Revised September 1970

7 DYNAMIC DERIVATIVES%

Dynamic derivatives are associated with those aerodynamic forces and moments caused by the
angular velocities and linear accelerations of the vehicle motion. A detailed discussion of wing,
body, and wing-body dynamic derivatives is given in the introduction to each of the subject
sections. In most cases the methods presented are based on theories that necessarily assume
attached-flow conditions. They are thus limited to low angles of attack except for high-aspectratio wings below stall at subsonic speeds.

"Ingeneral, dynamic derivatives for attached-flow conditions are smaller and of less importancethan those for separated-flow conditions. For separated-flow conditions the derivatives become
functions of the amplitude and frequency of oscillation. Such conditions exist on low-aspectratio wings and bodies at all speeds and at all angles of attack except for a narrow range aroundzero.
Becauoc of the limited amount of data at high angles of attack and the complexity of deriving
adequate generalized methods, particularly for the frequency and amplitude variations, nomethods are presented that cover separated-flow conditions. Rather, a literature summary ispresented in table 7-A and a bibliography on pages 7-1 through -10.
REFERENCES
1.

Anon.: A General Investigation of Hypersonic Stability and Control Under Trimmed Flight Conditions. Flight Sci. Lab.
R-63-011-104, 1964. (U)2.

Bauer, R. C.: An Investigation of the Influence of Several Shape Parameters on the Dynamic Stability of Ballistic Re-entry
Configurations. Phase I. Effect of Afterbody Configuration at Mach Numbers 0.5 Through 1.5. AEDC-TN-59-78, 1969. (U)3.

Beam, B. H.: The Effects of Oscillation Amplitude and Frequency on the Experimental Damping in Pitch of a Triangular
Wing Having an Aspect Ratio of 4. NACA RM A52G07, 19•52. (U)4.

Beam, B. H., Rued. V. D., and Lopez, A. E.: Wind-Tunnel Measurements at Subsonic Speeds of the Static andDynamic-Rotary Stability Derivatives of a Triangular-Wing Airplane Model Having a Triangular Vertical Tail.
NACA RM A55A28, 1955. (U)5.

Bennett, C. V., and Johnson, J. L.: Experimental Determination of the Damping in Roll and Aileron Rolling Effectivenessof Threw Wings Having 20, 420, and 620 Sweepback. NACA TN 1278, 1947. (U)6.

Sielat, R. P., and Wiley, H. G.: Dynamic Longitudinal and Directional Stability Derivatives for a 450 Sweptback-Wing
Airplane Model at Transonic Speeds. NASA TM X-39, 1959. (U)7.

Bird, J. D., Jaquet, B. M., and Cowan, J. W,: Effect of Fuselage and Tail Surfaces on Low-Speed Yawing Characteristics ofa Swept-Wing Model as Determined in Curved-Flow Test Section of Langley Stability Tunnel. NACA RM LBG13, 1948. (U)8.

Bird, J. D., Jaquet, B. M.,and Cowan, J. W.: Effect of Fuselage and Tail Surfacet on Low-Speed Yawing Characteristics ofa Swept-Wing Model as Determined in Curved-Flow Test Section of Langley Stability Tunnel. NACA TN 2483, 1951. (U)9.

Bird, J. D., Lichtenstein, J. H., and Jaquet, B. M.: Investigation of the Influence of Fuselage and Tail Surfaces on
Low.Speed Static Stability and Rolling Characteristics of a Swept-Wing Model. NACA TN 2741, 1952. (U)10.

Bird, J. D.: Some Theoretical Low-Speed Span Loading Characteristics
NACA TN 1839, 1949. (U)of Swept Wings in Roll and Sideslip.

11.

Bird, J. D.: Some Theoretical Low-Speed Span Loading Characteristics
NACA TR 969, 1950. (U)of Swept Wings in Roll and Sideslip.

7

"7-1

12.

Bird, J. D., and Jaquet, B. M.; A Study e.: the Use of Experimental Stability Derivatives in the Calculation of the Lateral
Disturbed Motions of a Swept-Wing Airplane arid Comparison with Flight Results. NACA TN 2013, 1960. (U)13.

Bird, J. D,, Lichtenstein, J. H., and Jaquet, B. M.: Investigation of the Influence of Fuselage and Tail Surfaces on
Low-Speed Static Stability and Rolling Characteristics of a Swept-Wing Model. NACA RM L0H1-, 1947. (U)14.

Bland, W. M., Jr.: Effect of Fuselage Interference on the Damping in Roll of Delta Wings of Aspect Ratio 4 in the Mach
Number Range Between 0.6 and 1.6 as Determined with Rocket-Propelled Vehicles, NACA RM L52E13, 1952. (U)15.

Bland, VI. M., Jr., and Dietz, A. E.: Some Effects of Fuselage Interference, Wing Interference, and Sweepback on theDamping in Roll of Untapered Wings as Determined by Techniques Employing Rocket-Propelled Vehicles.
NACA RM L51 D25, 1951. (U)11U. Blend, W. MN.,Jr., and Sandahl, C. A.: A Technique Utilizing Rocket-Propelled Test Veh:cles for the Measurement of theDamping in Roll of Sting-Mounted Models a-snd Some Initial Results for Delta ond Up,-vept Tapered Wings.
NACA RM L50D24, 1950. (U)17.

Bond R., and Packard, 0. B.: Unsteady Aerodynamic Forces on a Slender Bodýy of Revolution in Supersonic Flow.
NASA TrN D-859, 1961. (U)18.

Bradley, W. G.: Measurement o; the Lateral Stability Derivatives of a Morane-SaEnier v,.S 760 'Paris' Aircraft G-APRU.
CoA Report Aero. 1856 1965. (U)19.

Brewer, J. D., and Fisher, L. R.: Effect of Taper r'atk on the Low-.:o.,x: Rolling St&bility Derivatives of Swept andUnswept Wings of Aspect Ratio2.61, NACATN 2555,:1Q1.(U)20.

Brown, C. E. and Heinke, H. S., Jr.: Preliminary Wind-Tunnnl *rests of 'rriangular and Pactangular Wings in Steady Roll at
Mach Numbers of 1.62 and 1.92. NACA RM LBL30, 1949. (U)21.

Brown, C. E., and Adams, M, C.: Damping in Pitch and Roll of Triangula.. Wings at Supirrsonic Speeds. NACA TR 892,
1948. (U)22.

Buell, D. A., Reed, V. D., and Loper, A. E.: The Static and Dynamic-Rotary Stability Derivatives at Subsonic Speeds ofan Airplane Model with an Unswept Wing and a High Horizontal Tail. NACA HM A56104, 1956, (U)23.

Campbell, J. P., and Woodling, C. H.: Calculated Effects of the Lateral Acceleration Derivatives on the Dynamic Lateral
Stability of a Delta-Wing Airplane. NACA RM L54K26, 1955. (U)24.

Campbell, J. P., and Mathews, W. 0.; Experimental Determination of the Yawing Moment Due to Yawing Contributed by
the Wing, Fuselage, and Vertical Tail of a Midwing Airplane Model. NAC:A L-387, 1943, (U)25.

D. E.: Low-Speed Study of the Effect of Frequency on the Stability
Campbell, J. P., Johnson, J. L. Jr., and Her'..,Derivatives of Wings Oscillating in Yaw with Particular Reference to High Angle-of-Attack Conditions. NACA RM L55H05,
1955. (U)26.

Campbell, J. P., and Goodman, A.: A Semiempirical Method for Estimating the Rolling Moment Due to Yawing of
Airplanes. NACA TN 1984, 1949. (U)27,

Chapman, R., Jr., and Morrow, JI, D.: Longitudinal Stability and Drag Characteristics at Mach Numbers from 0.70 to 1.37
of Rocket-Propelled Models Having a Modified Triangular Wing. NACA RM L52A31, 1952. (U)28.

Clay, J. T.: Nose Bluntness, Cone Angle, and Mach Number Effects on the Stability Derivatives of Slender Cones.
ARL 67-0185, 1967. (U)29.

Cole, I. J., and Margolis, K.: Lift and Pitching Moment at Supersonic Speeds Due to Constant Vertical Acceleration forThin Sweptback Tapered Wings with Streamwise Tips. Supersonic Leading and Trailing Edges. NACA TN 3196, 1954. (U)30.

Conlin, L. T., and Orlik-Ru'ckemann, K. J.: Comparison of Some Experimental andof a Delta-Wing body Configurationi at Supersonic Speeds. NRC LR-266, 1959. (U)31.

Cox, A. P., and Butler, S. F. J.: Low-Speed Wind-Tunnel Measurements of Damping in Yaw (nfi) on an A.R.9 Jet-Flap
Model. ARC CP 869. 1967. (U)32.

Curtiss, H. C., Jr.: Some Basic Considerations Regarding the Longitudinal Dynamics of Aircraft and Helicopters. Princeton
Univ. Report 562, 1961. (U)T

heoretical Data on Damping-in-Roll

7-2..............
L,.-

J.L.~...~____-

i-

33.

D'Aiusolo, C. T,: Low-Amplitude Damping-in-Pitch Characteristics of Tailless Delta-Wing Body Combinations at Mach
Numbers from 0.80 to 1.35 as Obtained with Rocket-Powered Models. NACA RM L54D29, 1964, (U)34.

D'Aiutolo, C. T., and Parker, R. N.: Preliminary Investigation of the Low-Amplitude Damping in Pitch of Taillebs Deltaand Swept-Wing Configurations at Mach Numbers from 0.7 to 1.35. NACA RM L52G09, i952. JU)

35.

D'Aiutolo, C. T., ana Mason, H. P.: Preliminary Results of the Flight Investigation Between Mach Numbers of 0.80 and
1.36 of a Rocket-Powered Model of a Supersonic Airplane Configuratiot Having a Tapered Wing with Circular-Arc Section
and 400 Sweepbeck. NACA RM 501H29a, 1960. (U)36,

Delaney, B. R., and Thompson, W. E.: Dynamic Stability Characteristics in Pitch and in Yaw for ; Model of a
Variable-Sweep Supersonic Transport Configuration at Mach Numbers of 2.40, 2.98, and 3.60, NASA TM X-761, 1963. (U)37.

Diatz, A. E., and Edmondson, J, L.: The Damping in Roll of Rocket-Powered Test Vehicles Having Rectangular Wings
with a NACA 65-006 and Symmetrical Double-Wedge Airfoil Sections of Aspect Ratio 4.5. NACA RM LSOB10, 1950. (U)38.

Dorrance. W. H.: Nonsteady Supersonic Flow About Pointed Bodies of Revolution, Jour. of Aero. Si., Vol. 18, No. 8,
1951. (U)

"K7-3

39.

Dubose, H. C.: Static and Dynamic Stability of Blunt Bodies. AGARD Rapt. 347, 1961. (U)40.

Edmondson, J. L., and Sanders, E. C., Jr.; A Free-Flight Technique for Measuring Damping in Roll by Use ofRocket-Powered Models and Some Initial Results for Rectangular Wings. NACA RM L9101, 1949. (U)41.

Emerson, H. F., and Robison, R, C.; Experimental Wind-Tunnel Investigation
Characteristics of Two Wing-Body Combinations. NASA Memo 11-30-58A, 1958. (U)42,Ericsson, L. E., and Reding, J. P.: Aerodynamic Effects of Bulbous Bases. NASA CR-1339, 1969. (U)43.

Ericsson, L. E.: Effect of Boundary Layer Transition on Vehicle Dynamics. AIAA Paper 69-106, 1969. IU)

44.

Feigenbaum, D., and Goodman,
NACA RM L7E09, 1947. (U)A.:

Preliminary

Investigation

at Low Speeds of Swept Wings in

Rolling Flow.

46.

Fink, M. R., and Carroll,
UAL C910188-17, 1965. (U)B.:

Hypersonic

StaticDynamic

Configurations.

46.

Fisher, L. A.: Approximate Corrections for the Effects of Compressibility on the Subsonic Stability Derivatives of Swept
Wings. NACA TN 1854, 1949. (U)47.

Fisher, L. R., and Fletcher, H. S.: Effect of Lag of Sidewash on the Vertical-Tail Contribution to Oscillatory Damping in
Yaw of Airplane Models. NACA TN 3356, 1955. (U)48.

Fisher, L. R.: Equations and Charts for Determining the Hypersonic Stability Derivatives of Combinations of Cone
Frustums Computed by Newtonian Impact Theory. NASA TN 0-149, 1959. (U)49.

Fisher, L. R.: Experimental Determination of the Effects of Frequency and Amplitude on the Lateral Stability Derivatives
for a Delta, a Swept, and an Unswept Wing Oscillating in Yaw. NACA RM A56L19, 1956. (U)50.

Fisher, L. R.: Experimental Determination of the Effects of Frequency and Amplitude on the Lateral Stability Derivatives
for a Delta, a Swept, and an Unswept Wing Oscillating in Yaw. NACA TR 1357. 1958. (U)51.

Fisher, L. R.: Experimental Determination of the Effects of Frequency and Amplitude of Oscillation on the Roll-Stability
Derivatives for a 600 Delta-Wing Airplane Model. NASA TN D-232, 1960. (U)52.

Fisher, L, R., and Michael, W. H., Jr.: An Investigation of the Effect of Vertical-Fin Location and Area on Low-Speed
Lateral Stability Derivatives of a Semitailless Airplane Model. NACA RM L51 AlU, 1951. (U)63.

Fish.r, L. A., Lichtenstein, J. H., and Williams, K. D.: A Preliminary Investigation of the Effects of Frequency andAmplitude on an Unswept-Wing Model Oscillating in Roll. NACA TN 3554, 1956. (U)54.

Fletcher, H. S.: Low-Speed Experimental Determination of the Effects of Leading-Edge Radius and Profile Thickness on
Static and Oscillatory Lateral Stability Derivatives for a Delta Wing with 600 of Leading-Edge Sweep. NACA TN 4341,
1958. (U)J.

andStabilityof the TransonicofWing-Fuselage

Damping-in-Pitch

-

55.

Gillis, C. L., Peck, R. F., and Vitale, A. J.: Preliminary Results frorh a Free-Flight Investigation at Transonic andSupersonic Speeds of the Longitudinal Stability and Control Characteristics of an Airplane Configuratim with a Thin
Straight Wing of Aspect Ratio 3. NACA RM L9K25a, 1950. (U)56.

Gillis, C. L., and Chapman, R., Jr.: Summary of Pitch-Damping Derivatives of Complete Airplane and Missile Configuretions as Measured in Flight at Transonic and Supersonic Speeds. NACA RM L52K20, 1953, (U)57,

Gillis, C. L., and Vitale, A. J.: Wing-on and Wing-off Longitudinal Characteristics of an Airplane Configuration Having aThin Unswept Tapered Wing of Aspect Ratio 3, as Obtained from Rocket-Propelled Models at Mach Numbers from 0.8 to
IA. NACA RM 1501<16, 1951. (U)58.

Gilmore,59.

Goodman, A., and Thomas, D. F., Jr.: Effects of Wing Position and Fuselage Size on the Low-Speed S ASic and Rolling
Characteristics of a Delta-Wing Model. NACA TN 3063, 1954. (U)60.

Goodman, A., and Adair, G. H.: Estimation of the Damping in Roll of Wings Through the Normal Flight Range of Lift
NACA TN 1924, 1949. (U)61.

Goodman, A., and Fisher, L. R.: Investigation at Low Speeds of the Effect of Aspect Ratio and Sweep on Rolling
Stability Derivatives of Untapered Wings. NACA TN 1835, 1949. (U)62.

Goodman, A., and Fisher, L. R.: Investigation at Low Speeds of the Effect of Aspect Ratio and Sweep on Rolling
Stability Derivatives of Untapered Wings. NACA TR 968, 1950. (U)63.

Goodman, A., and Brewer, J. D.: Investigation at Low Speeds of the Effect of Aspect Ratio and Sweep on Static andYawing Stability Derivatives of Untaperod Wings. NACA TN 1669, 1948. (U)64.

Goodman, A., and Feigenbaum,
NACA RM L7109, 1948. (U)65.

Greenwood, G. H.: Free-Flight Measurements of the Dynamic Longitudinal-Stability Characteristics of a Wind TunnelInterference Model (M = 0.92 to 1.35). ARC CP 648, 1963. (U).Coefficient,

66.

A. W., at al.: Investigation
ASD-TDR-62-460, 1963. (U)of Dynamic

Stability

Derivatives of Vehicles Flying at Hypersonic Velocity.

"Stability

D.:

Preliminary

Investigation

at Low Speeds of Swept Wings

in

Yawing Flow.

Guyett, P. R., and Curran, J. D.: Aerodynamic Derivative Measurements on a Rectangular Wing of Aspect Ratio 3.3.
ARC R&M 3171, 1961. (U)67.

Harmon, S. M.: Stability Derivatives at Supersonic Speeds of Thin Rectangular Wings with Diagonals Ahead of Tip Mach
Lines. NACA TR 925, 1949. (U)"68. Harmon, S. M.: Stability Derivatives of Thin Rectangular Wings at Supersonic Speeds. NACA TN 17U6, 1948. (U)
69.

70.

Harmon, S. M., and Jeffreys, I.: Theoretical Lift and Damping in Roll of Thin Wings with Arbitrary Sweep and Taper at
Supersonic Speeds -- Supersonic Leading and Trailing Edges. NACA TN 2114, 1950. (U)Henderson,

A., Jr.:Pitching-Moment

Slender-Body Combination
TN 2553, 1951. (U)

"NACA
71.

-

and

Derivatives Cmq and Cm.

Approximate

Solutionsforat Supersonic Speeds for a Slender-Delta-Wing andBroad.Delta-Wing

andSlender-Body

Combinations.

Hewes, D. E.: Low-Speed Measurement of Static Stability and Damping Derivatives of.a 600 Delta-Wing Model for Anglesof Attack of 00 to 900. NACA RM L54G22a, 1954. (U)

"

"-*•"

72.

Hoerner, S.: Forces and Moments on a Yawed Airfoil. NACA TM 906, 1939, (U)73.

Holleman, E. C.: Longitudinal Frequency-Response and Stability Characteristics of the Douglas D-558-11
Determined from Transient Response to a Mach Number of 096. NACA RM L52E02, 1952. (U)74.

Holt, C. F., and Kron, G. J.; Delta Wing Stability in the Pitch Plane. AIAA Student Jour., Feb. 1965. (U)75.

Hopko, R. N.: A Flight Investigation of the Damping in Roll and Rolling Effectiveness Including Aeroelastic Effects ofRocket-Propeued Missile Models Havying Cruciform, Triangular, Interdigitated Wings and Tails. NACA RM L51D16,(U)"1951.

76.

*!

Airplane asHowe. R. M.: An Investigation of the Relative Effect of Stability Derivatives on the Dynamic Characteristics of the
F-10OA. WADC TN 57-297, 1957. (U)

7-4
---

-

.. .-

J

'1

77.

Hui, W. H.: Stability Derivatives of Sharp Wedges in Viscous Hypersonic Flow Including the Effects of Thickness and Wave
Reflection. Univ. of Southampton, AASU Rapt. 278, 1967. (U)78.

Hunt, G. K.: Free-Flight Model Measurements of the Dynamic Stability of a Supersonic Strike Aircraft (TSR2).
ARC CP918, 1967. (U)79.

Hunton, L. W., and Dew, J. K.: Measurements of the Damping in Roll of Large-Scale Swept-Forward and Swept-Back
Wings in Rolling Flow. NACA RM A7Dl1, 1947. (U)80.

Ishii, T., and Yanegizawa, M.: Measurerments of the Aerodynamic Derivatives of an Oscillating Biconvex-Flat Airfoil inSupersonic Flow at Mach Number 2 to 3. National Aerospace Lab. of Japan. TR-67T, 1963. (U)81.

Jaffe, P., and Prislin, R. H.; Effect of Boundary Layer Transition on Dynamic Stability Over Large Amplitudes ofOscillation. AIAA Paper 64427, 1964. (U)82.

Jaquet, B. M., and Brewer, J. D.: Effects of Various Outboard and Central Fins on Low-Speed Static-Stability and Rolling
Characteristics of a Triangular-Wing Model. NACA RM L9E18, 1949. (U)83.

Jaquet, B. M., and Fletcher, H. S.: Experimental Steady-State Yawing Derivatives of a 600 Delta-Wing Model as Affectedby Changes in Vertical Position of the Wing and in Ratio of Fuselage Diameter to Wing Span. NACA TN 3843, 1956. (U)84.

Jaciuet, B. M., and Brewer, J. D.: Low-Speed Static-Stability aid Rolling Characteristics of Low-Aspect-Ratio Wings of
Triangular and Modified Triangular Plan Forms. NACA RM L8L29, 1949. (U)85.

Jchnson, J. L., Jr.: Low-Speed Measurements of Rolling and Yawing Stability Derivatives of a 600 Delta-Wing Model,
NACA RM L54G27, 1954. (U)86.

Johnson, J. L., Jr.: Low-Speed Measurements of Static Stability, Damping in Yaw, and Damping in Roll of a Delta, a
Swept, and an Unswept Wing for Angles of Attack from 00 to 90 . NACA RM L56001, 1966, (U)87.

Jones, A. L., and Alksne, A.: The Damping Due to Roll of Triangular, Trapezoidal, and Related Plan Forms in SupersonicFlow. NACA TN 1548, 1948. (U)88.

Kemp, W. B., Jr., and Becht, R. E.: Damping-in-Pitch Characteristics at High Subsonic and Transonic Speeds of Four 350
Sweptback Wings. NACA L53G29a, 1953. (U)

.4

89.

Kenner, H., Ashley, H., and Stapleford, R. L.: Forces and Moments on Oscillating Slender Wing-Body Combinations at
Supersonic Speed. Part II - Applications and Comparison with Excperiment, OSR TN 58-114, MIT Fluid Dynamics
Research Group Rapt. 57-5, 1957. (U)90.

Kennet, H., and Ashley, H.: Review of Unsteady Aerodynamic Studies in the Fluid Dynamics Research Group, 1955-1958.
OSR TN 59-6, MIT Fluid Dynamics Research Group Rapt, 58-5, 1958. (U)91.

Kilgore, R. A., and Hillje, E. R.; Some Transonic Dynamic Longitudinal and Directional Stability Parameters of a Canard
Airplane Model Designed for Supersonic Flight, NASA TM X-633, 1961. (U)92.

Kind, R. J., and Orl.ik-R•ickemann, K.: Stability Derivatives of Sharp Cones in Viscous Hypersonic Flow. NRC LR-427,
1965. (U)93.

Kuhn, R. E., aid Myers, B. C., I1: Effects of Mach Number and Sweep on the Damping-in-Roll Characteristics of Wings ofAspect Ratio 4. NACA RM L9E10, 1949. (U)94.

Kuhn, R. E.: Notes on Damping in Roll and Load Distributions in Roll at High Angles of Attack and High Subsonic
Speed. NACA RM L53G13a, 1953. (U)95.

Kursn, R. E., and Wiggins, J. W.: Wind-Tunnel Investigation to Deteraine the Aerodynamic Characteristics in Steady Rollof a Model at High Subsonic Speeds. NACA RM L52K24, 1953. (U)96.

Larnpkin, B. A., and Tunnell, P. J.: Static and Dynamic Rotary Stability Derivatives of an Airplane Model with arn
Unswept Wing and a High Horizontal Tail at Mach Numbers of 25, 3.0, and 3.5. NACA RM A58F17, 1958. (U)97.

Landahl, M. T.: The Flow Around Oscillating Low Aspect Ratio Wings at Transonic Speeds. Royal Institute ofTechnology, KTH Aero TN 40, 1954. (U)98.

Larndahl, M. T.: Forces and Moments on Oscillating Slender Wing-Body Combinations of Sonic Speed. OSR TN 56-109,
MIT Fluid Dynamics Research Group Rapt. 56-1, 1956. (U)

7-5

Sij

0

9'Ji.

Lehrian, D. E.' Calculation of Stability Derivatives for Oscillating Wings. ARC R&M 2922, 19656. (U)100.

Letko. W., and Brewer, J, D.:' Effect of Airfoil Profile of Symmetrical Sections on the LowSpeed Rolling Derivatives of
450 Sweptback-Wing Models of Aspect Ratio 2.61 NACA L8L31a, 1949. (U)101.

Letko, W., and Jaquet, B. M.: Effect of Airfoil Profile of Symmetrical Sections on the Low-Speed Static-Stability andYawing Derivatives of 450 Sweptback Wing Models of Aspect Ratio 2,61. NACA RM L8H110, 1948. (U)102.

Letko, W., and Fletcher, H. S.: Effects of Frequency and Amplitude on the Yawing Derivatives of Triangular, Swept, andUnswept Wings and of a Triangular-Wing-Fuselage Combinaion With and Without a Triangular Tail Performing Sinuoidal
Yawing Oscillations. NACA TN 4390, 1958. (U)103.

Letko, W., and Wolhart, W. D.: Effect of Sweepback on the Low-Speed Static and Rolling Stability Derivatives of Thin
Tapered Wings of Aspect Ratio 4. NACA RM L9F14, 1949. (U)104.

Letko, W., and Cowan, J W.: Effect of Taper Ratio on Low-Speed Static and Yawing Stability Derivatives of 450
Sweptback Wings with Aspect Ratio of 2.61. NACA TN 1671, 1948. ,U)105.

Letko, W., and Riley, D. R.; Effect of an Unswept Wing on the Contributions of Unswept-Tail Configurations to the
Low-Speed Static- and Rolling-Stability Derivatives of a Midwing Airllane Model. NACA TN 2175, 1950. tU)

106.

Stability Characteristics of a Model Having a 450
Letko. W.: Effect of Vertical-Tail Area and Length on the Yawing
Sweptback Wing. NACA TN 2358, 1951. (U)S107.

Letko, W.: Experimental Determiaition at Subsonic Speeds of the Oscillatory and Static Lateral Stability Derivatives of aSeries of Delta Wings with Leading-Edge Sweep from 30*

to 86.5? NACA RM L57A30, 1957. (U)108.

Lichtenstein. J. H., and Williams, J. L.: Effect of Frequency of Sideslipping Motion on the Lateral Stability Derivatives ofa Typical Delta-Wing Airplane. NACA RM L57F07, 1957. (U)109.

Lichtenstein, J. H.: Effect of High-Lift Devices on the Low-Speed Static Lateral and Yawing Stability Characteristics of an
Untapered 450 Sweptback Wing, NACA TN 2689, 1952. (U)110.

Lockwood, V. E.: Damping-in-Roll Chaiaceteistics of a 42.7, Swepiback Wing as Deter-iied honi d Wihd-Tu•nel
Investigation of a Twisted Semispan Wing. NACA L9F15. 1949. (U)111.

Lockwood, V. E.: Effects of Sweep on the Damping-in-Roll Characteristics of Three Sweptback Wings Having an Aspect
Ratio of 4 at Transonic Speeds. NACA RM L50J19, 1950, (U)112.

Lornax, H., and Heaslet, M. A.: Damping-in-Roll Calculations for Slender Swept-Back Wings and Slender Wing-Body
Combinations. NACA TN 1950, 1949, (U)113.

Lopez, A. E., Buell, 0I. A., and Tinling, B. E.: The Static and Dynamic Rotary Stability Derivatives at Subsonic Speeds ofan Airplane Model Having Wing and Tail Surfaces Swept Back 450, NASA Memo 5-16-59A, 1950. (U)114.

MacLachlan, R., and Letko, W.: Correlation of Two Experimental Methods of Determining the Rolling Characteristics of
Unswept Wings. NACA TN 1309, 1947. fU)

115.

MacLachlan, H., and Fisher, L. R.: Wind-Tunnel Investigation at Low Speeds of the Pitching Derivatives of Untapered
Swept Wings. NACA RM LG319, 1948. (U)116.

Maggin, B., and Bennett, C. V.: Low-Speed Stability and Damping-in-Roll Characteristics of Some Highly Swept Wings.
NACA TN 1286, 1947. (U)117.

Malvestuto, F. S., Jr., arid Hoovor, D. M.: Lift and Pitching Derivatives of Thin Sweptback Tapered Wings with Streamwise
Tips and Subsonic Leading Edges at Supersonic Speeds. NACA TN 2nT(, 1951. (U)118.

Malvestuto, F. S., Jr., and Hoover, D. M.: Supersonic Lift sod Pitchiig Moment of Thin Swaptback Tapered Wings,'j Supersonic Trailing Edges. NACA TN 2315,
Produced by Constant Verticsl Acceleration .- Subsonic Leading Edges
1961. (U)119.

Malvestuto, F. S., Jr., Margolis, K., and Ribner, H. S.: Theoretical Lift and Darnping in Roll at Supersonic Speeds of Thin
Sweptback Tapered Wings with Streamwise Tips, Subsonic Leading Edges,. and Supersonic Trailing Edges. NACA TR 970,'*

1950. (U)

7-6

120.

Malvestut,, F. S., Jr., Margolis, K., and Ribrier, H. S.: Theoretical Lift and Damping in Roll of Thin Sweptback Wings ofArbitrary Taper and Sweep at Supersonic Speeds - Subsonic Leading Edges and Supersonic Trailing Edges. NACA
TN 1860, 1950. (U)121.

Malvestuto, F. S., Jr., a-id Margolis, K.: Theoretical Stability Derivatives of Thin Sweptback Wings Tapered to a Point with
Sweptforward Trailing Edges for a Limited Range of Supersonic Speeds. NACA TR 971, 1950. (U)122.

Mangler, K. W.: A Method of Calculating the Short-Period Longitudinal Stability Derivatives of a Wing in Linearized
Unsteady Compressible Flow. ARC R&Mvi2924, 1957. (U)123.

Mantle, P. J.: On the Rolling Motion of Low-Aspect-Ratio Delta Wings. Jour. of Aero. Sci., May 1961, (U)124.

Mantle, P. J.: Some Free Flight Results of the Lift Curve S.ope and Rolling Derivatives of Low Aspect Ratio Delta Wings.
Canadian Arm. Res. and Devel. Est. TM AB-67, 1960. (U)125,

Margolis, K., and Babbitt, P. J.: Theoretical Calculations of the Stability Derivatives at Supersonic Speeds for a High-Speed
Airplane Configuration. NACA RM L53G17, 1953. (U)126.

Martin, J. C., Margolis, K., and Jeffreys, I.: Calculation of Lift and Pitching Moments Due to Angle of Attack and Steady
Pitching Velocity at Supersonic Speeds for Thin Sweptback Tapered Wings with Strearnwise Tips and Supersonic Leading
and Trailing Edges. NACA TN 2699, 1952. (U)127.

Martina, A. P.: Method for Calculating the Rolling and Yawing Moments Due to Rolling for Unswept Wings With or
Without Flaps or Ailerons by Use of Nonlinear Section Lift Data. NACA TN 2937, 1953. (U)128.

McDearmon, R. W., and Heinke, H. S., Jr.: Investigations
Supersonic Speeds. NACA RM L53A13, 1953. (U)129.

Miles, J. W.: Quasi-Stationary Airfoil Theory in Compressible Flow. Jour. of Aero. Sci., Vol. 16, No. 8, August 1949. (U)130.

Mitcham, G. L., Crabill, N. L., and Stevens, J. E.: Flight Determination of the Drag and Longitudinal Stability and Control
Char.cteristics of a Rocket-Powered Model of a 60D Delta-Wing Airplane from Mach Numbers of 10.75 to 1.70. NACA
L51
5 14, 1951. (U)131.

Mitcham, G. L., and Blanchard, W. S., Jr.: Summary of the Aerodynamic Characteristics and Flying Qualities Obtainedqof the Damping in Roll of Swept and Tapered Wings atfrom Flights of Rocket-Propelled Models of an Airplane Configuration Incorporating a Sweptback Inversely Tapered Wing
at Transonic and Low Supersonic Speeds. NACA RM LSOG18a, 1950. IU)
132.

Myers, 6. C., II, and Kuhn, R. E.: High-Subsonic Damping-in-Roll Characteristics of a Wing with the Quarter-Chord Line
Swept Back 350 and with Aspect Ratio 3 and Taper Ratio 0.6. NACA RM L9C23, 1949. (U)133.

Nethaway, J. E,, and Clark, J.: Flight Tests to Investigate the Dynamic Lateral-Stability Characteristics of a 45-deg. Delta,
Cropped to Give Three Aspect Ratios. ARC R&.M 3243, 1962. (U)134.

Ohman, L. I-I.: A Surface Flow Solution and Stability Derivatives for Bodies of Revolution in Complex Supertonic Flow.
Part I - Theory and Some Representative Results. NRL LR-418, 1964. (U)135.

Ohman, L. H.: A Surface Flow Solution and Stability Derivatives for Bodies of Revolution in Complex Superspnic Flow.
Part II - riesults for Two Families of Bodies of Revolution for 1.1 < M <5. NRL LR-419, 1964. (U)136.

Orlik-Ruckemann, K. J.: Stability Derivatives of Sharp Wedges in Viscous .- ypersonic Flow. NRC LR-431, 1965. (U)

-

S137.

Orlik-Ruckemann, K. J., and Laberge, J. G.: Static and Dynamic Longitudinal Stability Characteristics of a Series of Delta
and Sweptback Wings at Supersonic Speeds. NRC LR-396, 1966. (U)

-

138.

Parks, J. H.: Longitudinal Aerodynamic Characteristics of a Model Airplane Configuration Equipped with a Scaled X-1
Airplane Wing. NACA RM 1,51LlOa, 1952. (U)1,39.

Parks, J. H., and Kohlet, A. B.: Longitudinal Stability, Trim, and Drag Characteristics of a Rocket-Propelled Model of an
Airplane Configuration Having a 450 Sweptback Wing and an Unswept Horizontal Tail, NACA RM L52F05, 1952. (U)140.

Pearson, H. A., and Jones, R. T.; Theoretical Stability and Control Characteristics of Wings with Various Amounts ofTaper and Twist. NACA TR 635, 1938. (U)

141.

Peck,

R. F., and Mitchell, J. L.: Rocket-Model

Investigation of Longitudinal Stability and Drag Characteristics of an

Airplane Configuration Having a 600 Delta Wing and High Unswept- Horizontal Tail. NACA RM L52KO4a, 1953. (U)
?7-

142.

Piland, R. 0.: Summary of the Theoretical Lift, Damping-in-Roll, and Center-of-Pressure Characteristics of Various Wing
Plan Forms at Supersonic Speeds. NACA TN 1977, 1949. (U)143.

Pinsker, W. J. G.: A Semiempirical Method for Estimating the Rotary Rolling Moment Derivatives of Swept and Slender
Wings. ARC CP 524, 1960. (U)144.

Platzer, M. F.. and Sherer, A. D.: Dynamic Stability Analysis of Bodies of Revolution in Supersonic Flow. AIAA Paper
67-607, 1967. (U)145.

Platzer, M. F., and Hoffman, G. H.: Quasi-Slender Body Theory for Slowly Oscillating Bodies of Revolution in SupersonicFlow. NASA TN D-3440, 1966. (U)146.

Polhamus, E. C.: A Simple Method of Estimating the Subsonic Lift and Damping in Rl11 of Sweptback Wings. NACA
TN 1862, 1949. (U)147.

Queijo, M. J and Jaquet, B. M.: Calculated Effects of Geometric Dihedral on the Low-Speed Rolling Derivatives of Swept
Wings, NACA TN 1732, 1948. (U)148,

Queijo, M. J., and Jaquet, B. M.: Investigation of Effects of Geometric Dihedral on Low-Speed Static Stability and Yawing
Characteristics of an Untapered 450 Sweptback-Wing Model of Aspect Ratio 2,61. NACA TN 1668, 1948. (U)149.

Queijo, M. J.. et al.: Preliminary Measurements of the Aerodynamic Yawing Derivatives of a Triangular, a Swept, and an
Unswept Wing Performing Pure Yawing Oscillations, with a Description of the Instrument Emp!oyed. NACA RM !55Lt4,
1956. (U)150.

Queijo, M. J., and Wells, E. G.: Wind-Tunnel Investigation of the Low-Speed Static and Rotary Stability Derivatives of a0.13-Scale Molxel of the Douglas D-558-11 Airplane in the Landing Configuration. NACA RM L52G07. 1952. (U)151.

Quinn. B.: The Hypersonic Roll Damping Derivative of a Blunt Cone. ARL 68-0137, 1968. (U)152.

Ribner, H.: The Stability Derivatives of Low-Aspect-Ratio Triangular Wings at Subsonic and Supersonic Speeds. NACA
TN 1423, 1947. (U)153.

Ribner, H. S., and Malvestuto, F. S., Jr.: Stability Derivatives of Triangula. Wings at Supersonic Speeds. NACA TR908,

1948. (U)

7-8

154.

Rightley, E. C.: Dynamic Stability Derivatives for a 10(0 Blunt Cone at Mach Numbers from 0.5 to 21. Sandia Corp.
Report SC-R-65-907, 1965. (U)156.

Ross, A. J.: The Lateral Oscillation cf Slender Aircraft. ARC CP 845, 1966. (U)156.

Sadoff, M., Ankenbruck, H. 0.. and O'Hare, W.: Stability and Control Measurements Ob:ained During USAF-NACA
Cooperative Flight-Test Program on the X-4 Airplane (USAF NO. 46-677). NACA RM A51H09, 1951. (U)157.

Sanders, E. C., Jr., and Edmondson, J. L.: Damping in Roll of Rocket-Powered Test Vehicles Having Swept, Tapered
Wings of Low Aspect Ratio. NACA RM L51G06, 1951. (U)158,

Sanders, E. C., Jr.: Damping in Roll of Straight and 45o Swept Wings of Various Taper Ratios Determined at High
Subsonic, Transonic, and Supersonic Speeds with Rocket-Powered Models. NACA RM L51H14, 19651. (U)159

Scruton, C., Woodgate, L., and Alexander. A. J.; Measurement of the Aerodynamic Derivatives for Swept Wings of Low
Aspect Ratio Describing Pitchinglnd
Plunging Oscillati~ns in Incompressible Flow. ARC RSM 292M, 1957. (U)1160.

Scruton, C., at al.: Meamurement of Pitching-Moment Derivatives for Aerofoils Oscillating in Two-Dimensional SupersonicFlow. ARC R&M 3234, 1962. (U)161.

Smith, C. B., and Beane,
1951. (U)162.

Statler, I.C., Tufts, O.B., and Hirtreiter, W. J.: The Development and Evaluation of the CAL/Air Force Dynamic
Wind-Tunnel Testing System. Part I - Description and Dynamic Tests of an F-80 Model. AFFDL-TR-66-153 Part I,
1967. (U)163.

Statler, I. C.: T?,ie Development and Evaluation of the CAL/Air Force Dynamic Wind-Tunnel Testing System. PartII - Dynamic Tests of an F-104 Model. AFFDL-TR466-153 Part II, 1967. (U)B. J.: Damping in Pitch of Bodies of Revolution at Supersonic Speeds. IAS Preprint 311,

*..

S

*•

.5**.

-

.,.

-

...

-.

Ilk,

,

1

•

164.

Statler, I.C.: Dynamic Stability at High Speeds from Unsteady Flow Theory. Jour of Aero Sci, Vol. 17, No. 4, 1950. (U)166.

Stone, 0. G., and Sandahl, C. A.: A Comparison of Two Techniques Utilizing
Determination of the Damping-in-Roll Derivative. NACA L51A16, 1961. (U)166.

Swanson, R. S., and Priddy, E. L.: Lifting-Surface-Theory Values of the Damping in Roll and of the Parameter Used in
Estimating Aileron Stick Forces. NACA L-63, 1945, (U)167.

Thomas, H, H. B. M., and Ross, A. J.; The Calculation of the Rotary Lateral Stability Derivatives of a Jet-Flapped Wing.
ARC R&M 3277, 1962. (U)168.

Thompson, J. S., Fail, R. A., and Inglesby, J. V,: Low Speed Wind Tunnel Measurements of the Oscillatory Lateral
Stability Derivatives for a Model of a Slender Aircraft (HP 115) Including the Effects of Frequency Parameter. RAE TR
69018, 1969. (U)169.

Tobak, M.; Damping in Pitch of Low-Aspect-Ratio Wings at Subsonic and Supersonic Speeds. NACA RM A52L04a,

170.

Tobak, M., and Lessing, H, C.: Estimation of Rotary Stability Derivatives at Subsonic and Transonic Speeds. AGARD
Report 343, 1961. 4U)

171.

Tobak, M., Reese, D. E. Jr., and Beam, B. H.: Experimental Damping in Pitch of 450 Triangular Wings. NACA RM
A50J26, 1960. (U)172.

Tobek, M., and Wehrend, W. R.: Stability Derivatives of Cones at Supersonic Speeds. NACA TN 3788, 1956, (U)

173.

Toll, T. A., and Qusijo, M. J.: Approximate Relations and Charts for Low-Speed Stability Derivatives of Swept Wings.
NACA TN 1581, 1948. (U)174.

Tosti, L. P.: Low-Speed Static Stability and Demping-in-Roll Characteristics of Some Swept and Unswept Low-AspectRatio Wings. NACA TN 1468, 1947. (U)176.

Triplett, W. C., and Van Dyke, R. D., Jr.: Preliminary Flight Investigation
Characteristics of a 350 Swept-Wing Airplane. NACA RM A5OJOge, 1950, (U)176.

Tucker, W. A., and Piland, R. 0.: Estimating of the Damping in Roll of Supersonic-Leading-Edge Wing-Body Combinetions. NACA TN 2151, 1950. (U)177.

Turner, K. J., Ross, A. J., 3nd Earley, G.: The Dynamic Stability Derivatives of a Slender Wing, a Comparison of Theory
with Free-Flight Model Tests at Near-Zero Lift, M - 8.0 to 2.4. ARC CP 995, 1968. (U)178.

Turner, K. J., and Hunt, G. K.: Free-Flight Measurements for the Transonic Roll-Damping Characteristics of Three Related
Wings of Aspect Ratio 223. ARC R&M 3274, 1962. (U)179.

Turner, K. j.; Measurements of Dynamic Stability from Three Simplified Free-Flight Models of a Supersonic Research
Aircraft (Bristol T.188) Over the Mach Number Range 1.2-2.6. ARC CP 816, 1965. (U)180.

Vitale, A. J., McFall, J. C,, Jr., and Morrow, J. D.: Longitudinal Stability and Drag Characteristics at Mach Numbers fromfor the1953. (U)0.75 to 1i5 of an Airplane Configuration Having a 3o Swept Wing of Aspect
Propelled Models. NACA L51KO6, 1952. (U)181.

Rocket-Propelled Vehiclesof the Dynamic Longitudinal-Stability

Ratio 2.24 as Obtained fromRocket-Walchner, 0., and Clay, J. T.: Hypersonic Stability Derivatives of Blunted Slender Cones. AIAA Journal, Vol 8, No. 4,
April 1965. (U)182.

Walker, H. J,., and Ballantyra, M. B,: Prossure Distribution and Damping in Steady Roll at Supersonic Mach Numbers of
Flat Swept-Back Wings with Subsonic Edges. NACA TN 2047, 1950. (U)183.

Webrend, W. R., Jr.: A Wind-Tunnel Investigation of the Effect of Changes in Base Contour on the Damping in Pitch of a
Blunted Cone NASA TN U-2062, 1963. (U)184.

Wehrend, W. R., Jr.: Wind-Tunnel Investigation of the Static and Dynamic Stability Characteristics of a 10o Semivertex
Angle Blunted Cone. NASA TN D-1202, 1962. (U)185.

Wiggins, J. W.: Wind-Tunnel Investigation of Effect of Sweep on Rolling Derivatives at Angles of Attack up to 13° end at
High Subsonic Mach Numbers, Including a Semiempiricel Method of Estimating the Rolling Derivatives. NACA Th 4185,
198. (U)r-'.-•'• " '• '•il ' '" J• •'ir • •ap•u:
u nn- ""- • : m•.

.

..."'• -

-

:• •

" '•

• • . . .

. " -•. ..

" • . . ..

.

7..
.

185.

Wiggins, J. W.: Wind-Tunnel Investigation at High Subsonic Speeds to Determine the Rolling Derivatives of Two
Wing-Fuselage Combinations Having Triangular Wings, Including a Seniempirical Method of Estimating the Rolling
Derivatives. NACA AM L53L18s, 1963. (U)187.

Williams, J. L.: Measured and Estimated Lateral Static and Rotary Derivatives of a 1/12-Scale Model of a High-Speed
Fighter Airplane with Unswept Wings. NACA AM L53K09, 1954. (U)188.

Wolhart, W. D.: Influence of Wing and Fuselage on the Vertical-Tail Contribution to the Low-Speed Rolling Derivatives ofMidwing Airplane Models with 450 Swept Back Surfaces. NACA TN 2587, 1951. (U)189

Wood, R. M., and Murphy, C. H.: Acrodyr amic Derivatives for Both Steady and Nonsteedy Moeion of Slender Bodies.
Jour, of Aero. Sci.. Vol. 22, No. 12, 1965. (U)190.

Woodgate, L., and Pugh, P. G.: Measurements of the Pitching-Moment Derivatives on a Sharp-Edged Delta Wing in
Incompressible Flow. ARC R&M 3379, 1963. (U)191.

Wright, B. R., and Brower, M. L.: Aerodynamic Damping and Oscillatory Stability in Pitch for a Model of a Typical
Subsonic Jet-Transport Airplane. NASA TN D-3159, 1966. (U)192.

Zartarian, G., and Ashley, H.: Forces and Moments on Oscillating Slender Wing-Body Combinations at Supersonic Speed.
Part I - Basic Theory. OSR TN 57-386, MIT Fluid Dynamics Research Group Rpt. 57-2, 1957. (U)TABLE 7-A
DYNAMIC DERIVATIVES

SRef.
2

Oerwtiw
Cm +C

Cmr + Cm. C%. Cnp,
C%-C.,CCnr,C

6

Cm + Cm
I.

x

IdegI
-

+ Cm

3Cmq,

4

A

-

M

Freq.

Ampi.

.S*1,5

X

X

Config.

4

45 LE

0

.10 -ý1.

x

x

We

2.2

60 LE

.03

.25-"96

X

X

WS

.

Low speed

45 c/4.2.70 -1.15

Cn.

10

2 c/4

52

421

2.5

62

-,

W

X

X

WBHV, BHV,
WBH, WaV-W. W

Cnr - Cn* coS %X
7Cy

8CnC,Cer

2.63

45 c/41.0.13

Cy.Cn, Cer
r r
r

2.61

45 c/4

1in

.13

W. wa

2,61

45 c/4

1.0

.17

W,
We

1.5,3.5,60, 30,45.60 c/4

0, .5,1.5

Low ipeedW

1.5,3.5,6.10

0, 30,45.60 c/4

0, .5, 1.0

LowspeedW

4.51

35 c/4

1.84

.17

We

9Cg Cn, .Cy

p
10

p

p

Cn (theor.)r

II

Cp ltheer.)
p

12

CnC C

C

Cnr' Cf CYr

7-10

[

3.

TABLE 7-A (CONTD)"Ref.

Derivative

A

(dog)

A

M

Freq.

Amni.

13

CR Cy.Cn
p
p p

2.A3

48 LE

1.0

.17

-

-

14

Cp

4

45 LE

0

-+ 1,-

W, we

15

Cep

3.7

0, 45 LE

1.0.- bw.e

W
WWa

16

Cp

4.0

45 LE

0

0 c/2

.5

-

-

1.1 -2A

X

-

0 c/4

.5, 1.0
5,25,! 0

.17

-

-

w

1.0,0

1.62,1.92

-

-

We

•,

Cm +Cm-19

C,C ,Cep
p p

""20

2.61

45 c/4
1.47-0,4.0

0 "*45LE
configuratioi)

21 C Lq !Cm.4m

rot ALEtan-I A
4-LE

0

Supronicn€

2.44

27 LE

.38

25 -+".94

X

X

Cir - CC , CCn.

2

OE

0

Cnp,

2.2

60 LE

Lowup.d-.W"C2p (thoor.)
22

Cm:

"":

,Cep, Cnp,

C

weW

WW8
W8-+1.7

PCCp"p,(12

Config,W

8, WB, WBV
WBHV,
BV,

Cr,. CR (theo,.)i
24

26

26

7

Cn

6

OLE

6.7

1.0

-

.40

60 LE
45 c/4
0 c/2

0
.25
.

Low speed

C2 - C24

231
2.61
3.0

C

1 "346LE0

1

Low s eCn - Cn

r

C

+C

i

2.61
4

45
45

2.31

52.2 c/4

02 1,350.6604

0

x

K

W

WLW BihO. .25.6q
29Cmq + C
Cn
q
CL.,C
(theor.)WB.70-1.374
7W
14,2

PA-3-20
PA -2

"*20

cot ALE .1 -400
P cot AL El-

30

2

63.4 LE

11 C

9.20

6 c/4

0.25,6_,76-

Suprsonrc

W

1.0

.

0

1.35-• 2.03

X

.40

Low speed

xw,

x

W eV

w e,

~WBV, WSHV
7-11

l

.1

rABLE 7-A (CONTO)
Rot.

Derivative

34

A

.

A

(dog)

A

M

C
Cmq
+ Cm

2
3

6343 LE
53.07 LE0.80 -1.35

-

-

We

Cm

.4
3

45 LE
37,51I0.6.

.7-"1.35

-

-

We

3.6

38.$

.4

4

40 €/4.5.80 " 1.36

1.56

70/75 LE-

2.4, 28,+ Cm

35

Cmq + Cmi36

Cmq +Cma,

-k

2

Cml

'

3.6

-

-

WS

X

X

WBHV,

q

Wey,
2

C2 . co ,a + k C.,
Cn -c

Config.

WBH

Cos 0ý

Cn •cos a+n ink2C

37Cp

4.5

CL.'"m.•CL'
38

0 LE

1.0

25-+1.45

-

-

We

-

Supersonic

X

X

B

-

.7-5.0XXa

1.85 -+ 1.40

-

-

We

.6-*1.18

X

X

We

-

.25"+

X

XA

-

5,6,7,8

1.0

.17

-

-

W

X

-

I, Wa

*Cm

(theo.)
3

+

9Cm

40

Cp

41

Cmq

-m-

63,4 LE

0

-

.4065

a,

Z;2 Cm43

Cmq + Cm.-

44-

Cp, Cnp, Cp
p.

...

p

+C

Cm

q0

16

2

"3

""mqmq

Cm.

0 LE

m

"Cry-

45

3.71

CL C
q

,

q

2.61

-45 LE

5.16

0

2.61

45

1.34

600

.4 (exposed)

0LE

1

4.8,8.08,10,

.aI

79

0

17

1.6

70

0

C2 .

Ce,'nCp

4,42,

30 c/4

-

0

1.0

50

-

0"1.0

--

-

-

W

-

-

We

o-1.o
.

30

CRp (theor.)
40

1.0, 5

30
45

47

7-12

L......

Cn

445, 0 c/4

.6

Low speed.-

r-;4.. -.

.r-•r-,- -rrl-,•'

".

- - r

....

. -'"...
..- ....

. •- •'""-.. -

-• -

1 "".

-

- "..

.,".

-

'

TABLE 7-A (CONTD)

48

CLq*Cm49

Cr.Cnr,(theor.)- CnC- -Cr r -.

Fre,4.

M

(dog)

A

Derivative

Rif.

Ampl.

Config.

iHypertonic

--

W

.13

2.31

60 LE

0

4

0, 45 c/4

,6

0

C r,.

2.31
4

60 LE
O, 45 c/4

0
.6

.13

X

X

w

51

cGnp,

2.31

60 LE

0

.13

X

X

W, WB

52

Cy

3.60

41.57 LE.455

.13.16

2 c/4

.5

.13

2.31

60 LE

0.3

3

16 c/4.49"*1.3 6

1,87-6.02

1.9- 60 c/4(18 configurations)0-1.63.6-*1.7

3

16c/4

.4

1.456

70 LE

0

8.08

2.31

60 LE

0

.17

CQ

C

W, W8

rp, pp53

Cyp, Cnp CQ,

54

Cn - Cn ,Cer - Cký

55

Cm

+ Cm

56

C7

+

57

Cm

+ Cm

58

CNq (theor.), C.nq,

X

we

x

W

.8 -

1.4

]

W S

WS

-

-

WB

x

8,W

Cmq + Cm•

59

,
CnC

C
p

60

•"

"

61

Cp
p

p

1.34 459-(22

OQpCnp. Cyp

= •€

."62

c/4

0 -60

1.3

W

I.

.13,.17

W

1f0

.17

0-'1.00-

configurations)
1.34
1451

0 LE
60

P.p

"4

W, WB

p

2.61

I45

0

5.16

0

.40LE

O~,nCy

-w

60
..

..

2.81
I

0
45

5
2.

0

6

.

7-13

6,i

--

v7

TABLE 7-A (CONTO)
Ref.

Osrivative

A

(deg)

?

M

Freq.

Amp|.

Config.

63

Cý,Cyr' Cnr

1.34

0 LE

1.0

.13

-

-

W

2.61

60
0

I.D

.13

r

r

I45
5.16

64

60
0

'5

1

60

Cn,..r
Cr2.61.- 45 LE

IN"

5.16

SCm +C(theor.)

2.61

45

1.34

60

2.83

45 c/2.333.92-1.35

0 LE

1.0

Supersonic

X

-

w

•q
1C24, Cmp, c~p.
CLI + .263 CL

aiother combinations
67

C Lq'C*CQC
Cq cmq'P,,y
Cn~p'c r'CYr' Cn r.

*A

i

1 -20

W
1•'

(theor.)6B% or C,". Cy
C.

69

%

P.

0"-20

Supersonic-(theor.)

2,4

63 LE
0-60

C

.*Cm: (theor.)

CR + Cqi wn a,
Cn•

Cn• cos 0

72

CQr.C

(zneor.)73

Cm + Cm

74

CLC

75

CG

p

7-14

(thoor.)

1.2 - 2.6

.5

2

53

.5

2

24

53

0-1.0

2

0coz A

I1

8.10, 12,16, 20

0

.5

0-10
PA - 3,4.5,6.

71

-

nr

2.4

70

-

Itheor.)Or Cnp

orQ yrW

0..25,.5.
.75,1.0

1.07

75 LE

2.31

I6)

1 -+3

0

1

5

4

45

1 - 1.8

6.93

30

1 - 1.5

2.2

60LE

0

6,1.25

0 LE

1.0

3.57

35 .3c.565.6-.96

2

60LE

0

.13

2.3

45,60 LE0

.8-12 RW8-

Lo" speed

W-

W

W aI

-

-

WB
M

F

TABLE 7-A (CONTD)
A(

,, f.

Derivative76

CLCm CLCm.•

a

qq
C

A

(dog)

-

-

-

7C -88 LE

M

Freq.

-

.6,1.2

-

-

WBHV

0

10,17

-

-

W

X

X

X

-

B

-

-

W, WB

-

-B

Am6WI.

Config.

aC,'CCnp

Yr'rr(theor.)
77

Crn

m
78

(theor.)Cmqq+ Cm

.C~p o
Cnr

1.96

-0.74

S3.12
C+mC4.69-45 c/4
4.62-0"

81

Cr

-+1.4

.38
40
.30

WVBHV

Low speed,high speed

.55

4.84

430

3.64

45A2

.44

+ Cm

2.01,3.02,4.56

q82

Cyp, Cnp,.CQp

2.31

60 LE

0.13

83

CYrCnCr

2.31

60 LE

0

.13

84

Cy, .Cnp CR

2.31
1.07

60 LE
75

0

.13

-

W

-

WB

454
3
2
I

hi8,
85

,Cn,

C,

C~p .Cnrpr0

LOw speed

2.31

60 LE

0

Low speed

2.61

45 c/4
2
0 c/

.25
.50

0 LE

1.0,0

Supersonic3

36 c/4.6

High sIubonic,
trmnsonic

2.309

60 LE

0

Supersonic

2.0

63.4 LE

0

Transonic

.904

62 LE

.418

.8,,9,1.0,1.2

3.0
87

8

p

(theor.)Cm + Cm

.58

60 LE

2.2

r

pp

0
.15
.36

q

"89 Cm + Cm.

a

q

X

W

-

W

-

W

-

WB

X

x

WO

X

x

WB

-

Cmq + Cm (theor.)90.

Cm4 + Cm,C91

+ CMi (ther.)Cmq +Cm.
Crir - Cn c lt,ait

5-

0

CLCm

-92

qI

e'.

~

8ti,.e8

B

"K

7-15

•

_..

,.

-•

,

-

-

.

-

-.--

-

-

-.-

-

-

-

-

-

-

TABLE 7-A (CONTU)

R-ef.

Derivative

93

Cp

94

C%

96
,:

Cn-

97

4

3.6, 32.6,46 c/4.6.4-".91.2-6-.91

-

WE

4

-

W

4

3.6,32.6,45c614

.6

6o LE

0

2

4

4 6

m&'
cos,

2.44

C,:

CR,sionQ,

Cn+ C.n,
si Ct

2.5, 3 0D,
3.5

X

X

W Be,
WHV

0

Transonic

X

X

W

0

Transonic

X

X

8, W,

1.0

75.9 LE

1,5
2.0

69.5
63.4

m

m 6t.theor.1

2,50

mb, CLq

C m +Cmmq,(lheor,)

LQm,

S2.0

L&,

mt

3.5

53.1
48.8

4,0

45

1.5

e69. LE

2 .0
25

6 3 .41.2

68.2 L

W8.

.143

I

S3.0

101

102

C%%

CrC

Cf

C QCr,C
r

103

104

C

.9."

68.

"3.0

10

W
w-

.38

C.CLq
a,
Cmq,

0

Config.

27 LE

+

q

.

Ampl,

.6

CL

L

Frvq,

c/4

C
0

'

M

+Cyp

Q C.
-C

C

(deg)

2.31
C

~9

A

.C

C

(yr, Cnr, C

Subsonic

Subsonic

X

x

x

x

-.. 9

x

x

.389

Subsonc

x

x

2,81

45 L

1.0

.17

2.61

45 LE

1.0

.13

-

2,31
4

60 LE
45,0 c/4

0.0

Low speedX4

46. c/4

.6

13

-w

45 c/4

1.0O .M0..25

.13

-

2.61

r

-

w

W

W

x

W, W

W
:r

105

106

107

C

C,

C
,n

Cp

nr, CrCR, -C 2

4

0/4

.6

,17

4

46 c/4

.6

.13

.53

8'2.5 LE
7b
60

0

.1

1.07

ell,

Cn

2.31

13

--

--

W, WB

W, WB

K

7-16

...................................................

TABLE 7-A (CONI'D

..

All
Ref.

Drstliw,A(do@)

2.CYr.
2.1

60LE

2.61

45 LE

I C; Cyp

M

Freq.

Ampl.

0

.13

X

X

1.0

.13

Config,.- r - CI
109

C , Cnr Cyr

110

Cp

4

42.7 LE

111

CRO

4

0 c/4
351

112

Cgp (theOr.)

113

Ck,,CnC•.cC

C,

niq

115

,

3.56

P

.61.15.6.,.,.,WB
2 5, .9, 295, 1.0,

46 LE

p

45c/4

W

0, "1.0

.3

F

Low

W. we

rWBH, .23-.94

x

X

8,w,
BN,

-C

+0C
N

m
CO - Cf.

114

Low

W

Cnr

" hif

WOHV,
BHV, BV

C

6.38

0 LE

1.0

6.

18.44

.333

OLE

1

.13

Lowspeed1

Cmq, CL

.3

4

2.611
6S16

Subsonic-w

0

2.61-5

2

42 LE

.793

3

-38, 42 LE

.707

5*

-30,42 LE.50

0A - 2A-2-'10
2.4

-

53 LE

.25,.50,.75
.5

lie

p

11p

S

q

CL (theor.

29

1
lie

C8 C.C 1 (•thaoi

119

Cep (theor.)

2.4

.3

PiA - 2A-10

ft cot ALE - 01

2.4
I ,6

"* .4
2,4

Ile

120

Cg(theor.)

2,4

"2,4
1(.,5A2,4
PA *2.3,4. 5.

N,

.5

1.5

.5

1.414

0-4.75

1.414

l

46-N

q2.4

Supenonic
Supersonic.0,.75-W

W

Supersonic

W

53 LE

1.2,+1.6

W

63

1.5

.25,

1

45` 556
63 LE

456-D
53
53
.Pcn A -O".0
0

1.414
.6

1.2-- I A

.5
.5

1,44
1.5

0..?
0,.25..50

1.414

W

7-17

TABLE 7-A (CONTD)"RO. Derivae
121

LCC

A

Cl

X

M

ALE 410Supersonic0

Sulbonic,
tansic,

-<co

.'A<
C q4
q

a

CLqo Cm,

Cmq + Cm(theow)

123

Cgp/(AR)

124

C

126

Cm

(AR)

C L,

128

129

cOt ALEtan-I 2LE
A

X

x

W

(thtor.l

1.2

2.6, 1,2

0

-

W

S,1,2

0

--

-

W

0

1.414

.25,_,7Supeorson i-

320

5

LE

oA - 3-+20

cot ALE

3.0

-

WOH

a
(theo,.)Cq

CQ•I Cnp (theor.)

1-o

0,

4

-

.6

-

.4

21.2"- 70 LE

0,25

3.4

W

1.0

9.021
1D2-

Cp
p

Cm

W

A

qA -A2-020
127-

t.upesonic

3

q
126

4

3

+C

Config.

4Cot ALE
4

I
CL., Cm

Ampl.

4

q

r. ,Cyr (themr.)

