1、R - -b- ti- ? =- - .- . _ - .E_ * - _.-8 *-L- -. - - . .a71 a71. TECHNICAL NOTESs NATIONAL ADVISORY COMMITTEE FOR AERONAUT ICS-b.No. 925: / “- “”- -.I. - . . . . .- .- . . .A LEASTSQUARES PROCqURE FOR THE SOLUT IOI?OFTHE LIFl!IHG-L IHE IbT7!EGRALEQUAT IONBy Francis B. Hildebrandliassachusetts Instit
2、pte of TechnologyJCLASSIFIEDMCUMENTThisdocumentcontainaelaanifiedinformationaffectingtheNationalBefenmeoftheUnitedStatenwithinthemeaning. ofthe EspionageAct,USC50:31and32. Itstranmis6ionortherevelationof itsoontentaA in anymannertoan unauthorized*. .*rsonisprohibitedbylaw.Infor-.mationsoclassifiedma
3、ybeimrt-i? edonlyto personsinthemiliteryand navalServicesoftheu-nitd* States,appropriatecivilianoffi-cersandemployeesof theFederalGovernmentwhohavea legitimateinteresttherein,md to UnitedStatescitizensof knownIqaltiyand discretionwho of neceseitymustbeinformed thereof.-. “.WashingtonFebruary 1?344.
4、 -,.,- - .-.-_- ._ .- -.-. - .-r-.- .-.-.- r. .-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-m.8 .“:l!llllmminllllll“-:”y:;y:;:.:”-,:,:,:,31 i760fi4338223 -: .- ,- ._- :-. ._.r.- . -.- .-.NATIONAL ABV ISORY COMM.?TTEE,FOR .J?3ROMAUTICS .-.-.-.
5、A LEAS TSQUARXS PROCEDURE “FOR TEE SOLU!3?01703TH23 LIF!IIN(+L IiW INTEGRLiL EQTJATION -.By Francis BIIildebrand - .-SIJMMfi”lshould be of a form readily adaptableto the approxmation of the function T, the characteristic behavior of which usually is known.In the present procedure an approximat
6、ion to Y(y)is assumed in the form. . Wlth the exception of the3+ fi ) An Yan (+7)nfirst term, the approxi-mating functions ae conventional ones employed isewhere. The cooffici.ent of 33, which is of the formrequired by equation (16), was originally chosen for usein cases when a(y) has a discontinuou
7、s first derivativeat tho root (e.g, in the case of a symmetrically linearangle of attack), since the contribution of this term tothe integral representing the induced angle of attack,has a discontinuous derivative at the root (Y= O), whilothe function itself has a continuous derivative at thispoint
8、 The function was, however, retained for use inthe more general case since itaproximating functions, being$!w th functions 1-y2figure l.)With the approximation ofline equation (5) becomescomplements the otherintermediate in behaviorand yadlya . (See aequation (17) the liftin*?.?.rProvided by IHSNot
9、 for ResaleNo reproduction or networking permitted without license from IHS-,-,-,. NACA Technical Note No”. .925 11+!320(+)+= ”+aa+A4”4+A”- - -,. . .,. . . . . . .-.s.(+(y ) Yj .T ” .TT -1 “ay” .Yn ,.- . . .,. .is .pi.ecewise constant in .theint,erval.Y.- .- .,. “1!lr(y)dy= -,y (48)b ;-,-.i “. -“- .
10、l+. r;+. , . .”=. - ,-. -. -5.sneeded. . _, ,.-,., ;, . ,t,.?q.-.:,-. .-.-,- - . -:-.:-1-2,.:;As an exam”ple, stippose.that the iiileron deflectionof a wtng is such thatrO -l+1and” it follows from equation (20) that . _ . .-AIIangle Of attack o“f-1-rad,ian-fro-m O “to-.489”ahd0 from0.489, to. 1 .s-
11、precried, sb ,hat, in th6 notat”io of”the. .examplo of the p.reced.”ingsectibu . , I _ _ .,. . ,. a = 0.489 (55) -!I!hoequation to be solved is thenTo(y) + 1 . df = cL*(y)C*(Y) =“ -1 dq yq(56)Wherea71Fe(y) = P(y) l?(y) (57).and P(y) and a*(y)(52),are defined in equations (50) andT“: -.,:Provided by
12、IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-26 NAOA Technical Note No. 925The func.%ions C*(Y)? P(y) and a*(y) are tabulated.at the nine points considered in table 3,If the operations indicated in equation (19) (whero a is replaced by a*) are carried out, eq
13、uation (21)takes the form “B A. AZ Ad A-.1919b 1,19190 -,09596 -,02399 -.01199-,07040 1,25021 -.07042 -,02800 -.01323.10621 1.29847 .00920 .03391 -,01759.31748 1,33285 .14545 -.02293 -.02469.54852 1,34659 .33665 -.03619 -, oaQQ4.77976 1.32734 .57247 .18323 ,03444,97608 1s25020 .82318 .45254 .223161.
