SFTE Reference Handbook Third Edition 2013

Society of Flight Test Engineers

Reference Handbook

Third Edition 2013

Page - i SFTE Reference Handbook Third Edition 2013

Society of Flight Test Engineers Reference Handbook

2013 Edition

Corporate support supplied by

Cessna Aircraft for printing the 2007 Edition

And

The National Test Pilot School

Contributing Authors Al Lawless (sections 1-8, 10-12, 15, 18) Greg Lewis (section 2.6) Bill Norton (sections 9, 13) Dan Hrehov (section 14) Steven Arney (section 16) John Minor (section 19) David Kidman, Christopher Moulder, Craig Stevens (section 17)

Edited by Lee Gardner & Darcy Painter 1998-2006 Harold Weaver 2006-2013

The SFTE handbook committee continually seeks corporate sponsors for this book and authors for new sections (including but not limited to INS, GPS, EMI/EMF, radar, avionics, R&M, E-O, human factors, orbital mechan- ics, armament)

Page - ii SFTE Reference Handbook Third Edition 2013

Publication Policy

Copyright (C) 2013 by Society Of Flight Test Engineers

All rights reserved. This Technical Handbook is for the exclusive use of the Society of Flight Test Engineers individual and Corporate Members. The Technical information contained herein may not be reproduced by any other individual or organization in any form without writ- ten permission from the Society of Flight Test Engineers. The Society reserves the exclusive right of publication.

For further information concerning the publication policy, write to:

Society of Flight Test Engineers 44814 N. Elm Avenue Lancaster, California 93534 USA

Or:

Contact the Society of Flight Test Engineers through their web site at www.sfte.org.

Please submit corrections or additions to

SFTE Handbook Committee 44814 N. Elm Avenue Lancaster, CA 93534 USA

Phone (661) 949-2095 Fax: (661) 949-2096 email: [email protected]

Page - iii SFTE Reference Handbook Third Edition 2013

SFTE Reference Handbook

Quick Index

Tab Section______

1 General Information 2 Mathematics 3 Earth and Atmosphere 4 Pitot Statics 5 Aerodynamics 6 Axis Systems and Transformations 7 Mass Properties 8 Motion/Vibration Analysis 9 Material Strength (Loads) 10 Reciprocating Engines 11 Propellers 12 Fixed-Wing Performance Standardization 13 Acoustics 14 Electromagnetic Compatibility 15 Handling Qualities 16 Rotary Wing 17 Gas Turbine Propulsion 18 Radio Communications 19 The Electromagnetic Spectrum

Page - iv SFTE Reference Handbook Third Edition 2013

SFTE Reference Handbook Complete Table of Contents Tab Section______1 General Information 1.1 Unit Conversions 1.2 Greek Alphabet 1.3 Greek Symbols Used for Aircraft 1.4 Common Subscripts 1.5 Common Abbreviations 1.6 Sign Conventions 1.7 Thermodynamic Relations 1.8 Mechanics Relations 1.9 International Phonetic Alphabet and Morse Code

2 Mathematics 2.1 Algebra 2.2 Geometry 2.3 Trigonometry 2.4 Matrix Algebra 2.5 Vector Algebra 2.6 Statistics 2.7 Standard Series 2.8 Derivative Table 2.9 Integral Table 2.10 Laplace Transform Table

3 Earth and Atmosphere 3.1 Universal Constants 3.2 Earth Properties 3.3 General Properties of Air 3.4 Standard Atmosphere 3.5 Sea States 3.6 Sunrise and Sunset Times 3.7 Crosswind Components 3.8 Geodetic Measurements

4 Pitot Statics 4.1 Subsonic Airspeed and Mach Equations 4.2 Scale Altitude (Compressibility) Correction 4.3 Subsonic Relations between Compressible and Incompressible Dynamic Pressure 4.4 Supersonic Airspeed and Mach Equations 4.5 Total Temperature Equation 4.6 Altimeter Equation 4.7 Position Error Test Methods 4.8 Position Error Certification Requirements 4.9 PEC Correction Process Flow Chart 4.10 Airspeed/Altitude/Mach Graphic Relation 4.11 Effect of Errors on Calibrated Airspeed and Altitude

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5 Aerodynamics 5.1 Dimensional Analysis Interpretations 5.2 General Aerodynamic Relations 5.3 Wing Design Effects on Lift Curve Slope 5.4 Elements of Drag 5.5 Aerodynamic Compressibility Relations 5.6 Drag Polars

6 Axis Systems and Transformations 6.1 Earth Axis System 6.2 Aircraft Axis System 6.3 Euler Angles 6.4 Flightpath Angles 6.5 Axis System Transformations

7 Mass Properties 7.1 Abbreviations and Terminology 7.2 Longitudinal and Lateral cg Measurement 7.3 Vertical cg Measurement 7.4 Moment & Product of Inertia Measurement

8 Motion/Vibration Analysis 8.1 Recurring Abbreviations 8.2 First Order Motion 8.3 Second Order Motion 8.4 Complex Plane 8.5 Parameter Conversions 8.6 Vibration Nomograph

9 Material Strength (Loads) 9.1 Terminology 9.2 Material Stress and Strain 9.3 V-n Diagram 9.4 Strain Gauges

10 Reciprocating Engines 10.1 Abbreviations and Terminology 10.2 Reciprocating Engine Modeling 10.3 Reciprocating Engine Power Standardization 10.4 FAA Approved Engine Temp. Corrections

11 Propellers 11.1 Abbreviations and Terminology 11.2 Propeller Geometry 11.3 Propeller Coefficients 11.4 Efficiency and States 11.5 Propeller Theory 11.6 Propeller Modeling 11.7 Propeller Flight Test

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12 Fixed-Wing Performance Standardization 12.1 Recurring Abbreviations 12.2 Standardization Techniques 12.3 Takeoff Distance 12.4 Landing Distance 12.5 Climb/Descent/Acceleration 12.6 Level Turn Performance 12.7 Reciprocating Engine Cruise Performance 12.8 Jet Aircraft Cruise Performance

13 Acoustics 13.1 Abbreviations and Terminology 13.2 Velocities, Spectrum, and Reference Levels 13.3 Pressure, Intensity 13.4 Weighting Curves 13.5 1/3 Octave Center Frequencies

14 Electromagnetic Compatibility (EMC) 14.1 Introduction 14.2 Abbreviations 14.3 Terms 14.4 Fundamentals 14.5 Electromagnetic Interference (EMI) 14.6 Testing 14.7 Lightening 14.8 High Intensity Radiated Fields (HIRF) 14.9 Precipitation Static (Pstatic) 14.10 Reference Material

15 Handling Qualities 15.1 Cooper-Harper Rating Related Figures

16 Rotary Wing 16.1 Principal Aeroderivatives 16.2 Forward Flight Static And Dynamic Stability

17 Gas Turbine Propulsion 17.1 Turbine Engine Basics 17.2 Propulsion System Analysis 17.3 Turbine Engine Operation 17.4 Additional Information

18 Radio Comunications

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19 The Electromagnetic Spectrum 19.1 Electromagnetic Waves 19.2 The Electromagnetic Spectrum

19.3 Radio Frequency Electromagnetic Radiation 19.4 Optical Frequency Electromagnetic Radiation 19.5 Atmospheric Transmission Windows

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Section 1 General Information Unit Conversion Website Link http://www.digitaldutch.com/atmoscalc/. 1.1 Unit Conversions Prefix Multipliers Angles Angular Acceleration Angular Velocity Area Density Electrical Quantities Energy / Work Force Illumination Inertia Length Linear Acceleration Mass Power Pressure Temperature Time Torque Velocity Viscosity Volume

1.2 Greek Alphabet 1.3 Greek Symbols used for Aircraft 1.4 Common Subscripts 1.5 Common Abbreviations 1.6 Sign Conventions 1.7 Thermodynamic Relations 1.8 Mechanics Relations

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1.1 Unit Conversions (references 1.1, 1.2)

Prefix Multipliers 1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 10 deka da 10-1 deci d 10-2 centi c 10-3 milli m 10-6 micro μ 10-9 nano n 10-12 pico p 10-15 femto f 10-18 atto a

Multiply by To Obtain

(Common FTE conversions in boldface) Angles circles 1 circumferences circles 12 signs circles 21,600 minutes circles 2π radians circles 360 degrees degrees .01111 quadrants degrees 3600 seconds degrees 60 minutes mils (Army) .05625 degrees mils (Navy) .05729 degrees quadrants 90 degrees radians 57.2958 degrees revolutions 360 degrees sphere 4π steradians #

#solid angle measurement

Angular rev/min2 0.001745 rad/sec2 Acceleration

Angular cycles/sec 6.2814 rads/sec Velocity rads/sec 0.1592 rev/sec (cycles/sec) rads/sec 9.549 rpm rad/sec 57.296 deg/sec rpm 0.01667 rev/sec

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Multiply by To Obtain

Area acres 43,560 ft2 ares 100 m2 barn 10-28 m2 centares 1 m2 circular mils 7.854 x 10-7 in2 cm2 100 mm2 ft2 144 in2 ft2 0.09290304 m2 in2 6.452 cm2 in2 106 mils2 m2 10.76 ft2 section 2,589,988.1 m2 st. mile2 27,780,000 ft2 st. mile2 2.590 km2 township 93,239,572 m2 yd2 9 ft2 yd2 0.8361 m2

Density * grams/cm3 0.03613 pounds/in3 grams/cm3 62.43 pounds/ft3 kg/m3 16.018463 pounds/ft3 slugs/ft3 515.4 kg/m3 pounds/in3 * 1728 pounds/ft3 slugs/ft3 1.94 grams/cm3 * Converting between force and mass (e.g. kg force to kg mass or pound force to pound mass) uses g = 32.174 ft/sec^2

Electrical amperes 0.1 abamperes Quantities amperes 1.0365x10-5 faradays/sec amperes 2.998x109 statamperes amperes.cicmil 1.973x105 amperes/cm2 ampere-hours 3,600 coulombs ampere-hours 1.079x1013 statcoulombs ampere turn/cm 1.257 gilberts/cm ampere turn/cm 1.257 oersteds coulombs 0.1 abcoulombs coulombs 6.243x1018 electronic charges coulombs 1.037x10-5 faradays coulombs 2.998x109 statcoulombs faradays 26.8 apmere-hours farads 10-9 abfarads farads 106 microfarads farads 8.986x1011 statfarads gausses 1 maxwells/cm2

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Multiply by To Obtain

Electrical gausses 6.452 lines/in2 Quantities gilberts 0.7958 ampere turns Cont. henries 109 abhenries henries 1.113x10-12 stathenries maxwells 1 lines oersteds 2.998x1010 statoersteds ohms 109 abohms ohms 1.113x1012 statohms ohm-cm 6.015x106 circ mil-ohms/ft volts 108 abvolts volts 0.003336 statvolts

Energy / Btu 1.055x1010 ergs Work Btu 1055.1 Joules (N-m) Btu 2.9302x10-4 kilowatt-hours Btu 251.99 calories (gram) Btu 778.03 foot-pounds calories 4.1868 watt-seconds calories 3.088 foot-pounds electron volt 1.519x10-22 Btu ergs 1 dyne-centimeters ergs 7.376x108 foot-pounds foot-pounds 1.3558 Joules (N-m) foot-pounds 3.766x10-7 kilowatt-hours foot-pounds 5.051x10-7 horsepower-hours hp-hours 0.7457 kilowatt-hours hp-hours 2546.1 Btu Joules 0.23889 calories Joules 1 Newton-meters Joules 1 watt-seconds Joules 107 ergs kilowatt-hours 3.6x106 Joules thermies 4.1868x106 Joules watt-second 0.73756 foot-pounds

Force dynes 3.597x10-5 ounces kilogram force 9.80665 Newtons kilopond force 9.80665 Newtons kip 4,448.221 Newtons Newtons 0.224808931 pounds Newtons 100,000 dynes

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Multiply by To Obtain

Forc eounce 20 pennyweights Cont. ounces (troy) 480 grains pennyweights 24 grains pound 12 ounces pounds 32.174 poundals pounds 4.4482216 Newtons pounds 5760 grains quintals (long) 112 pounds quintals (met.) 100 kilograms stones 14 pounds tons (long) 2,240 pounds tons (metric)* 1.102 tons (short) tons (short) 2000 pounds

Fuel gal 5.8 lbs (U.S. AV gas) gal 7.5 lbs ( U.S. oil) Liter (jet A) 0.812 kilograms Liter (jet A) 1.794 pounds

Note: Fuel densities are temperature dependent

Illumination candles 1 lumens/steradian candles/cm2 π lamberts candlepower 12.566 lumens foot-candles 1 lumens/ft2 foot-candles 10.764 lux foot-lamberts 1 lumen/ft2 lamberts 295.72 candles/ft2 lamberts 929.03 lumens/ft2 lumens 0.001496 watts lumens/in2 1 fots lumens/m2 1 lux lux 1 meter-candles lux 0.0001 fots meter-candles 1 lumens/m2 millilamberts 0.2957 candles/ft2 millilamberts 0.929 foot-lamberts milliphots 0.929 foot-candles milliphots 0.929 lumens/ft2 milliphots 10 meter-candles

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Multiply by To Obtain

Length angstroms 10-10 meters astronmcl units 1.496x1011 meters cable lengths 120 fathoms caliber 0.01 inches cubit 0.4572 meters fermi 10-15 meters fathoms 6 feet feet 12 inches furlongs 40 rods hands 4 inches inches 2.54 cm kilometers 3281 feet kilometers 0.53996 nautical miles leagues (U.S.) 3 nautical miles light years 5.88x1012 statute miles links (engnr’s) 12 inches links (srvyr’s) 7.92 inches meters 3.28084 feet meters 39.370079 inches microns 0.16 meters mils 0.001 inches nautical miles 1.15078 statute miles nautical miles 1,852 meters nautical miles 6,076.115486 feet paces 0.762 meters parsec 1.9163x1013 statute miles perch 5.0292 meters pica (printers) 0.0042175176 meters point (printers) 0.0003514598 meters pole (=rod) 5.0292 meters skein 109.728 meters statute miles 5,280 feet statute miles 1.609344 kilometers statute miles 8 furlongs yards 3 feet

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Multiply by To Obtain

Linear feet/sec2 1.09728 kilometers/hr/sec Acceleration feet/sec2 0.3048 meters/sec2 feet/sec2 0.6818 mph/sec g 32.174049 feet/sec2 g 9.80665 meters/sec2 gals (Galileo) 0.01 meters/sec2 knots/sec 1.6878 feet/sec2 meters/sec2 3.6 kilometers/hr/sec mph/sec 0.447 meters/sec2 mph/sec 1.609 kilometers/hr/sec

Mass* carats 200 milligrams grams 0.035274 ounces grains 6.479891x10-5 kilograms hndrdwght long 50.80 234544 kilograms hndrdwght shrt 45.359237 kilograms kilograms 0.06852 slugs kilograms 6.024x1026 atomic mass units kilograms 2.2046 pounds ounces (avd)* 28.349523125 grams ounces (troy)* 31.1034768 grams pounds (mass) 1 pounds (force) pounds (mass) 0.45359237 kilograms pounds (mass) 0.031081 slugs scuples (apoth) 0.0012959782 kilograms slugs 32.174 pounds slugs 14.594 kilograms tons (long) 1016.047 kilograms tons (assay) 0.02916 kilograms tons (metric) 1000 kilograms tons (short) 907.1847 kilograms * Converting between force and mass (e.g. kg force to kg mass or pound force to pound mass) uses g = 32.174 ft/sec^2

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Multiply by To Obtain

Moments gram-cm2 0.737x10-7 slug-ft2 of pound-ft2* 0.031081 slug-ft2 Inertia* slug-in2 0.0069444 slug-ft2 slug-ft2 1.3546 kg-m2 slug-ft2 32.174 pound-ft2 slug-ft2 12.00 pound-inch-sec2 slug-ft2 192.00 ounce-inch-sec2 * Converting between force and mass (e.g. kg force to kg mass or pound force to pound mass) uses g = 32.174 ft/sec^2

Power btu/min 0.01758 kilowatts calories(kg)/min 3087.46 foot-pounds/min ergs/sec 7.376x10-8 foot-pounds/sec ft(lbs)/min 2.260x10-5 kilowatts ft(lbs)/sec 0.07712 btu/min ft(lbs)/sec 1.356 watts hp 550 ft(lb)/sec hp 33,000 ft(lbs)/min hp 10.69 calories (kg)/min hp 745.7 watts [J/sec] hp (metric) 735.5 watts hp 1.1014 horsepower (metric) kilowatts 1.341 horsepower watts 107 ergs/sec watts 1 Joules/sec

Pressure atmospheres 14.696 pounds/in2 atmospheres 29.92 inches of Hg atmospheres 76 cm of Hg bars 1,000,000 dynes/cm2 bars 29.52 inches of Hg barye 0.1 Newtons/m2 dynes/cm2 10 Newtons/m2 2 inches of H2O 5.20237 pound/ft inches of Hg 70.72619 pounds/ft2 inches of Hg 0.491154 pounds/in2

inches of Hg 13.595 inches of H2O kiloPascals 100 bars hectoPascals 1 millibars millibars 0.02953 inches of Hg mm of Hg 0.019337 pounds/in2

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Multiply by To Obtain

Pressure mm of Hg 133.32 Newtons/m2 Cont. Pascals 1 Newton/m2 pieze 1000 Newtons/m2 pounds/ft2 0.01414 inches of Hg pounds/ft2 47.88 Newtons/m2 pounds/in2 2.036 inches of Hg 2 pounds/in 27.681 inches of H2O pounds/in2 6894.75728 Pascal torrs 133.32 Newtons/m2

Temperature Kelvin = oC+273.15o Rankin = oF + 459.67o oCentigrade = [oF − 32o] 5/9 oFahrenheit = (9/5)oC + 32

Time days (solar) 24 hours days (sidereal) 23.934 hours days (solar) 1.0027 days (sidereal) hours 60 minutes minutes 60 seconds months (sdrl) 27d + 7hr +43min +11.47sec months (lunar) 29d +12hr +44min + 2.78sec year 365.24219879 days

Torque * foot-pounds 1.3558 Newton-meters foot-pounds 0.1383 kilogram-meters ounce-inches 72.008 gram-centimeters pound-inches 1129800 dyne-centimeters * Converting between force and mass (e.g. kg force to kg mass or pound force to pound mass) uses g = 32.174 ft/sec^2

Velocity inches/sec 0.0254 meters/sec knots 1.68781 feet/sec km/hr 0.621371 mph km/hr 0.9113 feet/sec Knots (kts) 1.15078 mph Knots (kts) 1.852 km/hr Knots (kts) 0.51444 meters/sec meters/sec 3.281 ft/sec meters/sec 3.6 km/hr meters/sec 196,85 feet/min mph 1.466667 feet/sec

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Viscosity centistokes 10-6 m2/sec ft2/sec 0.0929 m2/sec pound sec/ ft2 47.880258 Newton secs/ m2 poise 0.1 Newton secs/ m2 rhe 10 m2/Newton second

