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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED May 1945 as Advance Restricted Report L5E02 PREDICTION OF MOTIONS OF AN AIRPLANE RESULTING FROM ABRUPT MOVNEMIT OF LATERAL OR DIRECTIONAL COiNROLS By Chester H. Wolowicz Langley Memrial Aeronautical Labaratory Langley Field, Va. . .. :."  N AC) r ..: : . " ; ,. .... t .: WASHINGTON NACA WARTIME REPORTS are reprints of papers originally Issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre viously held under a security status but are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. L 125 DOCUMENTS DEPARTMENT L o. L502 ARR No. L5Z02 J Digilized by Ihe Iniernel Archive in 2011 WillI lunding from University ol Florida, George A. Snmahers Libraries wiIll sLpport front LYRASIS and Ihe Sloan Foundalion hlp: www.archive.org details predictionofmnoliO1ang NACA ARR No. L5E02 2 ' iJATION4L ADVISORY Co TIiITTEE FOR ALEP;TAUTICS ADVANCE RESTRICTED REPORT PREDICTION OF O'ciO.Tza OP AiN AlPLATTE REZULTI;tUG FROI . ABRUPT I!OVEi2.E;T OF I.AT'TRAL OR DIR:CTIO'IAL CONTROLS By Chester H. 'iolowicz SU;: AaY A procedure is presented for determining the motions of an airplane resulting from the deflection of the lateral or directional controls for the case of non linear derivatives. The stepbystep integration on which the procedure is based Considers the rolling, the yawing, and the lateral accelerations computed from wind tunnel data as functions of the sideslip angle. A sample computation table is presented to illustrate the appli cation of the procedure. A comparison is made of different methods for calcu lating the disturbed .tiolns of an airplane resulting from an abrupt aileron movement. Experimental data, which were obtained front convent zal windtunnel tests of a model of a recent fighter airplane, are used in the co;n putations for comparing the various methods. The resulting solutions show that, for the case of nonlinear derivatives, the calculated motions are in better agreement with the results obtained from flight tests if the rolling and yawing accelerations computed from staticmodel tests are considered as functions of the sideslip angle. The lateral acceleration, which is often essumied to be negligible, should be considered. The variation of the rolling and yawing acctlerations resulting from ailuron movement probably should also be considered when sufficient data are available. The variation of the dynamic derivatives Lp, N Lr, and Nr should also be taken into account when sufficient dynamic test data are available. It is shown that the present stepbystep integration method is reliable for cases in which only the first quartercycle of the motion is required (for example, in cases in which the maximum value of the sideslip angle is desired for determining vurticaltail loads in rolling pullouts). For the range past the first quartercycle of the motion curve, the method requires, further refinements 2 NACA ARR Noe L5E02 such as those provided by the RungeKutta su mation method. The present stepbystep integration method may be applied to the solution of motions produced by rudder movements or by a combination of rudder and aileron movement, as well as to the solution of motions produced by ailerons alone. INTRODUCTION The increasing importance of predicting the flying qualities and maneuverability of an airplane has empha sized the need for a more accurate method of computing the lateral motion resulting from abrupt control movement. Increased speed and maneuverability have, in addition, made it necessary to predict the maximum sideslip angles in lateralcontrol maneuvers in order that maximum verticaltail loads may be estimated. Much work has been done on the subject of disturbed motions (references 1 to 5), but all the solutions deal with constant lateralstability derivatives. These treatments assume that the rollingmoment coefficient CL and the yawingmoment coefficient CO are linear func tions of the sideslip angle P, the rolling velocity p, and the yawing velocity r. Windtunnel tests, however, indicate that most presentday airplanes do not possess these linear variations of C, and Cn with p, since the degree of linearity is affected