NAcA L'U^ ARR No. L5E02 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIdNALLY ISSUED May 1914.5 as Advemce Eestrlcted Eeport L5E02 PKEDICnOH OF MOTIONS OF AN AIRPLANE RESULTING FRCM ABRUPT MOVEMEaST OF LATERAL OR DIRECTIONAL CONSOLS By Chester E. 'Wolowlcz Langley Memorial Aeronautical Laboratory Langley Field, Va. NACA 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. DOCUMENTS DEPARTMENT Digitized by tlie Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/predictionofmotiOOIang NACA AxRR No. L5S02 ^ ^^ ^ '^^^^ HATIOrliiL ADVISORY COin'vIITTEE FOR A}^R0:TAUTICS ADVANCE RESTRICTED REPORT PREDICTION OF liOTIONS OE' Ai! AlicPLAITE RECIJLTING PROM . ABRUPT MOVEMS.rJT OP I,ATt;,RAL OR DIRECTIOiVAL CONTROLS By Chester H. Wo3.owic2 SUMIvIAxRY A procedure Is presented for determining the motions of an airplane resultina, from the deflection of the lacersl or directional controls for the esse of non- linear derivatives. The step-by-step integi^ation on - ' which the procedure is based Considers the rolling, the yav;ing, and the lateral accelerations computed from wind- tunnel data as I'uncticns 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 r.otions of an airplane resulting from an abrupt aileron movement. Experimental daba, which v;ere obtained from conventional wind-tunnel tests of a model of a recent fighter airplane, are used in the com- putations for comparing the various methods, Tlie resulting solutions shov/ 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 static-model tests are considered as functions of the sideslip angle. The lateral acceleration, which is often ussiiir.ed to be negligible, should be considered. The variation of the rolling and ya-.ving accelerations resulting from aili^ron movement probably should also be considered when sufficient data are available. The variation of the dynamic derivatives L , N , L^, and N should also be taken into account when sufficient dynamic- test data are available. It is shown that the present step-by-step integration method is reliable for cases in v/hich only the first quarter-cycle of the motion is required (for example, in cases in which the maximiur. value of the sideslip angle is desired for determining vertical-tail loads in rolling pull-outs). For the range past the first quarter-cycle of the motion curve, the m.ethod requires, further refinements 2 NACA ARR No, L5E02 such as those provided by the Runge-Kutta stoeaatlon method. The present step-by-step integration method may be applied to the solution of motions produced by ruddej» movements or by a combination of rudder and aileron movement, as well as to the solution of motions produ««d by ailerons alone. INTRODUCTION The increasing importance of predicting the flying qualities and maneuverability of an airplane has empha- sized the lieed 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 maximijun sideslip angles in lateral-control maneuvers in order that maximum vertical-tail loads may be estimated. Much work has been done on the subject of distiorbed motions (references 1 to ^) , but all the solutions deal with constant lateral-stability derivatives. These treatments assume that th« rolling -mom.ent coefficient Cj and the yawing-moment coefficient Cj^ are linear func- tions of the sideslip angle p, the rolling velocity p, and the yawing velocity r. Wind- tunnel tests, however, indicate that most present-day airplanes do not possess these linear variations of C^ and C^^ v/ith 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 I4. inapplicable. The procedure for the solution v/ith nonlinear characteristics presented herein is a refinement and an expansion of the integration procedure of reference 1, »i/ith the wind-tunnel data available at the present time, only the linear and angular accelerations PYn, pLo, PN^, 6Le, and 5Ng may be determined as functions of the