Volume acre-feet 43,560 ft3 acre-feet 1,233 m3 acre-feet 3.259x105 gals (U.S.) barrels 31.5 gals (U.S.) board-feet 144 in3 bushels 1.244 ft3 bushels 32 quarts (dry) bushels 4 pecks cm3 0.001 liters cm3 0.03381 fluid ounces cm3 0.06102 in3 cord-feet 4x4x1 ft3 cords 128 ft3 cups 0.5 pints (liquid) dram (fluid) 3.69669x10-6 m3 ft3 0.0283167 m3 ft3 1728 in3 ft3 28.32 liters ft3 7.481 gals (U.S.) gals (Imperial) 1.2009 gals (U.S.) gals (Imperial) 277.42 in3 gals (U.K.) 4546.1 cm3 gals (U.S.) 231 in3 gals (U.S.) 0.003785 m3 gals (U.S.) 3.785 liters gals (U.S.) 4 quarts (liquid) gals (U.S.) 0.0238095 barrels (U.S.) gils 7.219 in3 hogshead 2 barrels in3 16.39 cm3 liters 0.02838 bushels liters 0.9081 quarts (dry) liters 1.057 quarts (liquid) liters 1000 cm3 liters 61.03 in3 m3 1.308 yd3

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Multiply by To Obtain

Volume m3 1000 liters Cont. m3 264.2 gals (U.S.) m3 35.314667 ft3 mil-feet (circ.) 0.0001545 cm3 ounces (U.K.) 28.413 cm3 ounces (U.S.) 29.574 cm3 pecks 8 quarts (dry) pecks 8.81 liters perches 0.7008 m3 perches 24.75 ft3 pints (dry) 33.60 in3 pints (liquid) 28.88 in3 pints (liquid) 4 gals quarts (dry) 1.164 quarts (liquid) quarts 2 pints register tons 100 ft3 shipping ton (U.S.) 40 ft3 shipping ton (Br.) 42 ft3 steres 1000 liters tablespoons 0.0625 cups teaspoons 0.3333 tablespoons

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1.2 Greek Alphabet

Α α Alpha Β β Beta Γ γ Gamma Δ δ Delta Ε ε Epsilon Ζ ζ Zeta Η η Eta Θ θ Theta Ι ι Iota Κ κ Kappa Λ λ Lambda Μ μ Mu Ν ν Nu Ξ ξ Xi Ο ο Omicron Π π Pi p ρ Rho Σ σ Sigma Τ τ Tau Υ υ Upsilon Φ φ Phi Χ χ Chi Ψ ψ Psi Ω ω Omega

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1.3 Greek Symbols Used for Aircraft

α angle of attack (degrees or radians) ατ tail angle of attack β angle of sideslip (degrees) γ flight path angle relative to horizontal γ specific heat ratio (1.4 for air) δ relative pressure ratio (Pa/Po) δa aileron deflection angle δr rudder deflection angle δe elevator deflection angle ε downwash angle at tail (degrees) ζ damping ratio η efficiency θ body axis/pitch angle θ relative temperature ratio, Ta/To ι angle of incidence ιF thrust angle of incidence ιT horizontal tail angle of incidence λ pressure lag constant Λ wing sweep angle μ coefficient of absolute viscosity = ρν μ Mach cone angle ν kinematic viscosity = μ/g π nondimensional parameter ρ density ρa ambient air density ρo standard atmospheric density (slugs/ft3 ) σ air density ratio (ρα /ρο) σcr critical density τ shear stress (pounds per square inch) psi τR Roll Mode Time Constant (sec) φ bank angle (degrees) ψ aircraft heading (degrees) ω frequency ω rotational velocity (radians per second) ωd damped natural frequency ωn natural undamped frequency

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1.4 Common Subscripts

a aileron a ambient alt at test altitude avg average c calibrated e elevator e equivalent E endurance leg of mission F final I initial i inbound leg of mission i indicated ic instrument corrected l subscript for coefficient of rolling moment m mission conditions m pitching moment n yawing moment O outbound leg of mission o sea-level standard day o sea level r reserve leg of mission r rudder S standard day s standard day at altitude SL sea level T True t test day

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1.5 Common Abbreviations

a lift curve slope a linear acceleration (ft/sec2 or m/sec2) a speed of sound A/A air-to-air a/c aircraft AAA anti aircraft artillery AC aerodynamic center ac alternating current ACM air combat maneuvering A/D analog to digital ADC air data computer ADC analog-to-digital converter ADF automatic direction finder ADI attitude direction indicator AFMC Air Force Materiel Command AFOTEC Air Force Operational Test and Evaluation Center A/G air-to-ground AGL above ground level AHRS attitude heading reference system AM amplitude modulation AOA angle of attack AOED age of ephemeris data APU auxiliary power unit AR air refuel (mode of flight) AR aspect ratio = b2 / S ARDP advanced radar data processor ARSP advanced radar signal processor ASPJ airborne self protection jammer ATC air traffic control avg average ax longitudinal acceleration ay lateral acceleration AZ azimuth b span of wing (feet) B/N bombadier/navigator bbl barrel BHP brake horsepower BICOMS bistatic coherent measurement system BID bus interface device BIT built-in test BSFC brake specific fuel consumption Btu British thermal unit BW bandwidth oC degrees centigrade...see T c brake specific fuel consumption (BSFC)

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c speed of light in a vacuum (186,282 miles/sec = 299,792,500 [m/s]) c mean aerodynamic chord (MAC) of a wing C/A coarse acquisition C/No carrier to noise ratio CADC central air data computer CARD cost analysis requirement document CD coefficient of drag CD i induced drag coefficient CD o zero lift drag coefficient (also parasitic drag coefficient for symmetric wing) CDI course deviation indicator CDMA code division multiplex access CDR critical design review CDRL contracts data requirement list CDU control display unit CEA circular error average CEP circular error probable Cf coefficient of friction CFE contractor furnished equipment CFT conformal fuel tank cg center of gravity (normally in % MAC) CH hinge moment coefficient cine cinetheodolite Cl rolling moment coefficient, airfoil section lift co efficient CL lift coefficient CLHQ closed loop handling qualities Clp roll damping coefficient Clr roll moment due to yaw rate coefficient Cm pitching moment coefficient CM moment coefficient cm centimeters cos cosine cot cotangent Clβ (dihedral) rolling moment due to sideslip Clδa aileron power coefficient Cmq pitch damping coefficient Cmα longitudinal static stability coefficient Cmδe elevator power coefficient Cn yawing moment coefficient Cnr yaw damping coefficient cnst constant Cnβ directional stability coefficient Cnδa adverse yaw coefficient Cnδr rudder power coefficient COTS commercial, off–the-shelf

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CP center of pressure CP propeller power coefficient CPU central processing unit cr wing root chord CRM crew resource management ct wing tip chord CTF combined test force CY calendar year CY side force coefficient CYβ side force due to sideslip coefficient CYδr side force due to rudder coefficient D diameter D drag D/A digital/analog DAC digital to analog converter DAPS data acquisition and processing system DARPA Defense Advanced Research Projects Agency db decibel DC direct current deg degrees DG directional gyro DGPS differential GPS DMA Defense Mapping Agency DME distance measuring equipment DoD Department of Defense DOP dilution of precision DSN defense switched network DT development test DTC data transfer cartridge DTIC Defense Technical Information Center e Oswald efficiency factor e natural mathematical constant = 2.718281828459 E energy E lift-to-drag ratio (CL/CD, L/D) EAS equivalent airspeed EC electronic combat ECCM electronic counter countermeasures ECM electronic countermeasures ECP engineering change proposal ECS environmental control system EGT exhaust gas temperature EL elevation ELINT electronic intelligence ELV expendable launch vehicle EM electromagnetic Emax maximum lift-to-drag ratio EMC electromagnetic compatibility EMI electromagnetic interference EMP electromagnetic pulse

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EO electro optical EOM equations of motion EPR engine pressure ratio EPROM electrically programmable read only memory Es specific energy ESA European Space Agency ESD Electronic Systems Division ESHP equivalent shaft horsepower ETA estimate time of arrival ETE estimate time en-route EW early warning EW electronic warfare oF degrees Fahrenheit f frequency...hertz (originally cycles per second) F.S. fuselage station Fa aileron force FAA Federal Aviation Administration FAR Federal Aviation Regulation FCF functional check flight FDC flight data computer Fe elevator force Fex excess thrust Fg gross thrust FL flight level Flip flight information publication FLIR forward-looking infra red FM frequency modulation FMC fully mission capable FMS flight management system FMS foreign military sales Fn net thrust Fn/δ corrected thrust parameter FOM figure of merit FOT&E follow-on test & evaluation FOUO for official use only FOV field of view fpm feet per minute fps feet per second FQT formal qualification test Fr rudder force FRD functional requirements document FRL fuselage reference line FRL force, rudder, left FRR force, rudder, right FRR flight readiness review FSD full scale development FSI full scale integration ft feet ft-lb English unit of work...foot-pound...

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fwd forward FY fiscal year g acceleration due to gravity at altitude G gravitational constant = 6.6732x10-11 [N m2/kg2] GAO Government Accounting Office GCA ground control approach GCI ground controlled intercept GDOP geometric dilution of precision GMT Greenwich mean time go standard acceleration due to gravity (sea level, 46 deg latitude) GPS global positioning system GS ground speed GSI glide slope indicator h % MAC H altitude HARM high-speed anti-radiation missile Hc calibrated altitude (assumed to be pressure altitude in flight test) HD density altitude HDDR high density digital recorder HDOP horizontal dilution of precision HF high frequency Hg mercury Hi indicated altitude hm stick-fixed maneuver point (%MAC) h'm stick-free maneuver point (%MAC) hn stick-fixed neutral point (%MAC) h'n stick-free neutral point (%MAC) hp horsepower hr hour hrs hours HSI horizontal situation indicator HUD head-up display HV host vehicle Hz hertz I/O input/output IAS indicated airspeed IAW in accordance with ICAO International Civilian Aviation Organization ICU interface computer unit ICBM intercontinental ballistic missile IFF identification friend or foe IFR instrument flight rules ILS instrument landing system IMC instrument meteorological conditions IMN indicated Mach number IMU inertial measuring unit in inch

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INS inertial navigation system INU inertial navigation unit IOC initial operational capability IOT&E initial operational test & evaluation IUGG International Union of Geodesy and Geographics Ix, Ix, Iz moments of inertia Ixy, Ixz, Iyz products of inertia J joules energy, (Newton-Meter) J propeller advance ratio J&S jamming and spoofing JCS Joint Chiefs of Staff K Kelvin (absolute temperature) K temperature probe recovery factor K, k1 constants KCAS knots calibrated airspeed KEAS knots equivalent airspeed kg kilogram, metric unit of mass KIAS knots indicated airspeed KISS keep it simple, stupid km kilometer KTAS knots true airspeed kt knots L Lift (lbs) l length L rolling moment L/D Lift-to-drag ratio LANTIRN low altitude navigation and targeting IR for night lat lateral lb pound lbf English unit of force, often just lb (pound) lbm English unit of mass, often just lb (slug) LCC life cycle cost LCD liquid crystal display LED light emitting diode LLH latitude, longitude, height ln natural log, log to the base e LO low observables Log common log, to the base 10 LOS line of sight lt distance from cg to tail's aerodynamic cent Lδa rolling moment due to aileron deflection M moment (ft-lbs) M Mach number m mass m meter (length) M pitching moment MAG magnetic MAP manifold pressure

Page 01- 20 SFTE Reference Handbook Third Edition 2013

mb millibar MCA minimum crossing altitude Mcr critical Mach number Md drag divergence Mach number Mac mean aerodynamic cord MGC mean geometric chord MHz megahertz mHZ millihertz Mic instrument-corrected Mach number MilSpec military specification MIL-STD military standard (publication) min minute (time) Mm millimeters MOA memorandum of agreement MOE measure of effectiveness MOP measures of performance MOU memorandum of understanding MP manifold pressure MSL mean sea level MTBF mean time between failures MTTR mean time to repair MX maintenance N newton (force) N rotational speed (RPM) n load factor (g's) N yawing moment N1 low pressure compressor speed N2 high pressure compressor speed NACA National Advisory Committee for Aeronautics NADC Naval Air Development Center NASA National Aeronautics and Space Administration NAV navigation NED North, East, Down NM, nm nautical mile (6080 feet) NOE nap-of-the-earth NOFORN not releasable to foreign nationals NOTAM notice to airmen NRC National Research Council (Canada) NWC Naval Weapons Center Nx longitudinal load factor (g's) Ny lateral load factor (g's) Nz normal load factor (g's) OAT outside air temperature OAT on aircraft test OEI One engine inoperative OPR Office of Primary Responsibility OSD Office of the Secretary of Defense OT&E operational test & evaluation p aircraft roll rate (degrees/sec)

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P pressure (N/m2 ,pounds per square inch) Pa ambient pressure PCM pulse code modulation P-code precision code PD pulse Doppler PDM pulse duration modulation PGM precision guided munitions PIO pilot induced oscillations Piw total thrust horsepower required Pk probability of kill PLF power for level flight 2 Po standard atmospheric pressure (2116.22 lb/ft ) POC point of contact Pp pitot pressure ppm parts per million Prop propeller Ps static pressure PS pulse search psf pounds per square foot psi pounds per square inch PT total pressure PW pulse width Q or q dynamic pressure = 0.5ρV 2 q aircraft pitch rate Q engine torque qc impact pressure (Pt − Pa) oR degrees Rankine = oF + 459.67 R perfect gas constant = 8314.34 [J/kmol K] r aircraft yaw rate (degrees/sec) R earth radius R range R&D research and development R&M reliability and maintainability R/C rate of climb rad radians Radar radio detection and ranging RAF resultant aerodynamic force RAM radar absorbing material RAT ram air turbine RCS radar cross section Re Reynolds number (dimensionless) REP range error probable RF range factor RLG ring laser gyro rms root mean square RNG range ROC rate of climb ROC required obstacle clearance RPM revolutions per minute (a.k.a. N)

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R/T receiver/transmitter RTO Rejected/refused takeoff RTO responsible test organization S wing area (ft2 or m2) Sa horizontal distance between liftoff and specified height or between specified height and touch down. SA selective availability SA situational awareness SE specific endurance sec seconds (time or angle) SFC specific fuel consumption Sg ground roll distance SHP shaft horsepower SI international system of units SIGINT signal intelligence sin sine SL sea level SLAM standoff land attack missile SLR side-looking radar S/N serial number S/N signal -to-noise ratio SOF special operations forces SOW stand-off weapon SR specific range SRB safety review board ST tail area std standard ST total takeoff or landing distance (Sa + Sg) STOL short takeoff and landing STOVL short takeoff and vertical landing T period of oscillation T temperature t thickness T, t time (sec) t/c thickness-to-chord ratio Ta ambient temperature TACAN tactical air navigation tan tangent Tas standard temperature at altitude TAS true airspeed TBD to be determined TD touchdown TED trailing edge down TEL trailing edge left TEMP test and evaluation master plan TER trailing edge right TEU trailing edge up

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TF terrain following THP Thrust Horsepower THPalt horsepower available at altitude THPmax maximum horsepower available THPmin minimum horsepower required THPSL horsepower required at sea level TIT turbine inlet temperature TM telemetry TMN true Mach number T/O takeoff o o To standard sea level temperature (59.0 F, 15 C) TO technical order TRB technical review board TRD technical requirements document TRP technical resources plan TSFC thrust specific fuel consumption TSPI time, space, position information Tt total temperature TV television T/W thrust to weight ratio TWT track while scan TWT traveling wave tube u velocity along aircraft's x-axis UAV uninhabited aerial vehicle UHF ultra high frequency UPT undergraduate pilot training USA US Army USAF US Air Force USCG US Coast Guard USMC US Marine Corps USN US Navy UT universal time UV ultraviolet v velocity along aircraft's lateral axis VH horizontal tail volume coefficient VV vertical tail volume coefficient V1 takeoff decision speed V2 takeoff safety speed VA design maneuvering speed VAC volts AC Vb buffet airspeed VB design speed for max gust intensity Vbr velocity for best range Vc calibrated airspeed VD design diving speed VDC volts DC VDOP vertical dilution of precision Ve equivalent velocity VFE maximum flap extended speed

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VFR visual flight rules Vg ground speed VHF very high frequency Vi indicated airspeed Vic indicated airspeed corrected for instrument error Viw velocity at sea level std day and std weight VLE max speed with landing gear extended VLO max speed while operating landing gear VLOF lift off speed VLSIC very large scale integrated circuit Vmc minimum directional control speed VMC visual meteorological conditions Vmca minimum directional control speed in the air Vmcg minimum directional control speed on the ground Vmo/Mmo maximum operating limit speed Vmu minimum unstick speed VNE never exceed velocity Vno max structural cruising speed Vopt optimum velocity for endurance flight VOR VHF omni-directional range VORTAC VHF omni-directional range Tactical Air Navi gation VPmin velocity for minimum power VPmin,SL velocity for minimum power at sea level VR rotation speed VS stall speed VS0 stall speed in landing configuration VS1 stall speed in some defined configuration VSTOL vertical/short takeoff and landing VT true airspeed VTOL vertical takeoff & landing VVI vertical velocity indicator VW wind velocity VX speed for best angle of climb VY speed for best rate of climb W weight w component of velocity along aircraft's Z-axis WDL weapon data link W/δ weight-to-pressure ratio Wf fuel weight WGS-84 World Geodetic System, 1984 WI watch item WOD word of day WOW weight on wheels WPT waypoint wrt with respect to

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W & f , corrected fuel flow parameter δ ϑ

W/S wing loading Wf fuel flow (lb/hr) x aircraft longitudinal axis, a line running through the nose & tail Xac distance from leading edge to aerodynamic cen ter Xlink cross link y aircraft lateral axis, a line running the wingtips Y force along y-axis Y-code encrypted P-code z aircraft vertical or yaw axis, a line perpendicular to the longitudinal and lat eral axes ΔHic altimeter instrument correction ΔHpc altimeter position error correction ΔPp pitot pressure error ΔPs static pressure error ΔVc scale attitude correction to airspeed ΔVic instrument correction to airspeed indicator ΔVpc correction for airspeed position error ∞ infinity, or freestream conditions

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1.6 Sign Conventions (reference 1.8)

Editor’s note There is near unanimous agreement on most sign conventions except for pilot inputs and control surface deflections. Although individual organizations generally are consistent in-house, confusion often arises when trying to mathematically translate inputs & deflections from one organization to another. This section documents the generally accepted “body axes” sign conventions then discusses the rationale for several view- points addressing the “inputs & deflections” debate. Below is the SFTE sign convention.

Wind Axes Sign Convention Winds are listed according to the direction they are coming from. Airports refer winds to magnetic North while winds at altitude are typically referred to true North. Headwind is true airspeed minus ground speed. (Vw=VT- Vg).

