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Body Axes and Velocities in Aerodynamics, Study notes of Dynamics

World Mission UniversityDynamics

The concepts of body axes, forces, moments, and velocities in the context of aerodynamics. It covers the transformation of velocities from inertial coordinates to body coordinates, the calculation of body angular velocities, and the resolution of forces from wind axes to local geocentric coordinates. The document also mentions the use of Euler angles and a functional flow diagram for a six-degree-of-freedom flat-planet option.

Typology: Study notes

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UNCLASSIFIED
AD
NUMBER
AD452592
NEW LIMITATION CHANGE
TO
Approved
for
public
release,
distribution
unlimited
FROM
Distribution
authorized
to
U.S.
Gov't.
agencies
and
their
contractors;
Administrative/Operational
Use;
Oct
1964.
Other
requests
shall
be
referred
to
Air
Force
Flight
Dynamics
Laboratory,
Wright-Patterson
AFB,
OH
45433.
AUTHORITY
AFFDL ltr,
29
Jan
1975
THIS
PAGE
IS
UNCLASSIFIED
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UNCLASSIFIED

AD NUMBER

AD

NEW LIMITATION CHANGE

TO

Approved for public release, distribution

unlimited

FROM

Distribution authorized to U.S. Gov't.

agencies and their contractors;

Administrative/Operational Use; Oct 1964.

Other requests shall be referred to Air

Force Flight Dynamics Laboratory,

Wright-Patterson AFB, OH 45433.

AUTHORITY

AFFDL ltr, 29 Jan 1975

THIS PAGE IS UNCLASSIFIED

  • -- " -^ "r•^.^ ,^ -'^ J.^ i

-. i^ i

SI.bGWO-REO -69ktR IzEi%. coMPUTER FIIHTP PROGRAM.^ STUDY

""":ý'P, "PtI- eRBLEM^ FORMULATION,

"WTECHNICAL DOCUMENTARY^ REPORT^ No.^ RTD-TDR-64-1,

PART I,^ VOLUME^ I
' \ OCTelIf•f 1964
-. AIR FORCE FLIGHT^ DYNAMICS^ LABOPATORY
RESEARCH AND^ TECHNOLOGY^ DIVISION
AIR FORCE SYSTEMS COMMAND
WRIGHT-PATTEISON AIR^ FORCE"BASE,^^01110

i5 1964^ S "";:~~~ "Project,-vo.^ 1431,^ T1'a'k^ -N.-143103~^ E^ i5^ I'll~os •t

ODe-IRA C

McDontell^ (repared^ underAircraft^ Contract, Corporation,St^ No.^ AVI^ 33T~S2 Lbuis,^ by

Missour.,
Robert C. ,ron,^ 1o1)irt^ V.^ Bruile,^ A.^ E.^ Combs,

Best Av ald 1C). Giffin, o nym•, •i•', •Best Available Copy

FOREWORD

by AF The^ research^ program^ summarized^ in^ this^ report^ was^ initiated^1 JunT^ 19b PattersonFlight Dynamics^ Laboratory,^ Research^ and^ Technology^ Division,^ Wright- Air Force Base, Ohio. (^) The research effort consisted (^) of converting RTD's Six-Degree-of-Freedom Flight Path (^) generalized computer program from (^) SOS to FORTRAN/FAp computer (^) laan.guage and was undejtaken as a tortion of the (^) study

conducted by McDonnell Aircraft Corporation under USAF Contract No. AF33(6>()-

8829 during the period (^1) June 196? to 31 December (^) 1963. This report, prepared by A. E. Combs, McDonnell (^) Aircraft Corporation, (^) is essentially the original

formulation report (WADD m-60-781, Part I) with the additions, modifications,

and corrections (^) made since its publication. (^) Mr. B. R. Benson of the (^) AF Flight Dynamics Laboratory (^) has been the Air (^) Force technical representative. This report was (^) prepared under Project (^) 1431, "Flight Path (^) Analysis", Task 143103, "Six-Degree-of-Freedom (^) Flight Path Analysis".