122

Freq.

ides)

-

-

W

1.62.1.93,2.41-W(28 configurations)(theor.)-Supemonic

X

x

W

.7b

1.70

-

-

We

q
130

Cmq +CM

2.31

60 LE

0

131

Cmq + Cm

3.07

40 c/2

1.628.55"-11.2

-

-

we

132

C3
lp

35C/4

.6

.4A.9l

-

-

We

133

CCn
t

,2"+.9

X

-

WBHV

-

1.1-45---

1.1-.5-,.-87,84,81 LE

0

9.17

-

-

W

3.

49.1 LE.072

1.22-'2,

X

X

W

,2

2.

60

|

5-1.t4

1.25

70.1

3

49.1

"

C

(thew.)

3U8
3.0
2.3

Cnr

134

CN4

N6,' Cmq,,

45 L.E02
147
.274

-

Cm q +-C.( (theor) a136

C Nq'C CNC.Cmq(theor.)136

Cm +Cm. (theor.)

137

C.CLm
6

7-18

i,

2.

60

1.25

7,0,01ITABLE 7-A (CONTD)
Rf.

Derivatile

13

Cm

A

+Cm.

6.02

0.c

.5

q
13D

Cmq

140

Cn

Cm

4

CR C

CRP

704.6
45

6.10,160 c4

1.50,.;.5

W LE

0

M

Freq,

Ampl.

Config,.O-+l.17

;c

x

-A

W

.63 -11.16

-

-

WI

-

_

(thoor.)
141

142

C

+C

2.31

Cfp (theor.)

2 "+8

.9-

1.37

-

Supersonic-0

Low ap~d--_-.7-8.2 ,-_-1.2-3,0

-

-

0-1.0

-

-

-

45 LE

1.0

Low speed

-

-

45 LE

1.0

.13

-

Low pwd-.13,.17-

0 "#69.4LE

0

1.0

-

We

W

(44 configurations)
143

Cp

144

C

C

145

.75,1.0+ CNmq

4 C-1.12(theor.)C (theor.)

1.0 -5CPCnC2
17

148

149

W

Cmq +Cm

CN. +CN146-(theor.),C

2.61

nr, Cyr

CnrC

2.61

Cr

C
150
Cnp .CQ~*CYr

0 -

c/4

2.31

O0LE

0

4.0

0, 45 c/4

.6

3.57

35 c/3

.565

-

W

W

.-

-

W

We

Cnr' CRr

151

152

C% (theor.).-

CLc; C mqCQCR.

CnqCnr. Cq

CYr

14

<.5

0

Subsonic

--

-

--

-

W

-W

w

X

B

Supersonic(theor,)
1 L3 , CL~~r Crc
f
Cn.

A
4--<

cot ALE 4Lit1

0

Supersonic

Varies.5"21

Cyp (theom.)

Cmq +VCmis

155

r

r r
Cr 'Cn (theor.)0

L o w V oee d I-W

X

i
W+fin

7-19

7 7-7Tý,-

.. ,

..-

7..-

-

-.-.-

..

-

--

TABLE 7-A (CONITD)

Dorivotive

Ad.

"16 Cm+q +SCCC m

3.6

167

4

Rof.

•3.5

16

2.9
I4

158

CRp

p
CL

159

q

M

41.57 LE.456

,4- ".88

42.7
16.5 LE,5,56.7-+1,•4

23.1

.41

37A, 46.7.60.9

3.5

63

.6
.26

6

46.2

.6

3.71

0, 45 c/4

1,.6,.3,0

1.2

68.2 LE

.143

1

0

WB

WB

.45

-

-

Subsonicx-

1.37 -2.43

x

-

A

S alon•€(theor.)

Cm

-

.389

14o4

181

Wa

0

1.6
1.32

'm

Config.

.rq

X.

w

B

-

-

q
102

C L ' c ma , cL q .cm q

163

CL1 .

m4' CLq. Cm
qq
q q

164

CL

c•

165

%

,56-• .8-

2.46

CLqCm

3.7

166

Ithoor.)167

C•,Cr. Cn,

WBHV

-

.70-+1.10

X

-

WBHV

-

.30,.50,,70,.75

x

X

WBH

1 .0

.6"+1.4

WB

-

_5_-

-

-

W

267

X

-

WBV

1.2-"1.9

-

-W

4,6,8

Cnr

(theor.)'Go

n. - Cn 0 cosa•

.92

74 LE

0

2

63.4 LE

0

4

45

1

3
3

45
19.1

.403
.407

.094-.

- CYo cosCy

4%
C~rr -C QIi•cos
Cn p + Cný sin O.
+ CyY4 sin (t,Cyp
Cep * Cg sn C

log

+ Cm

Cm

q

L

03

7-20
•--

-

"

-

-.-

--

-

-

TABLE 7,A (CONTD)

A

170

CLa'Cm6" CLC"C0<
Cin

q

+ Cm

a

Idog)

4

tn

PAM

--- E

Config.

1W

00 -1.001.16 - 1.70-

Supersoni,(theor.)"171

Cmq +Cr

0172

CL.' C

CL., CCmq

4

45 LE

(thebor.)

173

An'I,

WW8

-

hype"lOnic

C9 ,Cn .Cy

2.61

6.160
2.61

C~r;Cnr CY;'

160 Cm q +m2

4.79

+CC
176815C
C (theor.)
1748
R

2.8
-1

-45 LE

1.0

Low sWp

A1.65- 1.04

1J1.0

Low1. -

45

323 c/4

80.45,5/4SSQ

we

-

W.
-.

WB

11 co fiurtins
97

p
078

rR

2283

2 7.95, 53.55 LiE

P49.97

179

ina.1

173
ISO

C

!18'

C

186

CyP, Cn, Cp

,JU7
.066

0/3816 LE+C
Ctmq +Cm•

.8-

2,24

73. LE'22
60 c/4+1.

1
0.3 3.75

B
a

" 2,6
.
"+1.50

x

S i

w e

(theor.)aq

4

3.6. 32.6, 45, 60 c/4.6.5--.95

WB

4
2.31

45 LE

0

.5

.95

WB

5.90

0 LE

.473

.13.17

Wa

p pp
186

187

C9.P,CPlcy,.

CR..CYr Cnr'Cer

-

6"

J
7-21

TABLE 7-A (CONTO)

R f.

rirotiwA(doq)

188IyO

. Cn

4

45 d4

139

C LC

%

Cm
,

FreM

A rvl.

Config.

.17

-

-

W

Superonic--

0.09,.18

x

X

x

x

we

-

a . W,.6Cm
C(fheor.j

190

Cm

191

C

1.484

Q

+Cm.

aq

r.

L"

*.

.

7-22

41.5I37.5 LE(.33).20-..94

1.5

695 LE

0Supersonic

2.0
2.5

63.4
so

q

n- Ih°°r')qq+C Mit

[,Cm

7.035

.

W8

•.•. .• . .::... : .., :...4

..

..

..

:

1.•

.-

,- s- n" is

,"

"." "

"

P- P

" =

"

"

."
" •

...

.

...
i•,W~
.
...... ,....

0I

Revised September 1970

7.1 WING DYNAMIC DERIVATIVES

The methods presented in this section are to be used for the estimation of pitching, acceleration,
rolling, and'yawing dynamic derivatives of isolated lifting surfaces. The methods and charts
applicable to this section are based on lifting-surface theory for subsonic speeds and on linearizedtheory for supersonic speeds. The charts are thus limited to conditions for which the flow isessentially attached over the surface of the wing, i.e., the linear lift-curve range. This means thatat subsonic speeds the methods are valid for high-aspect-ratio wings up to stall angles of attack
but ace limited for low-aspect-ratio wings to low angles of attack.
For wings operating uwder conditions of partially separated flow, such as low-aspect-ratio wings
at moderate to high angles of attack, experimental data, e.g., references I and 2, show that
substantial nonlinearities exist in the dynamic derivative,-:. in, add,-tion, tests made by, wind-tunnel
oscillating-mode techniques show that the dynam.c derivatives are functions of both theamplitude and frequency of oscillation. In general, dynamic derivatives for separated-flow
conditions are larger and therefore more significant than for attached-flow conditions. Because of
the complexity of the separated-flow case, no quantitative information on these effects ispresented in the Datcom. Instead, a qualitative discussion of the characteristics of dynamic
derivatives under partially separated-flow conditions is given.

1 w 0'Flow

separation on wings at angles of attack below stall is discussed in Sections 4.1.3.3 and4.1.3.4. The dominant feature of flow over low-aspect-ratio or swept wings, i.e., wings for whichflow separation below stall is important, is the leading-edge vortex. The strength of theleading-edge vortex is determined by wing planform and airfoil leading-edge geometry.

m

In general, large leading-edge sweep angles and sharp airfoil sections are conducive to high vortex
strengths. The strength of the leading-edge vortex, in turn, determines the nature and magnitudeof the nonlinear static force, moment, and dynamic-derivative characteristics. The effects ofgeometry on the dynamic derivatives of a triangular wing at low speed are presented inreference 3.

r q'-Partially__dynamicL

separated flow over a wing causes the dynamic derivatives to be frequency-dependent.
The reason is that a change in wing attitude changes the boundary-layer conditions, which, inturn, alter the flow-separation pattern. The time required for the flow to adjust to a change in"attitude is appreciable, and the dynamic derivatives become functions of the rate changv' of
attitude (or frequency). The flow about wings with essentially attached flow is less dependent
upon the boundary layer and therefore adjusts more rapidly to attitude changes. Hence thederivatives for these conditions are not frequency-dependent over the practicalfrequency range.
For wings that exhibit nonlinearities in their static forces and moments (partially separated flow
"exists), the dynamic derivatives also depend upon the amplitude of the attitude changes. For
instance, sketch (a) ( reference 2) shows the rolling moment in sideslip for a triangular wing athigh angles of attack. It is clear that the rotary derivative C1r depends upon the amplitude ofoscillation about the Z-axis.

L

i•!

K-

7.1-1

.

p..7Fir.,r.r.-rrrrdr.....

.••

*r

:.

.

. -

..-

•

*.;--.
'•''••

.

••

"

'"

". -.

'

-'

- -.

- -

C-)
.81

SKETCH (a)REFERENCES
1.

Campbell, J. P., Johnson, J. L., Jr., end Hewes, D. E.: LoW-Speed Study of the Effect of Frequency on the Stability
Derivatives of Wings Oscillating in Yaw with Particular Reference to High Angle-of-Attack Conditions. NACA RM L55-06,

2.

Johnson, J. L., Jr.: Low-Speed Measuremrents of Static Stability, Damping in Yew, an--dDamping in Rail of a Delta, a
Swept, and an Unmavpt Wing for Angle of Attack from 00 to 90N. NACA RM L56B01, 1956. (U)3.

Fletcher, H. S.: Low-S•ped Exporimental Determination of the Effncts of Laeding-Edge Radius and Profile Thickness on
Sttic and Oscillatory Lttra! Stability Derivatives for a Delta Wing with 000 of Leing-Edg
1958. (U)Sweep. NACA TN 4341,

L.

S

7.1-2

*

-.

-I-

-

--

-

S

Revised September 1970

7. 1.1 WING PITCHING DERIVATIVES
7.1.1.1 WING PITCHING DERIVATIVE CLq

The wing pitching derivative CLq is generally small compared to other terms in the equations of
motion and is frequently neglected. However, methods are presented for determining CLq of thewing in subsonic and supersonic speed ranges. The supersonic value of CLq is used in estimatingsupersonic values of Cmq in Section 7.1.1.2.

"If the wing pitching derivative CLq is to be used in method 1 of Section 7.3.1.1 to obtain(CLq)WB, the exposed wing planform area should be used for all calculations in the D1atcon)
methods. Using the exposed planform area will yield CLq based on the product of exposed wingarea and exposed wing MAC, rather than the product of total wing area and 'Ving MAC isindicated.
DATCOM METHODS
A. SUBSONICThe equation for estimating the subsonic pitching derivative C' (derived in reference 1), based onq
the product of wing area and wing MAC SW dw, is given. by

C

Lq

+2

7.1 1.1-aawhere --can be expressed asc

7.1.1.1-b

a.c.

and
is the distance between the center of gravity and the aerodynamic center, positive foraerodynamic center behind center of gravity.

"xa'

(x~\

"C2\.Cr')4.1.4.2k--), the longitudinal distance from the wing leading-edge vertex toC)

the aerodynamic center measured in mean aerodynamic chords, positive aft.
xc'g.
--- ~
c

is the longitudinal distance from the wing leading-edge vertex to the center of gravity
measured in mean aerodynamic chords, positive aft.
7.1.1.1-1

L
i,

--

-

-

-

-

-

(Section 4.1.3.2) at the Mach number under consideration,slope
isthe wing
based
on thelift-curve
total wing
area.

Ca

Sample Problem
Given:cr

A

X= 0.68

4.0

46.30

(from planform geometry with c.g. at ý/4)

Xc.-1.04
Cr

CL

ALE

3.20 per rad=-- 1.18M = 0.20(Section 4.1.3.2)

Compute:xusing the method of Section 4.1.4.2

Calculate
cr
=

1.046

Atan ALE

4.18

tan ALE

S= 0.98
3/tan ALE

0.937

Xa.c.

- 1.05(figures 4.1.4.2-26d, -26e, -26f)

CrC,•culate the distance between the center of gravity and the aerodynamic center
"-

7

Xa.c.

xc.

Cr

7.1.1.1-2[(equation 7.1.1.l-b)

= (1.05 - 1.04) 1.18
= 0.0118
Solution:
Calculate the wing pitching derivativeCtq =+ 2 c'ý

T 2+ 2(0.0118

CL aC(equation 7.1.1.1-a)

3 2

. 0

-

(0.52) (3.20)-

1.66 per rad (based on SW c)

B. TRANSONIC

Pg

There are few data and no theory available on the derivative CLq in the transonic region. For
the purpose of the Datcom, it is suggested that equation 7.1.1.1-a be applied in the transonicregion.

C. SUPERSONICThe supersonic value of CLq , based on the product of wing area and wing MAC SW•ýw, is given byC

L C Lq'+2)N~C(per radian)

7.1.1.1 -c

where
x

aC"

x

xC "

7.1.1.1-b

andCL

is referred to body axis with the origin at the wing aerodynamic center and is obtained
as indicated below, based on the product of total wing area and wing MAC SW cw"

SCN

is the wing normal-force-curve slope (Section 4.1.3.2) at the Mach number underconsideration, based on the total wing -area (per radian).
7.1.1.1-3

--

-- -

Methods of estimating CLLq
1.

Wings with subsonic leading edges (0 cot ALE < 1.0)
For wings with subsonic leading edges, CLq is obtained by the method of reference 3 for
LX= 0 and by the method of reference 4 for X = 0.25 to 1.0. The following methods are
not valid if the Mach line from the trailing-edge vertex intersects the leading edge or if thewing-tip Mach lines intersect on the wings or intersect the opposite wing tips.
a.

Zero-taper-ratio wings (.

0)

CLq' is derived in reference 3 asL(d - Xac.)

11

CL
q

- A[3G(Q3C) F 3 (N) - 2E"(PC) F4 (N)1 + 2(N]+2aN2

cCN(per radian)(eain

7.1.1.1-d
whereE"(P3C) and G(3C) are obtained from figure 7.1.1.1-8.
F 3 (N) and F4 (N) are obtained from figure 7.1.1.1-9.
d is two-thirds the basic triangular wing root chord (d =(See sketch (a).)

A is the aspect ratio of the wing.

r
Cr

BASIC
TRIANGULAR
WING

Brj

_

0

*

S..SKETCH

L7.1.1.1-4

(a)

b.

0.25 to 1.0

Wings with X,

CLq' is derived in reference 4 as
"CL'

CL"

- 2

CN

(per radian)

7.1.1. 1-e

where
Z:

C Lq" is referred to body axis with the origin at the wing leading-edge vertex and is
q.obtained from figures 7.1.1.1-10a,7.1.1.1-10b,and 7.1.1,1-10c, for X=0.25,0.50,
and 0.75, respectively, and from the equations of reference 4 for X > 0.75.
2.

Wings with supersonic leading edges (, cot ALE > 1.0)
For wings with supersonic leading edges, CL q' is obtained from figures 7.1.1.1-11 a through7.1.1.1-1 1k. This method, derived in reference 5, is valid for the range of Mach numbers forwhich the Mach lines from the leading-edge vertex intersect the trailing edge. An additional
limitation is that the foremost Mach line from either wing tip may not intersect the remotehalf of the wing.
Sample Problem

Given:
C~r

A

p

A..

3.46

600

-

c.g. at -b = 16 ft

4

=

1.50

M

cr

9.25 ft

=

X

1.50

=

0

6.17 ft

Compute:Scot ALE

0.647 (subsonic leading edge)
4 cot ALE

-0.333A

N= 1E"(O3C)
E"G(C)G(PQ_.F3= 0.770= 0.570k= 0.5701== 0.913}"' (N)
F4(N)
0.913
F4

=

1.12

(figure 7.1.1.1-8)(figure 7.1.1 1-9)
7(N)

,.

N, ý

Obtain C\

from Section 4.1.3.2

o3/tan ALE

0.647

=

A tan ALE

6.0

tan ALE C= 5.40 per rad(figure 4.1.3.2-56a)

3.12 per rad

CN
NO

x
-

Obtainfrom Section 4.1.4.2

4i

ac
-"

0 .90

.a c

(figu re 4 .1.4 .2 -26 a)Cr

x a.c.

x
a-)

-c

x

(C)

1.3eq5in.. 11b(from Cplanform geometry with

c

*0.67

8.33

0 ( 1.35)(6.17)0

xa.c. 1.35

Calculate xds

3

r

(~

• .....

e.g.
at U/4)(equation 7.1.1.1l-b)

- .

bc r
-Xa..c.

: --XcC,.)

C,=(0.90 - 0.67) 1.50

i •?-Calculate=0.345d from the characteristics of the basic
AB-

b2•
I•1 b

=-

4
tan A L

=--

4
1.732

.
triangular wing (see sketch (a))= 2.3 1
.

2

rb

2b
b
AB

Cr

"C

2(16)

=

2.31

2

2

d = -c-13.85 ft=

(13.85)9.233 ft

Solution for Cq'
Lq(d - Xa.c.)

irA

CL=q

[3G(,3C) F 3 (N)-

2E"(3C) F4(N)] + 2(equation 7.1. 1.1-d)h!-=-•r(9.233

2- 8.33)(3.46) [3(0.57) (0.913) - 2(0.77) (1.12)J + 2

(

.23
6.17

(3.12)= -0.891 + 0.913=

0.022 per rad (based on SWw)

Solution for CL
Lq
Lq

CLq

2

y

N"

(equation 7.1.1.l-c)

q

-

=

0.022 + 2(0.345) 3.12

= 2.175 per rad(based on SW CU)

REFERENCES
1.

Tol, T., and Qusijo, M.. Approximate Relations and Charts for Low-Speed Stability Derivatives of Swept Wings. NACA
TN 1581, 1948. (U)2.

MacLachlan, R., and Fisher, L. R.: Wind-Tunnel Investigation at Low Speeds of the Pitching Derivatives of Untapered
Swept Wings. NACA RM L8G19, 1948. (U)3.

Malvestuto, F. S., Jr., and Mergolis, K.: Theoretical Stability Derivatives of Thin Sweptback Wings Tapered to a Point with
Sweptback or Sweptforwaid Trailing Edges for a Limited Range of Supersonic Spgds. NACA TR 971, 1950. (U)4.

Malvesturto, F. S., Jr., and Hoover, D. M.: Lift anl Pitching Derivatives of Thin Sweptbeck Tapered Wings with Streamw'se
Tips and Subsonic Leading Edges at Supersonic Speeds. NACA TN 2294, 1951. (U)5.

Martin, J. C., Morgolis, K,, and Jeffreys, I.: Calculation of Lift and Pitching Moments Due to Angle of Attack and Steady
Pitching Velocity at Supersonic Speeds for Thin Sweptback Tapered Wings with Streamwise Tips and Supersonic Leading
and Trailing Edges. NACA TN 2699, 1952. (U)S~7.1.1.1-7

StJPVRSONIC SPEEDS

-d-T

-71~

.66.
8COTACL00

FIGURE 7.1.1.1-8

ELLIPTIC INTEG~RAL FACTORS OF THE STABILITYDERIVATIVE-• -••1 •

,

,•,

-

;

:

• .

-•

•

-

,

•

:

,

c

•

-

-

- •

'

. -

•

SUPERSONIC SPEEDS

7.5-

I

7.06.5a
6.0/

5.5m/

/

/

(1- N)- a

4(N)

-4--

4.0--

III~iII
--

---

2.5

-.

'

~

.

-

-

-

--

1.2

M,1.0

2.!

N

7-.6

Ti~i~

4.

-1.0

-. 8

-. 6

-. 4

-. 2

0

TRAILING-EDGE-SWEEP PARAMETER, N

FIGURE 7.13.1.-9

F(N) FACTORSOF THE STABILITY

.6

.4

.2

.8

Lo.

1 .
A

OEfIVATIVE

7.1.1 .1-9p.

SUB3SONIC LEADING EDGE

32-

IJI(a)N

0.25

T-

i'--

LJ

I
-U

ISPROISUBSONIC
B
TE-

I-jTEj28"

A~I1il.24--/C20------

8Lq

12

-

4J
0

.2.4810/3COTAL(b) Nr0.50~-J---.---~SUPERSONIC TE

28

0

--

(petrtad)

-

-

12.1

4o-

0

.2

.

6.
/3COT AFIGURE 7.1.1.1-10

VARIATION OF /3CL

WIHL3OWIH1qOAE.

SUPERSONIC SPEEDSSUBSONIC LEADING EDGE

32

MT

0751

7

T ~

7~

SUBSONIC TE~SUPERSONICTE

~~

24

.A,6

-

~

(per cad)-

126-

4

4i

0

.2

.4

.6

.

.

83GOT ALk

FIGURE 7.1.1.1-1O(CONTD)

SUPERSONIC LEADINC, EDGE

r8---------

08

k 1

-

_

-

___

p..

F.

'3C L

r4

q

0

4

8

12

16

"29
COT

FI( UP E 7. 1.1.1-11
hW

VARIATION OF

28

2(/3COTALE(deg)

WITH COT-' (13COT AL'71.-I
l

32

36

40

44

48

SUPERSONIC SPEEDSSUPERSONIC LEAVING EDGE

--i
-~~i

i

-

-

-A

Vr-*r-r
V t..
4-

--

lpe

4-4

trd)

-4-

-t

-

-10,

t48&7 24 4 4_4

-~4

LIK
-14
(perO(ad)-I,---- 'COT
FIGURE(df

COT.1111(CONTO)~

1-1~I.2
6*(/3COTA, F)(deg)1PERSRONIC SPEED$"SUPERSONIC LEADING EDGE~(d) •, -0.25

.':

71

LL

(pet red)

_

-

-1.6"

_

--

-2.D
-2.4"

-3.2_

0

t

__

4

8

12

16

20

24

COT-1 (I3COTA L)

28

32

36

48

44

40

(dog)

0.

•

-12"

I

_

A,6

2-

---

_

.

12
FIGURE 7.[.1.1-11 (CONTD)

16

20
CTL

24

28

(7.

32

36

40

44

48

1. .(deal,SLIJiERSONIC WPEEDSSUPERSONIC LEADING EDGE.

.3,

k.

/

3

A/3

00

COT-'(,BCOT ALE) (des)A,8FIGURE 7.1.1.1-11 (CONTD)[S

7.1.1.1-14

74
SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE

63C

(per rad)

0~

12

20

1 6

COT1

-

FIGURE- 7.1.1.1--11 (CONTD)

228(,e COT A L E~(deg)

3

36

4

44

4-

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE
2()?=0. 50

-2-

4-2

-to.

(per
cad)I

V

-12-

-188

088
L

-22

L.

FIGRE1.1.1111.16TO

I

f

T

T.04v

SUP! RSONiC SPEEDSSUPER~SONIC LEADING EDGE
COT-

04

I.

2

12

t

--16-

(,600TAE) (deg)

-"
2

2

6t4

4

.-

-44.

2
L

7.

"SUERSONIC SPECDSSUPERSONIC LEADING EDGE
COT-' (,6 COT A LE) (deg)
0

-22

4

-6"

*

7.11.1-1

12

16

20

24

28

32

36

2

8

3

i. 6 z0- 1 0 -2---

L.4

8

----

---

----

----

2

---

6

----

2

-2

40

44

48

Reviwd Septem•mr 1970

7.1.1.2 WING PITCHING DERIVATIVE C"qWhen a wing rotates in pitch about a given center of gravity at an angular velocity 0 in afree-stream velocity 0.. , changes in local angle of attack are produced that are proportional tow/V (w is the local Vvertical
disturbance velocity). For wings having sweepback, an additional
increment in angle of attack is produced that is a function of spanwise wing station. These local
changes in angle of attack produce an effective angle-of-attack increment for the complete wingthat results in a pitching-moment increment. This pitching-moment increment is expressed as thewing contribution to the derivative Cmq.
If the wing pitching derivative Cmq is to be used in method 1 of Section 7.3.1.2 to obtain(CM, q ) w, the exposed wing planform area should be used for all calcu;ations in the Datcom

methods. Using the exposed planform area will yield Cmq based on the product of exposed wingarea and the square of exposed wing MAC, rather tIlhn the product of total wing area and thesquare of wing MAC as indicated.
DATCOM METHOD
A. SUBSONIC'i

The low-speed value (M - 0.2) of Cm q based on the product of total wing area and the square ofwing MAC Sw cw 2 , is given by
A [1 .+2(.g)]
(C

mq

m

)

M- 0.2-0.7 cj, cos Ac/ 4cQ

A

3 tan 2

Ac,4

/

+

24

A+6cosA,/(per radian)

where
-

A2cZ

A + 2 cos Ac/4

12A

is defined in Section 7.1) I
8

7.1.1.2-a.1.

is the wing sec ion lit. -.urve slope from Section 4.1.1, (per radian).

This equation is a modified foJrm of that derived in reference 1. The equation was modified inreference 2 by the empirical factor 0.7. It is strictly applicable to aspect ratios between I and 6.
For aspect ratios of about 10 or 12 the empirical factor should be approximately 0.9, but there
are no experimental data available to show how this empirical factor should vary for intermediate aspect ratios. It is suggested that a smooth fairing be used.
For higher subsonic speeds the derivative Cm q, based on tW'e product of wing area and the square.
of wing MAC SW cw 2, iL oI tamed by applying an approximate compressibility correction derived
in reference 3:

K

7.1.1.2-1

A 3 tan 2 Ac/ 4 + 3
AB + 6 cos Acl4
B
M>)C

B+c0
1
A3 tan2 AMl4-

q M> 0.2(C

A + 6 cos A,/4

7.1.1.2-bradian)B(per q )m
-+3

where
is obtained from equation 7.1.1.2-a.

(Cmq)
B

V/1 - M2 cos 2 At/4=

Sample Problem
Given: Same wing as in sample problem of paragraph A, Section 7.1.1.1.
CA

X = 0.68

4.0

=

r

450

Ac14

1.18

Additional Characteristics:
Smooth airfoil surface

Airfoil: 64-006

M0.2, 0.6

0.0118 (sample problem, paragraph A, Section 7.1.1.1)

Compute:cga

0.109 per deg

cos Ac/ 4

At M(table 4.1.1-B)

0.707; tan Ac/ 4- 0.6, B =1.0

M2 cos 2 Ac/

11/1 - (0.36) (0.50)

0.905

Solutiuo. (M = 0.2):

Ic\.~~
q M, 0.2

cA

cYA~2(
2 c A./4'/A 3 tan2 A,,,)

24

A+6

Ac14

8

(equation 7.1.1.2-a)"7.1.1.2-2--___77_7..

-(00118) + 2(0.0118)2--

4-"(-0.7)(0.109) (57.3) (0.707),+72(0.707)

'

J .

1(

24

:

4 )'

1

1

4+6(0.707)

8

-•=
+
-.-3.090
9 LS-i0.0247 +21 (• 64[5.414k8.242)24

8]= -3.090(0.00456 + 0.3235 + 0.125)=-1.400 per rad (based on Sw Z'w 2)

Solution (M = 0.6):
A3 tan2 A,,4(c)CAtM>

3B

AB + 6 cos AA
L0.2
A3 tan2 A,4M
A

6 cos' Ac/4(

(4)3 (1)3

4(0.905) + 6(0.707)

0.905(-1.40)+ 3

4 + 6(0.707)

L64-7.862+ 3.3104

8.242(8.143.)-(equation 7.1.1.2-b)' 3(4)3 (1)1)Cmq(- 1.40)+ 3(-1.40)-1.49 per rad (based on Sw ý 2 )7.1.1.2-3

q

B. TRANSONICAt transonic speeds the derivative Cmq based on the product of wing area and the square of wing
MAC Sw zw , is estimated by

C q+=(CL).-( CZ o 7 12. (CCmq[Cq)il

cr

m

)

.

--

)

+(C'q)Mcr(per radian)

Cn

\rj

7.1.1.2-c
*•

where
CL

is the wing lift-curve slope (Section 4.1.3.2) at the Mach number under consideration, based on the total wing area (per radian).

C
,
(C

*

1.2

is obtained from the supersonic method of paragraph C of this section, basedon the product of wing area and the square of wing MAC SW c 2 (per radian).
is obtained from equation 7.1.1.2-b at the critical Mach number, based on the
product of wing area and the square of wing MAC Sw w2 (per radian).

,q/

For this purpose the critical Mach number M., is taken equal toa the force-break Mach number Mtb
defined in paragraph B of Section 4.1.3.2.

C. SUPERSONICThe supersonic value of Cm,, referred to body axis and for any center-of-gravity location andbased on the product of wing area and the square of wing MAC Sw Zw 2, is given by

Cmq-- Cmq-(x)CLq (per radian)

7.1.1.2-d

where
CLq

q

is obtained from Section 7.1.1.1, based on the product of wing area and wing MAC(per radian).

-is defined in Section 7.1.1.1.

Cis referred to body axis with the origin at the wing aerodynamic center, based on the
product of wing area and the square of wing MAC SW Cw and obtained as indicatedbelow.

i

7.1.1.2-4

Methods for estimating Cm
Iq

Wings with subsonic leading edges (j3 cot ALE <S'"Forwings with subsonic leading edges, Cm,,'

1.0)

is obtained by the method of reference 4 for

X = 0 and by the method of reference 5 for X
0.25 to 1.0. The following methods are
not valid if the Mach line from the trailing-edge vertex intersects the leading edge or if thewing-tip Mach lines intersect on the wings or intersect the opposite wing tips.

"a.

Zero-taper-ratio wings (X = 0)
Cm q' is derived in reference 4 as

C

'

-

rF

3

N)6

3(N)

q)(
C

(per radian)+ 2

d - x.
--

F(PNC

"

CL q'

C

7.1.1.2-e

N

where
G(PQC) and E"(PC) are obtained from figure 7.1.1.i-8.
F 5 (N), F 7 (N), and FI(N) are obtained from figure 7.1.1.2-8.

is defined in paragraph A of Section 7.1.1.1.

-

SCN

is the wing normal-force-curve slope (Section 4.1.3.2) at the Mach number underconsideration, based on the total wing area (per radian).
d

is defined in paragraph C of Section 7. 1. 1. 1.

CL

is obtained from paragraph C of Section 7.1.1.1, based on the product of wingarea and wing MAC (per radian).

qb.

Wings with X = 0.25 to 1.0
Cm

q

'

is derived in reference 5 as

Cmq'-Cm q" +C(per radian)

7.1.1.2-f

7.1.1.2-5

where
C

"

Sq

CL

"

is obtained from paragraph C of Section 7.1.1.1, based on the product of wingarea and wing MAC (per radian).
is defined in paragraph A of Section 7.1.1.1.

X

2.

is referred to body axis with the origin at the wing leading-edge vertex and is
obtained from figures 7.1.1. 2 -9 a, 7.1.1.2-9b, and 7.1 1.2-qc,for X = 0.25, 0.50,
and 0.75, respectively, and from the equations of reference 5 for X > 0.75 (perradian).

Wings with supersonic leading edges (0 cot ALE > 1.0)
For wings with supersonic leading edges, C, 'is obtained from figures 7.1.1.2-10a through7.1.1.2-lOk.This method is derived in reference 6 and is valid for the range of Mach
numbers for which the Mach lines from the leading-edge vertex intersect the trailing edge.
An additional limitation is that the foremost Mach line from either wing tip may notintersect the remote half of the wing.
Sample Problem

Given: Same wing as in sample problem of paragraph C, Section 7.1.1.1.
A = 3.46

X

C.g. at4

0
M

ALE

600

1.50

1.50
6

b = 16ft

6.17 ftc7

9.25 ft

4
From sample problem of paragraph C, Section 7,1.1.1:
f cot ALE = 0.647 (subsonic leading edge)
N

=

0.333

E"(PQC)

0.770

G((IC)

5.70

CN

3.12 per rad

Sd-x.c.

"Cq

0.022 per radCLq2.175 per rad

Compute:

7.1.1.2-6

...

SF(N)

01
=0133

F 7 (N)F, (N)= -0.200= 0.660(figure 7.1.1.2-8)- 0.146-

0.345

Solution:

M
C~=

1

T

G(jC) F7 (N) + 1- E (PC)

irA

q3

.

CL q' + 2 (d(d,.,)-

LF I
1

(N)l

(equation 7.1.1.2-e)

Xac

0.1331[(057.16

S7r(3.46)

0.570) (-0.200) + L---

(0.770)3

160.660](0.146) (0.022) + 2(0.146)2 (3.12)= -2.038(-0.1140 + 0.8276) - 0.0032 + 0.1330= -1.325
CK.

per rad(ISq

mq= -1.325

CLq(equation 7.1.1.2-d)\CCI- (0.345) (2.175)= -2.075 per rad (based on SWzw 2 )REFERENCES
1.

Toil, T., and Queijo, M.: Approximate Relations and Charts for Low-Speed Stability Derivatives of Swapt Wings. NACA
TN 1581, 1948. (U)2.

MacLachlan, R., and Fisher, L. R.: Wind-Tunnel Investigation at Low Speeds of the Pitching Derivatives of Untapered
Swept Wings. NACA RM LSG19, 1948. (U)3.

Fisher, L. H.: Approximate Corrections for the Effects of Compressibility on the Subsonic Stability Derivatives of Swept
Wings. NACA T"N 1854, 1949. (U)4.

Malvestuto, F. S., Jr., and Margolis, K.: Theoretical Stability Derivatives of Thin Sweptbsck Wings Tapered to a Point with
Sweptback or Sweptforward Trailing Edges for a Limited Range of Supersonic. Speeds. NACA TR 971, 1950. (U)5.

Malvestuto, F. S., Jr., and Hoover, D. M.: Lift and Pitching Derivatives of Thin Sweptback Teaered Wings with Streamwise
Tips and Subsonic Leading Edges at Supersonic Speeds. NACA TN 2294, 1951. (U)6.

Martin, J., Margolis, K., and Jeffreys, I.: Calculation ot Lift and Pitching Moments Due to Angle of Attack and Steady
Pitching Velocity at Supersonic Speeds for Thin Sweptback Tapered Wings with Streamwise Tips and Supersonic Leading
and Trailing Edges. NACA TN 2699, 1952. (U).

7.1.1.2-7
l.t

SUPERSONIC SPEEDS

6.1

0-

T 1 ______

1.8

__

-~~~

jN)-

(I

---

FI 7 (N)F 7 (N)

I

ii

-E

-

F5(N)

1.2

-

- 2 -}10--F-f--

F~(N)

.8

.2

-

Fý(N)

0--(-N)

-10-8

-6

-. 4

-.2

TRAILING-EDGE-SWEEP

0.2

PARAMETER, N.4

I-

F(N) FACTORSOF THE STABILITY DERIVATIVE{.64 COT ALE
L

A

FIGURE 7.1.1.2-8

-

{

-71-

--

-

-

.8

-

1.0

.2

SUPERSONIC SPEED$SUBSONIC LEADING EDGE

140-]Ti

SUPERSONIC

II

----

100

"

--

-t--

-

g-f
(p errod)

O0N

-

40

-

-

200

.2.4

681.
83COT AL 6

V-7ri--

140-~~7-SUBSONICTE~SUPERSON~IC TF

800(per fad)

6

.2

K0

.4.6810
8 COT AL

FIGURE 7.1.1.2-9

VARIATION OF GCm

WITH 83COTA~
qL

E

SUPERSONIC SPEEDSSUBSONIC LEADING EDGE

140-

1

75
(; ,= 0.75

-

I

ii

100=

-suPERs-oN-TE---SP R•NCI

TONIC

TEl

i

t

---

(per
fad)60
2,"40

-

00

.6

.4

.2

0."-

1.0

.8

8 COT ALE
FIGURE 7.1.1.2-9 (CONTD)-.

,

iy-SUPERSONIC LEADING EDGE

'

1.5

A

--

j

/

,

1.2(per
t ad)

4

:=;22

;.,::v.

;:"

71/ .2-10.7 0
FIGURE 7.1.1.2-10

4

8
VARIATION OF

12
6Cm

16

2U
24
28
COT (/3COTAL.r (deg)
WITH COT-' (/8CT ALE)

32

36

40

44

48

S-T

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE

(b) X

24-

(per rad)

i i

_-

20-

_--

_

2

0AS

-

1

1-

-

-1

:. 1I2-4

7...2I-.;;;;f.

5
4

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE

1.6..-

--

-

0.2

1.4

-

1.3

.1

_

20

1.2

....

(per tad)

-

-

-

.

-

-

-

-

---

1.2,

.

---

-

__

-------

-.

.

+

11

_

__

40

48

COT' (/3 co AL ~deg)
LI

55
25_
(d) N=0.25

2.0

-,

-

.

..

-

7

-_

-

-

-_

--

---

-- --

--

-/

./-1

20
12

30"-

0

8

-

624

3!2

40

4

(per rad)F..11.-1

25-

(d)

I

025

200

COT'

FIGURE 7.1.1.2-10 (CONTD)(ACOTK) (dog).

,.

SUPERSONIC SPEEDS
3.0.

LE AING EDGE

-UPE-RSONIC--T-(e) k -0.50-

At3

2.24-----~(per rad)

F

-2

1.64-

r

----

-

1.2-

COT

0-

(f) X~ 0.501

-

60'

50
__

(per tad)

~1~

t-

~
--

(,83CUT A,~ (des)

-

-

--

-

_

--.--.-

3.---__

Y22_

-

-

-

12--

20

Ci8-16COT

FIGURE 7,1.1.2-10(CONTO)

24-32(/COT At.E (deg)

40

48

~- -

SUPERSONIC SPEE)6
SUPERSONIC LEADING EDGE

1,1.----------- t-t------

- .

.

-

t 2p2
- ..

.

- -- . -....-----.--.

.

--

4--

(Perird) 9.7-

I.............................

L
*.

(pe•-

-

-

-

-

-

~-•---- •_.- . .....

.

I..
COT' (613COTAAL)(dog)--.......

...

L

/

(h) --0.75I/

ILe
27

-

•

32mi"

--

-- --

--

-- -,-

--

--

--

--

____

"

,-,-

--I-

""

=----

--

=----------------.-

16-I(pat rad)

16,Z

, 1

2

24 -

/71.2-14

0

8

16

24
COT

FIGURE 7.1..L2-10

32

40

48

()3 CUT AL'r (dogi)(CONTD)=.

-.

I~ ~ r--

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE

(i \=0.75

1.7-

1.4

--

1.3----

Ile-.

(p7.1.1.2-1.2

-~

---

~

-

---

--

---

-

~-.7v.

""UPERSONIC SPEEDSSUPERSONIC LEADING EDGE
2.4,-

-

2.?-1

1

-

J

--

_

-

•l

.

2, ... . ..

6

2

1. iL

-

16

8

0

'

I

-

--

24

COT"' (,3COT A L) (del)

•I-

2

,

-

48

40

32
-

.]_

__

I _

I

-

.i

17'I-III(perrtad)

400

___l

30. .. .

.

-2.LI................
....

--

--

I

10•.

o

8

16

24

32

40

48

COT" (/3COT ALI) (deg)FIGURE 7.1.1.2-10

7.1 .1.2-16
S(CONTD).

-I

April 1978

7.1.1.3 WING PITCHING DERIVATIVE CDqqThis scction presents a method for estimating the wing contribution to the pitching derivative CDq
at subsonic speeds. This derivative is the change in drag coefficient due to a change in pitchingvelocity at a constant angle of attack and is defined as
CCqC

where CD is based on Sw.

-

In general, this derivative is small and has a negligible effect on longitudinal stability; hence, it isusually neglected.
A, SUBSONICThe method presented herein uses the Weissinger method of Reference I to calculate the section liftK

due to angle of attack, twist, and pitch rate. The lift is then rotated through the local downwash
angle due to each of these effects to produce a chordwise component of force.
DATCOM METHOD
Design charts for predicting the wing contribution to CD
0.2 and 0.8. The wing pitching derivative CjDq is given by

C~~~
C

)D

ICDI
. "0a+

are available only at Mach numbers of

(q
DC(O+
~qC~
(L

2V

F +

7.1.1.3-a

where

C

D0

o

is the contribution due to zero-angle-of-,attack loading obtained from Figures
7. 1. 1.3-3a through -3f as a function of wing aspect ratio, taper ratio,
and sweep.
isthe wing twist in degrees between the root and tip sections, negative for washout(see Figure 5.1.2.1-30b).

aCD
is the contribution due to angle of attack obtained from Figures 7.1.1.3- 7 a through3OF- 7f as a function of wing aspect ratio, taper ratio, and sweep.

at;

is the fuselage angle of attack in degrees.

'acJ)-cq(V Nis the contribution due to the rate of change of pitch obtained from Figures

7.1.1.3-12a through -12f as a function of wing aspect ratio, taper ratio, and sweep.

"r7.1.1.3-1.

qis the pitch rate in degrees per second,C:-

is the wing mean t *)dynarnic chord.

V

is the free-stream velocity.
Sample Problem

Given: The following wing-body configuration
Wing Characteristics;
A

35°

7

A = 0.25

ALE

225.0 ft

Sw = 3500 ft 2

0 = -5.0

aCF = 10q

Additional Characteristics:

M = 0.8
V = 796 ft/sec(h1

5 deg/sec

30,000 ft)

Compute:0.00036 per deg 2 (interpolated using Figures 7.1.1.3-3d, -3e, and -30)

CD

q-

aCo

q#

S..
2V

0.000876 per deg 2 (interpolated using Figures 7. 1.1.3- 7d, - 7 e, and=- 0.00061 per deg 2 (interpolated using Figures 7.1.1.3-12d, -12e, and -12f0.

(5)(2S)92(796). 0.0785 deg-

Solution:CDqCl)qq3( 0) + -- A-31a[; .

I~ \'-]_}q-A(Equation 7.1.1.3-a)(0.00036)(+5.0) + (0.00087(1)(1.0) + (0.0001)(0.0785)
0.0027,1 per deg
7.1.1.3-2- 7 f)

B. TRANSONICNo method is presented.
C. SUPERSONICNo method is presented.
REFERENCE
1.

U

Do Young, J., and Harper, C. W.: Theoretical Symmetric Span Loading at Subsonic Speeds for Wings Having Arbitrary
Planform. NACA TR 921, 1948. (U)

"M .2
ALE

J

.0006

0

=

[

.0004
C

7
D

(per deg 2 )S...~~.

0002 ..-

0

"

.2

.4

.6

.8

TAPER RATIO, X

FIGURE '7.1.1.3-3 THE ZERO-ANGLE-UF-A'ITACK LOADING CONTRIBUTION TO THE
WING PITCHING DERIVATIVE CD
q7,1.1.3-3

LI

M=.2
.000

300

__ALE

.0004 ()
CDq--

(per deg 2)

TAPER RATIO, X
M
AL.0004*-

t

=.2

600

______________

(C)

0002-

.A

-.

7I
18

0

..............

1..0
R RATO,-M = .8
ALE

=

0

(d)

.0008

.0006-(per deg 2 ) .0004'

0

.2

.4

.6

.8

1.0

TAPER RATIO, X

M

ALE = 300(e).o0oo4i.0004-

.8
_

---

A

4

CD

(per deg2)-•i~.0002-

, .70

.2

.4

.6.8

1.0

TAPER RATIO, X

FIGURE 7.1.1.3-3(CONTD)

7.1.1.3-5~..
- ,.~-a.-.---a~---.-.~'-----

.

~

a-

.

~.

-

-

-

-~

-

-

M=.8

ALE
_

600

_

_•04

.00021

c.
Dq

2)o 0

(perdeg.0002-.0004-

FIGURE 7.1.1.3. 3 (CONTD)

7.11.3-6.............
.2

.4

.6

APER RATIOd,

.8

--

1.0

M =.2".0 08ALE;(a)A.0006FlF

2)(per deg
.0004

•

w...•

•

10

m,,
'

L.

S~.0002-•

I..

0

.2

-..

.4

.6

-

-'-

.8

1.0

TAPER RATIO, X

FIGURE 7.1.1.3- 7 THE ANGLE-OF-AYTACK CONTRIBUTION TO THE WING PITCHINGDERIVATIVE CD
q

7.1.1.3-7

M)~.0008.

.2

ALE = 300

.000647FF(per deg2 )

1.0•.0002.
M2.

0,
0

.2

.4

.6

.8

1.0

.8

1.0

TAPER RATIO, X

M 1..2AL E=60°.0006, (c).0004.ý•acD(per deg2)0
FIGURE 7.1.1!.3- 7(CONTD)
7.1.1i.3-8

.2

.4

.6

TAPER RATIO, X

M= .8ALE.00 120(d).00 10 ..
10.0008-

TV .0006"

b~aa

>F

"-(perdeg2 ),.0004-.,,..•.0002

o.4

.2
"0

.4

.6

.8

1.0

TAPER RATIO, X

FIGURE 7.1.1.3- 7(CONTD)
7.1.1.3-9

II

M
ALE

e

.0010

f.

--

8
-

N4
A7

•

".008

10I

.0006
aaF
2

(per deg ).0004--

1.
0.2

.4

.6.8

TAPER RATIO, X*t

FIGURE 7.1.1.3- 7(CONTD)"

7.1.1.3-10

1.0

2

F.7

-

.-.

..

-- 600
ALE
M=.8

'0MO

acD
aF

.0004-

4(per deg2 )

1

41V
.8:

1.0

TAPER RATIO , X,

77

FIGURE 7.1.1.3- 7(CONTD),•-

7,1.1.3-11

i i llM

S(a)

ALE

= .2
0

dCD q

._. _ _ 1
.00024.)(per deg2 )

"0

.2

.4

.6

.8

.0

TAPER RATIO, X

L-'11"FIGURE7.1.1.3-12 THE PITCH-RATE CONTRIBUTION TO THE WING PITCHINGDERIVATIVE CD

r.ip"

7.1.1.3-12

SI

.0012

M

.2•

AL•

300

b

(b)A
-10

-

.0010-

.0008

--

/

F7

aCDq

S.0006.

(per deg2 )

4.0002.

:),

0
0.6

.8

1.0

TAPER RATIO, X

1

FIGURE 7.1.1.3-12 (CONTD)
7.1.1.3-13

"ALE.

.0045 -...-

[.

acoq

"

-

.

.-

.0034 .

3

---~

600

.-

/O

(per deg2 )

r_-

-•

----

--

--

TAPER RATIO, X

FIGURE 7,1331,2 (CONTD)
S-:",,.,-.

7- .1,.3-14..

M--.80.0008ALE:(d)-•.0006 -

A._

acD
ar-

.0004,(per deg2 )

0-

0

.2

.4

.6

.8

1.0

TAFER RATIO, X

4

FIGURE 7.1.1.3-12 (CONTD)
7.1,1.3-15

M
.0016

AL

( e)m-.8
toF 300

_

_

A

/I

•q
O1.00
028 - - -_

.0010

.

.

...

. ....
.

.

0

a/

ac

(per deg2 ).0006 -

..

--

.

..

.0 0 0 4-

_

_"_I

II
.4.

TAPFR RATIO, X
FIGURE 7.1.1.3-12 ((CONTrD)
7.1.1.3-16

.8

M=.
ALE =600~~~.005 :f)

•

(f,"

/A
01

.0033-

"

~

.C.

Sq/

~~(perdeg2)/

/

-

.0012

-4

.004

0

.2

.4

.6

.8

ED.'

TAPER RATIO, X!.

FIGURE 7.1.1.3-12 (CONTD)[_g.

7.1.1.3-17

Revised June 1969

7.1.2 WING ROLLING DERIVATIVES
7.1.2.1 WING ROLLING DERIVATIVE CyPThis section presents methods for estimating the wing contribution to the rolling derivative CyP at
subsonic and supersonic speeds. This derivative is the change in side-force coefficient with change inwing-tip helix angle and is expressed asacy

A. SUBSONIC
The wing rolling derivative Cyof rolling wings.

results from the angle-of-attack distribution and the tip-suction effects

The angle-of-attack distribution caused by rolling produces incremental changes in aerodynamic forces.
"For wings having sweep and/or dihedral, these incremental changes have components in the lateral
direction, causing a side force. The contribution to Cy due to the angle-of-attack distribution causedby rolling is derived in reference 1, based on simple sweep theory. This result is limited to swept,
untapered wings at low lift coefficients. The effect of taper ratio has been derived based on tc
experimental results presented in reference 2. The effect of dihedral is taken from reference 3. Although
the expression for the dihedn•' effect has been derived specifically for untapered wings, it should bereasonably reliable for wings o"' any taper ratio over the range of wing dihedral angles of practicalinterest.
The side force due to rolling of unswept wings is not accounted for by the theory of reference 1. Thisvalue is presumed to be caused by tip suction and is given by the empirical expression developed inreference 4. Experimental results show that the tip-suction effect is independent of sweep and varies
inversely as the aspect ratio.

-.

The method of reference 5 is applied to extrapolate the potential-flow values to high lift coefficients byusing experimental values of the lift and drag at high lift coefficients. If experimental lift and drag data
for the particular planform of interest are not available at the chosen Mach number, no attempt should
be made to estimate the variation of Cyp with lift coefficient. The negligible importance of thisderivative does not warrant the effort involved in estimating the wing lift and drag variation.
Furthermore, no known general method for estimating the variation of drag coefficient will give results
reliable enough to use in determining the correction factor for extrapolating the potential-flow values tohigner lift coefficients.

DATCOM METHOD
The variation of the wing rolling derivative CGy

with lift coefficient, based on the product of thewing area and wing span, is given by

e7.1.2.1-1

Kyp

CL

+

1C L/) L.O + (ACY )r7(per radian).1.2.1-aM

wherey(.3!)

is the slope of the side force due to rolling at zero lift given by

)
\-(CL

O

ML

A 4
AB+cosA/

Cy

c/4

c/4

L

AB +4 cos A c4

A +cos A,4

AB+4cosA

LCL)MC,

where BV--

M2 cos 2 AC/4

CLý

j C MLM"0

and

cyp14

CL=O
M
i0

CL
(ACy

is the slope of the low-speed side force due to rolling at zero
lift, obtained from figure 7.1.2.1-9 as a function of
aspect ratio, sweep of the quarter-chord, and taper ratio.
This chart has been derived by using the results of references 1, 2, and 4.
Equation 7.1.2.1-b modifies thelow-speed value by means of the Prandtl-Glauert rule to yieldapproximate corrections for the first-order three-dimensional
effects of compressible flow up to the critical Mach number.

is the wing lift coefficient.

)isle increment in Cy,1

(ACY)

due to dihedral given by[3 sin r (1 -2

sin

I) (C

)

(per radian)CLO

7.1.2.1-c

where

7.1.2.1-2

Fis the geometric dihedral angle in degrees, positive for the wing tip
above the plane of the root chord.

zis the vertical distance between the c.g and the wing root
quarter-chord point, positive for the c.g. above the wing root
chord. (This parameter is independent of angle of attack.)bis the wing span.

(CI

)

-

isthe roll-damping derivative of the wing without dihedral and at
c L 0 zero lift. This value is obtained from paragraph A of Section 7.1.2.2
and is given by (see equation 7.1.2.2-a)*( )3C/
C: /(per radian)

CL =0

K

is a dimensionless correction factor used to extrapolate the potential-flow values tohigh lift coefficients. At zero lift this factor is taken as 1.0. At lift coefficients otherthan zero this factor accounts for the variation of profile drag with lift coefficientand is given by

f--

a]

"a (CL tana--K

(CD - CD)

7.1.2.1-da•-(CL tana)-(---

Test values of lift and drag at the chosen Mach numbae for the particular planformof interest must be used in evaluating equation 7.1.2.1-d. The terms of equation
7.1.2.1-d are evaluated by taking the slopes of CL tan a, CD - CD0 ,and CL 2 /(irA), plotted versus angle of attack.

If reliable values of the static-force coefficients are available, the method should provide results within±20 percent accuracy throughout the lift-coefficient range to the st.ll.
Sample Problem
Given: The wing designated 32.6-4-0.6-006 of references 5 and 8.
Wing Characteristics:
A = 4.0

X = 0.60

Ar/ 4

32.60

=

0

S = 2.25 sq ft
Additional Characteristics:
M = 0.7
The following test values from reference 8 at M

0.7

CD0 = 0.01
CD0"The effact of profile drag on the roll damping at zero lift Is neglected.

7.1.2.1-3

CL

.05

1.

.2

.3

.4

.5

.6

.7

a

.75

1.50

2.95

4.30

5.60

7.20

8.80

11.20

CD

Oil

.0115

.015

.021

.033

.0535

.078

.145

Compute:Determine Cv p/CL at zero lift
P

(CY)
\ /L=0(figure 7.1.2.1-9)

0.42 per rad

=

M=O

B"c

1

4

'•-• A + 4 cos Ac/ 4= 1.116(4.0) (0.808) + 4 cos 32.60

AB + 4 cos Ac/ 4
-.

;"

"

(4.0) (0.808) + cos 32.60

AB + cos A 1/ 4
'A + cos Ac/ 4=

4.0 + cos 32.60

A + 4 cos A c/4

AB + cos A c/4

AB +4 cosAc/4

A + cos Ac/4

= 0.8414= (1.116)(0.8414)

A

As

S

0.808

(cos 32.60)2

4.0 + 4 cos 32.6 0

=

k.

(0.7)

AB + COS Al 4
A +4 COS Ac 4"- AB t 4 cos Ac/ 4 A + cos Ae/ 4

=

0.939

(equation 7.1.2.1-b)

pI

L

=C
0
14=0L

P 14ýY= (0.939) (0.42) =0.394

Determine the K factor-(CLtana) -(C

-D CD

(equation 7.1.2.1-d)K----a" (CL tana) -- a7.1.2.1-4

ArA)

.

-

Test

cL

(dog)

cc

.

-

V--

-

7--

ton Otton 02

CL tanc.C
D

T"r.s

- C0
D - 0.01

CL 2/IorA)(D 2 /(4,r)

.06

.75

.01309

.0007

.011

.001

.0002

.10

1.5

.02619

.0026

.0115

.0015

.0008

.20

2.95

.05613

.0103

.015

.005

.0032

.30

4.30

.07519

.0226

.021

.011

.0072

.40

5.60

.09805

.0392

.033

.023

.0127

.50

7.20

.1263

.0632

.0535

.0435

.019

.80

8.80

.1.548

.0929

.078

.068

.0287

.70

11.20

1980

.139S

14

.135

.oaDo

Plot CL tan a,

CD - CD 0 , and

CL 2 /(wA)

versus angle of attack (see sketch (a)).

.16
CL TAN a
.12

f(ci)

C:D -CDo
C

.08

0 1

0

2

4

6

8

10

12

c (deg)
SKETCH (a)7.1.2.1-5

2CL

L tana-'(C 0 -

K

0

L

o)

0

-

-

-

1.000

.05

0018

.00095

.0002

.531

.10

.0040

.0012

.00115

.982

.20

.0066

.0030

.0022

.818

.30

.0105

.0066

.00325

.538

.40

.0136

.0116

.0044

.228

.50

.0168

.0152

.0049

.135

.60

.0184

.0195

0052

-

083

.70

.01865

.0279

.0031

-

.596

Solution:

CY,

=K[(-

CLI + (ACY ) r)(equation 7.1.2. 1-a)M

K R(0. 3 9 4 ) CL] + 0 = 0. 3 9 4 KCL

cvyp(based on SWbW)
CL

K(plr rad)(eq. 7.1.2.1-a)

0

1.000

0

.01

.531

.0105

.10

.992

.0387

.20

R18

.0645

.30

.538

.0636

.40

.228

.0359

.60

.136

.0266

.60

- .083-. 0196.70--. 1641.595

The calculated results are compared with test values from reference 5 in sketch (b).
7.1.2.1-6

-

r.2-

J
-P.??.

-0

Test points-

Calculated

...

I

CY

0

(per rad)

C\

-. 1

-

.4

.2

0

.8

.6

C.L
SKETCH (b)B. TRANSONIC
No generalized method is available in the literature for estimating transonic values of the rollingderivative Cy P. Furthermore, no known e.xperimental results are available for this derivative at transonic
speeds.
C. SUPERSONICAt supersonic speeds a design chart based on theoreticw. calculations is presented for estimating therolling derivative Cy at low values of the lift coefficient. The design chart is based on the results ofreference 6 for wings with subsonic leading edges and supersonic trailing edges, and the results ofreference 7 for wings with supersonic leading edges and either subsonic or supersonic trailing edges. The
results of both references 6 and 7 are based on linearized-supersonic-flow theory and are therefore
restricted to thin, swept-back, tapered wings with streamwise tips. The lateral force due to rolling istaken as that arising entirely from suction forces on the wing edges. For wings with supersonic leadingedges no suction forces are induced along the leading edges, and the determination of Cy involves
only the unbalanced suction forces along the wing tips. Therefore, for zero-taper wings with supersonicleading edges the theory gives Cyp = 0.

-.

No experimental data are available for this derivative at supersonic speeds. Therefore, the validity oflinearized-supersonic-flow theory for estimating Cy1 cannot be determined.
DATCOM METHOD.--

The wing contribution to the rolling derivative Cy , at supersonic speeds and at low values of the lift
"coefficient is obtained from figure 7.1.2.1-10 as a function of the wing aspect ratio, taper ratio,leading-edge sweep, and Mach number.

7.1.2.1-7

l-" ,V.

,----

-

•..

-. •

---

•-

-

---

•

---.-

------I

Sample Problem
Given: Tapered, swept-back wing.
A = 3.22

A = 0.25

M =2.41;

2.19

ALE = 55.20

Compute:PA = (2.19) (3.22) = 7.07
Solution:C!

=0.50 per rad 2 (based on Swbw) (figure 7.1.2.1- 10)

REFERENCES

11. Toil, T. A., and Oueijo, M. J.: Approximate Relations and Charts for Low-Speed Stability and Control Derivatives of Swept Wings.
NACA TN 1581,1948. (U)2.

Brewer, J. D., and Fisher, L. R.: Effect of Taper Ratio on the Low-Speed Rolling Stability Derivatives of Swept and Unwrept Wings ofAspect Ratio 2.61. NACA TN 2555,1951. (U)a3 Quejo, M. J., and Jaquat, B. M.: Calculated Effects of Geometric Dihedral on the Low-Speed Rolling Derivatives of Swept Wings. NACA
TN 1732, 1948. (U)4.

Goodman, A., end Fisher, L. R.: Inves•igation at Low Speeds of the Effect of Aspect Ratio and Sweep on Rolling Stability Derivativesof Untaperod Wings. NACA TR 960,1950. (U)5.

Wlggins, J. W.: Wind-Tunnel Inveition of Effect of Sweep on Rolling Derivatives at Angles of Attack Up to 13 and at High Subsonic
Mach Numbers, Including a Semiempirkal Method of Ebtimatlng the Rolling Derivatives. NACA TN 4185, 1958. (U)"6. Margolis, K.: Theoretical Calculations of the Lateral Force and Yawing Moment Due to Rolling at Supersonlc Speeds for Sweptback"TapeedWings Wth Straanwtise Tips. Subsonic Loading Edges. NACA TN 2122, 1950. (U)7.

S8.