Body Axes Sign Convention The generally accepted body axes sign convention is based on the establishment of a three-dimensional axis sys- tem with the following properties: 1. It is right-handed orthogonal 2. Its origin is at the vehicle's reference center of gravity (defined by builder). 3. The axis system moves with the airframe.

y

x Aft view, looking forward Typical + Fuselage Buttock Reference line = 0 Station + Waterline + x y

z z

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Translational displacements, rates, accelerations, & forces are positive along the positive body axes di- rections. In spite of the simplicity of this logic, it is important to recognize that lift and normal load factor are positive in the negative z direction and the drag is positive in the negative x direction. Angular displacements, rates, accelerations & moments, are positive according to the “right hand rule” (a clock- wise rotation while looking in the direction of the positive axis) as shown in the figure.

The body axes, forces & translations along them, and moments & rotations about them are shown with arrows indicating the positive direction. Angular displacements, rates, accelerations & moments, are positive according to the “right hand rule” (a clock- wise rotation while looking in the direction of the positive axis) as shown in the figure.

The body axes, forces & translations along them, and moments & rotations about them are shown with arrows indicating the positive direction.

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Angle of attack is positive clockwise from the projection of the velocity vector on the xz plane to the reference x body axis. The angle of sideslip is positive clockwise from the xz plane to the velocity vector (wind in the pilot’s right ear).

Aircraft true heading is the angle between true North and the projection of the x-body axis onto the hori- zontal plane. Mag. heading refers to mag North

The velocity vector is measured relative to the air mass while the flightpath is measured relative to the ground. They are equivalent only when winds are zero. Flightpath heading angle (ground track heading) σg, is the horizontal angle between true North and the projec- tion of the flightpath on the horizontal plane. Positive rotation is from north to east.

Flightpath elevation angle; γ, is the vertical angle between the flightpath and the horizontal plane. Posi- tive rotation is up. During a descent, this parameter is commonly known as glide path angle. Flightpath bank angle; μ, is the angle between the plane formed by the velocity vector and the lift vector and the vertical plane containing the velocity vector. Positive rotation is clockwise about the velocity vector, looking forward.

Fuselage reference station (FRS), Water line (WL), and Buttock line (BL) are reference coordinates es- tablished by the design group.

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Summary of Generally Accepted Body Axes Sign Convention

Parameter Name Symbol Positive Direction

Translational Measurements

Longitudinal axis x from ref cg towards nose

Lateral axis y from reference cg towards right wing tip

Vertical axis z from reference cg towards vehicle bottom (body axis) Longitudinal velocity u along +x axis

Lateral velocity v along +y axis

Vertical velocity w along +z axis

Long. acceleration ax along + x axis

Lateral acceleration ay along +y axis

Vertical acceleration az along +z axis

Longitudinal load factor Nx along +x axis

Lateral load factor Ny along +y-axis

Normal load factor Nz along –z axis

Longitudinal force Fx along the +x axis

Lateral force Fy along the +y axis

Normal force Fz along the + z axis

Drag force D along the –x axis

Side force Y along the + y axis

Lift Force L along the –z axis

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Summary of Generally Accepted Body Axes Sign Convention

Parameter Name Symbol Positive Direction

Angular Measurements

Bank angle φ right wing down

Pitch angle θ nose-up

Heading ψ 0 North, +Eastward

Angle of attack α normal flight attitude

Angle of sideslip β “wind in the right ear”

Roll rate p right wing down

Pitch rate q nose up

Yaw rate r nose right

Roll moment L right wing down

Pitch moment M nose up

Yaw moment N nose right

Flightpath bank angle μ right wing down

Flightpath elevation γ climb

Flightpath heading σg 0 true North, + East- ward

Discussion of “Input & Deflection“ Conventions

The debate regarding proper inputs and deflections stems from the user’s viewpoint. From the body axis convention above, flight testers recognize that a climbing right turn generates positive angular measurements. Logically then, pull, right roll and right yaw pilot inputs and subsequent surface deflections should also be posi- tive. The traditional flight tester’s convention follows as “All input forces & displacements, surface deflections, and motions that cause a climbing right turn are positive.”

Due to differential nature of aileron deflections, they require more discussion. The flight tester’s logic implies (but does not dictate) positive deflections are right aileron up and left aileron down. It is, however, equally acceptable to assign downward (or upward) deflection as positive for both ailerons and calculate the dif- ference between the two as a measure of rolling moment.

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The rationale within the wind tunnel community is also logical: any control surface deflection that in- creases lift is positive. From this, positive deflections are trailing edge down (TED) for each: trailing edge flap, stabilizer, elevator, stabilator, rollervator, ruddervator, canard, aileron, flaperon, and all their tabs. Leading edge flap down is also positive. Similarly, since side force is positive to the right, then positive rudder and rudder tab deflections are trailing left (TEL). The only exception to this straightforward logic is for spoilers and speed brakes that extend only in one direction: this deflection is positive even though it might decrease the lift.

Since the above rationale defines downward deflection as positive for both ailerons, a measurement of rolling moments requires calculation of the differential aileron deflection. This rationale does not, however, spe- cifically dictate whether a “positive” differential deflection should generate right wing down (RWD) or left wing down (LWD) moments. Differential aileron can be calculated as either.

δ aR −δ aL δ aL −δ aR δ a = or δ a = 2 2

Selection of the RWD convention is obvious from the flight tester’s viewpoint since deflections that generate right rolls are positive. An alternative interpretation is that a positive differential aileron deflection is one that lifts the positive (right) wing lifts more than the left (LWD).

Another common convention for ailerons is one that gives the same sign to both ailerons for any input. The “right hand screw” convention is opposite to the flight tester’s convention, but may be more common:

δaR = +TED, δaL = +TEU.

The above wind tunnel rationale dictates only the polarity for individual control surface deflections, and leaves open the sign convention debate about controller (inceptor) input forces & displacements. One approach is that positive inputs should generate positive motions while an alternate approach is that positive inputs gener- ate positive surface deflections. Only the flight tester’s convention states that positive inputs yield positive mo- tions and deflections. All approaches are mathematically connected to the hinge moment sign convention dis- cussed below.

The simplest control surface hinge moment convention is that all positive hinge moments (generated by the pilot and the aerodynamics) move the surface in a positive direction, i.e., positive input forces yield positive deflections. This has different implications for the different sign conventions:

• According to the above flight tester’s sign convention, a positive pull force is required to generate a positive (TEU) elevator deflection (positive stick force generates a climb). • According to wind tunnel sign convention, a positive push force is required to generate a positive (TED) ele- vator deflection (positive stick force generates a dive).

The alternate viewpoint defines a positive inceptor hinge moment as one that opposes the aerodynamic mo- ments. In other words, a positive inceptor hinge moment moves the surface to a position which generates posi- tive aerodynamic hinge moments or “positive input forces & displacements generate negative surface deflec- tions.”

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Based on the above background, the SFTE technical council proposes the following standard convention for in- ceptor & surface forces & deflections:

• Due to its widespread use and its simple & robust nature, use the wind tunnel convention for control surface deflections. • Due to widespread test pilot & FTE familiarity and logical nature, use the flight tester’s convention that pos- itive inceptor forces & displacements generate a climbing right turn. • A fallout from these conventions is that positive inceptor hinge moments generate positive aerodynamic hinge moments (negative surface deflections). • Consistent use of the above logic requires that the calculated value for aileron deflection be negative for right wing down moments. Similarly, differential ruddervator deflections generating nose right yawing mo- ments should have negative values.

Conventions for Positive Control Surface Deflections

SFTE/ Parameter Symbol Flight Test Wind Tunnel

Horizontal Stabilizer δi TEU TED

Elevator δe TEU TED

Elev. Tab δet TED

δeL , δeR TEU TED Stabilators or Rollerva- tors, δ = (δ + δ )/2 average: e eR eL differential: Δδe = (δeR - δeL)/2

δvL , δvR TEU TED Elevons average: δv = (δvR + δvL)/2 differential

Δδv = (δvR - δvL)/2

δfR , δfL TED Flaperons or trailing edge flap = (δ + δ )/2 average: δf fR fL differential: Δδf = - (δfR - δfL)/2 = (δfR - δfL)/2

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Conventions for Positive Control Surface Deflections (Cont’d) SFTE/ Parameter Symbol Flight Test Wind Tunnel

δcL , δcR TED Canards

average: δc = (δcR + δcL)/2 differential Δδc = - (δcR - δcL)/2 = (δcR - δcL)/2

δlefL , δlefR TED Leading edge flap

Average: δlef = (δcR + δcL)/2 Differential: Δδlef = - (δcR - δcL)/2 = - (δcR - δcL)/2

δrvL , δrvR TEU TED Ruddervators = (δrvR + δrvL)/2 Average: δ rv Differential: = - (δrvR - δrvL)/2 Δδ rv

δaRTEU, δaL TEDor {δ δaL , δaR δaR, δaL TED aR, δaLTED} Ailerons Aileron Tab δ = (δaR +δaL)/2 δat TED Average: at

δa = - (δaR -δaL)/2} = (δaR -δaL)/2 *

δsL , δ sR Extended Spoilers average: = (δ +δ )/2 Differential: δs sR sL

Δds = (δsR -δsL)/2 = - (δsR -δsL)/2

Rudders δrR , δrL TER TEL Average: δr = (δrR +δ r L)/2

Rudder tab δrt TEL

Speed brake δsb Extended Conventions for Positive Inputs and Hinge Moments SFTE/ Parameter Symbol Flight Test Wind Tunnel Stick/Wheel F Pull Long Force e Stick/Wheel F Right Lateral Force a

Pedal Force Fr Right pedal push Stick/Wheel δs Aft Long. deflectn e

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Conventions for Positive Inputs and Hinge Moments (Cont’d)

Parameter Symbol Flight Test SFTE#

Stick/wheel δsa Right Lat. deflection

Pedal deflection δpR, δpL Right pedal push

Aerodynamic Hinge Chδ positive moments Moments Chα generate Chδο positive deflections Chδtab

Inceptor ChFe + moments generate + moments generate Hinge Moments ChFa + deflections - deflections ChFr

*The wind tunnel rationale does not inherently define the polarity for control surface differential deflections.

#The wind tunnel rationale does not specify a convention for positive inputs or hinge moments. Historically, Dutch, U.S. and some British aircraft use a climbing right turn, while it is a diving left turn for Canadian, Aus- tralian, and some British aircraft.

The SFTE Technical Council recognizes that several combinations of the above possibilities are currently in use around the world, and invites comments, additions, or corrections to the above summary and proposal. Alt- hough SFTE does not expect all organizations to adopt this standard, it still provides a cornerstone for reference purposes

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1.7 Thermodynamics Relations (references 1.3, 1.4, 1.5, 1.6)

A Process is an event with a redistribution of energy within a system.

A Reversible process is one that can be reversed such that the system returns to its original state (form, location & amount).

An Irreversible process cannot return to its original state due to heat flow from higher to lower temperatures, fluid turbulence, friction, or inelastic deformation. The change in entropy is non-zero.

An Isothermal process is one in which the temperature of the fluid is constant.

An Adiabatic process is one in which heat is not transferred to or from the fluid.

Work is the energy transfer by way of changing mechanical energy.

Heat is the energy transfer from one body to another by virtue of a temperature difference between them.

An Isentropic process has constant entropy.

Conduction is the energy transfer from a warmer body by tangible contact (transfer of some internal molecular kinetic energy).

Convection is the repositioning the energy of a fluid without state changes or energy transformations (e.g. heat- ed air moving from one room to another room).

Radiation is the energy transmission through space.

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A = area C = compressibility factor C = speed of sound E = u = specific internal energy (e.g. Btu /lb) H = specific enthalpy ≡ E + PV (e.g. Btu/lb) J = Joule’s equivalent 107 ergs = 778 ft-lb/Btu Q = energy supplied to a system or region as heat (e.g. Btu/lb) P = absolute pressure (e.g. lbs/ft2) V = specific volume (e.g. ft3/lb) W = work (+ if entering) V = velocity Δ = change ( final – initial value) Z = altitude dE + PdV S = specific entropy ≡ for a reversible process ∫ T

R = gas constant for each gas (for air = 287 J/[kg K] = 53.35 ft-lb/lbmR)

R = R[M] = universal gas constant = 8.314 kJ/[kmol K] = 1545 ft lb/[lbmol R] M = molar mass (for air = 28.97 kg/kmol) N = number of moles ρ = density

The First Law of Thermodynamics shows that the net amount of energy added to a system equals the net change in energy within the system (Principle of Conservation of Energy): W + Q = (E2 - E1)

The Second Law of Thermodynamics states that entropy increases during any irreversible process:

S2>S1

Ideal Gas Equation of State (a.k.a. Perfect gas law): PV=RT, P = ρRT, PV = mRT, PV = nRT

δ = σθ where δ Pa/Po, σ = ρa/ρo, θ = Ta/To

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Boyle’s Law states that when the temperature of a given mass of gas is held constant, then the volume and pres- sure vary inversely.

Charle’s Law states that when a volume of a given mass is held constant, then the change in pressure of the gas is proportional to the change in temperature.

Real Gas Relation: PV = CRT for reversible processes

W = −∫ PdV Q = TdS ∫ for reversible adiabatic process

γ P1 ⎡V2 ⎤ = ⎢ ⎥ P2 ⎣V1 ⎦ γ −1 ⎡ ⎤ T1 V2 = ⎢ ⎥ T2 ⎣V1 ⎦ γ −1 T ⎡ P ⎤ γ 1 = 1 ⎢ ⎥ T2 ⎣P2 ⎦ γ P1 ⎡ ρ1 ⎤ = ⎢ ⎥ P2 ⎣ρ2 ⎦

Steady Flow Energy Equation

2 2 V 1 V 2 Q + H + + Z =W + H + + Z 1 2g 1 2 2g 2

Bernoulli Equation:

2 2 ΔP V 2 −V 1 + + ΔZ = 0 ρg 2g

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Flow per Unit Area:

W γ P M = A R T ⎛ γ −1 2 ⎞ γ +1 ⎜1+ M ⎟ ⎝ 2 ⎠ 2()γ −1

Velocity of sound in a perfect gas: c = γgRT

Development of Specific Heat Relations

∂H specific heat at constant pressure c ≡ o p (for air = 1004.76 J/[kg K]) ∂T P

∂u specific heat at constant volume cv ≡ o ∂T v (for air = 717.986 J/[kg K])

c p κ = γ ≡ = ratio of specific heats c v

Enthalpy equation in differential form is: dH = du +d(PV) Substituting definitions and ideal gas law gives cp dT = cv dT + Rdt or cp = cv + R

κ R Rearranging gives c p = R and cv = κ −1 κ −1 Development of Poisson’s Equation:

1) From the 1st law: W+Q = E2-E1 2) Substitution for each term gives T dS – P dV = du 3) Divide through by T: dS = du/T + P dV/T

4) Recall du = cv dT and PV = RT

5) Substitution gives dS = cv dT/T + R dV/V 6) Assume constant specific heat and integrate:

T2 V2 s2 − s1 = cv ln + R ln T1 V1

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7) Assuming a reversible adiabatic process T V c ln 2 = −R ln 2 v T V 1 1 κ −1 R T ⎛ V ⎞ 8) Substitute to get: 2 = ⎜ 1 ⎟ cv = ⎜ ⎟ κ −1 T1 ⎝V2 ⎠

9) Differentiate H: dH = du + P dV + V dP 10) Substitution into step #2: T dS = dH-V dP

T2 P2 11) Integrate: s2 − s1 = c p ln + R ln T1 P1 T P c ln 2 = −R ln 2 12) Assuming a reversible adiabatic process: p T P 1 1 κ −1 κ κ 13) Substitute to get: T2 ⎛ P2 ⎞ c p = R = ⎜ ⎟ κ −1 ⎜ ⎟ T1 ⎝ P1 ⎠

κ P2 ⎛ V1 ⎞ κ 14) Combine steps #8, #13 to = ⎜ ⎟ get: or PV = const. P1 ⎝V2 ⎠

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1.8 Mechanics Relations

Abbreviations

a = linear acceleration = dV/dt

ar = centripetal (radial) acceleration

aT = tangential acceleration F = force g = acceleration due to gravity (32.174 ft/s2 = 9.80 meters/s2) G = moment H = angular momentum = Iw H = height Hp = horsepower (Hp = 550ft-lbs/sec) I = rotational moment of inertia (see section 10) J = impulse = change in momentum k = radius of gyration m = mass

Nr = radial load factor = ar/g P = power = dW/dt L = linear momentum = mV Q = moment (a.k.a. torque) r = radius S = distance, displacement s = seconds t = time V = true inertial velocity

Vo = initial inertial velocity 2 2 W = work = FS = ½ m [V - Vo ] q = angular displacement Vol = volume ω = angular velocity (radians/second) ώ = angular acceleration

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Newtons Laws 1st law (law of inertia): “Every body persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed on it.

