1、/ A/44_/- TR_ _E9NATIONAL ADVISORY COMMITTEEFOR AERONAUTICS. . ; .REPORT 40,:589I( %AN ANALYSIS OF LATERA_L STABILITY INPOWER-OFF FLIGHT WiTH CHARTSFOR USE IN DESIGNBy CHARLES H. ZIMMERMANLangley Memorial Aeronautical Laboratory131079-37 1_REPRODUCEDBYNATIONAL TECHNICALINFORMATION SERVICEU.S. DEPART
2、MENTOF COMMERCESPRINGFIELD,VA. 22161_J .J/iProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSHEADQUARTERS, NAVY BUILDING, WASHINGTON. D. C.LABORATORIES, LANGLEY FIELD, VA.“Created by _v_:_.Congres_ved March 3,
3、 1915_f_t_e superv_0n _nd direction 0f_he scientific study o(the vroblems _f_g_!i _:_a_l_e_p was increasecl; _929. _ Ti_e _ber_ _appointed by the President, and serve as such without compensatmn.JOSEPH S. AMES, Ph.D., Chairman,President, Johns Hopkins University, Baltimore, Md.DAVID W. TAYLOR, D. En
4、g., Vice Chairman.Washington, D. C.CHARLES Go ABBOT, Sc. D.,Secretary, Smithsonian Institution.LYMAN J. BRIGGS, Ph.D., k Director, National Bureau of Standards_BENJAMXN D. FOULOIS, Major General, United States Army,Chief of Air Corps, War Department. ,_WILLIS RAY GREGG) B. A, /Chief, United States W
5、eather Bureau.HARRY.F. GUGGENHEIM, M. A.,Port Washington, Long Island, N.Y. /ERNEST _; KING, Rear Admiral; United States Nav_,Chief, Bureau of Aeronautics, Navy Department.(AERODYNAIMICSPOWER PLANTS FOR AIRCRAFTAIRCRAFT STRUCTURES ANP MATERIALSCHARLES A. LINDBERGH, LL. D., _New York City. _ WILLIAM
6、P. MAcCRACKEN, Jr., Ph. B., _.!/:_Washington, D. C.AUGUSTINE W. ROBINS) Brig. Gen., United States Army,Chief, Mat6riel Division, Air Corps, Wright Field, Dayton,Ohio. ,_ -EUGENE L. VIDAL, C. E.,Director of Air Commerce, Department of Commerce.EDWARD P. WARNER, M.S., ,“_.,o,:Editor of Aviation, New Y
7、ork City. _ _:_R.D. WEYERRACHER, Commander, United States Na_)Bureau of Aeronautics, Navy Department. _ _,_:,ORVILLE WRIGHT, SC. D., _ _“:Dayton, Ohio; _. -Coordination of Research Needs of .Military and Civil Aviation!Preparation of Research ProgramsAllocation of ProblemsPrevention of DuplicationCo
8、nsideration of Inventions._J._;_.LANGLEY MEMORIAL AERONAUTIC_.L LABORATORYIIOFFICE OF AERONAUTICAL INTELLIGENCELANGLEY FIELD, VA. WASHINGTON, D. C.Unified conduct, for all agendes, of Collection, classification, compilation,scientific research on the funda:_ental _ .,._: and dissemination of scienti
9、fic and tech-problems of flight. _- “_ _ : _ ;:-_ _:i_ :_-:_ _, ./.mcal mformatmn on aeronautms.GEORGE W. LEWIS, Director of Aeronautical ResearchJo_ F. VICTORY, _eeretaryHENRY J. E. REID, Engineer in Charge, Langley Memorial Aeronautical Laboratory, Langley Field, Va.JOHN-J. Iw, Technical Assistant
10、 in Europe, Paris, France.,. . TECHNICAL COMMITTEESAIRCRAFT ACCIDENTSINVENTIONS AND DESIGNSProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N 0 T I C EAS
11、THIS DOCUMENT HAS BEENBEST COPY FURNISHED USAGENCY. ALTHOUGH IT ISTAIN PORTIONS ARE ILLEGIBLE, IT ISLEASED IN THE INTEREST OF MAKINGMUCH INFORMATION AS POSSIBLE.REPRODUCED FROM THEBY THE SPONSORINGRECOGNIZED THAT CER-BE ING RE-AVAILABLEProvided by IHSNot for ResaleNo reproduction or networking permi
12、tted without license from IHS-,-,-REPORT No. 589AN ANALYSIS OF LATERAL STABILITY IN POWER-OFF FLIGHTWITH CHARTS FOR USE IN DESIGNBy CHARLES H. ZIMMERMANSUMMARYThe aerodynamic and mass .factors governing lateralstability are discussed and formulas are given.for .theirestimation. Relatively .simple re
13、lationships between thegoverning factors and the resulting, stability characteristicsare presented. A series oJ charts is included with whichapproximate stability characteristics may be rapidly.estimated.The effects of the various governing Jactors upon the Istability characteristics are discussed i
14、n detail. It ispointed out that much additional research is necessary bothto correlate stability characteristics with riding, flying, andhandling, qualities and to provide suitable data.for accurateestimates o.f those characteristics of an airplane while it isin the design stage.INTRODUCTIONThe late
15、ral stability of airplanes has been the subjectof considerable mathematical treatment and manytheoretical analyses. (See references.) The main as-pects of the problem are therefore well known to stu-dents of the subject. Use of the mathematical theoryin design is, however, limited by practical diffi
16、culties inits application. Determination of numerical values forcertain of the aerodynamic quantities is difficult andthe results are uncertain. The required calculationsare extensive and must be carefully made to avoid erro-neous and confusing results.In this report lateral stability will be discus
17、sed andanalyzed in a way that, it is believed, _4ll aid in theacquisition of a working knowledge of the subject with-out long and intensive study. The classical equationshave been simplified as much as seems consistent withreasonable accuracy to permit rapid estimation of thestability characteristic
