Appendix A
Appendix A Spanwise Load Distribution Calculations of a Wing with Arbitrary Plan Form Using the Weissinger Method Summary The application of Weissinger's theory is shown for calculating the spanwise loading across a wing. From this procedure a spanwise load distribution can be calculated for a wing of arbitrary plan form from a 2-D lift curve slope and angle of zero lift. The geometry of the wing may be asymmetric about the centerline and can include sweep, taper, or twist. Partial span flaps may be modeled as discontinuities in twist.
Fowler motion can also be modeled as an increase in chord length. Lift, induced drag, pitching moment, and induced angle of attack may then be calculated.
Method The Weissinger method models the wing as a plate of zero thickness, but the planform and twist remain identical to the actual wing. The chordwise load distribution at each span station is concentrated into a lifting line located at the wing's quarter-chord line. The theory requires that the quarter chord line is straight, however a discontinuity is allowed at the plane of symmetry so that a swept wing may be considered.
The method assumes that the lifting line and its trailing vortex sheet are continuous. However, discrete values for the circulation strength are determined only at the span stations along the quarter-chord line corresponding with control points. The
numberof controlpoints,m, is specified,andeachis placedatthe three-quarter-chord
line. Mathematicallythis impliesthatthe lift curveslopeis 2n rad -1, and a correction
method must be incorporated for a lift curve slope that varies from the theoretical value.
The boundary conditions specify that the induced angle of attack due to the downwash of the trailing vortex is equal to the angle of attack of the plate. This ensures tangency of the flow to the plate at each control point.
Substituting the wing with a bound vortex system and applying the boundary conditions enables the formation of a set of simultaneous equations. Each of these Ctc equations is a function of the angle of attack of the wing, the load coefficient Gj - 2b at each span station j on the quarter-chord line, and the influence coefficients A_j which are only a function of geometry and relate the influence of the circulation at any point j along the lifting line to the downwash at any control point i. The distribution of the load Gj may then be calculated from the set of m simultaneous equations.
m (A-I) _i = EAi,jGj , i = 1,2,3 .... m j=l where W i (_i -- Vo ° - (Of.wing - (3£ o )i (A-2) Each equation gives the downwash angle at the control point rl = cosd_i resulting from the circulation effects of m points along the span.
The following trigonometric substitutions are used to define span locations at the control points or on the lifting line respectively, _ _hA. _ ....
C_ (A-3) (A-4) .... :ients are shown below.
(A-6) ?
- --gi,j (A-7) :_i 1-(-1) j-i (A-8) -.os_i) 2 2(m+l) --_nts are given as M _-i,ofj,o + Ci,M+lfj,M+ 1 + _--' Li.k fj,k) (A-9) k=l .... of span stations used to integrate the downwash and does not have - - L The subscripts 0 and M+ 1 denote values for k. Also note that =j (A-10) coefficients L_,k are defined as follows: (A-',:) kT_ (A-12) m+l for __< 0 1 _/[1 + (b / c)( rl + q) tanA]2 + (b / c)2 (rl - q) 2
-1}
Li'k (b/c)(rl-7) { = 1 +(b/c)(rl + q)tanA (A-13) 2tanA_/[1 + (b / c) r 1 tanA] 2 +(b/c)Zq 2 + [l+(b/c)(r I -q)tanA][l+(b/c)(r I +q)tanA] for _>_0 1 X/[1 + (b / c)([rl]- q)tanAl 2 + (b / c) 2 (r I - _)2
-1}
Lik = {
'
1 + (b / c)([rll- rl)tanA (A-14) The mathematical series coefficients fj.k are used in a numerical integration method and are shown below: kT_ (A-15) d_k = COSM+ 1 2 m (A-16) fjk -- _--_kt sin _tqb j c°s kt_b k ' m+l_=_ Sectional Lift curve Slope Correction The Weissinger method assumes that the lift curve slope is 27_ rad -I . This implies that the control points lie along the three-quarter-chord line. When considering airfoils with lift curve slopes that vary from this theoretical value (as a result of viscous, compressibility, or geometry effects), a correction method must be incorporated. This
involvesmovingthe controlpointsfrom thethree-quarter-chord line; but beforethis is
donea derivationof this locationwill be shown.