by such factors as the geometry of the airplane, the power, the type of pro peller, and the blade angle. Lack of mathematical equations for expressing the derivatives as functions of the motions makes the method of references 2 and 4 inapplicable. The procedure for the solution with nonlinear characteristics presented herein is a refinement and an expansion of the integration procedure of reference 1. with the windtunnel data available at the present time, only the linear and angular accelerations PYp, PLp, PN 6L6, and 6N6 may be determined as functions of the sideslip angle p. Lack of modeltest data for effects of the rate of roll p and the rate of yaw r still makes it necessary to deal with the dynamic deriva tives Lp, L1, and Np determined from theoretical :ACA ARF Po. 1,502 treatments (reference 3). The dna ,.ic derivative Nr is determined partly 'rorn windtunnel data and partly fro.; theoretical considerations (references 6 and 7). In the present report three previously established procedures, based upon constant derivatives, for deter .ninin; the disturbed motions of an airplane that result from abrupt aileron movemerit are co.;:rared with a stepby step integration procedure that considers accelerations, computed from windtunnel data, as functions of the side slip Cnrle p. This stepbystep intrt ration not only generally provides more accurate solutions for disturbed motions but also should prove usF.ful in determining the verticaltail loads resulting from rolling pullout maneuvers as discussed in reference 5. Unrublihed experimental data (fig. 1) obtained from conventional windtunnel tests of a nmoael of a recent fighter airplane are used in calculating the motions, and the results are compared with flight results. CCCFFICIENTS AND STOLS The coefficients and symbols used herein are referred to a system of axes in which the Zaxis is in the plane of syTuletrcy and perpendicular to the relative air stream, the Xaxis is in the ,lane of symnnetry and perpendicular to the Zaxis, and the Yaxis is perpendicular to the plane of sym,!metry. The coCfLicients and symbols are defined as follows: CL airplane lift coefficient (Lift qS ) CL, lift coefficient of ;in ACLf increment of lift coeffiL..ent resulting from flap deflection Cdw ;:rofiledrEg coefficient of wing ACdo increment of orofil.edrap coefficient caused by flap deflection ,l rollingmoment coefficient (Rollin.oment) n yawingomoment coefficient (Yawing moment NACA ARR No. L5EO2 C0 rollingmoment coefficient caused by aileron deflection Cn yawingmoment coefficient caused by aileron a deflection Cy lateralforce coefficient Lateral force b wing span, feet bf flap span, feet X taper ratio; ratio of tip chord to root chord A aspect ratio b distance from center of gravity to rudder hinge line, feet 6a aileron deflection, derees; used with subscripts L and R to refer to left and right ailerons, respectively 6f flap deflection, degrees 6r rudder deflection, degrees av angle of attack of vertical tail, degrees aa absolute angle of attack of wing treasured from zerolift line, degrees S angle of yaw, degrees (3 sideslip angle, radians except as otherwise indicated; considered in static windtunnel tests to be equal to t Cn6 rate of change of yawingmoment coefficient with r /C rudder deflection (6Cn 66r/ 6r inverse of rudder effectiveness parameter at constant lift I 3 Dvj IIACA iR: No. 1,502 5 7p rate of change of rollingmoment coefficient with wingtip helix anfil; 1I Cn rate of change of yawingmnoment coefficient with wingtip helix anglo  C7, rate of change of rollinemoment coefficient with rb 1 212 OC, rate of change of yawinnmoment coefficient with b /\ 2V rb Cn, rate of change of yawingmoment coefficient with angle of yaw \, ) Lp rate of change of rolling acceleration with rate of roll b qSb \ r 2V rk 2 Lr rate of change of rollin' acceleration with rate ofr~ b q Sb of yaw 'r 2Vr !p rate of change of yawing acceleration with rate of roll /Cnr b ) S rate of chan.e of yawing acceleration with rate of aw n kz2 MACA ARR Io. L5E02 6L5 rolling acceleration caused by control deflection, /(C q.Sb\ radians per second per second  (1n1.bFcriots a and r indicate aileron and rudder, respectively.) 