14、05476 1.05256 1.00377 ,82257 ,644961.02073 .80166 1.08951 1.25742 1,389351.852691.947841.940881.766571.09D23.49026.0527-.S?629-.32527.(58;The coefficients of the final set of linear aquations,obtained by taking the dot product of the auxillary matrixdefined in equation (25) into both sides of equati
15、on (58), .are then found, .B. A. Aa A4 Aa5.45683 7.35230 4.66690 3.11236 2,3.15147.35230 18.51243 5,71837 3.14707 2.127954.66690 5,71837 4.lli95 2+92894 2,262823*11236 3.14707 2.92894 2.42110 2.050022*31514 2=12795 2.26282 2.05002 1.,85Z66 I1.7626115.54154=.62091-.51871-,60723,(59)and the solution o
16、f the corresponding set of equationg ia33 = 4.20498 A. = 1.40506 Aa = -9.94467 -In table 4 and figure 4 the auxiliary lift function .F(y)i determined by the equation .F(y) = P(y) - Ye(y) .is presented in compalson with a solution given by Pearson(referenoe 4), Z%iS solution was obtained by using a t
17、en-term series of the type given in equation (17a) (omittingrProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I?ACA Technical Eote. No. 925 27.the first term) and determining the parameters by amethod given by Miss Lotz (reference 3). l?or furthercomp
18、arison there fS inolqded in.figure 4 a SOluti On o _tained by the present least-squares procedure with anapproximation of the type of equation (17), that is,without. using the additional approximating function .-P(y). Whilethe Pearson soution agrees closely wifhthe present solution over a large part
19、 of the span, itappears that even with a ten-term”approximat icn, thecorrect behavior of the lfift curve cannot be satisfactorily approximated near the end -of the flap except by theuse of a term similar to the function P. The valuesof the angle of attack corresponding to the present sou-tion and to
20、 the Pearson solution are compared with theprescribed values of a in table 5 and figure 5. “rHODITIED PROCEDURE TOE WINGS WITH DISCXMTINUOUSCHORD PARIATIONSuppose that the angle of attack a(y) is continuousand that the chord cy) is continuous except for finitejumps at the points y= an. Theri the fun
21、ct$on (Yy/c*(Y)has corresponding discontinuities of magnitude - !( an)whereYn1-C*(an+)Hence. since a is. . . . .1 1=O*(an+) c*(an) t61)c*(an) C*(an+)C*(an)continuous, theequation (5) must be continuous andf$ dT dq._=dq yqlefthand side ofthe functionmust have discontinuities of magnitude +YnY(an) at
22、 thepoints y = an. , . -.,If the function l?(y) is written in the formI(y) = Fl(y) Q(Y) (62) “-.,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.,.28whore .NACA Technical Note No: 925.-.(63) .%,.:.nd n (Y) is defined by equation (39) , the ccnstants
23、 u can ho determined so that thele.f.$-hand side of eua-tion (5.).is cotitinuous and consequently the function(Y) has a finite derivativ inside the “interval y l.l?or, accordiug to equation (40), if equation (62) is introducad into equation (5) the lefthand 6ide. of”th.re-sulting equation has, at e
24、ach point y = am a dificontinui-tyof magnitude.It follows tha”t thedis continuitf”es will disappear if theconstants IIn satisfy the equationswhere .il, m=nL(65),.-.The constants Kn are-thus .deter,ioined.,from quation(64) as linear combinations of-the values of Fl(y) atthe points of:discotii.inuity,
25、 “a71Qquat ion (6) can now be rf.tten in the fora .r “1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N-AGA TeChil”ikal-k.t“eNo. -9= “ ;,. “B.,Since the function lizy) has a finite derivative insidethe interval : /yl cl; an .“approxirn&*iO-nto (Y) c
26、an be.assumed In the” fa71rm gi.v”bn “by equat ion (17) and the termin brackets in equation (66) becomes a known llnear combi-nation of the parameters 3 aha A*a. If this term isevaliiatb at the points yk with the help of equation(40) and if the. st-,of coefficiets f B and Aan inthe resultant eressio
27、ns is mitten as a matrix, thematrix equation replacing equation (21.) iS obtain?d bysubtracting this matrix from. the left-hand side of .equation (19)* The east-squares procedure can then %e a”plied as before.As an example, suppose. that the cherd variation of asymmetrical wing has dis continuities
28、of equal magnitudeand opposite sign at the points y = +a .and writeY= c*( a+) - c*( a-) = _ c*(-a+) - C,*(-a) (6?)c*(-a+) c*(-a-)C*( a+) C*( a-)Then equat ion (6 becomesUSC), since I(y) ,is an even function of y and since17a(a) = 17_a(-a) = ora(-a) =-l_a(ti) = 2a ldg a!“(69)equation (64) becomes -so that J.-. _ -.1 -“ ,. ”-”Provided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-