18、s. Also included is a series ofcharts designed to facilitate the rapid estimation of theapproximate lateral-stability characteristics of airplanesthroughout the normal-flight range. It is hoped thatthese charts, together with those on longitudinal sta-bility presented in reference 1, will aid in put
19、ting theestimation of the complete stability characteristics ona practical basis.The material is presented in the following order: (1)A discussion of the aerodynamic and mass factors thatgovern the uncontrolled motion of the airplane togetherwith formulas for estimating these factors; (2) formulasfo
20、r estimating the stability characteristics of the uncon-trolled motion having given the governing factors; (3)charts for the rapid estimation of stability character-istics; (4) a discussion of the effects of the governingfactors upon the stability characteristics; (5) commentsand suggestions for fut
21、ure s.tudy; (6) a brief derivationof .the classica_ stability formulas (appendix I); (7) anaccurate semigraphical method for solving biquadraticswith a useful approximation based on this method(appendix II); and (8) a list of symbols and theirdefinitions (appendix III).FACTORS GOVERNING STABILITY Bo
22、th theory and experiment indicate that, withcertain exceptions, the uncontrolled motion of an air-plane can be divided into two independent phases.One_phase includes components of the motion that donot displace the plane of symmetry of the airplanefrom the plane with which it coincides during the st
23、eadymotion. Stability of this part of the motion is termed“longitudinal stability.“ The other phase of the com-plete motion includes all components that do displacethe plane of symmetry. This phase of the motion iscalled “lateral motion“ and its stability characteristics,“lateral stability.“ Althoug
24、h, in the past, referencehas frequently been made to directional stability asdistinguished from rolling stability (also called “lateral“stability), both theory and experiment indicate thatno such division is physically possible for the conven-tional airplane.The uncontrolled motion of an airplane qu
25、ite ob-viously depends upon the aerodynamic forces andmoments arising from any deviation from a steadystate together with the inertial forces and momentsaccompanying the accelerations coupled with thedeviations. The lateral motion is zero in steadyflight on a straight course. The components of later
26、almotion in unsteady flight are a linear Velocity v alongthe Y axis (see appendix I and fig. 1) and angularvelocities p and r about the X and Z axes, respectively.The forces and moments governing lateral motiontherefore arise from the aerodynamic reactions to the1Provided by IHSNot for ResaleNo repr
27、oduction or networking permitted without license from IHS-,-,-2 J REPORT NO. 589-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSvelocities v, p, and r (in the theoretical treatment aero-dynamic reactions are assumed to be unaffected byaccelerations) and the inertial reactions to the accelera-tions dr dr
28、; g sin cos 7, g sin _bsin 7, dp/dt, and dr/drwhere _ is the angle of roll, 7 is the angle of the flightpath, and _bis the angle of yaw.For convenience the components of the reactionsreferred to the coordinate axes are used rather thanthe resultant reaction. It appears, then, that a velocityv should
29、 result in a side force AY, a rolling moment AZand a yawing moment AN. Similarly there will beAs of 17, Z, and N corresponding to the rolling andyawing velocities p and r. The basis for the classicaltheory of stability is that the algebraic sum of thevalues of AY (for example) for a unit value of v
30、when pand r are zero, for a unit value of p when v and r arezero, and for a unit value of r when v and p are zero isequal to the value of AY when the total motion is theresultant of coexisting unit values of v, p, and r. It isfurther assumed that a reaction AY due to a disturb-ir_oF1OUR_ L-Angular a
31、nd vertical relationships in flight, power off.ance of velocity v is directly proportional to the magni-AY dYtude of v, that is =v_-_. This assumption is ad-mittedly an approximation but is valid, in general, forsmall values of the velocities of the disturbance. Onthis basis the aerodynamic reaction
32、 AY to a lateraldisturbance is- dY dY dYand similar expressions exist for AL and AN.As a matter of convenience it has been found desir-able to express the derivatives dY/dv, dL/dv, etc., interms of the nondimensional coefficients Cr, C_, and G,wherey1 2_pV SZlp V_SbNCn-_ 1_P V2Sb .In order to make t
33、he treatment entirely nondimensional,it is convenient to consider the ratios v/V, pb/2V, andrb/2V rather than v, p, and r. For small values v/Vis equal to 8, where t3is the angle of sideslip (in radians),and pb/2V is the difference (in radians) between theangle of attack at the center of gravity and