6F16 yawin; acceleration caused :)iT control deflection, /Cnqqb\ radian? per scconu nor second 2 (Subscripts a and r indicate aileron and rudder, respectively.) PLP rolling acceleration resulting from sideslip angle, radians per second per second 7,l  pDTN yawing acceleration rcisulting from sideslip angle, radilans per second iJpr second n b\ pY siQdeslippin:3 acceleration resulting from sideslip angle, fIet per second :er second (Cyv) dp d rolling angular acceleration, radians per second dt per second di yawing angular acceleration, radians per second dt per second do sideslipping velocity, radians per second dt dv  sideslipping acceleration, feet per second per dt second ZLn net induced rolling accelerations at t = n SNn net induced yawing accelerations at t = n JACA A": I:o. 15n02 7 p rolling velocity, radians per second except as otherwise indicated r yawing velocity, radians per second except as otherwise indicated / angle of roll, radians except as otherwise indicated P air density, slus per cubic feet V velocity along Xaxis, feet Ie second v sideslioping component of velocity, feet per second q dynamic pressure, pounds per square foot (V2) S wing Crea, square feet r mass of airplane, slugs kX ralius of yration about XxYis, feet k, radius of gyration about Zaxis, feet t time, seconds g gravitational acceleration (32.2 ft/sec2) Ko, ':f, 1, K2, K3 constants used in determinining U The subscripts n and n 1 denote values corresponding to the time t and to the immediately preceding time t At, respectively. PROCEDU.TE FOR C,'.DT'TITM LATERAL MOTIONS BY STEPBYSTEP INTEGRATION All the procedures considered for determination of disturbed motions are based upon the following vellknown dynamic lateralmotion equations.for level flight: d ( d = 6L + pLo + rL. + FpT (1) FACA ARR No. L5E02 dr = 61 + + + rU! + (Tp (2) dt dv = sin $ rV + ,(3) dt d = (4) dt V 11 (5) The individual terms in equations (1) and (2) rep resent the values of the instantaneous angular accelora tioas produced by the magnitude of the aerodynamic nmo.nents acting on the airplane at any given instant of time. The individual terms in equation (3) similarly represent the instantaneous lateral accelerations produced by the gravitational and aerodynamic forces. The instan taneous accelerations are in;de)enuent of the manner in which the aerodynamic moments and forces vary and are dependent only upon the instantrneous magnitudes of the moments and forces acting at any j.ivon tine. For the linear case, the acceleration terms such as pNp and PNp may be expressed as products of an angular displacement or velocity, as the case may be, and a con stant slope representing the acceleration caused by the disturbance per unit disturbed ;motion. Equations (1) to ()..) may therefore, be directly in::terated (reference 2). ror tte nonlinear case, direct integration is seldom possible. Wr'hen direct integration is not possible, the accelerations, such as CGN, P', and 5L6, determined from moJel experimental data, may be clotted as functions of 3; such a olot permits a solution for the nonlinear case of disturbed motions by the use of stepbystep inteTration or, when available, a differential analyzer. No variation of CL6 and 6N3 with p was considered for the airplane in the present report since no such experimental data were available. The appendix presents the data, the references, the calculations, and the information for curves such as figure 2 necessary for the formal stepbystep integration. The expression for Nr, as given in 9* " A APT Fo. L5E02 the aprtrndix and used in conjunction with the method of the pr.eent report, differs sli.itly from the expression given in reference 6 in that the first term of the equa tion for the determination of Cn in reference 6 1 b6 ( taO l on ttail of which represents the dnnpinr of the vertical tail brd is suitable for prDoplleroff conditions, has been replaced herein by the expression iL.6 Cn 65r b Cr v Analysis indicated that the rotation of the propeller slipstream and sidewash in model tests precluded a reliable SCn determination of the verti 'altall efTectiveless  when the expression of reference 6 was used. The expres sion givenr in the present report is more general and is suitable