Harmon, S. M., and Martin, J. C.: Theoretical Calculations of the Lateral Force and Yawing Moment Due to Rolling at Supersonic
Speeds for Sweptbeack Tapered Wings With Streanwisa Tips. Supersonic Leading Edges. NACA TN 2156,1950. (U)Wiggins, J. W., and Kuhn, R. E.: Wind-Tunnel I nvestigation of the Aerodynamic Characteristics in Pitch
at High Subsonlc Speeds. Sweep Serws. NACA RM L52D18, 1952.'(U)

7.1.2.1-8

of Wing-Fuaslage Combinations

____

____

7

SUBSONIC SPEEDS

\__"

--

70

65

60

1--

.--

_

-4

__-

45

-

0

-30

__

04

-45

A,/ 4 (deg)6--. 6-ASPECT RATIO, A--t__-_

_

\

._

10*

-

_

y (per1rad)
711
I

I

L'-

0.5
FIGURE 7.1.2.1-9,0
WING ROLLING DERIVATIVE C.

7.1.2.1-9

SUPERSONIC SPEEDS

t

.444

4n1

7.1.2.1-10

jRevised June 1969

7.1.2.2 WING ROLLING DERIVATIVE C1This section presents methods for estimating the wing contribution to the rolling derivative C1 P at
subsonic and supersonic speeds. This derivative is the change in rolling-moment coefficient with change

in wing-tip helix angle and is expressed as
a C1

"CP

pb

A. SUBSONIC
The wing rolling derivative C Ip at subsonic speeds is a function of the wing lift-curve slope, the wing
drag, and geometric dihedral. The effects of dihedral become significant when the displacement of therolling axis from the wing root chord is moderate or large.
For unswept wings of moderately high aspect ratio the lift-curve slope undergoes only small changes
throughout the lift-coefficient range, and the drag contribution is relatively unimportant. Therefore,
potential flow values of Cj p at zero lift are generally satisfactory at all lift coefficients below the stall
for these planforms.
'7

Several methods are available for estimating the potential flow value of Clj P. Reference 1 presents
design charts for Clp at zero lift of unswept wings based on lifting-line theory. An effectiveedge-velocity correction is applied to the lifting-line theory results of refcrcnce 1 in refeience 2. Finallythe results of reference 2 are modified for the effects of sweep in reference 3. A similar method that
accounts for the effects of sweep on the edge-velocity correction is presented in reference 4. Reference
5 presents a more rigorous method of estimating C
for wings of arbitrary planform at zero lift, basedon the simplified lifting-surface theory of Weissinger for determining the additional span loading due torolling. The Datcom method for estimating CjIP at zero lift is taken from reference 6. It is essentiallythat of reference 5 corrected for compressibility effects and extended to a wider range of planformparameters.
For wings of moderate to high aspect ratios and with moderate sweep, the value of C1 in the
nonlinear-lift range is estimated to a first approximation by assuming that variations in tlme tift-curve
slope will affect Clp in the same proportion as CL,,.
On low-aspect-ratio and/or highly swept wings, the flow separates and forms a stable leading-edge vortex
that is responsible for the generation of considerable ad&..tional lift on the outer portions of the wing.
The drag associated with these high lift coefficients cames significant changes in C1 P. The incrementin Cl due to drag is derived in reference 7, based on the strip-theory procedure of reference 3. Since
the effects of drag due to lift and of profile drag on the roll-damping derivative are not of equal
importafice, they are considered separately in determining the increment due to drag.
The correction for geometric dihedral is considered in detail in reference 8. Although the expression forthe dihedral effect has been derived specifically for untapered wings, it should be reasonably reliable forwings of any taper ratio over the range of wing dihedral angles of practical interest.
The Datcom method accounts for the variations in wing lift-curve slope, drag due to lift, and profile

7.1.2.2-1
,-. .

.

.

.

.

..

~

-

drag, as well as the effect of dihedral. The method requires knowledge of the variation of lift and dragover the angle-of-attack range to the stall for the particular configuration at the appropriate Mach
number. Therefore, this method is quite readily applied if experimental lift and drag data are available.
DATCOM METHOD
The value of the wing rolling derivative Cproductat a given lift coefficient at subsonic speeds, based on theof the wing area and the square otfthe wing span SWb 2, is given by

(we

IC

(C 1 )L+CLa)CL+ (AClP)drg(per radian)

7.1.2.2-a

where

/-P)c

CL0

is the roll-damping parameter at zero lift, obtained from figure 7.1.2.2-20 as a functionof A, and PA/K.
The parameter K is the ratio of the two-dimensional lift-curve slope at tile appropriate
Mach number to 21r/3; i.e., (cla) M/(2ir/P). The two-dimensional lift-curve slope isobtained from Section 4.1.1.2. For wings with'airfoil sections varying in a reasonably
linear manner with span, the average value of the lift-curve slopes of the root and tip
sections is adequate.
The parameter Aa is the compressible sweep parameter given as(tan AC/4'
AP(CLOC .

~aC~L

O

tan-

),where3

I ----M .

test data or estimated by using
is the wing lift-curve slope at zero lift, obtainedA fromof Section 4.1.3.2 at the appropriate
the straight-tapered wing method of paragraph
Mach number.

is the wing lift-curve slope at any lift coefficient below the stall, obtained from test

(CL)
CL

data or estimated by using the straight-tapered wing method of paragraph A of Section
4.1.3.3 at the appropriate Mach number.

is the dihedral-effect parameter given by
(

P)r.0

(Cip~r =7.1.2.2-2

2'

sin r + 3Q ~sin 2 1']7.1.2.2-b7

-iy -w~w---rr

r

.W

.

r

"r.Yr

-

, ww

••

.rr .•--

,.

••

•

where

--

p)

F

is the geometric dihedral angle, positive for the wing tip above the plane of
the root chord.

zis the vertical distance between the c.g. and the wing root chord, positive forthe c.g. above the root chord.

bis the wing span.

is the incre.ment in the roll-damping derivative due to drag, given by

DL

ACl)

2
C L2

I
-

C

(per radian)

7.1.2.2-c

where

CL2
CL

is the drag-due-to-lift roll-damping parameter obtained froom figure
7.1.2.2-24 as a function of A and Ac/4.

is the wing lift coefficient below the stall.

K

Co

is the profile or total zero-lift drag coefficient. If e7xperimental data
are not available, CDO may be estimated by the method of para-graph A of Section 4.1.5.1 at the appropriate Mach number.
This method includes the effects of compressibility and may be applied up to the critical Mach number.
(The drag-due-to-lift term does not include a Mach number correction; however, it is small except at
high CL where the Mach number is generally low.)" -;ases

The most important factor considered in this method is the variation of the wing lift-curve slope. If
reliable values of this parameter are. available ov& the lift-coefficient range, the method will in mostgive satisfactory results over that CL range for configurations with aspect ratios of approximately
2 or greater.
For wings of low aspect ratio and/or high sweep the accuracy of the method rapidly deteriorates withincreasing CL, even when experimental values of lift and drag are used. The error results from the factthat the high values of (C p)C L obtained from figure 7.1.2.2-24 for these configurations are notrealized in practice. Therefore, as Ct, increases the calculated values of the roll-damping derivative
become progressively smaller than those given by experiment.

.7

A comparison of the roll-damping derivative calculated by using this method with teat results ispresented as table 7.1.2.2-A. Experimental values of lift and drag have been used in evaluating theroll-damping derivative of all the configurations listed in the table. Several additional references
containing test results of the damping-in-roll characteristics of straight-tapered wings are lsted in table
7-A.
7.1.2.2-3

IL.

Sample Problem
Given: Model 8 of reference 17.
Wing Characteristics:

X =0.15

A = 3.00!==/436.90

A

Airfoil Characteristics:
Y90
NACA 0012 airfoil

Y99
=

2

1.448

= 0.2602

Additional Characteristics:
M = 0.13;

•

=

0.992

R

=

1.254 x 106

C=

0.036(testvalue)

The following test values from reference 17:

0

.1

.2

.3

.4

.5

.6

.7

.8

.0525

.0525

.0525

.053

.053

.054

.0541

.050

.030

C J
CL

I

Compute:/PC. %

Determine the roll-damping parameter at zero lift
Yo

Y9
99

Y90
tan

2

a

CL = 02

9-1.448.-0.26

0.132

-09

c.
- = 0 768 (figure 4.1.1.2-8a, extrapolated by plotting vs log1l R1 "(cq Cory= 6.88(figure 4.l.1.2-8b)

1.05 cho
3(Cl

(Cj)

,(cj)
,

theoWy(equation 4.1.1.2-a))hory'

1.05-(0.768) (6.88)

0.992

5.59

(cla)mt• =7.1.2.2-4

2r/2=

2w/0.992=0.883

5.59 per rad(3.0)
L3A (0.992)
=

AO

*DV

=

0,75

tan-'

a

tan-'

I CL.O
K )C

(

3.37

=

0.883

K

= -0.251 per rad009-92/= tan-' 0.7569 = 37.120(figure 7.1.2.2-20, interpolated)

Determine the dihedral-effect parameter(e-

1.0 (equation 7.1.2.2-b at1 = 0)

Determine the increment in roll damping due to drag (ACp)drag(Ci,)\C 2
C- 0.034 (figure 7.1.2.2-24)(C(ACLp)IAu g

CL2 -

CL 2

034

=-(-0.

6

CD 0 (equation 7.1.2.2-c)-

0.036 per rad

)CL 2

{

"L
CL

C2

0

0

0.1

0.01

-0.00480.2

0.04-0.0059

0.3

0.09-0.00760.4

0.16-0.0099

0.5

0.25-0.01300.6

0.36-0.0167

0.7

0.49-0.02120.8

0.64-0.0263L0o0.0045

-. 34

n
A.LCD

--0.0045

7.1.2.2-5

K
pk

.-

*

1.

-

-'

Solution:
I.-

L

"L

(PC1
cC

'C.

K ." "C

0.883

L

-10

0.992 (: C(~=-0.223)(equation 7.1.2.2-a)p)P0( C )=(-0.25 1)[

L(CLo. C

K

p

[1

(C )

\LQC

ACP n

.

e/CL

+~C C
n
(1.0) + (AC 1p)

~

(CL)

•(
i..-L

CLO

0C.2L L0

CL
L

"0

()CL

0C

(per red)-0.223 (ý)+()0.05251.00-0.0045-0.2275-0.0048-0.2278-0.0069--0.2289-..0.0076-0.2328-009-0.2361-0.0130-0.2427

0,063

1.01

0.064

1.03

0.6

"

(basednrod)(eq. 7.1.2.2-0)

L01per

0.4

_- -

(C)d
AC

(0)/0.0525

ps0.2
r0.5

__a

L~X
(
(test results)

0.1

0.3

C~

P_

-0.0167--0.2464

0.7

0.050

0.952

-0.0212-0.2335

0.8

0.030

0.571

--0.0263-0.1536

The calculated rewuts i'."are compaed0,4with test values in sketch0,009(a) and in table 7.1.2.2-A,

7.1.2.2-6

'~

-. 3

--

0 Test points
Calculated

-. 2

_

C,

.2

.4

CL

.6

.81.

SKETCH (a)B. TRANSONIC
There are no reliable methods for estimating the derivative Cl p in the transonic region. Although thisderivative might be expected to vary with Mach number in the same manner as the lift-curve slope, thistrend is not exhibited by experimental data. A considerable quantity of test data is available, however,and reference should be made to table 7-A.
C. SUPERSONICAt supersonic speeds design charts based on theoretical calculations are presented for estimating therolling derivative CIp of wings of vanishing thickness.

h

The design charts are those of reference 9 and are based on the results presented in the following
references:
Reference 10 - in the region of supersonic leading and trailing edges
Reference 11 - in the region of subsonic leading edges and supersonic trailing edges
Reference 12 - for values of P3A < 2
The results presented in references 10 and 11 are based on linearized supersonic-flow theory whil,. :ho;*(of reference 12 are based on slender-wing theory. The slender-wing-theory value of C/r
PA = 0 was used to establish a straight-line relationship between slender-wing theory and the low
limit of the linearized supersonic-flow theory. Thin airfoils have been assumed in these theori(.,Thickness effects are not important except for conditions where the Mach lines lie on or near the wif.
leading edge. Under these conditions the wing-leading-edge shock position is displaced forward from if',theoretical position by the finite thickness effects of the leading edge. Thlis displacement results insubstantial losses in normal-force-curve slope and consequently, in roll damping.
The empirical chart presented in Section 4.1.3.2 for determining the leading-edge-thickness effect on the.
"normal-force-curve slope of straight-tapered wings has been adopted in this section to determine tli
leading-edge-thickness effect on the roll damping. This chart is presented as figure 7.1 .2.2-27 in the forni--

7.1.2. 2-7

h.L

-

.

.

.

.

.

-

-

-

-

,0

,

of a ratio of the actual roll-damping derivative to the theoretical roll-damping derivative. For
straight-tapered wings with sharp leading edges, the airfoil nose semiwedge angle (measured normal to
the wing leading edge) determines the shock position relative to the wing. Experimental data indicatethat the parameter corresponding to the nose semiwedge angle is Ayn = Ay/cos ALE, where sy is thedifference between the upper-surface ordinates at the 6-percent- and 0.15-percent-chord stations. The
parameter Ay is presented for several airfoil shapes in figure 2.2.1-8. For double-wedge and biconvex
airfoils there is a linear relationship between AyL and the leading-edge semiwedge angle, given by
Ay 1=

5.85 tan 6L

Either Ayj or 5j1 may be used to calculate the thickness effects.
The Datcom" method is applicable to straight-tapered wings of arbitrary taper ratio with wing tips
parallel to the free stream and with subsonic or supersonic leading edges and supersonic trailing edges. A
further restriction is that the foremost Mach line from the tip may not intersect the remote lialf-wing.
Wings with inverse taper (A > 1) have not been considered. Wings with swept-forward leading edges areincluded through the use of the reversibility theorem (references 13 and 14). The reversibility theorem
states that the roll-damping derivative Clp of the wing in forward flight equals the roll-damping
derivative of the same wing in reverse flight.
DATCOM METHOD
The wing zontribution to the roll-damping derivative C1 at supersonic speeds, based on the productof wing arxa and the square of the wing span SW bw,,is given byCP) thory

e

CC

*

A

A

P

(CIP) theory(per radian)

7.1.2.2-d

whereis the wing aspect ratio.

A

is the theoretical roll-damping parameter obtained from figures 7.1.2.2-25a through.

*(C)theory
A

*

CL
*

7,1.2.2-,25e.
is the empirical thickness correction factor obtained from figure 7.1.2.2-27.

C1 )through 7 .1.2. 2 -25e represent an upper limit
The sonic trailng-edge boundaries on figures 7.1.2.2-25a below
the sonic trailing-edge boundary are for
Valuesderivatives.
ofthevaluestheoretical
for the truewings with subsonic trailing edges and are in violation of one of the basic assumptions of the theory.

For configurations with subsonic trailing edges (iCot ATE < 1) the design charts will overestimate theroll-damping derivative.

7.1.2.2-80,

It should be noted that the "kinks" in the curves of figures 7.1.2.2-25a through 7.1.2.2-25e correspondto the conditions of sonic leading or trailing edges. Experimental evidence shows that these "kinks" donot occur in practice.

"Comparisons of the supersonic roll-damping derivative in the linear-lift range calculated by this methodwith test results are presented in tables 7.1.2.2-B and 7.1.2.2-C. The configurations listed in table
7.1.2.2-B have supersonic leading edges, while those of table 7.1.2.2-C have either sonic or subsonicleading edges. The roll damping is predicted quite accurately by the Datcom method when the wingleading edges are supersonic. However, when the wing leading edges are sonic or subsonic, the calculatedroll damping is in almost all cases considerably greater than that given by experiment. Application of the"thickness correction factor presented as figure 7.1.2.2-27 improves the agreement between the calculated
and experimental values of roll damping in almost all cases presented in table 7.!.2.2-C, since this factor
represents a reduction in roll damping. For wings with subsonic leading edges the theoretical resultspresented in figures 7.1.2.2-25a through 7.1.2.2-25e show the poorest agreement with experiment at thelower values of PDot ALE for a given value of PA.

nplrobem
Given: Wing 14 of reference 23.
AL

=

600

Airfoil:Ae, 4 = 53.40

A = 3.12= 0.25Constant 3/16-in. thickness with symmetrical 50 bevel on all edges in a direction parallel tothe root chord.
B= 100M = 2.41; P = 2.19
Compute:PA = (2.19) (3.12) = 6.83
A tan A/2 = (3.12) (tan 53.40) = 4.20=

-0.0716 per rad(figure 7.1.2.2-25b)

A
P cot ALEt

2.19 (cot 600) = 1.265 (supersonic leading edge)ALE
tan 600=- 2= 0.791
2.19
P
C1 P
=

0.865

(figure 7.1.2.2-27)

7.1.2.2-9

Solution:

p

=

•

A theo°iJ A(C, P)theory(equation 7.1.2.2-d)- (-0.0716) (3.12) (0.865)=-0.193 per rad (based on Sw

This compares with a test result of -0.188 per radian from reference 23.
REFERENCES
1.

Pearson, H. A.. and Jones, R. T.: Theoretical Stability and Control Characteristics of Wings With Various Amounts of Taper and Twist.

NACA TR 635, 1938. (U)2.

Swanson, R. S., and Priddy, E. L.: Lifting-Surface-Theory Values of the Damping in Roll and of the Parameter Used in Estimating
Aileron Stick Forces. NACA WR L-53, 1945. (U)3.

Toll, T. A., and Queijo, M. J.: Approximate Relations and Charts for Low-Speed Stability Derivtives of Swept Wings. NACA TN 1581,
1948. (U)4.

Polhamus, E. C.: A Simple Method of Estimating the Subsonic Lift and Damping in Roll of Sweptback Wings. NACA TN 1862, 1949.
(U)

5.

Bird, J. D.: Some Theoretical Low-Speed Span-Loading Characteristics of Swept Wings in Roll end Sideslip. NACA TR 989, 1950. (U)6.

De Young, J.: Theoretical Antisymmetric Span Loading for Wings of Arbitrary Plan Form at Subsonic Speeds. NACA TR 1056, 1951.
(U)

7.

Goodman, A., and Adair, G. H.: Estimation of the Damping in Roll of Wings Through the Normal Flight Range of Lift Coefficient.
NACA TN 1924, 1949. (U)8.

Quaijo, M. J., and Jacluat, B. M.: Calculated Effects of Geometric Dihedral on the Low.Speed Rolling Derivatives of Swept Wings. NACA
TN 1732,1948, (U)9.

Anon: Royal Aeronauticel Society Date Sheets - Aerodynamics, Vol. III (Aircraft S.06.03,01), 1957. (U)10.

Harmon, S. M., and Jeffreys, I.: Theoretical Lift and Damping in Roll of Thin Wings With Arbitrary Sweep end Taper at Supersonic
Speeds. Supersonic Loading and Trailing Edges. NACA TN 2114, 1960. (U).11.

Malvestuto, F. S., Jr., Margolis, K., and Ribner, H. S.: Theoretical Lift and Damping in Roll at Supersonic Speeds of Thin Sweptbeck
Wings of Arbitrary Taper and Sweep at Supersonic Speeds. Subsonic Leading Edges and Supersonic Trailing Edges. NACA TR 970.
1950. (U)12.

Mangler. K. W.: Calculations of the Preasure Distribution Oer a Wingat Sonic Speeds. ARC R&M 2888, 1955. (U)13.

Harin,-,i S. M.: Theoretical Relations Between the Stability Derivatives of a Wing in Direct and in Reverse Supersonic Flow. NACA TN
1943, 1949. (U)14.

Brown, C. E.: The Reversibility Theorem for Thin Akrfoils in Subtonic and Supersonic Flow. NACA TN 1944, 1949. (U)15.

Letko, W., and Wolhert, W. D.: Effect of Sweepback on the Low-Speed Static and Rolling Stability Derivative of Thin Tapered Wings ofAspect Ratio 4. NACA RM LgF14, 1949. (U)16.

Jopuet, B. M., and grower, J. D.: Effects of Various Outboard and Central Fins on LowSprId Static-Stability and Rolling Characteristics
of a Triangular-Wing Model. NACA RM L9E 18, 1949. (U)

7.1.2.2-10

Wings of Triangular and17.

Jaquat, B. M., and Brower, J. 0.: Low-Speed Static-StabIlity and Rolling Characteristics Of LOW-AePect-Ratl
Modified Triangular Plan Forms. NACA RM L8L29, 1949. (U)18.

Bird, J!- D.. Lichttenstein, J. H., and Jaquet. B. M.-. Invetigetion of the Influence of Fuselage and Tail Surfaces on Low-Speed Static
Stability and Rolling Characteristics of a Swept-Wing Model. NACA TN 2741, 1952. (U)19.

Fisher, L. R.. and Michael, W. H., Jr.: An investiption of the Effect of Vertical-Pin Location endi Arm on Low-Speed Lateral Stability
Derhteives of aSamitailloss Airplane Model. NACA RM L51A1O, 1951. (U)20.

Goodman, A., and Thomats. 0. F.. Jr.' Effect of Wing Position and Fuselage Size on the Low-Speed Stetic and Rolling Stability
Characteristics of a Delta-Wing Model. NACA TR 1224,1955. (U)21.

Letko. W. and Riley, 0. R.: Effect of an Unawept Wing on the Contribution of Unsvwept-Tail Configurations to the Low-Speed Static
and Rolling-Stability DerihAtives of a Midwving Airplane Modal. NACA TN 2175, 1950. (U)22.

Wiggins, J. W., and Kuhn, R. E.: Wind-Tunne Investigation of the Aerodynamic Charactaristics in Pitch of Wing-Fuselage Combinations
at High Subsonic Spewds -Swoop Series. NACA RM 1-621318. 1952. (U)23.

Mc~oermon, R. W., and Heinke. H. S.. Jr.: I nvestigations of the Damping in Roll of Swept and Tapered Wings at Supersonic Speeds.
NACA RM L53A13, 1953. (U)24.

Bland. W. M., Jr., and Sandehi. C. A.: A Tedmqjtue Utilizing Rocket-Propelled Test Vehicles for the Measurement of the Damping in
Roil of Sting-Mounted Models and Some Initial Results for Delta and Unswept Tapered Wings. NACA RM L-501324, 1950. (U)25.

Bland, W. M., Jr., and Diatz, A. E.: Some Effects of Fuselage Interference, Wing Interference, and Sweepback on the Damping in Roll of
Untapered WingsasDetemined by Techniques Employing Rlocst.Propellsd Vehicles. NACA RM L511325. 1951. (U)TABLE 7.1.2.2-A
SUBSONIC WING ROLLING DERIVATIVE C1DATA SUMMARY

Ac4 Airfoil

R1CL&

r

D

CI

C

0

C.tlc.
(per rod)

Test(per red)

Percent-0.322-0.345

6.7

Ref.

A

X

(deingSection(dog)

M

X 10-6

CL(per deg)(test)

15

4.0

0.60

3.6

65AODS

0

0.13

0.72

0

0.065

0.1

0.072

-0.356-0.3460.2

0.072-0.357-0.347

-2 9

0.3

0.072

-0.367-0.348

-2.6

0.4

0.067

0.5

0.067

-0.334-0.334-0.355-0.370

5.91
9.7

0.6

0.063
0.047

-0.316--0.240-0.370-0.275

14.6
12.7

0

0.062

-0.306-0.330

6.7

0.1.

0.062

-0.06

-0.330

6.7

0.2

0.062

-0.309-0.336

8.0

0.3

0.066

-0.329-0.360

8.

0.4

0.066

-0.330-0.387

1.7

0.5

0.066

-0.332--0.370

10.3

0.6

0.061

-0.309-0.295

-4.7

0.7

0.046

-0.234 L-0.195-20.J0

I0.7
32.6

-

-

-

-I

-0.8,

1

0.027

(test)
0.033

0.033

-

-0.145-0.0B5

Error

-

2.9

.1

7.1.2.2-11

TABLE 7.1.2.2-A (CONTD)

L
Hof.

15

A

4.0

A

0.60

(dog)

""
AirfoilF.Sectioni(dog)

66Ao06

46.7

0

M

0.13

x 10-60.72.

416 2.31

0

52.2

65

.006.5

0

0.13

1.624

C

(per dog)

0

Ca8c.

C
Test

Percent(test)(per red)(per red)

Error

0

(test)
0.062

0.033

-0.287-0.300

4.3

0.1

0.062

-0.288-0.305,

5.6

0.2

0.062

-0.288-0.335

14.0

0,3

0.056

-0.309-0.378

18.3

0.4

0.066

-0.312-0.392

20.4

0.5

0.066

-0.315-0.382

17.5

0.6

0.059

-0.285-0.350

18.6

0,7

0.046

-0.230-0.297

22.6

0,8

0.036

-0.193-0.256

24.6

0

0.045

-0.172-0.165- 4.2

0.1-0.173-0.1751.1

0.2-0.176-0,1802.2

0.3-0.180-0.1800

0.4-0.106-0.182-2.2

0.5-0.195-0.182-7.1

0.6-0.20M-0.180-13.9

0.7-0.218-0.179-21.8-0.233-0.172-35.5-0.228-0.230

0.9

CL

0.020

0.8
17

30

2.31

0.115

0

36.9

52.2

0012

0

0.13

1.254

1.624

0.036

0

0.025

0.1

0.0625-0.228-0.2300.9

0.2

0.0525-0.229-0.232

1.3

0.3

0.053

-0.233-0.235

0.9

0.4

0.053

-0.235-0.232

-1.3

0.5

0.064

-0.243-0.232

-4.7

0.6

0.054

-0.246-0.228

-7.9

0.7

0.050

-0.234-0.198-18.1

0.8

0.030D0.5-0.090

-71.1

0

0."il

-0.168-0.150-12.0

0.1-0.169--0.10-12.7

0.2-0.172-0.150-14.7

0.3-0.176-0.149

-18.1

0.033

-0.143-28.0-0.191-0.138

-38.4
-66.6

0.4-0.183
0.5
0.6

0.0435-0.212-0.1280.7

0.0435-0.225-0.120-87.5-0.127

-92.9

0.81

7.1.2.2-12

0.046-0.245TABLE 7.1.2.2-A (CONTO)

CL

R!

CD
0

Ri.

A

k

ud

Setimm

(deg

M

31.6

4.0

is

2.61e

0.36

I.O

,M

45.0

CL

(esmt)

(tat)

1.23

0

0.056

0.037

-0.230-0.227- 5.3-0.239-0.230-3.9

0.2-0.240-0.230-4.3

0.3-0.241-0.235-2.6

0.4-0.243

-0.=2

-6.6

0.5

-0.245-0.200

-22.5

1.335

0121J.E)

0

0.17

1.40

3.60

0.481

36.16

0010.64

0

0.16

1.138

0.0546-0.242-0.160

-51.2

0.7

0.047

-0.214-0.117

-82.9

0.8

0.037

-0.178

-

-

0

0.036

-0.183-0.228

19.7

0.1

0.041

-0.208-0.221

5.9

0.2

0.044

-0.224-0.231a0

0.3-0.226-0.2322.6

0.4-0.229-0.222-3.2

0.5-0.233-0.228

-2.2

0.041

0.6

1

-0.239-0.242

1.7

0.7

0.043

-0.239-0.250

4.4

,0

0.063

-0.295-0.240-22.9

00.04W2-0.218-0.2305.2

0.1-0.218-0.2326.0

0.2-0.219-0.235

6.8
7.1

0.023

-0.223-0.2400.4

g-0.227-0.242

6.2

0.5

0.048

-0.245-0.256

4.3

0.6

0.050

-0.260-0.275

5.5

0.7

0.065

-0.339-0.318--6.6

0.8

0.074500.0515

0.1
0.2

0.055

0.018

0.3

'

1

Prcent
Error

0.6

0.3

19

Clp

x 10-4

0.1

2.0

CIP

Cac. rrest
Iper rmd)(per raoo-0.393-0.385-2.1-0.271-0.266--0.290-0.290-0.261-0.279

-11.1
-3.9

1.9

-0.292-0.290

-0.7

OA
0.5

0.056
0.054

-0.300-0.292-0.299-0.292

-0.3
0

0.6

0.046

-0.254-0.248

-2.4

0.7

0.035

-0.200-0.150

-33.3

1 0.8

0.017

-0.110-0.12512.0

1

7.1.2.2-13

!P

TABLE 7.1.2.2-A (CONTO)

a1Ip eC/D

AC0'(Per dg)

COIc.

Test

Percent(pe red)(per red)Error-0.170-0.16066.3

0.1-0.171-0,155-10.3

0.2-0.175-0.158-10.8

0.0455-0.193-0.165

-17.0

0.4

0.047

-0.205-0.165

-24,2

0.5

0.048

-0.218-0.162

-34.6

0.6

0.048

-0.228-o.155

-47.1

0.7

0.047

-0.236-0.150

-57.3

0.8

0.047

-0.250-0.157-59.2

00.06.1

A./4

Airfoil

r

R

A

X

(dog)

Section

IdOg

M

x 10-

CL

(tutl)

(to

20

2.31

0

52.4

AOo3

0

0.17

2.06

0

0.042

0.01h

I

0.3

21

4.0

%

0.6

0

9C5K~iP800.166.

0.88--0.317-0,3303.9

0.1-- 0.317-0.3406.8

0.2-0.318-0.37014.1-0.318-- 0.392

18.9-0.320-0.321-0.384-. 0.355

16.7
9.6

0.027

0.3
0.4
0.6

I

Average Error"'..22-n7.1.2.2-1 4-

e__. = 115%

TABLE 7.1.2.2-8
SUPERSONIC WING ROLLING DERIVATIVE

I

SUPERSONIC LEADING EDGESDATA SUMMARY

,,
Re.
A()
23

E
M

pcot

AL

3.07

0

60.0

2.41

1.27

1.90

0.26

60.0

2.41

1.27

1.91

0.26

55.0

1.93

1.16

2.41

1.54

1.93

1.15

2.41

1.54

2.37

0.25

55.0

3.12

0.25

60.0

2.41

1.27

3.22

0.25

55.2

1.93

1.15

0.26

46.0

3.2

1.83

2.41

1.53

1.62

1.27

2.34

2.31

-A
(pra)

Beveled10.0-.0.0536

10.0
8.75

-

Cole.
(per red)

Teot(per rad)

Percent
Ero

0.866-0.142-0.1392.2-0.0982

0.866-0.161-0.163-

1.2-0.11260.860-0.185-0.195-5.1-0.0980

0.913-0.171-0.188

-

9.0.

01 01

0.860

-0.207-0.210-1.4-0.0834

0.913-0.181-0.200

-

9.5

10.0

-0.07160.865--0.193-0.1882.7

8.75-0.08530.850--0.234-0.2254.0-0.0669

0.916-0.197-0.2117.08-0.10100.896-0.282-0.272

8.75

1(C 1 P)

theory

6.6

-

3.3

1.65

-0.08390.94-0.248-0.253

42.41

2.20

-0.06421.000-0.200-0.190-0.0972

0.860-0.153-0.160-4.4-0.0756

0,913-0.127-0.138-8.0-0.10700.863-0.166-0.185~-10.3-0.0960

0.902-0.158-0.176

-10.2

0.966

-0.132-0.138

-

4.,

0.860

-.0.164-0.170

-

3.5

0.913

-0.133-0.155-14.2-0.0882

0.896--0.203-0.210

-

3.3

-

8.0

0

65.0

1.93

1.16

8.75

-2,0

5.3

2.41

1.54

111.93

1.07

0

1.93

1.16

2.41

1.5-0,0625

1.62

1.27

1.93

1.65

-0.07900.945-0.172-0.1872.20-0.06101.000-0.141-0.140-0.0862

0.933-0,210-0.2290.996-0.180-0.1705.9

1,0M0-0.140-0.165

- 9.7

0

0

50.0

54.9

46.0

1.62

2.4 0

7.8

1.39
24 1.4-0.0765

2.1

--.

(dog)

311.93

1
1.82

Airfoilsection

8.7

7.06

6.57

-0.0615f62

1560

4041.93
I 2.4

j1.94
2.I

-0.0775

2.6-0.0699--

404

4

-00
j

0.7
-

8.3

______

7.1.2.2-15

TABLE 7.1.2.2-8 ICONTD)

ALE
f.

A

23

3.07

0

3

0

m

45.0

1.62

1.27

1.93

5

3.02

2.33

0.2

0.25

2.410
3.22

1.65

-0.0627

0.945-0.181-0.200

49.5

2.41

2.20

-0.0483

1.000-0.148-0.144

2.8

1.62

1.93

-0.0778

1.000-0.235-0.224

484

1.93

2.50

0.06131.000-0.185-0.178

309

2.41

3.34

-0.0480

1.000-0.145--0.134

8.2

1.62

1,93

-0.1292

1.000

1-0.243-0.257

S 1.932.50-0.11631.000-0.219-- 0.210

4.3-0.11630.895-0.243-0.240

1.3

33.4

33.5

I

46.0

0

04WOled

7.08

6.08

6.0

7.08

-5,4

1.62

1.27

1.93

1.65

0,10320.945-0.227-0.2352.41

2.00--0.0823

1.000-0.192-0.186

3,2

1.62

2.46

-0.1172

1.000-0.282--0,275

2.5

1.93

3.28

-0.0991000

0.23 -0

-

4.25
"2.41

-0.0780

1.000-0.188-0.1783.6-0.11851.000-0.281-0.267

5.2

5.64

5.65

-

3.4

70.225 1.2

1.62

2.46

0

1.93

3.19

-0.0998

1.000--0.237-0.218

8.7

I

2.41

1.5

.-0.0790

1.000-0.187-0.173

8.1

1.62

3.28

-0.0965

1.000-0.311-0.317

- 1.9

1.93

4.25
6.66

0 0775-0.0606

1.000
1.000

-0.250-0.195-0.220-6.175

1306
11.

50,0

2.41

1.27

10.0

-0.0651

0.865-0,131-0,138

5.1

2.37

0.25

2.8

27 4

21.2

S2.4'

"m2.32
0

7.36

2.81

0

55.0,

1.93

2.41

1.16

1.54

8.75

-0.0868-0.0611

0.861

0.013

-0.207-0.167 -0.160-0.167

24.0

3.40

0

50.0

1.62

1.07

7.8

-0.0778

0.865--0.226--0.20}6

10.2

;

I

1.93

1.39

I

0.05950.905-0.183-0.205

-10.7

•

•

2.41

1.84

-0.0453

0.966-0.149--0,156

4.5

m'[
*

7.1.2,2-16

6________T
int Pecent
Error(per rd)
1.9

S

-

CIC.
per red)-0.214

273

3.22
40

..-

-

d)

--0.218

VI

.

A
(per

0.895

1
1.88

Airfoil
Section-0.0795

1

K1

P cot

e

4.4

TABLE 7.12.2-B CONTO)

ALE(C ' P)th eory
CIPA(C)Cic.

Airfoil

Cl

c 'p
Tat

Prce

(per red)(per red)

Error

Hof.

A

X

(deg)

M

Cot ALE

SectionAy.

(per red)

24

4.00

0

45.0

1.485

1.097

Symmetricaldbl. wedge
t/c = 0.04

3.24

-0.0750.934-0.280

-0.20

0

4.00

0

46.0

1.485

1.097

7.3

-0.0750.867-0.260-0.230

13.0

4.00

0.50

9.5

1.70 .

8.21

Symmetricel
dbl. wedge
t/c = 0.09
Symmetical

2.67-0.0841.000-0.336-0.350

-

heory

4.0

dbl. wedge
t/c - 0.046

25

3.70

13.70

1.00

1.00

0

46.0

1.485

6.56

-0.1001.000-0.400-0.406

- 1.2

1.414

5.98

-0.1061.000-0.424-0.440

- 3.6

1.60

o

-0.1011.000-0.374-0.3429.4

1.50-0.1075

1.000-0.306-0.360

10.6

1.414

-0.11401.000-0.423-0.37612.5

1.30-0.1236

1.000-0,467-0.39017.2-0.101

0.812-0.303-0.310-0.1050.792-0.30B-0.307

1.60
M1.50

1.25
1.12

65A009

66A00

16.6

16.6

Avmege Error

--

2.3
0.3

&=6.2%
n7.1.2.2-17

TABLE 7.1.2.2-C
SUPER3ONICWING ROLLING DERIVATIVE C~r
SUBSONIC OR SONIC LEADING EDGEDATA SUMMARY

1p hoy
ALE

Ref.

-•

A

1

A
1.82
0

I-.47

(dog)
...70.0

P cot A

Airfoil.0,466

Section____Beveled

M
..
1.62

*

I

Flat Plate

S1.3

2.34

0

699

0
0.8W0

132
1 1.930.4665
0.605

.0

2.33

3.0•7

1.87

1.84

1.90

*7.1.2.2-18

0
0

0.25

0.25

0.25

(peor ed)Test(perrad)

Percent
Error-0.120

22.5

ICec,
1.000

-__________-7 -1o1.22

0-0.0795

0.922--0.134-0.11417.5-0.07690.820--0.115--0.09027.8-0.0707-0.0698

1.000
0.920

-0.165-0.150--0.092-.0.098

79.3
53.1

-0.0m

0.818

-0.128-0.08060.0-0.0797

0.914--0.175-0.140

25.0

0.595

1.93

0.770

-0.07700.850-0.162-0.11631.0

t .0m0-0.06830.800"--0.127--0.118

7.6

1,93

0,770

I1.62

0.795

80.0

2.41
11.62

1.000
0,735

.- 0.062

0.850.-0.173-0.11649.1

1.03-0.07520.840-0.207-0.1743

16.7

1.03

-0.0587-0.0752

0,800
0.880

-0.144.- 0.203-0.0705--0.109.- 0.174

0.820

1.

-0,17

-0.1581.

-0.0937

1,000-0.175-0.098

78.6

1.3
0I0

-,0.0943

0.922-0.162-0.116

3.

2.41

0.800

-0.09160.820-0.140-0.086

1.62

0.604

-0.1058

0.938-0.183-0.187

2.1

1.93

0.782

-0.1046

0.845-0.162-0.135

20.0

S2.41

1.000

-0.0988.807-0.147-0.113

30.1

1.62.

0.736

-0.11115

0.880-0.187-0.219

-14.6

1.93

0.955

-0,10970.820-0.171--0.176-

I 1.930.,9 5

70.0

0.406

60.0

1.52

.805

60..0

.7

Cp

1.62

•--2.41

3.07

e+

1.5

,600

2.41

2241

A
(Perrod)-0.0810

Cl

1.62

1.52

1.21

1.03

IC.8

2.0

1.91

0.25

55.0

1.62

oJego

0.90

-0.11176

0.840-0.188-0.222

-15.3

2.37

0.25

70.0

1.62

0.466

1.52

-0.08051.000--0.191-0.142

34.5

1.93

0.600

--0.0833

0.922-0.182-0.118

54.2

2.41

0.8W0

:-00220.820-0.160-0.09866C7_TABLE 7.122-C (CONTD)

Rot.

A

X

ALE
Id@0

M

pC" A LE

23

2.34

0.25

5.0

1.82

0.5

123

0.770

2A

1

(1• Am1m

1.22-0.034

000

*

)0070

Testred)(per

Percunt
Enw

0.940

-0210

-0.183

14.8

41.50

-0.186-0.12746.5

0600-0.163-0.127

28.3

12

O62

0.0

-0.1002

0.840

-0218

-0.221

- 1.4

3.06

0.25

05.1

1.62

0.502

1.22

-0.0620.950-0.239-0.155

54.2

1,

0.70

-0.08180850-0.213-0.132

61.4

7

0.300

-0.185-0.12064.2--0.000.800-0.248-0230

7.8-0.0700.820-0.222-0.2"58.3

1.000-0*75-G
1.03

312

0.25

6

1.2

0.75

1

1

1

113

0.995

3.22

025

12

1.2

GUS

0.30

-0.0043

0.840

-0256

-0226

12.3

1.83

0

1.

1.62

0.33

. 00

-*.10400.40-0.160-0.156

2.6

2.34

0

54.

1.62

0

O65

-0=67

0.3

-0.187-0.187

0

2.32

0

6.

12

0735

1.03

-0.00

0.3

-0.181-0.156

14.6

1

1

1

23

0.15

I

-0*5

0.20

-0.163-0.155

52

2.31

0

55

12

0.300

080

-0

0

0.840

-0.210-0.172

22.1

4.00

0

45

1.414

1.00

0.=32

-4

0.810

-0.300-0.21010.4-0.O00

0.343

-0233

-0230

25.2

-0.302-14.0-0.311_0.3

1

Sgmmmrol.~t/c .0.04

0

4.00

45

A14

10 ;dbL Syvmnmm8ica
cwwin

0.75

0I

3.WO
_

_

45.0
_

_

1.A14

1.000

1.74

I1

_

_

0 _03o_

GAO0S
_

_

i

I

t

_

)th my__ Ipm
CaIc.
red)

15.0

4t

I25

(

0.25

I

_

BMWled

Avi

2.37

2.

24

Airfoil
Sedioncs

czc,(ct),~v_-0.106_-0.10565
_

0.770

.,0o
, -0.312____

AmwwrqEror

-

20.4

7.1.2.2-19

SUBSONIC SPEEDS(a) X =0
P3A

CL10

88

-3.5712.52-2-

SUBSONIC SPEEDS

Li(b)?,= 0.25
P3A

CL

4

(per rad)b2.

7.1.2.2-21

SUBSONIC SPEEDS
Mc

X

0.5 0

101

FIGUE
7..2.20 (CNTD
7 1.2.2-2

(d) X-.

SUBSONIC S,'EEDS

1.0

10

4.51
____L__

0

SUBSONIC SPEEDS
(deg)
70
60
(C,)
C2
L

(per rad)

-2

-__

50

A

FIGURE 7.1.2.2 -24 DRAG-DUE-TO-LIFT ROLL-DAMPING PARAMETER

7.1.2.2-24

SUPERSONIC SPEEDS

k

-. 10

-. 03

/SONIC T. E.

-A(per rad)5-. 097s-. 06 - _

____

6

_

-. 05wc/2' (b).X.0.25

-. 14

SUPERSONIC SPEEDS---(c) X =0.50-. 3A TANA,/ 2

_

__

2/_
( ) -. 10
A4
-

.09

(per rad)-. 08 -

__

__

__

-. 07~

V.0
-.05

y

-.123

-. 11-Ki:

A -. 09---~(per rad)(COTD)AAFIGUR 7.12.2-2*.8 7A1.2.2-2-

10

SUPERSONIC SPEEDSA TAN A,,/ 2-. 12(e)X =1.

-_

7~I

7.12,25(CND
FIGUR

1.081.

-.06-.

)Ahe(...22
t

(CGU)

pOTD

C,
.8'

.8

-

.7-

*

.6 -*For

wedge leading edge onlyI
.
6
.
0 .2'

.2
.4
.6
TAN ALE~DAMPING-IN-ROLL CORRECTION FACTOR FOR
SONIC-LEADING-EDGE REGION
1.0

.8

(

#i/TAN AL

FIGURE 7.1.2.2-27

7.1.2.2-27

Revised June 1969

7.1.2.3 WING ROLLING DERIVATIVE Cnp
This section presents methods for estimating the wing contribution to the rolling derivative C. at
subsonic and supersonic speeds. This derivative is the change in yawing-moment coefficient with change
in wing-tip helix angle and is expressed as

acn
Cn.P

pb

A SUBSONIC
The wing rolling derivative Cn, results because the unsymmetrical lift distribution causes a difference indrag between the wing panels when the wing is rolling.
The method for estimating the wing rolling derivative Cn- , is derived from an analysis of references I
through 5. The method is applicable over the lift-coefficient range up to the stall, providing reliablevalues of lift and drag are available over this range.
The value of C. near zero-lift coefficient is the potential-flow value based on simple-sweep theory
from reference f. The effects of linear wing twist and symmetric flap deflection are taken fromreferences I and 3. Geometric dihedral also causes an increment in yawing moment that is associatedwith the increment in lateral force. The empirical results of reference 4 show that this increment is
independent of lift coefficient over the low to moderate lift-coefficient range and increases at the higher
vilues of lift coefficient. However, over the range of wing dihedral angles of practical interest theincrement in C,, due to dihedral is very small and may be neglected.
At moderate or high lift coefficients, a comparatively large change in Cnp/CL occurs, especially forswept wings, due to the rise in drag associated with the increase in lift. In references 2 and 5, methods
are presented for evaluating Ca over the lift-coefficient range up to the stall by using a correction
factor to account for the variation of profile drag with lift coefficient. Results obtained by using themethods of both references 2 and 5 for estimating C, over the lift-coefficient range have been
analyzed and the method of reference 5 selected for the Darcom.
Theoretically, the tip-suction contribution to the lateral force also contributes to the yawing moment.
Since this contribution is inversely proportional to aspect ratio, the increment in Cp due to tip suction
becomes quite significant for highly swept and/or low-aspect-ratio planforms. A comparison has been
made of C., calculated with and without the tip-suction effect of reference 5 with test results. In allcases better agreement was obtained when the tip-suction effects were neglected. The analysis indicates aloss in tip suction particularly at the higher lift coefficients. Therefore, the effect of tip suction has been
omitted from the Datcom.
If experimental lift and drag data for the particular planform of interest are not available at the chosen
Machi number, no attempt should be made to estimate the variation of Cp with lift coefficient. No
known general method for estimating the variation of drag coefficient will give results reliable enough to
use in determining the correction factor for extrapolating the potential-flow values to higher lift
coefficients.

7.1.2.3-I

"DATCOMMETHOD

S'

Tlh variation of the wing rolling derivative CA, with lift coefficient at subsonic speeds, based on the
product of the wing area and the square of the 'ing span SW b, is given by

CA
:

ae-

1

tan~
ce

K

\CC

"

A~n

CL] +

/
M

I

+

L

-f

(per radian)

7.1.2.3-a

where
C1

SCpmethodis the roll-damping derivative at the appropriate Mach number estimated by using theof paragraph A of Section 7.1.2.2.
is the angle of attack in degrees.

CL(is the lift coefficient.

n,,,)CL\L

L0'

is the slope of the yawing moment due to rolling at zero lift given by

AB + i(AB +cos A.
A+cs~ 4+ 4cosan/4/4'LA +2AP(A+c4A(AB+4cosAccos)tan 4

M(•-P)CL.
M

'CA

2

A, 4

2ttj,

j (C/4

CL__P .

.

7

M-•0

7.1.2.3-b
where B-M2 cos 2 Ac/4 and=/CL-)c=Lois the slope of the low-speed yawing moment due to iolling at zero lift
given byM-0

tan2 AR tanA(C+v)_\ ) CL

.0O

M-0

+6(A

6

+ cos Ac, 4 )A + 4 cos Ac 4

4A

+

12
(per radian)

7.1.2.3-c
7, 1.2.3-2where I is the distance from the center of gravity to the aerodynamic
center, positive when the a.c. is aft of the c.g. and c is the wing mean
aerodynamic chord. Equation 7.1.2.3-b modifies the low-speed value ofequation 7.1.2.3-c by means of the Prandtl-Glauert rule to yield approximate corrections for the first-order three-dimensional effects of compressible flow up to the critical Mach number.

-.

ac
ACS
is the effect of linear wing twist obtained from figure 7.1.2.3-12.

90

is the wing twist between the root and tip stations in degrees, negative for washout(see figure 7.1.2.3-12).
Ac P

is the effect of symmetric flap deflection obtained from figure 7.1.2.3-13.

act
6(

is the streamwise fap deflection in degrees.

(-)

isthe two-dimensional lift-effectiveness parameter as obtained from Section 6.1.1.1.

K

is a dimensioniess correction factor used to extrapolate the potential-flow values tohigh lift coefficients. This is the same correction factor used in Section 7.1.2.1 toaccount for the variation of profile drag with lift coefficient. At zero lift this factor
is taken as 1.0. At lift coefficients other than zero this factor accounts for theuistioe of profile drag with lift coefficient and is given by

a
K

a
tanc)

-

%( -CD0)

7.1.2.3-dS(C tan a) - z

Test values of lift and drag at the chosen Mach number for the particular planfornii
of interest must be used in evaluating equation 7.1.2.3-d. The terms of this equation
are evaluated by taking the slopes of CL tana, (CD - CD 0 ), and CL 2 /(wA), plottedversus angle of attack.
,If reliable values of the static-force coefficients are available, the method should provide results within±20 percent accuracy throughout the lift-coefficient range to the stall.

7.1.2.3-3

Sample Problem
Given: The wing designated 45-4.0-0.6-006 of references 5 and 8.
Wing Characteristics:

A=4.0

E

X= 0.6S = 2.25sqftb

NACA 65A006 Airfoil

Ac/ 4

0 =0

=450

3.0ft
Y

0

R

; 3.1 x 106[origin of moments (c.g.) located at xac.]

Additional Characteristics:
M = 0. 7 0; 3 = 0.714

The following values of a and CD are test results from reference 8. The variation of C1 P
with C, has been calculated using the method of paragraph A of Section 7.1.2.2.

CLC

0

.1

.2

.3

.4

.5

.6

.7

.75

a

01

1.70

3.30

4.90

6.55

8.00

9.60

11.80

13.20

CD

.011

.0121

.016

.024

.040

.063

.094

.138

170

C

-. 314

314

12

.326

325

- .280-. 181-. 156-.

Compute:Determine
the slope of the yawing moment due to rolling at zeru lifi•
\CL /)c.o
I(

tan A

1 A+6(A+cosA,/ 4 )kEAtan 2'A
c/4

CL

CL=
M=0

/14

12

+

+cs

M4.0+6(4.0+

Lan2 45°•,
cos,450) 0+-d11
6'•..r...7.1,2.3-4

S!

4.0 + 4 cos 450

i 4.0 + 6 (4.7071)+
64.0-+2.828
-

--.

5

M

rr

/

(equation 7.1.2.3-c)

M 2 cos 2 A,,

B =/1-

A + 4 cos Ac/ 4

AB+4cosAc/ 4AB(0.7)2 (0.7071)20.87

4.0 + (4.0) (0.7071)..
(4.0) (0,87) + (4.0)(0.7071)

6.828
. .
6,308

-

AB2+I (AB +cos A.4) tan2Ac14(4.0) (0.87) + -A + I (A + cos A!4 tan2 A¢
c42
2
c4

1082
1.

1(4.0) (0.87) + 0.70711 (1)4.0 + -L 14.0 + 0.70711 (1)5.76.354

0.877[AB+ I (AB +cosAc 4 )tan 2 Ac/4 -C
C

L

=

A.

AB +4 cosAc/ 4- (A +cos A~

tan 2 A 1 4(equation 7.1.2.3-b)'= (1,082) (0.877) (-0.155)•-'-0.147 per rad

Determine the K factor
8

n/c

(

-•(c taa) -

K =

(E

a

Tiaa

-D

(equation 7,1.2.3-d)

L P

CL.tnC00

Q

CL

(dog)

tan

0

0

0

0

.011

J

0

•

1 70

.02968.00297

.012

.001

00

.2

3,30

.05766.01153.016.000.. 0318.34.90.08573.02572.024.013.00716

A4

6.55

.1148

.04592.U40.029.01273

-

8.00

.1405

.07027.063.062.01990

.6

9.60

.1691

.1015

.094

.083

02860

.7

11.80

.2089

.1462

.138

.127

.03900

.

1
!3.20

.2346

.1760

.170

.159

.04476•p-o,1Test

7.1.2.3-5.
.. .

Plot CL tang

,

CD

-CD

CL 2 /(rA) versus angle of attack (see sketch (a)).

0 , and

.18

4

C TAN
7Q
.16-

CD -CD 0

.14-

.10-

.08-

K

.06-

2
CL

~.04

ir

.02

-'

0

2

4

6

8

10

12

a (deg)
SKETCH (a)*~~a/C ..car(cc)(ca&(c~~tanx)

~

L}

--

K

2\

0Aa\

To,

(.C)~

(eq. 7.1.2.3-d).10033

.0014

.0010

.826

.4

.01440

.0128

.0040

.154

.6

.01720.0173.00616

.6

.0195

.0195

'0051

.7

.0212

.0215

.0041

.0216

.0220

.3.005.09.00.8.7

7.1.2.3-6

WO3

--.

008

0
-.

018

14

Solution:
Ci
-

C tan
P P)
.

-

-e K -C,tan a p0\tane- K [-C1

C
L=

/

tana -(-0.147) CL

-C(D
Q

•c¢p

.

(per redi0-. 314

.1

-Cl

p

)

tan a 0K-tan

c I tancv(1-K)

basd on swtý

K

1-

0

1,00

0

0

0

0

-. 314-. 0093

.826

.174

.00162.0121-. 0105.2-. 313-. 0180,301.00542.0206-. 0152.3-. 312-. 0267

.517

.0138

.0213

-. 0075

846

.0317

.0091

..0226

0.0460

&99
.483

p

0,147K
0.147

.4

--.32

-. 0374.1,54.5-. 325-. 0467-. 008.6-. 280-. 0474

0

1.00

I
D460
.0474

.7

-. 181-. 0378-. 0181.018.75-. 156-. 0368,-. 0231.023

1.008.400.06

0,e•. 7.1.2.3.a
prad

0

.0474

.0385

-. 0019

.0404

.0374

-. 0025.0399

The calculated results are compared with test values from reference 5 in sketch

S.1K) -0.147 CL K

p00
CIP tana
Sec. 7.1.2.2
CL(equation 7.1.2.3-a)(b).

Testp o ints
CalculatedC.
(per rad)

-10
.2

.4

CL

.6

.8

SKETCH (b)7.1.2.3-7

i-

-

-

-

-

-

-

--

- - - --

- - - - -.

B . TRANSONIC
No generalized method is available in the literature for estimating transonic values of the rollingderivative Cnp. Furthermore, no known experimental results are available for this derivative at transonic
speeds.
C. SUPERSONICAt supersonic speeds design charts based on theoretical calculations are presented for estimating therolling derivative CP at low values of the lift coefficient.
The design charts are based on the results of reference 6 for wings with subsonic leading edges andsupersonic trailing edges, and on the results of reference 7 for wings with supersonic leading edges andeither subsonic or supersonic trailing edges. The results of both references 6 and 7 are based onlinearized-supersonic-flow theory and are therefore restricted to thin, swept-back, tapered wings withstreamwise tips. The yawing moment due to rolling is taken as that arising entirely from suction forceson the wing edges. For wings with supersonic leading edges no suction forces are induced along theleading edges; consequently, the determination of Cnp involves only the unbalanced suction forcesalong the wing tips. Therefore, no design chart is presented for zero-taper wings with supersonic leadingedges, since the theory gives C,, = 0.
The design charts for wings with subsonic leading edges give values of C,. that are referred to bodyaxes with the origin located at the wing apex. The design charts fur wings with supersonic leading edges
give values of Cnp that are referred to body axes with the origin located at the projection of theleading edge of the tip on the wing root chord. The Datcom method presents transformation fo1mulas
for conversion from body axes to stability axes with the origin located at an arbitrary distance from theleading edge.

r

No experimental data are available for this derivative at supersonic speeds. Therefore, the validity oflinearized-supersonic-flow theory for estimating CP cannot be determined.

DATCOM METHOD*•-

Subsonic Leading Edges (P cot ALE < 1)
For wings with subsonic leading edges the contribution to the rolling derivative C,,
speeds, and at low values of the lift coefficient is given by

Cn

n..

bo dyI)

body
axis

2 xcg
(
+

A(]+

Q

-

(C

-

c)

(per radian 2 )at supersonic

7.1.2.3-e

where
-'n"-

is the supersonic yawing moment due to rollingteterredorigin at the center of gravity.

7.1.2.3-8

to stability axes with the

(Cn

" od

is the supersonic yawing moment due to rolling referred to body axes with theorigin at the wing apex, given by

body
axis
,P .€PC0

C

+

=

body

1

axis

C

_

,2

(per radian 2 )7.1.2.3-f

2

where

(

)
1

is obtained from figure 7.1.2.3i-14a through 7.1.2.3 14d as a function of
A, P cot ALE, and taper ratio. For X = 0, (Ci/a) 1 =0.

Cn p-4( /2)2and(3.) are obtained from figure 7.1.2.3.16 as a function of

13cot ALE.
xC."

is the distance from the wing apex to the center of gravity in root chords, positive
when the e.g. is aft of the wing apex.

n•Cy

is the supersonic side force due to rolling obtained by using the method of paragraph
C of Section 7.1.2. 1.
is the supersonic roll-damping derivative obtained by using the method of paragraph
C of Section 7.1.2.2.

is the supersonic yaw-damping derivative. It is negligible except for very low-aspectratio wings. References are noted in paragraph C of Section 7.1.3.3 that outline
approximate methods that may be used to determine this derivative.
ais the angle of attack in radians.

Supersonic Leading Edges (P cot ALE > 1)
For wings with supersonic leading edges the contribution to the rolling derivative CP at supersonic
speeds and at low values of the lift coefficient is gi.'en byP--

a Pbody
xA

axis

L

+

X)

2 tan ALE]

-

--

C1

(per radian 2 )"Lody
7.1.2.3-g

where
'f-"

C

4"(".

is the supersonic yawing moment due to rolling referred to stability axes with theorigin at the center of gravity.

the supersonic yawing moment due to rolling referred to body axes with the,isorigin at the projection of the leading edge of the tip on the wing root chord. It is
obtained from figures 7.1.2.3-17a through 7.1.2.3-17d as a function of P3A, P3cot
ALE, and taper ratio. No design chart is presented for zero-taper wings
since (Cnpll/)body = 0 for these planforms. For wings with taper ratios less than

body
axis

axis
values of (Cnp/a!)bo~dy should be obtained by extrapolating values
0.25 O, < 0.25)
aXIS

from figures 7.1.2.3-17a through 7.1.2.3-17d.

The remaining terms in equation 7.1.2.3-g are defined under the subsonic-leading-edge case.
Sample Problem
Given: Tapered, swept-back wing

"-A

r

,.22

1
M = 2.41;

X = 0.25

ALE

55.20

2.19

Xc.g.

0.742

A• 2

= 46.80

Compute:1.522,•cot ALE= (2.19) (cot 55.20)

9A = (2.19) (3.22)

=

7.07

0.50 per rad 2

7=

-.-.

=

-0.0114(figure 7.1.2.1-10)

per rad 2(figure 7.1.2.3-17a)

body
axis

A tan Aq=

(3.22) (tan 46.80)""-0.066 per radA

7.1,2.3-10(supersonic leading edge)

=

3.433

(figure 7.1.2.2-25b)

C1

S

-'

P

- -0.213 per rad-- (-0.066) (3.22)

Solution:a!

[ 2 x....

(Cp)

Cn
=e

a

+

=

LA(1+X)

body

axis

= -01.014-0.0114AG +r 2(0.742)+

1 C_
A.L FC
S=+tantan AL I Cr
2
jP
-C

-

11

tan 55.20

24
L3.22(1 +0.25)+ [0.3687 - 0.71941 0.50 + 0.213= 0.026 per rad 2(-efened(equation 7.1.2.3-g

0.50 - (-0.213)

to stability axes with origi

at x,.,. and based on Swbwz)

REFERENCES

S1.

Toll, T. A., and Queljo, M. J.: App-roximete Relations and Charts for Law-Speed Stability and Control Derivatives of swept Wings.
NACA TN 1581, 1948. (U)2.

Goodman, A., and Fisher, L. A.: Investigation at Low Speeds of the Effect of Aspect Ratio and Sweep on Rolling Stability Derivativesof Untapered Wings. NACA TR 968, 1950. (U)3.

Pinsker, W.: The Aerodynamic Coefficients of Free Lateral Motion. DVL UM 1144/1 and 1144/2, 1943. (U)4.

Queljo, M. J., and Jaquet, B. M.; Calculated Effects of Geometric Dihedral on the Low-Speed Rolling Derivatives of Swept Wings. NACA
TN 1732,1948. (U)5.

Wiogins, J. W.: Wind-Tunnel Investigation of Effect of Sweep on Rolling Derivatives at High Angles of Attack up to 130 and at High
Subsonic Mach Numbers, Including a Semiempirical Method of Estimating the Rolling Derivatives. NACA TN 4185, 1958. (U)

6L

Margolis, K.: Theoretical Calculations of the Lateral Force and Yawing Moment Due to Rolling at Supersonic Speeds for Sweptback
Tapered Wings With Streamwise Tips. Subsonic Leading Edges. NACA TN 2122, 1950. (U)7.

Harmon, S. M., and Martin, J. C.: Theoretical Calculations of the Lateral Force and Yawing Moment Due to Rolling at Sopersonic
Speeds for Sweptback Tapered Wings With Strearnwise Tips. Supersonic Leading Edges. NACA TN 2156, 1950. (U)8.

Kuhn, R. E., and Wigins, J. W.: Wind-Tunno)l Investigation of the Aerodynamic Characteri3tics in Pitch of Wing-Fuselage Conminations
at High Subsonic Speeds. Aspect Ratio Series. NACA RM L52A29, 1952. (U)r.

7.1.2.3-11l-.,

SUBSONIC SPEEDS-. 00 10

-l-

-

NOTE:

-.

ACa per radian L.

-

0 in degrees-. 0008.

\1,\,-. 0006--. 0004---. 0002 ----___(rad!-dee)

110-1.0002 ---

\

.\

\\\--

.00104*FU2 FTAOTSPECTRAIO.A
WIN
SECTION
ZERO-LIFT LINE
0

2

46
ASPECT RATIO, A

8

10

12

FIGURE 7.1.2.3,-12 EFFECT OF WING TWIST ON WING ROLLING DERIVATIVE Cn
P

10

7.1.2.3-12

.002-SUBSONIC SPEEDS_____(a) -=0.4
bK(rad~eD

NOTE:
Ac

per radian.-

0--

.001

(b) --

=0.60

b

AC

f

-c.002-

-

--

_

-

.6 -

p.-0

.001---. 001--(c)=08

FIUE71..-3EFETO

FA

. 2

8-L
ASEC RAIOb
ELETO
N IGRLLNEIVTVC
7.1631'f0-------

t--~

~

-

--

-~-

,

.

SUPERSONIC SPEEDS_________

(a)),

0.25

_

-. 80- 2
1__

TAN ALFc)

-

ELE

I

o

-. 76

(per rad

2Z)ý
1.32

P COT ALE

7.2

(~

1..33

(pe

rad

4--±-

-

-

-

-

-- 2..

TAN'ALE()a3

SPEEDS_________SUPERSONIC

1_

3.

(per rad 2

-1.6

.3

.5

.4

.

7.
j3COT ALE

-2.8--TAN ALE(~l-2.4---

2(per rad )

-2.2-

--

2.08

SCOT AE
FIGURE 7.1.2.3-14 (CONTD).

SUPERSONIC SPEEDS-. 20-A (-P2

and
-. 08

-

-

ATAN2ALE((\ -.04

-

--

2

-

(per rad )

.3

.4

.5

-

*

----

.7

.6

.8

1.0

.9

COT ALEFIGURE 7.1.2.3-16 YAWING-MOMENT-DUE-TO-ROLLING COMPONENTSSUBSONIC LEADING EDGES

7.1.2.3-16

AND

-3

-. 16(a)X =-.14.

-.32 •

_

_

...

-

.

.

p1

iq-.10•SIU

SUPERSONIC SPEEDS
.

