1、GAINER and HOFFMANNATIONAL AERONAUTICS AND SPACE ADMINISTRATIONProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-fProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA SP-3070Thomas G. Gainer and Sherwoo
2、d HoffmanLangley Research CenterHampton, VirginiaPrepared by Langley Research CenterScientific and Technical Inormation OficeNATIONAL AERONAUTICS AND SPACE1972ADMINISTRATIONWashington, D.C.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-For sale by t
3、he National Technical Information Service, Springfield, Virginia 22151 - Price $3.00Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PREFACEA summary of equations often used in free-flight and wind-tunnel data reduction andanalysis is presented. Inclu
4、ded are transfer equations for accelerometer, rate-gyro, andangle-of-attack instrumentation; axes-system transfers of aerodynamic derivatives; andmethods for measuring moments.of inertia. In general, the equations are in a completeform; for example, those terms are retained that are missing when pla
5、nar symmetry isassumed for airplanes.iiiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLE OF CONTENTSPageiiiPREFACE .1INTRODUCTION 2SYMBOLS .SECTION
6、 I - EQUATIONS INVOLVING BASIC FLIGHT MEASUREMENTS . 17GENERAL AXES TRANSFORMATIONS FOR COMPONENTS OFACCELERATION, LINEAR VELOCITY, AND ANGULAR VELOCITY . 17Transfer From Vehicle Reference Axes to Instrument Axes 18Transfer From Instrument Axes to Vehicle Reference Axes 19CORRECTIONS TO ANGLES OF AT
7、TACK AND SIDESLIP 2121Angle of Attack Angle of Sideslip 21DETERMINATION OF FREE-STREAM MACH NUMBER . . . . 21DETERMINATION OF AERODYNAMIC FORCES AND MOMENTS FROMACCELEROMETER AND RATE-GYRO READINGS . 22DIRECT FLIGHT MEASUREMENTS OF LIFT AND DRAG 23SECTION II - TRANSFER OF AERODYNAMIC FORCE AND MOMEN
8、TCOEFFICIENTS AND DERIVATIVES TO ANOTHER REFERENCE25CENTER 25TRANSFORMATIONS FOR a,/3,V DERIVATIVES X-Axis Force Coefficients and Derivatives . . . . 25Y-Axis Force Coefficients and Derivatives 26Z-Axis Force Coefficients and Derivatives . . . . 27X-AXis Moment (Roll) Coefficients and Derivatives 27
9、Y-Axis Moment (Pitch) Coefficients and Derivatives . 28Z-AXis Moment (Yaw) Coefficients and Derivatives 29TRANSFORMATIONS FOR u,v,w DERIVATIVES . . 3030X-AXis Force Coefficients and Derivatives .Y-AXis Force Coefficients and Derivatives 31Z-Axis Force Coefficients and Derivatives 31X-Axis Moment (Ro
10、ll) Coefficients and Derivatives 32Y-Axis Moment (Pitch) Coefficients and Derivatives . 32Z-Axis Moment (Yaw)Coefficients and Derivatives 33vProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PageSECTION Ill - TRANSFER OF AERODYNAMIC FORCE AND MOMENTCOE
11、FFICIENTS AND DERIVATIVES TO A ROTATED AXES SYSTEM 34DESCRIPTIONS OF AXES SYSTEMS 34Body Axes “ “ “ “ . . . . . . _. 34Principal Axes “ “ “ . . . . . . 34Flight Stability Axes “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ .“ “ 35Wind-Tunnel Stability Axes “ “ * 35Wind Axes 35NOTES ON USE OF TRANSFORMATION EQUATIO
12、NS . 35DIRECT TRANSFORMATIONS 37Static Force and Moment Coefficients (Direct, Table III) 37Static-Stability Derivatives (Direct, Table III) 37Dynamic-Stability Derivatives (Direct, Table III) 39u,v,w Derivatives (Direct, Table HI) 41INVERSE TRANSFORMATIONS . 46Static Force and Moment Coefficients (I
13、nverse, Table IV) 46Static-Stability Derivatives (Inverse, TableIV) . . . . . . . . . . . . . . . . . 46Dynamic-Stability Derivatives (Inverse, Table IV) 48u,v,w Derivatives (Inverse, Table IV) . . . . . 50SECTION IV - TRANSFORMATION EQUATIONS FOR MOMENTSOF INERTIA . . . 55DIRECT TRANSFORMATIONS (TA
14、BLE III) 55INVERSE TRANSFORMATIONS (TABLE IV) . . . . . . 56SECTION V - EQUATIONS OF MOTION FOR SIX DEGREES OF FREEDOM 57FORCE EQUATIONS . 57X-Axis Forces . 57Y-Axis Forces “ “ “ * 58Z-Axis Forces . . . . 59MOMENT EQUATIONS . 59X-Axis Moment (Roll) . 59Y-Axis Moment (Pitch) 60Z-Axis Moment (Yaw) . 6
15、1AUXILIARY EQUATIONS . . 61Relationship Between Euler Angles and Angular Velocities . 61Vehicle Coordinates 62Trajectory Parameters (See Fig. 6) . 63Angle of Attack, Sideslip, and Relative Velocity (See Fig. 7) . 63viProvided by IHSNot for ResaleNo reproduction or networking permitted without licens
16、e from IHS-,-,-Page64Wind Corrections (See Fig. 8) Resolution of Engine Thrust and Torque Into Components AlongVehicle Reference Axes (See Fig. 9) . 64Components of Gravitational Acceleration Along X,Y,Z VehicleReference Axes With Earth-Oblateness Effects Included . 65APPENDIX A - SUMMARY OF FREQUEN