for Fnyl power and piol.ller arranj2ment. The values of " and K3 have not been solveO for in the appendix since they are used for flaps deflecte conditions and the airplane used in the present report was in the cruising configuration. After the calculations indicated in the appendix have been made and after curves such as figure 2 have beei plotted, the stepbystep integration form shown as taUle I may Oe used. In using the stepbystep integrstion, it may be desirable to use time increments of 1/10 second for coni putational convenience as well as for brevity of the solution combined with a fairly good degree of accuracy. The integration indicated in table I is based upon the suiruatlon process of solution of equations '1) to (5). This surr.nat.5on process, as used in table T, may be expressed as n = (in) t + Pn1 (6) nn "TACA ARP No. L5E02 Pn + Pn1 On = 2 At (O) ni tC)nl tr n dr n dt] r = )n1 n= (dtnl at + rn1 At + Pnl = L + PnLp + rnILr + (p~L = 6! +nl P o + rn1Nr + (P n1) = sin n V 1 (PY~)n1  rn + n1 V The subscripts n and n 1 denote values corresponding to the time t and to the irrediately preceding time t At, respect vely. The first step in using the stepbystep integration involves the insertion of values for the constant accelera tions and derivatives 5 L, 6N6, L,, Lr, Hp, and Nr in columns (5), (11), (20), (21), (24), and (25) in the underlined spaces provided in the headings of table I. The values in radian measure of the initial rate of roll p, the angle of bank 0, the rate of yaw r, and the angle of sideslip P should be inserted in columns (5), (8), (13), and (18) for t = 0. From curves such as figure 2, the values of pYI, PLr, and pNP should be determined for the valup of p at t = 0 (P = 0 in the present case). These values should be inserted in columns (14), (19), and (23) for t = 0. where (7) (8) (9) (10) (11) (12) i!JAA A.,: H!o. L5E02 11 Columns (9), (10), (15), (16), (20).to (22), and (24) to (26) may now be filled in for t = 0. Column (22) pro vides the inicuced rolling accelerations; column (26) pro vides the induced yavwing accelerations. The net instan taneous rolling and yawinc accelerations may now be determined for t = 0 by performing the computations indicated in columns (3) and (11). By repestin7 the procedure indicated in the headings of table I and by using. the :ariple values obtained for t = 0, the values of p, 0, r, and p are obtained for t = 0.1 secon.:J. After the value of p for t = 0.1 second has been obtained, corresponding values of PY, PL, and ip are determined and inserted in. columns (14), (19), and (25) for t = 0.1 second. The net induced accelerations ZLn and ZNn for t = 0.1 second may now be determirned (columns (22) and (26)) a&ii, as a result, the values in columns (3) and (11) r.sy be deter mined for t = 0.1 second. The remainder of table T for the other values of t may now be solved by repeaiting the procedure indicated in the hedirn.s and by using curves similar to figure 2. The angle of bank $ was determined by averaging, the rate of roll p (columns (5) to (7)). This averaging was not followed thr.ugh for sin / and for r in the determination of p, because it was thought desirable to maintain simplicity in the table and the errors intro duce'l. by a disreCrd of th.se averages are small and are within the accuracy of the data used for the c&lcsla tions in the a~.ncxi. The stenoyste .it :'atioin cr.sented herein is not limited t.o the soluition of motions rod'i.ced by silerons. Such i..te6riti)nm rray just as readily be applied to the solution of cdisturbad motions oroduceod by rudder movements nor by a combination oi ru'dder ad. aillron rrovermenrt. For the c2se of latarel disturbances caused by rudder alone, 6aLa and aN6a would be chjnIed to 5r.LA and 5r5 r. .'hen the stepbystep intear'ation is applied with variable derivat ves to flight conditions involvin.r accel erations greater than 1 g, the value of the airplane speed used should be the true airspeed V. The acceleration, however, must be considered in determining the airplane lift coefficient. The values of Cn and Ca (if an aileron movement is concerned) ar.d the derivatives correspond to the new lift coefficient. FACA ARR No. L5E02 Cri.,PAk;iT&; OF