2nd Law: “ The change in motion is proportional to the motive force impressed and is made in the direction of the straight line in which that force is impressed” (motion defined as velocity x quantity of matter or linear momentum, mV). dF = dmV/dt = (dm/dt) + (dV/dt) For constant mass in rectilinear motion: F = ma For constant mass distribution in curvilinear motion: G = ω& I

3rd Law: “Every action has an equal and opposite reaction; or, the mutual attraction of two bodies upon each other are always equal and directed to contrary parts.[opposite directions]”

Planar Kinetics, Work, Power and Energy Rectilinear motion Curvilinear motion displacement S angular displacement θ velocity V = dS/dt angular velocity ω = dθ/dt acceleration a = dV/dt angular acceleration = ω& d ω /dt

2 inertia m rotational inertiaI = ∫ r dm momentum L = mV angular momentum H = I ω& ω force F = ma torque Q = I work W = ∫ FdS work W = ∫ Qd θ power P = FV power P = Q ω kinetic energy ½ mV2 kinetic energy ½ I ω2 potential energy mgH n/a

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Planar Kinematics at Constant Acceleration

Rectilinear motion Curvilinear motion

V = Vo + at ω = ωo + t 2 2 2 2 ω& V = Vo + 2aS ω = ω o +2 θ 2 2 S = Vot + ½ at θ = ω ot + ½ ω& t

S = ½(V + Vo)t θ = ½( ω + ω o)t 2 2 2 2 S =(V −V0 )/2a θ =(ω −ω0 )/2ω&

−V + V 2 +2aS −ω + ω2 −2ωθ t = 0 0 t = 0 0 & a ω& 2(S −V t) 2(θ −ω t) 0 0 a = 2 ω& = 2 t t

Curvilinear motion with constant acceleration and radius: 2 r = V /gNr V = ωr NR = ar/g ω = gNr/V

ω = V& & R

2 2 ar = rω = V /r

a r = ω& r

Aircraft in level turn:

Nzw = load factor normal to flight path r = turn radius Ω = turn rate (rad/sec)

V 2 r = g N 2 − 1 () zw g ()N 2 − 1 ω = zw V

⎛ ω V ⎞ N zw = ⎜ ⎟ + 1 ⎝ g ⎠

V= inertial velocity

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Gyroscopic Motion (reference 1.7) for bodies spinning about an axisymmetric axis

ψ& = spin rate φ& = precession rate θ& = nutation rate

Iz = moment of inertia about spin axis

It = transverse moment of inertia about the spin point (perpendicular to spin axis)

Icg = moment of inertia about the cg (perpendicular to spin axis)

Mx = moment about spin point (acting along plane that defines θ) . . For steady precession (constant θ, φ , ψ )

2 ∑ M x = −Itφ& sin θ cosθ + I zφ&sin θ (φ&cosθ +ψ& )

For torque free motion (gravity is only external force)

Icg − I z ψ& = φ&cosθ I z

note that Icg>Iz yields regular precession

while Icg

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Section 1.9 International Phonetic Alphabet and Morse Code

A Alpha • ▬ ▬●●● B Bravo C Charlie ▬ • ▬ • D Delta ▬ • • E Echo ● F Foxtrot • • ▬ • G Golf ▬ ▬ • H Hotel • • • • I India • • J Juliet • ▬ ▬ ▬ ▬ • ▬ K Kilo L Lima • ▬ • • M Mike ▬ ▬ N November ▬ • ▬ ▬ ▬ O Oscar P Papa • ▬ ▬ • Q Quebec ▬ ▬ • ▬ R Romeo • ▬ • S Sierra • • • T Tango ▬ U Uniform • • ▬ V Victor • • • ▬ W Whiskey • ▬ ▬ X X-ray ▬ • • ▬ Y Yankee ▬ • ▬ ▬ Z Zulu ▬ ▬ • •

1 One • ▬ ▬ ▬ 2 Two • • ▬ ▬ ▬ 3 Three • • • ▬ ▬ 4 Four • • • • ▬ 5 Five • • • • • 6 Six ▬ • • • • 7 Seven ▬ ▬ • • • 8 Eight ▬ ▬ ▬ • • 9 Niner ▬ ▬ ▬ ▬ • 0 Zero ▬ ▬ ▬ ▬ ▬

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Section 1 References http://www.onlineconversion.com/

1.1 Anon., “Weight Engineers Handbook”, Society of Weight Engineers, P.O.Box 60024 Los Angeles, CA 90060,1976.

1.2 Anon., “Aeronautical Vestpocket Handbook”, United Technologies Pratt & Whitney Canada, 1000 Ma- rie Victorin Blvd. E. P.O.B. 10 Longueuil, Quebec Canada J4K 4X9.

1.3 Jones, J. P., Hawkins, G.A., “Engineering Thermodynamics” John Wiley & Sons, 1960.

1.4 Esbach, Ovid W., “Handbook of Engineering Fundamentals”, John Wiley and Sons Inc., 1963.

1.5 Potter, M.C., Somerton, C.W., “Engineering Thermodynamics” Shaum’s Outline Series, McGraw-Hill, Inc.,1993.

1.6 Abbott, M. M., Van Ness, H. C., “Thermodynamics”, Shaum’s Outline Series, McGraw-Hill, Inc., 1989.

1.7 Halliday, D., Resnick, R., “Fundamentals of Physics”, John Wiley & Sons, New York, 1981.

1.8 Roberts, S.C., Chapter 3 Aircraft Control Sytems , “Aircraft Flying Qualities Testing”, National Test Pilot School, 1997. P.O.B. 658, Mojave, CA, 93501.

1.9 Unit Conversion Website Link http://www.digitaldutch.com/atmoscalc/.

Page 01 - 47 SFTE Reference Handbook Third Edition 2013

NOTES

Page 01- 48 SFTE Reference Handbook Third Edition 2013

Section 2 Mathematics

2.1 Algebra Laws Identities Equations Interest & Annuities

2.2 Geometry

2.3 Trigonometry Graphs Identities Oblique Triangle Laws

2.4 Matrix Algebra

2.5 Vector Algebra

2.6 Statistics

2.7 Standard Series

2.8 Derivative Table

2.9 Integral Table

2.10 Laplace Transform Table

2.11 References

Page 02 - 1 SFTE Reference Handbook Third Edition 2013

Section 2.1 Algebra (reference 2.1)

LAWS commutative: a+b = b+a ab = ba associative: a+(b+c) = (a+b)+c distributive: a(b+c) = ab+ac

IDENTITIES exponents: axay = ax+y (ab) x = axbx (ax)y = axy amn = (am)n a 0 = 1 a ≠ 0 if

1 x a −x = = ⎛ 1 ⎞ x ⎜ ⎟ a ⎝ a ⎠ a x = a x− y a y x ab = [x a ][x b] x y y x y x a = a = ()a 1 a y = y a x x a y = y a x = ()y a x a y a = a (1 / x ) + (1 / y ) = xy a x + y

a + b = a + b + 2 ab

Page 02 - 2 SFTE Reference Handbook Third Edition 2013 logarithms: if M, N,b are positive and

logbb = 1

logb1 = 0

log b MN = log bM + log b N

log b [M/N] = log b M – log b N p log b M = p log b M

logb [1/M] = -log b M

q 1 log b M = log b M q log M logM = log M log c = c b c b log b c

examples: log 6.54 = .8156, log 6540 = log (6.54 x 103) = .8156 +3 = 3.8156 log .654 = log (6.54 x 10-1) = .8156 -1 = 9.8156 -10 log .000654 = log (6.54 x 10-4) = .8156 -4 = 6.8156 -10

calculate 68.31 x .2754: log 68.31 = 1.8354 log .2754 = -.56 1.8354 +(-.56) = 1.2745 log-1 1.2745 = 18.81

calculate [.6831]1.53: log .6831 = -.1655 1.53 x (-.1655) = -.253 ` log-1[-.253] = .5582

calculate [.6831]1/5: log .6831 = -.1655 1/5 x(-.1655) = -.0331 log-1 (-.0331) = .9266

solve for x in .6931x = 27.54: log[.6931x] = log 27.54 x log [.6931] = log 27.54 x = log 27.54 /log [.6931] =1.44/[-.1655] = -8.701

Page 02 - 3 SFTE Reference Handbook Third Edition 2013

EQUATIONS

Quadratic Equation: for ax2 + bx + c =0 (has two roots, both real or both complex)

2 − b ± b − 4ac x1,2 = 2a

Cubic Equation: for y3 + py2 + qy + r = 0 (has three roots, all real or one real & two complex) let y = x –(p/3) to rewrite equation in form of x3 + ax + b = 0 where a = (3q –p2)/3 and b = (2p3 – 9pq – 27r)/27

b b2 a3 let A = 3 − + + 2 4 27

b b2 a3 and B = 3 − − + 2 4 27

then x1 = A +B 1/2 x2 = -(A + B)/2 + {[-3] /2}(A – B) 1/2 x3 = -(A + B)/2 - {[-3] /2}(A – B) special cases… if (b2/4 + a3/27 <0), then the real roots are 1/2 o x1,2,3 = 2[-a/3] cos(φ/3 + 120 k) where k = 0,1,2 and cosφ = +[(b2/4)/(-a3/27)]1/ 2 if b<0 or cosφ = -[(b2/4)/(-a3/27)]1/ 2 if b>0 if (b2/4 + a3/27 >0) and a>0, the single real root is x = 2[a/3]1/2 cot(2φ) where tan(φ) = [tan(ψ)]1/3 and cot(2ψ) = +[(b2/4)/(-a3/27)]1/ 2 if b<0 or cot(2ψ) = -[(b2/4)/(-a3/27)]1/ 2 if b>0

Page 02 - 4 SFTE Reference Handbook Third Edition 2013 if (b2/4 + a3/27 =0), the three real roots are 1/2 /2 1 x1 = -2[-a/3] , x2 =x3 = +[-a/3] if b>0 1/2 1/2 or x1 = +2[-a/3] , x2 =x3 = -[-a/3] if b<0

Quartic (biquadratic) Equation: for y4 + py3 + qy2 + ry + s = 0 let [y = x-(p/4)] to rewrite equation as x4 + ax2 + bx + c =0 let l, m, n denote roots of the following resolvent cubic… t3 + at2/2 + (a2 –4c)t/16 – b2/64 = 0 the roots of the quartic are 1/2 1/2 1/2 x1 = +[l] + [m] + [n] 1/2 1/2 1/2 x2 = +[l] - [m] - [n] 1/2 1/2 1/2 x3 = -[l] + [m] - [n] 1/2 1/2 1/2 x4 = -[l] - [m] + [n]

Page 02 - 5 SFTE Reference Handbook Third Edition 2013

INTEREST AND ANNUITIES (reference 2.3) Amount:

P principal at i interest for n time accumulates to amount An:

simple interest: An = P(1 + ni) n at interest compounded each n interval: An = P(1 + i) nq at interest compounded q times per n interval: An = P(1 + r/q) where r is the nominal (quoted) rate of interest

Effective Interest: The rate per time period at which interest is earned during each pe riod is called the effective rate i. i = (1 + r/q)q -1

Solve above equations for P to determine investment required now to accumulate to amount An

True discount , D = An – P

Annuities: rent R is consistent payment at each period n

n let (1+ i) −1 sn ≡ i

1− (1+ i)−n and let rn ≡ i

then An = Rsn

log(An + R) − log R or n = log(1+ i)

present value of an annuity, A is the sum of the present values of all the future payments. A = Rrn

Monthly interet rate = MIR = (annual interest rate) /12

Month Term = # months in loan

Monthly payment = [amount financed]* [MIR/(1-{1+MIR}-#months)]

Final value (FV) of an investment is a function of the initial principal invested (P), interest rate (r –expressed as .05 for 5%, .1 for 10% etc.), time invested (Y- typically years), and compounding periods per year (n – typically =1 for yearly or =12 for monthly). FV = P (1 + r / n)Yn

Page 02 - 6 SFTE Reference Handbook Third Edition 2013

Section 2.2 Geometry (references 2.1, 2.2)

General definitions: A = area a = side length b = base length C = circumference D = diameter h = height n = number of sides R = radius V = volume x, y, z = distances along orthogonal coordinate system β = interior vertex angle triangle: A = bh/2 sum of interior angles = 180o rectangle: A = bh sum of interior angles = 360o parallelogram (opposite sides parallel): A = ah = absin β trapezoid (4 sides, 2 parallel): A = h(a+b)/2 pentagon, hexagon, and other n-sided polygons: A = {(na2)cot (180o/n)}/4 R = radius of circumscribed circle = {a csc(180o/n)}/2 r = radius of inscribed circle = {a cot(180o/n)}/2 β = 180o –(360o/n) sum of interior angles = n180o-360o

Page 02 - 7 SFTE Reference Handbook Third Edition 2013 circle: A = πR2 C = 2πR = πD perimeter of n-sided polygon inscribed within a circle = 2nRsin(π/n) area of circumscribed polygon = nR2tan(π/n) area of inscribed polygon = {nR2sin(2π/n)}/2 equation for a circle with center at (h,k): R2 = (x-h)2 + (y-k)2 ellipse: f = semimajor axis g = semiminor axis e = eccentricity = ([f2-g2]1/2)/f A = πef equation for ellipse with center at (h,k): (x-h)2/f2 + (y-k)2/g2 = 1 if major axis along x-axis or (y-k)2/f2 + (x-h)2/g2 = 1 if major axis along y-axis distance from center to either focus = [f2-g2]1/2 latus rectum = (2g2)/a parabola: p = distance from vertex to focus e = eccentricity = 1 equation for parabola with vertex at (h,k), focus at (h+p,k): (y-k)2 = 4j(x-h) if (j>0) equation for parabola with vertex at (h,k), focus at (h,k+p): (x-h)2 = 4j(y-k) if (j<0)

p Focus

Vertex

Directrix

Page 02 - 8 SFTE Reference Handbook Third Edition 2013

hyperbola: p = distance between center and vertex q = distance between center and conjugate axis e = eccentricity = ([p2+q2]1/2)/p equation for hyperbola centered at (h, k): (x-h)2/p2 – (y-k)2/q2 = 1 if (asymptotes slopes = +/- q/p) or (y-k)2/p2 – (x-h)2/q2 = 1 if (asymptotes slopes = +/- p/q)

p

asymptotes

q sphere: A = 4πR2 V = 4πR3/3 equation for sphere centered at origin: x2+y2+z2 = R2

torus: A= 4π2Rρ V = 2π2Rρ2 ρ = smaller radius

Page 02 - 9 SFTE Reference Handbook Third Edition 2013

Section 2.3 Trigonometery (references 2.1, 2.2)

For any right triangle with hypotenuse h, an acute angle α, side length o opposite from α, and side length a ad- jacent to α, the following terms are defined: sine α = sin α = o/h cosine α = cos α = a/h tangent α = tan α = o/a = sin α /cos α cotangent α = cot α = ctn α = a/o = 1/tan α = cos α /sin α secant α = sec α = h/a = 1/cos α cosecant α = csc α = h/o = 1/sin α exsecant α = exsec α = sec α -1 h versine α = vers α = 1-cos α o coversine α = covers α = 1-sin α α haversine α = hav α = (vers α )/2 a also defined are the following… hyperbolic sine of x = sinh x = (ex – e-x)/2 hyperbolic cosine of x = cosh x = (ex + e-x)/2 hyperbolic tangent of x = tanh x = sinh x/cosh x csch x = 1/sinh x sech x = 1/cosh x coth x = 1/tanh x

Identities Pythagorean Identities: sin2 α + cos2 α = 1 1 + tan2 α = sec2 α 1+ cot2 α = csc2 α

Half Angle Identities: sin [α/2] = +/- [(1 - cos α)/2]1/2 (negative if [α/2] is in quadrant III or IV) cos [α/2] = +/- [(1 + cos α)/2]1/2 (negative if [α/2] is in quadrant II or III) tan [α/2] = +/- [(1 - cos α)/(1+cos α)]1/2 (negative if [α/2] is in quadrant II or IV)

Page 02 - 10 SFTE Reference Handbook Third Edition 2013

Double-Angle Identities sin 2α = 2sin α cos α cos 2α = 2cos2 α –1 = 1-2sin2 α = cos2 α - sin2 α tan 2α = 2tan α /[1-tan2 α] n –Angle Identities: sin 3α == 3sin α – 4sin3 α cos 3α = 4cos3 α – 3cos α sin nα = 2sin (n-1)α cos α – sin (n-2)α cos nα = 2cos (n-1)α cos α – cos (n-2)α

Two-Angle Identities: sin (α+β) = sin α cos β + cos α sin β cos (α+β) = cos α cos β – sin α sin β tan (α+β) = [tan α + tan β]/[1- tan α tan β] sin (α-β) = sin α cos β - cos α sin β cos (α-β) = cos α cos β + sin α sin β tan (α-β) = [tan α - tan β]/[1+ tan α tan β]

Sum and Difference Identities: sin α + sin β = 2sin [(α+β)/2] cos [(α-β)/2] sin α – sin β = 2cos{(α+ β)/2] sin {(α- β)/2] cos α+ cos β = 2cos [(α+β)/2] sin [(α-β)/2] cos α -cos β = -2cos [(α+β)/2] sin [(α-β)/2] tan α + tan β = [sin (α + β)]/[cos α cos β] cot α + cot β = [sin (α + β)]/[sin α sin β] tan α - tan β = [sin (α - β)]/[cos α cos β] cot α - cot β = -[sin (α - β)]/[sin α sin β] sin2 α – sin2 β = sin (α+β) sin (α-β) cos2 α – cos2 β = -sin (α+β) sin (α-β) cos2 α – sin2 β = cos (α+β) cos (α-β)

Page 02 - 11 SFTE Reference Handbook Third Edition 2013

Power Identities: sin α sin β = [cos (α-β) - cos (α+β)]/2 cos α cos β = [cos (α-β) + cos (α+β)]/2 sin α cos β = [sin (α+β) + sin (α-β)]/2 cos α sin β = [sin (α+β) - sin (α-β)]/2 tan α cot α = sin α csc α = cos α sec α = 1 sin2 α = [1-cos2α]/2 cos2 α= [1+cos2α]/2 sin3 α = [3 sin α– sin 3α]/4 cos3 α = [3 cos α + cos 3α]/4 sin4 α = [3 - 4cos 2α + cos 4α]/8 cos4 α = [3 + 4cos 2α + cos 4α]/8 sin5 α = [10sin α –5sin 3α +sin5α]/16 cos5 α = [10cos α + 5cos 3α + cos5α]/16

OBLIQUE TRIANGLES (no right angle, angles A,B,C are opposite of legs a,b,c) Law of Sines: a/sin A = b/sinB = c/sinC

Law of Cosines: B 2 2 2 a = b + c – 2bc cos A c a b2 = a2 + c2 – 2ac cos B 2 2 2 c = a + b – 2ab cos C A C C = cos-1[(a2+b2-c2)/2ab] b

Law of Tangents: [a-b]/[a+b] = [tan ({a-b}/2]/ [tan ({a+b}/2]

Projection Formulas: a = b cos C + c cosB b =c cos A + a cos C c = a cos B + b cos A

Mollweide’s Check Formulas: [a-b]/c = [sin ({A-B}/2)]/ cos (C/2) [a+b]/c = [cos ({A-B}/2)]/ sin (C/2)

Page 02 - 12 SFTE Reference Handbook Third Edition 2013

Section 2.4 Matrix Algebra (reference 2.5)

Matrix multiplication can be defined for any two matrices only whe the number of columns of the first is equal to the number of rows of the second matrix. Multiplication is not defined for other matrices. [A][B] = [C] [aim][bmj] = [cij]

The product of a pair of, 2 × 2 matrices is:

⎡aa11 12 ⎤⎡bb11 12 ⎤ ⎡ab11 11+ ab 12 21 ab 11 12+ ab 12 22 ⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎣aa21 22 ⎦⎣bb21 22 ⎦ ⎣ab21 11++ ab 22 21 ab 21 12 ab 22 22 ⎦

The identity (or unit) matrix [I] occupies the same position in matrix algebra that the value of unity does in ordi- nary algebra. That is, for any matrix

10L 0 01 0 I = L MMOM 00 1 L

[A]: [I][A] = [A][I] = [A]

The identity [I] is a square matrix consisting of ones on the principle diagonal and zeros everywhere else; i.e.:

a ⋅ α−1 = α1−1 = α0 = 1

In the same way, the matrix [A]-1 is called the inverse matrix of [A] since:

[A][A]-1 = [A]-1[A] = [A]0 = [I]

Page 02 - 13 SFTE Reference Handbook Third Edition 2013

Cofactors and Determinates

aa11 12L a 1n

aa a Aa==21 22L 2 n ij MMOM

aann12L a nn

The signed minor, with the sign determined by the sum of the row and column, is called the cofactor of aij and is denoted by: i+j Aij = (−1) Μij

The value of the determinant is equal to the sum of the products of the elements of any single row or column and their respective cofactors.