sideslip angle p. Lack of model-test data for effects of the rate of roll p and the rate of yaw r still makes it necessary to deal v/ith the dynamic deriva- tives Lp, Lj., and Np determined from theoretical MAG A ARF No. I,5E02 ' 3 treatments (reference fi ) . The dynamic derivative Np is determined partly from wind-tunnel data and partly frora theoretical considerations (references 6 and 7)' Tn the present report three previously established prccedujres, hased upon constant derivatives, for deter- mining the disturbed motions of an airplane that result from abrupt aileron movement are compared v^^ith a step-by- step interrration procedure that considers accelerations, computed from wind-tunnel data, as functions of the side- slip angle p. This step-by- step inte.cration not only generally provides more accurate solutions for disturbed motions but also should prove useful in determining the vertical-tail loads resulting fron; rolling pull-out maneuvers as discussed in reference 5- Unpublished experimental data (fig. 1) obtained from conventional wind-tunnel tests of a model of a recent fighter airplane are used in calculating the motions, and the results are oomxpared with flight results. COEFPICISWTS A^T Sym.BCLS The coefficients and symbols used herein are referred to a system of axes in which the Z-axis is in the plane of symmetry and perpendicular to the relative air stream, the X-axis is in the ^jlane of sy/imetry and perpendicular to the Z-axis, and the Y-axis is perpendicular to the plane of symmetry. The cc -efficients and symbols are defined as follov/s? Ct airnlane lift coefficient (IlI^ L \^ qS CX^,^ lift coefficient of v/ing ACt,^ Increm.ent of lift coefficient resulting from flap deflection C^ profile -drag coefficient of wing ACf^ increment of profile -drag coefficient caused by f flap deflection -IT. , „„. „ , /Rollin'T moment\ Ci rolling-moment coefficient ^ ) '' " V qcb / On yawlng-moment coefficient ^ Ya-^-'ingjnoment \ I|. MCA ARR No. L5E02 C; rolllng'-tr.oment coefficient caused by aileron ''a deflection Cj-, yav'ing-rr.oment coefficient caused by aileron ^ deflection , . , „ ^^. . . /Lateral force \ C,r lateral-force coefficient ( ) Y \^ qS / b wing span, feet hf flap span, feet X. taper ratio; ratio of tip chord to root chord A aspect ratio I distance from center of gravity to rudder hinge line, feet Sg^ aileron deflection, de,~r3en: used with subscripts L and E to refer- to left and right ailerons, respectively 5^ flap deflection, degrees 5j, rudder deflection, degrees a angle of attack of vertical tail, degrees a^^ absolute angle of attack of wing rreasured from zero-lift line, degrees vl/ angle of yaw, degrees (3 sideslip angle, radians except as otherwise indicated; considered in static v/1nd-tunnel tests to be equal to -\|/ Cj-^ rate of change of yawing -moment coefficient with °r /dCA rudder deflection ( — —] 6j^ inverse of rudder effectiveness parameter at constant lift v^'^v/. NAG A i^RR No. L5E02 C-i rate of change of rolling -moment coefficient with v;ing-tip iiellx an^lo / — — Cn rate of chanre of vawin,s'>riionient c oof fie lent with wing-tip helix anplo f Cj rate of change of rollizig-mornent coefficient r 'n^. •4-1 ^^ With -— /3^ ■'It Cj^ rate of change of yav; in --moment coc;fficient with Si ^^^^' C„ rate of chance of 7awin'''->-io:,ient coefficient with ansle of yaw L„ rate of change of rolling acceleration with rate of roll fcy AJLSL\ Lj, rate of change of rolling acceleration with rate of yaw ^7 A^\ V^ rate of chanr^e of yawing acceleration vrlth rate / of roll ;-r,,4^^ LJ^ rate of change of yawing acceleration with rate b qSb' of yaw [C ^r 2^' , 2 6 MCA ARR i:o. L5E02 5L5 rolling acceleration caused Dy control deflection, /O^ qSb" radians per second per second / — Vjnkx (Subscriots a and r indicate aileron and rudder, respectively.) 