0.25

ax es

-.08 -

0r\--. 04 ---. 02-

0. 1

4

2

3

4

5

6

7

FCOT ALE

FIGURE 7.1.2.3-17 YAWING MOMENT DUE TO ROLLING - SUPERSONIC LEADING EDGES BODY AXES
7.1.2.3-17

SUPERSONIC SPEEDS

(b) X

0.50

-

P3A

~-.3

-

---

--

.

-

-

__._

.22

8 COT ALE""

S7

FIGURE 7.1.2.3-171 (CONTD)

1.2.3-18

-

.

•

_

_

-1

-2---7I_.

SUPERSONIC SPEEDS-(c) X

1

0.75

II

-. 4

"N

*

~~-.3

____

I

-_.-_..

1

-

-

'
13

4

67

/ COT ALE

FIGURE 7.1.2.3- 17 (CONTD)

7.1.2.3-19

SUPERSONIC SPEEDS(d)-.X

1.0

1PA

(__

-Cl~b

I.x

-. 2

•
2

~12
3

4
/ COT ALE

FIGURE 7.1.2.8-17 (CONTD)

7.1.2.3-20

i
5

6

7.1.

,"•- WN

Y

...

..

IVATIVES

7.1.3 WING YAWING DERIVATIVES
7.1.3.1 WING YAWING DERIVATIVE Cv

This section recommends methods for estimating the wing contribution to the yawing derivative
Cy, at subsonic and supersonic speeds. However, at subsonic, transonic, and supersonic speeds
no generalized methods are available for estimating the wing contribution to Cy ,. This derivativeis the change in side-force coefficient with variation in yawing velocity and is expressed as

aCv
C yr

-

a

rb "

rVv

A. SUBSONIC
The wing contribution to Cy, is best evaluated from available experimental data (see table 7-A
and references 1, 2, 3, and 4), since no generalized method is availab;e in the literature.

U- •However, a method is available in reference I for wings with a taper ratio of one and moderate
sweep at low subsonic speeds. The range and accuracy of th-is method arr limited and generally
inadequate for making reliable estimates of the wing contribution.
Since the wing contribution to Cy, is usually quite small in comparison to the vertical-tailcontribution, it is somiktimes neglected.
B. TRANSONIC
No generalized method is available in the literature for estimating transonic values of the wing
contribution to the yawing denvative Cy ,. Furthennore, there is a scarcity of experimental data
for this derivative at transonic speeds.
C. SUPERSONIC
No generaiized method is available in the literature for estimating supersordc values of the wing
contribution to the yawing derivative Cyr. A few theoretical methods are available for specificconfigurations (see table 7-A). Although the use of experimental data for a similar configuration
is preferable to theoretical methods, experimental data are so scarce that the use of the limited
theoretical methods becomes the only alternative for most configurations.

REFERENCES
1.

Toll, T. A., and Oueijo, M. J.: Approximate Relations end Charts for Low-Speed Stsbilit Derivetive&of Swept Winp. NACA
TN 1581, 1948. (U)

7.1.3.1-1

2.

OuGao, M. J., and Jmluet, B. M.: Investigation of Effect of Geomotric Diheur3l on Low-Sxod Static Stability and Yeaing
Chnlctrisbt

of an Untpered 450 Sweptmack-Wing Model of Aspact Ratio 2.61. NACA TN 1668, 1948. (U)3.

Goodman, A., and Brewer, J. D.: In tigptIon at Low Speeds of the Effect of Aqspct Ratio and Sweep on Static and Yawing
Stability Deratives of Unuaperad Winp. NACA TN 1689, 1948. (U)4.

Letko. W., and Cowen, J. W.: Effect of Taper Ratio on Low-Speed Static and Yawing Stability Derivatives of 45- Sweptback
Wing with Aspect Rato of 2.61. NACA TN 1671, 1948. (U)

7.1.3.1-2

Revind June 1969

7.1.3.2 WING YAWING DERIVATIVE C1,This section presents a method for estimating thei wing contribution to the yawing derivative C4 -at
subsonic speeds. No generalized methods are available for estimating C/, at transonic and supersonic
speeds; however, theoretical methods for determining this derivative at supersonic speeds for specialclasses of wing planforms are discussed. This derivative is the change in rolling-moment coefficient withchange in the yawing-velocity parameter and is expressed as

ac1

A. SUBSONIC
The wing yawing derivative C1, results from the lift differential between the wing panels when thevehicle is yawed about its vertical axis.
Over the range of lift coefficients for which C1, is linear with CL, the derivative Cj. for wings withoutgeometric dihedral, twist, or flaps is based on the lifting-line theory of reference I for aspect ratios
greater than three and on the experimental data of references 2 and 3 for aspect ratios less than three.
The increment in C1, due to geometric dihedral is taken from reference 4. The effects of symmetricflap deflection and wing twist are taken from reference 1.
In addition to the increments in C1, due to dihedral, twist, and flaps, an additional increment
in C1. arises due to Cy, if the center of gravity does not lie at the same height as the quarter-chordpoint of the wing MAC. This contribution is obtained from the expression(AClr)etorce =-C

where z is the vertical distance between the center of gravity and the quarter-chord point of the wing
MAC, positive for the c.g. above U/4. The side force due to yawing Cyr is small except at high angles
of attack. Therefore, this increment in C1r is omitted from the Datcom method.
The fore and aft movement of the center of gravity also affects C1,, but this effect is neglected becauseof the questionable accuracy of the basic effect of wing sweep.
The method of reference 5 is applied to extrapolate the potential-flow value of C4 to higher lift
coefficients. The method is semiempirical in that it requires test values to determine the correction
factors to be applied to the theory. Analysis of experimental data shows that the discrepancies between
theoretical and experimental values of C1. for wings are similar to the discrepancies between theoreticaland experimental values of the static derivative ClD, Based on this analysis, a correction factor is appliedin reference 5, which is the increi, -ntal value of C1. obtained by subtracting the experimental value
from the theoretical value.

7.1.3.2-1

Experimental data indicate that for unswept wings C11 is nearly proportional to the lift coefficien• until
maximum lift occurs. For sweptback wings, a linear variation is obtained over only a limited lift range,
which is reduced as sweep increases. At high lift coefficients C1, decreases, and for sweptback wings
may become negative near maximum lift.
If reliable values of the rolling moment due to sideslip Q, are available, the method should provide
results within ±20-percent accuracy over the lift-coefficient range for which C1. is approximately linearwith CL.

DATCOM METHOD
The variation of the wing yawing derivative C1 , with lift coefficient, based on the product of the wingarea and the square of the wing span Swb, , is given by

8k•-,

La

whereCC_r ](per radian)+MM

is the slope of the rolling moment due to yawing at zero lift given by

A(IB2)(=~

A(l -B)
1+L
2B(AB + 2cos A,/ 4)

tan 2 AX/AB + 2 cos A
c/4___c/4

AB+4 cos A,/ 4

8
7.1.3.2-b\L

A + 2 cos Ac/
tan2 AC/4
1+1A+4cos Ac./44
8where BCLc0

M=o7.1.3.2-2

r ++7 .1.3.2-aa[/l-M2 cos 2 Ac! 4 andis the slope of the low-speed rolling moment due to yawing at zero lift,
obtained from figure 7.1.3.2-10 as a function of aspect ratio, sweep of
the quarter-chord, and taper ratio. This chart has been derived by using
the results of references 1, 2, and 3. Equation 7.1.3.2-b modifies the lowspeed value by means of the Prandtl-Glauert rule to yield approximate
corrections for the first-order three-dimensional effects of compressible
flow up to the critical Mach number.

CLis the wing lift coefficient.

(AC, )
CL

is a semiempirical correction factor used to extrapolate the potential-flow values of
C 1, to higher lift coefficients. This parameter is given by

(AC1

)

= CL

CL )-

(Ca)t

(per radian)

7.1.3.2-c

where
Cl,

is the theoretical value of the slope of the rolling moment due to sideslip
at zero lift obtained by using the method of paragraph A of Section
5.1.2. 1, neglecting the effects of twist and dihedral.
In applying thismethod the compressibility correction to the sweep contribution should beconsidered.

-

CL

C,)

is the experimental value of the rolling moment due to sideslip at thetest appropriate Mach number.

Aq,
is the increment in Cr1 due to dihedral, given by

r

AC/r
r

r

'
=

-

urA sin Ac/4-v(per radian 2 )7.1.3.2-d

12 A + 4 cos Ac/4

is the geometric dihedral angle in radians, positive for the wing tip above the plane of
the root chord.

ACI
is the increment in CIr due to wing twist obtained from figurerg
Ois the wing twist between the root and tip sections in degrees, negative for washout(see figure 7.1.3.2-11).
AC 1

8f

7.1.3.2-11.

I

is the effect of symmetric flap deflection obtained from figure 7.1.3.2-12.

is the streamwise flap deflection in degrees.

7.1.3.2-3- .-.-.-.-.........

(-.)

is the two-dimensional lift-effectiveness parameter a6 obtained from Section 6.1.1. 1.
f

The expression given for the effect of dihedral (equation 7.1.3.2-d) is based on an extension of thesimple-sweep theory of reference 6. It has been shown by comparison with test data thai the increment
in C1r due to dihedral is underestimated by equation 7.1.3.2-d. However, it has not been possible toimprove upon the simple-sweep-theory result because of a lack of experimental data.
Furthermore, not enough data are available to examine the validity of the lifting-line-theory resultspresented for the increments in C4r due to either twist or flap deflection.
A comparison of the slope of the rolling moment due to yawing at zero lift, obtained by using figure
7.1.3.2-10, with test results is presented as table 7.1,3.2-A.
The sample problem illustrates the application of the method over the lift-coefficient range to the stall.

Sample Problem

Given: The sweptback, untapered wing of reference 4.
Wing Characteristics:
X = 1.0

A = 2.61

A1 c/ 4

AC/2

450

=

450

r

100

0

Additional Characteristics:
Low speed; M - 0
The following test values from reference 4:
..

L

0
CL0

.1
.

..
2

.

4

-. 0458 -. 1031 -. 140 -. 17T6-.2060(per rad)Compute:VC

0.419 per rad (figure7.1.3.2-10)

V /CL"7.1.3.2-4

0
M=0

.5

.6

.7

.8

1.
.9

1.0 1.1

235 -. 260 -. 274 -. 260 -. 211 -. 102 .0287

0Determine the theoretical value of Cia/CL (Section 5.1.2.1)=-0.0044 per deg

KM A ; 1.0I(figure 5.1.2.1-27)

0.0038 per deg-(figure 5.1.2.1-28b)(figure S.1.2.1-28a for M

C

0)

KMA

KM

cL/2AcI\L/A]\J+ 0 tan A c/4r)

0 tanlA c/4(equation 5.1.2. l-a)

Neglecting the effects of twist and dihedral

CI

FL A/2

C/A
LM
Km

C

= (-0.00 38)(1.0) + (-0.0044)"- -0.0082 per deg
0.470 per rad

-

(AC,)
I

=C

CLI

(equation 7.1.3.2-c)-(CI= C1, (-0.470)

--

(CI)

(see calculation table)

7.1.3.2-5

IAC

r
ACI

1

r A sin A,/4

12

A+4cosAc/ 4(equation 7.1.3.2-d)r"

1 vr(2.61) (0.707 1)
12 2.61 + 4(0.7071)

I vr(2.61) (sin 450)
12 2.61 +4cos45

00.0884 per rad 2=

Solution:
C

= CL

CL =0

(equation 7.1.3.2-a)

r

+)c r

)L

(AC

'+

L

M=0

C=L(0.419) + ICL (-0.470) LL
0.419 CL +-0.470 C
L

C[, )

C
L

\-CLT -C L(tes)M(per red)

0.419

0

-. 0458

.1
.2
.3

10

+ 0.0154
test

0

\ CL'

&"-

""CL

10

CIOts
008)5.-•(0.0884)
test57.3

L(eq.
7.1.3.2-c)\MrL./)L(m

G•(

on SWbw)(based(q7.1.3.2-a)+W& + 0.0154-0.470 ((per red)(per rod)

0

0

.0458

.0612

-. 1031-. 1400

.0419
.0838

-0.0470-0.09400.0561
0.04600.11340.1452-. 1760.1257-0.14100.03600.1761-. 2060.1676-0.18800.01800.2010.5-. 2350.20,15-0,235000.2249.6-. 2600.2514-0.2820-0.02200.2448.7-. 2740.2933-0.3290-0.06500.2537.8-. 2600.3352-0,3760-0.1160

0.2346.9-. 2110.3771-0,4230-0.21200.1805

1.0-. 1020.4190-0.4700-. 0.3680

0.0664

1.1

0287

.4609

-0,5170-0.5457--0.0694

7.1.3.2-6

The calculated results are compared with test values from reference 4 in sketch (a).

.3-

W.
U

0

Test points
Calculated

.2 o-

IQ

S:i-i-

Ci(per rad)

4(9
.2

.4

6.81.0C\

12SKETCH (a)B. TRANSONIC0

In the transonic speed regime no theoretical methods are available for estimating the wing yawing
derivat ve C,,. Furthermore, no known experimental results are available for this derivative at transonic
speeds.
C. SUPERSONIC
No g i.eral method is available for evaluating the wing contribution to the yawing derivative C1. atsupersonic speeds. However, methods are presented in references 7, 8, and 9 for evaluating C1. ,forspecial classes of wing planforms. The results presented in these references are based on supersonic linear
theory. The methods are restricted to estimating C1. over a limited range of Mach numbers forzero-thickness wings with no dihedra!. Furthermore, the methods are considered tentative, since the
spanwise variation of Mach number in the case of yawing has been neglected.

7.1.3.2-7

;•3" •-

3 ......

.

3i

./

" •

.

.

.

•

.

.

.

.

.

..

REFERENCES

1.

Pinsker, W.: The Aerodynamic Coefficients of Free Lateral Motion. DVL UM 1144/1 and UM 1144/2, 1943. (U)2.

Goodman, A., and Brewer, J. D.: Investigation at Low Speeds of the Effect of Aspect Ratio and Sweep on Static and Yawing Stability
Derivatives of Untapered Wings. NACA TN 166, 1948. (U)3.

Letko, W., and Cowen, J. W.: Effect of Taper Ratio on Low-Spied Static and Yawing Stability Derivatives of 450 Sweptback Wings With
Aspect Ratio of 2.61. NACA TN 1671,1948. (U)4.

Queijo, M. J., and Jaquet, B. M.: Investigation of Effects of Gebmetric Dihedral on Low-Speed Static Stability and Yawing
Characteristics of an Untepered 460 Sweptback-Wing Model of Aspect Ratio 2.61. NACA TN 1668, 1948. (U)5.

Campbell, J. P., and Goodman, A.: A Semlempirical Method for Estimating the Rolling Moment Due to Yawing of Airplanes. NACA TN

1984,1949.(U)6.

Toll, T. A., and Queijo, M. J.: Approximate Rela-.ons and Charts for Low-Speed Stability Derivatives of Swept Wings. NACA TN 1581,
1948. (U)7.

Ribner, H. S., and Malvestuto, F. S., Jr.: Stability Derivatives of Triangular Wings at Supersonic Speeds. NACA TR 908, 1948. (U)8.

Harmon, S. M.: Stability Derivatives at Supersonic Speeds of Thin Rectangular Wings With Diagonals Ahead of Tip Mach Lines. NACA
TR 925, 1949. (U)9.

Malvestuto, F. S., Jr., and Margolis, K.; Theoretical Stability Derivatives of Thin Sweptback Wings Tapered to a Point With Sweptback or
Sweptforward Trailing Edges for a Limited Range of Supersonic Speeds. NACA TR 971, 1950. (U)10.

Lichtanstein, J. H.: efflect of l4igh-LifLt Devika on the Low-Spned Static Latetal aýd Yawing Stability Chlaracteristics of an Untapered
450 Sweptback Wing. NACA TN 2689, 1962. (U)11.

Letko, W.: Effect of Vertical-Tail Area and Length on the Yawing Stability Characteristics of a Model Having a 450 Sweptbsck Wing.
NACA TN 2358,1951. (U)12.

Williams, J. L.: Measured and Estimated Lateral Static and Rntary Derivatives of a 1/12-Scale Model of a High.Speed Fighter Airplanewith Unawept Wings. NACA RM L53KO9, 1954. (U)13.

Goodman, A., and Feigenbaum, D.: Preliminary Investigation at Low Speeds of Swept Wings in Yawing Flow. NACA RM L7109, 1948.
(U)

14..,

Letko, W., and Jaquet, B. M.: Effect of Airfoil Profile of Symmetrical Sections on the Low-Speod Static-Stability and Yawing
Derivatives of 450 Sweptback Wing Models of Aspect Ratio 2.61. NACA RM LBH1O. 1948. (U)15,

Bkid, J. D., Jaquet, B. M., and Cowan, J. W.: Effect of Fuselage and Tail Surfaces on Low-Speed Yawing Characteristics of a Swept-Wing
Model as Determinad in Curved-Flow Test Section of Langley Stability Tunnel. NACA TN 2483, 1951. (U)16.

Jlaquet. B. M., and Fletcher, H. S.: Experimnntal Steady-State Yawing Derivativ.'s of a 600 Delta-Wing Model as Affected by Changes in
Vertical Poition of the Wing and in Ratio of Fuselaeg Diameter to Wing Span. NACA TN 3843, 1956. (U)

7.1.3.2-8, .

TABLE 7.1.3.2-A
SUBSONIC WING ROLLING MOMENT DUE TO YAWINGDATA SUMMARY/rCL

I•- iCl

C r/CL

Ac/ 4

CaIc.

Test

Percentx(dog)(per red)(per rad)

Error

2.61

1.0

45.0

0419

0.415

1.0

11

4.00

0.60

46.0

0.413

0.396

4.3

2

5.90

0.473

--3.5

0.240

0.225

6.7

Ref.

A

10

13

1.34

1.0

60.0

0.445

0.470

5.3

5.16

1.0

0

0,260

0.220

18.2

1.34

1.0

a

0.195

0.161

21.1

46.0

0.353

0.350

0,9

0

0.230

0.260

-11.5

60.0

6.525

0.558

- 5.9

45.0

0.475

0.480

- 1.0

0.596

0.550

8.4

45.0

0.365

0.300

21.7

45.0

0.302

0.315

- 41

52.2

0.295

0.185

3.5

2.61

5.16

S60.0
5

2.61

0.50

*.0.252.31

0Averup Error --

-

8.1%

n

7.1.3.2-9

SUBSONIC SPEEDS

1/4(e)TAPER
RATIO

0

15

xI

60

45

30)__

.62

CL

=0

4

6'

8=0

ASPEC
A RATO,.2.*4

FIGURE 7.1.3,2-10

Lbr

WING YAWING DERIVATIVE

5.

C1

SUBSONIC SPEEDS
NOTE:
._

.004

r

ACr

per radian0

in degrees

_

_

__

.003

.00 2
r-egROOT-SECJION
ZERO-LIFT LINE

.001•

TIP-SECTION
LINE-;,-•ZERO-LIFT

--

0

2

4

6

8

10

"ASPECT RATIO, A

FIGURE 7.1.3.2-11 EFFECT OF WING TWIST ON WING YAWING DERIVATIVE
Cl

7.1.3.2-11

SUBSONIC SPEEDSr

ASPECT RATIO, A
0

1

2

3

4

6

8

10

ARATIO

/'b/

0

.002

.004

.06

0.2

(rad -deg)

FLAPS IN PERCENT SEMISPAN,

per radian

bf in degrees

AC
1f
NOTE:

[AC

-

r

f

FIGURE 7.1.3.2-12
7.1.3.2-12..

1.

LOCATION OF INBOARD AND OUTBOARD

AC,

NO2I

4.6'

1

1

1

AC
1J
acr

,
outboard

EFFECT OF FLAPS ON jLG)f jinboard/

Revised June 1969

7.1.3.3 WING YAWING DERIVATIVE Car

This section presents a method for estimating the wing contribution to the yawing derivative C 3 r at
subsonic speeds. No generalized methods are available for estimating C.r at transonic and supersonic
speeds; however, theoretical methods for determining this derivative at supersonic speeds for specialclasses of wing planforms are discussed. This derivative is the change in yawing-moment coefficient withchange in the yawing-velocity parameter. It is commonly referred to as the yaw damping and isexpressed as

Can

nr a(

rb

A. SUBSONIC
The wing yawing dteivative Ca, is due to the antisymmetrical lift and drag distributions over the wing
resulting from the yawing velocity.
The wing contribution to the yaw damping in the range of lift coefficients for which Car varies linearlywith CL is composed of two major contributions; namely, that resulting from the drag due to lift andthat resulting from the profile drag. The contribution resulting from the drag due to lift is given to afirst approximation by the simple-sweep-theory result of reference 1. It is a negative quantity except for
highly swept wings in which case it can become positive.
The increment in C,,r due to profile drag is also taken from reference 1. Although the spanwise
distribution of profile drag is required for an accurate determination of the effect of profile dragon C,,, the profile drag has been assumed constant over the wing surface in the analysis reported inreference I. This approximation greatly simplifies the analysis, since it allows the profile-drag effect tobe expressed as a function of only the wing geometry.
Flaps and wing twist will also induce increments in Car, primarily as a result of their influencing the lift
distribution across the span. However, the technique of the superposition of lift distribution proportional to angle of attack, which was used to determine the effects of either flaps and/or twist on therotary derivatives C.P and CQ,, cannot be applied in this case. The contribution of the drag-due-to-lift
component to C., increases as the square of the angle of attack. Therefore, a breakdown of the lift
distribution proportional to angle of attack is not possible. No methods are available in the literature for
estimating the effects of flaps or wing twist on the wing contribution to C,,. Ftirthermore, not enoughtest data are available to allow derivation of an empirical method.
The wing side force due to yawing Cy, will also produce a yawing moment when the center of gravity
does not lie at the same longitudinal station as the quarter-chord of the wing MAC. However, the sideforce due to yawing is small except at high angles of attack; consequently, this increment in Cn, isomitted from the Datcom.
Experimental data indicate that for unswept wings the yaw damping is nearly proportional to the lift
coefficient until maximum lift occurs. On the other hand, for sweptback wings linear variationsof Car are obtained over only a limited lift coefficient range, which is reduced as sweep and/or aspectratio in=tease. Experimental data also show that, in general, the yaw damping of a sweptback wing
7.1.3.3-1

"* 7--- -./ --

i=

-.-

7 .-

*

-"

-.

--

-.

-.

N•

r %"

"-

h

r.-'

-

-

.•--

•

.

.. ''

L... .

changes sign and becomes positive at some moderate lift coefficient. The lift coefficient at which thischange in sign occurs is reduced as wing sweep and/or aspect ratio increase.
Results obtained by using the Datcom method agree reasonably well with test data over the range of lift
coefficients for which C'Ir varies linearly with CL.
Since the wing contribution to the total Cn, of the airplane is small, no method has been developed toaccount for the effects of compressibility. For the purpose of the Datcom the effects of compressibility
are accounted for by evaluating the wing profile-drag coefficient at the desired Mach number,
Experimental data show that the effect of wing dihedral on the yaw damping is negligible.
Because of the insignificance of the wing contribution to the total yaw damping and the approximate
nature of the Datcom method, the method is applicable to wings with twist and/or symmetric flap
deflection as well as to plain wings,
DATCOM METHOD
The variation of the wing yawing derivative Cn, with lift coefficient at subsonic speeds, based on the
product of the wing area and the square of the wing span Sbb, is given by
VC

1 =QL)CH.

L Z jr)C(per radian)

7.1.3.3-a

where
CL

is the wing lift coefficient.

C-Z

is the low-speed drag-due-to-lift yaw-damp. ,arameterobtained from figure 7.1.3.3-6
as a function of wing aspect ratio, taper ratio, sweepback, and c.g. position.

Cn-

is the low-speed profile-drag yaw-damping I-

CD0a function of the wing aspect ratio, sweep-back, and c.g. position.

C0

is the wing profile drag coefficient evaluated at the appropriate Mach number. For thisapplication Co 0 is assumed to be the profile drag associated with the theoretical ideal

meter obtained from figure 7.1.3.3-7 asdrag due to lift and is given by

CD

C=

C
CD

C 2
L
A

where CD is the total drag coefficient at a given lift coefficient, obtained from experimental data.
7.1.3.3-2

-.

-

-" -

Sample Problem"Given: The delta-wing model of references 5 and 6.
Wing Characteristics:
A=2.31

= 0

=

52.40

0;0

:/U =0 (c.g. at U/4)
Additional Characteristics:
M = 0.13

The following test values from reference 6.

K

C

0

.1

.2

CD

.017

.020

.029

.047

.074

.5

6

.7

.8

.105

.141

.184

.235

Compute:C nr-

0.0080 per rad(figure 7.1.3.3-6)

r- = -0.68 per rad(figure 7.1.3.3-7)

CL2

C
-CD0

Solution:

L

Cn

CL2

CDir

(equation 7.1.3.3-a)CD

0.008 CL 2 + (-0.68) CDS+

7.1.3.3-3

-

.

-

.

.

.

I

0C

IrL 2/c

('I)
L
CL

(j 2

(0002)
CL(based onSWbý)2-CL /W )

CC

Q

()

(D

Dw

2.1

1-06)4-)

r

.3)

_

C0

(eq. 7.1.3.3-a)®+G2MA)(tesD
Q)(-0.56)6)(per red)

0

0

0

.017

0

.0170

-. 0116-. 0116.1.01.00006.020.00138.0186-. 0127-. 0126

.2

.04

.0003

.029

.00651.0236-. 0160-. 0157

.3

.09

.0007

.047

.01240.0346-. 0236-. 02_n

.4

.16

.0013

.074

.0220

.0520

-. 0364-. 0331

.25

.0020

.106

.0344

.0706

-. 0480-. 0460

.6

.36

.0029

.141

.0496

.0914

-. 0621-,0692

.7

.49

.0039

.184

.0675

.1165

-. 0792-. 0753

.8

.64

.0051

.235

.0882

.1468

-.08

-.0947

.5

i

/C 0_

The calculated results are compared with test values from reference 5 in sketch (d).

-. 2

,

0 Test points-Calculated•nC(per rad)

0

.2

.4
CL
SKETCH (a)"7.1.3.3-4.6.8

B. TRANSONICIn the transonic speed regime no theoretical methods are available for estimating the wing yawing
derivative. C,,~. Furthermore, no known experimental results are available for this derivative at transonic

speeds.
C. SUPERSONIC
No general method is available for evaluating the wing contribution to the yawing derivative C., atsupersonic speeds. However, methods are presented in references 2, 3, and 4 for evaluating C. r for
special classes of wing planforms. The results presented in these references are based on supersonic linear
theory. The methods are restricted to estimating Ct over a limited range of Mach numbers forzero-thickness wings with no dihedral. Furthermore, the methods are considered tentative, since the
spanwise variation of Mach number in the case of yawing has been neglectedt.

REFERENCES
elomwan Owte foruLwaeeed tabiiy Dwlwmlvu of qwpt Whip. NACA TN 1561,
App xroxhiinta
1. ToN. T.A.. and Omelo.U. J.:
1194113(.11W
2.

Rlbnar, 14.S.. and Minemsetuo. F. 5, *r.: Stablity Darlivatvaof Trlangmahr Whipat Superbonic Speeds. NACA TR 908.1948. (U)3I

Hs m ~.S. M.: Subllity DOn katisTR 925.1949. (USupersnic Speeds of Thin Rectenguiler Whipg With Diagonals Ahead of Tip Mach Linee. NACA

Wi~ip Tapered to aPoint With Svweptbadc
Jr., and Margolle K.: Theoretim Stl~ty Dwletkat of Thin *ap*ec
4. Mahweauto F.S..
and ewapormwod Trailing Edges for a Uinitad Rang of Suparanic Speeds. NACA TN 971,1950.1WU
5.

iquet,UB. M., and Fi ch1r. H. L.: Experbn-- Ctsady~uta Yamf Dariwesuof aGV Delta-Winip Model at Affected by Chianges in
to Wing Sma. NACA TN 11143,19%61.11U
Vertloal Potion of the Wing and inRatio of Fuselage Oh wote

IL Goodeman. A.. end Thomma 0. F., Jr.: Effects of Wing Position aid Fuselage flin on the Low.Speed Static and Rolling Stablity
Cheracerlmatcof a Deft.Whg Modal. MACA TN 1224.1955.1WU
7.

Wi~llens. J. L- Mumeoed and Estimated Lateral Staic: and Rotaly Dwkrlesie of a 11124cob Model of a High~peed Fighter Airplaen
With UwAeps Whip. MACA RM LU3IC, 1914. 111.1

8.

Goodmant.

A.. and Imvwe. J. 0.: I - dip iln at Low Seed of the Effect of AwAct Ratio and 8, L

Derliethmuof Utapad Whip. NACA TN im,

on Static and Yavuing Stability

1943.41.0

9.

Sad.J. 0.. JmuS. .
, ands" Cmn. J.W.: Effect f , I s-e
and TallTunnel.
Surfem(U)
TN 2483,1951.
NACA
on Low
StabiitModelal-Deterisnedh CwuWd.bw Tom Section of Langley
ftaed Y awli
ngCh vc ter ilce of a Swept.W l "'10,

Latko. W., and JesuS* IL U.: Efifetmof Aitoll Prof lb of Syuwiierilul Sections on the Low~peed Statlotability and Yawning
B0I. 1943. (WDerluiwstofl411P S g- 'cWingModalsof Awect Ratio 2.61. NACA ~41.1M

111. Latito. W.: Effectof VstlinTdI ieand Lwathion theYmkA
MACA TN 23. 1961. It)

8tllty Cheraehim offa ModellHmhn a 450 &vepttzocWing.

7.1.3.3-5

SU BSONI( SPEI)DS
e14I(deg)

x ..
k

_-

TAPER

.

..

x

_

___..

_

_

-

_

--.

RATIO
1.0

1

1

TAPER RATIO

-

F0

.

..

j

I NOTE: ]

1.0

..

__

_ __-__.__ -.
--_
K is distancefrom the c.g. to the a~c., positive for the a.c. aft of the eg

C is the mean aerodynamic chord.

0.

21-Z
11,/L
(per rad)j

TAPER

.

0.4
___

(deg)

____

±60

.....

.

.

..

.

.

RATIO
1.0
1

2

FIGURE 7 .j.3.3-,,7..

ASPECT RATIO, A

is0(per rad)-LOW-SP'ý.ED DRAG-DUE-TO-LIFT YAW-DAMPING PARAMET'ER-6----

-

j

141

00'

---

SUBSONIC SPEEDSASPECT RATIO, A
2
4
6

.

10

A= 4
(deg)
J0

(a)--0
c

8

l

Cnr

5

CD
(per rad)i-1.2•ASPECT
0
2
(b)7

RATIO, A
6

4

0.2

8

to

lAc/4

r0

ASPECT RATIO, A
CD

Kx(per rad)c0- -(eC)ASPECT RATIO, A
4
6

20'(c)-x=0.4

810

IAc/4--- r

1- (d eg)

Cn•
..

//i--

CD0
-. 8
(per rad)

._

[ . .
_ - --

S-L

so5_-'- " 6 0

-1.2 oi is the distance from the c.g. to the a.c., positive for the a.c. aft of the c~g.
Z is the wing mean aerodynamic chord.
FIGURE 7.1.3.3-7 LOW-SPEED PROFILE-DRAG YAW-DAMPING PARAMETER
NOTE:

7.1.3.3-7

"'-•"- 't •'-• -¸••¸'
,

.

.

=Y¶•,

F

.r -.': o

,.

r *

1

•

-, .ig
'

1'

*

q *

E,

Y,

W

..
...

S -

I*

I'

*

-

*.....

1

Revoyud September 1970

7.1.4 WING ACCELERATION DERIVATIVES
7.1.4.1 WING ACCELERATION DERIVATIVE CL&

Methods are presented for estimating the wing contribution to -he derivative CL at low angles ofthe subsonic and low transonic speed ranges and for planfornis
attack for a triangular planformwith the leading edge swept back and the trailing edge swept back or swept forward in the
supersonic speed range. In addition, the supersonic results are directly applicable to wings withsweptforward leading edges, in view of the reversibility theorem (see reference 5). This derivative isused in estimating Cm&, in Section 7.1.4.2.
If the wing acceleration derivative CL& is to be used in method I of Section 7.3.4.1 to obtain(CL&)WB, the exposed wing planform area should be used for all calculations in the Datcom
methods. Using the exposed planform area will yield CL& based on the product of the exposedwing area and the exposed wing MAC, rather than the product of wing area and wing MAC as
indicated.
DATCOM METHODS

A. SUBSONICAn equation for estimating the subsonic acceleration derivative CL & of a triangular wing (derived inreference 1), based on the product of wing area and wing MAC SwCw, is given byCL.

1.5(c/

CL+ 3 CL(g)7.1.4.1-a(per radian)

where
CLLa
..

is the wing lift-curve slope (Section 4.1.3.2) at the Mach number under considewation,
based on the total wing area (per radian).

r

is obtained from Section 4.1.4.2.

-

Cr

CL (g) is the lift-coefficient correction term obtained from figure 7.1.4.1-6 (per radian).
Because of the restrictions placed on the lift-coefficient correction terni, this method is valid only
for 0 < OA < 4.
Explicit expressions for estimating CL , of other wing planforms in the subsonic region are notavailable at this time.

F

7.1.•4.1-1
-

---

---

---.

-J

Sample Problem

Given:
A

4.0

?,=0

ALE

45-

CL

4.0perrad (Section 4.13.2)

M = 0.6

Compute:
=-M2
0.80

/=

1.
L

tan AEttIULE$/tan ALE=

0.80

PA = 3.20
A tan ALEx

4.0

- 0.570(figure 4.1.4.2-26a)Cr-P2 CL(g)- 0.1245 per rad

irA/2(figure 7.1.4.1-6)

CL(g) = -1.22 per rad

Solution:
0xC=

1.5 (Ic-•"-)

CL+ 3 CS()(equation 7.1.4.1-a)= (1.5) (0.570) (4.0) + (3) (-1.22)= -0.240 per rad (based on Sw
W)

B. TRANSONIC
The value of CL& of a triangular wing from the critical Mach number to M

1.0 is given by themethod of paragraph A, provided 0 < PA < 4.

7.1.4.1-2
- .

.

-

-.

.

1

There is no general theory available in the literature that gives the transonic values of Cr&,eitlher
for additional w'ng-geometry parameters or for Mach numbers greater than 1.0. Furthermore,
there is a scarcity of test data in the transonic region for any wing planform.
C. SUPERSONICK.

The supersonic value of CL&, based on the product of wing area and wing MAC SwEw, is
del-ived in references 2 and 3 for wings with subsonic leading edges and in reference 4 for wings
with supersonic leading edges.
I.

Wings with subsonic leading edges (P3cot ALE < 1.0)
For wings with subsonic leading edges, CL& is obtained by the method of reference 2 for
X =- 0 and by the method of reference 3 for X ý 0.25 to 1.0. The following methods are
not valid if the Mach line from the vertex of the trailing edge inttrsects the leading edge orif the wing-tip Mach lines intersect on the wings or intersect the opposite wing tips.
a.

Zero-taper-ratio wings (K = 0)
CL& is derived in reference 2 as

CLM
a

[3GOC) F 3 (N) + 2E"(OC) F 2 (N) +-m2

E"(3C) FI(N)](per radian)

7.1.4.1-b

where
E"(PQC) and G(JC) are obtained from figure 7.1.1.1-8.
F,(N), F 2 (N), and F 3 (N) are obtained from figure 7.1.4.1-7.
b.

Wings with A r- 0.25 to 1.0
CL.

is derived in reference 3 as
M2
CL

-

ICL)I

-

2

(c 19

(per radian)

7.1.4.1-c

where(CL J)
X

and (CL.)are obtained from figures 7.1.4.1-8a through 7.1.4.l-8f for

0.25, 0.50, and 0.75 and from the equations of reference 3 for X > 0.75.

7.1.4.1-3

2.

Wings witth supersonic leading edges (0 cot ALE > 1.0)
For wings with supersonic leading edges, CL& (derived in reference 4) is given by equation

7.1.4.1-c with (Cia)I and(CL&)2 obtained from figures 7.1.4.1-1 Ia through 7.1.4.1-11o.
Figures 7. 1.4.1-1 la through 7.1.4.1--I 1o are valid for the range of Mach numbers for which
the Mach lines from the leading-edge vertex intersect the trailing edge. An additional
limitation is that the foremost Mach line from either wing tip may not intersect the remotehalf-wing.
Sample Problem
I.

Wing with subsonic leading edge

Given:
A = 5.80

=00

ALE

60 0

M

=.50

Compute:*F_~

cot ALE

l

--

=

1.12

0.5774=0.647

ý4cot ALE4

cot ALEi1=0.602
A
E"(3C)= 0.770

N

GOC)=0.570F) (N)=

F 2 (N)
F3 (N)

1.090=0.907-8(figure 7.1.1.1-8)

0.520(figure 7.1.4.1-7)

Solution:
r

CL

7rAM'
202
232

[I 3G3C) F 3 (N)2E'(OC)F(N)2+ m2 E"(3C) F (N)-(equation 7.1.4. 1-b)ir(5.80) (2.25)
2(1.25)7.1.4.1-4

3(0.570) (0.907)+ 2(0.770) (1.090) +(0.770) (0.520
2.25

= -5.22 ir(-1.551 + 1.679 + 0.178)-2.

-5.02 per rad (based on Swc)

Wing with supersonic leading edge

Given:

ALI= 0.25A =4.0=450

M =2.0Compute:v M2=-1=

1.732

PA = 6.928
cot ALE = 1.00

P cot ALE
cot-

qT

=

1.732

(( cot ALE) = 300

P (CLJ) =-0.390 per rad
(CL.)

=

(figure 7.1.4.1-11d)-0.225 per radO(CL,) = 4.200perrad(figure 7.1.4.1-lif)

2.425 per r(C) J

Solution:
.M2MI

CL

(Ct.)

-

4

--

2 (CLt)(equation 7.1.4.1-c)1(-0,225)-• - (2.425)-1.108 per rad (based an S•w)
7.1.4.1-5

REFERENCES
1.

Tobak, M., and Lessing, H. C.: Estimation of Rotary Stability Derivatives at Subsonic and Transonic Speeds. Agard Report
No. 343, Apr 1961. (U)2Malvestuto, F. S., Jr., and Margolis, K.: Theoretical Stability Derivatives of Thin Sweptback Wings Tapered to a Point with
Sweptback or Sweptforward Trailing Edges for a Limited Range of Supersonic Speeds. NACA IR 971, 1950. (U)3.

Malvestuto, F. S., Jr., and Hoover, D. M.: Supersonic Lift and Pitching Moment of Thin Sweptback Tapered Wings Produced
by Constant Vertical Acceleration. Subsonic Leading Edges and Supersonic Trailing Edges. NACA TN 2315, Mar 1951.. (U)4.

Cole, I. J., and Margolis. K.: Lift and Pitching Moment at Supersonic Speeds due to Constant Vertical Acceleration for Thin
Sweptback Tapered Wingswith Streamwise Tips. Supersonic Leading and Trailing Edges. NACA TN 3196, Jut 1954. (U)5.

Harmon, S. M.: Theoretical Relations Between the Stabiiity Derivatives of a Wing in Direct and Reverse Supersonic Flow.
NACA TN 1943, 1949. (U)SUBSONIC SPEEDS

.12'

_7Z

7TA/2----------------(per red)j.04"-'

-.

.02-

0

FIGURE 7.1.4..1-6

2

3

VARIATION OF LIFT-COEFFICIENT CORRECTIONTERM WITH '6A.

TRIANGULAR WING. 0<J3A<4.

7I1.4.-6

GA

4

"

-i

SUPERSONIC SPEEDS

7.Oml

)

-- 1

7.56.5m..

5.0.

.5.5--. . .

..

..

.

.

. .

.

-

4.5m

F.(N)

...

-F2.(N)4.---------------

F~

-

I...

(1N)

. .---

II

"

3.5-

-"

-

-

2.5-

2.0

-

--

-

-

-

f--r-

--

-

1.0----------------iF,(N)-0,- -10

-

-. 8

-

-. 6

-. 4

I

-I

-. 2

0

.2

.4

.6

.8

1.0

4 C-OT ALEA
FIGURE 7.1.4.1-7 F(N) FACTORS OF THE STABILITY DERIVATIVETRAILING-EDGE-SWEEP PARAMETER, N

1

7.1,4.1-7

•.•;

• -•

. ••-

-

j

.

,

•

•

,

•

•

-,

.

-.

.

-

-

.•,-

-.

.

,

-

-

- .

r-•

,-

,

SUPERSONIC SPEEDSSUBSONIC LEADING EDGE
(p)

0.2 5

(a)

•_ :0.25,SASONIC

• -

0

LE

6A.
3

_

SONIC TE-'N\

-7.

F-

v

.52

,----I[.~.I.

.60

.68

.76

.84

~

-

.92

1.0

/8COT ALE(b) X =0.25
6.

___

__

5.

__

5

S(CLN.Ia

----

3.0"

*

(per fad)

'__1_

3.5v

3.0

.44

FIGURE 7.1.4.1-*

7.1,4.I-8

4__

__

__-

_

.52

.60

VARIATION OF 3(CL 'LIAND /3(CL ')a

.68

.76

WITH /3 COT ALF

.84

LEF
SNIr.
n

.92

1.0

-.

• ,

SUPERSONIC WEEDSSUBSONIC LEADING EDGE

(c)

[

.5

SONIC LE

2
-0

-

-2

SONIC TE ,-,6..

--•CL.)I,.

(po(ad) 4w

"\

.----

-

.5

(pd)-4, 1.5

-

-

-

SONIC TE

8 -:-

-

3-__
31_

010

1o

ml

.60

.64

.68

.72

.72

.76

.80

.84

S84.2w.92.88

COT AL.
-..-......
J-..........--.._
.S
0. 50

I ""

.8A = - •(d)

4.0Cd
7m

3.8'

ooSONIC TE\" - -SONIC

(p•

L 2 6
A.4

3.,

/;(e
L°')"

3.2"

d

_

.60

TE-

-A

8.A

---

10

.10

6

5

.. J

K-

-

-

3

.80

!

..
--

-8

_

_

4

.76

1.0

SONIC LE--\,,.,

.64

.72

.96

COT AL.

0--1

FIGURE 7.1.4.1-8 (CONTD)

7.1.4.1-9

SUPERSONIC SPEEDSSUBSONIC LEADING EDGE(e) ?•=0.75
3_-8(pc' rad),ONCT

I~~~~~CO
_

_

_

__

-

- -

_

I

_

LIO

-6

.76

.80

.84

.88
7 COT ALE

.92

.96

1.0

-__

(f) X= 0.75
6.6

SONIC rE--

6

5.8
,ACLJ.)(porad)5.___~SONIC LE''o

/
r
3.4•
.76

.- 0

2
.88
,6COT AL.

FIGURE 7.1.4.1-8 (CONTD)'7..4.1-10

.92

1.0

SUPERSONIC SPEEDST~.8

SUPE~RSONIC LEADING EDGE

T-~-

.4

1

3

(CL-4

FIN

P-2,0
4

0

12

A214

32
COTl (,5 COT AL

40

48

(del)

2..

7(14.-1

"~JERSOtdICSPEEDSSUPERSONIC LEADING EDGE

7. .0Ib

A(c) X o0
202

2

60
.6

3.921
_

4.4-

3.9 -,6-

6.-

39 --

-

12-6

-

T

3,d2-g1

LO

5.6-

COTc

44

4.04.-124

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE
COT- (COT ALE) (deg)h~0.

(per rad.)

1.6

4.02

7.2

4.00

6.8

3.98

-

I

-

78.4

T"/

3

h6.
c-

-

--

*

--

--

~

-

.!

7.6

--

20

4 .

A o/
2.J

-

3.94u-----)BICL )I

(Pei rad)
5.6-

3.92--

4.4w

3.88m-..-

-

-

-

-

-

-.

FIGURE 7.1.4.1-11 (CONTD)

I>

-

.

----

of(,6COT A~

COT..1-1(dog)

7.1.4.1-13

SUIPERSONIC WPEEDSSUPERSONIC LEADING EDGE

I.4

(per rod'

44
50CTA.(d

COT

'o.

wo
.2*

I6

(po fad-IUR 71.4111(ONO

7.1.4.1-1

CO 'LC

TA a

&

1

-.

1

.

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE(I) X 0.50
810

4.05

7.5-

4.00o.

6.5-

3.90,

-

-

:-w l-O--- */CL~a_(per rod)

4.5-

COT-

ir

-

ON

(A COT AL 9) (deg)-SUPERSONIC SPEEDSSUPER-SON-IC LEADING EDGE

N
-

0.75,
i-'------

---

-

t----

-4

-

--

-

--

-V-

-4--

(per
fad)

=2~20
-2

--- 1.3'-'

0i f2 617147

-2

0616
COT

6.0

1

24

32404

(BCOT ALE)(des)-

-

-

Z

fill-

i30.7

5.5-

COT' (m8OTLC

V7FIGURE. 7.1.4116CO

)L

6cg

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE

1.2-

T. .-

T.

(i) N = 1.0

2+-.-

i .

---

--

..

.

4I

t•CLi)l(W efrd)

)

----

5_

-20----

4w_ _ _ _L
•

.

..

-IN

-1.2o

N--

3644 40
48

I

-l

16

2.2'

12-

0.

L

-

20

-

40

- ••

.441-

...

48

:

~....'--------.

-C"-L---

V

32

-•

---

j

-3.

24
coT"- (,acoT A,_,) (dog)

-

--"-L

25,X
-.4-

tJ

2...

...

4 ID"--I-_.

-

i

'

20

-~

--

7.1 .4.1,17os240--

s

24

24•

1%74
COT- ( # COT A. K)(dox)'048

FIGURE 7.1,4.1-11 (CONTO)

7.1.4.1 - 17

SLIPURSONIC SPEEDSSUPERSONIC LEADING EDGE

4.4

11a..

0-

4.2

n-

8A

6-

r5

$- 3.8.-1-1

'

Z,-

7 ..

Revised September 1970

7.1.4.2 WING ACCELERATION DERIVATIVE C,6a

Methods are presented for estimating the wing contribution to the derivative Ca for atriangular planform in the subsonic and low transonic speed ranges and for planforms with theleading edge swept back and the trailing edge swept back or swept forward in the supersonic
speed range.
If the wing acceleration derivative Cma&is to be used in method 1 of Section 7.3.4.2 to obtain(C a&)WB , the exposed wing planform area should be used for a!) calculations in the Datcom
methods. Using the exposed planform area will yield Cm& based on the product of the exposedwing area and the square of the exposed wing MAC, rather than the product of the wing area
and the square of the wing MAC as indicated.

DATCOM METHODS
A. SUBSONICThe subsonic value of Cm

based on the product of wing area and the square of wing MAC

Wde
wr 2, referred to body axis and for any center-of-gravity location is given by
=C.&
Cm +Q

-) CL 6 (per radian)

7.1.4.2-a

where
Cm al

is referred to body axis with the origin at the wing leading-edge vertex and is obtained
as indicated below (per radian).
is the longitudinal distance from the wing leading-edge vertex to the center ofgr vity,
measure I in mean aerodynamic chords, positive aft.

CL

is obtained from Section 7.1.4.1, based on the product of wing area and wing MAC(per radian).

A method of estimating the subsonic acceleration derivative C
reference 1 as

C

4

8
3"

9

8
c

c,42

" of a triangular wing is derived in

C-L

+

Cm(g)

o

(per radian)

7.1.4.2-b732~7.1 .4.2-1

whereCLis the wing'lift-curve slope (Section 4.1.3.2) at the Mach number under consideration, based on the total wing area (pex radian).

*i

X..
-

is obtained from Section 4.1.4.2.

C 0(g)

is the pitching-moment-coefficient correction term obtained from figure(per radian).

7.1.4.2-8

Because of restrictions placed on the pitching-moment-coefficient correction term, equation
7.1.4.2-b is valid only for 0 < OA <4.
Explicit expressions for estimating Cm& of other wing planforms in the subsonic region are notavailable. An approximation may be made by subtracting the value of Cm q (Section 7.1.1.2)
from the appropriate test value of total pitch damping (Cmq + Cm&) taken from table 7-A. Tests
indicate that the relative importance of body damping of a conventional configuration in
subsonic flow is small; therefore the use of wing-body test data of total pitch damping in thisregion is acceptable for an approximation, provided the test data are for a conventional design.
Sample Problem
Given: Same wing as in sample problem of paragraph A, Section 7.1.4.1
A

4.0

X= 0

-0.75

ALE

CL a

= 450

(from planform geometry with c.g. at ý/4)

4.0 per radM(Section 4.1.3.2)

0.6=

From sample problem oi paragraph A, Section 7.1.4.1

Xe-==0.5700.8

OA

Compute:P2 C(g)
0.071 per rad

7~ A/22

C

7. 1.4.2-2(g)=

0.697 per rad(figure 7.1.4.2-8)

3.20

CL&

=

-0,240 per radSolution:
81

"(C

CL
CL+

Xa.c.

3
-

9
"2

Co(g)

(equation 7.1.4.2-b)81

9(0.570)2 (4.0) + - (0.697)
32
2
-3

-

Ix

Cm

-

CM

0.153 per rad+)CL(equation 7.1.4.2-a)- -0.153 + (0.75)(-0.240)-0.333 per rad (based on Sw•)

B. TRANSONIC
The variation of Cm& of a triangular wing from the critical Mach number to M = 1.0 is given by
the method of paragraph A, provided 0 < PA < 4.
There is no generalized theory available in the literature that gives the transonic values of Cm a,either for additional wing-geometry parameters or for Mach numbers greater than 1.0. An
approximation may be made by subtracting the value of Cq (Section 7.1.1.2) from theappropriate test value of total pitch damping (Cmq +Cm&) taken from table 7-A. Although theimportance of body damping in this region is unknown because of a lack of experimental data,wing-body test data of total pitch damping will probably have to be utilized if an approximationof Cm& is to be made, simply because of the lack of sufficient wing-alone test data. This methodof analysis gives only a rough approximation and, is limited to conventional designs where test
data are available.
C. SUPERSONICThe supersonic value of Cm&, based on the product of wing area and the square of wing MAC
SwZw 2 , referred to body axis and for any center-of-gravity location, is given by equation
7.1.4.2-a, i.e.,
Cm

Cm"

+

CL.

(per radian)

where the parameters are defined in paragraph A and the supersonic value of C,1
as indicated below.

is obtained

7.1.4.2-3

M'?thods for estimating Cm 1"
Wings with subsonic leading edges (13 cot ALE < 1.0)
For wings with subsonic leading edges, Cm,&" is obtained by the method of reference 2 for
X = 0 and by the method of reference 3 for X = 0.25 to 1.0. The following methods are
not valid if the Mach line from the vertex of the trailing edge intersects the leading edge orif the wing-tip Mach lines intersect on the wings or intersect the opposite wing tips.

a.

Zero-taper-ratio wings (A = 0)
C M is derived in reference 2 as

~FS
(N)

p

2

Cm3AM
16P

G(Q3C) F(N) + 1

2

16AM2
92
9+

E(FPC)

1 1 (N)J3F[E"iC)F(N)]

16p32

-)

"c

FN-

C

;

(per radian) 7.1.4.2-c
where
E"Q3C) and G(J3C) are obtained from figure 7.1.1.1-8.
F 5 (N), F 7 (N), and F, 1 (N) are obtained from figure 7.1.1.2-8.
F 6 (N) and F 8 (N) are obtained from figure 7.1.4.2-9.

b.

2= -C- c(see sketch (a),dis two-thirds the basic triangular wing root chord, dSection 7.1.1.1).

CL&

is obtained from paragraph C of Sectior 7.1.4.1, based on the product of wingarea and wing MAC.

Wings with A = 0.25 to 1.0

6-I

Cm,

is derived in reference 3 as

Cm2

7.1.4.2-4(Cm- )-•(Cm(per radian)

7.1.4.2-d"where(m)and (C')

are obtained, from figures 7.1.4.2-l1Oa through7.42-1O

for X = 0.25, 0.50, and 0.75 and from the equations of reference 3 for X > 0.75.
i2.

Wings with supersonic leading edges (g cot ALE > 1.0)
is derived in reference 4 as

For wings with supersonic leading edges Cm."
2
"M

(M2(

""Cm

7.1.4.2-e(per radian(

-

where
(Cm.)C and (Cm.) are obtained from figures 7 .1.4 .2 -13a through 7.1. 4 .2-13p.

Figures 7.1.4.2-13a through 7.1.4.2-13p are valid for the range of Mach numbers for which
the Mach lines from the leading-edge vertex intersect the trailing edge. An additional
limitation is that the foremost Mach line from either wing tip may not intersect the remotehalf-wing.

Sample Problems
1.

Wing with subsonic leading edge

Given: Same wing as in sample problem 1 of paragraph C, Section 7.1.4.1.
X

A =5.80

0

ALE

600

c.g. at 4

b

16 ft

M

1.50

xcg
=

1.50 (from planform geometry with c.g. at Z/4)

From sample problem 1 of paragraph C, Section 7.1.4.1:

F

1.12
E"(43C) = 0,770

1 cot ALE = 0.647
G(QC)

=

0.570

N= 0,602
CL.

=-5.02 per rad

7.1.4.2-5

Compute:
Fs(N) = 0.251
F 7 (N)(figure 7.1.1.2-8)= -1.620= 0.395

L -FII(N)•(figure 7.1.4.2-9)

F 6 (N) = -2.600
F8 (N) = 0.880

Obtain d from the characteristics of the basic triangular wing (see sketch (a), Section 7. 1. 1.1).
AB

4tan ALE

Cr B=

2b
AB

13.85 ft

2

d-

2.31

9.233 ft

3

'B=

5.517 ft-,:•,2bc r -A= 3.678 ft

Solution:

K'+

--

3Am2
[16q

(N) +

16

E "(-PC)

16AM 2

7rA

9q32

16932

F

N
-

(I)

(equation 7.1.4.2-c)_

3vr(5.80) (2.25)
3+1.(..8)9(1.25)(9.233)(3.678)7.1.4.2-6)16(1.25)25 0.570) (-1.620) + 16
3(-5.02)(0.21)1(0.770) (0-=35)[(0.770 (0.880)] + 16(1.25)[(0.770) (-2.600)]

-6.15(-0.9234 + 2.610) + 12.576 - 1.824 + 12.602=

12.98 per radCMCm." +=

CL.

'(equation 7.1.4.2-a)" 12.98 + (1.5) (--5.02)
5.45 per rad (based on Sw=

2.

SGiven:Wing with supersonic leading edge
Same wing as in sample problem 2 of paragraph C, Section 7.1.4.1.

A =4.0= 0.25

ALE =450

c.g. at -

M

2.0

Xcg.

0.964 (from planform geometry with c.g. at d/4)
From sample problem 2 of paragraph C, Section 7.1.4.1:
P-

1.732

PA

cot- 1 (•cot A

6.928) = 300

t cot ALE
CL.

=

1.732

--1.108 per rad

Compute:(Cm1) = 11.80 perrad

*

(m.)

(figure 7.1.4.2-13d)

6.813 per rad-5.530 per rad=(figure 7.1.4.2-130

S(Cm)per rad*2-3.193

M(Cm.

Solution:"'"Cm,"

- (C

-

(I

+

m.)

(equation 7.1.4.2-.)

7.1.4.2-7

.

•

.

--

--

.

.

g

44
33

+

- -(6.813)=9.08-- 7.45=1.63

per racix=C~.
i---~-CL.
a +
C

ai
=

1(-3.193)-(equation 7.1.4.2-a)a1.63 +(0.964) (-1.108)=0.562.....

4q

per rad (based on SWý,z 2 )13)+..
.....

..

1.

-..

193)

-

*-

-

REFERENCES
I Tobak, M., and Lassing, H. C.: Estimation of Rotary Stability Derivatives ut Subsonic and Transonic Speeds. AGARD
0i
Report 343, 1961. (U)2.

Malvestuto, F, S., Jr., and Margolis, K.: Theoretical Stability Derivatives of Thin Sweptback Wings Tapered to a Point with
Sweptback or Sweptforward Trailing Edges for a Limited Range of Supersonic Speeds. NACA TR 971, 1950. (U)3.

Malvestuto, F. S., Jr., and Hoover, D. M.: Supersonic Lift and Pitching Moment of Thin Sweptback Tapered Wings
Produced by Constant Vertical Acceieration. Subsonic Leading Edges and Supersonic Trailing Edges. NACA TN 2315,
1951. ( U)

4.

Cole, 1. J., and Margolis, rZ.: Lift and Pitching Moment at Supersonic Speeds dlut. to Constant Vertic~al Acceleration forThin Seeeptback Tapered Wings with Streamnwise Tips. Supersonic Loading and Trailing Edges. NACA TN 3196, 1954. (U)SUBSONIC SPEEDS

S*.
~.

.08--

..~

-

1a
m0 (g
7rA/2J(perrrd)

1)4

x¢.CL

C*=

(qain

..

2-.02---

0-

-

00.4

1

-

-

FIGURE 7.1.4.2-8 VARIATION OF PITCHING-MOMENT COEFFICIENT CORRECTION TERMTRIANGULAR WING 0 < 63A< 4.

7.1.4.2-8--

2

WITH A1A.

-- - -

-

-

- .-

-- .

.- .

.

•,o 7.

_

.

-,-

.

.,

.--- -

-.

.-.

•

-

,

.

.. .

.

..

SUPERSONIC SPEEDS

-

-

-

/

__

--

/

2.

(N
N),

0.-

-

1.4

-

-

i .8
(N)F(N

-ý

---

-

-

_-_/

-4

-1.0

,,I

--

--

-. 8

.8

-.

-.6

-. 4

-. 2

0O

2

.4

TRAILING-EDGE-SWEEP PARAMETER, N

1

.6

1.0

8

.

"4COTALA

FIGURE 7.1.4.2-9

F(N) FACTORS OF THE STABILITY DERIVATIVE

7.1..4.2-9

L

--

"

-

-

~

. .

.

. .

-

.-

.-

-

.

- -

SUPERSONIC SPEEDSLEADING EDGE

28

-SUBSONIC

(a)

SONIC

. -- 0.25|I

24,

0.25

/310

20

12

.

..--.

SO--I
..

-----

-

-- .

8.--.

(per radl

0-4--

-

.44

.76

.68

.60

.52

-

. -

-.

iJi

,

.

.

,;2

.4

02COT ALE

7-- -i-10o:•. . .. ...... 1-.. . .
-K .-4.

..

(ea)-12'--

.I

*---

_

-- •

]....__...

-

i~~ONI
__6_

l____

/

-__

_

--

S6 T

_14

__

-

-

-

L

SONIE
-

-20

.68

FIGURL 7.1.4.2-10 VARIATION OF

p.

7.i.4.2-1O

8(Cm.)AND ,8(C,PCOTA.,

WITH

.76

ACOTrAL

.84

.92

1.00

SUPERSONIC SPEEDSSUBSONIC LEADING EDGE

32

0
.