17、TLY USED FORMS OF AXESTRANSFORMATIONS AND EQUATIONS OF MOTION 67EULER ANGLE TRANSFORMATION BETWEEN TWO ORTHOGONALAXES SYSTEMS 67Direct Transformation 67Inverse Transformation 67TRANSFORMATIONS FOR ACCELEROMETER AND RATE-GYROMEASUREMENTS 68Case I - Orthogonal Instrument Axes; No Restrictions on68Misa
18、linement Angles Case II - Nonorthogonal Instrument Axes; Small Misalinement Angles . . . 69SPECIAL FORMS OF TRANSFORMATIONS FOR AERODYNAMIC FORCEAND MOMENT COEFFICIENTS AND STABILITY DERIVATIVES 70Simplified Forms for Transferring Coefficients and Derivatives toAnother Reference Center . 70Transfer
19、of Coefficients and Derivatives From Body to Wind-Tunnel72Stability Axes .Transfer of Coefficients and Derivatives From Wind-TunnelStability to Body Axes . 73TRANSFORMATION EQUATIONS FOR MOMENTS AND PRODUCTSOF INERTIA 75Body to Flight Stability Axes 75Body to Principal Axes 75Flight Stability to Bod
20、y Axes . 75Flight Stability to Principal Axes 75Principal to Body Axes 76Principal to Flight Stability Axes 76SPECIAL FORMS OF EQUATIONS OF MOTION 76Coupled Linear Equations of Motion . 76Uncoupled Equations of Motion 78Wind-Axes Equations for a Point Mass . 79viiProvided by IHSNot for ResaleNo repr
21、oduction or networking permitted without license from IHS-,-,-PageAPPENDIX B - DERIVATION OF EQUATIONS PRESENTED IN SECTION ITFOR TRANSFER OF COEFFICIENTS TO NEW REFERENCE CENTERAPPENDIX C - METHODS OF MEASURING CENTER-OF-GRAVITYLOCATIONS AND MOMENTS OF INERTIA OF MODELS AND FLIGHT8OVEHICLES 84CENTE
22、R-OF-GRAVITY LOCATION 84Longitudinal c.g. Location 84Vertical c.g. Location . . . . . . . 84MOMENTS OF INERTIA . . . . 84Compound-Pendulum Method (Fig. ii) . . . . . . . 84Spring Method 85Spring Method for Full-Scale Vehicles . 85Torsion-Pendulum Method 86Multifilar-Pendulum Methods . 86APPENDIX D -
23、 DETERMINATION OF LONGITUDINAL AND LATERALSTABILITY DERIVATIVES BY USING SIMPLIFIED LINEAR ANALYSIS 87APPENDIX E - USE OF DIRECTION COSINES AND QUATERNIONS INMOTION CALCULATIONS 90DIRECTION-COSINE METHOD . “ “ “ “ “ , 90QUATERNION METHOD 92Basic Quaternion Relationships . 92Relationship Between Eule
24、r Parameters and Euler Angles . 93APPENDIXF - SCALING PARAMETERS . . . . . 94DYNAMIC-SCALING PARAMETERS 94AERODYNAMIC-SCALING PARAMETERS 94AEROE LASTIC-SCALING PARAMETERS 96COMMENTS ON SCALING PROCEDURES . . . . . 96Dynamic Scaling . 96Aerodynamic Scaling . 96REFERENCES 98BIBLIOGRAPHY ON METHODS OF
25、ANALYZING FLIGHT DATA 99TABLES I - CONVERSION OF U.S. CUSTOMARY UNITS TO SI UNITS .H- RELATIONSHIPS BETWEEN ol,_,V AND u,v,w DERIVATIVES .III- ANGLE DESIGNATIONS FOR DIRECT TRANSFORMATIONS .102102103104viiiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-
26、,-,-PageIV - ANGLE DESIGNATIONS FOR INVERSE TRANSFORMATIONS 105V - DESIGNATIONS OF FORCE AND MOMENT COEFFICIENTS FORDIFFERENT AXES SYSTEMS VI - SCALE FACTORS FOR DYNAMIC SCALING .FIGURES 106IQ71081 - Systems of vehicle reference axes and instrument axes . 1082 - Axes systems for transfer from vehicl
27、e c.g.to new reference centerby equations of section II 1093 - Systems of vehicle reference axes, including body, principal, wind,flightstability,and wind-tunnel stability 1094 - Relationship between earth-centered inertial axes, gravity axes, andvehicle reference axes . 1105 - Systems of gravity ax
28、es and vehicle reference axes . 1116 - Relation of heading angle H and flight-path angle 7 to earth-centered inertial axes 1127 - Resolution of relative velocity into components along vehiclebody axes 1138 - Directions of surface and geostrophic winds . 1149 - Alinement with respect to vehicle refer
29、ence axes of thrust and torquedue to rotating mass of engine . 11510 - Determinatiori of vertical location of center of gravity 11511 - Measurement of moment of inertia by compound-pendulum method . 11612 - Measurement of moment of inertia by spring method 11613 - Determination of vertical location
30、of center of gravity and rollingmoments of inertia for full-scale airplanes. (Reproducedfrom ref. 3.) . 11714 - Methods of measuring yawing moments of inertia 118(a) Torsion pendulum 118(b) Bifilar pendulum . 11815 - Damped angle-of-attack oscillation assumed in analysis of appendix D . . . 119ixPro
31、vided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INTRODUCTIONThe equations in this report are the coordinate transformation and motion equationsused in the
32、various tasks associated with free-flight and wind-tunnel data reduction andanalysis. These tasks range from reducing flight data to calculating the motions on a digital or analog computer and to applying various techniques for analyzing the data, suchas in references 1 and 2.While many publications