5Fc, yav;in,g acceleration caused by control deflection, radian? per secona ner second ( - dp dt dr dt dt dv dt rnk^^ ■; (Subscripts a and r indicate aileron and rudder, respectively.) pLg rolling acceleration resulting from sideslip angle, radians per second per second /C7 ~^~~~n] PNq yavrlng acceleration re.^:ulting from sideslip angle, radiems per second per second fC-^^ .A pY^ sidesliijping acceleration resulting from sideslip angle, feet per second per second (Cy — ro],ling angular accel«ratlon, radians per second per second yawing angular acceleration, radians per second per second sideslipping velocity, radians per second sideslipping accelert'tion, feet per second per second ELj^ net induced rolling accelerations at t = n ZNf^ net induced yawing accelerations at t = n "MACA ARR No. L5S02 7 p rolling valoclty, radians per second except as otherwise indicated r yawing velocity, radians per second except as otherwise indicated ^ angle of roll, rt^dians except as otherwise indicated p air density, slugs po'- cubic feet V velocity along X-axis, fset per second V sideslioping component of velocity, feet per second q dynamic pressure, pounds per square foct (tP'^ ) S wing area, square feet m mass of airplane, slugs ky radius of gyration about X-axis, feet k™ radius of gyration about Z-axis, feet t time, seconds g gravitational acceleration (52.2 ft/sec j Kq, r^, K^ J K^ , Kz constants used in determining Nj, The subscripts n and n - 1 denote values corresponding to the tim_e t end to the immediately preceding time t - At, respectively. PROCEDURE FOR CO?.-TT'-tIKG LATERAL MOTIONS 3Y STEP- BY- STEP INTEGRATION All the procedures considered for determ.ination of disturbed motions are based upon the following well-known dynamic lateral-motion equations .for level flight; |^= 5L5 + pT^ + rL^ + PLp (1) dt 8 FACA ARR Uo, L5E02 5N5 + p?Tp + rVr> + ,FNp (2) sin (?f - rV + CY^ (5) (1+) (5) The individual terms In equations (1) and (2) rep- resent the values of the instantaneous angular accelora- tions produced by the magnitude of the aerodynariic moments acting on the airplane at any given instant of time. The individual terras in equation (3) similarly represent the instantaneous lateral accelerations produced by the gravitational and aerodynamic forces. The instan- taneous accelerations are independent of the manner in vvhioh the aerodynamic moments ana forces vary and are dependent only upon the instantaneous magnitudes of the moments and forces acting at any <::ivon time. dv dt d(^ dt - P P V ■tr For the linear case, the acceleration terms such as j3N-3 and pNp may he expressed as products of an angular a con- by the IS ri) Terence 2 ) . ppig ana piVp may no expressea as proaucts oi an displacement or velocity, as the case may be, and stant slope representing the acceleration caused t disturbance per unit disturbed motion. Equation; to (h.) may therefore be directly integrated (refe ■"^or the nonlinear case, direct integration is seldom possible. When direct integration is not possible, the accelerations, such as Qlln, PYo, and 5L5, determined from mo:3el experimental data, may be plotted as functions of P; such a plot permits a solution for the nonlinear case of disturbed motions by the use of step-by-step integration or, when available, a differential analyzer. No variation of 5Lg and 6Nq vvith p v;as considered for the airplane in the present report since no such experimental data "vere available. The appendix presents the data, the references, the calculations, and the information for curves such as figure 2 necessary for the formal step-by-step integration. The expression for N^, as given in f-rCA AT^?t Fo. L5E02 the apoendix and used in conluncti.on with the method of the present report, differs slightly from the expression given in reference 6 in that the first terir of the equa- tion for the deterri;ination of 0^ in reference 6 \ ^tail