The velocity inducedby aninfinite vortex of strengthF anddistance h is given as
F
w - (A-17)
2nh
andthe sectionallift coefficientis relatedto thecirculationstrengthasfollows: 1 2 (A-18) pVooF = _ pVoocCi Rearranging these two equations and combining them yields wh (A-19) C I = 4n Vo_ c w Since ot = _ the above equation can be rewritten as C t h - 4n- (A-20) (_ C h 1 If the lift curve slope is assumed to be 2n then it can be shown that - must be c Then c 2 " the point at which no flow passes through the plate occurs at -_ c from the lifting line or at the three-quarter-chord line.
If the lift curve slope deviates from the theoretical value each control point location must be modified. This means that each control point must be moved forward of the three-quarter-chord line if the slope is less than 2n rad -1 or must be moved rearward if b it is greater. This variation can also be modeled as a change in the value of -- by the Cv C icXcxp at the ratio of the experimental lift curve slope to the theoretical lift curve slope --
2 /13
desired Mach number, M, where 13= x/i-- M 2 . The modified geometry can be expressed as b b (--) mod = k _ -- (A-21) Cv Cv where C letexp (A-22) k_- 2n/]3 Caution should be taken when large deviations in the lift curve slope are used. The method does not rigorously allow such modifications, and the accuracy of the results may deteriorate for large angles of sweep.
Compressibility Correction Compressibility effects are included with the aid of the Prandtl-Glauert rule. It approximates the effects of compressibility as an increase in the local chord and sweep angle by the factor - b 1 b (A-23)
-13cv
. -l tanA (A-24) AI3 = tan t-if--) Thus the span loading for the original plan form at a given Mach number can be calculated for an incompressible flow using a modified geometry.
Aerodynamic Characteristics The circulation strength is calculated directly from the dimensionless load coefficient and has the dimensions [fl2/s].
(A-25) F v = bV_oG The sectional lift coefficient is also calculated from the load coefficient as b (A-26) Cl_ =2--Gv Cv The total lift coefficient is determined from reAR _'_Gv sind?_ (A-27) m¥i o, The induced angle of attack at the quarter-chord point of station v is given by m (A-28) oti_ = b_G_ - _-"/b_nG n n=l where the primed summation sign indicates that the value for n = v is not summed. The total induced drag is calculated from nAR m (A-29) Coi -- m---+l _-Gv_iv sin d_v V: l
Appendix B
Appendix B
Weight Estimation of Other Control Surfaces Like leading and trailing-edge device weights, the estimation of aileron and spoiler weights is based on historical data of existing aircraft. Similarly, the total system weights are broken up into components of surface, support, and actuation. The weight estimation equations given below are only a function of the planform area of the representative control surface.
Aileron Weight Surface Weight: W_urf = 5.5. Sail (B- 1) Support Weight: Wsupp = 0.74. Sail (B-2) Actuation Weight: Wact = 4.3" Sai I (B-3) Total Aileron Weight: Wai t = Wsorf + Wsupp + Wact (B-4) Spoiler Weight Surface Weight: (B-5) Wsurf = 110+ 1.65.Ss0 Support Weight: (B-6) Wsupp = 1.3.Ssp Actuation Weight: (B-7) Wact = 270+ 1.35-Ssp Total Spoiler Weight: (B-8) Wsp = Wsurf + Wsupp + Wac t
Appendix C
Appendix C
High-Lift Weight Data of Various Aircraft
(All weights are in pounds.)
