1、NATIONALADVISORYCOMMITTEEFOR AERONAUTICSTECHNICAL NOTE 3754METHOD FOR CALCULATING THE CHARACTERISTICSADUTCH ROLL MOTION OF AN AIRPLANEBy Bernard B. K1.swansLangley Aeronautical LaboratoryLangley Field, Va.WashingtonOctober 1956ratic characteristic equation.Provided by IHSNot for ResaleNo reproductio
2、n or networking permitted without license from IHS-,-,-Computationalcussed. Numericaltive airplanes arebCLaclClr =Cn)P “- Cn Kxs2 + C2pKxz - =P + (9)that before the addition the rolling-momentequationK= and the yawing-mcnnentequation was multiplied byIt shouldbe notedwas multiplied byKxs2. This proc
3、edure simplified considerably the resulting expressionbecause it eliminated the complex parsmeter /$ as a factor of thevalues are not in close agree-ment - that of numerical error and that of failure to converge.Infrequently, convergence does not occw (e.g., static directionallyunstable airplanes an
4、d light airplanes that have extreme dsmping in yaw),and the conventionalmethod must be used to evaluate the variables.The equations of the iterativemethod have been derived from theconventionalequations of motion without any additional.assumptions. Asa result, it is impossible for the iterativemetho
5、d to converge onanswers different from those calculatedby the conventionalmethod.Uses for the method.- The proposed iterativemethod is designed tocircumvent the necessity for the solution of the biquadratic characteris-tic equation. The biquadratic is in effect replacedby a quadratic withcomplex coe
6、fficients (eq. (9) that may be easily iterated. Thus, theiterativemethod is somewhat simpler mathematically snd requires onlyabout one-third the computational time of the conventionalmethod. Inaddition, evaluation of the ratios of roll to yaw and sideslip to yawwhich are necessary to describe adequa
7、tely the Dutch roll is inherentwith the iterative method.Equation (9) may be rewritten as1 (2P Kzs2Kxs2 - KW2)( )Cn KY(S2+ CzPKn $p. =0I-1(I-2)(2p Kzs2s2 - 2%)Equation (12) is an extremely interesting equation in that it contains allthe terms in a thee-degree-of-freedom system that affect the Dutch
8、roll,expressed as a quadratic in the operatorof the important stability derivatives onseen more clearly from equation (12) thanistic equation.D. Primary effects of variationthe period and damping can befrom the fourth-order character-Provided by IHSNot for ResaleNo reproduction or networking permitt
9、ed without license from IHS-,-,-10REPRESENTATIVESOLUTIONSNACA TN 3754Table 11 presents the characteristicsof four representativeair-planes. These airplaneswere exsmined by means of the simple iterativemethod to determine their Dutch roll characteristics.Example solutionsby the iterativemethod are gi
10、ven in table 111.A conventionalmodern bomber, an extreme-altitudefighter, and a sonicinterceptor represent typical solutions for present and proposed air-planes, and a hypothetical delta-wing light airplane in the landing con-dition represents an atypical solution in which the iterativemethoddoes no
11、t converge. The high rate of convergenceof the typicalsolutions of the Dutch roll characteristicsis apparent. The atypicalcase illustrates the smple warning (lack of convergence)that is presentwhen the iterative solution should not be used.CONCLUSIONSA simple method for calculating the characteristi
12、csof the Dutchroll motion of an airplsne has been obtainedby arranging the lateral .equations of motion in such form and order that an iterative process isquickly convergent. The iterativemethod is believed to be particularlyuseful when no extensive cumputing facilities are available, although it vm
13、ay be used to reduce co?rputationaltime on any te of digital cmnputingequipment. Experience gained from the calculation of the Dutch rollcharacteristicsof more than 70 airplanes by the iterativemethod ascompared with the conventionalmethod has indicated that1. About one-third the computationaltime i
14、s required for theiterativemethod as was required for the conventionalmethod.2. The arithmeticalprocesses are shpler in the iterativemethodbecause the need for the solution of a bi.quadraticequation is avoided.3. In the iterativemethod the arithmeticalprocesses are iterativewhich permit running nume
15、rical checks.4. Evaluationof the ratios of roll to yaw and sideslip to yaw isinherent in the iterative process.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 37545. Primary effects of aerodynamic derivatives on the Dutch rollroots can be mor