...............i'-tnd 'c--- D.b. ,esid,: 0L Trle

System ccornology (^) Division for contributions (^) to the original analytical for~iu- lat.Lon and to the (^) following members (^) of the McDonnell (^) Automation Center: Messrs. (^) F. W. Seubert (^) and N. E. Usher (^) for design and modification (^) of the computing program, (^) and R. F. Vorwald (^) for further modification, (^) correction, and conversion (^) of the machine language. For ease (^) of reading, the (^) documentation of (^) this project has been (^) prepared in several (^) parts. The total (^) documentation is summarizee (^) as follows: Part (^) I Volume 1 - Basic (^) Problem Formulation Volume 2 - Structural (^) Loads Formulation

Voluie 3 - Optimization Problem Formulation

Part II Volume (^) I User's. Manual (^) for Part I, Volume (^) 1

"Volume 2 - User's Manual for Part I, Volu,<. 2

I,-Volume 3 - User's. Mrnual for Part 1, Volume 3

Best Available (^) Copy B t A.

ABSTRACT

A trajectory computation program^ is^ (^ iscribed^ for^ determuin.a.ag^ vehicle^ per- formance throughout the^ entire^ flight^ regime^ of^ speed^ and^ altitude^ in^ the atmosphere and^ graviLy^ field^ of^ a^ noy.-ispherical^ rotatin6.g^ planet.^ The^ program isfreedom formulated problem for to Feventhe two-degreeoptions^ of pointvarying mass^ refinvn.iL problem. fromA reverse^ the^ six-degree-of- option for the aerodynamic analysis of light test data,^ a^ punched^ card^ output,^ and^ a nemi-aUtorlatic computational tie to^ an^ interplanetary^ trajectory^ computer program are included. The program is specifically oriented for^ computation on the IBM 7090/7094 digital^ computer^ using^ the^ FAP/FORTRAN2^ machine^ language.. This teclhaical documentary report has oe;-n^ v,'i-ewed^ Lnd^ is^ approved.

J- (^) L4d/'f.P I., ChiefAF Flight Flight Dynamics Mechfuici Labor& D! (^) xv" , U.aoi-

liii

(^1) 3.4 Interpianetary Trajectory (^) Problem Coordinate 67 Trazisfowmations (^) -

3.4.1 The Coordinates of the Interplanetary Tiajectory 67
Problem

3.4.2 TheFreedom Inertial Problem Coordinates^ uf^ the^ Six-Degree-of-^68

3.4.3 Astronomical Angles Required for the Coordinate 68
Transformation

r 3.4.4 Transformation From Interplanetary to the Six- 68

Degree-of-Freedom Inertial Coordinate System

3 3.4.5 Transformation From the Six-Degree-of-Freedom 70

to Interplanetary Coordinates
  1. VEHICLE CHARACTERISTICS (^74)
4.1 Aerodynamic Coefficients 74

4.1.L (^) Form of Data Input (^74) 4.1.2 Flight Path and (^) Vehicle Types (^75)

4.1.3 Error Constants 78

4.2 Thrust and Fuel. Flow Data (^) 78 4.2.1 Data Iuputs (^78) 4.2.3 Error Constants (^82) 4.3 (^) Physical Characteristics 8I 4-3.1 1i..,Re (^) Categories of Phyic!-tl Chwaeuteristics (^82) J.3 Errorfcrcnco Constants^ Weight 84 84 4.4 Sta!es and Staging (^84)

lL 5I~rryt Anal -:se 86

4.5.1 4.5.2 (^) Tlu•,c~.1y:.',ic .t (^) and FuelData Flow CL-',zacteristics Br; 4.5.3 (^) Vehicle Physical Characteristics 87