FROCDFUL.:' FCF COMP;'PTT'; I4 ATE.AL DISTTJ ..UATJ "'he charactris'ti.c curves ohtainF.c by the stepby ste Int ratio. rre coMo:rEd in fijires 3 to 6 with the res t.i s obtained fro.! ac.t,t' fli_ht tests, with the metnorl of diff:rrcntia'L. operators (reference 2), which is an eyant solution dealinfi :;,ith constant slopes; and rith "n acs'rov.l imte nielytical solutlmn in which constant slooes are also used (referencnu !) and which is applicable only to the solution of tLhe sile.slip anrle. '.Vhcn the maciriu. sUid lsl p anle r ,r:s cobyl.ted by the approuimate method of reference ', the coputed value was found to be 57.50, ; which ices nob compare with the 1IL determined fro.n flight tests. '.'hen the Vdlue of Cn was considered equal to Ca + ( , the conmptel value of the maximum sideslir anrle was determined to be 1r'.20, which is still rather hih. T1,e prsent procedure provides the most accurate correlation v.'ith flisit test results for all the mctions consider d. It should be n.t:Cd that the refine,:ent used in the present re,.ort or te determination of' ,Jr was not used in tht a:'.licatirn o; the methods io refrecnces 2 and 4. It should also tb: noted that v 'V, '" rich is considered equal to the value of p in radians in all ti procedures, is in its rtrictect sense equal to ban p. Thu assumption TT that p = lads to much lar.Cr errors for large values than I'or fnall values of p. For example, consideration of thevr two sources f error r'ducev the maximum side slip nngle of 20, shown'm f'o the a,prnximate procedure of reference I, to a value of 560. rho I.mproved method in consideri:w I accounted for 00, wbhreas the other 270 were A.ccoanted for by th3 fact ;that v/V was considered equal to *ar p. In the care of the tepbystep pro cedurc of the prudent report, thr .:anxirmum sid3slip angle would hlve been equal to about 250 if Nr had been deter mined by the method of rafcrc.nco 6. If v/7 had been considered equal to tan 2, thu maximum sideslip angle by t!i: step' tcp 'iethod would havL been reduced about . V. ,,. NACA .?;r No. L5202 For solutions involving the ,.Ecsu!iption of linear slopes, the slopes u cid in the jrcsent problem were arbi traril measured through *: = 0. If the more usual practice of selecting the avcra e sloocs over a wider ran:r. of yaw angles had been employed, the calculated results vwo'ld have approached more closely the results of tho var.iableslope method. cFr cases in which vertical tail loads in highspcod dives are of primary concern, however, small anglos of sideslip may be critical, and consideration of average slopes over a wide rangc of yaw ingles ray be unwise. It a:.!:.'a:s th'rOfore that, although thc rrcvious proccdurLs may be recson ,ly reliable in a number of instances in which the charactcriEtic Ci, C,, and Cv curves possess ap prcxi.:iatoly linear relation ships, nonlinear characteristics occur with sufficient frLqt'rncy to make the general use of thb nonlinear step bystep procedure desirable. In order to determine the importance of the lateral acceleration term 1Y the '.esent prceecure was repeated ;:ith = Although the resultin curves indicate that the influence of (iY for the subject airp~ .r.e was not very lar'e, the eff: t of pYg may be niore significant for cther types of airplane and there fore s'."ould not be neglected. A co.::.rison of the stepbystE, solution using cons. !rt s.ycps with t.he ctho l .'f dI ffi.er.ntial operators (ref cr.ice 2) indicated that vwlies obtained by tle step bystmp solu.tio;' tendd to Jdeviate a little mor. from flight te:t results tl.an the values obtained by the opera tional meeLhod The stepbystei solution, for this particular compmxrison, apparently gives a sideslip angle appro:;.iratcly 20 larger than the operational procedure of reference 2. The tendency of 'hCe stepbystep solution in tC'^ linear case to deviate a 1:,.ttle more from flight tests than a direct integration procAdure may reasonably be )resumed to persist in the a, location of the stepby step solution to the ncnlinear ca&e, as in the present ieport. Further refinement of the stepbystep procedure may therefore be expected to provide correspondingly closer aprecrent with flihr'it,. The r i.neFutta surmation method (rcferences 3 and 0) provides such refinciments of proce:.ures. The stebystep procedure as outlined in the prnorent report, I vevcer, is believed to provide sufficient engineering acc:.'