llU.6r- /g„ - C„ ' ^' •' on tail off; which represents '►'he danpinn; of the vertical tail and is suitable for propeller-of f conditions, has oeen rerjlaced herein by the expression -llU.6-- Gn. ^r b "^S^ ^a^. Anf-lysis indicated that the rotation of the propeller sliDstream and sioewash in model tests precluded a reliable determination of the vertical-tall effectiveness v'hen the expression of reference 6 was used. The expres- sion given in the present report is more general and is suitable foi- any power and propeller arrangement. The values of ^> , X^ > and K, have not been ' ^ :? solved for in the appendix since they are used for flaps- deflected conditions and the airplejie used in the present report was in the crul sing configuration. After the calculations indicated in the appendix have been m.ade and after curves such as figure 2 have been plotted, the step-by-step integration form shown as table I m.ay be used. In using the step-by-step integration, it may be desirable to use time incremients of 1/10 second for com- putational convenience as well as for brevity of the solution comibined with a fairly good degree of accuracy. The integration indicated in table I is based upon the s'junmation process of solution of equations (1) to (5)" This sumuTiation process, as used in table T, may be expressed as 'dp - ^ n - 1 10 ?!ACA ARR Fo. L5302 , Pn + Pn-1 4 = ^ it + 4.^ (7) Pn = (flX.^ " ^ Pn-1 <9) where (f X., = ^"5 ^ Pn-l^^o + -n-l^V - (P^V),.i ^''^ Vcit/ 1 V ^n-1 n-1 Y (12) The subscripts n and. n-1 denote values corresponding to the time t and to the iirarediatel^/ preceding time t - At, respectively. The first step in using the step-by-step integration involves the insertion of values for the constant accelera- tions and derivatives 5Lc, SK^ , L , L^,, ¥ , and Np in colur.ns (3), (11), ^20), (21), (2k) > 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 banV: ^, the rate of yaw r, and the angle of sideslip p should be inserted in columns (5)> (8)> (15), and (18) for t = 0. From curves such as figure 2, the values of ?^u, PL^, and pWg should be determined for the value of p at t ~ ( 6 = in the present case). These values should be inserted in columns ( II4.) , (19) > and (25) for t = 0. MO A ARR No. L5E02 11 3olun-ais (9), (10), (15), a6), (20) to (2?), andf2k) to (26) nay now be Ijlled in lor t = 0. Colujai (22) pro- vides the incuced rolling acoelerations ; colujrji (26) pro- vides the induced yav.'ing eccelerations. The net instan- taneous rolling and yawing accelerations may now be determined for t = by performing, the comoutatlons indicated in colunins (3) and (11). ^o' 3y repeating the procedure indicated in the heading of table I and by using the sample values obtained for t = 0, the values of p, ^, r, and p are obtained for t ~ 0.1 second. After the value of p for t = 0..1 second has been obtained, corresponding values of pYpj PL,^, and pNp, are determined and inserted in columns {lk)\ (1.9), and (25) for t = 0.1 second. The net induced accelerations EL^ and SNj^ for t = 0.1 second may nov/ be determined (columns {22} and (26)) and, as a result, the values in columns (3) and (11) Tiay be deter- mined for t = 0.1 second. The remainder of table T for the other values of t m.ay now be solved by repeating the procedure indicated in the headings and by using curves sin)ilar to figure 2. The angle of banV" was determined by averaging the rate of roll p (columns (5) to (?)). This averaging was not followed through 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- duced by a disregard of those averages are small and are within the accuracy of the data used for the calcula- tions in the aopendix. The step-by-step inte. ration presented herein is not limdted to the solution of motions prodviced by ailerons. Such integration may just as readily be applied to the solution of disturbed motions produced by rudder movements or by a combination of rudder and^ aileron movem.ent. For the case of lateral disturbances caused by rudder alone, Sg^Lg and GaNg^ would be changed to SpLg and S^Nq • '■.