known structural materials desige ,th gs i dn overload factor gravs i a , - methods. readee 1'4Th referres i r completa o dt detailed ean d itational acceleration, JJL is the mass of the aircraft, and p is derivation in Ref. 5. wing loading. The approximate value of effective airfoil thick- The geometry of a subsonic aircraft wing with span b is ness coefficient ET is shown in Fig. 1. We define the inboard wing as the part of the wing between the fuselage joints. It is assumed that the 4!T, 1.1 1.2 •• 1.4 (4) inboard wing section is not tapered and not swept. The mid- 2T2 + T3 board wine pare th wing f gth o tbetwees i fuselage nth e joint ane sweedth p joint e outboardTh . wing lies betweee th n where Tl is the first spar height, T2 is the maximum airfoil swee wine p th join gd considee tipan t W . r thaaircrafe th t t height, and T3 is the most rearward spar height. The values has no inboard wing concentrated loads, has nm /-numbered of actual stresses averaged over the lower panel volume an< tne midboard wing concentrated loads, and has n0 /-numbered ^act lower ^ value f actuao s l stresses averaged ovee th r outboard wing concentrated loads. The relative mass of each upper panel volume cr can be obtained from the statistics l j j actuppcr loa \ms di r ?mo c c (i.e. mase , m'th r m botf o s co i c h sym- of an aircraft with approximately the same mass, size, and metric loads located in both parts of the wing). The relative manufacturing quality. These values also can be estimated as coordinat concentratea f o e d load (absolute r ZZ'o J coordinate divided by half-span) is z'c or z{. = Slowe . r = ^tt upper ( C\ The estimated relative mass of the wing structure is "act lower £. fc > "act upper i » > \J) ^s/lower man 'Vs/uppcr man

mw = ktm(m mri mai mfla (1) wher coefficiene eth t &lowee 5/lowcth s i r panel ultimate stress Slower divided by the permissible fatigue lower panel stress, formulae Th estimato et e relative masse ribf so s mrib, ailerons s u r kiupper i PP^ panel ultimate stress o-Muppcr divided by the per- mail, load-free skin raflapd skan , s m presentee flaar p Refsn di . missible fatique upper pane e manufacturinl th stres d an s g 1,2, and 4. The twist moment factor ktm is coefficient kman has expert value within bounds presented in e valuTablTh f fc. eo 1 e decreases whe e dimensionnth s Q.015V]4( 12A,+ ) man + 1 k = and mass of aircraft are increased. If the absolute root wing tm 1 -I( - A,)cosA (2) thickness T does not exist, it can be inferred from the overall geometrs ya where A is the aspect ratio, A, is the taper ratio to wing tip (wing tip chord divided by wing root chord), and A is the 2t half-chord wing sweep. T = r (A, - z,) - zf) Receive , 19916 y ; Ma daccepte r publicatiofo d , 199128 y . nMa (6) . AndersonCopyrighJ . . W Udi V d . 199nS an © t . y 1Publisheb y db e Americath n Institut f Aeronauticeo Astronauticsd an s , Inc. with where t is the root relative thickness; \ is the taper ratio to permission. r s * Visiting Research Scholar, Aerospace Engineering Department. the sweep joint (chord at sweep joint divided by root chord); Permanent affiliation: Graduate Student, Moscow Aviation Institute, zs is the relative coordinate of sweep joint; and zyis the relative Moscow, Russia. coordinate of fuselage joint. The coefficients KMo, KMm, and tProfessor of Aerospace Engineering. Senior Member AIAA. KMi in Eq. (3) are 726 J. AIRCRAFT : 4 ENGINEERIN . , VOLNO , 29 . G NOTES

2 - Z. ~ z,) h, h] 3 *:*,M„ = h? ° (h, - h,)cos A0 3(* - .h, T - 2

1- A , m

, ,( A - lA + r) 9z ,- z,)-f1 . 6(z2( - ,+ z f)

zs(l - A,) + A, H- 1 (1 + ifc)(zs - zf) + (ifc + i/f,)(l - zs) 9zf + 6(zs - zf) + 2(1

/7-z-T(*.-*.) + *ATI

» (l-)cosA

9zf -f 6(z5 - zf) + 2(1 - z,) - zf)

2 - 2* r*, + ,(A. - 2A A )?