28

-

24

.-2E
.60

_3SONIC LE-

SON.C

-

_

~~~~

24

..

.

.

I-

=72

68

.64

L16-

/ C m .)

12 ....

12(p e r la d )

8.

.

3

.

..

.

.

. .

..

.. .---- 1"- " ' '

-

40
-4

-

-

.72

.76

-4=

J

i

.80

.88

.84
6 COT A

"p

S(d)

J

,j

.96

.92

2

1.00

-

050
0.

-------108-- 12 m .

----

...

. -

-

.

A3Cii),(per rad)-14

SONIC TE__ __-

--

---

_-

-_

-_

SII
-168

"-16SON-I-

--

____

__"

-20,
-6
-22

SON IC T E

-

-

-__

-7
-26'1.0.6
.72

6
.76

.80

.84

.88

.92

.96

1.00

8 COT ALE

FIGURE 7.1,4.2-10(CONTO)

7.1.4.2-11

I•RE RS NIC UPlEEliV,SUBSONIC LEADING FDGE
32

s

_

ICL

()X 0.75

K---

24

.

(Pat a"o)

-10

-

20m

12
SONIC TE-N

4w

.4o
2o

I• I

-- 6

---------

....

5

..4

.80

i

SONIC T E--

3A,z-j-----6--- -- -

1..0

6

12

"•
-

--

!X, 0.75____.___-12"------__(per (ad) -14.-O

,9

-__

.0

-16,

N---t;---q------

-22
-

-(

)

I
-4-

SO I -..--.---T

-26"
.76

.80

•
.84

7.1.4.2-12•-"-f I--.----

.96

1.00

I
-3 COT AL

FIGURE 7.1.4.2-10 (CONTD)

--.

-

,.

-

SUPERSONIC PEEDSSUPERSONIC LEADING EDGE

14..7

1316

--ii l12

3A -. 20

-..

_-'-

--.

..

!,

-, -. -.

.

-I 6,t

d'

- --

fi

5iiii~

3T4(per 14d)

X

8

24

0816

70.

COT - (6 COr A,.,) (deg)

....

4048

32

..

20-

1

--

60

S/3(C

I

4

"

1

I

A

501I-

.

. .

..

. .

m .)'
(pel rad)

!

~30',

401j

24

(deg

32

218

40

44

46

8048
COT
FIGURE 7.1.4.2-13

3(f3COTAL -)de

VARIATION OF / (Cm), AND 6I(Cm )2 WITH COT(3COTALE)

7,1.4.2-13

a

1-.4

SIJPFRSOEIC WEEDS
SUPERSONIC LEADING EDGE

L

r!~~t
T

-I-

(per fad)

)320

\1

-2

624

0

COT
(d) d\0.25I

40

YIiZ

I

i-

I

II
8

1261

100

(pei

K.-FIGURE

32(,8 COT ALE(doe)r48

OTr'(,6COTA, )(dollI
7.1.4.2-13 (CONTD)

7.1.4.2144
SJ

48

"SUERSONIC9-EOS

SUPERSONIC LEADI 'G EDGE(e) 0o.25
60

16

140

2

50

40

20-

-

i3A

20

-

-

-

-

-

-

44

-

-

-

-

10,1

Aý,

424_!

4

COT

FIGUR

-

0...251(B3COT AL.) (deg)ICOTTOF1

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE
; 0.5o0
A)

(

IT

,

--

t-4-+

--1•I --- --

14

--

}

i

'_-i__-, i • i

III

u

I

I

-'

-- pi

I" I

,

/I L/ 1-'-/

(pe i Fad) 10

8..-

•

;

"'•"
-.

i-

'

Y ~izziz./
280.

60

200m

(,6 COT

(h) I

-

i

/3A=2.

/11i~z--- A --

4COT

,0

I

L E (deg)----- - 7 --

0 0

-

2
1
(pe

3

C)50.

adi)70)

1600

-

I '

24f,

.

-,

S/"
30

20k-

0
10,

JiL 1LV1----/--....------.
0.o..:/

F...

28

22
COT

FIGURE

7.1.4.2-16

7.1.4,2-13(CONTD)38(,e3COT

ALE) (deg)40

44

SUPERSONIC SPEEDSSUPER~SONIC LEADING EDGE

~-5

-

12
A~13A

Moo6
L12
2BO

COT 1

L

dg

FI-UE .14-1(CN)720.21SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE

70

-~-1

280-----6---------

63A=20240Am)50 .

-

-

-

-

-

0t

a

*

(per rad)01
0)

COT-' (,6 COTA L ) (deg)FIGURE7.I.42-i3

0C.N5D

7PA.H.2-1
-~Zz

3

C

C

.

---

-

~

-

-

-

SUPERSONIC SPEEDS

35~~~~~

SUPERSONIC LEADING EDGEr

TT T h
3A

- -

Ti--1

20

(per rad)15

22

5.4g

0

1

240
COT' (/63COT AL(deg)

7.1.0-1

SUPERSONIC SPEEDSSUPERSONIC LEADING EDGE
-2w.

-10

-5-I

CO

(0) CO1L.0(dgI.L(per rad)

2081

24
2
LfE
COT" (,BCOTA

71..-20-

32408
(dog)

April 1978

7.1.4.3 WING DERIVATIVE CD;
This section presents a method for estimating the wing contribution to the derivative CD, at
subsonic speeds. This derivative is the change in drag coefficient due to a change in & at a constant
pitch rate and is defined asaCD&/--'where C is based onSw.

CD-

k2V/
In general, this derivative is small and has a negligible effect on longitudinal stability; hence, it isusually neglected.
A. SUBSONICDATCOM METHOD
From the two-dimensional unsteady-flow theory of Garrick (Reference 1), the perturbation section
drag coefficient of an oscillating surface can be approximated by
cd =

d2

7rT

c

V

[I + 2C(k)] a&

7.1.4.3-a

where
cd

is the drag coefficient per unit span.

cis the wing chord.

V

is the free-stream velocityC(k)is Theodorsen's function, defined as
7.1.4.3-b

C(k) = F(k) + iG(k)where
F(k), G(k)are Theodorsen's functions obtained from Table 7.1.4.3-A as a function of
the reduced frequency k, where k is defined as"k= --

7.1.4.3-c

2V
where c and V are as defined above, and w is the angular velocity (rad/sec)

7.1.4.3-1

-Only one part of the complex equation is needed. It is arbitrary whether to use the real or
imaginary part. If the wing oscillation can be defined as""
4•c=et&•

cos Ot + ia,1 sin wt=

7.1.4.3-d

where

-

a

is the complex form of the angle of attack of oscillation

aC1

is the amplitude of the oscillation (radians)t

is the timethen by substitution and expansion the imaginary part of Equation '7.1.4.3-a can be zhown to beequal to the following:
Cd

-

7r c-

_a2cowtl + 2F)(COS2 (.)t -t

7.1.4.3-e2wt) -AG si, wt Cos wt *where all the terms are defined above and where the oscAkation il- give, by C

al si

ot.

The three-dimensional drag may be obtained by
C

.,

2

J

-

bw dy

7.1.4.3-f

where

:

Cd

is the two-dimensional drag from Equation 7.1.4.3-e.

cis the local wing chord.

Sw

is the wing reference area.

If the derivative Coa is desired, variations in CD may be calculated for different values of &. For oc
as defined by Equation 7.1.4.3-d, the imaginary part of & (to be used in conjunction with Equation
7.1.4.3-e) may be expressed as
a o Wcos Wt

7.1.4.3-g

where the terms are defined above.
*The expression in brackets could diso be written as [(1 4 2F) cos 2w*t - 26 sin 2wtl, which would indicate that Cd varies at twice"theangular velocity.

7.1.4.3-2'

Then CDý may be approximated by(CD)t2 - (CD)t

3CD

CD

7.1.4.3-h

[(•)2

.'"i

(TVV

]V2
2Ot

--

where
CD

is the three-dimensional drag coefficient obtained by application of Equation 7.1.4.3-f
the rate of change of angle of attack obtained by application of Equation 7.1.4.3-g

and the subscript tn refers to a specified time such that t 2 - t1 is the time interval over which CD
and &are evaluated.
Sample Problem
Given: The following wing-body combination
Wing Characteristics:

r.

A = 4

X

A= 0

Sw = 64.0 ft 2

1.0

c = 4.0ft

Additional Characteristics:Vw = 27r rad/sec

200 ft/sec

a1 = 10a = c (cos wt + i sin wt)
Compute cd at t

1 sec:

Find the reduced frequency kk =(Equation 7.1.4.3-c)

2V.)(2r(4.0)
2(200)-

0.0628 rad

GF 0.887,5 J
G =-0.1451)

Table 7.1.4.3-A (linear interpolation)-C = -2+ 2F) (costct- sin wt) - 4G sin wt cos wtj'

(Equation 7.1.4.3-e)

7.1.4.3-3

""~

:

""

-

i

~

-"

-.---

--

--

--.-

*,.~

---

211r

!2 ()57r.0(_•(5_.3=

I +2(0.8876)](1.0 -0)-

2 r [I1+

4(-0.1451)(0)(142(0.8876)]1.}

0.000167a,1co
W cos Wt
1

(Equation 7.1.4.3-g)(21r)(1)

0.110

CD

=

2

f

wdy

(Equation 7.1.4.3-0

WS

f

b

Sc

c2 c bS2

2

Cd(RectwingSw

bc)

CD = 0.000167

Compute cd for t

1.1 sec.
I- W

cd=

7r 14.0\

G1+ 2F)(cos2 tot - sinllt2 wttt) - 4G sin

1

2

27r [1 + 2(0.8876)] [(0.8090)2 - (0.5878)214(-0.1451)(0.8090)(0.5878)}- 0.0000681=

acos Wt__1-

27r(0.809)= 0.0887

7.1.4.3-4

cos(Equation 7.1.4.3-g)(Equation 7.1.4.3-e)

fb/2

cd c

CD 2 fcSw

dy(Equation 7.1.4.3-f)bb-,2 22
bc

Cd

fSW
SCdC

b

S
2
Sw
CDI

CdC

2

(Rect Wing Sw

bc)

0.0000681

Solution:3

CD

C&• a\2/
C
-& a

(CD)t

[(&)t

- (CD)t

- (&

]

C_

(Equation 7.1.4.3-h)(0.0000681 - 0.000167)(0.0887 - 0.110)

4.02(200)= 0.464 per rad= 0.0081 per deg
B. TRANSONICNo method is presented.
C. SUPERSONICNo method is presented.
REFERENCE
1.

Garrick, I. E.: Propulsion or a Flapping and Oscillating Airfoil. NACA TR 567, 1936. (U)

7.1.4.3-5

0]

TABLE 7.1.4.3-A
THEODORSEN'S FUNCTIONS

k

*

S

7.1.4.3-6-- GF
0.50000

10.00

0.50060.02066.00

0.50170.02064.00

0 50370.03053.00

0.50630,04002.00

0.51290.05771.50

0.52100.07361.20

0.53000.08771.00

0.53940.10030.80

0.5541,0,11650.66

0.56990.13080.60

0.57880.13780.56

0.58560.14280.50

0.59790.1507044

0.61300.15920.40

0.62500.1650(1.34

0.64690.17390.30

0,66500.17930.24

0.69890.18620.20

0.72760.18860.16

0.76280.18760.12

0.80630,18010.10

0.83200.17230.08

0.86040.16040.06

0.89200,14260.05

0.90900.13050.04

0.92670.11600.025

0.95450.08720.01

0.9824

0,0482

0

1.000

0

Revised September 1970

7.2 BODY DYNAMIC DERIVATIVES

The -methods presented in this section are for estimating pitching and acceleration dynamic
derivatives of isolated bodies. The methods and charts of the subsonic, transonic, and supersonic
speed ranges are based on a combination of slender-body theory and the theories used in
predicting the body-lift-curve slope and pitching-moment-curve slope in Sections 4.2.1.1 and4.2.2.1, respectively. Newtonian impact theory is used in the hypersonic speed range. The
methods are restricted to angles of attack near zero and should yield values suitable for first
approximations to dynamic stability.
No test data are available on body dynamic derivatives. Therefore, all theoretical methods must
be considered tentative until compared with experimental results. A brief discussion of available
theoretical methods is presented.
The starting point of almost all theories is the well-known linearized potential equation. Various
methods based on linear theory have been developed for obtaining the flow field about bodies insupersonic flow. The: problem of determining the dynamic stability derivatives for bodies has
been treated principally within the assumptions of slender-body theory. The application of
Munk's slender-body theory to the calculation of the aerodynamic coefficients describing steadymotion has been made by a number of authors (seL references I through 4). In reference 5,
slender-body .values of the aerodynamic coefficients associated with nonsteady angle of attack
have been obtained as a by-product of a linear analysis of the potential equation for nonsteadysupersonic flow. However, approximations made in the analysis effectively limit its application tobodies of vanishingly small thickness. The problem of determining the dynamic derivatives for asmooth slender body of arbitrary cross section performing slow maneuvers is treated within theassumptions of slender-body theory by Sacks in reference 6. This approach is novel in slenderbody theory in that the squared terms in the pressure relation for slender oonfigurations are
retained and all motions of the configuration are treated simultaneously. However, the derivatives
are obtained in terms of the mapping functions of the cross sections and are to6 complex forinclusion in the Datcom.

*

A method is developed for estimating aerodynamic loads on slender, symmetrical configurations
performing small lateral oscillations of limited reduced frequency in sonic and supersonic flow inreferences 7 and 8, respectively. This method is an extension of an iterative technique originallyproposed by Adams and Sears in reference 9, and is a combination of first-order and secondorder cross-flow solutions. The results consist of slender-body-theory terms plus higher-ordereffects of fineness ratio. For sonic flow about a body of revolution, reference 7 shows that byretaining only the first-order terms in reduced frequency the pitching derivatives are given bysimple slender-body theory; whereas the acceleration derivatives are influenced by the secondorder terms and contain logarithms of the reduced frequency. Under certain limitations on the
rapidity of the oscillations, all derivatives in supersonic flow prove to be independent of changes
in reduced frequency; consequently the method can be applied to slow time-dependent motions.
At present this method is limited to slender bodies of revolution, and in most cases requires aconsiderable amount of mathematical manipulation to obtain a solution. This method can beapplied to bodies of more general cross section; however, the practicality of such an analysis
would depend on the possibity of solving the integrals that appear.
7.2-1

In reference 10 an attempt has been made to overcome the slender-body limitation by adapting
hybrid theory to the calculation of the body dynamic derivatives. (This method, derived by
Van Dyke in reference 11, has proved successful in calculations for the static aerodynamic
derivatives.) This method is used to predict the dynamic derivatives of a cone, and the results are
consistent with those obtained by impact theory at the higher Mach numbers. In order to extend
this method to bodies of more general shape, it is necessary to satisfy the boundary conditions
corresponding to the specified body. Unfortunately, the analytical expressions required to do sohave been found only for the cone.

:1

The application of simple slender-body theory appears to be the only method of solution
warranted at the present time, in view of the large effects of viscosity on the forces acting overslender bodies, the mathematical complexity involved in solutioa of the linearized equations forgeneral planforms, and the lack of test data.
Since extensive use is made of slender-body-theory results throughout this section, the term"slender" is clarified.* Tsien, in reference 2, pointed out that slender-body theory applies to theflow about inclined, pointed projectiles at supersonic speeds. Subsequently, Jones, inreference 12, indicated that his slender-wing theory (slender-body theory extended to wings)
applies to both subsonic and supersonic speeds, at least for pointed planforms. Actually, themeaning of the term "slender" is somewhat different in the various speed regimes. Forsupersonic Mach numbers "slender" implies that the body lies well within the Mach cone fromthe body apex. This leads directly to the limitation to pointed bodies. It also leads to theconclusion that relatively blunt bodies may qualify as "slender" at low supersonic speeds; while
for hypersonic speeds the method must fail for practical shapes. At subsonic speeds the term"slender" becomes less restrictive as the Mach number increases from 0 to 1, until at sonic speed
all bodies become slender. Thus, slender-body theory is seen to apply to at jeast some body
shapes throughout the speed range from low subsonic to hypersonic.
REFERENCES
1.

Munk, M. M.: The Aerodynamic Forces on Airship Hulls. NACA TR 184, 1924. (U)Z,

Tsien, H, S.; Supersonic Flow Over an Inclined Body of Revolution. Jour. Aeto. Sci., Vol. 5, No. 12, October 193& MU

3.

Laitorm, E, V.: Lineaized Sbsonic and Supersonic Flow About Inclined Slender Bodies of Revolution. Jour. Aero. Sol.,
Vol. 14, No. 11, November 1947. (U)4.

Tobak, M., Reese, D. E., Jr., and Boon, B. H.: Experimental Damping In Pitch of 450 Triangular Wings. NACA RM
ASOJ.26, 1950. (U)5,Dorrence, W. H.: Nonsteedy Supersonic !ri.w About Pointed Boulies of Revolution. Jour. Aero. Sci., Vol. 18, No. 8,
August 1951. (U)S

Sacks, A. H.: Aerodynamic Forces, Moments, and Stability DOrivetNss for Slerile Bodies of General Cross Section. NACA
TN 3283, 1954. MU

7.

Lendahl, M. T.: Forces and Moments on Oscillating Slender Wing-Body Combinations at Sonic Speel. MIT Fluid Dynemics
Research Group Report No. 56-1, 1956. (U)Zartarion, G., and Ashley, H.: Forces and Moments on Oscillating Slender Win-r4Body Combinsaltos at Supersonic Speed.
Part I - Basic Theory. MIT Fluid Dynamics Research Group Report No. 67-2, 1967. (U)'This discussion is mesentially quoted 'rom reference 9

7.2-2

'5)

-

'.

9.

Adams, M. C., and Sears, W. A.: Slander-Body Theory - Review and Extension. Jour. Aero. Sol., Vol. 20, No. 2, February
1953. (U)10.

Tobok, M., and Wehrend, W. R,: Stability Derivative of Conn at Supersonic Spoeas. NACA TN 3788, 1966. (U)11.

Van Dyke, M. D.: First- and Second-Order Thiory of ;5uriairson•c Flow Put Bodies of Revolution. Jour. Aero. Sci., Vol.
18, No. 3, March 1951. (U)

12.

Jone,

R. T.: Propertie of Low Aspect Ratio Wings at Spoods Below and Above the Sped of Sound. NACA TR 835,

1948. (U)

17

7.2-3

Revised Septembe 1970

7.2.1 BODY PITCHING DERIVATIVES
7.2.1.1 BODY PITCHING DERIVATIVE CLq"The pitching derivative CLq is a measure of the lift proouced by rotational motion of theairframe about a spanwhe axis. This derivative is generally small compared to other terms in theequations of motion and is frequently neglected. However, methods are presented for
determining the body contribution to CL q in the subsonic, transonic, supersonic, and hypersonic
speed ranges. The value of CN in the hypersonic speed range is used to obtain the value of Cm q
in the hypersonic speed range mn Section 7.2.1.2.
In the subsonic, transonic, and supersonic speed ranges the Datcom methods are based on therelatively simple results derived from slender-body theory and the assumption that a relationshipof corresponding slender-body derivatives may be employed with reasonable accuracy to the caseof steady pitching in a manner similar to that of reference 1. This approach to the calculation ofbody dynamic derivatives has been applied with reasonable success by Walker and Wolowicz inreference 2. It was shown in reference I that, although slender-body theory alone does notaccurately predict the characteristics of nonslender configurations, the ratio of corresponding
slender-body derivatives may be employed with reasonable accuracy in predicting the staticforces on nonslender configurations. The body contribution to CLq is thus given as the productof the lift-curv slope CL, and the ratio of slender-body derivatives, Le.,CL q -(LO
L .y w

blender-body theory states that body force characteristics are independent of Mach number. The
effect of Mach number is taken into account by the static force coefficient. Therefore, thelimitations of these methods are determined by the limitations of the methods employed indetermining tfie static derivative CL, in the various speed regimes. Experimental data should be
used for the body lift-curve slope when available.
SA.

SUBSONIC
There is no explicit method available in the literature for obtaining the body dynamic derivatives
for general planforns.

"The method presented is based on the application of the results of slender-body theory in themanner previously discussed.

DATCOM METHOD-The body contribution to CL.q, based on body base area and body length and referred to the"center of rotation, is given by
7.2.1.1-1

CLq

=

2C

7.2.1.1-a(per radian)

-

where
is the body lift-curve slope from paragraph A of Section 4.2.1.1 multiplied by

CLa(VB 2/3/Sb) (per radian).
xm

is the longitudinal distance from the body vertex to the center of rotation, positive aft.

krB

is the length of the body.

VB

is the total body volume from Section 2.3.

Sb

is the body base area.
Sample Problem

Given:
6d

*1_

X

XO

+ 6.5dV.

d1.0 ft2

B =XM

6.0 ftFineness Ratio

=

6.0

xm

3.8 ft0.548 per rad (based on VB 213 ) (Section 4.2.1.1)

CL
V

/3 =

2.47 sq ft (Sect+.'on 2.3)=-

0.785 sq ft

Compute:•2.475).(VB) 2/CLI,

7.2. 1. -2=(0.548)i-(VS=(0;548)0.7i=

1.724 per rad (based on Sb)

3.8

0.367

Solution:

"Lq

0L

KB/

(equation 7 .2.1,l-a)= 2(1.724) (0.367)

1.265 per rad (based on Sb B)
B. TRANSONIC
The linearization of the transonic flow problem has been accomplished by Landahl, inreference 3, by introducing a small amount of unsteadiness into the motion. This theoreticalmethod is briefly discussed in Section 7.2. By neglecting the second-order effects of reduced
frequency the pitching derivatives of a slender body of revolution are those given by slenderbody theory. Since slender-body theory does not predict a dependence on configuration
parameters, this method cannot be expected to give reasonable approximations for the pitching
derivatives of nonslender configurations. Therefore, the method of paragraph A is applied
throughout the transonic speed range.
C. SUPERSONIC
Several of the theoretical methods that have been developed for estimating the body pitching
derivative CN in the supersonic speed range are briefly discussed in Section 7.2. The available
theoretical methods are limited to simple slender-body theory, theories treated within theassumptions of slender-body theory, and hybrid theory. The method presented here is based on
simple slender-body theory. Theories treated within the assumption of slender-bodv theory aremathematically complex and restricted to speciftc body shapes; therefore, no general quantitative
results are presented.
SThe supersonic 4 Nq for cones can be estimated by the hybrid theory solutions presented inreference 4.
DATCOM METHOD
The method presented here for determining the body contribution to CNq, based on thecone-cylinder or ogiee-cylinder maximum frontal area and body length and referred to the centerof rotation, is the same as that of paragraph A and is given by equation 7.2.1.1-a, i.e.,

"CN

CN

(per radian)"7.2.1.1-3--

L•--'-"- -..-

.-

-.

"

*.

where CN. is the body normal-force-curve slope from paragraph C of Section 4.2.1,1, evaluatedat the appropriate Mach number and based on the cone-cylinder or ogive-cylinder maximum"frontal area (per radian).
D. HYPERSONIC
Simple Newtonian theory is used in this section to estimate the contribution of cone frustum
bodies, with or without spherical noses, to the derivative CNq. Newtonian theory is discussed inparagraph D, Section 4.2.1.1.
D•ATCOM METHOD
Charts taken from reference 5, based on simple Newtonian theory, are presented for determining
CNq of spherical segments and cone frustums at small angles of attack.
The coefficients of these charts are referred to the body base area and base diameter and to a
moment center at the forward face of the segment. By proper use of the data presented, thetotal CNq may be determined for bodies composed of multiple cone frustums with or without
spherically blunted noses.
The Newtonian value of the derivative CNq for a complex body is obtained as follows:
Step 1.

Compute CNq' for each body segment about its front face using figures 7.2.1.1-9a and7.2.1.l-9b.

Step 2.

Transfer the individual derivatives CNQ to a common reference axis by applying thefollowing transfer equation to each body segment
Cq =Cq-2(.!)C,,7.2. 1. 1-b

where

Step 3.

'"CNis the normal-force-curve slope for each segment based on individual baseareas from paragraph D, Section 4.2. 1. 1.

CNqis the pitching derivative for each segment based on individual base areas
and base diameters and referred to a moment center at the forward faceof the segment, from figures 7.2.1.1-9a and 7.2.1.1-9b.

nis the distance from the face of a given frustum to the desired moment
reference axis of the configuration, positive aft.

dis the base diameter of a given frustum.

The transferred derivatives of the individual body segments are converted to a common
reference area and diameter and added. The total derivative is given by

7.2.1.1-4-

rn

v.A -~ --

-

-.

-

. .

-

J

"CNq

7.2. 1.1-c

Nq

Sample Problem
Given: Same multiple-segment body as sample problem of paragraph D of Sections 4.2.1.1 and4.2.2.1.

dla 2

12 a8

dsa4

4

=0.55 ft (distance from moment reference center to body nose)to

Forward cone frustum

Spherical segment
Q-- 0.18 ft

r=

0.36 ftd,

0.62 ft

a2 = 0.62 ft
0.180

ft

d2 = 1.20ft
2=

072ft

02 = 22.50

Rear cone frustum

CylinderL

a3 = 1.20ft

a4 = 1.20 ft

d3= 1.20Oft

d4~ =23 = 1.20ft

k4

1.368 ft

= 0.96 ft

04 = 50"03=0

7.2.1.1-5•ii•!: , ,.•---• . .

...

.

__

- - --- -.

-~ -- - -Compute:

Spherical segment
2Rl/d,=£1 /r,

CNq=

0.865 per rad (figure 7.2.1.1-9b)

n,

=-o

dI

;a

= 0.50

2

ft

-0.55

=

= 0.62 ft

0.75 per rad (figure 4 .2.1.1-23)

CN

--

CN

/n,•l

CNa
No

2-

'

(equation 7.2, 1. 1l-b))/ 0.75. 0.865 - 2\

= 2.195 per rad (basedon

4)

Forward cone frustum... ...

a2 /d 2 = X2=0.517= 0. 1CN(figure 7.2.1.1-9a)

1.22 per rad

'=

q2

:'.

= -(V0 +

n2

-0.73

.)

(figure 4.2.1.1-26); 1.250 per rad

CN
N U2

=

(equation 7.2.1.1-b)

C

CNq2- 2CN

1.250

1.22- -2
\1.20/

7.2.1.1-6
.

.

,.

s±

.-

,

-

-.

,

a f

f

t

a

a

r.a.k

~

2.7 pe1
274 er ad (based on-

4$i

Cylindera3 /d 3 = X3

CN

1.0

= 0 (figure 7 .2 .1.1-9a)'n-(320 +Q1

CN

=

+R2)

=

0 (figure 4.2.1.1-26)

a 3
CN

3

CN ' -2

q3

-1.45

q3

d

=

0.877

N

a

(equation 7.2.1.1-b)-0

Rear cone frustum
a4id4

=4X

Cxq'

n4

n

0.36 per rad (figure 7.2.1.1-9a); - (R~o+ 1 + 2 + 3) = - . +2R1 +
x223
-2.65

CN

=

0.450 per rad(figure 4.2.1.1-26)

an4
CN

N

2(2
C
\d 4 /'aCq'q4(equation 7.2.1.1-b)4

1-2.65\0.36

2-0.457.2.1.1-7-.

Solution:
Converting the derivative for each segment to a common reference area and diameter, the basearea and diameter of the rear cone frustum, and addingCN(equation 7.2.1.1-c)(d)o.1.3686,

IIj4q
q

2.

ý

+ 2.74

2. 1950.36)3=+ C.q(\2)+ C~q(\)

C~q5

C.q

+ C.q

1,.20o\.
(1.20)33(4q+ 0 + (2.10)

0.2043 + 1.8494 + 2.10

4.154 per rad(based on -REFERENCES
1.

Pitts, W. C., Nielsen, J. N., and Kaattari, G. E.: Lift and Center of Pressure of Wing-Body.Tail Combinations at Subsonic,
Transonic, and Supersonic Speeds. NACA TR 1307, 1959. (U)2.

Walker, H. J., and Wclowicz, C. H.: Theoretical Stability Derivatives for the X-15 Remsrch Airplane at Supersonic andHypersonic Speeds Including a Comparison with Wind-Tunnel Results. NASA TM X-287, 1960. (U)3.

Landfhl, M. T.:. Forces and Moments on Oscillating Slendar Wing-Body Combinations at Sonic Speed. MIT Fluid Dynamics
Research Group Report No. 56-1. 1956. (U)4.

Tobak, M., and Wehrend, W. R.. Stability Derivatives of Cones at Supersonic Speeds. NACA TN 3788, 1966. (U)5.

Fisher, L. R.: Equations and Charts forning the Hypersonic Stability Derivatives of Cone Frustums Computed by
Newtonian Impact Theory. NASA TN D-149. 195W. (U)

7.2.1.1-8

ImI
44

MH'PERSUNIC SPEEDS-

-

3.6-1

3.2

7

-

-

\X\1
-\

2.C
4

q

.

2.4w

•='-

N.EWTONIAN THEORY-C.,BASED 0ONBASE AREA-

2- AND BASE DIAMETER

1.6-

O.4

0
FIGURE 7,2 1.1-9.

10

20

30
40
50
60
SEMICONE ANGLE, 9 (do#)
PITCHING UERIVATIVE C.q FOR CONE FRUSTUMS

70

80

90

.8

.9

q

111

-

-

q

C.Sq

(Wefad)ISI~~I q?

I

NEW'I"ONIAN 1 HEORY
.2

-

-

C, BASED ON BASE AREA
-AND BASE DIAMETERFIGURE 7.2.1.1-9b

3

.

.1

o'

"

.4

.66

.05

.7

L.o

~NOSE BLUNTNESS, 21/d.

PITCHING DERIVATIVE Cq

FOR SPHERICAL SEGM.NT

NqI

",

i .. : •-

• ,.

, .

•-

. .•.

_ ._ .

7.2.1.1-9.

,. . .

. ." .

. . . ' ,.

. ...

_

,.

. .

.,

r.'

,r-

*wn

-

- --

-,

-

- .

-. - -

,l

-

-

----

-

.-

,

-

.--I-•-v--• - -. ,

--

.

-

-w

-.-

-

,~

•-

-

RevW4 September 1970

7.2.1.2 BODY PITCHING DERIVATIVE Cm,

The derivative Cmq is a measure of the pitching moment produced by rotational motion of theairframe about a spanwise axis and is commonly referred to as the pitch-damping derivative.
Methods are presented for determining the body contribution to this derivative in the subsonic,transonic, supersonic, and hypersonic speed ranges.

.

In the subsonic, transonic, and supersonic speed ranges the Datcom methods are based on theassumption that was made in regard to the body contribution to the derivative CLq' andthe general discussion of Section 7.2.1.1 is directly applicable here.

S.sameThe body contribution to Cmq is expressed as

'4m

Cu

q
dender-bodylthearyThe limitations of these methods are determined by the limitations of the methods employed indetermining the static derivative Cm. in the various speed regimes. Experimental data should bebq-t"

used for the body pitching-moment-curve slope when available.
A. SUBSONICThe comments of paragraph A of Section 7.2.1.1 are directly applicable here.
DATCOM METHOD
The body contribution to Cmq, based on body base area and the square of body length andreferred to the center of rotation, is given by

Xm\2

Cm q -

V

•C

ma

V

a
r

7.2.1.2-a

Vs

~Skb

£B

where
Cma

is the body pitching-moment-curve

slope from paragraph A of Section 4.2.2.1,

multiplied by V,/(SbR£B).
V3

a

3.--.

is the total body volume from Section 2.3.

.7.2.1.2-1

Sb

is the body base art.

x,is the longitudinal distance from tlic vertex to the cexitroid of the volume and isgiven byf0

S(x) x dx

7.2.1.2-b

VB
where
S(x)

is the body cross-sectional area at any station. It is not feasible to present
generalized design charts of xc, but the equation can be integrated forany arbitrary body of revolution.

P13

is the length of the body.

xm

is the longitudinal distance from the body vertex to the center of rotation,
positive aft.

Sample Problem
Given: Same ogive-cylindei conligniration as sample problem of paragraph A rf S"%tion 7.2.1.1.

~~Od
Ax.5

f

666

d

1.0 ftx=x= 3.6 ft

6.0 ft

2.5 ft(k 2 - kj) = 0.863

7.2.1.2-2s=(Section 4.2.1.1)

Fineness Ratio = 6.0

xm=

3.8 ftSb

VB=

3.88 cuft- 0.785 sqft

Compute:
Determine Cmxx

Sttion

d

MAx
dx x

-(x

I
dx

-Nx) .x
n

1

0.4

0.126

0,126

3.467

0,4368

2
3

0.68
0.89

0.363
0.622

0.237
0,259

3.028
2.539

0.71760.6576

4
1x

0.98
1.0

0.754
0.785

0.132
0.031

2,04G
1,549

0.2701
0.0480,

x0

1.0

0.785

0

0.75

0

Xo

dS
x

2(k2 -kI )

O

Sxo

L

x"

(O
(xm

- x) Ax = 2.1301

x) Ax. (equation 4.2.2.1-a)(2) (0.863) (2.1301)
3.88

0.948 per rad (based on V.)

r
Cm

=

0.948)= (0.948) \ _-.-/-

0.781 per rad (bsed on SbRB)*x is token to b• at dw centwr of vilume of each body wstont.

7.2.1.2-3

0Determine x

Station

d

Six)

x*

8(x) x Ax

1
2

0.4
0.68

0.126
0.363

0.333
0.772

0,02100.1400

3
4
5

089
0.98
1.00

0.622
0.754
0.785

1.261
1.754
2.251

0.39200.66150.8836

6
7
3
9

1.00
1.00
1.00
1.00

0.785
0.785
0.785
0.785

2.75
3.25
3.75
4.25

1,07951,27551.47201,6680

10
11

1.00
1.00

0.785
0.785

4.75
5.25

1.8646
2.06M5

12

1.00

0.785

5.75

2,2670I._..

RB

S(x)x Ax = 13.775

X

JB

.

SVB

s

"Ru£

13.775
=
3.88

-

-

(equation 7.2.1.2-b)

SW x dx= 3.55 ft(3.88)(0.785) (6.0)

=1

6.0

= 0.824

0.367

L
Solution:
(1

Cm

2 CM

m2

VB

X
XX

b
i-

Sb QB

*
abnw of volume of each body *Waent.
"x it taken to be * the6

7.2.1.2-4m(equation 7.2.1.2-a)3.8'~
T.O)( 0.824) 3.5(0.367)2
6.0=(2) (0.781)(2) 0.78)~ 0.367 '- 0.824.2=0. 135 - (0.824) (-0.0417)1-0.457

L[0.693]=1.562_.0 547J= -0.579 per rad (based on SbRB 2 )B. TRANSONICThe comments of paragraph B of Section 7.2.1.1 are directly applicaoie here. The method ofparagraph A is applicable throughout the transonic speed range.
C. SUPERSONIC
The theoretical methods applicable for the estimation of the body contribution to the derivative
CM q parallel those of the body contribution to the derivative CLq.
Therefore, the general comments of paragraph C of Section 7.2.1.1 are also applicable here.
The supersonic Cmq for cones can be estimated from the hybrid theory solutions presented inreference 1.
DATCOM METHOD
The method presented here for the body contribution to Cmq, based on body base area and thesquare of body length and referred to the center of rotation, is the same as that of paragraph A
and is given by equation 7.2.1.2-a, i.e.,C

B- SbQ B

=2C
2 C

( I

l

VB

xB

S(1--

Sb B

)

where Cmc is the body pitching-moment-cur'e slope from paragraph C of Section 4.2.2.1
evaluated at the appropriate Mach number.
7.2.1.2-5

L

I

•

f

• • '

| • ' l •j,3•: " 1:1%

•

"n "

•

p

" • - m l

•

•.

. ..

-

-

.

...

..

.

. .

.

. .

.

.

rg

D. HYPERSONIC
Simple Newtonian theory is used in this section to estimate the derivative Cmq of cone frustum
bodies with or without spherical noses. Newtonian theory is discussed in paragraph D, Section
4.2.1.1.
DATCOM METHOD
Charts based on simple Newtonian theory (reference 2) are presented for determining Cmq ofspherical segments and cone frustums at small angles of attack. The coefficients of these charts
are referred to the base area and the square of the base diameter and to a moment center at theforward face of the segment. By proper use of the data presented, the total Cmq may be
determined for bodies composed of multiple cone frustums with or without spherically blunted
noses.

The Newtonian value of the stability derivative Cmq for a complex body is obtained as follows:
Step 1.

Step 2.

Compute Cm,'for each body segment about its own front face using figures 7.2.1.2-12
and 7.2.1.2-13b if the body has a spherically blunted nose,and figures 7.2.1.2-12 and7.2.1.2-13a if the body nose is a cone frustum.
Transfer the individual derivatives Cmq' to a common moment center by applying thefolowing axis transfer equation to each body segment:

C,2n
C

rnq

q

Mq

2

d

n
Cm-'
+-CNq
m
C

d

q

/nd2
21-2

d/

C

N a

7.2.1.2.<c

where
CNa

is the normal-force-curve slope for each segment based on individual baseareas from S('ction 4.2.1.1.

CNq'is the pitching derivative for each segment based on individual base areas
and base diameters and referred to a moment center at the forward faceof the segment. This derivative is obtained from paragraph D of Section
7.2.1.1.

Cm)'is the pitching-moment-curve slope for each segment based on individual
base areas and base diameters and referred to a moment center at theforward face of the segment. This derivative is obtained from paragraph D
of Section 4.2.2.1.

Cnlq'

7.2.1.2-6

is the pitching derivative for each body segment based on individual baseareas and the square of base diamete-s and referred to a moment center atthe forward-.

.

face of the segment. If a complex body consists ofcombinations of cone frustums, the derivative for the first frustum must.

be obtained from figure 7.2.1.2-13a, which accounts for the front facebeing exposed to the air stream. If the body has a spherically blunted
nose, the derivative of the nose is obtained from figure 7.2.1.2-13b. For
subsequent frustums the derivatives are obtained from figure 7.2.1.2-12.
Step 3.

The transferred derivatives of the individual body segments are added after being
converted to a common reference area and squared diameter. The total derivative of
the individual body segments is given by3(Cq)nZnn

Cm q =(±d)

7.2.1.2-d

Sample Problem
Given: Same multiple-segrnient body as sample problems of paragraph D of Sections 4.2.1.1, 4.2.2.1,
and 7.2.1.1.

19

d8.
d,33

-0

dFa4

0.55 ft (distance from moment reference center to body nose)
Spherical segment

Forward cone frustum

1 =0.18 ft

a2 = 0.62 ft

r=

0.36 ft

d2

1.20ftd=0.62 ft

R2

0.72 ft

02

=

22.50

7.2.1.2-7

Rear cone frustum

Cylindera3a4 = 1.20 ft- 1.20 ft

d3 = 1.20ft

d4 = 1.368 ft

1.20 ft

24 = 0.96 ft

R3 =

04 = 50

03 = 0Compute:Spherical segment= 21/r1 = 0.50

221/dc-0.50 perrad (figure 7.2.1.2-13b)

C mq1'= -0.55 ft

nl =d=

a2 = 0.62 ft

CN

=

C

' = -. 0.430 per rad (figure 4.2.2.1-20b)0.75 per rad (figure 4.2.1.1-23)

CN ql

0.865 per rad (figure 7.2.1.1-9b)

'=

-2-C
Cm qlI =Cm
qd qm

7.2.1.2-8

,l

(

CN

(equation 7.2.1.2-c)qd=-0.50 - (-1.774) (-0.43) + (-0.887) (0.865) - (1.574) (0.75)=-0.50 - 0.763 - 0.767 - 1.180=-3210 per rad (based on -4di)

Forward cone frustum
a2 /d 2

-

+ nC

=

2

=

0.517

Cm'= --0.68 per rad(figure 7.2.1.2-12)q2n=+R 1 ) =

-(Ro

-0.73

CN2 = 1.250 per rad

C m2'=-0.590
CN

=
q2

(figure 4.2.1.1-26)

per rad(figure 4.2.2.1-20a)

1.22 per rad(figure 7.2.1.1-9a)

C

c

q2

'
q

d2

+=
+
a2

(equation 7.2.1.2-c)
T2

Cq2

2

CN

-

-0.68

- (-1.217) (-0.590) + (-0.608) (1.22) - (0.74) (1.25)

-

-0.68

- 0.718 - 0.742 - 0.925-3.065 per rad (based on -Cylinder
=

X3

=

1.0

S
Cm

0 (figure 7.2.1.2-12)

'

q3

L"3

A-(o + R1

=

-1.45

CSN 3 = 0 (figure 4.2.1.1-26)
0
C

CN q
C

= 0 (figure 4.2.2.1-20a)3-= 0 (figure 7.2.1.1-9a)=0

7.2.1.2-9

Rear cone frustum*.0.877

a4 /d 4

?4X

Cm

-0.250 per rad'(figure 7.2.1.2-12)

q4

"•4 =
"C

0+ R

(figure 4.2.2.1-20a)-0.19 per radCMCN(figure 4.2.1.1-26)=0.450 per radaN
4

0.36 per rad

'

-2.65

=

(figure 7.2.1.1-9a)

""4
C

=C

q

- Cm
2=
-2q•_-m+ d C Nq44-(equation 7.2.1.2-c)

-dqN

-2

- -0.25 - (-3.87) (-0.19) + (-1.94) (0.36) - (7.53) (0.45)--0.25 - 0.735 - 0.698 - 3.388--5.072 per rad (based on

Solution:
Converting the derivative for each segment to a common reference area and square of the diameter, the base area and diameter of the rear cone frustum, and adding gives

q

1 0d.

-3.210 t-- .62

Cm+Cm

2

3

4 / 1

)-3.065(-1.36(368,)
\321.

36-8f

""-0.1354 - 1.8148 - 5.072""-7.C22 per rad (based on 2

7.2.1.2-10

+

q3 (d

)

+ 0 + (-5.072)(equation 7.2.1.2-d)

"REFERENCES
1.

Tobek, M., and Wshr-nd, W. R.: Stability Derivatives of Cones at Supersonic Speleds. NACA TN 3788, 1956. (U)2.

Fisher, L. R.: Equations and Charts for Determining the Hypersonic Stability Derivatives of Cone Frustums Computed by
Navtonian Impact Theory. NASA TN D-149, 1959. (U)

k,

L7-

L-

7.2.1,2-1Il

HYPERSONIC SPEEDS

per rad -2.0

-

ANGE,
SE~cmON

7.2.1.2.12

7.2.1.2-12

PITCHING DERIVATIVE CýAND
EOTE SQUH AEDAEE-40-FIGUREON BASEAREDUE TO INCLINED

SIDES OF CONE FRUSTUMS

,

.

' •

.

.

..

.

7

-

-

-

-

-

HYPERSONIC SPEEDS

.

inI Y

!-l 1tTHESUAREOF

IA

NEWTONIAN THEORY-3.2w.

"I

II

/•

II

BASED ONSQUAREWE AREAAeNDTHE
OF

-Clq'

I'

q

0

-

20

C,

C'

0

.1

.2

.3

90

FOR CONE FRUSTUMSNEWTONIAN THEORY

-d

80

70

60
SC
30
40
SEMICONE ANGLE, 0 (dog)

PITCHING DERIVATIVEFIGURE 7.21.2-13&

Per r

10

THE BASE DIAMETER

1

BASED ON BASE AREA]-ANDTHE SQUARE OF
THE BASE DIAMETER

.4

.5

.6

2

w

.

.7

d

.8

.9

1.0

"NOSE BLUNTNESS, 2 li/ds
FIGURE 7.2.1.2-13b

PITCHING DERIVATIVECFOR SPHERICAL SEGMWNTS

7.2.1.2-13

-..
:
Ri4nd September 1970

7.2.2 BODY ACCELERATION DERIVATIVES
7.2.2.1 BODY ACCELERATION DERIVATIVE CL&
Methods are presented for determining the body contribution to the derivative CL4 in thesubsonic, transonic, and supersonic speed ranges.
Datcom methodb are based on the relatively simple results derived from slender-body theory in amanner similar to that used to predict the body pitching derivatives. It is assumed that a
relationship of corresponding slender-body derivatives may be applied with reasonable accuracy
to the case of vertical acceleration in a manner similar to that of reference 1. This approach tothe calculation of body dynamic derivatives has been applied with reasonable success by Walker
and Wolowicz in reference 2. It was shown in reference I that, although slender-body theory
alone does not accurately predict the characteristics of nonslender bodies, the ratio ofcorresponding slender-body derivatives may be used with reasonable accuracy to predict thestatic forces on nonslender configurations. The body contribution to CLj is then given as the
product of the lift-curve slope CLa and the ratio of slender-body derivatives, i.e.,

=LZ

CL()

Since slender-body theory states that body force characteristics are independent of Mach
number, the limitations of these methods are determined by the limitations of the methods usedin determining the derivative CL., in the various speed regimes. Experimental data should be used
for the body lift-curve slope when available.
A. SUBSONIC
K
7

There is no explicit method available in the iiterature for the estimation of body acceleration
derivatives in the subsonic speed range. The method presented below is based on the applicationof slender-body theory in the manner previously discussed.
DATCOM METHOD
The body contribution to CLa, based on body base area and body length, is given by

CL

=

2 CL

-

(per radian)

7.2.2.1-a

7.2.2.1-1

whereCLoC

is the body lift-,,urve slope from paragraphA of Section 4.2.1.1 multiplied by
S(VB 2 / 3 /Sb), based (n Sb (per radian).

VB

is the total body v, ,lume from Section 2.3.

Sb

is the body base area.

IB

is the length of the body.

Sample Problem"Given: Same ogive-cylinder configuration as sample problem of paragraph A of Section 7.2.1.1.

"2,5d

d

I ft

V

3.88 cu ft (Section 2.3)

CL

1.724 per rad (based on Sb, Section 7.2.1.1)

R13

6.0 ftFineness Ratio

S

Compute:
YB
V

;..

•

(3.88)-== 0.824(0.785) (6.0)

Sb B

Solution:CL- 2 C1B(equation 7.2.2.1-a)cx.

2(1.724) (0.824)= 2.84 per rad (based on S k)r

7.2.2.1-2= 0.785 sq ft

6.0

B. TRANSONICThe only available method for determining the body dynamic stability derivatives at transonic
speeds is that of Landahl in reference 3. However, Landahl's results for the acceleration
derivatives are frequency-dependent and no quantitative information is presented. Since slenderbody theory states that body force characteristics are independent of Mach number, the methodof paragraph A is applicable throughout the transonic speed range.
C. SUPERSONIC
Several of the theoretical methods available for estimating the body acceleration derivatives inthe supersonic speed range are briefly discussed in Section 7.2. These include simple slender-bodytheory, theories treated within the assumptions of slender-body theory, and hybrid theory. The
method presented here is based on the results of simple slender-body theory. Theories treatedwithin the assumptions of slender-body theory are mathematically complex and restricted to
specific planforms; therefore, no general quantitative information of their results is presented.
The supersonic CN& of cones may be obtained from the hybrid theory solution of reference 4.
DATCOM METHOD
The method presented here for determining the body contribution to CN6&, based on body basearea and body length, is the same as that of paragraph A and is given by equation 7.2.2.1-a, i.e.,CN.

= 2 CNa (-B)

where CNe is the body normal-force-curve slope from paragraph C of Section 4.2.1.1, evaluatedat the appropriate Mach number. The other parameters are defined in paragraph A of thissection.
D HYPERSONICThe body contribution to the derivative CN&, in the hypersonic speed range is shown inreference 5 to be zero when determined by the Newtonian impact theory.
REFERENCES
1.

Pitt$, W. C., Nielsen, J. N., and Kaattari, G. E.: Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic.
Tranaonic. and Supersanic Speeds. NACA TN 1307, 1957. 4U)

2.

Walker, H. J., and Wolowicz, C. H.: Theoretical Stability Derivtives for the X-15 Research Airplane at Supersonic andHypersonic Speeds Including a Comparison with Wind-Tunnel Reults. NASA TM X-287, 1980. (U)3.

Landahl, M. T.: Forces and Moments on Oscillating Slender Wing-Body Combinations at Sonic Speed. MIT Fluid Dynamic
Raeearch Group Report No. 56-1, 1956. (U)

7.2.2.1-3

,.

Tohak, M., and Wshrenl, W. R.: Stability Darivatives of a Cone at Supersonic Spead.

5.

Fisher, L R.; Equations and Charts for Datermlning the Hypersonic Stability Derivatives of Cone Frustums Computed by
Newmonian Impxct Theory. NASA TN D-149, 1950. (U)

7.2.2.1.4

NACA TN 3788, 1966. (U)Revied Septmber 1970

7.2.2.2 BODY ACCELERATION DERIVATIVE Cm&

Methods are presented for determining the body contribution to the acceleration derivative Cm&.

in the subsonic, transonic, and supersonic speed ranges.
Datcom methods are based on the same assumptions that were made in regard to the estimationof the body contribution to the derivative CL&, and the general discussion of Section 7.2.2.1 isdirectly applicable here. The body contribution to Cm& is expressed as

ca =

Ci. a

yt w

The limitations of these methods are determined by the limitations of the methods used inestimating the static derivative Cma in the various speed regimes. Experimental data should be
used for the body pitching-moment-curve slope when available.
A. SUBSONICThe comments of Paragraph A of Section 7.2.2.1 are" directly applicable here.
DATCOM METHOD
The body -contribution to Cm &, based on body base area and the square of the body length andreferred to an arbitrary moment center, is given by

V
Cm

Cm

a

2B

~xm~
V3

7.2.2.2-a

where
Cme

is the body pitching-moment-curve slope from ParagraphA of Section 4.2.2.1,
multiplied by VB/(SbRB).

xC

is the longitudinal distance from the vertex to the centroid of the volume and isgiven by Fquation7.2.1.2-b, i.e.,

09

S(x) x dx

7.2.2.2-1-xm

is the longitudinal distance from the body vertex to the center of rotation, positive
aft.

RB

is the length of the body.

VB

is the total body volume from Section 2.3.

Sb

is the body base area.

Sample Problem
Given: Same ogive-cylinder configuration as sample problems of Paragraph A of Sections

7.2.1.2

and 7.2.2.1.

2.5d

4."t-- = 71

.LI

V 6.5d
1

•

d
Fi.

1 ft

m

6.0 ftB=

Fineness Ratio

.

6.0

Xm

= 3.8 Ift
In1

-,

.

-

0.824 (Section 7.2.2.1)

rS " iS .

0.781 per rad"'a
C = 3.55 ft
I.!

*

)

(Section 7.2.1.2)- .Solution:
-

3.8-

0.367

7.2.2.2-2

.

2

mf,

a

(1

£[)

VB

.2 t 3.55

2(0.781) L(Equation 7,2.2.2-a)
3.8 1

0.367 - 0.824

J

1.56?(-0.0343)-

0.117 per rad (based on SbR2)

B. TRANSONICThe comments of Paragraph B of Section 7.2.2.1 are directly applicable here. The method ofParagraph A is applicable throughout the transor.ic speed range.

C. SUPERSONIC
The theoretical methods applicable for the estimation of the body coatribution to the derivative
Cma parallel taose of the body contribution to the derivative CL6. Therefore, the generalcomments of Paragraph C of Section 7.2.2.1 are also applicable here.
The supersonic Cm& for a cone can be estimated by the hybrid theory solutions presented in
Reference 1.

K•

DATCOM METHOD
The method presented here for the body contribution to Cm,; based on body base area and thesquare of the body length and referred to an arbitrary moment center, is the same as that of
Pparagraph A and is given by Equation 7.2.2.2-a, i.e.,

C

2C

()

2.2-

72,2.2.-3

where Cma is the body pitching-moment-curve slope from ParagraphC of Section 4.2.2.1,
evaluated at the appropriate Mach number. The other parameters are defined in Paragraph A ofthis section.
D. HYPERSONICThe body contribution to the dertvative Cn& in the hypersonic speed range is shown inReference I to be zero when determined by the Newtonian impact theory.
REFERENCE

1.

Tobak, M., and Wehrend, W. R.: Stability Derivatives of a Cone at Supersonic Speeds. NACA TN 3788, 1956. (U)-V~~

7.2 .2.2-4

II

Revied September 1970

7.3 WING-BODY DYNAMIC DERIVAi VES
7.3.1 WING-BODY PITCHING DERIVATIVES"7.3.1.1 WING-BODY ?ITCHING DERIVATIVE CLq
The information contained in this section is for estimating the pitching derivative CLq Of
wing-body configurittions at low angles of attack. In general, it consists of a synthesis of materialpresented in other sections; however, the method of application is new.
The Datcom methods are based on the assumption that the mutual interferences that occur
between components for angle-of-attack variations, determined in references I and 2 andpresented in Section 4.3.1.2, may be employed with reasonable accuracy to the case of steadypitching. This approach to the calculation of wing-body pitching derivatives has been used withreasonable success by Walker and Wolowicz in reference 3.
A. SUBSONIC
Two methods are presented for determining the pitching derivative CLq of a wing-bodyconfiguration, differing only in their treatment of the mutual interference effects.
DATCOM METHODS
Method I
For wing-body configurations similar to sketch (d) of Section 4.3.1.2, the pitching derivative
CLa, based on the area and mean aerodynamic chord of the total panel and referred to a
moment center at the assumed center of gravity or center of rotation, is given by("KW(B)+ KB<)w](CLq++(L'q\

S.3.1.1-awhere
Kw(B) and KB(w) are the appropriate wing-body interference factors obtaint.d from Section
4.3.1.2.
(CLq),. is the contribution of the exposed panel to the pitching derivative CLq, obtained

from Section 7.1.1.1. (See Section 4.3.1.2 for the definition of exposed surfaces.)(CLq)B is the contribution of the body to the pitching deiivative CLq,
Section 7.2.1.1.

iobtained from

7 .3.1.1-1

is the ratio of the exposed to the total panel area.

S
S

is the ratio of the body base area to the total panel area.

-7C

is the ratio of the mean aerodynamic chord of the exposed panel to the mean
aerodynamic chord of the total panel.

_is the ratio of the body length to the mean aerodynamic chord of the total panel.

c

Moment transfer calculations are included as an integral part of the wing and body derivative
estimation methods of Sections 7.1.1.1 and 7.2.1.1, respectively.
Method 2
For wing-body configurations similar to sketch (c) of Section 4.3.1.2, the pitching derivat.ve
CLq, based on tht. area 4,.u mean aerodynamic chord of the total panel and referred to a
moment center at the assumed center of gravity or center of rotation, is given by(c\\C Lqw

K(WB) (CL\ + (L\J
qW+

B(qf)

7.3.1.1 -b
1

where
K(wD)

is the wing-body interference factor, obtained from figure 4.3.1.2-12c of Section

4.3.1.2.
.. (CLWis the contribution of the total panel to the pitching derivative CL q obtained from
Section 7.1.1.1.

r

S_"as
.

The remaining terms are defined in method I above. Moment transfer calculations are included
an integral part of the wing and body derivative estimation methods of Sections 7.1.1.1 and7.2. 1. 1, respectively.

Sample Problemt-

Method 1
Given: An ogive-cylinder-body-triangular-wing configurationS"

7.3.1.1-2

r

-

•

-

<

•- ,

-

o

.• - •-- ., -

'

-

"

•

.

"

"

- " -

-

.

-

XM

7

'm

ýA LE

..4 EXPOSED PANEL

13

Wing Characteristics:
Total Panel
A

=5.0

=

0

A,

= 38.670

S = 414.0 sq ft

be

iw

0

c=

Ac/2=

18.20 ft=

Xe = 0=5.0

A

b = 45.50 ft-

Exposed Panel

38.37 ft
21.80

So = 294.50 sq ft
Cr12,133 ft

15.375 ft=

10.25"ft

NACA 66-206 airfoil section

Body Characteristics:'

25.0 ft

LB = 77.20 ft

d7.127 ft

The following ratios based on total panel dimensions:-Sl"=
0.7113

-

0.845

-Sb

=

0.0963- 6.363

7.3.1.1-.1"Additional Characteristics:
43.70 ft (moment center at ý/4 of total panel)x=

M = 0.60

d
0.80

j3

0.157

b

Compute:Step 1.

Pitching derivative CL

for exposed panel (Section 7.1,1..1)

6.19 per rad (table 4.1.1-B): -yr27r=0.985+ 2ttan•

0.985

K

(C)
A € = 0.90 per radr[0.64+0.161"'

4.54(figure 4.1.3.2-49): 4.50 per rad(CL)Q
tan ALE

1.0
A tan ALE =4.0

( ' )~rC

0=0.560(figure 4.1.4.2-26a)C(/- = 0.408

Cre(from planform geometry of exposed panel with c.g. at Z/4 of total
panel)

S.(equation 7.1.1.1-b)

L 1

7.3.1.1-4>

(-0.--

"

X

X)

~

(C~q~
=(-h
+

()

CL)

=

(0.560 - 0.408) 1.50

=

0.228

(equation 7. 1.1.1-a)(0.956) (4.50)=

4.30 per rad (based on S,ý,)
Step 2.

Wing-body interference factors (Section 4.3.1.2)

Kw(B
KB)
Step 3.
kd-= 1.125=0.215(figure 4.3.1.2-10)

Pitching derivative CL
77.20
7.127

- 10.83(k2 - k1 ) = 0.947
x=

R

=

for body (Section 7.2.1.1)(figure 4.2.1.1-20a)

25.0ft

xi
25.0
~£9
77.2

- 0.324-B~X
x=

42.46 ft

SO = S,
IN

25.0

7-27
7.127 =3.508

--d

--VN
VN

= 39.88 sqft

=0.535 (figure 2.3-5, ogive)-

SoN

VN = (0.535) (39.88) (25.0) = 533.4 cu ft

7.3.1.1-5

=

V5

(Ru

-9

+ VN

N) S

(30.88) + 533.4= (77.2 -'25.0)= 2615.1 cu ft
VB2/ 3189.8 sq ft=

2(k2 - k,)So

(CL)

=

V21

4.2.1.1-a)0(equationr2(0.947) (39.88)
189.8=.(CLa.)

0.398 per rad (based on VB 2 /3 )=0.398

V 2/3
Sb

0.398

189.8"
139.1s8/-

1.894 per rad (based on Sb)b

L(~ = 1-

(C)

=

0.434

(equation 7.2.1.1-a)

2 (C)= 2(1.894) (0.434)
1.644 per rad (based on Sb Q1)
Solution:(CLa)W 5=+C[KW(B) +)CL JW

ZLq (equation 7.3.1.1-a).+-J(1.1^3 - 0.215) (0.7113) (0.845) (4.30) + (1.644) (0.0963) (6.363)= 3.46 + 1.007= 4.467 per rad(based on the area and mean aerodynamic chord of the total
panel and referred to a moment center at ý/4 of the total pane!)

7.3.1.1-6

"B.TRANSONIC
The aerodynamic interference effects for slender wing-body configurations are relatively
insensitive to Mach number; consequently, the slender-body interference factors of Section
4.3.1.2 should give reasonable results. For nonslender configurations, transonic interference
effects can become quite large and sensitive to minor changes in local contour.
DATCOM METHODS
It is recommended that the methods presented in paragraph A for estimating the pitching
derivative CLq of a wing-body configuration be applied to the transonic speed regime. Careshould be taken to estimate the contributions of the lifting panel and body at the appropriate
Mach number. The interference factors should be obtained from paragraph C, Section 4.3.1.2.
C. SUPERSONIC

L"

•

--..

The information included in the Datcom accounts for most of the mutual interferences that
occur between components of a wing-body configuration at supersonic speeds. These interference
effects have been accounted for by the slender-body interference factors of Section 4.3.1.2.
DATCOM METHODS-The

methods presented in paragraph A for estimating the pitching derivative CLq Of a wing-bodyconfiguration are also applicable to the supersonic speed range. Care should be taken to estimate
the contributions of the lifting panel and body at the appropriate Mach number.
Sample Problem
Method 1
Given: Same configuration as sample problem of paragraph A
M = 1.4

1=0.98Compute:Step 1.

Pitching derivative CL

for exposed panel (Section 7.1.1.1)

tan ALE=0.816

4.0

At tan ALE="3(CNj)4.0 per rad(figure 4.1.3.2-56a)

7.3.1.1-7

4.08 per rad

(CN)

- -

=

0.67

(figure 4 .1.4. 2 -26a)•xc'g'=

0.408(from planform geometry of exposed panel with c.g, at U/4 of total
panel)

1**-

(

[

-

,car
C (X

_

(eqiiation 7.1.1.1-b)e"- c---/

vi cot ALE--(0.67_Xc..g.

0.408) 1.50

=0,3931.225 (supersonic leading edge)

3Ae =4.90
cot- 1( cot ALE) ==0 (figure 7.1. 1.1 -11a)=) (CLq(Cy.

39.230

+~2

(CNe

)

(equation 7. 1.1. 1-c)

= 0 + 2(0.393) (4.08)= 3.21 per rad (based on V.)

Step 2.

Wing-body interference factors (Section 4.3.1.2)

KW(B)= 1.125:d(0.98) (7.127)
15.375CreK(

7.3.1.1-8

CN)(figure 4.3.1.2-10)(X, + 1)=0.4551.)]

3.90(figure 4.3.1.2-11a)

01

+ b

[~ (

+1)

CN

2,•.

3.90
21.51
K

"'w)

(0.98) (4.08) (1) (6.38--

1)

2 1.51- 0.181

Pitching derivative CL for body (Section 7.2.1.1)

Step 3.

f77.2

fqfN

- 25.0

£A

fA

=- RN

= - 25.0

di

7.127

7.32

=.23.51-= 2.09SfA /fN

'Q

i]

-~.

f

13 0.98
0-98

fN

3.51

- 0.279(figure 4.2.1.1-2 1a, extrapolated)

2.74 per rad

CN)

-

=

(Cq)

0.434

(sample problem. oaragraph A)(equation 7.2.1.1 -a)

2 (CN)~= 2(2.74) (0.434)= 2.378 per rad (based on SbRB)
Solution:(Cjq)ý

1KWB

KB(W)l(-)()Cq+ (CLq)B( ) (c(equation 7.3.1. 1-a)4-

(1.125 + 0.181) (0.7113) (0.845) (3.21) + (2.378) (0.0963) (6.363)= 2.520 + 1.457- 3.977 per radI(based on the area and mean aerodynamic chord of the total
panel and referred to a moment center at 6/4 of the total panel)

7.3.1.1-9

*1

REFERENCES

p

r

1r

.

Pitts, W., Neilsen, J., and Kaattari, G.: Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic,
and Supersonic Speeds. NACA TR 1307, 1959. (U)2.

Spreiter, J.: The Aerodynamic Forces on Slender Plane, end Cruciform-Wing and Body Combinations. NACA TR 962,
1950. (U)3.

alker, H., and Wolowicz, C.: Theoretical Stability Derivatives fo. the X-15 Research Airplane at Supersonic andHypersonic Speeds Including a Comparison with Wind-Tunnel Results. NASA TM X-287, 1960. (U)Revised September 1970

7.3.1.2 WING-BODY PITCHING DERIVATIVE Cmq

The information contained in this section is for estimating me pitching derivative Cmq ofwing-body configurations at low angles of attack. The derivative Cmq is a measure of thepitcning moment produced by rotatiorial motion about a spanwise axis and is commonly referred
to as the pitch-damping derivative,
The Datcorn methods are based on the same assumption that was made in regard to the pitching
derivative CLq of a wing-body configuration, and the general discussion of Section 7.3.1.1 isdirectly applicable here.
A. SUBSONIC
Two methods are presented for determining the pitching derivative Cmq of a wing-bodyconfiguration, differing only in their treatment of the mutual interference effects.
DATCOM METHODS
Method 1
6

For wing-body configurations similar to sketch (d) of Section 4.3.1.2, the pitching derivative
Cmq, based on the area and the square of the mean aerodynamic chord of the total panel andreferred to a moment center at the assumed center of gravity or center of rotation, is given by(CmR cn[W(a)KBjw)](Cm\-\

7.3.1.2-a

where(Cmq)c is the contribution of the exposed panel to the pitching derivative C,,, obtained

from Section 7.1.1.2. (See Setion 4.3.1.2 for the definition of exposed surfaces.)(Cmq)8 is the contribution of the body to the pitching d ;ivative Crmq, obtained from
Section 7.2.1.2.
The remaining terms are defined in paragraph A of Section 7.3.1.1. Moment transfer calculations
are included as an integral part of the wing- and body-derivative estimation methods of Sections
7.1.1.2 and 7.2.1.2, respectively.

"Method 2
For wing-body configurations similar to sketch (c) of Section 4.3.1.2, the pitching derivative
Cnq, based on the area and the square of the mean aerodynamic chord of the total panel andreferred to a moment center at the assumed center of gravity or center of rotation, is given by

7.3.1.2-1--

S/+ (Cm qS--

7.3.1.2-b)

where.
(Cm q)w is the contribution of the total panel to the pitching derivative Cm q, obtained from
Section 7.1.1.2.
The remaining terms are defined in method I above and paragraph A of Section 7.3. 1. 1. Momenttransfer calculations are included as an integral part of the wing- and body-derivative estimation
methods of Sections 7.1.1.2 and 7.2.1.2, respectively.
Sample Problem
Method I
Given: Same configuration as sample problem of paragraph A, Section 7.3.1.1. The characteristics
are repeated below.
Wing Characteristics:
Total Panel

Exposed Panel

A =5.0X = 0

Ae =5.0

X. = 0ALE

S = 414.0 sq ft

be = 38.37 ft

So = 294.50 sq ft

cr--

Ac14

c,

= 15.375 ft

z=

10.25 ft

38.670:'iw

0b - 45.50 ft

18.20 ft

30.960== 12.133 ft

NACA 66-206 airfoil section
Body Characteristics:£=

25.0 ft=

d = 7,127 ft

77.20 ft

The following ratios based on total panel dimensions:Se
S 0.7113=

0.7140VS

S 0.0963

7.3.1.2-2=40.49*'*-Additional

Characteristics:.

M = 0.60

xm = 43.70 ft (moment center at Z/4 of total panel)

S-=0.80db0=0.157

Compute:Step 1.

Pitching derivative Cmfor exposed panel (Section 7.1.1.2)
q

ca
-)

6.19 per rad}=2(sample

problem, paragraph A, Section 7.3.1.1)

cs A,, 4 = 0.8576

tan Ac/4 = 0.600
B =

/1-M

2

cos2 Ac,4" 0.8575

rA
I(C

.

2 ]a

-0.7 cg, cos Ac/4

cI+24
As +2 cos Ac/ 4(A+ 6a 0 cs
Ac/4) +-"+

(equation 7.1.1.2-a)

5 (0.228) + 2(0.228)21--

0.7(6.19) (0.8576)

•5

[

4+ 5+

12S

2(0.576+ 2(0.8576)

6(0.8576).[

7.3.1.2-3

=372[1.090

1

/45

11

6.715

24

10. 146)-8-3.72(0.472 1)=-1.76 per rad (based on So 02)

H

A e3 tan 2 A c/4

'. '

+

04

ABB+6cosAc/ 4

A 3 tan 2 AS

e3 tnAc/4
SCm~e

-

B
Cmq)M

.e

(equation 7.1.1.2-b)

3

A +6cosA, 4(5)3 (0.6)2(5) (0.8575) + 6(0.8576)

30.8575(-1.76)(5)3 (0.6)2
5 + 6(0.8576) + 3

,i ..

(4.77
+ 3.499~~~~~4,,435+ -3 -Sk"- .6.,,.="269(-1.76)k7.435)-1.96 per rad (based on Seie2)=

Wing-body interference factors (Section 4.3.1.2)

Step 2.

Kw(B) = 1.125)(sample problem, paragraph A, Section 7.3.1.1)..

K•t "=

0.215
for body (Section 7.2.1.2)

Pitching derivative Cm"Step 3.

q",0 1 a' 34

067a9x

.

") •

__x_

7.3.1.2-4
I.

-. "L •, _

.

.

.

"•

•

" •

-

" -

-

-• -

-

Ax = 2.5 ft
From sample problem, paragraph A, Section 7.3.1.1:(k 2 - k 1 ) = 0.947

V4=2615,1 cu ft

x, = 25.0 ft

x= 42.46 ft

39.88 sq ft-Sb

Determine (Cm a)BdS

Station

ft

I1 ft

1
2

1.137

1,016

1.016

42.033

42.71

2.W0

5.300

4.293

39.787170.81

3
4

3.725
4.650

10.89816.982

5.5w9
6.064

37.37634.904208.89212.36

6
6
7

5.450
6.150
6.625

23.32829.70634.472

6.346
6.378
4.766

32.41729.92527.425206.72190.86130.71

8
9
x

6.950
7.100
7,127

37.93739.592
39.8C0

3.405
1.685
0.288

24.94022.44619.949

86.42
37.15
5.74

x0

7.127

39.880

0

9.970

0

dSx
AS
ASx

dx

dS
x

(xx*N
m

dx

)A

m

dS
x-0

(C

X
5O

2(k 2 - k)
2"

)

0

dS(
x

(xm

2(0.947) ( 1291)=- 2 970.935-""T . (Xm -x)AX

x) Ax

=

1291

(equation 4.2.2.1-a)

per rad (based on VB)

Z76 15.1

(Cm)
\ý

VB
(0.935) -b-=

a/BSbB(2615.1)= 0.935(21.1)(39.88) (77.2)

0.794 per rad (based on SbQ)

"x is taken to be at the center of vau'm. of each body SMOnt.

7.3.1.2-5

Determine x

Station

d

S(x)

ft

sq ft

x*

SWxl x Ax

1

1.137

1.016

1.667

4.23

2

2.600

5.309

3.913

51.94

3

3.725

10.8986.324

172.30

4
5

4.650
5.450

16.982•23.328

8.796
11.283373.43658.02

6
7

6,150
6.625

29,70634.472

13,77516.27b1023.00
1402.58
1 779.248

6.950

37.93718.7609

7.10O039.59221.2542103.72

10

7.127
S7.12739,88039.88023.75161.1002 367.97
106,377.

ZX- S(x) xAx
fO

xC
i.•

i

S(x) x dxV116,3132615.144.48 ft

116,313=X(equation 7.2.1.2.b)

2615.1B

0.4(39.88) (77.2-)

S k•
S..

=

0.434

(sample problem, paragraph A, Section 7.3.1.1)

;mb

-

2'

(C)

=2(

(

~

[

~

~

(0.434)2 - 0.849(44.48 -2(0.794)

3•4

7212a

-(equation

L4.8-4.0)l
43.71

77.20

.849
0.4(0. -4-*x is taken to be at the center of volume of each body sagent.

"7.3.1,2-6

Kr'.. •'" ';

,;s

•

-

,

•

•,

•...•

-

2 .

....

"...

....

•..

..

.

.

. . .-. .

....
.

.:.

.

'.

.

.

1.59 [ 0-1.59(2--0"849(0.0101)](0.1798-0,689 per rad (based on Sb B2)-

Solution:(Mq)WB=[W(B)

+

B(&

f)\)

(Cq, + (Cmt)B

KS T3(equation 7.3.1.2-a)= (1.125 + 0.215) (0.7113) (0.7140) (-1.96) + (-0.689) (0.0963) (40.49)--1.334 - 2.687= -4.021 per rad (based on the area and the square of the mean aerodynamic
chord of the total panel and referred to a moment center at E/4of the total panel)ý-.
B. TRANSONICThe comments of paragraph B of Section 7.3.1.1 are directly applicable here.
DATCOM METHODS
It is recommended that the methods presented in paragraph A for estimating the pitching
derivative Cmq of a wing-body configuration be applied to the transonic speed regime, Careshould be taken to estimate the contributions of the lifting panel and body at the appropriate
Mach number. The interference factors should be obtained from paragraph C, Section 4.3.1.2.
C. SUPERSONICThe comments of paragraph C of Section 7.3.1.1 are directly applicable here.
DATCOM METHODS
The methods presented in paragraph A for estimating the pitching derivative Cmq of a wing-bodyconfiguration are also applicable to the supersonic speed range. Care should be taken to estimate
the contributions of the lifting pane! and body at the appropriate Mach number.

"7.3.1.2-7

S--

Sample Problem
Method 1
Given: Same configuration as sample problem of paragraph C, Section 7.3.1.1 and paragraph A ofthis section.
(3=0.98

M = 1.4
Compute:for exposed panel (Section 7.1.1.2)

Pitching derivative Cm

Step 1.

q

)

0.393

"-cot ALE

4_1.225 (supersonic leading edge)

A0 p = 4.90

(sample problem paragraph C, Section
7.3.1.1)

cot-' 1 ( cot ALE)

mq
*•'

=

39.230(Lq

3.21 per rad.-- (Cm)=

1.018 per rad(Cmq').

=-1.039 per rad(Cm q)C
Je

=

(Cmq)
(C
Je

-

-

-1.039-De(figure 7.1.1.2-10a)Le(equation 7.1.1.2-d)(0.393) (3.21)"=.-2.301per rad (based on S Z 2)

Step 2.

Wing-body interference factors (Section 4.3.1.2)

Kw(B)7.1KB2) 8181

i)

7.3.1.2-8= 1.125=(sample problem, paragraph C, Section 7.3.1.1))

Step 3.

Pitching derivative CmqfN

for body (Section 7,2.1.2)= 3.51

f4 /f?=

2.09"(sample problem, paragraph C, Section 7.3. 1. 1)
fN = 0.279(CN )2.74 per rad=

xC= 44.48 ft

VB
0.849

=

Sb Q

(sample problem, paragraph A)

1 / -0.434

XC.,,.

- 0.195

xM

2B(figure 4.2.2.148a, extrapolated)

43.70
B

(Cm)

77.20

-0.566=(equation 4 .2.2.1-c)(CN)= (0.566 -- 0.195) 2.74

1.017 per rad (based on Sb B)-'BSb(Cmq)B

=

2(Cm)

RD

£

-RI

1

(equation 7.2.1.2-a)

Sb R

7.3.1.2-94

(0.434)2=2(.017)-(0.849)0.434 -

1--

I

0.849

0.1798\-0.415/= 2.034 I%. 17)= -0.881

per rad (based on SbRB 2)

Solution:'•,,+•-,
TSe(C°,)B I\t/(°)W.,[KW(D)KB~ W i-). +
(nq

(Cq)

+

(equation 7.3.1.2-a)=(1.125 + 0.181) (0.7113) (0.7140) (-2.301) + (-0.881) (0.0963) (40.49)-- 1.526 - 3.435=-4.961 per rad (based on the area and the square of the mean aerodynamic
chord of the total panel and referred to a moment center at Z/4of the total panel)

hi

i'O"
K.-

7.3.1,2-10

RevidiJune 1969

7.3.2.1 WING-BODY ROLLING DERIVATIVE CyPThis settion presents methods for estimating the wing-body contribution to the rolling derivative Cy at subsonic and supersonic speeds. This derivative is the change in side-force coefficient withchange in the wing-tip helix angle.
A. SUBSONIC
Experimental evidence indicates that the effect of the fuselage on the side force due to rolling is
negligible throughout the angle-of-attack range up to the stall. Therefore, the method of Section 7.1.2.1
for estimating the wing-alone derivative at subsonic speeds is also applicable to wing-body configurations.
Available test data are limited to wing-body configurations with body diameters less than about 30
percent of the wing span. Therefore, the application of the wing-alone method is limited to valuesof d/b < 0.3.
The method consists of determining the value of Cy at zero lift and extrapolating this result to highlift coefficients by using experimental values of lift and drag at high lift coefficients. The method also
includes the effect of geometric dihedral.

DATCOM METHOD
The variation of the wing-body rolling derivative Cy with lift coefficient at subsonic speeds isobtained by using the method of paragraph A of Section 7.1.2.1. In using this method experimental
values of wing-body lift and drag are used in evaluating the dimensionless correction factor K. Thismethod is limited to configurations with body diameters less than about 30 percent of the wing span,i.e., d/b < 0.3.
If experimental wing-body lift and drag data are not available, wing-alone test data may be used. Ifexperimental lift and drag data are not available, no attempt should be made to extrapolate the zero-lift
value to higher' lift coefficients. The negligible importance of the derivative does not wzrrant thecomplexities involved in estimating the lift and drag characteristics. Furthermore, no known general
method for estimating the variation of the drag coefficient will give results reliable enough to use indetermining the correction factor for extrapolating the potential-flow values to higher lift coefficients.
The sample problem presented at the conclusion of paragraph A of Section 7.1.2.1 illustrates the use ofthis method.
B. TRANSONIC
No generalized method is available in the literature for estimating transonic values of the rollingderivative Cy p. Furthermore, no known experimental results are available for this derivative at transonic
speeds.
C. SUPERSONIC
No generalized method is available in the literature i'or estimating the effect of the fuselage on therolling derivative Cy
For the purposes of the Datcom the fuselage effect is considered to be negligible for wing-body

7.3.2.1-Ir

configurations with body diameters less than abort 30 percent of the wing span. Therefore, forconfigurations with d/b < 0.3 the wing-body rolling derivative Cy is estimated by the wing-alone
method of paragraph C of Section 7.1.2. 1.
No known experimental data are available for this derivative at supersonic speeds. Therefore, the validityof the Datcorn application cannoL be determined.
DATCOM METHOD
The wing-body rolling derivative Cy at supersonic speeds and at low values of the lift coefficient isobtained by using the method of paragraph C of Section 7.1.2.1.
This method is limited to configurations with body diameters less than about 30 percent of the wing
span, i.e., d/b < 0.3.
The sample problem at the conclusion of paragraph C of Section 7.1.2.1 illustrates the use of thismethod.

72

I-I

I.I

K'.'-

Revised June 1969

7.3.2.2 WING-BODY ROLLING DERIVATIVE Clp

This section presents methods for estimating the wing-body contribution to the Tolling derivative Cp at
subsonic and supersonic speeds. Mhis derivative is the change in rolling-moment coefficient with change
in the wing-tip helix angle and is commonly referred to as the roll-damping derivative.
A. SUBSONIC
Experimental evidence indicates that in general for configurations with the ratio of maximum body
diameter to wing span less than about 30 percent, i.e., d/b 1 0.3, the effect of the fuselage on the roll
damping is negligible over the angle-of-attack range up to the stall. Therefore, the method of Section
7.1.2.2 for estimating the wing-alone derivative at subsonic speeds is applied in this section to estimate
the wing-body roll damping at subsonic speeds.
The method consists of first determining the value of C1P at zero lift based on simplified lifting-surface
theory, then assuming that variations in the lift-curve slope will affect C1 in the same proportionas CL 2 , and finally accounting for the drag associated with the additional lift on the outer portion of
the wing resulting from flow separation and the attendant formation of a stable leading-edge vortex. The
method also accounts for geometric dihedral.
The Datcom method accounts for the variations in wing-body lift-curve slope, drag due to lift, profile
drag, and the effect of dihedral. The method requires knowledge of the variation of lift and drag over
the angle-of-attack range to the stall. Therefore, the method is quite rc adily applied if experimental liftand drag data are available.
This method is restricted to configurations with values of d/b < 0.3.
Although all the bodies of the wing-body configurations analyzed were bodies of circular cross section,
the method should give reasonable values for configurations with noncircular cross-section bodies,
provided reliable values of the wing-body lift-curve-slope variation are available.
DATCOM METHOD
The variation of the wing-body rolling derivative Cwith lift coefficient at subsonic speeds is obtained"by using the method of Paragraph A of Section 17.1.2.2. In using this method experimental values ofwing-body lift and drag are used in evaluating the variations of lift-curve slope and profile drag.
The limitations of the method as applied to wing-alone configurations are equally applicable to themethod when applied to a wing-body configuration.
A comparison of the roll-damping derivative calculated by using this method with test results ispresented as Table 7.3.2.2-A. Experimental values of wing-body lift and drag have been used in