4.5.5 4-.5.4^ AutopilotAdditional Errors^ Functions 87

4.5.6 Atmospheric Density Error 88^8

5. VEHICLE ENVIRONMENT 89

5.1 Atmospheres (^89)

5.1.15.1.2 T,imitations 1959 ARDC Model Atmosphere 89
5.1.3 Accuracy 91
5.2 Winds Aloft 92

V

6.1.6 6.1.7 RollControl Rate Surface Channel Deflections^110 11l

vi

 - 5.35,4 Gravity - Local-Geocentric to Geodetic Coor.inaters.
    • --1WE,-5.4.2 5.4.1 -LatitudeFlight-Path Angles.
    • REEM, 5.4.3 Geodetic Altitude i - 6. AUTOPILOTS AND FLIGHT-PLAN PROGRANMERS - 6.1 Typical Autopilot
      • F - 6.1.1 Description of Flight Control System - 6.1.2 Control System Input Data Simulations - 6.1.3 Pitch Control" Channel - 6.1.56.1.4 AzimuthBody Angular Control Rates Channel and Accelerations - 6.2 6.1.8 Cornputational Flow Diagram - 6.2 Plight-Plan Programr.er - (. 6.2.2 2,1 D,;cussion"Mght,- : :.f Pro-).-er Selected Control•light-Plan Coremanda Soquence
        • 4 6.2.3 FlightTorquing Plan Commands Proýirammer to J'itch^10 o- Rate Programmed Gyro with^1 - (•,3 Structural Teme'erature Limiting Pri ft ,tnd Bir._s - 6.3.1 6.3 • 2 Temperature,ExampLeJli Fovmuli: Limitinp tior' Problem lormulatIon - 6.3"3 Discussion - 7. AERODYNAMIC 1'4EATING SUBPROGRAMS - T.1 Thin-Skin of Attack Temperature of Arbitrary Wedge at Angle - 7.2 7.3 EquilibriumLocal Flow ConditionsStagnation -Poinu Temperature - 8. INTERPLANETARY TRAJECTORY COMPUTATIONS - 9. AUXILIARY COMPUTATIONS - 9.19.2 Planet-SurfaceGreat Circle Range Referenced Range - 9.3 Down and Cross Range - 9.4 Theoretical Burnout Velocity and Losses
            1. INITIALIZATION AND COMPUTATION

K. (^) LUST8T1IQO I- Liz 7 ~IFigure 2.1 Generalized Inertial and (^) Body-Axes Coordinate Systems... 3,1 Relationship Between Inertial, Geocentric, Local- f Geocentric,^ and^ Body^ Coordinates.^.^.^.^.^.^.^.^.^.^.^.^.^.^ L 3.2 Intermediate Coordinate (^) System Transformation from Inertial to Local.-Geocentric Coordinates .......... (^16) 3.3 (^) Final Rotation in Transformation from Inertial to (^) Local- Geocentric Coordinates ...................... (^17) 3.4 (^) Relation Between Body Axes, Local-Geocentric, and (^) Inertial Coordinates for Motion in Equatorial Plane ......... (^22) 3.5 Unit Sphere Diagram (^) for Lateral Motion Coordinate Tmans-

formation ................ ...... . 27

3.6 Relation Between Local-Geocentric, Inertial, (^) and Earth- Referenced Coordinates for Point-Mass (^) Problems. ....... 30

qelationshipS3.7 Between Local-Geocentric Axes and Wind Axes. 31

3.8 .'fIatinnship Between Body Axes and Wind Axes......... 33

3.9 Relationship Between Body Axes and Vertical Wind Axes with
Zero Body Roll Angle.. ................... 35

3.10 Wind Components (^) for u PoinL Mass Analysis.......... 37 3.11 uieLlonal (^) Fluw Dlagam - Flatform Angles for Six-.Degree- of-Freedom Oblate Rotating Planet Option. (^) ........ 42 3.12 Unit (^) Sphere for Yaw-Pitch-Roll Sequence of Rotation.... (^41) 3.13 Unit Sphere (^) for Pitch-Yaw-Roll Sequence of Rotation.... 44 3.14 Unit Sphere for Pitch-Roll-Yaw (^) Sequence of Rotation.... 46 3.15 (^) Relation of Platform and Local-Geocentric Horizon Coordinates (^49) 3.16 Functional (^) Flow Diagram - Platform Angles for Six-Degree- of-Freedom Flat-Planet Option (^) ............... 53 3.17 Platform (^) Coordinate System Inertially Fixed at Launch Site 57 3.18 Platform Coordinate (^) System Torque at a Constant Rate....* 57