■hen the step-by-step integration is applied with variable derivatives to flight conditions involving accel- erations greater than 1 g, the value of the airplane speed used should be the true airspeed V. The acceleration, however, imist be consic^ered in determining the airplane lift coefficient. The values of C-n and C7 ^a t-a (if an aileron m.ovement is concerned) and the derivatives correspond to the new lift coefficient. 1?. FACA ARR :io. L5EC2 COIwPARIsOK OF FROCFlDlRJ;^^ FOR COMPTJTING LATERAL DIST^Jl-fRANCES "'he characteristic curves obtainecl by the ,step-by- step integration are com'oared in figures 5 to 6 with the resaltp obtained froin actuar:. flight tests; with the method of differencial operutors (reference 2), v/hich is an e:;'act solntion cealin^r with constant slopes; and with pn approxlnate analytical solution in which constant plooes are also used f reference I;) and m^hich is applicable only to the solution of the sideslip angle. IVhon the maxin-iori. sideslip anrrle was coTipi-ited by the approximate method of reference S, the coTiputed value was found to be y? .^"^ , which does not compare with the l6^ detemined frora flirrht tests. When the value of 0-^ v.'as considered equal to C + ^-, the cornpv-ted value of the •nazimum a ■^■'p'^^'' sideslip angle was determined to be li.9.2°, which is still rather higlj.. The present procedure pi-ovides the most accurate correlation with flight test results for all the motions considered. It should be noted that the refinement used in the preser.t report for the determination of j\fp was not used in the application of the methods of I'eferences 2 and Ij.. It shou.ld also be noted that v/v, which is considered equal to the value of p in radians in all the procedures, is in its strictest sense equal to tan p. The assumption that p = — loads to much lar.'^or errors for large values than I'or email values of p. For example, consideration of these two sources of error reduces the maximum side- slip angle of 9?-°, shown fo"-' the approximate procedure of reference L, to a value of '^6'^. The improved method In considering F accounted for '^'^ , whereas the other 27° were accounted for by the fact that v/v was considered equal to tan 6. In the case of the ctep-b^-step pro- cedure of the present report, the rcaxim.um sideslip angle v;ould have been equal to about 2^° if N^ had been deter- mined by the method of reference 6. If v/v had been considered equal to tan n, thu maximum sideslip angle by tri"_ stop-by- 5--;tep method would havo been reduced about ■?• • k IfAGA ARR IIo. L5S02 13 Per solxjtloiip involving;; tho assumption of linear slopoG, the slopes used in the ;oros'jnt problem v;ere arbi- trarily mcasTjred through \\t = 0*-', If the raore usual practice of selecting the avera.^e slopes over a -.vider range of yav; angles had been employed, the calculated results would have approached nore closely the results of t •'' variable-slope method. ?cr casts in v/hich vertical tail loads in hir^h- speed dives are of primary concern, however, snail angles of sideslip --nay be critical, and consideration of average slopes over a wide range of yavj angles maybe unv/isc. It appears thurefore that, although the prcviou.s proccdiores raay be reasonably reliable in a number of instances in which the characteristic Cj , C^, and r!y ci-ixves poss.'.