, z,( A - lA, + ) 6(zs - zf) + 2(1 - z,)

(1 - z,)(l - A,) :,(! - A,) + A, +

2(1 - z,)m:

,)(! - z,) 9zr + 6(zs - zf) + 2(1 - z,)

h mfu 3V' (- z.i )

- A,) + A, - zr) 6(zs - zf) + 2(1 - z,)/

m - 2 «i *i - - A. /z+ —1 )s(

- 2A,) z zz ~ f\~ ~ f s z,(l - A,) + A, + 1

2 (z. - - z,)(^ +*)(z. - zf) + (1 - z,)

',)(! - z,)

2 - z,)(z1 (z z- 2( f+ z,),)s + (7) 9zf + 6(z, - zf) + 2(1 - z.) . AIRCRAFTJ : 4 ENGINEERIN . , VOLNO , 29 . G NOTES 727

Table 1 Bounds on manufacturing coefficient Mashinostroenie, Moscow, 1983 (in Russian). 3Udin, S. V., and Anderson, W. J., "A Method of Wing Mass *1 Stepped thickness (rather than tapered) 0.1 . .0.13 Formula Derivation," Univ f Michigano . , Aerospace Engineering k2 Dead joint mass penalty 0.153 0. . . Rept. SM 90.1, Nov. 1990. 4 k3 Standard thicknes f websso , ribsothed an , r . 1 0. .0.13 Udin, S. V., and Anderson, W. J., "The Complete Derivation of elements Wing Mass Formula for Twin Fuselage Aircraft," Univ. of Michigan, k4 Joint fittings and joint defects 0.1. .0.15 Aerospace Engineering Rept. SM 90.2, Nov. 1990. 5 k5 Plus toleerances 0.04. .0.09 Udin, S. V., and Anderson, W. J., "The Complete Derivation of k6 Manufacturing thicknesses 0.03 . . 0.05 Wing Mass Formula for Subsonic Aircraft," Univ. of Michigan, Aer- k7 Breakdown joint mass penalty 0.1. .0.2 ospace Engineering Rept. SM 91.1, April 1991. 1.62 . . 2.05 kman = * + *1 + ^2 + ^3 + ^4 + ^5 + ^6 + ^7