33、 contain a number of these equations, no one contains allthat are usually needed in a complicated aerodynamic analysis; even the more nearly com-plete reports (refs. 3 and 4, for example) omit the equations for transferring aerodynamicstability derivatives from one moment reference to another. Moreo
34、ver, in most casesthe equations are simplified, when they are presented, by assumptions such as smallangles of attack, zero sideslip, and small perturbation motions. Expanded forms of manyof the equations, on the other hand, are needed in special problems that may arise. Forexample, parawing vehicle
35、s, which have their center of gravity located well below the wingsurface, require the expanded forms of the axes transformations when data measured abouta point on the wing are to be transferred to the center of gravity; reentry motion studiessometimes involve large-amplitude motions so the complete
36、 forms of the transformations,without the assumptions of small angles of attack or sideslip, are needed. The engineerworking on any of these special problems usually has to derive these equations himself,and this can be time consuming.The purpose of this report is to provide the basic equations from
37、 which many of theequations needed in a particular analysis can be generated. A comprehensive summaryof the basic axes transformation and motion equations is included, with most of thesegiven in their expanded, most general forms. Once these expanded forms are available,the simpler forms can be writ
38、ten out fairly easily, and yet the general forms are herewhen needed for special cases.The general forms presented include axes transformations that enable transfer backand forth between any of the five axes systems that are encountered in aerodynamic analy-sis. Equations of motion are presented tha
39、t enable calculation of motions anywhere in thevicinity of the earth. Special problems are also considered; since flight instruments,such as accelerometers or rate gyros, are not always alined along mutually perpendicularaxes, the procedure for correcting instrument readings for nonorthogonal alinem
40、ents isoutlined.In addition to these general forms, many of the simplified forms used frequently inpractical applications are summarized in appendix A.Other relationships are presented in appendixes B to F. For example, appendix Csummarizes methods used to measure moments of inertia of models and fu
41、ll-scaleProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-vehicles; appendix E discusses the use of the direction-cosine and the quaternion methods,often used in place of Euler angles in specifying vehicle alinement; appendix F discussesthe scaling par
42、ameters used in model testing. However, throughout this paper, the empha-sis is on providing the basic equations. For discussions of their development and of theprocedures used in their application, the user should turn to general published works onflight-motion analysis. A comprehensive bibliograph
43、y of these works is provided _andincludes textbooks and reports dealing with stability, control, and performance as wellas reports discussing various techniques for extracting stability derivatives from flightdata.SYMBOLSThroughout this paper, symbols are defined in terms of SI Units with equivalent
44、 U.S.Customary Units given parenthetically. Factors for converting from U.S. Customary toSI Units are given in table I.AA _eneralized angle of attack, defined for various axes systems in tables IIIand IV, rad (deg)ross-sectional area in eq. (F-11) of appendix F, m 2 (ft 2)angle between surface wind
45、vector and plane of local horizontal, measuredperpendicular to plane of local horizontal (fig. 8), rad (deg)AD acceleration along flight path, g units (lg = 9.807 m/sec 2)A L acceleration in lift direction, g units (lg = 9.807 m/sec 2)Ax,cg,Ay,cg,Az,cg components of acceleration along X,Y,Z vehicle
46、reference axesat c.g., respectively, g units (lg = 9.807 m/sec2)AX ,i,Ay ,i,AZ ,i components of acceleration indicated by accelerometers alongXi,Yi,Z i instrument axes, respectively, g units(lg = 9.807 m/sec2)a speed of sound, m/sec (ft/sec)a T damping constant defined by eq. (D-4) of appendix D, di
47、mensionlessdiameter of circle around which wires or rods are attached in bifilar ortrifilar methods of measuring moments of inertia in eq. (C-9) ofappendix C (see, also, fig. 14(b), m (in.)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-a ea sBS tbCCACCCD!CDCLCL,oC/,C m ,C