16、e readily seen by use of the iterativemethod.Langley Aeronautical Laboratory,National Advisory Committee for Aeronautics,Langley Field, Vs., June 8, 1956.11Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-:, 12., NACA TN 3754REFERENCES .1. Jones, Robe
17、rt T.: A SimplifiedAlication of the Method of Operatorsto the Calculation of DisturbedMotions of an Airplane. NACA .Rep. 560, 1936.2. Phillips, Willism H.: Appreciation and Prediction of Flying Qualities.NACA Rep. 927, lgkg. (SupersedesNACA TN 1670.)3. Zimmerman, Charles H.: An Analysis of Lateral S
18、tability in Power-OffFlight With Charts for Use in Design. NACA Rep. 589, 1937.4. Stertiield, Leonard end Gates, Ordway B., Jr.: A SimplifiedMethodfor the Determination end Analysis of the Neutral-Lateral-Oscillatory-Stability Boundary. NACA Rep. 943, 1949. (SupersedesNACA TN 1727.)j.Mokrzycki, G. A
19、.: Application of the Laplace Transformationto theSolution of the Lateral and Longitudinal StabilityEqpatione. NACATN2002, 1950.6. Campbell, JohnP., and McKinney, Marion O.: Summary of Methods forCalculating Dynemic Lateral Stability and Response and for Estimating -Lateral Stability Derivatives. NA
20、CA Rep. 1098, 1952. (SupersedesXACATN 2409.) .7. Liddell, Charles J., Jr.j Creer, Brent Y., and Van Dyke,Rudolph D., Jr.: A Flight Study of Requirements for SatisfactoryLateral Oscillatory Characteristicsof Fighter Aircraft. NACARMA51E16, 1951.8. Sternfield, L.: A Vector Method Approach to the Analy
21、sis of theDynamic Lateral Stability of Aircraft. Jour. Aero. Sci., vol. 21!no. 4, Apr. 1954, pp. 251-256.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3754 13TM13LEIRANGE OF AIRPIANE VARIABLES CONSIDERED FOR WHICHSATISFACTORY SOLUTIONSWERE
22、OBTAINEDBY MEANS OF ITEIUTIVE METHODPrimary variable Numerical range%s2 . O. 0.2 to 1.0KZ2P“” 12 to 275q, deg -5 to 5c -1.2 to 3.6r*”*”C%*”* -0.70 to 0.90Czr . . . . . . . -2.90 tO 3.10cl . . . . a71 . . -7.30 to 0.20PCn . . . . . . .P 0.03tO 0.25cl , a71 . . a71 0 a71 -0.22 to oP.4Provided by IHSNo
23、t for ResaleNo reproduction or networking permitted without license from IHS-,-,-14TABLE 11CHARACTERISTICSOF IIEI?KESENTATIVEAIRPLANES USED IN CALCULATIONSNACA TN 3734Kx2 . . . . . . .Kz2 . . . . . . .KXZ . . . . . . .h, ftv, ft/sec , a71 , .b, ft . . . . .Cy . . a71 . . . .Pc* . . . . . . .PCyr , .
24、 . , , . .C2P , . , . . . .cp . . . . a71 . a71cl . . . . . . .rC*P . . . . . . .c%”c%”I AirplaneBmiber0.03110.072031.830.44335,000700116-0.6100-0.14-0,440 a71 1490.12-0.0276-0.156Extreme-altitmdefighter0.01560.1560.00201820.4950,000776-o .;:o0-0.18-0,330.230.25-0,050-0.069Sonicinterceptor0.0100.065
25、-0.00465.40.32332,00069136.6-0.2800-0.08-0.150.400.028-0.20-0.40otheticaldelta-wingairplane inlandingcondition0.0300.0679-0.03011.851.099.:38.1-0.28600-0.0373-0,020.600.0573-oa71 20-1.10.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3754 15
26、TABLE IIIREPRESENTATIVESOLUTIONSAssumptions AiqpleneConventionalbomber/CnD=i . o.1620i2&2w PI* DIteratedvalms-1.755 - 1.253i -1.053+ o.o1317i -0.00491+ o.1679i-I.840-1.235i -1.053+ o.o177i -0.00441+ o.1679i-1.825-1.2k3i -1.053+ o.01692i -0.00452+ o.1679i, Exact values-1.825- 1.242i -1.053+ ().0162i
27、-0.00447+ o.1679iExtreme-altitudefighterWY I NV D/CnD=i = o.0664i2Czs2Iteratedvalues-6.23 - 2.95i -1.c62+ o.mo%i 0.00271+o.0665i-6.7-2.84i-.064+ O.Oi-6.17-2.85i 0.00256+ o.0665i-1.064+o.097oi0.00257+ o.0665iExact Vd.U.eS-6.17-2.85i I -1.064+ o.0970i 0.00258+ o.0665iI Sonic interceptor II I m I D II
28、Iteratedvalues IF -2.58-2.27i -1.102+ 0.07001 -0.01393+ 0.11981-4.41-2.54iD = L = 0.0574i -1.045+o.0775i -0.00852+ o.1N32i-4.33-2.46iWZS2 -1.047+ o.0749i -0.00886+ 0.u.8MExact valuesI -4”33-2”46iI -1.047+0.075M l-o.00885+0.1181iIothetical delta-winglightairplanein landingconditionI W4 I PI* I D1 IIt
29、eratedvaluesr-L.724+ 0.2253. -0.970+ o.323iCn -o.H345+ o.u23i =o.1887i -1.411+ o.231i -0.78L+o.0862i -0.1498+0.0286iD=i -1.039+0.234i -0.764-o.0227i -o.1389+ o.000617iW7S2Exact values-1.722+ o.589i -0.451+o.385i -0.0647+ 0.0822LProvided by IHSNot for ResaleNo reproduction or networking permitted wit
30、hout license from IHS-,-,-16 NACA TN 3754Principal axis 17x+a71Wind directionWind direction Y*Azimuth referenceFigure l.- Stability axes system employed with positive direction offorces, moments, end displacements shown.,.*NACA -Laiwley Field, VA.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