Arbitrarily expanding about the first row of a 3 x 3 matrix gives the determinant:

aa22 23 aa21 22 aa21 22 |A| = a11A11 + a12A12 + a13A13 = a11(++−+111) a12 ( ) a13 ( ) aa32 33 aa31 32 aa31 32

which expands to give the final solution:

|A| = a11(a22 a33 − a23 a32) − a12(a21 a33 − a23 a31) + a13(a21 a32 − a22 a31)

Page 02 - 14 SFTE Reference Handbook Third Edition 2013

There is a straightforward four-step method for computing the inverse of a given matrix [A]:

Step 1 Compute the determinant of [A]. This determinant is written |A|. If the determinant is zero or does not exist, the matrix [A] is defined as singular and an inverse cannot be found.

Step 2 Transpose matrix [A]. The resultant matrix is written [A]T.

Step 3 Replace each element aij of the transposed matrix by its cofactor Aij. This resulting matrix is de- fined as the adjoint of matrix [A] and is written Adj[A]. Step 4 Divide the adjoint matrix by the scalar value of the determinant of [A] which was computed in Step 1. The resulting matrix is the inverse and is written [A]-1.

Example: Given the following set of simultaneous equations, solve for x1, x2, and x3. 3x1 + 2x2 − 2x3 = y1 − x1 + x2 + 4x3 = y2 2x1 − 3x2 + 4x3 = y3

This set of equations can be written as: [A] [x] = [y]

⎡ 3 2 − 2 ⎤ ⎡ x1 ⎤ ⎡ y1 ⎤ ⎢− 1 1 4 ⎥ ⎢ x ⎥ = ⎢ y ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎣⎢ 2 − 3 4 ⎦⎥ ⎣⎢ x 3 ⎦⎥ ⎣⎢ y 3 ⎦⎥ and solved as follows: [x] = [A]−1 [y]

Thus, the system can be solved for the values of x1, x2, and x3 by computing the inverse of [A].

Step 1. Compute the determinant of [A]. Expanding about the first row |A| = 3(4 + 12) − 2 (−4 −8) −2 (3 − 2) |A| = 48 + 24 −2 = 70

Page 02 - 15 SFTE Reference Handbook Third Edition 2013

Step 2. Transpose [A]. ⎡ 3 − 1 2 ⎤ T ⎢ ⎥ []A = ⎢ 2 1 − 3⎥

⎣⎢− 2 4 4 ⎦⎥

Step 3. Determine the adjoint matrix by replacing each element in [A]T by its Cofactor.

⎡ 1 −3 2 −3 2 1 ⎤ ⎢ − ⎥ 4 4 −2 4 −2 4 ⎢ ⎥ ⎡16 −2 10 ⎤ ⎢ −1 2 3 2 3 −1 ⎥ ⎢ ⎥ adj []A = ⎢− − ⎥ = 12 16 −10 4 4 −2 4 −2 4 ⎢ ⎥ ⎢ ⎥ ⎢ 1 13 5 ⎥ ⎢ −1 2 3 2 3 −1 ⎥ ⎣ ⎦ ⎢ − ⎥ ⎣ 1 −3 2 −3 2 1 ⎦

Step 4. Divide by the scalar value of the determinant of [A] which was computed as 70 in Step 1.

⎡16 − 2 10 ⎤ 1 []A −1 = ⎢12 16 − 10 ⎥ 70 ⎢ ⎥ ⎣⎢ 1 13 5 ⎦⎥

if y1 = 1, y2 = 13, and y3 = 8

⎡x1⎤ ⎡16 −2 10⎤⎡1⎤ 1 ⎢x ⎥= ⎢12 16 −10⎥⎢13⎥ ⎢ 2⎥ 70⎢ ⎥⎢ ⎥ ⎣⎢x3⎦⎥ ⎣⎢1 13 5 ⎦⎥⎣⎢8⎦⎥

1 70 x = ()16 − 26 + 80 = = 1 1 70 70 1 140 x = ()12 + 208 − 80 = = 2 2 70 70 1 210 x = ()1 + 169 + 40 = = 3 3 70 70

Page 02 - 16 SFTE Reference Handbook Third Edition 2013

Cramer’s Rule

Given matrices A{x} = {b}

If the det (D) of a matrix (A) exists, and Dr is the det of the matrix obtained from A by replacing the rth column of A by the column {b}, then the solution to (1) is xr = Dr/D r = 1,2,…,n

Example of Cramer’s Rule

⎡ 1 0 2⎤ ⎡ x ⎤ ⎡ 6 ⎤ 1 ⎢− 3 4 6⎥ ⎢ x ⎥ = ⎢30 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥

⎣⎢ − 1 − 2 3⎦⎥ ⎣⎢ x3 ⎦⎥ ⎣⎢ 8 ⎦⎥

⎡ 1 0 2 ⎤ ⎡ 6 0 2⎤ ⎢ ⎥ ⎢ ⎥ A = − 3 4 6 A1 = 30 4 6 ⎢ ⎥ ⎢ ⎥ ⎣⎢ − 1 − 2 3 ⎦⎥ ⎣⎢ 8 − 2 3⎦⎥ ⎡ 1 6 2⎤ ⎡ 1 0 6 ⎤ A = ⎢− 3 30 6 ⎥ A = ⎢− 3 4 30 ⎥ 2 ⎢ ⎥ 3 ⎢ ⎥ ⎣⎢ − 1 8 3⎦⎥ ⎣⎢ − 1 − 2 8 ⎦⎥

det()A − 40 −10 det(A ) 72 18 det(A ) 152 38 x = 1 = = , x = 2 = = , x = 3 = = 1 det()A 44 11 2 det()A 44 11 3 det()A 44 11

Page 02 - 17 SFTE Reference Handbook Third Edition 2013

Section 2.5 VectorAlgebra (reference 2.5)

Addition

A + B B A A AB+

B ABC++

BC+ AB+ C

A . B

Subtraction

−B A AB−

B

Page 02 - 18 SFTE Reference Handbook Third Edition 2013

mA = Am Commutative

m()nA = ()mn A Associative

()m + n A = mA + nA Distributive m()A + B = mA + mB Distributive

P(x,y,z)

A

ai x $ akz $ y

ajy $

x

Dot Product A = a iˆ + a ˆj + a kˆ x y z 2 2 2 A = a x + a y + a z

A ⋅ B = A cosθ

Page 02 - 19 SFTE Reference Handbook Third Edition 2013

A

θ

B Acosθ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i⋅i =j⋅j=k ⋅k =1 i ⋅j=j⋅k=k ⋅i =0

Vector Product

ijk$$$ aayz aaxzaaxy AB×=aaaxyz =i$$ +−( 1) j + k$ bbyz bbxzbbxy bbbxyz

A x B = A B sin θ UAB=×

u$ A θ B

i$ × i$ = $j × $j = k$ × k$ = 0

i$ × $j = k$ $j × k$ = i$ k$ × i$ = $j

$j × i$ = − k$ k$ × $j = − i$ i$ × k$ = − $j

Page 02 - 20 SFTE Reference Handbook Third Edition 2013

Vector Differentiation

d ()A + B dA dB = + Distributi ve derivative dt dt dt d ()A ⋅ B dB dA = A ⋅ + ⋅ B Dot product derivative dt dt dt d ()A × B dB dA = A × + × B Cross product derivative dt dt dt d dB df ()t []f ()t B = f ()t + B Scalar vec tor product derivative dt dt dt

The first derivative of a position vector is a vector tangential to the trajectory with a magnitude equal to the speed of the particle.

Motion of a point using two reference systems.

Reference A can be considered the inertial frame while Rotation of the B reference relative to the A reference must be con- sidered when observing motion wrt the A reference system.

z Reference A

Trajectory C y r x R ω P z ρ Reference B V y x

Note: Unit vectors are along the B system axes. Subscripts denote reference system. Reference B can be equivalent to a maneuvering aircraft. ρ = xiˆ + yˆj + zkˆ ⎛ dρ ⎞ ˆ ⎜ ⎟ = x&iˆ + y&ˆj + z&k ⎝ dt ⎠ B

⎛ dρ ⎞ ˆ ⎛ ˆ& ⎞ ⎜ ⎟ = ()x&iˆ + y&ˆj + z&k + ⎜ xiˆ& + yˆ&j + zk ⎟ ⎝ dt ⎠ A ⎝ ⎠

Page 02 - 21 SFTE Reference Handbook Third Edition 2013

xiˆ& + yˆ&j + zkˆ& = x()ω × iˆ + y(ω × ˆj)+ z(ω × kˆ)= (ω × xiˆ)+ (ω × yˆj)+ ()ω × zkˆ = ω × ()xiˆ + yˆj + zkˆ = ω × ρ ⎛ dρ ⎞ ⎛ dρ ⎞ ⎜ ⎟ = ⎜ ⎟ + ω × ρ ⎝ dt ⎠ A ⎝ dt ⎠ B The velocities of the particle P relative to the A and to the B references are, respectively:

⎛ dr ⎞ ⎛ dρ ⎞ VA = ⎜ ⎟ VB = ⎜ ⎟ ⎝ dt ⎠A ⎝ dt ⎠B

These velocities can be related by noting that: r = R + ρ

Differentiating with respect to time for the A reference,

⎛ dr ⎞ ⎛ dR ⎞ ⎛ dρ ⎞ ⎜ ⎟ = V A = ⎜ ⎟ + ⎜ ⎟ ⎝ dt ⎠ A ⎝ dt ⎠ A ⎝ dt ⎠ A

dR The term ()dt A is the velocity of the origin of the B reference relative to the A reference, R& . The term dρ can be replaced with the above equation, () ()dt A dρ and denoting ()dt B simply as VB the above expression then becomes :

V = V + R& + ω × ρ A B

The term is the “transport velocity” and is the only velocity R& + ω × ρ if point P is rigidly attached to reference B.

Page 02 - 22 SFTE Reference Handbook Third Edition 2013

To get acceleration wrt A, differentiate:

•• ⎛ dVA ⎞ ⎛ dVB ⎞ ⎡ d ⎤ a A = ⎜ ⎟ = ⎜ ⎟ + R + ⎢ ()ω × ρ ⎥ ⎝ dt ⎠ A ⎝ dt ⎠ A ⎣ dt ⎦ A •• ⎛ dVB ⎞ ⎛ dρ ⎞ ⎛ dω ⎞ use product rule to get... a A = ⎜ ⎟ + R + ω × ⎜ ⎟ + ⎜ ⎟ × ρ ⎝ dt ⎠ A ⎝ dt ⎠ A ⎝ dt ⎠ A ⎛ dρ ⎞ ⎛ dρ ⎞ where ⎜ ⎟ = ⎜ ⎟ + ω × ρ B ⎝ dt ⎠ A ⎝ dt ⎠ B

⎛ dVB ⎞ ⎛ dVB ⎞ and similarly ⎜ ⎟ = ⎜ ⎟ + ω ×V ⎝ dt ⎠ A ⎝ dt ⎠ B

Combining gives the acceleration of point P relative to reference A

•• ⎛ dVB ⎞ ⎛ dρ ⎞ ⎛ dω ⎞ aA = ⎜ ⎟ +ω ×VB + R +ω ×⎜ ⎟ +ω ×()ω × ρ + ⎜ ⎟ × ρ ⎝ dt ⎠B ⎝ dt ⎠B ⎝ dt ⎠ A

dVB dρ dω Noting that ()dt B is aB ; ()dt B is VB ; and ()dt A is ω& , rearranging terms gives : ••

aA = aB + 2()ω ×VB + R + (ω& × ρ ) +ω ×()ω × ρ where ω × ()ω × ρ is the centripetal acceleration,

2()ω ×VB is the Coriolis acceleration, and

•• R+(ω& ×ρ)+ω ×()ω ×ρ is the transport acceleration and is the only acceleration if point P is rigidly attached to reference B.

Page 02 - 23 SFTE Reference Handbook Third Edition 2013

Motion of a point using one reference system. Reference A can be considered the inertial frame while The body can be equivalent to a maneuvering aircraft.

V

ω z ρ a P

y

x

ρ& = ω × ρ •• ρ = ω × ()ω × ρ + ω& × ρ

ρ& = Vb −Va

Vb = Va + ω × ρ

ab = aa + ω × ()ω × ρ + ω& × ρ

Page 02 - 24 SFTE Reference Handbook Third Edition 2013

Section 2.6 Statistics (reference 2.6) Definitions: Population: The set of all possible observations Sample: Any subset of a population Homogeneous Sample: The sample comes from 1 population only Random Sample: Equal probability of selecting any member of the population Independence (of events A and B): P(A and B) = P(A)*P(B)

1 n Sample and Population Mean (Average value): μ = x = ∑xi N i=1

Mode (Most commonly occurring value in a sample) Median (middle value, 50th percentile. Half of the sample values are greater and half are smaller)

Deviation (from the mean value): d i = xi − x

1 N Population Variance (from the mean value): 2 2 σ = ∑ di N i=1

N Population Standard Deviation (from the mean value): 1 2 σ = ∑ di N i=1

N 1 2 Sample Standard Deviation (from the mean value): s = ∑ di N −1 i=1

Discrete Probability Distributions: Binomial: N independent events, each having probability p of success, and1-p of failure. For example, tossing a fair coin N times where p = the probability of getting a head on any toss. If the random variable x indicates the number of heads in N=2 tosses, then P(x=0) = 1/4, P(x=1) = 1/2, P(x=2) = 1/4. If N=4, then the probabilities are illus trated in the following graph:

Page 02 - 25 SFTE Reference Handbook Third Edition 2013

0.6 0.4 0.2 P(x=X)

0 01234

As N approaches infinity ...

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 1 2 3 4 5 -5 -4 -3 -2 -1 -0 0.5 1.5 2.5 3.5 4.5 -4.5 -3.5 -2.5 -1.5 -0.5

So, the binomial distribution is the discrete case of the Normal distribution.

Continuous Distributions: As the number of samples increases and the width of the Discrete sample intervals shrink to zero, discrete distributions become continuous.

P(x=X) = 0 Must talk about intervals, e.g. P(a < x < b)

2 −( x−μ ) 1 2 The Normal Distribution: f (x) = e 2σ 2πσ

Page 02 - 26 SFTE Reference Handbook Third Edition 2013

Normal Distribution:

Normal Probability Mass Function

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 1 4 -5 -2 0.1 0.4 0.7 1.3 1.6 1.9 2.2 2.5 2.8 3.1 3.4 3.7 4.3 4.6 4.9 -4.7 -4.4 -4.1 -3.8 -3.5 -3.2 -2.9 -2.6 -2.3 -1.7 -1.4 -1.1 -0.8 -0.5 -0.2

Normal Cumulative Distribution Function

1.2

1

0.8

0.6

0.4

0.2

0

-5 1 2 3 -2 7 8 .7 1 2 .1 4 -4.7 -4.4 -4. -3.8 -3.5 -3. -2.9 -2.6 -2. -1. -1.4 -1.1 -0. -0.5 -0.2 0.1 0.4 0 1.3 1.6 1.9 2. 2.5 2.8 3 3.4 3.7 4.3 4.6 4.9

Page 02 - 27 SFTE Reference Handbook Third Edition 2013

The Standard Normal Distribution:

μ = 0,σ = 1

x − μ 1 z = ,dz = dx σ σ

− z 2 b 1 P(a < z < b) = e 2 dz ∫a 2π

Error Probable: An error budget that would contain half of the population data points. Assumes that events are independ- ent and identically distributed (iid). Also assumes N is large (greater than 30), or population is normally distributed.

0.4

0.35

0.3

0.25 ±0. 674σ 0.2

0.15

AREA = 0.5 0.1

0.05

0 1 2 3 4 5 -5 -4 -3 -2 -1 -0 0.5 1.5 2.5 3.5 4.5 -4.5 -3.5 -2.5 -1.5 -0.5

Circular Error Probable – the 2 Dimensional Case (X error and Y error):

σ x If σ x < σ y and ≤ 0.28 then CEP = 0.562σ x + 0.615σ y σ y

σ y If σ x > σ y and ≤ 0.28 then CEP = 0.615σ x + 0.562σ y σ x

Otherwise CEP = 0.5887 (σ x + σ y )

Page 02 - 28 SFTE Reference Handbook Third Edition 2013

Confidence Intervals: In practice, we take a sample from population. The sample mean and variance will differ from the population mean and variance. Confidence Intervals express how certain we are that the population statistics lie in a region around the sample statistics.

2 Central Limit Theorem: Given a population Normally distributed, ( μ , σ ) then the distribution of successive sample means from samples of n observations

2 Approaches a Normal distribution with parameters (μ , σ ) n

We want 1- α level of confidence that a region around our sample mean value contains the actual population mean. AREA = 1−α Pz()−<<=− xz 1 α 1−−αα1 0.4 2 2

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 1 2 3 4 5 -5 -4 -3 -2 -1 -0 0.5 1.5 2.5 3.5 4.5 -4.5 -3.5 -2.5 -1.5 -0.5 α α AREA = 2 AREA = 2 x − μ z = σ N σ σ P(x − z < μ < x + z ) = 1−α 1−α 1−α 2 N 2 N

Page 02 - 29 SFTE Reference Handbook Third Edition 2013

If n < 30, we must use Student’s T Distribution instead of the Standard Normal

s s P(x − t < μ < x + t ) = 1− α n,1−α n,1−α 2 n 2 n

Determining Sample Size: For the population mean to fall into an interval defined by

σ σ (x − z ) < μ < (x + z ) 1−α 1−α 2 N 2 N σ μ − x ≤ z 1−α 2 N

Where μ − x is the accuracy desired (or the error that can be tolerated).

Since the sample size decision must be made prior to the test, an estimate must be made for the population standard devia- tion. Using the estimate we can solve for N

z σ 2 1−α 2 N ≥ error

Hypothesis Testing: Begins with an assumption (hypothesis), usually about the underlying population distribution of some measured quantity or computed error. Select values for the hypothesis and alternate hypothesis(es) that partition the sample space. Collect N samples of the population test statistic or parameter. There are two types of errors: Type 1 errors reject the hypothesis when it is true; Type II accept the hypothesis when in is false.