viii

3.19 Functiontol Flow DLagram - Platform Angle for Three-Degree- • -

of-Freedom Longitudinal Computation ... ..... 58

3.20 Accelerometer with Sensitive AxiB Aligned with Local-

Geocentric Vertical ..... ................. 59
3.21 Inertial and Earth-Referenced Coordinate Systems.. ... 65

3.22 A Unit Sphere Showing Transformation from an Interplanetary Trajectory Problem to the (^) Six-Degree-of-Freedom Problem Inertial Coordinates ....................^^69 3.23 Unit Sphere Diagram Showing the Transformation from the Six-Degree-of-Freedom Problem to an Interplanetary Trajec- tory Problem... ............. ........ 71

4.1 Curve Fit Non.-Linear Aerodynamic Characteristic.... .. 75

4.2 Solution of Aerodynamic Forces and Moments Subprogram • • • 77

4.3 Thrust and Fuel Flow Subprogram ........ #.. 81
4.4 Thrust and Fuel Flow Subprogram (Multi-Engine Rocket,

Uncontrolled (^) Thrust)....... ......... .... 83

-5 Vehicle Pnysic9l Chnrncteristics Subprogram ..... ..... 85

5.1 Functional Flow Diagram - Winds-Aloft (^) Subprogram ........

5.2 Planct-Oblateness Effect on Latitude and Altitude. .. .. 96

. DfferencP a Ftuction B(etweenof, •eocentric Geodetic Latitude and Geocentric -i ^A..t.tude,.... Latitudes .... as 00

5.4 Relation of Geodetic and Geocentric Horizons.. ...... 100
6.1 Control System Functional Block Diagram....... .. 104

6e2 (^) Control-Surface Arrangement and Definition of Surface Deflections (^) ..... ........................ 112

6.3 Control System Computational Flow Diagram ............ 114

6.4 Six-Degree-of-Freedom Flight-Path Study Flight Progrartimer

Subprogram Control Functions ....... ................ 116

6.5 Flight Plan (4) - Programmed Wind-Axis Normal Load Factor. 120
6.6 Flight Plan (5) Programmed Flight-Path Angle .......... 13
ix

•°"one i^ Body^ Geometry^ for^ Thrusting^ Rocket^ with ÷ Changing Mass. .. .. .. .. .. .. .. ... 17 STWO I^ Rotating-Machinery^ Axes^ System^.^ ..^ ..^ .....^178

. •at, SFour^^1 ComparisonElevated^ ofTemperature^ Deflections^ for^ of^ Severala^ Cantilever^ Materials^ Beam^.^102 S2 Variation of the Effect of Static Aeroelasticity I • on the Control Derivative^ of^ a^ Typical^ Missile

÷with Dynamic^ Pressure^ ..........^193

r •Six 1 Normalized Body-Beniding Mode Shapes ..^ ..^.^ ..^198

S2 Aeroelastic Functional Block Diagram. ......^^200

Xi

SYMBOLS -AND NON4EXNCIATURE - -- *- ~-

Freedom^ The Flight-Patbsymbols^ and nomenclature Study computer^ us~edprogram^ in^ theaxe^ farajulatý.~iosummarized in^ thethis^ section.U-~-4ere~
Otandard symbole, currently in u~e in the fields to which they are applied,
have been used whenever such use does not result in corfl~icts, Duli~city of