evaluating the roll-damping derivative of all the configurations listed in the table.
If test data for wing-body lift and drag are not available, the wing-alone result should be used for tne
wing-body roll damping of configurations with d/b < 0.3. In this case wing-alone test data are preferred,
but if they are not availablethe wing-alone lift and drag should be estimated as outlined in Section 7.1.2.2.
The Datcom methods for estimating the wing-body lift and drag variations with angle of attack have notbeen substantiated and, therefore, should not be used in evaluating the wing-body roll damping.
7.3.2.2-1

The sample problem presented at the conclusion of Paragraph A of Section 7.1.2.2 illustrates the use ofthis method.
B. TRANSONIC
There are no reliable methods for estimating the derivative Cl in the transonic region. Although thisderivative might be expected to vary with Mach number in the same manner as lift-curve slope, thistrend is not exhibited by experimental data. A considerable quantity of wing-body test data is available,
however, and reference should be made to Table 7-A.
C. SUPERSONIC-

Reference I presents a linear-theory method for estimating the roll damping of thin supersonicleading-edge triangular and rectangular wings mounted on cylindrical bodies. The roll damping of thinsubsonic-leading-edge triangular wings mounted on cylindrical bodies is presented in Reference 2 based on
slender-wing theory. These theoretical results show that for most of the range of variables the net effectof the body is to decrease the roll damping. The body effect is small for body diameters less than about
30 prcent of the wing span (d/b < 0.3) for both triangular and rectangular wing configurations.
However, the body effect increases rapidly as the ratio of body diameter to wing span increases beyond
this vaiue. Unfortunately, at the higher values of d/b the theory does rot show the same trend with
Mach number for the body effects of configurations with triangular and rectangular wings. At thesehigher values of d/b the body effect increases with increasing Mach number for configurations withrectangular wings; whereas, the body effect decreases with increasing Mach number for configurations
with triangular wings. Therefore, at higher values of d/b the theoretical results for triangular andrectangular wings cannot be applied to indicate the general trend of the body effect for configurations
with other planforms.
The Datcom uses a design chart, based on the theoretical results of References 1 and 2 and a limited
amount of experimental data, to account fo1 the effect of a circular body on the roll damping of awing-body configuration at zero lift. This chart is applicable to all straight-tapered wing configurations atvalues of d/b < 0.3, but is restricted to triangul r wing configurations at higher values of d/b.
DATCOM METHOD
The wing-body roll-damping derivative at supersonic speeds, based on the product of the wing area andthe square of the wing span SW bjw, is given by
C1(C/P)W
"

O

(Cp)

(Clow

p

(C 1 )(per radian)

7.3.2.2-a

P dlb=O

where (Ct,)w is the wing contribution to the roll-damping derivative at supersonic speeds, obtained byusing the method of Paragraph C of Section 7.1.2.2, including the effects of wing thickness, andCt
C

L7.3.2.2-2/b-is the semiempirical body-correction factor obtained from Figure 7.3.2.2-13 as afunction of thc ratio of maximum body diameter to wing span and the Mach
number normal to the leading edge.

Figure 7.3.2.2-13 is applicable to all straight-tapered wing configurations with circular bodies over therange 0 < d/b < 0.3. For values of d/b > 0.3 this chart is applicable only to configurations consisting of triangular wings mounted or cyhindrical bodies.
Experimental results cc,:o,, ted in Reference 15 for triangular wing-body configuratVns
0.4 and 0.6, ;tLAd ror subsonic, sonic, and supersonic leading edges, show good agreementwith d/b
with the values given by !-;i.ra-e 7.3.2.2-13.
No experirr.ental dz;ti uae presently available to substantiate the theoretical results of Reference I for
rectangulaxr wing-oy configurations with values of d/b > 0.3.
As noted atx'qz, the triangular- and rectangular-wing results of References I and 2 at valuesof d/b > 0.3 c.<nnot be used to indicate the general trend of the body effects for other planforms.
The supersonic roll-damping derivative m the linear-lift range calculated by this method is comparedwith test results in Table 7.3.2.2-B. Although all configurations listed in Table 7.3.2.2-B have cylindricalbodies, the method may be applied to configurations having bodies which are not true cylinders, butwhich are almost cylindrical.
Sample Problem
A triangular wing-body configuration of Referencel4.

gALE

A = 4.05= 44.90

Ac/ 2 = 26.70

0.20

M = 1.92 ; 3 = 1.639

d/b

N= 0

Airfoil: Hexagon, 61 = 4.90

Compute:
0A = (1.639) (4.05)
A tan At/,
(C)

=

6.64

=

(4.05) (tan 26.70) = 2.04-

- 0.0505 per radA(Figure7.1. 2 .2 -25a)

Scot ALE = (1.639) (cot 44.90) = 1.642(supersonic leading edge)tan 44.90

tan ALE-

6
S1.6390.608

Cv=0.970(¶igure7.1.2.2-27)(C 1 )

7.3.2.2-3.K

-.. -7.....

7

Cl

.

7W_6

0.970

.

.

-i

-

,

-"

q-...-..

.

(Figure7.3.2. 2 -13)(C,)

Solution:

L-o.osos] (4.05)C,(0.970)

-:--

K

1W
(Cp)

-

radA d/b0-0.198Aper\L= (Cp)=(CP(Equation 7 .3. 2 .2 -a)(-0.198) 0.970= -0.192 per rad(based on Swbý)This compares with a test value of -0.182 per radian from Reference 14.

REFERENCES

1.

Tucker, W. A., and Piland, R. 0.: Estimation of the Damping in Roll of Superaonlc-Laeding-Edgs Wing-Body Combinations. NACA TN

2.

Lomax, H., and Heeslet, M. A.: Damping in Roll Calculations for Slender 9Swpt-Back Wings and Slender Wing-Body Combinations,
NACA TN 1950,1949. •U)

3.

ueijo, M. J., and Wells, E. G,: Wind-Tunnel Investigation of the Low-Speed Static and Rotary Stability Derivatives of a 0.13-Scale
Model of the Douglas D-558-11 Airplane in the Landing Configuration. NACA AM L62G07, 1962. (U)4,Kuhn, R. E., and Wiggins, J. W.: Wind-Tunnel Invedtgation to Determine the Aerodynamic Characteristics in Steady Roll of a Model atHigh Subsonic Speeds. NACA AM L52K24, 1953. (U)5.

Wiggins, J. W.: Wind-Tunnel Investigation at High Subsonic Speeds to Determine the Rolling Derivatives of Two Wing-Fuselage
Combinations Having Triangular Wings, Including a SemiempirIcl Method of Estimaing the Rolling Derivatives. NACA RM L63L18e,
1954. (U)6

Myers, B. C., II, and Kuhn, R. E.: High-SubsonIc DampIng-In-Roll Characteristics of a Wing With the Quarter-Chofd Line Swept Back
35P and With Aspect Ratio 3 and Taper Ratio 0.6. NACA AM L9C23, 1949. (U)7.

Fisher, L, R., and Michael, W. H., Jr.: An Investigation of the Effect of Vewtical-Fin Location and An. on Low-Speed Lateral Stability
Derivatives of a Semitailleas Airplane Modal. NACA AM L51A10, 1951. (U)2151,1960. (U)

7.3.2.24
-

-

--

--

--

-

U.

Bird. J. 0.. Lichmtenstein, J. H.. and Jaquet. 3. M.: Inv illation of the Influence of Fuseloae and Tal Surfaces on Low-Speed Stat~c
Stability and Rolling Characteristicsof a Svept-Wing Model. NACA TN 2741. 1952. (U)9.

Goodmn.n A.. and Thomaes. 0. F., Jr.: Effects of Wing Position and Fuselage Size on the Low-Speed Static and Rolling Stability
Choracteristics of a Delta-Win Model. NACA TR 1224. 1955. (U)10.

Letko. W., and Riley. 0. R.: Effect of an Uemgep Wing on tte Contribution of Unwa~pt-Tail Configurations to the Low-Speed Smaicand Rolling.tsblity Deri-etivesof a Midwhn Airplane 111del. NACA TN 2175,19W0. (U)11.

Lateral Static and Rotary Derivatives of a 1112-Scale Model of.a High-Speed Fighter Airplane
Willianm J. L: Moeameed and EstinuWith Unmuept Wings NACA RM LS3KOS. 1964. (U)12.

L~ockwood, V. E.: Effects of Swesep on the Darplng-ln-Roll Characteristics of Three Siweptback Wings Having an Aspect Ratio of 4 it
Tranioic Speads NACA RM 11.JI0.11911650. (U)13.

Jaciust,. M., and BrowserJ. 0.: Effectsof Various Oudboad and Central Fins on Low-Speed Static-Stabdl ty and Rolling Characteristics
of a Trlenguler-Winll 1Mode.
NACA RM L908,.1949. (U)14.

Broom,

C. E., ntanHeine. H4.&., Jr.: Priliminery Wind-Tunnel Teom of Triangular and Rectangular Wings in Steady RoNl at Mach

NWumbtsof 1.62 and 1.92. NACA TN 3740.1966 1U)

15.

Bed6.MJ.adl.
A.:onEfcto
uegIneeraWingilntefetso
vWnSwoopback on the Domilng in Rolt of
Untapered Wintp as Determined by Tedmniques Embioing Rodwet-Pope lld Vehidr. NACA RM L51 D25.1951. (U)I6& Blend, W, .M.Jr.:tFifd of Fuulop Interfarenee an the Domping in RoN of Delta Wlrgof Awect Ratio 4 In the Mach Numbter Range
BetewenO~and1.68sOlermlnei WkhP
ROd.1-Popumd Vehicle.NACA R L62t1E3. 1952, (UI
17.

Bland, W. K., Jr., aOd Sandash. C. A.- A Tedutiquit UtIllizIng Po"e.Prped Taom
Vehicles for the Measuremnent of the Daoming In
RoN1'
otf
kegMousfd MadebOWand nw Initial RAIks for Dela eOW UnaePt Tapered Whin. NACA RM L50024, oti0. (U)

7.3.2.2-5

r-

K>0

cl

0*Ca

00

In

I

0,

0
e4
C

N

_

N.
d

_

d

ho

La

in

o

oo

LL
>

o

M
C

0

000p

0

>.

Kc

-z

4<

7jw

C)L

0~0

7.3.2.2-6

-

1t

L

I.

.. II

MI

.

M

d

Ud

000000a0

.'.

S00

.

0

a

0

00aa0

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0000

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000

0

00

00

0

CI4

d

C.

0

0

0

0

0

"2-

7-30 2

7.3.2.2-7

rI

&I

i

0o

ca

7.3.2.2-8

_

_

_

A

ci

m

_

I

_

_

rI

o

ofo

0

0

0

d

"0

0

0

0oo

0

a

"

0

0

0

0

a,

0

VIA

.-

.

~

_

_

_

_

_

I,

_

_7.3.2.2__

_

C

K.

~

N

O~

~

(fl

a-I

_

_

I

_

I

0

_

__

d,
-.

-. jI

Ni0

I

.-

-

Liih

CiU

7.3.2.2-1

-

-

-4

Id

1%

L

N

6

m~

wiC N

k

0

m.

0D

0

%nCf

wO

IIL

0

-

00

0

0

0

Q

-4

cm

.u.U

00

z

-j

aU

c

eg

om
C.

N

0

0

0-

0

a-

0

0

0

00

o.3.2.0-10Ln %0

W

(0

in

~

Wn

a

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-

T

Ii

g~

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04

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U

7.32.-1

c

11CA

SUPERSONIC SPEEDS

I

•-

.8 .

. ..

# ~COT ALF.

- -1- --

. .

-- 0I COT A LE

: 1.0

C

(cp)d/b"
Cl•
0

I

(C,)

,4 ....

III

COT ALE > 1.0./0.

___ (Supersonic L.E. -/___-------.-------

I---

~ALL

TRIANGULAR

PLANFORMS

WINGS ONLY--

x\.2-.4--

1.0
d/b

FIGURE '7.3.2.2-13 EFFECI OF THE FUSELAGE ON ROLL DAMPING

7

.

*1
7.3.2.2-13

Revised June 1969

7.3.2.3 WING-BODY ROLLING DERIVATIVE Cnp
This section presents methods for estimating the wing-body contribution to the rolling derivative Cnp at
subsonic and supersonic speeds. This derivative is the change in yawing-moment coefficient with change
in wing-tip helix angle.
A. SUBSONIC
Experimental evidence indicates that the effect of the fuselage on the yawing moment due to rolling is
negligible throughout the angle-of-attack range up to the stall. Therefore, the method of Section 7.1.2.3
for estimating the wing-alone derivative at subsonic speeds is also applicable to wing-body configurations.
Available test data are limited to wing-body configurations with body diameters less than about 30
percent of the wing span. Therefore, the application of the wing-alone method is limited to valuesof d/b < 0.3.
The method consists of determining the value of Cp at zero lift and extrapolating the result to highlift coefficients by using experimental values of lift and drag at high lift coefficients. The method also
includes the effects of wing twist and symmetric flap deflection.
DATCOM METHOD
The variation of the wing-body rolling derivative Cn, with' lift coefficient at subsonic speeds is obtainedby using the method of Paragraph A of Section 7F1.2.3. In using this method experimental values ofwing-body lift and drag are used in evaluating the dimensionless correction factor K. This method islimited to configurations with body diameters less than about 30 percent of the wing span, i.e.,d/b < 0.3.
If experimental wing-body lift and drag data are nct available, wing-alone test data may be used todetermine the wing-body rolling derivative CnP. No attemipt should be made to extrapolate the zero-lift
value to higher lift coefficients using estimated lift and drag results. No known general method for
estimating the variation of drag coefficient will give results reliable enough to use in determining thecorrection factor for extrapolating the potential-flow values to higher lift coefficients;
The sample problem at the conclusion of Paragraph A of Section 7.1.2.3 illustrates the use of thismethod.
B. TRANSONIC
No generalized mrethoC is available in the literature for estimating transonic values of the rollingderivative Cnp. Furthermore, no known experimental results are available for this derivative at transonic
speeds.
C. SUPERSONIC
No generalized method is available in the literature for estimating the effect of the fuselage on therolling derivative Cnp at supersonic speeds.
For the purpose of the Datcom the fuselage effect is considered to be negligible for wing-bodyconfigurations with body diameters less than about 30 percent of the wing span. Therefore, for7.3.2.3-1

configurations with d/b • 0.3 the wing-body rolling derivative C,,p is estimated by the wing-alone
method of Paragraph C of Section 7.1.2.3.
No known experimental data are available for this derivative at supersonic speeds. Therefore, the validityof the Datcom application cannot be determined.
DATCOM METHOD"The wing-body rolling derivative Cn at supersonic speeds in the range of lift coefficients forwhich Cnp varies linearly with CL is obtained by using the method of Paragraph C of Section 7.1.2.3.
AThis method is limited to configurations with body diameters less than about 30 percent of the wing
span, i.e., df-b < 0.3.

S

7.3.2.3-2

.

•

.

,..

.-.

Revised September 1970

"7.3.3WING-BODY YAWING DERIVATIVES
7.3.3.1 WING-BODY YAWING DERIVATIVE Cy,

This section recommends methods for estimating the wing-body contribution to the. yawing
derivative. Cy, at subsonic speeds. However, at subsonic, transonic, and supersonic speeds nogeneralized methods are available for estimating the wing-body contribution to the yawing
derivative Cy,. This derivative is the change in side-force coefficient with variation in yawing
velocity.
A.. SUBSONIC
Experimental evidence indicates that the effect of the fuselage on the side force due to yawing is
negligible throughout the angle-of-attack range up to the stall. Therefore, the recommendationsof Section 7.1.3.1 for estimating the wing-alone derivative at subsonic speeds are also applicable
to wing-body configurations.
The recommendations of Section 7.1.3.1 consist of using available experimental data, whenapplicable, or the method of reference 1 to estimate CYr.
B. TRANSONIC
No generalized method is available in the literature for estimating transonic values of thewing-body contribution to the yawing derivative Cyr. Furthermore, there is a scarcity ofexperimental data for this derivative at transonic speeds.
C. SUPERSONIC
No generalized method is available in the literature for estimating supersonic values of thewing-body contribution to the yawing derivative Cyr. Furthermore, there is a scarcity ofexperirnen.•; data for this derivative at supersonic speeds.

REFERENCE
1.

Tall, T. A., and Queijo, M. J.: Approximate Relations and Chats for Low-Speed Stability Derivatives of Swept Wings. NACA
TN 1581, 1948. (U)73

7.3.3.1-1

7.3.3.2 WING-BODY YAWING DERIVATIVE C1,"This section presents a method for estimating the wing-body contribution to the yawing
S7.

derivative C(r at subsonic speeds. However, at transonic and supersonic speeds no generalized
methods are available for estimating the wing-body contribution to the yawing derivative Clr.
This derivative is the change in rolling-moment coefficient with change in the yawing velocity.
A. SUBSONIC
Experimental evidence indicates that the effect of the fuselage on the rolling moment due toyawing is negligible. Therefore, the method of Section 7.1.3.2 for estimating the wing-alone
derivative at subsonic speeds is also applicable to wing-body configurations. Available test data
are limited to wing-body configurations with body diameters less than about 30 percent of thewing span. Comsequently, the application of the wing-alone method to estimate the wing-bodycontribution to Cjr is limited to values of d/b < 0.3.
DATCOM METHOD:u'O,The variation of the wing-body yawing derivative CI, is obtained by using the method of Section
7.1.3.2. The method consists of determining the value of Clr at zero lift and extending this value
to high lift coefficients by using a serniempirical correction factor and test data for CIa. The
method also includes the effect of geometric dihedral, wing twist, and flap deflection. The
method is limited to configurations with body diameters less than about 30 percent of the wing
span; i.e., d/b < 0.3.
The sample problem presented at the conclusion of paragraph A of Section 7.1.3.2 illustrates theuse of this method.
B. TRANSONIC
No generalized method is available in the literature for estimating transonic values of thethe yawing
derivative
Cl,. Furthermore, there is a scarcity ofwing-body contribution
experimental
data for this toderivativeat transonic
speeds.
C. SUPERSONIC

F4

No generalized method is available in the literature for estimating supersonic values of thewing-body contribution to the yawing derivative Clr. Furthermore, there is a scarcity ofexperimental data for this derivative at supersonic speeds.

7.3.3.2-1

%•*

Revised September 1970

7.3.3.3 WING-BODY YAWING DERIVATIVE Cn 1
6This section presents a method for estimating the wing-body contribution to the yawing
derivative Cnr at subsonic speeds. However, at transonic and supersonic speeds no generalizedL-methods are available for estimating the wi-..body contribution to the yawing derivative Cn,.
This derivative is the change in yawing-mor .,A coefficient with change in the yawing-velocity
parameter and is commonly referred to as the yaw damping.
A. SUBSONIC
In general, the wing-fuselage contribution to damping in yaw is small in comparison to thevertical-tail contribution. The contribution from the fuselage depends upe' the relative size ofthe fuselage in comparison to the wing. For configurations in which the .aselage is large relative.
to the wing, the fuselage contribution is important. Fuselages having flat sides or having a
flattened cross section with the major axis vertical can also make a significant contribution(reference 1). Experimental data have also shown that a flattened cross-section fuselage with the
major axis horizontal can have negative damping in yaw at moderate and high angles of attack.
No generalized method exists in the literature for predicting the fuselage contribution to yaw
damping. Therefore, for the purposes of the Datcom, fuselage effects are considered to be
negligible. Consequently, the Datcom method of Section 7.1.3.3 for estimating the wing-alone
derivative at subsonic speeds is also applicable to wing-body configurations.
DATCOM METHOD
The estimated value of the wing-body yawing derivative C,, is obtained by using the method ofSection 7.1.3.3. In using this method experimental values of wing profile drag coefficient,evaluated at the appropriate Mach number, are used in conjunction with the lift coefficient.
These coefficients are used to evaluate the two major contributions to C,
namely, that
resulting from the drag due to lift and that resulting from the profile drag. The method does notaccount for the effects of wing flaps, wing twist, or wing dihedral.
The sample problem presented at the conclusion of paragraph A of Section 7.1.3.3 illustrates theuse of this method.
B. TRANSONIC
N3 generalized method is available in the literature for estimating transonic values of thewing-body contribution to the yawing derivative Cnr. Furthermore, there is a scarcity ofexperimental data for this derivative at transonic speeds.
C. SUPERSONICL

No generalized method is available in the literature for estimating supersonic values of thewing-body contribution to the yawing derivative Cr. Furthermore, there is a scarcity ofexperimental data for this derivative at supersonic speeds.

7.3.3.3-1

REFERENCE
1.

* -7.3.3.3-2.IL0'Campbell, J. P., and McKinney, M. 0.: Sumnmary of Methods for Calculating Dynamic Lateral Stability and Response and for
Estimating Lateral Stability Derivatives NACA TR 1098. 1962. (U)Revised September 1970

7:3.4 WING-BODY ACCELERATION DERIVATIVES
7.3.4.1 WING-BODY ACCELERATION DERIVATIVE CLS

The information contained in this section is for estimating the acceleration derivative CL& of
wing-body configurations at low angles of attack. In general, it consists of a synthesis of materialpresented in other sections; however, the method of application is new.
Datcom methods are based on the relatively simple results derived from slender-body theory in amanner similar to that used to predict the wing-body pitching derivatives. It is assumed that themutual interferences that occur between components for angle-of-attack variations, determined inreferences I and 2 and presented in Section 4.3.1.2, may be applied with reasonable accuracy tothe case of normal acceleration. This approach to the calculation of wing-body acceleration
derivatives has been applied with reasonable success by Walker and Wolowicz in reference 3.
A. SUBSONIC

Two methods are presented for determining the acceleration derivative CL& 1f a wing-bodyconfiguration, differing only in their treatment of the mutual interference effects.
DATCOM METHODS
Method I
For wing-body configurations similar to sketch (d) of Section 4.3.1.2, the acceleration derivative
CL&,j based on the area and mean aerodynamic chord of the total panel, is given by(C.)wl=[KW(B) + K ~

(?
E*

L)(L

().

7.3.4. 1-awhere(CL&4 is the contribution of the exposed panel to the acceleration derivative CL&, obtained
from Section 7.1.4. 1. (See Section 4.3.1.2 for the definition of exposed surfaces.)(CL&) is the contribution of the body to the acceleration derivative CL&, obtained from
Section 7.2.2.1.
The remaining terms are defined in paragraph A of Section 7.3.1.1.
Method 2
For wing-body configurations similar to sketch (c) of Section 4.3.1.2, the acceleration derivative
CL&, based on the area and mean aerodynamic chord of the total panel, is given by
7.3.4.1-1

K(fl) (La)W(CL&)W+ (CL)JO

7.3.4. 1-b

C

where (Cyj)w is the contribution of the total panel to the acceleration derivative CL&, obtained
from Section 7.1.4.1, and the remaining terms are defined in paragraph A of Section 7.3.1.1 andmethod 1 above.

Sample Problem
Method 1
Given: Same configuration as sample problems of paragraph A, Sections 7.3.1.1 and 7.3.1.2. Someof the characteristics are repeated below.
The following ratios based on total panel dimensions:
Se
S

c
c --

- 0.7113

0.845

Sb
S-

0.0963RB = 6.363-

Additional Characteristics:dM

p=0.80

0.60

A

5.0

-=

0.157

B= 7 7 .2 ft

Compute:Step 1.

Acceleration derivative CL . for exposed panel (Section 7.1.4.1)

(Ž±)

0.560

=

(sample problem, paragraph A, Section 7.3.1.1)
4.50 per rad(CILa)

PAC

4.0

C
CL
irA/2

CL(g)
7.3.4.1-2)=

0.140 per rad-1.718 per rad(figure 7.1.4.1-6)

(L&.

1.5

(CLj) + 3 CL(g)(equation 7.1.4.1-a)- (1.5) (0.560) (4.50) + 3(-1.718)= 3.780 - 5.154-1.374 per rad (based on S, Z)
Step 2.

Wing-body interference factors (Section 4.3.1.2)"Kw(B) " 1.125(sample problem, paragraph A, Section 7.3.1.1)
KD(w) = 0.215
Acceleration derivative CL& for body (Section 7.2.2.1)

Step 3.
Va= 2615.1 cu ft(sample problem,

Sb = 39.88 sq ft

paragraph A,
Section 7.3.1.1)

1.894 per rad (based on Sb)(CLC)B

VB\

2615.10.849(39.88) (77.20)

"

- 2-

(equation 7.2.2. 1-a)-2(1.894) (0.849)-

3.216 per rad (based on S Re)

Solution:*(~~CL&)"IKW(Bj) + KB~,I(-w

..

(CL.)

(CLi)B~~(equation 7.3.4.1-a)= (1.125 + 0.215) (0.7113) (0.845) (-1.374) + (3.216) (0.0963) (6.363)
7.3.4.1-3

--1.107 + 1.971= 0.864 per rad.

(based on the area and mean aerodynamic criord of the total
panel)

B. TRANSONIC
The aerodynamic interference effects for slender wing-body configurations are relatively
insensitive to Mach number; consequently, the slender-body interference factors of Section
4.3.1.2 should give reasonable results. For nonslender configurations transonic interference
effects can become quite large and sensitive to minor changes in local contour.

-

K>".

DATCOM METHODS
It is recommended that the methods presented in paragraph A for estimating the acceleration
derivative CL& of a wing-body configuration be applied to the transonic speed range. Care should
be taken to estimate the contributions of the lifting panel and body at the appropriate Mach
number. The interference factors should be obtained from paragraph C, Section 4.3.l.2.
C, SUPERSONIC
The information included in the Datcom accounts for most of the mutual interferences that
occur between components of a wing-body configuration at supersonic speeds. These interference
effects are accounted for by the slender-body interference factors of Section 4.3.1.2.
DATCOM METHODS,"-

The methods presented in paragraph A for estimating the acceleration derivative CL& of awing-body configuration are also applicable to the supersonic speed range. Care should be taken
to estimate the contributions to the lifting panel and body at the appropriate Mach number.

Sample Problem"Method I
Given: Same configuration as sample problems of paragraph C, Sections 7.3.1.1 and 7.3.1.2, andparagraph A of this section.
d
M =1.4/3=0.98-- =0. 157
b•Compute:
Step 1.

7.3.4.1-4

Acceleration derivative CN . for exposed panel (Section 7.1.4.1)

'3 cot ALE

1.225 (supersonic leading edge)=(sample problem, paragraph C,
Section 7.3.1.1)

PAC= 4.90"cot-1'

cot ALE)=ff[(CN.)]

2)J

9[(CN

SQT•

39.23°(figure 7.1.4.1-11a)

0

=4.00

(figure 7.1. 4 .1-I1c)[(CN&)2.

- 4.08 per rad(CN)e=() [(CN ),]-

N /e .

N&

S\o.4/(o)-(-)[(CN.)](equation 7.1.4.1-c)

1 [ CN1((4.08)= -4.25 per rad (based on S
Step 2.

Wing-body interference factors (Section 4.3.1.2)

KW(B)

=

1.125

KB()

=

0,181

(sample problem, paragraph C, Section 7.3.1.1)

Step 3.
VB
Q-

Acceleration derivative CN. for body (Section 7.2.2.1)

0.849(sample problem, paragraph A)

b •B

(CN)

=

2.74 per rad(sample problem, paragraph C, Section 7.3.1.1)

7.3.4.1-5

(CNa)B2(CN )BV"(equation 7.2.2.1-a)= 2(2.74) (0.849)= 4.653 per rad (based on Sb2B)
Solution:[KW(D) + KB )(NC &)IB/(W /\T/+ (CN)

Ts/(equation 7.3.4,1-a)=(1.125 + 0.181) (0.7113) (0.845) (-4.25) + (4.653) (0.0963) (6.363)=-3.336 + 2.851=--0.485 per rad (based on the area and mean aerodynamic chord of the total
panel)REFERENCES
1.

Pitt$, W., Nielsen, J., and Kaattari, G.: Lift and Center of Pressure of Wing.Body-Tail Combinations at Subsonic, Trarnsonic,
end Supersonic Speeds. NACA TR 1307, 1959. (U)2.

Spreiter, J.: The Aerodynamic Forces on Slender Plane-and Cruciform-Wing and Body Combinations. NACA TR 962,
1950. (U)3.

Walker, H., and Wolowicz, C.: Theoretical Stability Derivatives for the X-15 Research Airplane at Supersoni(. andHypersonic Speeds Including a Comparison with Wind-Tunnel Reslits. NASA TM X-287, 1960. (U)L

7.3.4.1-6
7

RevAsd Seplember 1970

7.3.4.2 WING-BODY ACCELERATION DERIVATIVE C,..

S ,wing-body

The information contained in this section is for estimating the acceleration derivative Cm& ofconfigurations at low angles of attack.
The Datcom methods are based on the same assumption that was made in regard !0 theacceleration derivative CL& of a wing-body configuration, and the general discussion of Section
7.3.4.1 is directly applicable here.
A. SUBSONIC
Two methods are presented for determining the accelerattivn derivative Ca& of a wing-bodyconfiguration, differing only in their treatment of the mutual interference effects.
DATCOM METHODS
Method 1
For wing-body configurations similar to sketch (d) of Section 4.3.1.2, the acceleration derivative
Cm&, based on the area and the square of the mean aerodynamic chord of the total pan',1 andreferred to a moment center at the assumed center of gravity, is given by[Kw(B) + K(Cmi)W

)

-/I

C

)

(Ci&) + (C/)

7.3.4.2-awhere(Cm,&)•.

is the contribution of the exposed panel to the acceleration derivative Cm&,obtained from Section 7.1.4.2. (See Section 4.3.1.2 for the definition of exposed
surfaces.)(Cm &)Bis the contribution of the body to the acceleration derivative Cm,&obtained from
Section 7.2.2.2.

The remaining terms are defined in paragraph A of Section 7.3.1.1. Moment transfer calculations
are included as an integral part of the wing- and body-derivative estimation methods of Sections
7.1.4.2 and 7.2.2.2, respectively.
Method 2
For wing-body configurations similar to sketch (c) of Section 4.3.1.2, the acceleration derivative
Cm6, based on the area and the square of the mean aerodynamic chord of the total panel andreferred to a moment center at the assumed center of gravity, is given by(m-(C &)+w(Cm &)(S)(

7.3.4.2-b)-7.3.4.2-1

where (Cm&)w is the coniribution of the total panel to the acceleration derivative Cm,, and theremaining terms ate defined in paragraph A of Section 7.3.1.1 and method I above. Momenttransfer calculations are included as an integral part of the wing- and body-derivative estimation ,.
methods of Sections 7.1.4.2 and 7.2.2.2, respectively.
'1

Sample Problem
Method I
Given: Same configuration as sample problems of paragraph A, Sections 7.3.1.1, 7.3.1.2, and7.3.4.1. Some of the characteristics are repeated below.
The following ratios based on total panel dimensions,
/

.sc

c.

2

1.50

0.7140.0.7113/R)-Sb = 0.0963= 40.49

Additional Characteristics:
dJ

m

A,

M
- 0.60

5.0
xM

0.80

b

--

0.157

Ra

77.20 ft

= 43.70 ft (moment center at ý/4 of total panel)Compute:Step I.
X•.

Acceleration derivative Cm. for exposed panel (Section 7.1.4.2)
0

0.560

*=

(sample problem, pararaph A, Section 7.3.1.!).(xc.•= 0.408(C.)*7.3.4.2-2

4.50 per rad

(jgi)c(0.408) (1.50)

X7e~7

0.612

=-1.374 p(, rad (sample problem, paragraph A, Section 7.3.4.1)
(L6~)
I3Ae

=4.0

gA2C
=0.0767 per rad

Cm0 (g) =0.941

p'er rad27(Cm I)p

/a\

-

--

(figure 7.1.4.2-8)

27

(CL)

3C 0 g(equation 7.1 .4.2-b)(0.560)2 (4.50) + 3(0.941)-2.381 + 2.823==0.442 per rad

(Cm.

=

C~

(C+

(equation 7.1.4.2-a)

0.442 + (0.612) (-1.374)-0.399 per rad (based on S' E 2)

Step 2.

Wing-body interference factors (Section 4.3.1.2)

Kw)

1,125

KBw

0.215

(sample problem, paragraph A, Section 7.3. 1.1)

7.3.4.2-3

Step 3.

Accel1eration derivative Cmfor body (Section 7.2.2.2)=0.849

Sb B

(I

=C

(sample problem, paragraph A, Section 7.3.11.2)

0.434

-B

44.48 ftSC11)

0.794 per rad2(Cm)B

(Cm

(-x)
VB

c

SbB

B

m

(equation 7.2.2.2-a)AI.L108944.48 - 43.70\
720=2(0.794)-0.415]1

per rad (based on Sb B2 )=-0.0328

Solution:(cm.)[KW(B)+KBwI(m)+()()(\(equation 7.3.4.2-a)=(1.

=-0.272

125 + 0.215) (0.7113) (0.7140) (-0.399) + (-0.0328) (0.0963) (40.49)
-

0.128

7.3.4.2-4

IL.

r

-

-0.400 per rad (based on the area and the square of the mean aerodynamic
chord of the total panel and referred to a moment center at Z/4of the total panel)

B. TRANSONICThe comments of paragraph B of Section 7.3.4.1 are directly applicable here.
DATCOM METHODS
It is recommended that the methods presented in paragraph A for estimating the acceleration
derivative Cm& of a wing-body configuration be applied to the transonic speed regime. Careshould be taken to estimate the contributions of the lifting panel and body at the appropriate
Mach number. The interference factors should be obtained from paragraph C, Section 4.3.1.2.

C. SUPERSONICThe comments of paragraph C of Section 7.3.4.1 are directly applicable here.
DATCOM METHODS
The methods presented in paragraph A for estimating the acceleration derivative Cm& of awing-body configuration are also applicable to the supersonic speed range. Care should be taken
to estimate the contributions of the lifting panel and body at the appropriate Mach number.
Sample Problem
Method 1
Given: Same configuration as sample problems of paragraph C, Sections 7.3.1.1, 7.3.1.2, and7.3.4. 1, and paragraph A of this section
M

r,

(3

1.4

d

0.98

0.157

b

"Compute:
Step 1.

Acceleration derivative Cm.

for exposed panel (Section 7.1.4.2)(3cot ALE = 1.225 (supersonic leading edge)cot 1 (j3 cot ALE)..

PAO

39.230(saniple problem, paragraph C,
4Section 7.3.1.1)

4.90

7.3.4.2-5

Ix .

(----)=0.612(CN.)=ýrcfnC

1, e=-4.25 per rad

9.05(Cm~e)1=

Fl[(Cm.)]= -4.52[(C(sample problem, paragraph A)(sample problem, paragraph C, Section 7.3.4.1)(figure 7 .1. 4 . 2 -13a)

9.235 per rad)(figure 7.1.4.2-13c)-4.612 per rad

(" =

/M2\

)(C)]

+

M

+ 1) [Cm&)]C.4.2-e)

(1.42

/1.4\

~0.98
+ 1
I (9.235) + 0.982
0-98

-09

=

(equation(-4.612)18.85 - 14.03= 4.82 per rad(Cm)(Cm.')\-(-!.

(CL.)

4.82 + (0.612) (-4.25)(equation 7.1.4.2-a)(CN. is assumed to be equal to CLa)

2.219 per rad (based on Se• 2 )44

Step 2.

Wing-body interference factors (Section 4.3.1.2)

Kw(D)= 1.125](sample problem, paragraph C, Section 7.3.!.1)

KB(w) = 0.181

7.3.4.2-6

4-

Acceleration derivative Cm. for body (Section 7.2.2.2)

Step 3.

= 0.849-

sb2(sample problem, paragraph A, Section 7.3.1.2)= 0.434-44.48 ft
"xc=

(Cm

)

= 1.017 per rad(sample problem, paragraph C, Section 7.3.1.2)

VB(Cm.j)=

2(Cm,)

x~n

(Cx

/R

SA

(equation 7.2.2.2-a)

VB

' ~x

44.48 -43.70)1'[0.89, - 77.'20
2(.07)(0.849)
0.434 - 0.849= 2(1.017)

2.034

r(0.849) (0.0101)-0.415= -0.0420 per rad (based on Sb s 2)

"Solution:(Cm ),.

= IKW,, ,+ K,, S

. + (C m.)
(C )j

a

(equation 7.3.4.2,a)= (1.125 + 0.181) (0.7113) (0.7140) (2.219) + (-0.0420) (0.0963) (40.49)= 1.472 - 0.164= 1.308 per Tad(based on the area and the square of the mean aerodynamic
chord of the total panel and referred to a moment center at Z/4of the total panel)
7.3.4.2-7

Re4ed September 1970

7.4 WING-BODY-TAIL DYNAMIC DERIVATIVES
7.4.1 WING-BODY-TAIL PITCHING DERIVATIVES
7.4.1.1 WING-BODY-TAIL PITCHING DERIVATIVE CLq

The information contained in this section is for estimating the nondimensional pitching derivative
CLq of wing-body-tail combinations at low angles of attack. This derivative represents the lift
due to pitching velocity at a constant angle of attack and is defined as
_

CL

CLL
q

aCL

2-•E

In general, the methods presented consist of a synthesis of material presented in other sections,although some new information is presented.
The derivative is presented in a manner similar to that used in reference I to calculate the lift ofa wing-body-tail combination. The complete derivative is the sum of contributions of individualcomponents, treated as isolated surfaces or bodies, and mutual interference effects. The mutualinterference effects are assumed to correspond to those due to angle-of-attack variations,established in references 1 and 2 and presented in Section 4.3.1.2.
The horizontal-tail contribution to the derivative CLq is based on the assumption that
instantaneous forces on the tail correspond to the instantaneous angle of attack. This assumption
is also the basis used in estimating the horizontal-tail contribution to the derivative Cmq in
Section 7.4.1.2. The effect of this assumption is presented in numerous aerodynamic texts, forexample, reference 3. The effects of CLq are generally small and frequently neglected in dynamic
analyses.
A. SUBSONIC
Two methods are presented for determining the pitching derivative CLq of wing-body-tailcombinations, differing only in their treatment of the effect of the flow field of the forwardsurface on the aft surface.
DATCOM METHODS
Method 1. (b'/b" > 1.5)
For configurations in which the span of .the forward surface is large compared to that of the aftsurface, the following approach can be used. This mhethod is to be used when the ratio of theforward-panel span to the aft-panel span is 1.5 or greater. The equation of the nondimensional
pitching derivative CLq of a wing-body-tail configuration, based on the area and mean
7.4.1.1-1

aerodynamic chord of the total forward panel and referred to a moment center at the assumed
center of gravity or center of rotation, is given by
+ K
KwB)

(CL

7.4.1. -a

where the primed quantities refer to the forward panel, the double-primed quantities refer to theaft panel, and the subscript e refers to the exposed panel. (See Section 4.3.1.2 for the definitionof exposed surfaces.)(CLq)WB

is the contribution of the wing-body configuration to the pitching
derivative CL., obtained from Section 7.3.1.1.

[Kw(B) + KB(w)]"is the appropriate wing-body interference factor obtained from
Section 4.3.1.2 for the aft panel.

(CLoe"

is the lift-curve slope of the exposed aft panel, obtained from Section
4.1.3.2.
is the ratio of the exposed aft panel planform area to the total

SI
xc.:.- x"

pianform area of the forward panel.
is the distance parallel to the longitudinal axis between the moment
reference center (usually the vehicle center of gravity) and thequarter-chord point of the mean aerodynamic chord of the total afthorizontal panel.
This parameter is approximated in Section 4.5.2.1 as x-x"
- "C(equation 4.5.2.1-e) where xc 4 and 2" are defined in figures 4.5.2.1-7a
and -7b.

q"

q
qoo

is the average dynamic-pressure ratio acting on the aft surface,obtained from Section 4.4.1.

Method 2. (b'/b" < 1.5)
For configurations in which the span of the forward surface is approximately equal to or lessthan that of the aft surface, the vortex shed from the forward surface interacts directly with theaft surface and the resulting interference effects must be accounted for in the tail terms. Thismethod is to be used when the ratio of the forward-panel span to the aft-panel span is less than
1.5. The equation for the nondimensional pitching derivative CLq of a wing-body-tail configuration, based on the area and mean aerodynamic chord of the total forward panel and referred toa moment center at the assumed center of gravity or center of rotation, is given by
7.4.1.1-2

x
CL q = (CLq)

-x

+ 2.

{[Kw(B) + KBLe)]Se°q"((Le.-.
('))e+ (CLO))W"(v)

7 .4.1. I-b

The parameters in the first two terms on the right-hand side are the same as those in equation
7.4.1.1-a. The last term represents the effect of the forward-panel vortices on the aft panel and isgiven by=22-(CLC()w(.(L)
L

L

(

.2

)

'W'(W"))

7.4.1.1-c

whered' and d"

are the body diameters at the midchord points of the MAC of the forward andaft surfaces, respectively.

b"#

is the span of the total aft panel.

(-!2!-)

is obtained from figure 4.4.1 -71 as a function of forward-panel geometry.

"lvWW(W")

is the vrtex interference factor, obtained from Section 4.4.1.

Ae"is the aspect ratio of the exposed aft panel.

(CL4Ceis the lift-curve slope of the exposed forward panel, obtained from Section

4.1.3.2 (per radian).
(Cte)e°'-is the lift-curve slope of the exposed aft panel, obtained from Section 4.1.3.2(per radian).
The remaining terms are defined in method 1 above.
The quantities (CL.)e' and (CLje" in the last term of equation 7.4.1.1-b must be expressed inradians. If the result of this term is desired per degree, the conversion must be applied after theterm is evaluated.
Because of the similarity of the two methods a sample problem of method 2 is not included.
However, evaluation of the term (CLa)w"(v) for a wing-body-tail configuration is presented in
Section 4.5.1.1.
7.4.1.1-3

Sample Problem
Method I
Given:

An ogive-cylinder-body--triangular-wing and triangular-tail configuration. The wing-bodyconfiguration is the same as that of the sample problem of Sections 7.3.1 and 7.3.4. Manyof the characteristics are repeated below.

L E.

rDT
PAAt

ce~~

b

EXPOSEDPANELS

Although a tail has been added to the wing-body configuration of Sections 7.3.1 and 7.3.4, forthe sake of simplicity the vehicle center of gravity is taken at *'/4. Therefore, x.g - x" = 2",
the longitudinal distance from the quarter-chord point of the MAC of the forward surface to thequarter-chord point of the MAC of the aft surface.

S

7.4.1.14

*Wing Characteristics:
Total Panel

Exposed PanelA' =5.0

A' =0

Ae'

=5.0

A

=E
=

1"

=

0

be'

=

i=

0

cr

=

18.20 ft

38.670b'= 45.50 ft'S'= 414.0sqft

38.37 ft

Xe'

=0

Se'

=

294.50sqft
15.375 ft

Cr'=

Z' = 12.133 ft

de'= 10.25 ft

NACA 66-206 airfoil section

Horizontal-Tail Characteristics:
Total Panel=4.0

Exposed PanelX" = 0

A,"

=4.0

Xe

=0

A"LE = 450

r"

be"

= 12.788 ft

Se"= 40.90 sq fti"= 0c," = 9.958 ftb"= 19.915 ft

V" = 6.638 ft

A"

= 0

A,= 26.560= 6.394 ft
e"

4.263 ft

S" = 99.16 sq ft
NACA 66-206 airfoil section

Body Characteristics:
RN = 25.0 ft

B = 77.20 ft

d = 7.127 ft

The following ratios based on total forward-panel dimensions:
R,,

Se"

"-2.26

S4- 0.0988

71
7.4.1.1-5

•

~~

~-

.-.-.

.--

.

- .

+

,,

_

,'.

-

,

.

-

-

.

.

i

Additional Characteristics:
5.16 x 107 (based onhH

7.18 ftd"-2.794R

k"

27.425 ft

M

0.60

Sea levelx

18.32 ft*

(3

0.80

Smooth surfaces

4..'

Swet

a' =40

.

--

2

Sref
Compute:Step 1.

Wing-body CLLq (Section 7.3.1.1)(CLq)w"+-

KB(W)Ii\'•'=Kw(B)

JWH

K

j-C

()ý

Lq)€'

+(CLq)B
+

s

e)(Cq

-

(equation 7.3.1 .1-a)-'\]\~q)WB
Step 2.

4.467 per rad (sample problem, paragraph A, Section 7.3.1.1)

Lift-curve slope for the exposed horizontal tail panel (Section 4.1.3.2)

6.19 per rad(table 4.1.1-B)

CRa

S-2=

0.985

",+,A,"[lA".1/
2 + tan2 A" 112
2
X.mL
tan"

"A" = 1.00

A~CL.).

4.0
41/[0.64 + 0.25

3.83

0.985

(figure 4.1.3.2-49)

4.00 per rad

I.I*x is the longitudinal distance measred from the wing root trailing edge aft. For this problem it is taken equal to k2 defined in
Section 4.4.1.

7.4.1.1-6
L--,,

Tail-body interference factors (Section 4.3.1.2)

Step 3.

d°

d

7.127
7.12

= 0.358b

19.915KW(.)"- 1.315

Kg(w)" -0.550(fgure 4.3.1.2-10)

Dynamic pressure ratio (Section 4.4.1)

Step 4.

"Obtainvalue at a' - 40
CL'0.391 (sample problem, paragraph A, Section 7.4.4.1)

-

=

e

1.62 CLA(equation 4.4.1-k)

1.62 (0.391)
5T= 2.3°-==tan-4tan -t+yee-a'=18.32 (0.3581) -Cf = 0.00236

CDo

= Cf

"L+

) +

cos A(t/€C).
.RI-' = 1.14

21.40 (mee sketch (d),ction 4.4.1)

21.4+2.3-4 a 19.70

z - x tan (- + e - a')--18.32i~x(equation 4.4.1 -)6.56(figure 4.1.5.1-26)

1+ L(L) + 100 (10) REL3)] = 1.072(equation 4.1.5.14)(figure 4.1.5.1-2 ka)= cosA.4S. = cos 23.76* = 0.915(figure 4.1.5.1-28b)7.4.1.1-7

CD=(0.00236)(1.072) (1.14) (2)2.42 (CD/Aq=0.00577

)/2

(Do)

(equation 4.4. 1-rn)+ 0.302.42 (0.00577)1/2
1.51 + 0.30

(5- + 0.715ý (equation 4.4.1-j)

0.68

UD

. 0

=

=0.68

*z

ý0.00577 (1.51 + 0.15)

6.56/12.133/Z=zw zw /Z

=0.0666=81

0.0666(equation 4.4.1-n)=(-.COS2~q,q

I,=(0.

(8.12) (57.3)

10291) C2[(.4=(0.102)(0.9834)2

0-.099

*

-=

1-

q.

-L.

(equation 4.4.1-o)q=1-0.099=0.901

Solutionl:

CLq(CL)~ + 2 [KW(B) + KDW]()(q~~. x).i(LJW

F(equation

7.4.1.1 -a)

J)J

=

4.467 + 2(1.315 + 0.550) (0.0988) (2.26) (0.901) (4.00)"4.467 + 3.002=

7.47 per rad (based on the area and mean aerodynamic chord of the total panel and"referred to a moment center at F'/4)

B. TRANSONIC-*

No accurate methods are available for estimating the characteristics of isolated panels or the
dynamic-pressure ratio in the transonic speed regime. The aerodynamic interference "KI" factors
for slender configurations are relatively insensitive to Mach number; however, for nonslenderconfigurations transonic interference effects can become quite large and sensitive to minorchanges in local contour.
DATCOM METHODS
It is recommended that the methods presented in paragraph A above be applied directly to thetransonic speed regime. Care should be taken to estimate the various parameters of equations
7.4.1.1-a and -b at the appropriate Mach number. The interference "K" factors should beobtained from paragraph C, Section 4.3.1.2.
C. SUPERSONIC
The information included in the Datcom accounts for most of the mutual interferences that

occur between components of wing-body-tail configurations at supersonic speeds.
DATCOM METHODS
The methods presented in paragraph A above are also applicable to the supersonic speed range.
Care should be taken to estimate the various parameters of equations 7.4.1.1-a and -b at theappropriate Mach number. Method 3 of paragraph C of Section 4.4.1 should be used to evaluate
the last term of equation 7.4.1.1-b.

Sample Problem
Method 1"Given: Same configuration as sample problem of paragraph A. Some of the characteristics arerepeated below.
The following ratios based on total forward-panel dimensions:=2.26£,=2.26-

So."S'

0.09884.
7.4.1.1-9

Additional Characteristics:c•' = 40

M = 1.40

0.98

A"

Sea level

1.2 x 108 (based on C)

= 450

"

16
A' = 5.0

K -

3 (page 4.1.5.1-1.6)

Smooth surfaces

cg at Z'/4
Compute:2

Step 1.

Wing-body CCLq (Section 7.3.1.1)(cL 9 )WB

KW(B) + KB~)(w='('(~~'+(~)(equation 7.3.1 .1-a)(C)

Step 2.

3.977 per rad (sample problem, paragraph C, Section 7.3.1.1)=

Lift-curve slope for the exposed horizontal-tail panel (Section 4.1.3.2)___-

tan ALESA"tan

=0.98

4.0

A"E

4.025

LE

A'tanA(figure 4.1.3.2-56a)'04.025 per radC )e"Stp 3.

Tail-body interference factors (Section 4.3.1.2)

d"0.358 (sample problem, paragraph A)

b"

1.31'

""W(B)"
O3d"
Cr

7.4.1.1-10-(figure 4.3.1.2-10)

0.98 (7.127)
--

6.394

1.092

cot A"E

0.98

=

=

2.85

(figure 4.3.1.2-11a)0.98(4.025) (1) (1.794)

1)

+

2.85

7.076

KB(w)

-- 1

+ I)

N)" ("

KB(W)"==7.076

0.4028

Dynamic pressure ratio (Section 4.4.1)

Step 4.

Obtain at a'

Viscous flow field= 40(table 4.1.1-B)

a0 ' =-1.60

CN

0.07-12 per deg

CN(o'- a')Swc
C -wetCff S~eCD f= 0.071214 - (-1.6)1

=

0.399

(figure 4.1.5.1-26)

Cf = 0.0018

C(sample problem, paragraph C, Section 7.3.1.1)(equation 4.1.5.1-i)= (0.0018) (2) = 0.0036(0.98) (1.25) = 1.225

j cot ALE

(supersonic leading edge)

bw

K

CD

-

2

-f

-)
\

(equation 4.1.5.1-k)

16-3(0.98) (0.06)2 (1)

;
-

CD

0.0196

=C
0

Df

+C

(equation 4.1.5. 1-h)DW

" 0.0036 + 0.0196 = 0.0232
i~f7.4.

1.1-112.42 (co0)1/2(equation 4.4.1-m)

0.30
C

2.42(0.0232)1/2=

N= 0.204

1.51 + 0.30
Wz-

=0.68

(equation 4.4.1 -j)[CDO (X +0.15)

0.68 v/(0.0232) (1.51 + 0.15)
0.13341.62 CN(equation 4.4.1-k)

A
iTA'

e

(1.62) (0.399)
51r=(57.3)

2.360

y =21.40(sample problem, paragraph A)y,+e-a' = 21.4+2.36-4 = 19.76'

z = x tan (-' + e - a')(equation 4.4.1-9)= 18.32(0.3592) = 6.581
z

z/

zw

z/c"
w

q

-

-

6.581/12.133

q- (k-/ ) co522z
2Zw)=(0.204) cos 2= 4.070.1334(equation 4.4.1-n)

3k.-4)(4.07) (57.3)(0.204) (0.9943)2 = 0.20

7.4.1.1-12

- .. T-

..

'

..

.-'

.5

"

q

'

'

"

"

"

i

%

'

i "

• - .

""

"'

A
1 - L--(equation 4.4.1-o)

1.0-0.20 = 0.80

Solution:

CLq

,,(C_

=(CL)W+ 21KW(B) + K (W
!I"(~)(~-q,5 -X).qNe.

C)(equation 7.4.1 .1-a)= 3.977 + 2(1.315 + 0.4028) (0.0988) (2.26) (0.80) (4.025)=

3.977 + 2.47

= 6.447 per rad(based on the area and mean aerodynamic chord of the total panel
and referred to a moment center at E°/4)

REFERENCES
1.

Pitts, W., Neilsn. J., end Keattwai, G.: Lift and Center of Presure of Wing.Body-Tail Conbinations at Subsonic, Transonic,
and Supersonic Speeds. NACA TR 1307, 1910. (U)2.

Sprciw. J.: The Aerodynamic Forces on Slander Plarne-end CruciformWing and Body Combinhtions, NACA TR 962,

195). (U.)
3.

Et#in, 0.: Dynamics of Flight. John Wiley and Sons, Inc., New York, 1959. (U)

7.4.1.1-13

Revined September 1970

7.4.1.2 WING-BODY-TML PITCHING DERIVATIVE Cmq
The information contained in this section is for estimating the nondimensional pitching derivative
Cm q of wing-body-tail combinations at low angles of attack. The derivative Cmq is the change inpitching-moment coefficient with varying pitch velocity and is commonly referred to as thepitch-damping derivative. It is expressed as

acm
C=
mq -

This derivative is very important in longitudinal dynamics, since it represents a major portion of
the damping of the short-period mode for conventional aircraft.
The Datcom methods are based on the same assumptions that were made for the total pitching
derivative CLq of wing-body-tail combinations, and the general discussion of Section 7.4.1.1 isdirectly applicable here.
A. SUBSONIC
Two methods are presented for determining the pitching derivative Cmq of wing-body-tailcombinations, differing only in their treatment of the effect of the flow field of the forwardsurface on the aft surface.

DATCOM METHODS
Method 1. (b'/b" > 1.5)
For configurations in which the span of the forward surface is large compared to that of the aftsurface, the following approach can be used. This method is to be used when the ratio of theforward-panel span to the aft-panel span is 1.5 or greater. The equation for the nondimensional
pitching derivative Cmq of a wing-body-tail configuration, based on the area and the square of
the mean aerodynamic chord of the total forward panel and referred to a moment center at theassumed center of gravity or center of rotation, is given by

M
q

Cmq=(

q)~-

2 1KwB

KwJ

+S'c
L(xu

q")
()

(W)e
C)(
L~

7.4.1.2-a

where the primed quantities refer to the forward panel, the double-primed quantities refer to theaft panel, and the subscript e refers to the exposed panel. (See Section 4.3.1.2 for the definitionof exposed surfaces.)
7.4.1.2-1

'74n.

is the contribution of the wing-body configuration to the pitching derivative
Cmq, obtained from Section 7.3.1.2.

(Cmq)WB

The remaining terms are defined in paragraph A of Section 7.4. 1. 1.
71

Method 2. (b'/b" < 1.5)
For configurations in which the span of the forward surface is approximately equal to or lessthan that of the aft surface, the vortex shed from the forward surface interacts directly with theaft surface, and the resulting interference effects must be accounted for in the tail terms. Thismethod is to be used when the ratio of the forward-panel span to the aft-panel span is less than
1.5. The equation for the nondimensional pitching derivative Cm q of a wing-body-tail configuration, based on the area and the square of the mean aerodynamic chord of the total forward
panel and referred to a moment center at the assumed center of gravity or center of rotation, isgiven byCm-=m(Cmq) w

[Kw(B)(CL.

KB(w)]+ (CL)"

7.4.1.2-b

All the above terms are defined in paragraph A of Section 7.4.1.1 and method 1 above.
Because of the similarity of the two methods, a sample problem for method 2 is not included.
However, evaluation of the term (CLa)w"(U) for a wing-body-tail configiiration is presented in
Section 4.5.1.1.

Sample Problem
Method 1"Given: Same configuration as sample problem of paragraph A, Section 7.4.1.1. Some of the"characteristics are repeated below.
The following ratios based on total forward panel dimensions:
S'"

2"
S•

C -~. =2.26

•. ,

fS'

0.0988= 0.80a'

Additional Characteristics:
I"IM

7.4.1.2-2

=

0.60

40

cg at '/4

Compute:
Step 1.

(Section 7.3.1.2)

Wing-body Cmq

S(C) per rad (sample problem, paragraph A, Section 7.3.1.2)=-4.021

Step 2.

Lift-curve slope for the exposed horizontal tail (Section 4.1.3.2)(CL)"

Step 3.

=4.0 per rad(sample problem, paragraph A, Section 7.4.1.!)

Tail-body interference factors (Section 4.3.1.2)

Kw(B)"

1.315

K

0.550

(sample problem, paragraph A, Section 7.4.1.1)
LKW)"

-Step

1

4.

Dynamic pressure ratio (Section 4.4.1)

q.

q 0.90)(sample problem, paragraph A, Section 7.4..1.1)(M q)WB

2fW

Solution:

Cmq

M
B BWJ

S)x

5

(equation 7.4.1.2-a)2(1.315 + 0.550) (0.0988) (2.26)2 (0.901) (4.0)--4.021--4.021 - 6.78"

-10.80 per radri-(based on the area and the square of the mean aerodynamic
chord of the total forward panel and referred to a moment center
ata

4.B.

'/4)

TRANSONICThe comments of paragraph B of Section 7.4.1.1 are directly applicable here.
7.4.1.2-3

-

,--

-

-

-

DATCOM METHODS
It is recommended that the methods presented in paragraph A above be applied to the transonic
speed regime. Care should be taken to estimate the parameters of equations 7.4.1.2-a and -b at
the appropriate Mach number. The interference "K" factors should be obtained fromparagraph C, Section 4.3.1.2.
C. SUPERSONICThe comments of paragraph C of Section 7.4.1.1 are directly applicable here.
DATCOM METHODS
The methods presented in paragraph A above are also applicable to the supersonic speed range.
Care should be taken to estimate the parameters of equations 7.4.1.2-a and -b at the appropriate
Mach number. Method 3 of paragraph C of S,:ction 4.4.1 should be used to evaluate the last termof equation 7.4.1.2-b.
Sample Problem
Method 1
Given: Same configuration as sample problem of paragraph C Section 7of this section. Some of the characteristics are repeated below.

paragraph A

The following ratios based on total forward panel dimensions:
S'°T

'

S'

=0.0988-c'= 2.26

Additional Characteristics:= 0.98M = 1.40= 40

cg at ý'/4

Compute:Step 1.

Wing-body C1n(Section 7.3.1.2)
qCm q) wB

Step 2.

(sample problem, paragraph C, Section 7.3.1.2)

Lift-curve slope for the exposed horizontal tail (Section 4.1.3.2)(CN)'

7.4.1.2-4= -4.961 per rad

4.025 per rad (sample problem, paragraph C, Section 7.4.1.1)"

**..-.-

Step 3.

Tail-body interference factors (Section 4.3.1.2)

Kw(B)" = 1.315•(sample problem, paragraph C, Section 7.4.1.1).KBW)= 0.4028"Step4.

Dynamic pressure ratio (Section 4.4.1)=

0.80 (sample problem, paragraph C. Section 7.4.1.1)

ItSolution:
Cm

Cq~

2[KW(B) + KW(B)IJ-(t

xJ

•(N.
(equation 7.4.1.2-a)=-4.961 - 2(1.315 + 0.4028) (0.0988) (2.26)2 (0.80) (4.025)=-4.961 - 5.583=-10.54 per rad(based on the area and the square of the mean aerodynamic
chord of the total forward panel and referred to a moment centerat Z/4)A.

7.4.1.2-5

April 1978

7.4.1.3 WING-BODY-TAIL PITCHING DERIVATIVE CDqThis section presents a method for estimating the wing-body-tail derivative CDq at subsonic speeds.
"This derivative is the change in the drag coefficient due to a change in pitching velocity at aconstant angle of attack and is defined as
-•

aCD
ac

,where CD

CDq

sed on S.

In general, this derivative is small and has a negligible effect on longitudinal stability; hence, it isusually neglected.
A. SUBSONIC
The wing-body-tail derivative is obtained by adding the horizontal-tail contribution to thewing-alone contribution developed in Section 7.1.1.3. The body contribution is negligible and has
been neglected. The horlzontal-tail contribution to CDq was computed from the tail-damping angle"ofattack and also from wing-induced downwash at the tail due to pitch rate. The horizontal-tail lift
was taken to act normal to the local flow direction at the tail to produce a force component in the
direction of the free-stream flow.

"DATCOMMETHOD
The wing-body-tail derivative CD is given by
q

aCD
CDq

=CDq

Dq

where
CD

is the wing contribution from Section 7.1.1.3-a~CDqoH)is the horizontal-tail contribution due to zero-angle-of-attack loading given by
=- L

/H

q0\H
C

4
.....

SH\[S

2

(QH

COS aF +zH
Z

w zH si
+-

nF

2 Eqq
H3

(per degree)

7.4.1.3 -b\lf1/\c7.4.1.3-1

whereCLis the horizontal-tail lift-curve slope (based on SH) obtained fromtest data or Section 4.1.3.2 (per degree).

iHis the incidence of the horizontal tail with respect to the fuselage
reference line in degrees.

SH

is the horizontal-tail reference area.

Sw

is the wing reference area.
aeis the variation in downwash with respect to pitch rate, eq

Eq(at ý/4 of the horizontal tail). Design charts are presented in Figures
7.4.1.3 -4a through -4x for Mach numbers of 0.2 and 0.8. These
charts are presented as a function of wing geometry and the spanwise
location of the horizontal-tail MAC relative to the wing span.
QH

is the distance from the moment reference center to the centet-ofpressure location of the horizontal stabilizer, measured parallel to thebody center line. For Datcom purposes, the horizontal-tail center-ofpressure location is assumed to be at 7H /4,

ZH

is the distance from the moment reference center to the center-ofpressure location of the vertical stabilizer, measured normal to thebody center line, positive for the stabilizer above the body.

U

iis the fuselage angle of attack.

cis the wing MAC.

6is the downwash of the wing at E/4 of the horizontal tail, excluding
the contribution due to pitch rate. This may be obtained from test
data or from Section 4.4.1.

acDis the change in horizontal-tail contribution with angle of attack, obtained by
aCD
aOtF

S

CL

2(HF]

Swc57.3

where all the terms are defined above.

7.4,1.3-2

7.4.1.3-c

Sample Problem
Given: Same configuration as sample problem of Paragraph A of Section 7.1.1.3.
Tail Characteristics:
CLaH

0.08 per deg

~H

Iideg

RK~£160.0Oft

L~

ZH=8.0Oft

SW
-=03

e

1.6 deg

U! .=0.19
Compute:Eq(interpolated using Figures 7.4.1 .3 -4p, -4q, and -4r)=-0.02

8 Cu4q

2 2

aM

4008
-.

-

S

(Rcosa,573

c1+Z sin aF)](Equation 7.4. 1.3-c)(0.99985) + (8.0) (0.01 745)IC2[(60.0)

91.

[-0.02---

=

25.0

'

0.0549perdeg 2

.a4
:2~~

L

(Sapl ProblemnPargrphA1ecio371..357.

7.4..3--0029prKg
Solution:__

*~

C

CD

(X

++

=0.00272+ 0.00549 -(0.00249) (1.0)

0.0O057 2per ceg

B. TRANSONICNo method is presented.
*

.~

C.

SUPERSONICNo method is presented.
M=.
N= 0
ALE 0
2.0-------------------------

1.6-

Elq

01

b/2

FIGURE 7.4.1.3-4 VARIATION IN DOWNWASH WITH JPITCH RATE
7.4.1.3-4(Equ~ation 7.4.1.3 -a)-

M
X~
AL E

1.2

__

-

--

.8-

-.4-

q4.0.3-

1.~

2.

-

-

A

.2
0
30()

M

=.2

0
ALE =600
X=

L) E-

2.0

1.6

74-

1.2-

F.1

L

--

04.6

/Y

H

2.1-

.3C.4.5D.

///
FIUR

-1.2"

-2.0
F•IGURE 7.4.1.3-4 (CONTD)
7.4.1.3 -61

M =.2LL

2.0- (d)AL

-~

1.6'-

0

-

I-4Z

I7

FIGURE
FIUR 7.,4.3-1.3-7D.4.1734.1.3N-7

M =.2
X = .25
1.6-

300

ALE

(e)

-----

.8

-

.4--

E7

"--. 4'

-. 8

00

-

-

-10

FIGURE 7.4.1.3-4 (CONTD)
S

7.4.1.3-8

-

-

-

--

--

-

.

-

-

-

777".

-

•.

-

-

M - .2
X = .25
ALE

600

2.0.

1.6

. .. .

•

-..

-

1.2-

A

.8

-

-

.4-

-

0

1

.4

.6

Xb/2
r41

2,

4.5.

7

-./,----- /----- -<- - - -FIGURE 7.4.1.3-4 (CON-D))
7.4.1.3-9

M = .2.5
ALE=~~2.0--

*1

0

__

(g)

6

1.6

1.2-

q

.8

,

-

0.1

.2

yeH

YZH
b/2

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-10

.3

.4

.5

.6

M - .2

"6'
1.6-

X .5
"AE=30 0

(h)

-1.2

.4

.
-1.6

--

/7"-'

b/2

-1.6

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-11

M= .2N = .5

ALE =600

2.0

-

-

1.2.-2

/

----

A

.8

-

--

.4-

--

0

.1

-

fq

-.4-----/

'W

.2

y
b/2

7/

-2.0FIGURE 7.4.1.3-4 (CONTD)

7.4.1.3-12

-

3

/

.4

5

.6

V -'-

.4-,1

.••,

r

c ,-"•. ----..

o••-.-r
..-

'

-•

.

•

+

.-.

.

.- -

....

.+ .

..

i

M =.2
A,= I
ALE= 0

2.0-

S~(j)

1..2A

Eq-----------------------------

.84

Ii

01

.

.2

yZ-H

.3

.4

.5

.6

'

bj/2

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-13

. ..

. .6

M

ALE

1.6-

1.2-

-

-

-

-

-

.2

3200-

--

AI

p.4

eq-

-

.-

-

---

..--

-

-

-

.

M

=

.2

ALE = 600

1.6-":,

-..

1.2-

...

A

~~.4-1

b

"

1

0q

0

.2

y-

.3

.5

.6

CH

-.4

.8-

1

---

-

-

-i-1

-

-1.1
•-1.6'-

-

-

..

-'I

-

-,

-i

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-15

flii

,,I

M =.8

K>-

2.0-

()ALE

1.2'eq.4

FIGURE 7.4.1.3-4 (CONTD)
7.4 1.3-16

0

II

M=.8

x=0

(n)

~1.2-•

a

.87.

0-

eq

.

-

-

-

-

-

.

0.l

.;2

y-a

.3

.,4

.5

.

b/2

.4

.... [...-.

-1.6

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-17

M =.8

(0
A LE

2.0

600

II
A)
-.401

-.4

I-2-

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-18

.2 .6

V.1

M=.8
X
A
2.0

.-

.25
=0

LE

(p)

1.6

k

-

--

4.4-

0

0

.1.2

:

Y-

.3

.t.5

.6

b/2

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-19

M =.8
X= .25
ALE =300

1.6

(q)

_

_

A

1.2 -

.8-_

eq

3

h0

.,.1

?

•

3

.4'

b /2

-160

.

-

.,__,--].

FIGURE 7.4.1.3-4 (CONTD)
4M* 3-20

__

.5S.

M=.8
X .25
ALE =600

__

I

__//_

A

1..6

.4

I

___

-. 8

___

_

,

-1.6

-2.0

.1

2b!.

FIGU RE 7.4.1.3-4 (CONTD)
7.4.1.3-21

M =.8
=.5

--

2.0

-__

ALE: = 0

-

1.6

1.2-

••

.%0

0

.1

.2

Y H

b/2

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-22

.3

.4

.5.6

M=.8
A = .5
1.6

1.2

ALE3

0

(t)

-

.8

-

7/

0"
o -r1

.2/

/
.3 Y-ý

17
.4

.5

.6

b/

-1.2

iI,

'v4

-1.6-

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3 -23

M=.8
= .5
ALE=

600

"1.6-

1.2----

.8

•q

0

__-

A

-

-

-

,

_

-

_

I

.41

0

.

...
.........

4

.2

.5

YC-H

b/2 /

/7/
- I/

I I

-•,oT
-2,0-

FIGURE 7.4.1.3-4 (CONTD)

7.4.1.3-24

A

--

.6

M

•.

..

=.8

= 0

.ALE

2.0 -(v)

1.6

-

1.2

Eq

A
.-.-.-

1---,

---

0I

Y'YT

.4

10

FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-25

M =.8=1.0

ALE= AP
1.6

(W)

1.2

---A

I

0

.3

41

.8
-

-. 4 -

*

/

T/

•.

-1.2

FIGURE 7.4.1.3-4 (CONTD)
S

b/2

7.4.1.3-26

7

10-

.5.

M = .8, -- 1.0
ALE= 600
2.0

-1.6

4e

0

-1.26

I.Ki/•

H~-2.0---FIGURE 7.4.1.3-4 (CONTD)
7.4.1.3-27

Re'ised September 1970

7.4.2 WING-BODY-TAIL ROLLING DERIVATIVES
7.4.2.1 WING-BODY-TAIL ROLLING DERIVATIVE CySThis section presents methods for esti~nating the nondimensional rolling derivative Cyp of
wing-body-tail combinations at subsonic speeds. However, at transonic and supersonic speeds nogeneralized methods are available for estimating the wing-oody-tail rolling derivative Cy p. The
derivative Cy is the change in side-force coefficient with change in wing-tip helix angle and isexpressed asCyp=

In general, the Datcom methods consist of a synthesis of material presented in Sections 7.1.2.1 and7.3.2.1, and the vertical-tail contribution based on the methods of reference I.
The derivative Cy p arises mainly from the wing and vertical tail. At high angles of attack thevertical-tail contribution predominates. The resultant side force on the vertical tail is generated
when the aircraft has a rolling velocity p about its longitudinal body axis and the vertical tail islocated either above or below the longitudinal axis. Generally Cy is of little importance inplateral dynamics, hence is frequently neglected.
The side force at the vertical tail is created by the effective angle of attack due to the rolling
velocity p and the sidewash generated from the wing and fuselage. The sidewash at the vertical
tail can significantly alter the tail contribution. This effect is discussed more fully in reference 2.
Studies have indicated that the effect of the sidewash varies considerably with tail size, location,and to some extent with wing planform.
A. SUBSONIC
Two methods are presented for determining the rolling derivativ'e Cy p of the wing-body-tailcombination,. differing only in their treatment of wing sidewash on the vertical tail. The first
method- is applicable to conventionally located vertical tails, and the second method applies totails located directly above, or above and slightly behind the wing. Both methods are based on
the assumption that the vertical-tail contribution to Cy , is zero at a = 0 and varies with angle ofattack.
For an, isolated-tail configuration the vertical-tail value of Cy

is approximated by
7

7.4.2.1.1

For a conventionally located vertical tail the effect of wing sidewash on the vertical tail has been
approximately accounted for by

\ -z
b

//

For configurations with tails located directly above or slightly behind the wing, the effect ofwing sidewash has been approximated by using the average of the isolated-tail and conventionaltail values.
DATCOM METHODS
Method I
For conventionally located vertical tails, the equation for the nondimensional rolling derivative
Cy of a wing-body configuration, based on the product of wing area and span Sw bw, is given
by
y P

CY P

( CY ) W +
(r)w
+ 2

[brl
bA

y
( #)

WBH)
W H (per radian)

7.4.2. 1-a

where

(tVp

bw

)WB

is the wing-body contribution to Cy
obtained from test data or
Section 7.3.2.1 and based on the product of wing area and span (perradian),is the wing span.
is the distance from the moment reference center to the center-ofpressure location of the vertical stabilizer, measured normal to the body
center line, positive for the stabilizer above the body. For Datcom
purposes, the vertical-tail center-of-pressure location is assumed to be
the quarter-chord point of the MAC of the total added panel.

Rp

is the distance from the moment reference center to the center-ofpressure location of the vertical stabilizer, measured parallel to the body
center line. For Datcom purposes, the vertical-tail center-of-pressure
location is assumed to be the quarter-chord point of the MAC of thetotal added panel.

z

is the vertical distance of the vertical-tail center-of-pressure location
above or below the moment-reference-center location. This value must
be calculated for each angle of attack (see sketch (a)). From sketch (a),z may be expressed asz

7.4.2.1-2

zP Cos a - VP sin ct

7.4.2. 1-b

('acy O)V(WBH)

5.3.1.1,
Sectiondata orshould
derivativesideslip
tail-bodyisbasedthe onthe wing
area (perradian).from
Thistest
derivative
includetheend-plate effects of the horizontal tail.

f P

VERTICAL-TAIL
CENTER.OF-PRESSURE
LOCATION

SKETCH (a)Method 2
For vertical tails located either directly above, Cr above and slightly behind the wing, theequation for the nondimensional rolling derivative Cyp of a wing-body-tail configuration, basedon the product of wing area and span SW bw, is given by
CYp-- (Cp)wB +ICy')W"H) (per radian)

7.4.2.1-c

where the components are described in method I above.
Sample Problem
Method I
Gi -n: Model of reference 3

a.. OFHORIZONTAL
TAIL

7.4.2.1-3

Wing Parameters:Sw= 428.0 sq in.

X = 0.565

3.57

bw=

38.84 in.

Aw

Ac/ 4

= 36.2'

zw

= 0

A

3.59

:

Horizontal-Tail Parameters:

bH = 18.66 in.

SH = 97.10 sq in.
X=

0,50

-ZH

=

6.60 in.

Vertical-Tail Parameters:
140,5 sq in.

S

X = 0.176= 5.0 in.

zbV

12.68 in.

Av

cv

11.0 in.

Ac/ 2

RP

24.3 in.

x

1.15

=450

- 7.25 in.

Additional Parameters:M.r.8.0'
2r,
*..,

CYP)

0.17

0.52 per rad (test data)d = 7.5 in.

= 4.0 in.

Compute:Calculate the tail-body sideslip derivative (zCy )BH) from Section 5.3.1.1
.1

12.68

bv
2r 1

4

AV (B)
A

Av

-

x

12.68
- 7.2511.0
7.4.2.1-4

1.29(figure 5.3,1.1-22a)

-6.60

Z11

bV

=3.17

-

- -

-

0.52

-0.659•.

Av (OiB)AV= 0.90

SH
-=

KH

=

Aeff

(figure 5.3.1.1-22b)

97.1
. •
140.5

= 0.691

0.76(figure 5.3 .1.1-22 c)

A• ,

AV

1 + K1 [Y!

tAv (B)= (1.29) (1.15)(equation 5.3.1.1-a)1(

11 + 0.76 [0.90 - 1.01,= (1.29) (1.15) (0.924)
=

1
K

1.37
(assumed)=

A,,ff [2.

a2 A t ]/2=(1.37)

10.971 +11 1/ 2S+= (1.37) (1.404)= 1.923
CL

CL

=

1.315

=

(1.315)(1.37)(figure 4.1.3.2-49)

1.80 per rad (based on Sv)

/

q,
Sq

0.724 + 3.06
.06&,rISw+=

0.724 + 0.72
3.06
.06++(140.5)/(428.0)
1.807

=

0.724 + 0.556 + 0.0321

=

1.312

0.4 z

0.009A(equation 5.4 .1-a)(0.4) 0 + (0.009) (3.57)
7.5

7.4.2.1-5

LI

k = 0.94(figure 5.3.1.1-22d)\CI(WB)

(

-k C
(C)

V(1i

aa\ qV
-

q1

SV
-

Si

(equation 5.3.1.1-b)

140.5
=

-(0.94) (1.80) (1.312) 48
428.0= -0.729 per rad (based on Sw)
Calculate the vertical distance z of the vertical-tail center-of-pressure location above or below
the moment-reference-center locationz ==cos 0i - 2 sin a(equation 7.4.2.1-b)(5.0) (0.9903) - (24.3) (0.1392)= 4.95 - 3.38

1.57 in.
Calculate the rolling derivative Cy

Cy

(Cy

+2

0.52 + 2
=

[

p

- -p

38.84

for the wing-body-tail configurationC)(equation 7 .4.2.1 -a)(-0.729)0.52 + (2) (---0.0883) (--0.729)
0.52 + 0.129=

0.649 per rad (based on Swbiw)

This compares with a test value of 0.62 per radian from reference 3.
B. TRANSONIC
No generalized method is available in the literature for estimating transonic values Jf the rollingderivative Cy-¢. Furthermore, only limited experimental data a,.' available for this derivative attransonic speeds (see table 7-A).

7.4.2.1-6

L

C. SUPERSONIC
No generalized method is available in the literature for estimating supersonic values of the rollingderivative Cy P. Furthermore, there is a scarcity of experimental data for this d ývative at
supersonic speeds.

For the purposes of the Datcom the fuselage effect is considered to be negligible for wing-body"configurations with body diameters less than about 30 percent of the wing span. Therefore, forconfigurations with d/b < 0.3, the wing-body rolling derivative Cy
is estimated by thewing-alone method of paragraph C of Section 7.1.2.1.
Methods are presented in reference 4 for evaluating the vertical-tail contribution to Cyv The
stability derivatives presented therein are derived by using supersonic linearized theory forfamilies of thin, isolated vertical tails performing steady rolling motions. Vertical-tail families(half-delta and rectangular planforms) are considered over a broad Mach number range. Also
considered are vertical tails with arbitrary sweepback and taper ratio at Mach numbers for which
both the leading edge and trailing edge of the tail are supersonic, and triangular vertical tails witha subsonic leading edge.and a supersonic trailing edge.

REFERENCES
1.

0-i

Ca•npbell,

J, P., and McKinney, M. 0.: Summary of Methods for Calculating Dynamic Lateral Stability and Response andfor Estimating Lateral Stability Derivatives. NACA TR 1098, 1952. (U)2.

Michael, W. H. Jr.: Analysis of the Effects of Wing Interference on the Tail Contribution: to the Rolling Derivatives.
NACA TR 10M6, 1952. (U)3.

Queijo, M. J., and Wells, E. G.: Wind-Tunnel Investigation of the LowSpeed Static end Rotary Stability Derivatives of a0.13-Scale Model of the Douglas 0-558-11 Airplane in the Landing Configuration. NACA RM L52G07, 1952. (U)4.

Margolis, K., and Bobbitt, P. J.: Theoretical Calculations of the Pressures, Forces, and Moments at Supersonic Speeds due
to Various Lateral Motions Acting on Thin Isolated Vertical Tails. NACA TR 1268, 1956. (U)