Table 2 Sample calculations for current transports

Aircraft , kg actual formula Error, % Whirl-Flutter Stabilit Pushea f yo r B-727-100 72,600 0.111 0.1109 - 0.02 Configuration in Nonuniform Flow B-747-100 322,000 0.122 0.1248 2.3 DC-9-30 49,000 0.106 0.0946 -10.8 DC-10-10 195,000 0.114 0.1048 -8.1 F. Nitzsche* and E. A. Rodriguest Dc-10-30 252,000 0.106 0.1109 4.6 EMBRAER—Empresa Brasileira de Aerondutica, A-300-B2 137,700 0.145 0.1337 - 7.8 C-5A 349,000 0.13 0.1344 3.4 S.A., Sao Jose dos Campos, 12225, Brazil C-130E 68,700 0.0772 0.0759 -1.7 Tu-154 90,000 0.102 0.1066 4.6 An-10 54,000 0.0981 0.1028 4.8 An-22 250,000 0.119 0.1226 9.5 Introduction An-24 21,000 0.1142 0.1196 4.7 N the new aircraft configurations with pusher power plants I1-76T 171,000 0.121 0.1237 2.2 I located at the rear fuselage, the whirl-flutter stability char- acteristics of the engine-propeller installation may be affected by the nonuniform freeflow induced by the aircraft compo- wher half-chore th A s mi d swee midboarf po d winge ; th A s 0i nents positioned upstream in the flowfield. Assuming that the half-chord sweep of outboard wing; m* is the previously it- flowfield may still be considered stationary, the effect of the erated or expert-estimated relative wing structure mass; mfu aforementioned interferenc s investigatedi e . Under thi- as s relative isth e fuel mass; hs = thicknese tsth \ s/ts i r s tapeo t r sumpton, the propeller aerodynamics may be adjusted to be- sweep joint; ht = t,\Jtr is the thickness taper to wing tip; \fis com periodiea c functio aeroelastie th f timo nd ean c system wine h= th sg\s i s airfoil area sweee tapeth o pt r joint; s ti s stability may be studied using any theory developed to analyze the sweep joint relative thickness; tt is the wing tip relative periodic systems e presenth n I . t work, Floquet-Liapunov's thickness wine &h^d th = g s an i ; t airfoil area tape wino rt g theorem1 could be applied efficiently to evaluate the stability formule tipTh . a canno nontaperea usee b r t dfo d win- gbe characteristic actuan a f so l configuration with many interfer- cause division by zero will occur if hs = 1 and hs = ht. If the ence elements. sweeo n win s pgha joint suggestes i t i , d that Basic Formulation 1 -A,. 9 The equation of motion of the aeroelastic system is cast in 2 - 2zf the state vector form 1 -A. h=l- (8) A(t)y= y (1) 2 - 2z/ 2z, where A(t) is the matrix of influence coefficients containing Neither simplification r dat r masno s fo a f existino s g wings e informatioth n about bot e dynamic e aerodyth h th d an -s have been applied formule th o s , a completely correspondo st 5 namics of the system. For a given set of initial conditions, the the concept model. solutio f thino s syste f ordinarmo y differential equationy sma be written as Applications The newly derived formula has been used to study the y(t) = (2) transport aircraf e detaileTabln i tTh . e2 d spreadsheet calculations are presente Refn di . 5. Value actuae th f so l stresses where 3>(f, f0) is recognized to be the transition matrix relating were estimated on the basis of engineering judgment. The e statth ee instan th vecto s initiat it a t o y rt /l valu t fa e(). accuracy of the formula is within [-10.8, +9.5]% and the Floquet's theorem states tha orden i t completelo rt y evaluate root-mean-square error is 5.9%. This is sufficient for most stabilite th y characteristic periodia f so c system t wili , l suffice preliminary design purposes. The formula can also be improved by using statistical data. For comparison, a previously derived formula for twin fuselage aircraft4 can be applied to a conventional aircraft. The Presented as Paper 90-1162 at the AIAA/ASME/ASCE/AHS/ASC 31st Structures, Structural Dynamics and Materials Conference, Long results yield accuracy of [-13.1, +11.7]% with root-mean- Beach , ApriCA , l 2-4, 1990; received Nov , 199023 . ; revisio- nre square erro 7.0%of r accuratas presen. Thithe not eas sis t ceived Sept , 19912 . ; accepte r publicatiofo d n Sept , 199126 . . formula. Copyrigh e America th 199© t y 0b n Institut f Aeronautico e d an s Astronautics, Inc. All rights reserved. References *Senior Engineer, Aeronautical Engineering Division; currently Postdoctoral Fellow, DLR/Institut fuer Aeroelastik, Bunsenstrasse 'Sheinin, V. M., and Kozlowsky, V. L, Weight Design and Effec- 10, W-3400 Goettingen, Germany. Member AIAA. tivity of Passenger Aircraft, ed.d 2n , Mashinostroenie, Moscow, 1984 tStaff Engineer, Aeronautical Engineering Division; currently (in Russian). Graduate Student, Purdue Univ., Schoo f Aeronautico l Astrod san - 2Eger, S. M., Mishin, W. F., Liseitsev, N. K., et al. Aircraft Design, nautics, West Lafayette 47907N I , .