-ss approximately lintjar relt.tion- ships, nonlinear characteristics occur with sufficient frequency to make the general use of the nonlinear stop- by-step procedure desirable. In order to determine '".he iT.portance of the lateral- acceleration term BY^, the present procedure was repeated with p.Y^ = t?. Although the resulting curves Indicate that the influence of pY^_, for trie subject airplane was not very large, the effect of pYp, may be more significant for other types of airplane and there- fore should not be neglected. A coinparison of the step-by-step solution using constant slopes with the method of differential operators (reference 2) indicated that values obtained by the step- by-step solution tended to deviate a little more from flight test results than the values obtained by the opera- tional method. The step-by-step solution, for this particular comparison, apparently gives a sideslip ang].e appro::imately 2'^ larger than the operational procedure of reference 2. The tendency of 'che step-by-step solution in the linear case to deviate a little more from flight tests than a direct integration procedure may reasonably be presumed to persist in the application of the step-by- step solution to the nonlinear case, as in the present report. Further refinement of the step-by-step procedure may therefore be expected to provide correspondingly closer agreem.ent with flight. The Rvinge-Kutta summation method (references and 9) provides such refinements of procedures. The sten-by-step procediuT'O as outlined in the present report, n.ovever , is believed to provide sufficient engineering accu'acyy v/hen no more than the f iT:-s eu?.Pter-Gycle of ohe mo :; ion j.s re-.uired. lij. FACA ARR IIo. L5E02 Although 5L5 and 5N5 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 L.., N„, L^, and Np 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 dynamic -model tests are available, hov/- ever, these values must be presumed constant for lack of more complete information. Other possible sources of discrepancy between calculations and flight results are the assumptions of level flight, constant normal accel- eration, constant speed, and instantaneous control deflection. For practical purposes, however, it was not believed necessary to take these factors into account. CONCLIJSIO'mS A procedure based upon step-by-step integration is presented for determining the disturbed motions of an airplane resulting from the deflection of the lateral or directional controls for the case of nonlinear derivatives. A comparison of the stop-by-step procedure with other methods indicated the following conclusions; 1. Tlie calculated disturbed motions of an airplane resulting from abrupt control movement v;ill be in better agreement v/ith the results obtained from flight tests If the variation of the experimentally determ.ined rolling, yawing, and sideslipping accelerations (3Lg, pNn, and pYj3 \7ith the angle of sideslip l3 is considered. The sideslipping acceleration t3Y;-.,, which is often assum.ed negligible, should be considered. The variation of the rolling and yawing accelerations Qa^e ^^^'^ ^a^s resulting from aileron movement probably should also be considered when sufficient data are available. The variation of the dynamic derivatives Lp, Np, Lj,, and F^ should also be taken into account when sufficient dynamic-test data are available. NACA ARR Wo. L5302 15 2. Tae value of the maximum sideslip axigle for use in the deterrtilnaoion of the vertical-tail loads in rolling pull-out maneuvers should be obtained by using the step- bj-step integration method. 5. The step-by-steji intO£-ration may be applied to the solution of motions produced by rudder movem.ents or by a combination rudder and aileron movement, as well as to the solution of motions produced Dy ailerons alone v;hen only the first' quarter-cycle of the motion is desired Langley Ivlemcrial Aeroriautical Laboratory National Advisory Comi.iitteo for Aeronautics Langley Field, Va. l6 KiCA .iRR No. L5E02 APPENDIX DETER!'TFATTON OF DYNAMIC LATERAL MOTIONS OF A FIGHTER ATPJ>LANE DUE TO ABRUPT AILERON MOVE^fENT Data r equired . - For the fighter airDlane used in the Illustration, the data required for the determination of dyna-'rj.c lateral motions resulting from abrupt aileron movei^ent are as follows: 'do,. 