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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.
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i5 1964^ S "";:~~~ "Project,-vo.^ 1431,^ T1'a'k^ -N.-143103~^ E^ i5^ I'll~os •t
Best Av ald 1C). Giffin, o nym•, •i•', •Best Available Copy
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
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
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".
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
Part II Volume (^) I User's. Manual (^) for Part I, Volume (^) 1
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-
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(^1) 3.4 Interpianetary Trajectory (^) Problem Coordinate 67 Trazisfowmations (^) -
4.1.L (^) Form of Data Input (^74) 4.1.2 Flight Path and (^) Vehicle Types (^75)
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)
4.5.1 4.5.2 (^) Tlu•,c~.1y:.',ic .t (^) and FuelData Flow CL-',zacteristics Br; 4.5.3 (^) Vehicle Physical Characteristics 87
5.1 Atmospheres (^89)
V
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- 5.35,4 Gravity - Local-Geocentric to Geodetic Coor.inaters.
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-
3.6 Relation Between Local-Geocentric, Inertial, (^) and Earth- Referenced Coordinates for Point-Mass (^) Problems. ....... 30
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
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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
Uncontrolled (^) Thrust)....... ......... .... 83
5.1 Functional Flow Diagram - Winds-Aloft (^) Subprogram ........
. DfferencP a Ftuction B(etweenof, •eocentric Geodetic Latitude and Geocentric -i ^A..t.tude,.... Latitudes .... as 00
6e2 (^) Control-Surface Arrangement and Definition of Surface Deflections (^) ..... ........................ 112
•°"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
r •Six 1 Normalized Body-Beniding Mode Shapes ..^ ..^.^ ..^198
Xi
(^5) by symbols the prog,-amhas been have allowed unique for derivation symbols assigned. purposes; The however,engineering all notation andquantities-computed the
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
Fligty,-Plan Pro ramnmer and Autopilot. .. .............. .=41ii Forces and Moments^ ..^ ................^ ..................^ xxxv GeoifhvsIcalfData .. .............. ...................... xxxvi Linear Velocities. .. ...... ........... ..... xxvii Pcoi~tion Data. .. ............ .......................... (^) xxxviii Miscellaneous. .. .............. .................... xxxix
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C~jq magus^ ForceWi^ m^ Coefficient^ =^ CNf/^ (P41/9%) C" 1magnus moment^ Coefficienlt^ (Pali/2va)
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
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
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CN bc/6 (A,12 per^ radian
c 6 2C y0(f(3d/2Va)OXcG -^ per^ radian^ per^ foot YJ•X y^ 2-^ G
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
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.I' CnOCn/'05r2^ nSr^^ r0 6Cn/65r^ -^ - perper^ degreedegree2- Cn2 Cnn/o - per degree^2 aOpr 0 22 Cn/pr - per degree 2
feet 2 /pound
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
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i• •"SYNB01S^ DEWINITION^ AND^ UNITS
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
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