One-Sided Test 0.4

0.35

0.3

0.25

0.2

0.15

0.1 α

0.05

0 1 2 3 4 5 -5 -4 -3 -2 -1 -0 0.5 1.5 2.5 3.5 4.5 -4.5 -3.5 -2.5 -1.5 -0.5 Reject System Accept System z 0

Page 02 - 30 SFTE Reference Handbook Third Edition 2013 Two-Sided Test 0.4

0.35

0.3

0.25

0.2 0.15 α 0.1 2 0.05

0 1 2 3 4 5 -5 -4 -3 -2 -1 -0 0.5 1.5 2.5 3.5 4.5 -4.5 -3.5 -2.5 -1.5 -0.5 Reject SystemAccept System Reject System

-z0 z0

0.08 Type I and Type II Errors

0.07

0.06

0.05

0.04 β 0.03 α 0.02

0.01

0 50 55 60 65 70 75 80 85 90 95 100

Page 02 - 31 SFTE Reference Handbook Third Edition 2013

x2 − nx 2 Large Samples, Unknown Variance use s = ∑ for σ n −1

x − μ′ z′ = σ

n

(μ − μ′) z′ = z + σ

n

x − μ t = Small Samples, Unknown Variance use: s n −1

x − μ′ t′ = s

n −1

μ − μ′ t′ = t + s

n −1

Adjusting α and β

Adjust the size of the Error we wish to Detect Change the sample size n

Page 02 - 32 SFTE Reference Handbook Third Edition 2013

H0 :Tj = 0,∀j Normal Equations: n k n k n k n k ∑∑Xij =∑∑m+ ∑∑t j = nkm+ n ∑t j , but ∑t j = 0 i==11j i==11j i == 11j j = 1 j=1 n k so ∑∑Xij =nkm i==11j n n n ∑Xij = ∑m+∑t j = nm+ ntj i=1 i=1 i=1

m is the least squares estimate of tj is the least squares estimate of Tj

n k n n SSr (m,t j ) = m∑∑X ij + ∑t j ∑X ij i==11j j=1 i = 1

Assuming H 0 is True, the model is :

X ij = μ + ε ij n k ′ ′ SSr (m ) = m ∑∑X ij i==11j

Between Treatments : SSr (m,t j ) − SSr (m′) n k 2 SSe = ∑∑X ij − SSr (m,t j ) i==11j

SSt (k −1) Test Statistic is : Fk −1,(n−1)k = SSe ((n −1)k)

Page 02 - 33 SFTE Reference Handbook Third Edition 2013

2.7 Standard Series (reference 2.4) Taylor’s

2 3 ( n −1) x − a ()x − a (x − a ) (x − a ) f ( x ) = f (a ) + f ' (a ) + f ' ' (a ) + f ' ' ' (a ) + ... f ( n −1) (a ) + R 1 2! 3! ()n − 1 ! n

Maclaurin’s (Taylor series with a = 0 ):

x (x)2 (x)3 (x)(n−1) f (x) = f (0) + f '(0) + f ''(0) + f '''(0) + ... f (n−1) (0) + R 1 2! 3! ()n −1 ! n

Binomial: n(n −1) n(n −1)(n − 2) ()a + x n = a n + na n−1x + a n−2 x 2 + a n−3 x3 + .... []x 2 < a 2 2! 3!

Exponential:

2 3 x (x ln a) (x ln a) a = 1 + x ln a + + + .... 2! 3!

x 2 x 3 x 4 e x = 1 + x + + + + .... 2! 3! 4!

1 x 2 x 4 (e x + e − x ) = 1 + + + .... 2 2! 4!

1 x 3 x 5 (e x − e − x ) = x + + + .... 2 3! 5! 4 6 8 2 x x x e − x = 1 − x 2 + − + .... 2! 3! 4!

Logarithmic:

1 1 ln x = (x −1) − (x −1)2 + (x −1)3 −.... []0 < x < 2 2 3

(x −1) 1 x −1 1 x −1 ⎡ 1⎤ ln x = − ( )2 + ( )3 −.... x > x 2 x 3 x ⎣⎢ 2⎦⎥

3 5 ⎡ x −1 1 ⎛ x −1⎞ 1 ⎛ x −1⎞ ⎤ ln x = 2⎢ − ⎜ ⎟ + ⎜ ⎟ +....⎥ []0 < x ⎣⎢ x +1 3 ⎝ x +1⎠ 5 ⎝ x +1⎠ ⎦⎥

Page 02 - 34 SFTE Reference Handbook Third Edition 2013

Trigonometric:

x3 x5 x7 sin x = x − + − +..... 3! 5! 7! x2 x4 x6 cos x =1− + − +..... 2! 4! 6! 3 5 7 9 2 x 2x 17x 62x ⎡ 2 π ⎤ tan x = x + + + + +..... ⎢x < ⎥ 3 15 315 2835 ⎣ 4 ⎦ 1 x3 1⋅3 x5 1⋅3⋅5 x7 sin−1 x = x + + + +...... []x2 <1 2 3 2⋅4 5 2⋅4⋅6 7! x3 x5 x7 tan−1 x = x − + − +...... []x2 ≤1 3 5 7 x2 x4 x6 lnsin x = ln x − − − −...... []x2 < π 2 6 180 2835 2 4 6 8 2 x x x 17x ⎡ 2 π ⎤ ln cos x = − − − − ..... ⎢x < ⎥ 2 12 45 2520 ⎣ 4 ⎦ 2 4 6 2 x 7x 62x ⎡ 2 π ⎤ ln tan x = ln x + + + −..... ⎢x < ⎥ 3 90 2835 ⎣ 4 ⎦ x2 3x4 8x5 3x6 esin x =1+ x + − − + +..... 2! 4! 5! 6! x2 4x4 31x6 ecos x = e(1− + − +.....) 2! 4! 6! 2 3 4 5 2 tan x x 3x 9x 37x ⎡ 2 π ⎤ e =1+ x + + + + +..... ⎢x < ⎥ 2! 3! 4! 5! ⎣ 4 ⎦

Page 02 - 35 SFTE Reference Handbook Third Edition 2013

Section 2.8 Derivative Table (references 2.2, 2.3)

[x is the independent variable; u and v are dependent on x; w is dependent on u; a and n are constants; log is common logarithm; ln is logarithm to the base e]

da = 0 dx d(ax) = a dx dx n = nx n−1 dx d(u + v) du dv = + dx dx dx d(uv) dv du = u + v dx dx dx d(u / v) 1 ⎛ du dv ⎞ = ⎜v − u ⎟ dx v 2 ⎝ dx dx ⎠ dw dw du = dx du dx du n du = nu n−1 dx dx d ln x 1 = dx x d ln u 1 du = dx u dx d log u log e du = dx u dx

Page 02 - 36 SFTE Reference Handbook Third Edition 2013

de x = e x dx da x = a x ln a dx da u du = a u ln a dx dx du v du dv = vu v −1 + u v ln u dx dx dx d sin x d sin u du = cos x or = cos u dx dx dx d cos x d cos u du = − sin x or = − sin u dx dx dx d tan x d tan u du = sec 2 x or = sec 2 u dx dx dx d sec x d sec u du = sec x tan x or = sec u tan u dx dx dx d cot x d tan u du = − csc 2 x or = − csc 2 u dx dx dx d sin −1 x 1 d sin −1 u 1 du = or = dx 1 − x 2 dx 1 − u 2 dx d cos −1 x − 1 d cos −1 u − 1 du = or = dx 1 − x 2 dx 1 − u 2 dx d tan −1 x 1 d tan −1 u 1 du = or = dx 1 + x 2 dx 1 + u 2 dx d cot −1 x − 1 d cot −1 u − 1 du = or = dx 1 + x 2 dx 1 + u 2 dx

Page 02 - 37 SFTE Reference Handbook Third Edition 2013

Section 2.9 Integral Table (references 2.2, 2.3) x is any variable, u is any function of x, a & b are arbitrary constants.

The constant of integration, c, has been omitted from this table but should be added to the result of every integration.

Fundamental Integrals ∫ adx = ax ∫∫af (x)dx = a f (x)dx ∫∫(u + v)dx = ∫ udx + vdx ∫ udv = uv − ∫ vdu udv du dx = uv − v dx ∫∫dx dx xn+1 xndx = , n ≠ −1 ∫ n +1 ∫ x−1dx = ln x dx w(u)dx = w(u) u ∫∫du dx 1 x = tan −1 ∫ a2 + x2 a a dx x = sin −1 ∫ 2 2 a − x a dx = ln(x − x2 ± a 2 ) ∫ 2 2 x ± a 1 ⎛ u ⎞ a2 − u 2 du = ⎜u a2 − u 2 + a2 sin −1 ⎟ ∫ 2 ⎝ a ⎠ du 1 u = tan −1 a > 0 ∫ u 2 + a 2 a a

Page 02 - 38 SFTE Reference Handbook Third Edition 2013

Expressions containing exponential and logarithmic functions

dx = ln x ∫ x ∫ e xdx = e x eax eax dx = ∫ a ax ax b b dx = ∫ a ln b ∫ ln xdx = x ln x − x bu bu du = ∫ ln u eax xeax dx = (ax −1) ∫ a 2 xbax bax xbax dx = − ∫ a ln b a 2 (ln b)2 eax x2eax dx = ()a 2 x 2 − 2ax + 2 ∫ a3 ∫ ln axdx = x ln ax − x x 2 x 2 x ln axdx = ln ax − ∫ 2 4 x3 x3 x2 ln axdx = ln ax − ∫ 3 9 ∫ ()ln ax 2 dx = x ()ln ax 2 − 2x ln ax + 2x dx = ln()ln ax ∫ x ln ax x n 1 e y dy dx = y = (n +1) ln ax ∫∫ln ax a n+1 y

Page 02 - 39 SFTE Reference Handbook Third Edition 2013

Expressions containing trigonometric functions

∫ sin xdx = − cos x ∫ cos xdx = sin x ∫ tan xdx = − ln(cos x) ∫ cot xdx = ln(sin x) ∫ sec xdx = ln(sec x + tan x) ∫ cscudu = ln(cscu − cot u) 1 1 sin 2 udu = u − sin u cosu ∫ 2 2 1 1 cos2 udu = u + sin u cosu ∫ 2 2 ∫ csc2 udu = − cot u ∫ tan 2udu = tan u − u ∫ cot 2 udu = − cot u − u 1 sin axdx = − cos ax ∫ a x sin 2ax sin 2 axdx = − ∫ 2 4a dx 1 ax = ln tan ∫ sin ax a 2 dx 1 = − cot ax ∫ sin 2 ax a dx 1 ⎛ π ax ⎞ = m tan⎜ m ⎟ ∫ 1± sin ax a ⎝ 4 2 ⎠ 1 sin x cos xdx = sin 2 x ∫ 2

Page 02 - 40 SFTE Reference Handbook Third Edition 2013

Section 2.10 Laplace Table (references 2.2, 2.3)

time domain f(t) frequency domain F(s)=L{f(t)} 1 (step function) 1/s (s>0) t 1/s2 (s>0)

n−1 ( n − 1 )! t s n (s>0)

1 −3 t π s 2 (s>0) 2

1 − 1 t π s 2 (s>0)

(1)(3)(5)...(2n −1) π −n− 1 tn-1/2 (n=1,2,…) s 2 2 n (s>0) 1 eat s − a (s>a) 1 at te () s − a 2 (s>a)

n −1 ! n-1 ax ( ) t e (n=1, 2, …) n ()s − a (s>a) sin at a 2 2 s + a (s>0) s cos at (s>0) s 2 + a 2 a ebt sin at (s?b) ()s − b 2 + a 2

s − b ebt cos at (s>b) ()s − b 2 + a 2 2as x sin ax 2 (s>a) ()s 2 − a 2

2 2 x cos ax s − a (s > 0) 2 ()s 2 + a 2

a sinh at ( s > a ) s 2 − a 2

Page 02 - 41 SFTE Reference Handbook Third Edition 2013

s cosh at (s > a ) s2 − a2 ssin b + a cosb sin (at + b) s 2 + a2 s cosb − asin b cos(at + b) s2 + a2 eat − ebt 1 a − b ()()s − a s − b

at bt ae − be s a − b ()()s − a s − b δ (impulse function) 1 1 square wave, period =2c −cs s()1+ e

1− e−cs triangular wave, period = 2c 2 −cs s ()1+ e at for 0 ≤ t < c a(1+ cs − e−cs ) sawtooth wave, period = c 2 cs s ()1− e 2abs sin at sin bt []s 2 + ()a + b 2 []s 2 + ()a − b 2

1− cosat 1 a2 s()s2 + a2 at − sin at 1 a3 s2 ()s2 + a2 sin at − at cos at 1 2 2a3 ()s 2 + a 2

Page 02 - 42 SFTE Reference Handbook Third Edition 2013

Section 2.11 References

2.1 Burington, Richard S., “Handbook of Mathematical Tables and Formulas”, McGraw-Hill Inc., 1973.

2.2 Ayers, F., Moyer, R., “Trigonometry”, Schaum’s Outline series, McGraw-Hill Inc., 1990.

2.3 Esbach, Ovid W., “Handbook of Engineering Fundamentals”, John Wiley and Sons Inc., 1963.

2.4 Hudson, Ralph G., “The Engineers’ Manual”, John Wiley and Sons Inc., 1944.

2.5 Jones, G., Chapter 14, Vectors and Matrices , from “Chapter 14, Vectors and Matrices , from “ Flying Qualities Testing, Vol II” , National; Test Pilot School, P.O. Box 658, Mojave CA, 93501.

2.6 Flying Qualities Testing, Vol II”, 1997, National Test Pilot School, P.O. Box 658, Mojave CA, 93501.

2.7 Lewis, G., Chapter 2, Data Analysis , from “Crew station Evaluation and Data Analysis, Vol IV”, 1997, National Test Pilot School, P.O. Box 658, Mojave CA, 93501.

Page 02 - 43 SFTE Reference Handbook Third Edition 2013

NOTES

Page 02 - 44 SFTE Reference Handbook Third Edition 2013

Section 3 Universe/Earth/Atmospheric Properties

3.1 Universal Constants Newtonian Gravity

3.2 Earth Properties Centrifugal relief from gravity Altitude effect on gravitational acceleration Actual gravitational pull on an aircraft Gravity influence on cruise performance

3.3 General Properties of Air Composition of air Viscosities of Air Psychrometric Chart

3.4 Standard Atmosphere Divisions of the Atmosphere Altitude Definitions Atmosphere Assumptions Standard Day Sea Level Conditions 1976 Standard Atmosphere Equations Standard Atmosphere Graph & Tables

3.5 Sea States

3.6 Sunrise and Sunset Times

3.7 Crosswind Components

3.8 Geodetic Measurements

3.9 References

Page 03 - 1 SFTE Reference Handbook Third Edition 2013

Section 3 Recurring Nomenclature

Hp = pressure altitude. The pressure associate with geopotential altitude on a standard day. T = absolute temperature (Rankin or Kelvin)

TR = absolute temperature, Rankin scale

To = standard day seal level absolute temperature P = ambient pressure

Po= standard day seal level ambient pressure ρ = ambient density

ρ ο= standard day seal level ambient density

δ = P/Po = atmospheric pressure/std day sea level pressure

θ = T/To = atmospheric absolute temp / std day sea level absolute temp

σ = ρ /ρ ο = atmospheric density/std day sea level density g = acceleration due to gravity go = standard earth acceleration due to gravity ao = speed of sound at std day sea level temperature

Page 03 - 2 SFTE Reference Handbook Third Edition 2013

Section 3.1 Universal Constants (reference 3.1)

23 Avogadros number, No 6.022169 x 10 molecules/mole Boltzmann constant, k 1.380 x 10-23 J/oK electron charge, e 1,602 x 10-19 coulomb -31 electron mass, me 9.109 x 10 kg gas constant, R 8.31434 J/oK mole gravitational constant, G 6.673 x 10-11 Nm2/kg2 -27 neutron mass, mn 1.674 x 10 kg Planck constant, h 6.625 x 10-34 J sec -27 proton mass, mp 1.672 x 10 kg speed of light in a vacuum, c 2.998 x 108 m/sec -27 unified atomic mass constant, mu 1.660 x 10 kg volume of ideal gas (std temp & press) 2.241 x 10 m3/mol

Newtonian Gravity The gravitaional field (g) near any mass can be calculated as

GM g = ()R 2 A where G is the universal gravitational constant and RA is the absolute distance from the center of mass M

Page 03 - 3 SFTE Reference Handbook Third Edition 2013

Section 3.2 Earth Properties (references 3.9.2, 3.9.3 )

2 2 Std Earth gravitational acceleration, go = 9.8066 m/s = 32.174 ft/s mass = 5.98333 × 1024 kg = 13.22 x 1024 lb rotation rate, ω = 7.292115 × 10−5 rad/sec average density = 5.522 g/cm3 = 344.7 lb/ft3 radius average, Ravg = 6,367,444 m = 3956.538 st. miles = 20,890,522 ft radius at the equator (Re) is 6,378,137 m (±2) radius at the poles Rp = 6,356,752 [m] radius as a function of latitude, φ (assumes perfect ellipsoid):

− 1 2 2 2 ⎡⎛ cosφ ⎞ ⎛ sinφ ⎞ ⎤ R = ⎢⎜ ⎟ +⎜ ⎟ ⎥ ⎢⎜ R ⎟ ⎜ R ⎟ ⎥ ⎣⎝ e ⎠ ⎝ p ⎠ ⎦

Centrifugal Relief from Gravity The earth's "normal" gravity field includes both the Newtonian Law and a correction for the centrifugal force caused by the earth's rotation. The centrifugal relief correction is

V 2 (R ω)2 ΔCR = − = − x = R ω 2 R R x x x where ω is the earth’s rotation rate and Rx is the perpendicular distance from the earth's axis to the surface and can be calculated as Rx =R cos φ (see figure below).

Rx φ = deg latitude R

For any centrifugal relief calculations associated with aircraft performance, it is sufficiently exact (g ±0.00004 m/s2) to use the average earth radius. An aircraft flying eastward contributes to centrifugal relief while a west- bound aircraft diminishes it.

Page 03 - 4 SFTE Reference Handbook Third Edition 2013

The International Association of Geodesy publishes the following equation (accurate to 0.005%) to calculate local sea level gravity including the effects of centrifugal relief for any point fixed to the earth's surface

2 2 m glsl = 9.780327(1+ 0.00530224 sin φ − 0.000058sin 2φ)[ 2 ] s The above equation is tabulated below for quick reference.

Latitude Normal glocal (deg) 2 2 (m/s ) (ft/s ) 0 9.780327 32.088

15 9.783659 32.098 30 9.792866 32.188

45 9.805689 32.171 60 9.818795 32.214

75 9.828569 32.249 90 9.832185 32.258

The standard acceleration (go) corresponds to a latitude of 46.0625°. glsl at the equator and the poles varies ±0.27% from go.

Altitude Effect on Gravitational Acceleration RA is the sum of the earth's local radius and the geometric distance (hG) above the surface: RA = R + hG

Gravitational acceleration at any geometric altitude:

h g alt G 2 (1000 ft) g lsl ⎛ R ⎞ g = g ⎜ ⎟ 0 1 alt sls ⎜ ⎟ ⎝ R + hG ⎠ 10 0.99904 20 0.99809 40 0.99618 60 0.99428 80 0.99238 100 0.99049

Page 03 - 5 SFTE Reference Handbook Third Edition 2013

Actual Gravitational Pull on an Aircraft Adding a centrifugal relief correction due to the aircraft’s velocity, a complete calculation for its gravita- tional acceleration is

2 2 ⎡ R ⎤ ⎛ V sinσ ⎞ g = g +ω 2 R cosφ − ⎜ω + G ⎟ ()R + h cosφ A/C []lsl ⎢ ⎥ ⎜ ⎟ G ⎣ R + hG ⎦ ⎝ R + hG ⎠ where VG = ground speed and σ = ground track angle (0° = true North, 90° = East, etc.).