'bf, percent b 66 LC(^ b, ft [l2. 83 Cd. 0.01 ^f ^ 0.50 Ci^ 0.01+ A 5.5 C^ -0.0065 I, ft 20.5 On. 5v, .... 0.00ll;7l; 5a^, deg 12-75 V, fps lL2.2 5a^, deg -17 q, Ib/sq ft . . . 2i^.09 5f, deg IT., slugs 558 ^a^ deg 17.1 S, sq ft 35]+ Lw "•'^" "X f Cl_, I.i42 k^^, sq ft . . . . 33.52 ^Ct k^^, sq ft 57.9 Landing gear Retracted The value of 5^ = -2.0 is determined from, reference 10. ^a, Procedure . - From reference 3> determine % - -0.0655 Cj^ = 0.308 FACA Ar.r; No. L5r^02 17 Then, from reference 6, deterrrane K - -0.9 5 :: " ° 2 + 2\ ■0.27U9 Frorr reference 7> deterir.ln^ K, - -0.0202 Compute the following: _ b qSh 5pl a=^5j _qSb_ = -I.85I+ l.llj.96 5«N, s^^S, aSh = -0.161 = -0.103 T rr r b qSb - 1.5$ Nr = Of qPb _b_ 1 2 2V = -0.3108 l8 NACA ARR ITo. L5S02 From wind-tunnel data for the configuration con- sidersd ffig. 1), ^:)lot the following against p or -'4^: qSh -z P J. n - u V ~~ h ■' III The values of Fo, Fo , and Ft heve not been solved for since they are used for flaps-deflected conditions and the airiolane used in the present report vas in the cruising configuration. Tor flaps-deflected conditions, Kf may te deter:nined frorr: the following formula from referonce 6; '^-f The values of Fp reference 7- -0.5 ^f k - 3—(i - K 2 + 2\ ?.nd F, m.ay be determined from Althou^'-h tho values of n„ and Hi in the present % ''r report have been dotormlnod solely from the ciir-ves of rofwrence 3> it may bo desir.^ble in some cases to incl\ide the effects of the vortical tail by use of tho method of reforonco 11. FAG A ARR No. L5E02 I9 REFERENCES 1. Weick, Fred E. , and Jones, Robert T.: The Effect of Lateral Controls in Producing Ivlotion of an Airplane as Coirputed from wind- Tunnel Data. IJACA Rep. No. 57O > 1956. 2. Jones, Robert T.: A Simplified A^opll cation of the Method of Operators to the Calculation of Disturbed Motions of an Airplane. NACA Rep. No. 56O, I956. 5. Pearson, Henry A., and Jones, Fobert T. : Theoretical Stability and Control Characteristics of ''.'ings with Various Amounts of Taoer and Twist. NACA Rep. No. 655, 1953. I4.. Kayten, Gerald G. : Analysis of Wind-Tunnal Stability and Control Tests in.T'^i'ms of 1 lying 3,ualities of Full-scale Airplanes. NACA A"^P No. 5J22, 19i4.5 . 5. Gllruth, Robert ^. : : Analysis of Yertical-Teil Loads in Rolling Pull-Out Maneuvers. NACA CB No. l1|H11^., I9UI.'.. 6. Campbell, John P., and Mathews, Tard C: Experimental Determination of the Yawing iVloment Due to Yawing Contributed by the V-ingp Fuselage, and Vertical Tail of a Midwing Airplane ivlodel. NACA ARR No. 3F28, 191+5. 7. Harmon, Sidney M. : Determination of the Damping Moment in Yawing for Taoered V.'ings with Parti al-Spsn Flaps. NACA ARR Ko. 3H2 5, 19i|5. 8. Levy, rl. , and Baggott, E. A.: Numerical Studies in Differential Equations. Vol.1, v.'atts & Co. (London), 193i+. 9. Scarborough, James 3.: Numerical Mathematical Analysis. The Johns Hopkins Press (Baltimore), 1930' 10. Ames, Milton B. , Jr., and Sears, Richard I.: Determina- tion of Control-Surface Characteristics from NACA Plain-FlaD and Tab Data. NACA Rep. No. 721, 19i|.l. 11. BaiTiber, Millard, J.: Effect of Some Present-Day Airplane Design Trends ^'U Reciuireruents for Lateral Stability. NACA TN No. 8li4., 19i|l. NACA ARR No. L5E02 20 O ^ a 5 t-l i Ir. C > < ii Q 1 I r O Q 8 ^* C n B ■> o Q "2 1 s o I o 3 5 N O i ■a • + < 5 o u o 1 1 5 1 1 I ■o c ■0|t3 X 00 o 1 5 i iS ♦ + N i" 5 1 5 r 1 ■ o tOI> o o ■ SI * 10 » * ^ ON G c c> ft 1 8 3 s s 1 o « ^ M ^ 03 S * + o * 1 * N ^ 5 h 1 t- ^ < X 1 1 1 1 5 1 vO la 1>I + ITS (M 5 ■0 On > 10 te\ o • > « 1 c ft ♦ *- ^ ^ ^ 5; s n 5 r4 o 5 o o ^ 5 Q § r-t 4 4 CO X iTN o ^ ^ N c — C •) X c> ^ 5 N OJ a * c Q. II =f Ul + "+ »fN o 1 0\ i "3 •o ! 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