34、 the angleof attack at the wing tip. Since the velocity at thewing tip is V+rb/2 the value rb/2V is the ratio of theportion of the velocity at the tip due to rotation to thevelocityat the center of gravity. Expressed in thisway, the lateral-force coefficient due to lateral motion isACt= fldd_V-_ pb
35、dCr rb dCr 2Vd2P_ _2Vd-_vand similar expressions exist for ACt and AG,.rbSince dOr/d2T_bV and dGr/d_-_are small, they are gen-erally neglected, leaving the following aerodynamicfactors to be considered:1. Those depending on sideslip: dGr/dfl, dCddt3, anddG./d#.2. Those depending on rolling velocity:
36、 dCdd andrb3. Those depending on yawing velocity: dCdd,z- _ anddO. “ rb,/au_In addition to the aerodynamic factors, others thatdepend on the amount and the distribution of the massof the airplane must be considered. The importantmass factors, expressed nondimensionally, are _, b/kx,and b/kz. The rel
37、ative density factor g is equal tom/pSb and may be considered as being proportional tothe ratio of the mass of the airplane to the mass of airinfluenced by it in traveling one chord length. Under13.1 (W/S)standard conditions _- bAERODYNAMIC FACTORSLateral force due to sideslip.-The rate of change of
38、lateral-force coefficient with angle of sideslip dCr/d3 Canbe accurately determined only by measurement in awind tunnel. Assuming the wind-tunnel data to havebeen obtained in terms of angle of yaw _ in degrees, thevalue of dCr/dfl is -57.3 (dCr/d_b), since t3isin radiansand opposite in sign to _b. I
39、n wind-tunnel practice,cross-wind force rather than lateral force is usuallymeasured. In such cases dCr/d_ can be determinedfrom the relationshipdot dO. _CProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.AN ANALYSIS OF LATERAL STABILITY IN POWER-OFF
40、FLIGHT 3(which follows from the fact that Cr=G cos /9-CD sin /3).Diehll gives (reference 2, pp. 254-255) an approxa-mate, empirical value ofdG bll= =0.12 N (2)where 11 is the over-all length. This formula is usefttlwhen wind-tunnel data are not available. -Rolling moment due to sideslip.-The rate of
41、 changeof rolling-moment coefficient with sideslip dOddB mustalso be measured in a wind tunnel if accurate values aredesired. Some systematic research has shown the effectof dihedral and tip shape on the value of dCdd/3 for thewing alone (reference 3) but very little is known aboutthe effect of fuse
42、lage interference. In certain experi-ments (data unpublished) a model having a wing withno dihedral mounted in a high-wing position gave avalue of dOddfl corresponding to 5o of positive dihedralfor the wing alone. The same model with the wingmounted in a low-wing position gave a very erratic.40.35.3
43、0.15./0.o5-.025-8IiI.i_ “ _O (12)Failure to meet the first of these conditions results in“spiral divergence,“ a form of divergence in which theFor purposes of approximate estimation using thevalues for the derivatives given in equation (10),equations (11) and (12) become4(dCdd_)(dC/d_v)_CL(dCn/d/3)
44、(13)and- CL(dCdd3) +3.2 (dCJd_) _0 (14)If the contributions of the wings, the fuselage, andrbinterference effects upon dC,/d_ and upon dC_/d_ areneglected, equation (13) further simplifies to-dCdd_ C_ b8 1 (15/This latter equation is, however, an oversimplificationfor any but the most approximate an
45、alyses.Formulas for estimating the damping of an oscilla-tion.-The number of seconds required for an oscillationto damp to one-half its original amplitude isT- - 0.313_/_ CLi- (16)where i“ is the damping coefficientThe time to dampto any other proportion of the original amplitude isgiven byT_ = T lo
46、g,n-0.693 (17)where n is the desired proportion, such as _l or _/. To afairly close approximation (=t= 15 percent)l_n,- l,np q_ _zn_(- n,)i-n,-lp) (-n,-lp) 2 2 221o n.-_-2n.l, I2 (_n _12,) z (18)z/ jIn equation (18) the terms in the first pair of bracketsare those which make _ more negative, i. e.,
47、decreasethe time required for the oscillation to damp; the termsin the second pair of brackets are those which make_ less negative.If the values from equations (10) are used, equation(18) becomes131079-37_2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-
48、,-,-8Since 8REPORT NO. 589-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS25.6 -_b“_“-0.07- 6.4+8( d_V)- .J-rb“+ 6.4+8 -_b“ 2+ / dC_ 6.4/dC,_dC_ (19)-_2 v/is small compared with 6.4 a furthersimplification is obtained by letting 8 a _, where_ _-Vdit appears in the denominator, equal 0.84. On this basisdC./ dO. _-= - 0.07- 3.5 _/d-_-_-_- 0.14 C,.2- 1.24_dVz_ dV, _ dO,dO, +0.3 tt-_-r _ _ _ (20)2(-_)_ _ _Formula (20) will lead to fairly large errors if the air-plane departs very far from the average. The error isroughly on a percentage basis so that/for small valuesof damping approachin
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