.acy wihu;: no more tLan t'ie i".."'" 1 !',.'.. r. .. i? 1" i .* r '. '..'"1 ( '," .",.j .' ITACA ARR Ho. L5E02 Although 6L6 and 56', were considered constants in the preceding example, further analysis indicated that the rolling and yawing accelerations resulting from aileron deflection should also be considered functions of p for a greater degree of accuracy. It is quite possible that Lp, Np, Lp, and N are not constant as ordinarily assumed and as assumed in the present report. If these parameters are not constant, some of the dis crepancy that still exists between flight test results and the present method would be explained. Until experi mental data from dynamicmodel tests are available, how ever, these values must be presuzed constant for lack of more complete information. Other possible sources of discrepancy between calculations and flight results are the assujTirtions of level flight, constant normal accel eration, constant seed, and instantaneous control deflection. For practical purposes, however, it was not believc.d necessary to tale thjse factors into account. A rocedurc base upon stepbystep integration is presented for drctermining the disturbed motions of an sirlanc resulting from the *lfl:cti.or of the lateral or dirJcticonal controls for th. c:ise of.. nonlinear derivatives. A comparison of the stp'byst.': p7roc.du're with other methods indicated the following conclusions: .. The calculated disturbed motions of an airplane resulr.in;, from abrupt control moveeient will be in better agreement iwith the results obtained from flight tests if the variation of the experimenantallyJ dcter'ined rolling, yawini, and sideslipping accelerations PLp, PNp, and pYP with the angle of sideslip p is considered. The sides lippinr acceleration ,3Y which is often assumed negligible, should be considered. The variation of the rolling and yawing accelerations aL6a and 5aNLa resulting from aileron movement probably should also be considered when sufficient data are available. The variation of the dynamic derivatives I, ip, Ly, and 'Tr should also be taken into account when sufficient dynamictest data are available. ITACA RR ITo. L5S02 15 2. The value of the maximum sideslip angle for use in the determination of the verticaltail loads in rolling pullout maneuvers should be obtained by using the step bystep integration method. 3. The stepbystep integration may be applied to the solution of motions produced by rudder movements or by a combination rudder and aileron movement, as well as to the solution of motions pro 'uced by ailerons alone when only tht first' qr,.rt;rcycle of the motion is desired. Langley Memorial Aeronautical Laboratory nationala l A.visory Comnittee for Aeronautics Langly Field, Va. 16 TACA .,RR No. L5E02 APPENDIX DETER'MINATTON OF DYNAITC LATEPAL P.OTIONE OF A FT.HTER ATPPLAITE DUE TO ABRUPT AILEROI MOVEMENT Data re 'iired. F)r the fighter airplane used in the illustration, the data required for the determination of dynamic lateral motions resulting from abrupt aileron movement are as follows. b, ft. ... 2.83 Cd . bf, percent b 66 ACd . Of S. 0.50 Ca . A . 5 5 C . 1, ft . 20.5 Cn5r . r a, 5ar, des .. 12.75 V, fps . 5ap, deg ........ 17 q, lb/sq ft 5f, deg .. 0 r, slugs . a,, deg 17.1 S, sq ft . CL . 1.42 kX2, sq ft . CL, ... k sq ft . ACL .. ......... 0 Z2 sq ft. Landing gear . The value of Procedure. From . 0.01 ... O 0.04 . .0.0065 * .o.oo67 S0.001474 . 1L2.2 S. 24.09 . 358 334 ...3352 . 33.52 . 57.9 . .. Retracted b5 = 2.0 is determined from reference 10. reference 3, determine C, = o.L25 Cnp = 0.0655 Cr =O.5308 :.A A.1 No. L5 02 Then, from reference 6, determine = 0.5335 l o 2 + 2X = 0.2749 From reference 7, determine K1 = 0.O022 Comrpute the following: L z b qib Lp= Lp 2V xP2 2 = 1..354 IT = C r  b qSb L b r r 2V e = 1.55 !,= = 11.6Cn 5 !'r : Ii.6 5 r rav 6 aL a a bk 2" 6' = na mk" = 0.1 + Cd, + Kf Ltor dow f dof 1+ lC 2 6 2 ACLf T + 5 ( L+ ) k q 2 :W  f = o.5108 18 NACA ARR iHo. L'E02 From windtunnrel d ata for the configuration con sider.ed fig. 1), nlot the following asainst P or W: j3L, qSO qs I)i The ,alues of 7o, F2, a.nd Y3 hi:ve ILOt been solved for 2Ine thiy sre L.sed for flisp'leflectel conditions and the airrlane used in the res ent report *.,s in the cruising coil'i ~grati on. Fr lepsdeflected conditions, K,. m:.: t determined fron the following formula from reference 6: f = .