7.4.2.1-7

.

• .- ;-.

Revised September 1970

7.4.2.2 WING-BODY-TAIL ROLLING DERIVATIVE C1p

This section presents methods for estimating the nondimensional rolling derivative C1p ofwing-body-tail combinations at subsonic speeds. However, at transonic and supersonic speeds no"generalized methods are available for estimating the wing-body-tail rolling derivative ClP. The
derivative Cl p is the change in rolling-moment coefficient with change in wing-tip helix angle andis commonly referred to as the roll-damping derivative. It is expressed as

aCt

I!' .]Clp-a(~

In general, the Datcom' methods consist of a synthesis of material presented in Sections 7.1.2.2 and7.3.2.2, and the vertical-tail contribution based on the methods of reference 1.

014

The derivative Cr1 is important in lateral dynamics, since it determines the damping-in-rollcharacteristics of the vehicle. The derivative is composed of contributions, negative in sign, fromthe wing, the horizontal tail, and the vertical tail, with the main contribution coming from thewing. The contribution from the vertical tail is usually negligible at low and moderate angles ofattack. However, the vertical-tail contribution can become significant at high angles of attack,when the effective moment arm of the tail (z/bw) becomes a large negative value.
The rolling wing produces a sidewash at the vertical tail, which can significantly alter thevertical-tail contribution. This effect is discussed more fully in reference 2. Studies have indicatedthat the effect of the sidewash varies considerably with tail size, tail location, and to some extent
with wing planform.
Conventional horizontal-tail effects on Clp are usually small and oftern neglected, althoughunusually large horizontal tails can contribute significantly (see references 1 and 3). The
horizontal-tail contribution is obtained by using the horizontal-tail geometry and the method ofSection 7.1.2.2. The value from Section 7.1.2.2 is then multiplied by a constant, which accounts
for the flow rotation produced by the wing, and by the appropriate geometrical parameters torefer the result to the proper reference base.

r tA.
,

SUBSONIC
Two methods are presented for determining the rolling derivative Clp of the wing-body-tailcombination, differing only in their treatment of wing sidewash on the vertical tail. The first
method is applicable to conventionally located vertical tails, and the second method applies totails located directly above, or above and slightly behind the wing. Both methods are based on
the assumption that the vertical-tail contribution to Clp is zero at a = 0 and varies with angle ofattack.
7.4.2.2-1

L

*

--

~a

I-~

-

.

.

-

.

.

For an isolated-tail configufation the vertical-tail value of CIP

is approximated by

2ciP)~ v= 2( _~ (ACY() H

For a conventionally located vertical tail the effect of wing sidewash on the vertical tail has been
approximately accounted for by(Cit)V=2b2) V(WBH)(ACY

For configurations with tails located directly above or slightly behind the wing, the effect ofwing sidewash has been approximated by using the average of the isolated-tail and conventionaltail values.

DATCOM METHODS
Method I
For conventionally located vertical tails, the equation for the nondimensional rolling derivative
ClP of a wing-body-tail configuration, based on the product of wing area and the square of wing
span Sw bw 2, is given by(CP)H+,

0.5 (7lp)H kSW \bW

\bw

NH

[bw])

(per radian)

7.4.2.2-at

where(CIp)wBis the wing-body contribution to Cl , obtained from test data or
Secticn 7.1.2.2 and based on the product of wing area and the squareof wing span (per radian).

CI

is the horizontal-tail contribution obtained from test data or Section
7.1.2.2, based on the horizontal-tail geometry 'per radian).

Si
S

(b.il

W
\bw/

is the ratio of the horizontal-tail area to the wing area times the squareof the ratio of the horizontal-tail span to the wing span.
is the vertical distance of the vertical-tail center-of-pressure location
above or below the moment-reference-center location. This value must
be calculated for each angle of attack. (See sketch (a) in Section
7.4.2.1.) From equation 7.4.2.1-b, z is expressed asz = zp Cos a

7.4 .2...-2[a7-41,•

Qp sin ap
A/R £pisthe distance from the moment reference center to the center-ofpressure location of the vertical stabilizer, measured parallel to the body-center line (see sketch (a) in Section 7.4.2.1). For Datcom purposes, thevertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

Zp

is the distance from the moment reference center to the center ofpressure of the vertical stabilizer, measured normal to the body center
line, positive for the stabilizer above the body. For Datcom purposes,
the vertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.
is the tail-body sideslip derivative obtained from test data or Section

SACY)v(wBH)5.3.1.1. It includes the end-plate effects of the horizontal tail and isbased on the wing area (per radian).
Method 2
For vertical tails located either directly above, or above and slightly behind the wing, theequation of the nondimensional rolling derivative Cjp of a wing-body-tail configuration, based o"the product of wing area and the square of wing span Sw bw 2, is given by

~'q ~
,I

,

(,)

+

(C/,),(CSW)Wb

W

b;)

+

(2b
(per radian)

7.4.2.2-b

where the components are described in method 1 above.
.iOeProblem
Method I
Given: Model of reference 4
a.c. OFHORIZONTAL
TAIL

H.

7.4.2.2-3

Wing Parameters:
Sw

1.90sq ft

bw= 2.16 ft

F = --i0o0.94 ft

=

Aw

2.44

X

0.38

A

18.0,

z

= 0

2.97

X

0.31

Horizontal-Tail Parameters:SH

0.48 sq ft

Ac 14

10.50

b

=

1.20ftt-= 0.05

r = 0

AH

=

-zH

Z = 0.44 ft=

8.30in.

Vertical-Tail Parameters:
Sv=

0.56sq ft

bv=

9.0 in.

cLi= 0.42 ft

Z = 0.87 ft

1.25 ft

x = 0.25 ft

p=

AV

=

0.86

AC/ 2 = 23.250

Xv=

0.37

zP

0.308 ft.

=

Additional Parameters:
ot

4.00

(C1 ) w

M = 0.25

pc
P WB

CL = 0.3RQ

1.5 x 106-0.30 per rad (test data)
2r,

4.88 in.

Compute:Calculate(C1P)Ifor the horizontal tail from Section 7.1.2.2

*

(B
! 1

)

(:CL

cL-

o

L)

(C )

(

+ (AC P ,)L O CI P)r(equation 7.1.2.2-a)(tanAc)3

7.4.2.2-4

J1 - M2=/ -(.25)2

=

0.968

0.1853

t
AO=
A
tan-

0.968)/= tan-' (0.1914) = 10.80.: = 1 (assumed)

3AH(0.968) (2.97)

K

1.0

=

-0.245(figure 7.! .2.2-20b)

(CL)

(CL.)
CL
C)

= 1.0(assumed)

0'

(C-p)

=I (no dihedral)( 1(Cl)•LCL

2

CL22

~

(CIp)rDL/daCg((Cp)DL= 0.015 per rad
CL2":•'•~~~~~

=

=

1°

(equation 7 .1.2 .2 -c)

D(figure 7.1.2.2-24)Lf)l/+10__41

CD 0

£

--

Swet

S--

++100+RL.S.

=(equation 4.1.5.1-a)

0.44ftSk = 0.05 x 'i0Q

(0.44)(12)k

5 x 10-sin(table 4.1.5.1-A, assume polished wood surface)=1.056 x 101

Cutoff Reynolds number, RR

7 x 106(figure 4.1.5.1-27)
7.4.2.2-5

1.5 x 106

Test Reynolds number, R2(figure 4.1.5.1-26)

Cf

0.00410L

1.2 (assume (t/c)max is located at x

,0

c

[i tL

cos 0

cos A (t/Cmax=e-2.0 (assumed)

0

fe

1.0

Sref =

bortizontij"=(0.00410) (1.06) (1.067) (2.0)-

0.00927(c)k I

ACtPp d-ag

(equation 4.1.5.1-a)(per radian)(equation 7.1.2.2-c)(Cip)r-o

(AC

1

L

-Cp)D

S

".S.

+ 100

IL+ L

CD

(igure 4.1.5.1-28a)(figure 4.1.5.1-28b)

1.067

RLS

Sref

=1.06

+100

.50c)

C
L
CL28

_ --

CD0

L

0.009278=(0.015) (0.3)2=

0.001350 -0.001159=

0.000191 per rad

-

Sc
(C)

P)HK

p C
C C

L(3(Cip)f(PCLQ CL=

0

,u

(per
(g radian)(equation 7.1.2.2-a)

7.4.2.2-6

(C)t(--0.245)(1) (1) + 0.000191= -0.253 + 0.000191-0.2,53 per rad (based on product of horizontal tail area and the squareof tail span

SHbH 2 )Calculate tLe vertical distance z of the vertical-tail center-of-pressure location above or below
the moment-reference-center locationg'z = zv cos a - ki::sin ot(equation 7.4.2.1-b)=(0.308) (0.9976) - (1.25) (0,06976)=

0.220 ftCalculate the tail-body sideslip derivative
•"bv

(ACY)

from Section 5.3.1.1

9.0
AV)
2r,

1.84

=

4.88

Av B
1.63

-

-

(figure 5.3.1.1-22a)

Av
zH
bv

_

!,.,

x
e~v

-G

--8.309.0-----0.922

--

0.25
E-.
0.42

.

0.595

A V (HB)
-__,,

L.

=

1.39

(figure 5.3.1.1-22b)

AV (B)

1..

SH

S"-

-.. = -0.48
&S v0
.5 6

= 0. 85 7

KH(figure 5.3.1.1-22c)

0.85

7.4.2.2-7r.

.l.nt[K

AV (B)AVIB
A~ AV=Aeff

H [vLVy(B.....
I + KHAV(1.63) (0.86)=(equation 5.3.1.1-a)

1V()
1l.39-11}

1 + 0.85(1.63) (0.86) (1.33)
1.86
K=

1 (assumed)+ tan 2 A20ta'
/2

-ff

=

1.86 [0.9375 + (0.4296)2)1/2

=

1.97

1.86 (1.06)

CL
-

t

Acff

CL

=

1.30

=

(1.30) (1.86)=

2.42 per rad (based on Sv)\(figure 4 1.3.2-49)qv
0.724 + 3.06

.

I1+

q.)

Sv/Sw
I+ cos/AC140.724 + 3.06

k

=

0.75

0.56/1.90
1.9511=0.724 + 0.462 + 0.022

=

1.208

zw
+ 0.009Ad+ 0.4 -

(equation 5.4.1-a)+ 0 + (0.003-,) (2 .)(figure 5.3.1.1-22d)(ACy)-k (CLa)+= (0.75) (2.42) (1.208)

iip1.90--(equation 5.3.1.-b)

0.56

0.646 per rad (based on Sw)

7.4.2.2-8.. - i~-.-9(C,P)W+ 0.5~ 1

(C.

(bj

tW2w

t)

bw

As

(equation 7.4.2.2-a)/0.48\/_1.20\ 2

22-0.308]

J (-0.646)-0.30 + (0.5) (-0.253)
2 /0.22\
\.6/

\

2.16

8

1.2

= -0.30 - 0.00986 - 0.00536= -0.315 per rad (based on the product of wing area and the square of wing spanS

2)This compares with a test value of -0.33 per radian from reference 4.

B. TRANSONIC
No generalized reliable method is available in the literature for estimating transonic values of theroll-damping derivative Clz. However, a considerable quantity of test data is available andreference should be made to tablt 7-A.
C. SUPERSONIC
No generalized method is available in the literature for estimating supersonic values of theroll-damping derivative ClP.
The wing-body roll-damping derivative at supersonic speeds is estimated by the wing-bodymethod of paragraph C of Section 7.3.2.2.
Methods are presented in reference 5 for evaluating the vertical-tail contribution to CIP. The
stability derivatives presented therein are derived by using supersonic linearized theory forfamilies of thin isolated vertical tails performing steady rolling motions. Vertical-tail families(half-delta and rectangular planforms) are considered over a broad Mach number range. Also
considered are vertical tails with arbitrary sweepback and taper ratio at Mach numbers for which
both the leading edge and trailing edge of the tail are supersonic, Ind triangular vertical tails witha subsonic leading edge and a supersonic trailing edge.
7.4.2.2-S

REFERENCES
1.

2.

Cmnpbell, J. P., end McKinnay, M. 0.: Summary of Methods for Calculating Dynamic Lateral Stability and Response andfor Estimating Lateral Stability Derivatives. NACA TR 1098, 1952. lU)
Michael, W. H. Jr.: Analysis of the Effects of Wing Interference on the Tail Contributions to the Rolling Derivatives.

NACA TR 1066, 1952. (U)3.

Booth, K. W.: Effect of Horizontal-Tail Chord on the Calculated Subsonic Span Loads and Stability Derivatives of Isolated
Unswept Tail Assemblies in Sideslip and Steady Roll. NASA Memo 4.1.69L, 1969. (U)4.

Buell, D, A., Reed, V. D)., and Lopez, A. E.: The Static and Dynamic-Rotary Stability Derivatives at Subsonic Speeds ofan Airplane Model with an Unswept Wing and a High Horizontal Tail, NACA RM A56104, 1956. (U)5.

Margulis, K., and Bobbitt, P. J.: Theoretical Calculations of the Pressures, Forces, and Moments at Supersonic Speeds due
to Various Lateral Motions Acting on Thin Isolated Vertical Tails. NACA fR 126,8 1956. (U)

7.4.2.2-10

Reviod September 1970

7.4.2.3 WING-BODY-TAIL ROLLING DERIVATIVE Cp

This section presents methods for estimating the nondimensional rolling derivative C,, ofwing-body-tail combinations at subsonic speeds. However, at transonic and supersonic speeds nogeneralized methods are available for estimating the wing-body-tail rolling derivative CP. This
derivative is the change in yawing-moment coefficient with change in wing-tip helix angle and isexpressed as

"Cnp

n

b

In general, the Datcom methods consist of a synthesis of material presented in Sections 7.1.2.3 and7.3.2.3, and thc vertical-tail contribution based on the methods of reference 1.
Contributions to this derivative arise from two sources, the wing and the vertical tail. The wing
contribution is usually negative; whereas the tail contribution may be positive or negative
depending on vertical-tail geometry, sidewash, and equlibrium angle of attack.
The rolling wing produces a sidewash at the vertical tail, which can significantly alter thevertical-tail contribution. This effect is discussed more fully in reference 2. Studies have indicatedthat the effect of the sidewash varies considerably with tail size, tail location, and to some extent
with wing planform.
The derivative Cnp is important in lateral dynamics because of its influence on Dutch-roll
damping. Although for most configurations Cn, is negative, positive values of Cnp are desired toincrease the Dutch-roll damping characteristics.
A. SUBSONIC
Two methods are presented for determining the rolling derivative Cnp of the wing-body-tai
combination, differing only in their treatment of wing sidewash on the vertical tail. The first
method is applicable to conventionally located vertical tails, and the seccxd method applies totails located directly above, or above and slightly behind the wing.
Both methods are based on the assumption that the vertical-tail contribution to Cnp is zero ata = (, and varies with angle of attack.
For an isolated-tail configuration the vertical-tail value of Cnp is approximated By
2 z )) (AC')p

7.4.2.3-1

For a conventionally located vertical tail the effect of wing sidewash on the vertical tail has been
approximately accounted for byn)=(Ac,"),. For configurations with tails

2located directly above or slightly behind the wing, the effect of wing sidewash has been
approximated by using the average of the isolated-tail and conventional-tail values.
DATCOM METHODS
Method 1
For conventionally located vertical tails, the equation for the nondimensional rolling derivative
Cp of a wing-body-tail configuration, based on the product of wing area and the square of wing
span Sw bw 2 , is given by[A

Asinai (WDH) (per radian)
(ACv"

)

(2,, cos a +

C

Sl a

bwZP
7.4.2.3-a

However, if test data for (ACng)p of the empennage are available, the above equation can be
rewritten to include the effective tail length, i.e.,=n(cap)W + 2

this!,z]&Acn) P(per radian)

7.4.2.3-b

where

rf

(cnp )wiis the wing-body contribution to Cp,, obtained from test data or
Section 7.3.2.3 and based on the product of wing area and the squareof wing span (per radian).

2P

is the distance from the moment reference center to the center-ofpressure location of the vertical stabilizer, measured parallel to the body
center line (see sketch (a) in Section 7.4.2.1). For Datcom purposes, thevertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

-"bw

is the wing span.

z

is the vertical distance of the vertical-tail center-of-pressure location"above or below the moment-reference-center location. This value must
be calculated for each angle of attack. (See sketch (a) in Section
7.4.2.1.) From equation 7.4.2,1-b, z can be expressed asz

FeD

7.4.2.3-2
2-.

=

Z. Cosa

RP sin a

K.pressureZp

is the distance from the moment reference center to the center ofof the vertical stabilizer, measured normal to the body center
line, positive for the stabilizer above the body. For Datcom purposes,
the vertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

tAi)v(WBH'(ACyis the tail-body sideslip derivative from test data or Section 5.3.1.1. It
includes the end-plate effects of the horizontal tail and is based on thewing area (per radian).

Cn'P

is the tail-body sideslip derivative from force-test data, where p refers topanels present in the empennage; It is based on the product of wingarea and span (per radian).

Method 2
For vertical tails located either directly above, or above and slightly behind the wing, theequation for the nondimensional rolling derivative Cap of a wing-body-tail configuration, basedon the product of wing area and the square of wing span SW bw 2, is given by
C

np

=

(Cn

--

V

O

P sinP]

J

w

[2z

1

zp]

J

((A
AC

)

(per radian)

Y~JW
D(WBH)#L
7.4.2.3-c

However, if test data for (AC,,'of the empennage are available, the equation can be. rewritten to&-alde the effective tail length, i.e.,(c~~) [2

z~

(-cz

S=

(per radian)

7.4.2.3-dwhere the components are described in method 1 above.
Sample Problem
Method 1

aLc. OFHORIZONTAL

Given: Model of reference 3

TAIL-

x

_

__,1L

-~7.4.2.3-3

j

rt.

_' ." •

•.

.

.

.

pa

.-a-?
*, r ',

.

-..*
"

~

"*-- *- "**'-r"-"-

*.

.- t*

.*

.,

.

"

'

" ¸

.

Wing farameters:

Swbw= 1.90 sq ft

=

S0.38

2.44

Aw

2.16ft

A,/4

18.00

Horizontal-Tail Parameters:
S

0.48 sq ft

=

=

bH

=

1.20 ft

0.31

AH

-zH

2.97

= 8.30 in.

Vertical-Tail Parameters:
0.56 sq ft

Sv

bv

9.0 in.

XV
X: 0.37cv= 0.42 ftz

0.308 ft

R=

1.25 ft

AV= 0.86
AC/ 2= 23.250

x = 0.25 ft

Additional Parameters:
(Cj

0.25

M

6.00

WIB

= 0.02 per rad (test data)

2r 1 = 4.88 in.
Compute:Calculate thlý tail-body sideslip derivative

S

Cy

from Section 5.3.1.1

1.84

9.0
4.88

2r1
Av isj(figure 5.3.1.1-22a)

1.63

AV
z,

8.30

-

9_-0.922
9.0

Vbv
x

0.25

cv

0.42

... .

. 5

0.595

7.4.2,3-4t-* * -.

*.

..

r

r
A V(HB)

-

AV
A (B)
SH

0.48
S-

Sv
KH

:"Af
eff

= 0.857

0.56(t5.;,e S.3.1.1-22c)

0.85

=

(figure 5.3.1.1-22b)= 1.39

.

.

V

. A,

I + K-"(1.63) (0,86) 11 + 0.85-:

(equation 5.3.1.1-a)

AV (B)[1.39-

14(1.63) (0.86) (1.33)
1.86

-

S=

1 (assumed)+ tan__[Aeff

A cI2

1.86 [0.9375 + (0.4296)2]

11121/

2~c/ p + ta'

I:A

1.8 6 (1 .06 )1/2- 1.9 7

CLa
Aeff

CL

1.30

=

(figure 4.13.2-49)(1.30) (1.86)

=

au
-

Sv
/ zw

2.42 per rad (based on Sve)+ q= 0.724

3.061cosA 4 0.4

0.72

0.5Ac4-+ 0.009A(equation 5.4.1-a)/0,56/1.90)
S0.724+ 3.060 + (0.009) (2.44)= 0.724 + 0.462 +0.022-

k = 0.75

1.208(figure 5.3.1.1-22d)

7.4.2.3-5
_U" '

. -'

"

"•

t'

-m

.

. ', 2 1"•

_.

_.__:

...

.....

..

......

...

...

. . .

.

.

.

(ACv )V(WBH)

= -k

C/

YP)cj)C

La)

/

y

\ qv

(equation 5.3.1.1-b)

q.0 SW+-(0.75) (2.42)

Sw

2

0.56

(11.908/-0.646 per rad (based on Sw)
Calculate the vertical distance z of the vertical-tail center-of-pressure location above or below the
moment-reference-center locationz

cos a -

sin a(equation 7.4.2.1.b)- (0.308) (0.9945) -(1.25) (0.1045)- 0.176 ft

[z-

2

(c

C

)V

-

~

Cos a+zZSinl a) [zpp(equation 7.4.2.3-a)=6(0.02)=

0.02 ---

2[(1.25) (0.9945) + (0.308) (0.1045)1

216 (1.275)

-2i

6/

.16

(-.

6 4 6)(-0,646)= 0.02 - 0.0466=-0.0266 per rad (based on Sw bw 2 )This compares with a test value of -0.048 per radian from reference 3.

B. TRANSONIC
No geineralized method is available in the literature for estimating transonic values of the rollingderivative Cn p.,

C. SUPERSONIC
No generalized method is available in the literature for estimating supersonic values of the rollingderivative CnP.

7.4.2.3-6

A

For the purpose of the Datcom the fuselage effect is considered to be negligible for wing-bodyconfigurations with body diameters less than about 30 percent of the wing span. Therefore, forconfigurations

with d/b <0.3, the wing-body rolling derivative Cnp is estimated by thewing-alone method of paragraph C of Section 7.1.2.3.
Methods are presented in reference 4 for evaluating the vertical-tail contribution to Cn P. The
stability derivatives presented therein are derived by using supersonic linearized theory forfamilies of thin isolated vertical tails performing steady rolling motions. Vertical-tail families(half-delta and rectangular planforms) are considered over a broad Mach number range. Alsoi

considered are vertical tails with arbitrary sweepback and taper ratio at Mach numbers for which
both the leading edge and trailing edge of the tail are supersonic, and triangular vertical tails witha subsonic leading edge and a supersonic trailing edge.
.•-

REFERENCES

i.

Campbell, J. P., ad McKinney, M. 0.: Summary of Methods for Calculating Dynamic Lateral Stability and Response andfor Estimating Lateral Stability Derivatives. NACA TR 1098, 1952. (U)2-Michael, W. H. Jr.: Analysis of the Effects of Wing Interference on the Tail Contributions to the Rolling Derivatives.
NACA TR 1086, 1952. (U)3.

Buell, 0. A., Roed, V. D., and Lopez, A. E.: The Static and Dynamic-Rotary Stability Derivatives at Subsonic Speeds ofan Airplane Model with an Unwapt Wing and a High Horizontal Tail. NACA 19M A56104, 1956. (U)4.

Magolis, K., and Bobbitt, P. J.: Theoretical Calculations of the Pressures, Forces, and Moments at Supersonic Speeds due
to Various Lateral Motions Acting on Thin Isolated Vertical TaiIls NACA TR 1268, 1956. (U)

.,2.

'I-,

7 4 2.-

Revised September 1970

7.4.3 WING-BODY-TAIL YAWING DERIVATIVES
7.4.3.1 WING.BODY.TAIL YAWING DERIVATIVE Cy,"This section presents a method for estimating the nondimensional yawing derivative Cy. ofwing-body-tail combinations at subsonic speeds. However, at transonic and supersonic speeds nogeneralized methods are available for estimating the wing-body-tail yawing derivative Cy r This
derivative is the change in side-force coefficient with variation in yawing velocity and isexpressed as
CY

Cy

=~

aCy
b

In general, the Datcom method consists of a synthesis of material presented in Sections 7.1.3.1 and7.3.3. 1, and the vertical-tail contribution based on the methods of reference 1.
Contributions to the derivative Cy arise from two sources, the wing and the vertical tail. The
vertical-tail contribution, which constitutes the major portion, is small and positive in sign.
Generally Cy, is of little importance in lateral dynamics, hence it is frequently neglected.
A. SUBSONICr.

For the oscillatory mode, the effects due to lag of sidewash in free oscillation are important andhence should be considered. However, no generalized method is available in the literature toaccount for oscillating sidewash effects on Cy ; therefore, only the aperiodic mode of Cy r ispresented here.
In the equation for estimating the yawing derivative Cy ,, the sideslip-derivative contribution of
the vertical tail should include the end-plate effects of the horizontal tail.
DATCOM METHOD
The equation for the riondimensional yawing derivative Cyr of a wing-body-tail configuration,
based on the product of wing area and wing span Sw bw, is given by2
7.4,3.1-a)v(w)(per radian)bw (Rp cos Of+
sinr ) ýC(Sil
C(Cr)w3

However, if test data for (A C,)

of the empennage are available, the above equation can be

rewritten to include the effective tail length, i.e.,
Cyr(Cyv)wB+ 2 (AC.)p(per radian)

7.4.3. 1-b

7.4.3.1-1

where
CY )wBis the wing-body contribution to CYr obtained from test data or byusing the recommendations of Section 7.1.3.1, based on the product ofwing area and wing span (per radian).

bw

is the wing span.

RP

is the distance from the rmoment reference center to the center-ofpressure location of the vertical stabilizer, measured parallel to the body
center line. (See sketch (a) in Section 7.4.2.1.) For Datcom purposes,
the vertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel,zp

is the distance from the moment reference center to the center ofpressure of the vertical stabilizer, measured normal to the body center
line, positive for the stabilizer above the body. For Datcom purposes,
the vertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

ACya)VtWBH)

is the tail-body sideslip derivative obtained from test data or Section
5.3.1.1. It includes the end-plate effects of the horizcontal tail and isbased on the wing area (per radian).

ACn ) pis the tail-body sideslip derivative from test data where p refers topanels in the empennage. It is based on the product of wing area andspan (per radian).
Sample Problem

Given:

Model of reference 2
a.c. OFHORIZONTAL
TAIL

a

rip

800

"7.4.3.1-2

L'.•.-.

,•",(Cvr)= -0.10 per rad (test data)

YW)WB

(ACn

= 0.42 per rad (test data).,

Compute:
Calculate the yawing derivative(Cyr )

C

+2

Cy

cn)

for the wing-body-tail configuration(equation 7.4.3.1I-b)--0.10 + 2 (0.42)=

0.74 per rad (based on SW bw)"Thiscompares with a test value of 0.50 per radian from reference 2.
B. TRANSONIC
No generalized method is available in the literature for estimating transonic values of the yawing
derivative Cyr. Furthermore, there is a scarcity of experimental data for this derivative attransonic speeds.
C. SUPERSONIC'

No generalized method is available in the literature for estimating supersonic values of theyawing derivative Cyr. Furthermore, there is a scarcity of experimental data for this derivative at
supersonic speeds.

REFERENCES
1.

Campbell, J. P., and McKinney, M. 0.: Sumrnmry of Methods for Calculating Dynuimic Lateral Stability and Response and for
Estimating Lateral Stability Derivatives. NACA TR 1098, 1952. (U)2.

Outijo, M. J., and Wells, E. G.: Wind-Tunnel I nvestigtion of the Low-Speed Static and Rotary Stability Derivatives of a0.13-Scale Model of the Douglas D-6584I Airplane in the Landing Configuration. NACA RM L52G07, 1962. (U)

7.4.3.1-3

Reviaed September 1970

"7.4.3.2 WING-BODY-TAIL YAWING DERIVATIVE C1,This section presents a method for estimating the nondimensional yawing derivative C1 ofwing-body-tail combinations at subsonic speeds. However, at transonic and supersonic speeds nogeneralized methods are available for estimating the yawing derivative Cii. This derivative is thechange in rolling-moment coefficient with change in the yawing velocity and is expressed as

ac,
Cirr

in general, the Datcom method consists of a synthesis of material presented in Sections 7.1.3.2 and7.3.3.2, and the vertical-tail contribution based on the methods of reference 1.

Contributions to this derivative arise from two sources, the wing and the vertical tail. The wing
contribution constitutes the major portion and is positive in sign. The lesser contribution of thevertical tail can be positive or negative, depending upon tail equilibrium angle of attack. This
derivative is not of primary importance; however, it is not neglected in lateral dynamic
calculations.
A. SUBSONICThe method presented here is based on the assumption that the wing and fuselage interference
effects on tail effectiveness can be determined floin geometric dimensions and the sideslip
derivatives. This is particularly true if experimental test data are available for
given below.
(ACIt) P in the equations.a&Y-p/VtwH)

or

DATCOM METHOD

The equation for the nondimensional yawing derivative CQ of a wing-body-tail configuration,
based on the product of the wing area and the square of wing span Sw bw 2 , is given by2a•) /z cos at 2 iAC~
7.4.32-aa)n CYO)kzCos - 2P
(2, Cos a + zP sin
(Cl,)
Cr

However, if test data for (AC,) , are available, the above equation can be rewritten to
approximate the effective vertical-tail center-of-pressure lo.ation (the height of the body centerline).
-(R, Cos a+2"P7.4.3.2-b

sin a)

ivH)
are available, *he above equation can be.(CYV0)
if test data for (AC 0',)P and.
rewritten to include the total effective vertical-tail center-of-pressure location.

7.4.3.2-1
.

.

---

C

+2

( tc p (ActtS) 2(AC )(per7.4.3.2-c(per radian)

where(Cir)wD

is the wing-body contribution to C4, obtained from test data or Section
7.3.3.2, based on the product of wing area and the square of wing span(per r.,dian).

bw

is the wing span.

Rp

is the distance from the moment reference center to the center-ofpressure location of the vertical stabilizer, measured parallel to the body
center line. (See sketch (a) in Section 7.4.2.1.) For Datcom purposes,
the vertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

zp

is the distance from the moment reference center to the center ofpressure of the vertical stabilizer, measured normal to the body center
line, positive for the stabilizer above the body. For Datcom purposes,
the vertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

(AC V)(wBH)

is the tail-body sideslip derivative obtained from test data or Section
5.3.1.1. It includes the end-plate effects of the horizontal tail and isbased on the wing area (per radian).

(ACIt) P

is the tail-body sideslip derivative from test data, where p refers to thepanels in the empennage. It is based on the product of wing area andspan (per rwdian).
is the tail-body sideslip derivative from test data, where p refers to thepanels in the empennage. It is based on the product of wing area andspan (per radian).
Sample Problem

Given: Model of reference 2,-7.4.3.2-2

x.c. OFHORIZONTAL
TAIL

p.

a

6.00

bw

38.84 in.

24.3 in.

z=

5.0 in.

per rad (test data)(CY •v(-0.64= 0.10 per rad (test data)

(CF)W

RP

Compute:C1. for the wing-body-tail configuration
Calculate the yawing derivative
2C,(C )wB=w (-k cos a+Z sina) (zcosa- -sina)(ACy )vwB)
7,4.3.2-a).•(equation-.

= 0 .10- (38.84)'k(5.0) (0.00451,(24.3) (0,0045) + (5.0) (0.1045)((24.3)(0.1045)- 0.64)0.10+0.051=

0.151 per rad (based on Swbw 2 )This compares with a test value of 0.15 per radian from reference 2.
B. TRANSONIC
No generalized method is available in the literature for estimating transonic values of the yawing
derivative Cir. Furthermore, there is a scarcity of experimental data for this derivative attransonic speeds.
C. SUPERSONIC
No generalized method is available in the literature for estimating supersonic values of theyawing derivative Cr. Furthermore, there is a scarcity of experimental data for this derivative at"supersonic speeds.

REFERENCES

f|

1.

Cmnipbell, J. P., and Goodman. A.: A Serniempirical Method for Estimating the Rolling Moment Due to Yaing of Airplanes.
NACATN 1984, 1949. (U)2.

Queijo M. J., and Wells, E. G.: Wind.Tunnel investigation of the Low-Speed Static end Rotarm Stailty Derivatives of a0.13-Scale Model of the Douglas D658-11 Airplane in the Landing Configuration. NACA RM L52G07, 1962. (U)

7.4.3.2-3

Revised September 1970

7.4.3.3 WING.BODY-TAIL YAWING DERIVATIVE CA,This section presents a method for estimating the nondimensional yawing derivative Cnr ofwing-body-tail combinations at subsonic speeds. However, at transonic and supersonic speeds nogeneralized methods are available for estimating the wing-body-tail yawing derivative Ca . This
derivative is the change in yawing-moment coefficient with change in the yawing-velocity
parameter. It is commonly referred to as the yaw damping and is expressed as
'

CA

ra

\rb -/

In general, the Datcom method consists of a synthesis of material presented in Sections 7.1.3.3 and7.3.3.3, and the vertical-tail contribution based on the methods of reference 1.

Contributions to this derivative arise from the wing, the fuselage, and the vertical tail. The
vertical-tail contribution usually constitutes the major portion and is negative in sign (positive
damping).
The derivative CA, is very important in lateral dynamics because of the important contribution it
makes to the damping of the Dutch-roll oscillatory mode. Its contribution to the spiral-mode
damping is also important. It is desirable to have a large negative value of Cnr for each mode.
A. SUBSONICFor the oscillatory mode, the effects due to lag of sidewash in free oscillation are important andhence should be considered, However, no generalized method is available in the literature toaccount for oscillating sidewash effects on CA,; therefore, only the aperiodic mode of Cnr ispresented here.
In the equation for determining the yawflq,; dcrivative Cer, the sideslip-derivative contribution of
the vertical tail should include the end-plate efft cts of the horizontal tail.
DATCOM METHOD.

The equation for the nondimensional yawing derivative CA, of a wing-body-tail configuration,
based on the product of wing area and the square of wing span Sw bw 2, is given by=(Cn)w+

However, if test data for AC,,

Cn•= (CAcosoi+zp sin/

(ACY

V

(per radian)

7.4.3.3-a

are available, the above equation can be expressed asi-w
t2(per radian)

7.4.3.3-b7.4.3_3-1

where(C"(s thpewing-body contribu tion to Qr obtained from test data or Section
7.3.3.3, based on the prodi,,t of wing area and the square of wing spanl

W0(per radian).
bw

is the wing span.

RP

is the distance from the moment reference -enter to the centem-ofpressure location of the vertical stabilizer, measured parallel to the body
center line. (See sketch (a) in Section 7.4.2.1.) For Datcorn purposes,
the vertical-tail center-ol-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

TP

is the distance from the moment reference center to the center ofpressure of the vertical stabilizer, measured normal to the body center
line, positive for the stabilizer above the body. For Datcom purposes, thevertical-tail center-of-pressure location is assumed to be the quarterchord point of the MAC of the total added panel.

(AC)the tail-body sideslip derivative from test data or Section 5.3.1.1 andisis basedon the wing area (per radian).

V
V(WBH)

ACyis the tail-body sideslip derivative from test data, where p refers to thepanels in the empennage. It is based on the product of wing area andspan (per radian).

)

Sample Problem
Given: Model of reference 2

aHR.
OF
HORIZONTALTAIL -b,:---

4 ~~~~-____-----~"(Cr/)
(1C

4
BMW"l

7.4.3.3-20.)0.43 per rad (test data)(AC Y)v-K~C-0.15 per i,--0.64 per

ktest dat.
(test data)[.Compute:"Calculate the yawing derivative C, for the wing-body-tail configuration

""=(C
= (

)

+2

(equation 7.4.3.3-b)

wr•Y

V(WBH)(0,43)2= -0.15 + (2) (-0.64)"= -0.15 - 0.578= -0.728 per rad (based on Swbw2
This compares with a test value of --0.60 per radian from reference 2.

B. TRANSONIC

!1i

FiI

No generalized method is available in the literature for estimating transonic values of the yawing
derivative C,,. Furthermore, there is a scarcity of experimental data for this derivative attransonic speeds.

S-

C. SUPERSONIC.'No

generalized method is available in the literature for estimating supersonic values- of theyawing derivative Ca.. Furthermore, there is a scarcity of experimental data for this derivative at
supersonic speeds.

•

r

V

REFERENCES

S1.

Campbell, J. P,, and McKinney, M. 0.: Summary of Methods for Calculating Dynamic Lateral Stability and Response andfor Estimating Lateral Stability Derivatives. NACA TR 1098, 1952. (U)2.

Queijo, M. J., and Wells, E. G.: Wind-Tunnel Investiglaion of ltu Low-Speed Static and Rotary Stability Derivatives of a0.13-Scale Model of th Douglas D-558-11 Aiprlane in the Landing Configuration. NACA RM L52G07. 1962. (U)

7.4.-3jRevised September 1970

7.4.4 WING-BODY-TAIL ACCELERATION DERIVATIVES
7.4.4.1 WING-BODY-TAIL ACCELERATION DERIVATIVE CL&

The information contained in this section is for-estimating the nondimensional acceleration
derivative CL& of wing-body-tail combinations at low angles of attack. This derivative is thechange in lift coefficient with rate of change of angle of attack and is expressed as

CLaCL

In general, the methods presented consist of a synthesis of material presented in other sections,although some new information is presented.
This derivative is presented in a manner similar to that used in reference I to calculate the lift ofa wing-body-tail combination. The complete derivative is the sum of contributions of individualcomponents, treated as isolated surfaces or bodies, and mutual interference effects. The mutualinterference effects are assumed to correspond to those due to angle-of-attack variations,established in references 1 and 2 and presented in Section 4.3.1.2.
The horizontal-tail contribution to the derivative CL& is based on the concept of the lag of thedownwash. The nonstationary character of the lift response of the tail to changes in tail angle ofattack is neglected, and the result is attributed entirely to the fact that the downwash at the taildoes not respond instantaneously to changes in wing angle of attack. This concept is also thebasis used in estimating the horizontal-tail contribution to the derivative Cma in Section 7.4.4.2.
The result of this concept is presented in numerous aerodynamic texts, for example, reference 3.
The effect of CL& on longitudinal stability is usually unimportant and is therefore frequently
neglected in dynamic analyses.
A. SUBSONIC
Two methods are presented for determining the acceleration derivative CL& of wing-body-tailcombinations, differing only in their treatment of the effect of the flow field of the forwardsurface on the aft surface

DATCOM METHODS
Method 1. (b'/b" > 1.5)
For configurations in which the span of the forward surface is large compared to that of the aftsurface, the following approach can be used. This method is to be used when the ratio of theforward-panel span to the aft-panel span is 1.5 or greater. The equation of the nondimensional
acceleration derivative CLd of a wing-body-tail configuration, based on the area and mean
7.4.4.1-1

aerodynamic chord of the total forward panel and referred to a moment center at the assumed
center of gravity or center of rotation, is given by

C

C )

+

2 [Kw()

x¢

+KB

x)

C)I

7.4.4.1-a
where the primed quantities refer to the forward panel, the double-primed quantities refer to theaft panel, and the subscript e refers to the exposed panel. (See Section 4.3.1.2 for the definitionof exposed surfaces.)(CL a)WB

is the contribution of the wing-body configuration to the acceleration derivative
CL&, obtained from Section 7.3.4.1.

8•

is the downwash gradient averaged over the aft surface, obtained from Section
4.4.1.

The remaining terms are defined in paragraph A of Section 7.4.1 .1.
D

Method 2. (b'/b" < 1.5)
For configurations in which the span of the forward surface is approximately equal to or lessthan that of the aft surface, the vortex shed from the forward surface interacts directly with theaft surface and the resulting interference effects must be accounted for in the tail terms. Thismethod is to be used when the ratio of the forward-panel span to the aft-panel span is less than
1.5. The equation for the nondimensional acceleration derivative CL& of a wing-body-tailconfiguration, based on the area and mean aerodynamic chord of the total forward panel andreferred to a moment center at the assumed center of gravity or center of rotation, is given by

rg,
C.9

.

2

S(

J

7.4.4. 1-b

All the terms are defined in paragraph A of Section 7.4.1.1 and method 1 above.
Because of the similarity of the two methods a sample problem for method 2 is not included.
However, evaluation of the term (CIC)w"(V,) for a wing-body-tail configuration is presented in
Section 4.5.1.1.
7.4.4.1-2

Sample Problem
Method I
Given: Same configuration as sample problems of paragraph A, Sections 7.4.1.1 and 7.4.1.2.
• iSome of the characteristics are repeated below. Note that for the sake of simplicity thevehicle center of gravity has been taken at Z'/4 and x,.g.- x'" R".
The following ratios based on total forward panel dimensions:

S2

:"b'

0.4377

9"

3Q

2.26

-

-

b'

0.4026b-Se"

0.2667-c'eff

0.8053

b'

S'

-

hH

- b'

0.15780.0988Additional Characteristics:
hH

7.18 feet£eff =

22

3==

M = 0.60

18.32 feet

= 0.8012.133 feet

' =

V'

27.425 feetSA'/430.970R =- 5.16 x 107 (based on
Sea levela' = 40

Smooth surfacesr

NACA 66-206 airfoil sectionsoCompute:Step 1.

Wing-body CL.

-(CL wB

Step 2.

=(Section 7.3.4.1)

0.864 per rad

Lift-curve slope for the exposed horizontal-tail panel
4.0 per rad

Step 3.

(sample problem, paragraph A, Section 7.3.4.1)(sample problem, paragiaph A, Section 7.4.1.1)

Tail-body interference factors

Kw(B)"

1.315,KH(w'

0.550.

(Section 4.1.3.2)(Section 4.3.1.2)(sample problem, paragraph A, Section 7.4.1.1)

7.4.4.1-3

II

Step 4.

Dynamic pressure ratio

qI

0.901

-

q.
Step 5.

(sample problem, paragraph A, Section 7.4.1.1)

Downwash parameter

Obtain value for o'
4.4.1 -g
ao'

-1.6°

=

(Section 4.4,1)

4' at M = 0.2 and correct for Mach number effects by equation=(table 4.1. 1-B)

<10.4003tan Ae/2

A' (

2

1)

S--32+tan'

05[0.98 + (0.4003)2]1/2

0.985

La
CL

CLa

= 4.00 per radCIa'(figure 4.1.3.2-49)

0.80

C AA

=

=

CLmax

0-

=

21.00

-ao0

0.0698 per deg
ai0 ')

CL'-

ao

a CL

=

0.0698 [4 - (-1.6)](Section 4.1.3.4)

4--(-1.6)=

21.0 - (-1.6)

max
A'ff

-

1.0

A'
(figure 4.4.1-66)
b'eff
b' b = 1.0

A'f
b'ef

7.4.41-4
S= 5.43

A¢/2)54
K-,:,(Section 4.4.1)

5.0=

45.50 ft

= 0.248=0.391

=0.40(figure 4.4.1-67)
•

Ay=

0.49

1.10

(figure 2.2.1-8)

Type of flow separation: Leading-edge separation is predominant

0.4 1CL\ahH(22 + 23) (a= 7.18 -(18.32

A-"/+ 12.133)

4
4

7.18 -(30.453)

0.56 A'=(0.0596)

=

b
/

tan P

-

2

(equation 4.4. 1-d)(0.41) (0.391)(3.14) (5.0))

45.50
(0)
2

5.37

0.56(5.0)0.391'-- 7.16 (See page 4.4.1-6)
0.3

CL'
,
Lru

[0.78 + 0.10 (X' - 0.4) + 0.003 A' 4 bIb=(figure 4.4.1-68a)(Ae/ 4 in deg)(equation 4.4.1 -f)(0.833) (45.50)= 37.90 ft= b ff -(boff-

22R

,

b
"ru)

45.5 -(45.5

=

45.5-

bH
b"
- =
b4
b.

-

I1/2
(equation 4.4.1-e)b'tr37.9) L

11/2
F36.64- .1)

L(45.) (7.16)j

2.5 = 43.0_

19.915

4-3.0

43.0

= 0.463

7.4.4.1i-5ta
2a_2(5.37)43.0

bv= 0.25- 0.93•low(figure 4.4.1-68b)

I'a)

--

CL)

(0.93) (0.49)=

0.4557wv

M0.=

4.50 (sample problem, paragraph A, Section 7.3.1. 1)

speed
C')

=

)M
0 .6 aa0)owCl\a4M---''(Lc'M=0"w(equation 4.4. l-g)

speed
spccd

= (0.4557)(0.513)-

Solution:

CL(CL)WB,+

2[KW(B)+

KB(W)]

g)

T

(CLe

(equation 7.4.4. 1-a)

S0.864 + 2(1.315 + 0.550) (0.0988) (2.26) (0.513) (0.901) (4.0)

7.4.4.1-6=

0.864 + 1.540=

2.404 per rad(based on the area and mean aerodynamic chord of the total
forward panel and referred to a moment center at g74)

K.
B. TRANSONIC
No accurate methods are available for estimating the characteristics of isolated panels, thedynamic- pressure ratio, or the downwash gradient in the transonic speed regime. The aerodynamic interference "K" factors for slender configurations are relatively insensitive to Mach
number; however, for nonslender configurations transonic interference effects can become quite
large and sensitive to minor changes in local contour.
DATCOM METHODS
It is recommended that the methods presented in paragraph A above be applied directly to thetransonic speed regime. Care should be taken to estimate the various parameters of equations
7.4.4.1-a and -b at the appropriate Mach number. The interference "K" factors should beobtained from paragraph C, Section 4.3.1.2.
C. SUPERSONIC
The information included in the Datcom accounts for most of the mutual interferences that
occur between components of wing-body-tail configurations at supersonic speeds.

DATCOM METHODS-

The methods presented in paragraph A above are also applicable to the supersonic speed range.
Care should be taken to estimate the various parameters of equations 7.4.4.1-a and -b at theappropriate Mach number. Method 3 of paragraph C of Section 4.4.1 should be used to evaluate
the last term of equation 7.4.4.1-b.

Sample Problem
Method I
Given: Same configuration as sample problems of paragraph C, Sections 7.4.1.1 and 7.4.1.2, andparagraph A of this section. Some of the characteristics are repeated below.
The following ratios based on total forward panel dimensions:-= 2.26

S'-

0.0988Additional Characteristics:
h11 = 7.18 ft

M = 1.40

A'= 5.0

p

0.98

= 40
cg at -'/4
7.4.4.1-7

Compute:(Section 7.3.4.1)

Wing-body CN.

Step I.

(CN.

KKB(W )(\j

=KW(B)

L)CC.
(equation 7.3.4.1-a)

(CN)

=

-0.485 per rad (sample problem, paragraph C, Section 7.3.4.1)

Lift-curve slope for the exposed horizontal-tail panel (Section 4.1.3.2)

Step 2.

" = 4.025 per rad (sample problem, paragraph C, Section 7.4.1.1)

C

Tail-body interference factors (Section 4.3.1.2)

Step 3.
Kw('

1.315

K B(w

= 0.4028(sample problem, paragraph C, Section 7.4. 1.1)

Dynamic pressure ratio (Section 4.4.1)

Step 4.
q,

__
q•

0.80

-

Step 5.

(sample problem, paragraph C, Section 7.4.1.1)

Downwash parameter (Section 4.4.1)

Obtain value in the plane of symmetry at

9Z__

3652 ft

x, y, and z referred to wind axes:

X

7.4.4.1-8

h1 sin ot + (36.52) cos c

=

36.93

y =0 (plane of symmetry)hH 4:os U -(36.52)

z

4615

sin a

2z

2x

1.656

-

• " i-9/b'y-=0.2029WbW

0

A'= 4.90
2h

=

1.0*

(figure 4.4.1-74a)

2z_

2h

2 +
2z
b= +

2h
'b'

_b4e
ff

o'3

(a'in rad)0.2029 + (1.0) (0.0684)--

0.27130.29*(figure 4.4.1,-76a)

I(

Solution:

(

)

~

x

~

405

...

a

(equation 7.4.4. l-a)

.•
I.-.

)

=-0.485 + 2(1.315 + 0.4028) (0.0988) (2.26) (0.80) (0.29) (4.025),"=-0.485 +.0.716= 0.231 per rad(based on the area and mean aerodynamic chord of the total forward
panel and referred to a moment center at C'/4)
REFERENCES

1.

Pitts, W., Neilson, J., and Kaettari, G.: Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic,
and Supersonic Speeds. NACA TR 1307, 1959. (U)2.

Spreiter, J.: The Aerodynamic Forces on Slender Plane- and Cruciform-Wing and Body Combinations. NACA TR 962,
1950. (U)3,

Etkin, B.: Dynamics of Flight. John Wiley and Sons, Inc., Now York, 1959. (U)*The data obtained from figures 4.4.1-74a and 4.4.1-76a (or triangular plartforms have been extrapolated to pA' = 4.90. (Seereference 14, Section 4.4.1.)

74.4.1-9.
.-.

Revised September 1970

7.4.4.2 WING-BODY-TAIL ACCELERATION DERIVATIVE Cm'&

The information contained in this section is for estimating the nondimensional acceleration
derivative Cm& of wing-body-tail combinations at low angles of attack. This derivative is thechange in pitching-moment coefficient with rate of change of angle of attack and is expressed bya*This derivative is important in longitudinal dynamics, since it is involved in the damping of theshort-period mode.
The Datcom methods are based on the same assumptions that were made for the total pitching
derivative CL& of the wing-body-tail combinations, and the general discussion of Section 7.4.4.1
is directly applicable here.
A. SUBSONIC
Two methods are presented for determining the acceleration derivative Cm& of wing-body-tailcombinations, differing only in their treatment of the effect of the flow field of the forwardsurface on the aft surface.

DATCOM METHODS
Method 1. (b'/b" >' 1.5)
For configurations in which the span of the forward surface is large compared to that of the aftsurface, the following approach can be used. This method is to be used when the ratio of theforward-panel span to the aft-panel span is 1.5 or greater. The equation for the nondimensional
acceleration derivative C,&of a wing-body-tail configuration, based on the area and the squareof the mean .e-rodynamicchord of the total forward panel and referred to a moment center at
the assurned center of gravity or center of rotation, is given by(C=-

2[I-C,) +x)

2

A••XLz•-(Xcui
C \Le
qK)1

7.4.4.2-a
where the primed quantities refer to the forward panel, the double-primed quantities refer to theaft panel, and the subscript e refers to the exposed panel. (See Section 4.3.1.2 for the definitionof exposed surfaces.)
7.4.4.2-1

is the contribution of the wing-body configuration to the acceleration derivative(Crn&)wB

Cm a, obtained from Section 7.3.4.2.

is the downwash gradient averaged over the aft panel, obtained from Section
4.4.1.

--.

The remaining terms are defined in paragraph A of Section 7.4.1.1.
Method 2. (b'/b" < 1.5)
For configurations in which the span of the forward surface is approximately equal to or lessthan that of the aft surface, the vortex shed from the forward surface interacts directly with theaft surface, and the resulting interference effects must be accounted for in the tail- or aftsurface-contribution terms. This method is to be used when the ratio of the forward-panel spanto the aft-panel span is less than 1.5. The equation for the nondimensional acceleration derivativeC,,j of a wing-body-tail configuration, based on the area and the square of the mean aerodynamic chord of the total forward panel and referred to a moment center at the assumed
center of gravity or center of rotation, is given byCC,,, )WB+ 2

7.4.4.2-b

C.-,.7
All the above terms are defined in paragraph A of Section 7.4.1.1 and method I above.
Because of the similarity of the two methods a sample problem for method 2 is not included.
However, evaluation of the term (CL a)w"(v) for a wing-body-tail configuration is presented in
Section 4.5. 1.1.
Sample Problem
Method I
Given: Same configuration as sample problems of paragraph A, Sections 7.4.1.1, 7.4.1 .2, and"7.4.4.1. Some of the characteristics are repeated below.
The following ratios based on total forward-panel dimensions:
Se

c

-

2.26

S

0.0988-Additional Characteristics:
M

0.60

7.4.4.2-2
KgI

b

0.80

o'

4'

cg at

/4

Compute:

Step I

Wing-body Cm.

(Section 7.3.4.2)[KW(B) + KB(W)](Cm.)(-)Q--)/

Cm.)e + (Cm)

(')

C)

(equation 7.3.4.2-a)= -0.400 per tad (sample problem, paragraph A, Section 7.3.4.2)

Step 2.

Lift-cuive slope for the exposed horizontal-tail panel (Section 4.1.3.2)
L(C)Step 3.

= 4.0 per rad (sample problem, paragraph A, Section 7.4.1.1)

Tail-body interference factors (Section 4.3.1.2)

KW(B)

=

1.315

KB(w)

= 0.550(sample problem, paragraph A, Section 7.4.1.1)'W -Step 4.

Dynamic pressure ratio (Section 4.4.1)-

Step 5.

0.901 (sample problem, paragraph A, Section 7.4.1.1)

Downwash parameter (Section 4.4.1)0(.513(sample problem, paragraph A, Section 7.4.4.1)

Solution:

Cm= C
Cm)WB

2 [KW(B)+ KB(W)I

(S.
)

C~

actL)(C(equation 7.4.4.2-a)-0.400 - 2(1.315 + 0.550) (0.0988) (2.26)2 (0.901) (0.513) (4.0)= --0.400 -- 3.48

= -3.88 per rad(based on the area and the square of the mean aerodynamic chordof the total forward panel and referred to a moment center at E'/4)
7A.4.2-3

B. TRANSONICThe comments of paragraph B of Section 7.4.4.1 are directly apphcable here.
DATCOM METHODS
It is recommended that the methods presented in paragraph A above be applied to the transonic
speed regime. Care should be taken to estimate the parameters of equationis 7.4,4.2-a and -b atfactors should be obtained fromthe appropriate Mach number. The interference "K"paragraph C, Section 4.3.1.2.
C. SUPERSONICThe comments of paragraph C of Section 7.4.4.1 are directly applicable here.
DATCOM METHODS
The methods presented in paragraph A above are also applicable to the supersonic speed range.
Care should be taken to estimate the parameters of equations 7.4.4.2-a and -b at the atppropriate
Mach number. Method 3 of paragraph C of Section 4.4.1 should be used to evaluate the last termof equation 7.4.4.2-b.
Sample Problem
Method I

Given: Same configuration as sample problems of paragraph C, Sections 7.4.1.1, 7.4.1.2, and7.4.4.1, and paragraph A of this section. Some of the characteristics are repeated below.
The following ratios based on total forward-panel dimensions:
-

S "

•.,
2.26

0.0988S,c
Additional Characteristics:
M

1.4

cg at ý/4

40
4'

0,98

Compute:Step 1.

Wing-body CM, (Section 7.3.4.2)

(M
C )WB

KW(B) +K 1 ~ 3 '(Se~)(cr.6)+()B1(equation 7.3.4.2-a)

7.4.4.2-4

1.308 per rad (sample problem, paragraph C, Section 7.3.4.2)(Cma)

Lift-curve slope for the exposed horizontal-tail panel (Section 4.1.3.2)

Step 2.

)

CN

Step 3.

4.025 per rad (sample problem, paragraph C, Section 7.4.1 .1)

Tail-body interference factors (Section 4.3.1.2)

KW(B)= 1.315(sample problem, paragraph C, Section 7.4.I.1)

KB(W)0.4028

Dynamic pressure ratio (Section 4.4.1)

Step 4.

0.80 (sample problem, paragraph C, Section 7.4.1.1)-

Step 5.

Downwash parameter (Section 4.4.1)=-0.29(sample problem, paragraph C, Section 7.4.4.1)

Solution:=(C

21KW(B)

KB(W)
K

(

(equation 7.4.4.2-a)4=

1.308 - 2(1.315 -i 0.4028) (0.0988) (2.26)2 (0.80) (0.29) (4.025)=

1.308 - 1.619= -0.3 1 per rad(based on the area and the square of the mean aerodynamic chordof the total forward panel and referred to a moment center at U'/4)

7.4.4.2-5

April 1978

7.4.4.3 WING-BODY-TAIL DERIVATIVE CD&
This section presents a method for estimating the wing-body-tail derivative CD& at subsonic speeds.
This derivative is the change in the drag coefficient due to a change in & at a constant pitch rate andis defined as

aCD
CD.

I where CD is based on Sw-

In general, this derivative is small and has a negligible effect on longitudinal stability; hence, it isusually neglected.
A. SUBSONIC
The wing contribution to CD& can be estimated using unsteady-flow theory. The body contribution
is small and has been neglected. The tail contribution is computed from conventional downwash-t-ag
theory. The horizontal-tail lift due to & was taken to act normal to the local flow direction at thetail, to produce a fora.. component in the direction of the free-stream.
DATCOM METHOD
The wing-body-tail derivative is given by
S

(CD)

+(CD)

+

aC.

7.4.4.3-a

whereis the wing contribution to CD. obtained from Section 7.1.4.3.

(C&CD

wa

*(C0) is the horizontal-tail contribution independent of angle of attack, obtained by

I
(CD)&

4

O)ý57.3

SIf (Q.-CO
coaF

CL

1HS W

sin cda'F

E/2

-

au'

(2e0 - ill

7.4.4.3-b

where
CL

aH

is the horizontal-tail lift-curve slope (based on SH. obtained from test data orfrom Section 4.1.3.2 (per degree).
is the horizontal-tail reference area.

*S14

SSw

is the wing reference area.
7.4.4.3-1

I'

RHH

is the distance from the momen reference center to the center-of-pressure
location of the horizontal stabilizer, measured parallel to the body center line.
For Datcom purposes, the horizonta!-tail center-of-pressure location is assumed
to be at the quarter-chord point of the MAC of the total added panel.

zH

is the distance from the moment reference center tc the center-of-pressure
location of the horizontal stabilizer, measured normal to the body center line,
positive for the stabilizer above the body. For Datcom purposes, thehorizontal-tail center-of-pressure location is assumed to be at the quarter-chordpoint of the MAC of the total added panel.
is the fuselage angleof attack.
F!

ae

--

is the downwash gradient at E/4 of the horizontal tail, obtained from test dataor from Section 4.4.1.
is the wing MAC.

C0

is the downwash angle at Z/4 of the horizontal tail at aF

iH

is the horizontal-tail incidence with respect to the fuselage reference line.

- raý-

3C.1
is the change in horizontal-tail contribution with angle of attack, obtained by

0.

H

(

SH

1

HCo s UF + H sinF(2

7 .4 .4 .3-c

where all of the terms are defined above.
Sample Problem
Given: Same configuration as sample problem of Paragraph A of Section 7.1.4.3.
Tail Characteristics:

RH = 8ft

ZH = 0

E0

iH

= 0

(RH

COS a'=

10

16 ft 2SH

= 0.05 per deg

CL

o

='

0.32

Compute:
(

D

( D0

5
H.4

g 7.4.4.3-2

•0

lCLn

H

C
.3
L-

cse0SWl-

U1 sin

c F)

3C

-

(2

i11)

(Equation 7.4.4.3-b)

i? ....

:

Mc

q

-'

~~16.0) [(8,0)(0.9998) + 0] 03)2(.)0

1
(

53

-

4/2

64.0

(o.32)2(1.o) -- 00.000558

aC~~\
acF/

57.3

S11
aH SW

.3 (0,05)
=

oa

i

+ 1

cF

-/2

'6--". 0

" 4/2

E

(2

k

1)

(Equation 7.4.4.3-c)(0.32)[2(0.32) - I]

-0.000100
Solution:
i~~~

CD&

(CD)

(CD&# W
,0.0085(CDCD+ (CD)
CDo0.

D.
+-t-.\ 801Ca(F- )Hper deg(Sample Problem, Paragraph A, Section 7.1.4.3)

F(Equation 7.4.4.3-a)

0.0085 + 0.000558 + (-0.000100) 1.0

0.00896 per degree
B. TRANSONICNo method is presented.
C. SUPERSONICNo method is presented.

I

7.4.4.3-3a'April 1978

7.4.4.4 WING-BODY-TAIL DERIVATIVE Cy.
This section presents a method for estimating the contribution of the vertical tail, in the presence of
the wing and body, to the derivative Cy4 at subsonic speeds. This derivative is the change inside-force coefficient with variations in the rate of change of sideslip angle at a constant yaw rateand is defined as
CY.

=

acy
_

, where Cy is based on Sw.

In general, at low to moderate angles of attack, this derivative is small and has a negligible effect onlateral stability; hence, it is usually neglected.

A. SUBSONIC
A wing contribution to Cyý can be evaluated by using unsteady-flow theory, but at low tomoderate angles of attack it is generally considered small and is neglected. At low angles of attackand for attached-flow conditions, the largest contributor to Cyv is the vertical tail. The method
herein applies a sidewash-lag theory in an analogous manner to the downwash-lag theory that is usedin finding the horizontal.-tail contribution to CL&,. The body contribution is small and has been
neglected,

411-

For a brief discussion of the physical flow phenomena at high angles of attack, i.e., leading-edge
vortex sheets and flow separation, and a comprehensive bibliography on related subject matter, thereader is referred to Reference 1.
DATCOM METHOD
Design charts for predicting the change of wing sidewash angle with the change of sideslip angle arepresented as a function of wing geometry, i.e., aspect ratio, sweep, and taper ratio, at Mach numbers
0.2 and 0.8. These design charts were generated from wing loadings in sideslip by using the theory
presented in References 2 and 3.

l

The vertical-tail contribution to the derivative Cy4 at low to moderate angles of attack is given byCy

u Sv ( P cos a F + z sin aF)

2 CL

0a

Sw

744.7.4.4.4-a

bw

where
CL

Sor,

aV

is the lift-curve slope of the vertical tail obtained from test data or Section 4.1.3.2.
is the change of sidewash angle (due only to the wing, i.e., no sidewash due tofuselage cross flow) with respect to the change in the sideslip angle. This factor canbe estimated by
7.4.4.4-1

a
-

~-

"

~

~-

-.-

O+01
P3O
WB

or
13 = o/3jF +03

57.3'P '.F

7.4.4.4-b*

where
0a

is the sidewash contribution due to angle of attack, obtained from Figures
Zv

7. 4 .4 .4-6a through -6p as a function of - and wing geometry. zv is theb/2
distance between the wing Z/4 point and the vertical-stabilizer center-ofpressure location, measured normal to the free-stream direction. For Datcom.
purposes the vertical-tail center-of-pressure location is assumed to be at thequarter-chord point of the MAC of the total added panel.
= z, cos Ot - V sin a-

7.4.4.4-c

where zP and P are defined below.
is the fuselage angle of attack in degrees.
%aris the sidewash contribution due to wing dihedral, obtained from Figures
7.4.4.4-22a through -22d as a function of -r

and wing geometry.

is the wing dihedral in degrees.
is 4 thesidewash contribution due to wing twist, obtained from Figures
7. .4 .4-26a through -26p as
a function of
and wing geometry.
-ois the wing twist in degrees between the root and tip sections, negative forwashout (see Figure 5.1.2.1-30b).

a0B

is the sidewash contribution due to the body effect on the wing loading, It is

O3WB

ZV

presented as a function of -7 and wing geometry, at three different bodyradius-to-wing-span ratios of 0.06, 0.12, and 0.24. In addition, the wing
position on the body has an effect on this term. However, for Datcom considerations a low-wing position was assumed as shown in Sketch (a). For ahigh-wing position as shown in Sketch(b), the sign of ain Figures
7.4.4.4-42a through -42p becomes negative.

°WB*A more accurate result can be'obtained from the equations

J

C'local cv dZ

07.4.4.4:-2

i*

-•;+:

:+

•

•'-

.'

",,•

•'

.+'•

-'.S-•'+.•

".

J

+

,-

.-..-....-..-.

-..

-

"

-

-

-'

-

Iz

SKETCH (a)42high-wing position as shown in sketch (b), the sign of j wB in Figures 7.4.4.4-42a through - p
becomes negative.

rz

SKETCH (b)7.4.4.4-3

"h

-,

-

-

-

-

-

-

-

,

;j

Sv

is the vertical-tail area, consistent with the vertical-tail area used to define
CL
aV

S

is the wing reference area.

kP

is the distance from the wing quarter-chord point to the center-of-pressure
location of the vertical stabilizer, measured parallel to the body center line. For
Datcom purposes, the vertical-tail center-of-pressure location is assumes to beat the quarter-chord point of the MAC of the total added panel, (See Sketch (a)of Section 7.4.2.1.).zis the distance from the wing quarter-chord point to the center-of-pressure
location of the vertical stabilizer, measured normal to the body center line,
positive for the stabilizer above the body. For Datcom purposes, thevertical-tail center-of-pressure location is assumed to be at the quarter-chordpoint of the MAC of the total added panel.

bw

is the wing span.

-. =Sample Problem

Given: The following wing-body-tail configuration
Wing Characteritics:
A

7

Sw = 3500 ft2

X = 0.25

ALE = 350

Low wingbw

0 =p = 30r5=

156.52 ft

Vertical-Tail Characteristics:
CL

=

Sv

0.055 per deg=

600ft 2zv
= 0.242b/2
Additional Characteristics:k=

M =0.860.0 ftzP=

20.OftBody Radius
=0.12
b/2
_______-

r.

i a;

7.4.4.4-4

a

=

jO

Compute:
Deteriaine.

uat + -• P -a=.

0.F 57.3(Figure 4.4.4.4.-22d)

Y.=-0.56a=or-0.0113(Figure 7.4.4.4-26k)

0.07(Figure 7.4.4.4-42k)=

n= (-0.013)(1) + (0.
-(Equation 7.4.4.4-b)(Figure 7.4.4.4-6k)-0.013=0+
06~ 'WB3- (-0.0113)(-5) + (0.07)-0.013 - 0.0293 - 0.0565 + 0.07

= -0.0288

Solution:
Sv ( 2 P COS aF + zp sillF)

C.2C =
2(0.055)(-0.0288)=(Equation 7.4.4.4-a)b

600I (60)(0.9998) + (20)(0.01745)j-0.000209 per deg

B. TRANSONICNo method is prqsented.

C. SUPERSONICNo method is presented.
REFERFNCES
1.

Coo, P. L., Jr., (Graham, A. B., and Chambers, J. R: Summary of Information on Low-Speed Lateral-Directional Derivatives
Due to Rate of Change of Sideslip 0. NASA TN D-7972, 1975. (U)2.

-Do Young, J., and Harper, C. W.: Theoretical Symmetric Spar, Londing at Subsonic Speeds for Wings Having Arbitrary
Planform, NACA TA 921, 1948. (U)3.

Qusijo, M. J.: Theoretical Span Loading Distribution and Rolling Moments for Sideslipping Wings of Arbitrary Planform inincompreshible;clow. NACA TR 1259, 1956. (U)

7.4.4.4-5
Ma.-q---

.

t-.

M .2
X= 0

---%024- .,020.

(ALF

(a)

0

-

-- :0---- --- ------_-

-

.

-;016

....

"-.012

--

-:008

-

-

(per deg)-. 004A
I
-

~4
.6

[

.004

.0

/b/21

•

.008

.012

.016
FIGURE 7.4.4.4-6 SIDEWASH CONTRIBUTION DUE TO ANGLE OF AITACK*

7.4-4.4-6

.

:-.

.-

.

"M= .2;k =0

ALE =350

S

-:08,

-

-

-

--

-

--

'-09

_

-

--07,

"-.06- -~-.05 ------

(per deg)

-

-

.04

-

-

-

-

-.03m

.02

-

A

0 1•

-

S

-

-

.2

"0

-

.4

Zv

--

-- _

.6

.8

1.0

b/2

(i

FIGURE 7.4.4.4-6 (CONTD)
7.4.4.4-7
-

.

M

=

.2

X= .25
ALE =0.020- (c)I----.016--.012--,004.--A(per deg)

-,

0.

!

a=-=-

Zv
b/2

.004

.008--

.012

6--

.016t

FIGURE 7.4.4.4-6 (CONTD)

7.4.4.4-8

6

.8

1.0

.2

M

= .25
ALE = 350

(d)

-%09'

.;08*-

'"

-

-

__

-

-

--:06"

S.-.05(per deg)

-02

--

0

.2

.4

Zv

.6

.8

1.0

b/2
FIGURE 7.4.4.4-6 (CONTD)
7.4.4.4-9

r

..

•

-

....

L

.

-

--

-.

r

.

M= .5

A

-012 -----. 004-0,O-----(per deg)

0

/4

[ -'

ý7
_1
.2

--

.4

zv

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hl/2

.

.8
.

.

.

l
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.

.004-

.008-

-

.012

.016

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-

FIGURE 7.4.4.4-6 (CONTD)

7.4.4.4-10

-

-

-

...

•: _- • '•ij: •"...........• :I•:
'•

- • •

''

'i: . .'- •

•

•

..

. ..

.

". ..

• - • • •-• - . . "• •

M =.2
X= .5
ALE

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350

-

-08

---

-.07.

(per deg)-. 03-

-4

IA

__

F44
0

.2

.4

zv

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.8

1.0

b/2
FICURE 7.4.4.4-6 (CONTD)
7.4.4.4-11

-

M

.2

A= 0
ALE

•.0 2 4 -

0

'

(g)

-7020 -

-016

-

-

-..

-

'.- 12---

i-•

.•.

~(per A"
deg)

-4J0

-

-

.2iI

0

.8 1.0
b/2-

-,-

---

.004--.008*-

FIGURE 7.4.4.4-6 (CONTD)'

7.4.4.4-12

.,r

-"

M
X.

-. 10

ALE-

.2
1
350

(h)

--.09=-.08 =-. 07-

-06

•'6

•,

-

i H

-

-705 -(perdeg)

-04

-

.

..

A-

-

""102

0.-o
0

-,

-

.2

.4

Zv

.6

.8

1.0

b/2
FIGURE 7.4.4.4-6 (CONTD)
7.4.4.4-13

M =.8x=0-. 024-

AIE =0i--020 --. 016-. 012-i-. 7008

_

:004'
..

'13•

S~

.

-

-

-

.. .

_

(per deg)

.
.004-

. 16

FIGURE 7.4.4.4-6 (CONTP)
7.4.4.4-14

4

.6

S-

.8

1.0

M=.8
;=0
.10

-09

-ALE

35o

-

.

'--.08-

__

-:07.

-

--706.

o",~

-

____

(per deg)

.04

41

81

b/2
FIGURE 7.4.4.4-6 (CONTD)

7.4.4.4-15

M =.8
X =.25-. 02~
(k)
-

_

-

0

ALE

-:016--h-'-_A-. 012-------------------008--------------------(per deg)

-

-

---

m

-

-

z

.0084---FIGURE 7.4.4.4-6 (CONTD)

-

-

-

_

-

-

-

M =.8.25
ALE = 350

(0"

.--

T05

-

-

(per deg)

04

-702

--

-

-

-

_

-

A

-

0.2

.4

zv

.6

.8

1.0

b/2
FIGURE 7.4.4.4-6 (CONTD)
7.4.4.4-17

M =.8
X=.5-. 024rm

ALE.0O16.-

-

____

0

--

-

-:008 •'

o,, -. 004 -____(per deg)

4

0

.

4

Z _v .6

b/2
.004-

.008-

.012-

~~.016'

HIGURE 7.4.4.4-6 (CONTD)
7.4.4.4-18-

A.0•12 •

*

-

.18

1.0

-~L

M

.8

~350

',-

-.08 --707--

-

-

-

-

-

-

-

-

-

-

-

(per deg)-:03-----02 -

A
I

-

---

b/2
FIGURE 7.4.4.4-6 (CONTD)
4

7.4.4.4-19

M=.8

x= lI

[.._

Ao)

-. 024•0

A"-020-

'-

.016.

"12-

_-

,

--

--008
(per deg).004 -..

.0084-.012I

FIGURE 7.4.4.4-6'(CONTD)
7.4.4.4-20--

• .•,''rv..

..

'

-"

- "U~>- -• w--.

- rr• .--" ---- " "-...

•"

•-

-...

.

.

..

.

.

M=.8
X=l1

.

- o.(p) ..

-:.09

ALE =350-.

_ .08..

':•'•?

-07-

6

-

-

.o-05
(per deg)

04o.-

-

\

-i.03

402.\.-

--

,

•i

"l

I

•-n
-•

.2

.0

FIGURE 7.4.4.4-6 (CONTD)
•i

.4

Zv

.6

.8

1.0

7.4.4.4-21

-1,2(pot 4e).

1 ,0

b/2
~1GUUE
*~

7-4.4.4-22

4,44-,22 SIDEWNASH CONTRI~uTiQN DI~EODHDA

M =.2

-•2.0

ALE

= 350

(b)

-18

-

-1.2

-

-

--

--

]*

---

(perdeg)_l.O-

-. 8

--

*

.44

0

.2

.4

z

.6

.8

1.0

b/2

"FIGUkE7.4.4.4-22 (CONTD)
7.4.4.4-23

M =.8

-1.84

--

-1.2

-

UPI-

(pet-deg) -1.0

-

--

-

--

0-

0
FIGURE 7.4.4.4-22 (CONTD)*

7.4.4.4-24.2.4.10

b/ 2

-

-

-

-A

M = .8
ALE

= 350

(d)A

-2.0

-1.-

-1.2

-

...

(per deg)_,. 0

-.6

0I

0

.2

.4

zv

.6

.8

1.0

b/2
FIGURE 7.4.4.4-22 (CONTD)
7.4.4.4-25

M =.2
AF = 0At-.0o2o

-

_

_

1t

_

_

0

I

_

(per deg)

-006

-...-

..

-n004.

-. 0020~b/2
FIGURE 7.4.4.4-26 SIDEWASH CONTRIBUTION DUE TO WING TWIST

7.4.4.4-26j

M = .2
X=0
ALE =35

-.09-

-

".08

-- 07---.06-000

:05
.-

-

-

-

-

(per deg)

.04 -

.03 -

-.02 --

\-

.01-

0

.2

.4

Zv
b/2

.6

.8

1 ,0

FIGURE 7.4.4.4-26 (CONTD)
7.4.4.4-27

M

.2

ALE

0

X =.25

(c)2

-.

...

-

-

-. 012-. 016 --

.

-

_
A

"-7014•]"7.012°o -'=010.-

-

_

-

-

(per deg)•.008 ..

.

-:.006-- 004-

4

-

-7002

____

___

-

__

01

0

FIGURE 7.4.4.4-26
7.4.4.4-28(CONTD)

..

4

Zv

.6

.8

1.0

.,.

.

," • ,

--

-.

.

.

,

. .

- -

-7

p7-7

,

-7

-7

M

.2
X = .25

ALE =

350

(d)

S~.-09,-.07Q.

CP -.ii-07---- --:06--------------1(per dcg)

.

.044
.:03-

*

-1

.02--

-

\

01
0

.2

.4

Zv

.6

.8

1.0

b/ 2
FIGURE 7.4.4.4-26 (CONTD)

4

7.4.4.4-29

'.-

M
X

2
.5

ALE

0

..

-. 020 !(e)

-018

•

-

-016

--

_

A
--.014--°3 o -010(per deg)

-

.008-

--

--.006 -

_

----

---

\-

4.o

--

0V-;. .6---.
[ 7"••,••": • .•• 'i , •'02 • N -' "7,'_.,..
•.4" :..
b/2
FIGURE 7.4.4.4-26 (CONTD)
7.4.4.4-30-. -.. -- 1.0 _••.. . .. - .-. . . .
.8

M =.2=.5.-

ALE = 350

-10o (f)".09 --.

-.0----08•-. 07----

G°o

-'.05

--

_

(per deg)

.04

A

-

..03 -

0

-

-

.2

.4

Zv

.6

.8

1.0

b/2
FIGURE 7.4.4.4-26 (CONTD)

7.4.4.4-31

M =.2
X=

(gALE

--l

(g)

1

= 0
--

-

-.09

"--081

0

4

-

-

--

-

°o --105(per deg)

-. 04

A,1
.01

0\

0

.2

.4

Zv

b/2
FIGURE 7.4.4.4-26 (CONTD)
7.4.4.4-32

.6

.8

1.0

'M

ALE= 350

-10

h-),

-

I

- 0- """

"

.2

-

-

-

-.068/

m'

°0o

•

--.05

(per deg)ASA--.04I

II--.03 -•'
-.

-.;02

-

_

-

-

•.-

-

0
0

.2

.4

Zv

.6

8

1.0

b/2
FIGURE 7.4.4.4-26 (CONTD)
7.4.4.4-33

.....

k~.r

"M= .8":',,'"

ALE = 0-. 010A--.:012a~-. 018

--

-010

-

-

(per deg)-- 008--.006 (./2"-.00"4

0:0
0

.2

.4

zv
b/2

FIGURE 7.4.4.4-26 (CONTrD)
7.4.4.4,34

.6

.8

1.0

M =.8N=0A,,= 35-

...

-09--

.08 ---.

-:07-

ý06ii..
ago

-- 05

-

(per deg)-. 04 -•--2,.

-03

A

-r

i

.-

..

o

=,

--

0

.2

.4

Zv

.6

"

.8

- 1.0

b/2
FIGURE 7.4.4.4-26 (CONiD)
7.4.4.4-35
I--

M =.8.01-"-.'.008p%

16(per deg)-k)".,--.0

-

-

.....
"01

.2

.401

81.

0.
b/2
FIGURE 7.4.4.4-26 (CONTrD)
7.4.4.4-36

M

=.8

X = .25
AL-L = 350-09 .-~-7.08

"-

--

-06

age

-~05m-(pfr deg)

ýA-

-. 03

.02

44

b/2
FIGURE 7.4.4.4-26 (GONTD)

7.4.4;4-37

M = .8),.5.-020o-__-. 016

.

-0.14

-A

~~-tio

ALE=0
0.
~

-

-

A

.

-01.....

-

-

-

.

(per deg).008'-

-

-

-

-006

-

-A0

0

-

00

.2.4

Zv
_

b/2
FIGURE 7.4.4.4-26 (CONTD)
7.4.4.4-38

.6

-•

-

.8

1.0

M

.8

A. X =.5
ALE
=3-5°

(n)

-:09-

-07
--70

O0p

0

L

--

- .05Lo(per deg)

t

--

-.043

S'

2'

.4

,L

.6

.8

1.0

b/2
FIGURE 7.4.4.4-26 (CONTO)
7.4.4.4-39

10-ALE
(o)

0:6

M =.8
\= !

-.

=0

-. 09

084,
-.

-077

°0o

--.05

-

-_

_

-

-

(per deg)

*

\\

0- 0

.2

.4_v

.6.1.0b/2

FIGURE 7.4.4.4-26 (CONTD)
7.4.4.4-40-_

M =.8

1

X=

S-ALE

=3 50

-08-

-07

-

-

(per deg)

•,i\A

-. 04

-

-

--

_

_

.o03--

--

0

0

.2

.4

Zv

.6

.8

1.0

b/2
FIGURE 7.4.4.4-26 (CONTD)
7.4.4.441

M=.2
BODY RADIUS = .06

b/2

0 < A< 350

.20-.

.15--

3

OWB

.10

-

.05 -..-..

.

A

/4
010

;.m
.

0•
..

0

.2

.4

z--v

.6

.8

1.0

b/2
FIGURE 7.4.4.4-42 SIDEWASH CONTRIBUTION DUE TO BODY EFFECT
7.4.4.4-42

Im

j

M .2
BODY RADIUS
b/2
.20•

.12

0_ A<35
,
-0

• Ii

SI

A

0 .

.

.4-.-6-

-

0

.

b/2
FIGURE 7.4.4.4-42 (CONTD)
7.4.4.4-43

"-

, --

-

-

.

-

,.

M=.2
BODY RADIUS

.24

b/2
X= 0
.20.

0

-LE

-

-

-

-

-

-

3

P WB

.101

.05*-

10

0r
0
FIGURE 7.4.4.4-42 (CONTD)

7.4.4.4-44

Li.

.2

.4

z
b/2

.6

.8

1.0

r

.

-

.

"

-.

A
.2
BODY RADIUS = .24

b/2

X= 0
.20-

350

ALE

(d).

.15*

-

X

S,,

i0
0
FIGURE 7.4.4.4-42 (CONTD)
4'

.2

.6

.4

.8

1.0

b/2
7.4.4.4-45

M =.2
BODY RADIUS

=.24

b/2
.25

X

.20-AL

0

aWB

10

01

00

.6

0.2'
b,'2
FIGURE 7.4.4.4-42 (CONTO)
7.4.4.4-46

.8

1.0

M=.2

BODY RADIUS

.24

b/2
S= .25
AALE=35
ý 350

.20-

15

Lj

-

..