Gravity Influence on Aircraft Cruise Performance Even at the same altitude, changes in gravity due to latitude or centrifugal relief directly alter the required lift, drag, and fuel flow. For example, with sufficiently precise instrumentation, data collected heading West could show about 0.5% more drag and fuel flow than data collected heading East (centrifugal relief effect). After deter- mining test and standard (or mission) values for g, flight test values for CL, CD, drag, and fuel flow can be cor- rected to standard as follows:

N z ⎡ g ⎤ C = C ωstd std Lstd Lt ⎢ ⎥ N z g A/C ωt ⎣ ⎦

(C )2 C = Lstd Dstd πARe

ΔD = Dstd − Dt = qS [C D − C D ] istd it W& = W& + ΔD ⋅TFSC f std ft

where Nz = normal load factor,

CL = lift coefficient, CD = drag coefficient, AR = aspect ratio, e = Oswald efficiency factor, ΔD change in drag force, TSFC = thrust specific fuel consumption, and

W& f std = standard day fuel flow

Page 03 - 6 SFTE Reference Handbook Third Edition 2013

Section 3.3 General Properties of Air (reference 3.9.1)

Gas constant, R = 53.35 ft lb/R lbm = 287.074 J/kg K = 1716 lb(ft)/slgs(R) = 3089.7 lb(ft)/slgs(K)

½ Speed of sound = ao(θ) 1/2 = 49.02 (TR) ft/sec 1/2 = 33.42 (TR) miles/hr 1/2 = 29.04 (TR) knots 1/2 = 20.05 (TR) m/sec

Density, ρ = .0023769 slug/ft3 = 1.225 kg/m3 (at 15o C)

2 4 Specific weight, gρ = .07647 sec /ft

o Specific heat capacity at 59 F (=To) at constant pressure, cp = .240 BTU/lb R = 1004.76 J/kg K at constant volume, cv = .1715 BTU/lb R = 717.986 J/kg K specific heat ratio, γ = {cp / cv } =1.4

Normal Composition of clean, dry atmospheric air near sea level

Nitrogen, N2 78.084 % by volume

Oxygen, O2 20.948 % Argon, A 0.934 %

Carbon Dioxide, CO2 0.031 % Neon, Ne 0.002 % total 99.9988 % plus traces of helium, kryton, xenon, hydrogen, methane, nitrous oxide, ozone, sulfur dioxide, nitrogen dioxide, ammonia, carbon monoxide, and iodine.

Viscosities of Air 7.3025x10−7 (T )3/ 2 Coefficient of Viscosity, μ = R lb/ft sec c T +198.72 R

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μ 2 Kinematic viscosity, v = c ft /sec gρ

⎡ ⎛ 734.7 ⎞⎤ 3/ 2 ⎜ ⎟ −10 2 Absolute Viscosity, lb μ = ρv = ⎢.317(TR ) ⎜ ⎟⎥x10 sec/ft ⎣ ⎝ TR + 216 ⎠⎦

Atmospheric Viscosity (U.S. Standard Atmosphere)

Pressure Altitude Kinematic Viscosity Absolute Viscosity ft υ ( ft2/sec) μ (lb sec/ft2)

0 1.572 x 10-4 3.737 x 10-7

5,000 1.776 3.638

10,000 2.013 3.538

15,000 2.293 3.435

20,000 2.625 3.330

25,000 3.019 3.224

30,000 3.493 3.115

35,000 4.065 3.004

40,000 5.074 2.981

45,000 6.453 2.982

50,000 8.206 2.983

55,000 10.44 2.985

60,000 13.27 2.986

70,000 21.69 3.005

80,000 35.75 3.043

90,000 58.53 3.080

100,000 95.19 3.118

150,000 1066 3.572

200,000 6880 3.435

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Psychrometric Chart for Seal Level Barometric Pressure

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Section 3.4 Standard Atmosphere

Divisions of the Atmosphere

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Constantly changing atmospheric conditions cannot be duplicated at will to provide the exact environment in which a flight takes place. A standard atmosphere provides a common basis to relate all flight test, wind tunnel results, aircraft design and general performance. Several models of “standard atmosphere” exist with minor dif- ferences based on mathematical constants used in the calculations.

Geometric altitude, hG , is defined as the height of an aircraft above sea level (also called tapeline altitude)

Absolute altitude, ha, is defined as the height of an aircraft above the center of the earth: (geometric altitude + radius of the earth).

Geopotential altitude, h, is required because g changes with height.If potential energy is calculated using sea level weight (WSL = mgo) instead of actual weight (W = mg), then the altitude must be lower. W hG = WSL h

Pressure altitude, Hp is the altitude, on a standard day, at which the test day pressure would be found

Density altitude is the altitude, on a standard day, at which the test day density would be found

Temperature altitude is the altitude, on a standard day, at which the test day temperature would be found

Assumptions on which the standard atmosphere is built

1. The air is dry (only 0.4% per volume of water vapor)

2. The air is a perfect gas and obeys the equation of state, P = ρgRT where R = 53.35 ft lb/oR lbm

3. The gravitational field decreases with altitude

4. Hydrostatic equilibrium exists (Δp= −ρgoΔh)

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Standard Day Sea Level Atmospheric Conditions

2 2 Po = 2116.22 lb/ft = 14.696 lb/in = 29.921 in Hg = 1013.25 HPa (mb) = 101325 Pa

o o To = 288.15 K = 518.67 R = 59 F = 15 C

3 3 3 o ρo = 0.0023769 slgs/ft = 0.07647 lbm/in = 1.255 kg/m (at 15 C )

ao = 1116.45 ft/sec = 661.478 KTAS = 761.14 mph = 340.294 m/sec

2 2 go = 32.174 ft/sec = 9.80665 m/sec

L = standard temperature lapse rate = 0.0019812 K /ft

1976 U.S Standard Atmosphere Equations Troposhere - below 36,089 ft (11,000 m) < 22636 Pa θ = 1 − (L/Το ) h = 1-(6.8755856 x 10-6) h σ = θn-1 δ = θn where n = 5.255876, h = geopotential altitude (ft) Stratosphere- between 36,089 ft and 65,616 ft (20,000 m) the standard day temperature is a constant 216.65 K, therefore: θ = 0.751865 .−0.000048063 [h −36 ,089 ] σ = .297076 e δ = .223361 e .− 0.000048063 [h − 36 ,089 ]

The above relations characterize the standard atmosphere table in this handbook. They may be re-written to solve for pressure altitude (Hp) for any ambient pressure. Below the tropopause (ambient pressure greater than 472.683 psf or 22632 Pa) 0.1902632 -6 Hp [ft] = [1-(Pa/Po) ]/[6.8755856 x 10 ]

In the troposphere (ambient pressure between 114.347 and 472.683 psf or between 5475 and 22632.1 Pascal)

Hp [ft] = 36089+[ln(Pa/Po)+1.498966]/ 0.000048063

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1976 U.S. Standard Atmosphere Graph

Thermospher

300 Thermosphere Mesopause

Speed of sound

Mesosphere

200 Temperature

Stratopause Geometric altitude (feet X 1000)

100 Stratasp here

50 Density Tropopause Pressure Troposphere

0 150 200 250 300 K(tem p) 288 0 1000 2000 3000 lb/ft2 (pressure) 2116 0 0.001 0.002 0.003 Slug/ ft3 (density) 0.002377 650 800 1000 1150 ft/sec (speed of sound) 1117

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Standard Atmosphere Calculator Website Link http://www.digitaldutch.com/atmoscalc/

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Section 3.5 Sea States (ref 3.3) Sea Stale International Swell Scale

Wave Height, Code Sea Crest to Trough (ft)

0 Calm 0

1 Smooth Less than 1

2 Slight 1-3

3 Moderate 3-5

4 Rough 5-8

5 Very rough 8-12

6 High 12-20

7 Very high 20-40

8 Mountainous 40+

9 Confused, Used as additional description 1-8

Code Swell In Open Sea

0 None low

1 Short or average

2 Long

3 Short Moderate height

4 Average

5 Long

6 Short heavy

7 Average

8 Long

9 Confused, Used as additional description 1-8

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Section 3.6 Sunrise Sunset Times 18/ Dec / 4/ Dec / 72 deg 70 deg 60 deg 50 deg 40 deg 30 deg 20 deg 20/ Nov / 6/ Nov / 23/ Oct/ 9/ Oct/ / p 25/ Se / p 11/ Se / g 28/ Au / g 14/ Au 31/ Jul/ 17/ Jul/ 3/ Jul/ 19/ Jun/ 5/ Jun/ / y Sunset Times Sunrise Times 22/ Ma / y 8/ Ma r/ p 24/ A r/ p 10/ A 27/ Mar / 13/ Mar / Subtract 4 minutes per degree longitude east of Prime Meridian (+4 if (+4 west) Meridian per degree longitude of east Prime 4 minutes Subtract 27/ Feb/ 13/ Feb/ 30/ Jan/ 16/ Jan/ 2/ Jan/ 19/ Dec /

0:00 9:00 8:00 7:00 6:00 5:00 4:00 3:00 2:00 1:00 0:00

23:00 22:00 21:00 20:00 19:00 18:00 17:00 16:00 15:00 14:00 13:00 12:00 11:00 10:00 UTC time UTC

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Section 3.7 Crosswind Components

40 Crosswind Components X-wind = (wind speed)*(sin Ψ) 35

30

25

20

Headwind Component 15

10 40 35 30 25 20 5 15 10 5

0 0 5 10 15 20 25 30 35 40 Crosswind Component

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Section 3.8 Geodetic Measurements

Acronyms, Abbreviations and Symbols

DGPS Differential Global Positioning System ECEF Earth Centered Earth Fixed coordinate system GPS Global Positioning System INS Inertial Navigation System WGS84 World Geodetic System 1984 a Earth’s semi-major axis radius b Earth’s semi-minor axis radius D Great circle distance between two points e eccentricity of the Earth square f Earth’s flatness factor h geodetic height N radius of curvature in prime vertical P radius of curvature in prime vertical

→ Vector from earth center extending to coordinates r Earth’s radius X ECEF x coordinate Y ECEF y coordinate Z ECEF z coordinate φ Geodetic latitude

ϕ Angle between the two → vectors originating at the Earth’s center and extending to their respective coordinates at the start and end points. λ Geodetic longitude ψ Runway heading with respect to true North.

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Earth Modeling

The Geodetic System defines the location of any point relative to the earth using latitude, longitude and height (Figure 3.8-1, point P). Longitude and latitude are expressed in degrees, minutes, seconds. Longitude lines extend 180 degrees from the Prime Meridian, run north to south, and converge at the poles. Latitude lines are parallel to the equator and extend 90°.

Figure 3.8-1 Geodetic Coordinate System

The 1984 world geodetic system, WGS84, models the earth’s surface as an oblate spheroid - an ellipsoid rotated about its semi-minor axis. In this model, used by global positioning systems, the earth’s semi-major axis, a is 6,378,137.0 meters and the semi-minor axis, b is 6,356,752.314 meters. The flatness factor ( f) is defined as: For the WGS84 model, f =1/298.257223563

Because the earth is not perfectly spherical, there are various methods for defining latitude. Unlike the geocentric latitude which uses the earth’s center for determining a point’s latitude, the geodetic latitude (used herein) is the angle between the equatorial plane and a line normal to the reference ellipsoid . Figure 3.8-1 exaggerates this with a normal line being well offset from the earth’s center. This definition leads to a degree of latitude being longer at the pole than at the equator: 111,694 m (60.3 nm) vs. 110,574 m (59.7 nm).

The geodetic longitude of a point is the angle between a reference plane and a plane passing through the point, both planes being perpendicular to the equatorial plane.

Mathematically, the geodetic surface is a smooth ellipsoid modeling the earth’s surface. Clearly, the topography (actual surface height) deviates from this model whenever land is above or below sea level. Less evident is that the actual sea level also deviates from the geodetic model due to local changes in the earth’s gravity. Specifically,

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SFTE Reference Handbook Third Edition 2013 mass variations caused by changes in earth density and topography, such as mountains or trenches, change local gravity vectors and therefore sea level relative to the ellipsoid.

Reference to Mean Sea Level (MSL) served as the traditional way to express topographic or bathymetric height. Geodesists once considered the sea in balance with the earth's gravity and formed a perfectly regular figure. MSL is usually described as a tidal datum that is the arithmetic mean of hourly water elevations observed over a 19- year (Metonic) cycle. This definition averages out tidal highs and lows caused by the changing effects of the gravitational forces from the moon and sun. MSL defines the zero elevation (vertical datum) for a local area. Because the sea surface conforms to the earth's gravitational field, MSL also has slight hills and valleys similar to the land surface but much smoother. Zero elevation as defined by one nation is often not the same zero elevation defined by another, thus locally defined vertical datums differ from each other.

The Geoid is the equipotential surface in the earth’s gravity field that coincides most closely with the mean sea level extended continuously under the continents. In other words, it approximates the level of any non-flowing water connected (actually or theoretically) to the seas by waterway or via trenches or tunnels. The geoid surface undulates relative to the geodetic ellipsoid and is perpendicular to the local gravity vector – as seen with a plumb line. Similarly, a spirit level defines the local surface parallel to the geoid, which is tangent to the local horizon. Because the geoid is an equipotential surface, it is the best datum for measuring potential energy and is the true zero surface for measuring elevations. Previously, there was no way to accurately measure the geoid, so heights were measured relative to the similar MSL. EGM96 (Earth Gravity Model 1996) represents the best geoid model currently available and shows smoothly changing surface undulations ranging from +85 to -107 meters relative to the WGS84 ellipsoid.

The geoid surface cannot be directly observed, thus heights above or below it can't be directly measured. Instead the geoid surface is modeled mathematically using gravitational measurements. Although for practical purposes, at the coastline the geoid and MSL surfaces are assumed to be essentially the same, at some spots the geoid can actually differ from MSL by several meters .

Figure 3.8-2 Height Comparisons

Ellipsoidal height (h) is the same as geodetic height and is the geometric distance between an object and the earth ellipsoid (Figure 3.8-2). This may be a GPS output.

Geoid height (N) is the height of the geoid above or below the ellipsoid. Some GPS devices output this undulation value in the data stream.

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Orthometric height (H) is the geometric distance along a gravitational force line from a given point P to the geoid. This is essentially the conventional height measurement because the geoid approximates MSL- the traditional method for determining height.

Modern GPS units typically include a geoid model (e.g. EGM-96) that provides N (geoid height over the WGS ellipsoid) at the current position. Such a unit can provide the height above geoid. If GPS height output is only available relative to the ellipsoid (h), then traditional Orthometric height (H) above the geoid can be obtained by subtracting the geoid height above the ellipsoid. [http://www.esri.com/news/arcuser/0703/geoid1of3.html]

ECEF Transformations

For the purpose of performance, navigation, or noise analysis, flight testers may require distances between two points (the shortest being along the great circle arc) and the average heading of that arc. Calculating these from typical Geodetic System Lat/Long inputs requires conversion to the Earth Centered Earth Fixed (ECEF) coordinate system as shown in Figure 3.8-3.

Figure 3.8-3 Earth Centered Earth Fixed Coordinate System

The ECEF coordinate system is a Cartesian system with the origin at the earth’s center. In this system, the X-axis is defined by the intersection of the Prime Meridian and equatorial planes. The Z-axis goes through the North Pole. The Y-axis completes a right-handed orthogonal system by a plane 90 degrees east of the X-axis and its intersection with the equator.

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Geodetic System (lat/long/height) data converts to ECEF as follows:

x = ( N + h)∙cos( φ)∙cos( λ) y = ( N + h)∙cos( φ)∙sin( λ) 2 z = ( N∙(1-e ) + h) ∙sin( φ) where, x = ECEF coordinate parallel to the X-axis y = ECEF coordinate parallel to the Y-axis z = ECEF coordinate parallel to the Z-axis φ = geodetic latitude λ = geodetic longitude h = height above geodetic (ellipsoid) surface N = Normal radius of curvature; distance from earth axis to any point on the geodetic surface at that latitude (extension of r to axis shown in Figure 3.8-3). 1 ∙ where, a = semi-major axis radius (6,378,137 m; 20,925,647 ft)

2 e = eccentricity squared; = 0.00669438002290 (Earth, per WGS84) . 1 2 ∙

Also useful is M = Meridian radius of curvature; distance from earth axis to any point on the geodetic surface at that longitude.

1 1 ∙ .

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Great Circle Calculations

Any plane passing through the center of a spheroid traces a Great Circle around the perimeter of that spheroid. The shortest distance between two points on the surface is that portion of the great circle arc encompassing both points (Figure 3.8-3).

Figure 3.8-3 Great Circle Arc

Except when moving around the equator, navigating along a great circle route has the characteristic of intercepting longitude lines at different angles. In other words, the heading (or bearing) changes along the route. Analysis shows bearing change along a great circle route:

• Is never greater than the longitude difference between the end points.

• Approaches the value of the longitude change as the final latitude approaches a pole (regardless of initial latitude).

• Is smallest when the final latitude is at the equator (for this case, bearing change ≈ longitude change x initial latitude/100).

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Calculate the great circle distance (D) between points (subscripts 1 and 2) as

,

→ ∙→ ∙ cos ∙ ∙ ∙

∙ ∙ ∙ ∙

∙ where P = distance from earth center to any point (including height above the spheroid surface).

→ = Vector from the Earth’s center to point P.

ϕ = Angle between the two → vectors

provides equations an online tool for calculating great circle initial & final bearings (headings).

For shorter distances typical of local flight testing, the Great Circle model matches the following Two-dimensional approximations .

Distance North-South (Northing): dy = N ∙ sin(∆φ)

Earth’s radius East-West: r = N∙cos(φ)

Distance East-West (Easting): dx = r ∙sin(∆λ)

2-D distance between two points: Heading between two points (relative to true north) ψ = arctan (dy/dx)

An error analysis of the above 2-D heading approximation shows it consistently lies between the initial and final headings transpiring during great circle navigation and is closest to the final heading. This occurs when considering up to 10 degrees longitude change and is therefore sufficiently accurate for lesser changes that arise in flight testing (e.g. radial from a navigation transmitter).

An error analysis of the above 2-D distance approximation shows accuracy within 0.6% of the great circle distance when changing latitude and longitude 1 degree, and within 3% when changing latitude and longitude 10 degrees. It is accurate to within 0.4% when changing only latitude or longitude 10 degrees.

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Local Distance Transformation

Latitude φ, longitude λ, and height (typical GPS output data) can be transformed into rectangular (X,Y,Z) coordinates. The following presents a method for applying this to two different coordinate systems, both with the X-Y axes defining the horizontal plane. This is useful when working with local distances associated with typical flight testing such as noise measurement, local navigation, or field performance.