*55(t  f \b 2 + 2% The values of Y2 and YF msy be determined from reference 7. Although th,~ valur. of i, and C in the present runort lhrve b,.cn d.c tormincd ] :'ly from th'. curvJs of r_ ,"' . nc 5, it I:n. bo desir ..ble i so rji cases to include the ffcc ts of the vertical tail us' s of thu rmvthod of rcfjrJcc 11. NACA ARR No. L5E02 FVE RE'CES 1. Weick, Fred E., an. Jones, Robert T.: The Effect of Lateral Controls in Producing '.otion of an Airplane as Corputed from WindTunnel Data. NACA Pep. Nlo. 570, 1936. 2. Jones, Robrt T.: A Sipolified Application of the method of Operators to the Calculation of Disturbed Motions of an Airplane. NACA Pep. IHo. 560, 1936. 5. Pearson, Henry A., and Jones, Fobert T.: Theoretical Stability ani Control CnarFr,teristice of ,'ings with Varicus Amounts of Taper an. T wist. NACA Pep. No. 635, 1953. )4. ;syten, Gerald G.: Analysis of 'WindTunnl3 Stability and Control Tests in T l'e s l llying Qualities of FullScale Airclanes. ACA . F No. 5J22, 1945. 5. Gilruth, Robert T.:: Analysis o' VerticalT&il Loads in Polling Pull7ut Y'.aneuvers. NACA CB INo. L4F14, 19L.. 6. Campbell, John P., and Mathews, '.'ard 0.: Experimental Determination of the Yawing Moment Due to Yawing Contributed by the Wing, Fuselage, and Vertical Tail of a :M.idwing Airplane !K:odel. NACA ARP No. 3F28, 1943. 7. Harmon, Sidney ':;.: Determination of the Damping Mor.ent In Yawing for Tanered 'ings with PartialSpan Flaps. NACA APR Iro. 3H25, 1945. 8. Lew, F. and Baggott, E. A.: Numerical Studies in Differential Equations. Vol.T, 'atts & Co. (London), 1954. 9. Scarborough, James 3.: Numerical Mathematical Analysis. The Johns Hopkins Press (Baltimore), 1950. 10. Ames, Milton B., Jr., and Sears, 5ichard I.: Determina tion of ControlSurface Characteristics from NACA PlainFlas and Tab Data. NACA Rep. No. 721, 1941. 11. Damber, Millard, J.: Effect of Some PresentDay Airplane Design Trends n Requirements for Lateral Stability. NACA TN Fo. 814, 1941. NACA ARR No. L5E02 20 a, a Io  r '" 1I I0 0 ^ I ~ 64 . s C , C1 C10 SI r, C, s 0 I C a *o .4 S " a ~ ~ ~ ~ ~ ~ ~ ^ ^ ^ :..i a ' i 5 S ? , J f ^ ^ ^ '' ! ' t    u, NACA ARR No. L5E02 .4 .4 0.4 .04 C) S.04 0* 0 40 30 20 10 0 /0 0 Angle of yaw, V, deg 30 40 FIgure 1. Var/atfon of the drectional and /dera/ coefficienhs C, Cn,, and Cy, with angle ofyaw 1Y determined from feit of a ,ccle model of a fighter airplane the Zanrley 7bv/lOf'nt funmel. Cruising configurafion;C/.42; ,=:qOf:0 . Fi g. 1 NACA ARR No. L5E02 4 6  / ! J5des/lp angle, deg NATlOIAL ADv1~ORv COMMITTEE jOh ArhONAUUICS figure 2. Curves of the acceleraf/ons 0L, (N , and a Y aj functhonS of the sides/p ang/e (3 determined from tefts of a s6cale model of a fighter airplane m the Loarey 7by/Ofoo/ funnel 6: 0 6a=0. 1.. Z I? O I, Os  <. 3  Fig. 2 NACA ARR No. L5E02 Fig. 3 ~ cga. li I K lzb i Itt =====^1 S k T i Iji^^  ^ I 9 I iiX i     '0 _i ") ' S1 __ I / _ ____ I'^ 1  AI V \ ./ ' coJ  . ^ 5 _____ ____ ___^ ^__ 't" 1  t R I __ go 0 0 ia Q (0 o (D~~~~t co i' n v 'n ^ NACA ARR No. L5E02 Q) "c, ~C D( ~E I I I 111 1 1 1 1  . I i   _ \C A'  ^ '  r' a        ^ ^       P m'^mi^^^^mz < to Q o 10 r ds,.,6'6p 'd 'Xpn0l&A, bul11oy I I I I Ii __ _ 8 5, /S i r s4 :^"=^,ij '4 c3 c^ o Fig. 4 Y1 *g S sjc o <' t < a 0^ K I;; P :^ ^^ .^ ^.i StZ NACA ARR No. L5E02 Fig. 5 , I _ / .0 .c_ _1 S  L N'  _ __ 0 I4 1 Itss t :4 q) aII ~04 I V NACA ARR No. L5E02 lii i _ __I _ \ === ,"\==^   2^ ^^  / __ __ V _Xl4 _: _i__ r IV I I 'b "U I 4 s 'k Sr SO H 8^ s I ^1 Q,'. 4 .'JiI l ^1 Co i SI I I I I I I '0 N '  76a _______ __ 1  i^ ^t7~~ i^^ j '.30)o/1a 6U MDA ^ R1 s rr "*i^ 0 ^ '1 NE s :&. <3 _Q ~N c00 (I O 18 a1 ' NM 2 uJq Fig. 6 UNIVERSITY OF FLORIDA 3 1262 08104 996 6 '' * ' UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 117011 GAINESVILLE, FL 326117011 USA '*1. ,( a iy 
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