Traillng-Edge Device Weight Data Weight Boenag 727-200 Boeing 737-200 Boeing 747-2 I P Boellg 757-200 Boemg 767 DC-8 Model O2F NE)-83 MD- I 1 Sarfa_ f,57 _,_ I 1_ ]53 432 785 2491 1713 5516 1518 _g_2 842 18_kl gJ 478 2285 129! I_)_ 1885 341_J • _ppofts 139l 76'_ 17E 411 473 884 516_ 2732 751c_ 1549 1427 979 24_ 98? 83! ]Of_ 191 1816 Fakings 291 51_ 1118 164 1319 (AI 158_ 392 159 170 329 -- -- -- 5¶ 191 213 Contmb 9"- 722 21"_: 722 -- -- 3_ 24] 977 Total 412- 2555 Ib78_ 4181 ]_I 1_1 631_ 17%A Nk_lted Ate& S-m 28: 175 84_ 327 173 236 4_) 4509' 2101 f_2 Wdght / S_ Satiate 4 2t, i 4 49 651 464 5.56 357 44l 5Ill, 6 I! 5/.2 SuppoNs 606i 505 #85 474 825 4 i5 588 233 (19a 272 C)anlgol. i 12! 4 13 25( 221 081 I 14 (18_ Total I 4 68, 14 (_) 19 8_ 1239 8 14 8 4_ 9 44 Ltadlng-Edge Device Weight Data Weight Boell_ 727-200 B_ein 8 737-200 Boe mlg 747-21P Boem 8 757-200 Boeing 767 DC-8 Model 62F MD-83 MD- 11 "'-_ I ..... t .......... I ...... I ........ I ..... I ........ I ..... I ......... t..... I ......... I ..... q ......... q .... I ....... I .... I '_ 25_ "_H 752 I27 272 597 652 652l 423 1591 2/114 Sudao_ 3')9 37_ I1_; 150'_ 5'): {) (I 0 93"_ 937 48_ l&]4 232_ Suppoll 214 214 It I(X) I(X) t)l: "_1_ Conlro[s _48 232 t25_ 148 I(Xt3: -- -- 1461 total t 314 731 ] 555 745 2592 _rs_ I_ _6 1208 I1( 12fl& 1(I) 3 3716 471_ Nested Area. Su_ 55 14! 2{x) 24 3 76 3 100.6 12_ 32( 5_1 -- 5-_) 422 4_ 4.z, SuJfaoe 455 34_ 37(, 523 356 397 311 34_ 33_ 776 -- 776 4_ 494 49_ 01_ I _ 107 Oflh ] 31 099 1) 74 2.1_ 171 Coatlmb 174 -- -- 231 27% g]o -- -- 31( Total {, 57 727 7 _)_ Iqzed Structure Weight Data Tr_tg Edge 648 37_) 1833 1351 871 I Weight 13-727 U-737 B-747 13-757 []-767 DC-_ MD43 MD-II I ,.-_ting Edge 817 4_12 _l(X) yr7 2213 Aileron Weight Data Weight [3-727 B-717 13-747 []-757 B-767 IX'-8 MD41 MD-II S*u dac_ 34. 137 'A_ 25Jt_ 359 I3 1(}2] Suppect 4_ 16 Ifd, 3861 2295!
Chat I_*lt 29 226 IIX_) 42_ 1097_ 5P,2_ Ailk_O0 At¢_. S.,* _7 2(I 9 226 48 1246 i6l_ 3_ 1871 SUp_OflS 07_I 1159 073 0()] lO_ I_ Contmb 5111 8411 446 264 28% 3 It Spoiler Weight Data Wetght B-727 I [3-717 [3-747 [3-757 1]-767 IX. "_g MD-_3 MD-]I _.faoe 33; 156 582 33' 391 ]32. T, 1751_ Total 71); -146 1(,53 4fdl 5! 11135 9_ l°._t ted Area. S,p 791 45 I _34 256 r_.4 4g_ _ M : lY
Wdghl I S., I
Suffaoe 4 [61 3 46 ] 91 o(X 26_ Supports 11013 0 t_l I 29 3 E I 2_ Contmh 576 6 43 2.24 9 4! 361
Appendix D
Appendix D Database Formulation The first step in integrating a database into the high-lift module is to formulate the data based on important parameters. This has already been done, and the empirical equations can be found in the documentation. The second step is to determine appropriate empirical coefficients for the formulas. This has been completed for leading- edge devices and single-slotted trailing-edge flaps but can be done for other configurations by finding the empirical coefficients which minimize the error between the predictions and the actual data. In other words, iterate on the empirical coefficients to minimize the RMS: RMS = X/average(predicted values - database values) 2 The empirical equations for Z, ACdmi n, ACImi n, mkp, Cis,nax, and Xcp are given in equations 11, 26, 27, 28, 62, and 41 respectively and must be fit to a new configuration data set in order to integrate the new configuration. The database values of these parameters are determined from the CFD data as follows: ACITE (Find Z which accurately predicts _8 = C _ C letclean _TE -- C cts.)
ACIT E = C I - Clclean