(^5) by symbols the prog,-amhas been have allowed unique for derivation symbols assigned. purposes; The however,engineering all notation andquantities-computed the

normal and def'initions units for haveeach beenquantity subdivided are included according with to the usage definition. as follows: The symbols

uate gory (^) EaMS Aerodynamics..... .. .. .. .. .. .. .. .. .xiii Aerodynamic Heating .. .. .. .. .. .. .... XX Angular Position Data. .. ............ .................. xxii Angular Velocities. .. .. .. .. ..... .. .. .. Xxiv Atmos3phere Data. .. .......... .......................... XXV Axes Systeis .. .............. ................ .. xxvi. Body Physical Oa~a........... .... ...* xxv.Lli Direction Cosines. .. .................. ...... .... xxx

Eng~ine Data.. ................................ ...... xxxi.

Fligty,-Plan Pro ramnmer and Autopilot. .. .............. .=41ii Forces and Moments^ ..^ ................^ ..................^ xxxv GeoifhvsIcalfData .. .............. ...................... xxxvi Linear Velocities. .. ...... ........... ..... xxvii Pcoi~tion Data. .. ............ .......................... (^) xxxviii Miscellaneous. .. .............. .................... xxxix

xi:

C~jq magus^ ForceWi^ m^ Coefficient^ =^ CNf/^ (P41/9%) C" 1magnus moment^ Coefficienlt^ (Pali/2va)

CAo CA^ at^ a^ p^ =^^00 -^ dimensionles

CA- CA/oa^ -^ per^ degree SCA• -^ per^ degree^2

•:.-:•:CAO5 C'p 6CA/•P - per^ degree

CAp 2 6CA/Z0f^2 -^ per^ degree^

2 -C- CAB;:A16A-q OCA/O5...^ -^ per^ degree 6C/,02^2 CA 8 2OA/O•q^ -perdegree

.- q^2 CAPC4^ -^ per^ deree^2

CAoq o^ 2A/cU;oq^ -^ per^ degree CAPB~q C~^2 CAfJB~q^ -^ per^ degree^2 (CA)^0 CA^ at^ 5r^ "^ -^ dimensionless CN° CN^ at^ a^ =^ =^^0 -^ dimensionl.ess CN a;CN/6a -^ per^ degree CNC? CN/O6^ -^ per^ degree CN OCN[ufp^ -^ per^ degree CNsq •CN/f5q^ -^ per^ degree CNFq CCNf65q - per (^) degree cN/0I62 - per^ degree CN~2 (^) -CNCfcco-.CN0Oq -^ per^ degree 2 uCN/O "^ -^ per^ degree CNPB~q 62Ccpuq^ prdge C(&CN/0(aIVa) -^ per^ radian 0C/Z (d•a/2Va)6xC.G.^ -^ per^ radian per^ foot

xiv

S-• • '' &•i'/ .. =. 5 •. ..... •. ...... .;. = .- • --- _- ..-- .r .... - .•. • • • .... = ; --• • - •

CN bc/6 (A,12 per^ radian

CNq 2 ..^ per^ radian^ per^ foot

-. (CN)5 0 CD c^^ at^^5 -0^ =^ dimensionless

c C at a = - dimensionless

•.a^ c^ ycy/I^ -^ per^ degree
  • Y2 OCy -^ per^ degree^2 c Cy•/b - per degree V 6cy/^2 -^ per^ degree^2 CYtr OCy/Zbr^ -^ per^ degrue yOc./r82 N^ - per degree^2 CyaB^6 2Cy^ 0/b^ r^ -^ per^ degree
Cy( P^ ou a^ 'uii4^ -^ per^ degree
c Yf~~r o 2C^ YfdPr^5202 -per^ dcgrecc
c. Ucy/u(ýAi2/2Va) - per radian

c 6 2C y0(f(3d/2Va)OXcG -^ per^ radian^ per^ foot YJ•X y^ 2-^ G

c yr 0- cyid(rdo/2Va) -^ per^ radian

Cyrx ý^2 Cy/o(rdP/2Va)0xC.G.^ -^ per^ radian^ pcr^ foot (Cy)6=0 Cy at bp = 8 =r 0 - dimensionless Clo C1^ at^ a^ =^ P^ =^^00 -^ dimensionless Cl, 6C-t 1 oo•^ -^ per^ degree Cld c.2 a- per degree Cl•PCIjbP - per degree

Cl fOC)( 2 1 r^^0 -^ per^ degree^2
Cl6p ,CIr5p^ -^ dýgz'•u2er
c:b 2 Zc E,6/2 - per degree

xv

SYMBOLS DEFINITION AND UNITS.