-

~~~~WB-

--

.10

-

I--

4.

---

-

o---7

\ \4 -

I-.
-

0

0

FIGURE 7.4.4.4-42 (CONTD)

-

--

1.2

.4

.6

.8

1.0

b/2
7.4.4.4-47

I

[.-

M .2
BODY RADIUS
b/2
.20

0

ALL

--

=.24

I.0

.150--

-

.05-

-

.10

-

--

-

-

-\-0

01

0

r.

FIGURE 7.4.4.4-42 (CONTD)
744.4448

.2

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z

6

.8

1.0

M =.2

BODY RADIUS
b/2
A
ALE

.20

.15

=.24

.5
=350

(h

-

---

-

-

-

PWB

.10

-

---

-

A

os-I

FIGURE 7.4.4.4-42 (CONIt

bI)
7.4.4.4-49

M .2
BODY RADIUS
b/2
0

ALE

OWB

-

.10-

-

-

-

.05-

47

00

I IGURE 7.4.4.4-42 (CONID)*

74.4.4-SO

-

-

A

b/2

=.214

-

-

-

-

MM .2
BODY RADIUS

.24

b/2

A= I

ALE =350
.20

I

.15

1

--

J)i

J

.1o. .,-,

\

\\'4

".05

.2

.4.6

0

.2 4

Zv

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6

.81.0

b/-2
FIGURE 7,4.4.4-42 (CONTD)
7.4.4.4-51

M.8
BODY RADIUS = .06

0 < A <350

.!5-

.10

OWB3

.05-

A

4

O.

00

0.2

10

.4

'v
b/2

FIGURE 7.4.4.4-42 (CONTD)
7,4.4.4-52

.6

.

1.0

M=.8
BODY RADIUS = .12

"b/2
.20

-

-

0

A < 35 0

.4

Zv
b/2

.15-

ýWB

L.1

1

.05

K

01,
0

.2

.6

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1.0

FIGURE 7.4.4.4-42 (CONTD)
7.4.4.4-53

M.8
BODY RADIUS

.24

b/ 2
= 0
ALE

0

.20-

.15•

P~WB

to-t

.05

.

A
I

0,
0
FIGURE 7.4.4.4-42 (CONTD)
7.4.4.4-54

.2

.4

v
.6
b-/2

.8

1.0

1I
M =.8
BODY RADIUS

=

.24

b/2
X= 0
LE
IA =350

(n)
.20-

.15-

"A1WB

i/.05

A
-

'2i0

.2

.4

zv

.8

.6

!.0

b/2

FIGURE 7.4.4.4-42 (CONTD)

7.4.4.4-55

~ ~

-

~

-

-

-

.,

M =.8
BODY RADIUS

=.24

b/2
X=.2 5.20.-Eu15-------------------------

k0

I

r,

FIGURE 7.4.4.4-42 (CONTD)

L

7.4.4.4-56

.2

.4 ZV

.6

.81.

L]

M= .8
BODY RADIUS

=

.24

b/2

S=

.25
NAE

.15.--

= 350

-

PWB

---

L

.- A
"A

- --

-. .

-

-

k7N.

FIGURE 7.4.4.4-42 (CONTD)

b/2'7.4.4,4-57

M.8
BODY RADIUS
b/2
= .5
ALE

0

(q)

.20•

-

.24

-\_

--

-

OWB

.1o.\•

A.05.--

.6

zv

.2

.4

zv

b/2
FIGURE 7.4.4.4-42 (CONTD
7.4.4.4-58

.8

1.0

M=.8
BODY RADIUS = .24b/2,x = .5
3-5o
ALE

-,.

(r)
2.20

t

1

-

.. .

1I-

_-

00WB

\\1
.05'-

- -1 -

\4

104

0

.2

.4

Z

.8

.6

1.0

b/2
FIGURE 7.4.,,,.. .

(CONTD)
0

.2.4

-

Zv-~

.6

-

.8

1-.- 0.-.

. - . .--7.4.4.4-S9

M=.8
BODY RADIUS

.24

b/2
ALE

T

0

1

20.

L

_______,___

F'\'_\\
3

' WB

.10

A
.05 -

'-

-

010

-

0

FIGURE 7.4.4.4-42 (CONTD)
7.4.4.4-60

.2

.4

v

.6

.8

1.0

M
.8
BODY RADIUS = .24

b/2,.

. •

kX 1.0

AL= 350-.

20.15' -

-.-

__

A

.05

S0

Li
.

.2
FIGURE 7.4.4.4-42 (CONTD)
•-•

.4

Zv

.6

.8

1.0
7.44.26
7 .4.4 .4 -6 1

Apri] 1978

7.4.4.5 WING-BODY-TAIL DERIVATIVE C,
This section presents a method for estimating the contribution of the vertical tail, in the presence of
the wing and body, to the derivative C14 at subsonic speeds. This derivative is the change inrolling-moment coefficient with variation in the rate of change of sideslip angle at a constant yaw
rate and is defined as
act
S-

.
where C, is based on Sw bw/fOb\WWIn general, at low to moderate angles of attack, this derivative is small and has a negligible effect onlateral stability; hence, it is usually neglected.
A. SUBSONIC
The wing contribution to C1ý can be evaluated by using unsteady-flow theory, but at low tomoderate angles of attack it is generally considered small and is neglected. At low angles of attackand for attached-flow conditions, the largest contributor to Cl is the vertical tail. The rolling
moment produced on the airframe by the vertical tail is due to the sidewash. time-lag effects fromthe wing. The body contribution is small and has been neglected.
For a brief discussion of the physical flow phenomena at high angles of attack, i.e., leading-edge
vortex sheets and flow separation, and a comprehensive bibliography on related subject matter, thereader is referred to Reference 1. Reference I also discusses the inadequacy of osci!lating-airfoil
theory and sidewash-lag theory for predicting the vertical-tail contribution to C1 ý at high angles ofattack. However, a modified flow-field-lag theory is discussed in Refeience 1 that appears to givequalitative agreement with experimental data for a current twin-jet fighter configuration.
DATCOM METHOD
The vertical-tail contribution to the derivative Clý at low to moderate angles of attack is given by
Ct•Cv

Cb = C(Z

coaF~ -z PCos 1tF)

sin

7.4.4.5-a

where
Cy.

is the vertical-tail contribution to the derivative Cs, obtained from Section 7.4.4.4.

zP

is the distance from the wing quarter-chord point to the center-of-pressure locationof the vertical stabilizer, measured normal to the body center line, positive for thestabilizer above the body. For Datcom purposes, the vertical-tail center-of-pressure
location is assumed to be the quarter-chord point of the MAC of the total added
panel.

7.4.4.5-1

is the distance from the wing quarter-chord point to the center-of-pressure locationof the vertical stabilizer, measured parallel to the body center line. For Datcom

QP

purposes, the vertical-tail center-of-pressure location is assumed to be at thequarter-chord point of the MAC of the total added panel. (See Sketch (a) of Section
7.4.2.1.)
oeF

is the fuselage angle of attack.

bw

is the wing span.

Sample Problem
Given: Same configuration as sample problem of Paragraph A of Section 7.4.4.4.
Compute:Cy.

=-0.000209
CC
C1A

C1.

(Sample Problem, Paragraph A, Section 7.4.4.4)

COS atF b

sin(0.00209)(20)(0.9998)
C1.
= (-0.000209)[i5(Equation 7.4.4.5-a)(60)(0.0175)]
6 -.5-I= -0.0000253 per deg

B. TRANSONICNo method is presented.
C. SUPERSONICNo method is presented.

REFERENCE
1.

Coe, P. L.. Jr., Graham, A. B., and Chambers, J. R.: Summary of Information on Low-Speed Lateral-Directional Derivatives
Due to Rate of Change of Sideslip J. NASA TN D-7972, 1975. (U)

7.4.4.5-2

April 1978

7.4.4.6 WING-BODY-TAIL DERIVATIVE Cn.
This section presents a method for estimating the contribution of the vertical tail, in the presence of
the wing and body, to the derivative Cn at subsonic speeds. This derivative is the change inyawing-moment coefficient with variation in the rate of change of sideslip angle at a constant yaw
rate and is defined as.C,where Cn is based on Sw bw.

-

For most configurations at low to moderate angles of attack, Cný is rather small and can beneglected in lateral dynamic calculations. However, at high angles of attack for swept- anddelta-wing configurations, Cný can approach the magnitude of Cn, and consequently have large
effects on the calculated dynamic stability of these configurations.
A. SUBSONIC
The wing contribution to Cn can be evaluated by using unsteady-flow theory, but at low tomoderate angles of attack it is generally considered small and is neglected. At low angles of attackand for attached-flow conditions, the largest contributor to Cn• is the vertical tail. The yawing
moment produced en the airframe by the vertical tail is due to the sidewash time-lag effects fromthe wing. The body contribution is small and has been neglected.

For a brief discussion of the physical flow phenomena at high angles of attack, i.e., leading-edge
vortex sheets and flow separation, and a comprehensive bibliography on related subject matter, thereader is referred to Reference 1. Reference 1 also discusses the inadequacy of oscillating-airfoil
theory and sidewash-lag theory for predicting the vertical-tail contribution to Cn at high angles ofattack. However, a modified flov.-field-lag theory is discussed in Reference 1 that appears to givequalitative agreement with experimental data for a current twin-jet fighter configuration.
DATCOM METHOD
The vertical-tail contribution to the derivative Cný at low to moderate angles of attack is given by

Cnt

=

-Cyý
•F

Cos

bw
+

s

7.4.4.6-a

where
Cy.

is the vertical-tail contribution to the derivative Cy, obtained from Section 7.4.4.4.

Qp

is the distance from the moment reference center to the center-of-pressure locationof the vertical stabilizer, measured parallel to the body center line. (See Sketch (a) in
Section 7.4.2.1.) For Datcom purposes the vertical-tail center-of-pressure loc;ition isassumed to be at the quarter-chord point of the MAC of the total added panel.
7.4.4.6-1

zP

is the distance from the moment reference center to t
centerie of pressure of thevertical stabilizer, measured normal to the body center line, positive for thestabilizer above the body. For Datcom purposes, the vertical-tail center-of-pressure
location is a,.sumedto be at the quarter-chord point of the MAC of the total added
panel.

bw

is the wing span.

K.

is the fuselage angle of attack.
Sample Problem

Given: Same configuration as sample problem of Paragraph A of Section 7.4,4.4.
Compute:Cyx=-0.000209

C

/

Y"-

C=

(Sample Problem, Paragraph A, Section 7.4.4.4): ZI, psin aFý

cos'. c-(---0.000209

±

0 bw/

)

(Equation 7.4.4.6-a)= (60)(0.9998) + (20)(0.01745)
156.52-.

+=0.0000805 per deg

B. TRANSONICNo method is presented.
C. SUPERSONICNo method is presented.
REFERENCE
1.

Coe, P. L., Jr., Graham, A. B., and Chambers, J. R.: Summary of Information on Low-Speed Lateral-Directionai Darivatives
Due to Rate of Change of Sideslip/j. NASA TN D-7972, 1975. (U)

7.4,4.6-2/