.I' CnOCn/'05r2^ nSr^^ r0 6Cn/65r^ -^ - perper^ degreedegree2- Cn2 Cnn/o - per degree^2 aOpr 0 22 Cn/pr - per degree 2

  • ~ ~ OnCn/00ý5rCn^5 j^ SCCnr n~aObr^ 2C^^ -^ perper^ degreedegree2^2 -e n, ,Cn/d ('de/•-Wfa) --^ ,.^ radian 2 .Cn (od2/2Va)6XC.G. -^ per^ radian^ per^ foot Cnr OCnfc(rd2/2Va)^ -^ per^ radian Cnrx o^2 Cnf(L(92/2Va)UxCG^ -^ per^ radian^ per^ foot (C) a 5 = = S = 00 - dimensionless Aerothermoelastic Coefficients P 1 Firsý Order Elastic Coefficient in^ C'^ Equation feet /pound A 2 Senopd Order Elastic Coefficient in CAp' Equation - A3 First Order Elastic Coefficient in CM' Equaion - Atee/puund A4 Second Order Elastic Coefficient in CA'b Equation - feet4/pound 2 (

A 5 First Order Elastic Coefficient in C'a Equation -

feet 2 /pound

A6 Second feet 4 /pound Order 2 Elastic Coefficicnt in C, Equation

A7 First Order Elastic Coefficient in CN'q Equation feet2/pound AB Secondfeet 4/pound Order 2 Elastic Coefficient in C1~ Equation- A9 First Order Elastic Coefficient in C'. Equat ion - feet 2 /pound A 10 Second Order Elastic Coefficient in C Equation feet 4 /pound^2 yE

xvii

i• •"SYNB01S^ DEWINITION^ AND^ UNITS

F_ -All^ First'feet^2 /poundOrder^ Elastic^ Coeffictent^ in-Cyot^ Equation^ -

SA12 Becopd Order Elastic Coefficient in Cyla Equation-^ :

feet4/pound 2 A First^ Order^ Elastic^ Coefficient^ in^ Cl^ Equation- 13 feet^2 /pound A14 Second Order Elastic Coefficient in C Equation - feet 4 /pound 2 A- 5 First Order (^) Elastic Coefficient in C15p Equation - feet 2 / oiilnd AI 6 Secopd Order Elastic Coefficient^ in^ l^ Equation^ - feet4/pound 2 -A1 First Order Elastic Coefficient in C' Equation - feet 2 /pound A38 Second (^) Order Elastic Coeffici.-nt in Cm• Equation - fcel4-pound A1 9 First (^) Order Elastic Coefficient in Cmbq Equation - fecet 2 /powid A',)0 Secondfeet 4 /pound2 Order Elastic Coefficieat in Ct Equation A2 1 First Order Elastic Coefficient in C.' Equation - feet 2 /pound A2 2 Secondfeet4 /pound2-P Order Elastic Coefflcibnt in C"_ Equation - First Order Elastic^ Coefficient^ in^ Cn-r^ Equabion^ - A2 3 feet 2 /pownd P'2_4 Second Order Flastic CoefficienL in Cn1r' Equation - feet ./Pound^2 r El Error Multiplier for CN - dimensionless E2 Incremental Error in CW - dimensionless C3 Error Multiplier for CA - dimensionless C4 Incremental Error in CA - dimensionless

£5 Error Miultiplier for Cy - dimensionless

xvili