Figure 3.8-4a shows a case where the XY coordinate system aligns with the latitude & longitude grid. Figure 3.8- 4b shows a case where the XY coordinate system aligns with a runway, with Y=0 defining the centerline. In both cases, a designated primary reference datum [φ0, λ0], such as the runway centerline threshold, coincides with X=0, Y=0. With the example shown in Figure 3.8-4b, the opposite end of the runway centerline [φ1, λ1], coincides with X= runway length and Y= 0.

Figure 3.8-4: Local XY Coordinate System Aligned with (a)φ,λ Grid; (b) Runway

Because each degree of latitude change is not always exactly 60 nm and because the spacing between degrees of longitude changes markedly with latitude, converting from degrees latitude & longitude change to distance requires scaling factors. First select equatorial distances for each degree change

φ_scale{φ=0} = 110,574 m = 362,776.6 ft = 59.70518 nm

λ_scale{φ=0} = 111,319.5 m = 365,221.4 ft = 60.10772 nm

Use average latitude to calculate latitude and longitude scaling factors

scale ∅_scale∅ scale _∅∙∅ ∅_ . λ_ ∙ ∙

Calculate X and Y components of distances aligned with the latitude & longitude grid (Figure 3.8-4a)

Y = φ _scale·(φ 1 - φ 0) X = λ_scale·(λ1 - λ0)

The 2-D local (vice great circle) distance between any two points on the XY plane is

D = (X2 + Y2)1/2

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For field performance work, it is preferable to align X & Y with the runway as shown in Figure 3.8-4b. To convert from grid-aligned to runway-aligned coordinates, apply a rotation matrix that ensures X remains positive when going from point [φ 0, λ0] towards [φ 1, λ1]. The rotation angle, ψ, is positive counter-clockwise from true East to the direction of the runway. [Note: do not confuse ψ with runway heading measured relative to magnetic north]. The function ψ = atan2(X, Y) returns rotation angles from - π to π, so that rotating to headings between 90° and 270° results in negative rotation angles.

For any point [φ, λ], calculate runway-aligned X & Y distances using

X = φ _scale·( φ - φ 0)·sin(ψ) + λ_scale·(λ - λ0)·cos(ψ) Y = φ _scale·( φ - φ 0)·cos(ψ) – λ_scale·(λ - λ0)·sin(ψ)

This provides a right-handed rectangular coordinate system where X is positive from [φ 0, λ0] towards [φ1, λ1] and Y is positive left of the runway centerline (Figure 3.8-4b).

For multiple tests from a given runway, it is convenient to define the following constants from the above equations K = φ _scale—sin(ψ) 1 K = λ_scale—cos(ψ) 2 K = φ _scale—cos(ψ) 3 K4 = λ_scale—sin(ψ)

The overall conversion from [φ, λ] to [X, Y] then reduces to

X = K1·( φ - φ 0) + K2·(λ - λ0) Y = K3·( φ - φ 0) – K4·(λ - λ0)

Calculating aircraft height Z above the XY plane requires defining where the XY plane lies. An analyst may define Z=0 at some arbitrary height (i.e. GPS altitude at the beginning of a maneuver) and consider only changes from that reference.

For field performance, it is typical to use the runway altitude as the reference. Because runway altitudes vary however, height should be surveyed and modeled or tabulated as a function runway centerline position, H{X}. For best accuracy, the runway survey accounts for GPS antenna height above the surface. Airplane height above the XY plane (Z) is then Z = ζ – H{X} where ζ is the test GPS antenna’s altitude.

Note: Analysts can determine Z using either Orthometric height above the geoid (H) or above the . geodetic surface (h) – as long as the runway surface model H{X} uses the same reference

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Section 3.9 References

3.1 Anon., “Aeronautical Vestpocket Handbook” ,Part No. P&W 079500, United Technologies Pratt & Whitney, Canada, 1990.

3.2 Lawless, Alan. R. et al, “Aerodynamics for Flight Testers”, National Test Pilot School, P.O. Box 658, Mojave CA, 93501, 1999.

3.3 Denno, Richard R., et al “AIAA Aerospace Design Engineers Guide” ISBN 0-930403-21-5, AIAA, 1987.

3.4 Global Positioning System Overview, Peter H. Dana, Department of Geography, University of Texas at Austin, 1994. ( www.colorado.edu/geography/gcraft/notes/gps/gps_f.html )

3.5 Charles D Ghilani, Penn State College of Engineering, 2008 (http://surveying.wb.psu.edu/sur351/georef/ Ellip4.htm)

3.6 Standard Atmosphere Calculator Website Link http://www.digitaldutch.com/atmoscalc/ .

3.7 Movable Type Scripts. [Calculate distance, bearing and more between Latitude/Longitude points ]

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NOTES

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Section 4 Pitot Statics

4.1 Subsonic Airspeed and Mach Equations 4.2 Subsonic Scale Altitude (Compressibility) Correction Chart 4.3 Subsonic Relations Between Compressible and Incompressible Dynamic Pressure 4.4 Supersonic Airspeed and Mach Equations 4.5 Total Temperature Equation 4.6 Altimeter Equation 4.7 PEC Test Methods 4.7.1 Tower Fly-by 4.7.2 Ground Course Method 4.7.3 Trailing Bomb, Cone Method 4.7.4 GPS Methods 4.8 Postion Error Correction Certification Requirements 4.9 PEC Correction Process Flow Chart 4.10 Airspeed/Altitude/Mach Graphic Relations 4.11 Effect of Errors on Calibrated Airspeed and Altitude

Editor's Note In an effort to reduce confusion and conflict regarding pitot and static pressure nomenclature, SFTE has elected to change two definitions and symbols since the first edition of this handbook was released. Henceforth, ΔPs shall indicate static pressure ERROR (ΔPs = Ps – Pa) and ΔPT shall indicate total (i.e. pitot) pressure ERROR (ΔPT = Pp – PT). This nomenclature eliminates the ΔPp symbol and confusion as to whether it indicates position error or pitot error.

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Section 4 Common Nomenclature a = speed of sound ao = speed of sound at sea level on a std day M = Mach number

Pa =ambient pressure

Po = ambient pressure at sea level on a std day (=2116.2 lb/ft2 = 29.92 in Hg)

Pp = pitot pressure corrected for instrument error only

Ps = static pressure (indicated at static port)

PT = total pressure q = incompressible dynamic pressure qc = compressible differential pressure (PT-Pa) qcic = instrument corrected differential pressure (= Pp-Ps)

Ta = ambient temperature (absolute scale)

To =ambient temperature at sea level on a std day (=288.15 oK = 15 oC = 518.7 oR = 59 oF)

TT = total temperature (absolute scale)

Vc = calibrated airspeed

Ve = equivlent airspeed

Vg = ground speed

Vi = indicated airspeed

Vic = instrument corrected airspeed

VT = true airspeed

ΔHic = altimeter instrument correction

ΔHpc = altimeter position error correction

ΔPD = dynamic pressure error (= PT-ΔPs)

ΔPT = total (pitot) pressure error (= PP - PT)

ΔPs =static pressure error (= Ps - Pa)

ΔVic = airspeed instrument correction

ΔVpc =airspeed position error correction

δ = pressure ratio between ambient and sea level std (= Pa / Po)

θ = temperature ratio between ambient and sea level std (= Ta / To) 3 ρo =ambient density at sea level on a std day (=.002377 slg/ft )

σ = density ratio between ambient and sea level std (= ρa /ρo) γ = ratio of specific heats (= 1.4 for air)

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Section 4.1 Subsonic Airspeed and Mach Equations True Airspeed

1   γ −1   2    γ   2γ Pa PT − Pa  V T =    + 1 − 1   γ − 1 ρ   P    a  a  

Equivalent Airspeed

(= VT equation with assumption of std day sea level density)

 2     7  Pa PT − Pa ρ a V e = 7   + 1 − 1  = V T = V T σ ρ   P   ρ o  a  o

Calibrated Airspeed

(= Ve equation with assumption of std day sea level pressure)

1   γ −1  2    γ   2γ Po PT − Pa  Vc =    +1 −1 γ −1 ρ o   Po     

P q .2857 7 o c +1 -1 ρ P ǰ o ƴʢ o ʣ Ƹ

Applying British units (lb/ft2) and converting from ft/sec to knots yields

2 7  PT − Pa  (kts) Vc = 1479  +1 −1  2116 

Mach Number

γ −1   2   γ    7  VT 2  PT − Pa   qc  M = =  + 1 −1 = 5  + 1 −1 a γ −1  P     P    a   a 

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Section 4.2 Scale Altitude (Compressibility ) Correction

The name comes from the fact that although the equivalent airspeed equation does correct for compressibility, the sea level pressure assumption used for calibrated airspeed makes the compressibility correction valid only for that (sea level) pressure. Above sea level, the calibrated airspeed must be re-scaled for pressure effects on compressibility. The mathematical method for determining Ve from Vc is to first solve the calibrated airspeed equation for qc 3.5 2 ⎡⎛ ρ V ⎞ ⎤ q = P ⎢⎜ o c +1⎟ −1⎥ c ο ⎜ P 7 ⎟ ⎣⎢⎝ o ⎠ ⎦⎥

Next, substitute this value and the ambient pressure (Pa) into the equivalent airspeed equation. (qc = PT - Pa )

2 ⎛ 7 ⎞ Pa ⎜⎡qc ⎤ ⎟ Ve = 7 ⎢ +1⎥ −1 ρ ⎜ P ⎟ o ⎝⎣ a ⎦ ⎠

The adjacent chart does this graphically for all subsonic airspeeds.

Ve = Vc +ΔVc

Page 04 - 4 SFTE Reference Handbook Third Edition 2013 00 700 1.0 Mach 1.0 0.9 Mach 0.8 Mach 0.7 Mach 0.6 Mach 0.5 Mach Vc, Knots CalibratedAirspeed 0.3 Mach 0.4 Mach Subsonic Scale Altitude (Compressibility) Correction +1)^0.28571-1])^0.5 a P / qc *[( 100 200 300 400 500 6

a P /1479.2)^2+1]^3.5-1)*101325 c ) c a V V - , P =4.647*( e =([( c e c V q V = Below subsonic knots equations use and airpseeds for pressures for pascals ( q c V 0 Δ -5

-10 -15 -20 -25 -30

Correction to be added to Vc to obtain Ve (equivalent airspeed) (equivalent Ve obtain to Vc to added be to Correction

Δ Vc (kts), (kts), Vc

Page 04 - 5 SFTE Reference Handbook Third Edition 2013

Section 4.3 Subsonic Relations Between Compressible and Incompressible Dynamic Pressure

For constant density (incompressible) flow Bernoulli’s equation reduces to

2 2q

VT = ()PT − Pa = ρa ρa

Where incompressible dynamic pressure q is defined as PT-Pa . As airflow speed increases, its density at the stagnation point increases thereby increasing the sensed pressure. The ratio between compressible & incompressible dynamic pressure can be written as a function of Mach num- ber 2 4 6 ⎡ M M M ⎤ qc = q⎢1+ + + +...... ⎥ ⎣ 4 40 1600 ⎦ True dynamic pressure q (as used in modeling) is defined in dimensional analysis as:

1 2 1 2 q = /2 ρaVT = /2 ρoVe

This value for q should not be confused with compressible (a.k.a. impact or differential) pressure, qc (= PT -Pa)

2 ⎛ 7 ⎞ 1 Pa ⎜ ⎡ qc ⎤ ⎟ q = ρo 7 ⎢ +1⎥ −1 2 P ⎜ P ⎟ o ⎝ ⎣ a ⎦ ⎠

Page 04 - 6 SFTE Reference Handbook Third Edition 2013 1 Poly . ( qc / q) Poly . ( q/qc ) vs. M Mach Number Mach Dynamic PressureDynamic Compressible : Pressure Dynamic 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.90 0.85 0.80 0.75

Page 04 - 7 SFTE Reference Handbook Third Edition 2013

Section 4.4 Supersonic Airspeed and Mach Equations

PT’ denotes pitot pressure behind shock wave

True Airspeed

1 γ −1 γ ⎡ ⎤ 2 ⎢ ⎥ P ' − P q ⎡ γ + 1 ⎛ V ⎞ ⎤ γ −1 1 T a = c = ⎢ ⎜ ⎟ ⎥ ⎢ ⎥ − 1 ⎢ 2 ⎥ Pa Pa ⎢ 2 ⎝ a ⎠ ⎥ 2 γ ⎛ V ⎞ γ − 1 ⎣ ⎦ ⎢ ⎜ ⎟ − ⎥ γ + 1 ⎝ a ⎠ γ + 1 ⎣ ⎦

Equivalent Airspeed (valid if qc/Pa >0.892929158)

7 ⎡ V ⎤ 166.92⎢ e ⎥ qc ⎣ao δ ⎦ = −1 P 2 a ⎡ V ⎤ 2.5 (7⎢ e ⎥ −1) a δ ⎣ o ⎦

Calibrated Airspeed (valid if Vc>ao)

7 ⎡Vc ⎤ 166.92⎢ ⎥ q a c = ⎣ o ⎦ −1 P 2 o ⎡Vc ⎤ 2.5 (7⎢ ⎥ −1) ⎣ao ⎦

Mach Number q 166.92[M ]7 c = −1 2 2.5 Pa (7[]M −1)

Page 04 - 8 SFTE Reference Handbook Third Edition 2013

Section 4.5 Total Temperature Equation

Since stagnation exists at the probe, it absorbs the energy of the air

ρ P T V ρT PT VT TT

Temperature Probe V 2 γ P γ P Apply Bernoulli: + ⋅ s = ⋅ p 2 γ −1 ρs γ −1 ρp

P 2 also /ρ = RT and a = γRT T ⎛ γ −1⎞ ∴ T =1+⎜ ⎟M 2 Ta ⎝ 2 ⎠ Use K ( probe recovery factor) to account for heat losses:

TT ⎡ K()γ−1 2⎤ = ⎢1+ M ⎥ Ta ⎣ 2 ⎦

During position error flight testing, measure Ti From Vc and Hpc determine M

2 Ti + ∆Tic = TT = Tα + Tα KM

2 plot Ti ~ M

TT

γ −1 Ta slope= T K 2 a

M2

Page 04 - 9 SFTE Reference Handbook Third Edition 2013

Section 4.6 Altimeter Equation

-6 5.256 Pa = Po (1-6.87535 x 10 H) below 36,089 ft

-.00004806[H-36,089] Pa = Po (.22335) e above 36,089 ft

Hi Indicated pressure altitude (29.92” Hg) ∆ Hic Instrument error correction Hic Altimeter corrected of instrument error ∆ Hpc Position error correction Hc Calibrated pressure altitude

Page 04 - 10 SFTE Reference Handbook Third Edition 2013

Section 4.7 Position Error Test Methods

4.7.1 Fly-by As depicted below, the flyby method originally used some sort of viewing platform with surveyed distances and a grid or other device for determining the aircraft’s relative angle above the platform’s altimeter. This information combined to give the aircraft’s actual pressure altitude. Modern methods replace the tower system with a radar altimeter or GPS unit to determine tapeline height above the flyby line (Hg). This geometric height is converted to a pressure altitude change using a temperature correction. When added to the aircraft’s pressure altitude on the runway, this change provides the actual pressure altitude during the flyby (Actual Hc = runway pressure altitude +

Hg(Ts/Tt).

Assumptions

1. No errors in total head. 2. Constant height runs 3. Surveyed course

 T  Actual H = H +Dtanθ ⋅ s  c ctower    Tt 

∆H pc = Actual Hc −()Hi + ∆Hic

∆P = − ρg∆H S pc

∆PS = q c − q cic ∆P = 1 ρ V 2 − 1 ρ V 2 (low Mach only) S 2 0 c 2 0 ic

Solve for Vc

∆V = V − V pc c ic

See flowchart for high mach or PT≠0 cases.

Page 04 - 11 SFTE Reference Handbook Third Edition 2013

Error Analysis

t 30 0 kno 1. to d ot rspee .5 kn Ai to 0 20 peed (sea level ft) Airs

10

Vic (knots) Cumulative Height Error Error Height Cumulative 100 200 300 400

Note: A check on basic instrument calibration is easily accomplished using a “ground block” where a parked test aircraft compares altimeters with tower. Any error can be treated as a bias.

This altitude-based Test method determines altimeter corrections and therefore static error directly. Accurately converting this static source error to an airspeed correction also requires knowledge of the test pitot tube's total pressure error (ΔPT ). This can be determined either through direct comparison with a calibrated noseboom pitot pressure or from one of the air- speed-based methods that directly yield airspeed corrections (pace, ground course, GPS). In lieu of these options, the pitot error may assumed to be zero, but this will reduce confidence in the calculated value for airspeed correction.

Page 04 - 12 SFTE Reference Handbook Third Edition 2013

Section 4.7.2 Ground Course Method

V i Δ S + ΔV ic = V ic V i + ΔV pc T i

= V c ÷ σ

= V T ΔS = ΔT

Fly known course at constant Vi Elapsed time = ΔT, ∴V = ΔS T ΔT

Use H i and Ti to compute Ve = VT ( σ ) = Vc for low altitude.

Correct Vi for instrument error corrections (ΔVic ) using

Vic = Vi + ΔVic

ΔVpc = Vc −Vic

To determine altimeter error assume ΔPT = 0 1 ρ [V 2 −V 2 ] = +ΔP 2 0 c ic S

ΔPS = +ρgΔH ΔP − ΔP If ΔP ≠ 0, then ΔH = + S T T pc ρg

Page 04 - 13 SFTE Reference Handbook Third Edition 2013

1 Error resulting from a /2 sec timing error 3 0.5 sec timing error e rs 2 2 mile course ou c ile m 3 e ours ile c 1 5 m Airspeed (kts) Error 0 50 100 150 200 250 Airspeed (kts)

Error resulting from wind changes

3 Wind speed increasing from 10 kts to 15 kts between runs in opposite directions 2

1kt wind, 180o direction change 1

Airspeed Error (kts) 0 50 100 150 200 250 Airspeed (kts)

This airspeed-based Test method determines airspeed corrections directly. Accurately converting this airspeed error to a static source error requires knowledge of the test pitot tube's total pressure error (ΔPT ). This can be determined using one of the altitude-based methods that directly yield altitude corrections (tower fly-by, trailing cone or bomb). In lieu of these options, the pitot error may assumed to be zero, but this will reduce confidence in the calculated values for static pressure error and altimeter. correction

Page 04 - 14 SFTE Reference Handbook Third Edition 2013

Section 4.7.3 Trailing Cone/Bomb Method

Static holes P sref

1. Measure PS (ref) away from pressure field of aircraft 2. Cone is used to stabilize static line 3. No speed limitations 4. Inexpensive—can be trailed on landing 5. Consider lag effects during rapid altitude changes

ΔPS = ΔPS A / C −ΔPS REF ΔP ΔH = + S = altimeter correction pc ρg

ΔPT − ΔPS = ΔPD = qic − qc assuming ΔPT = 0, M < .2

If pitot errors do exist, then they must be included in calculations for ΔVpc (see flowchart)

Using a trailing cone during stall testing may give airspeed errors due to lag errors during the deceleration.

This altitude-based Test method determines altimeter corrections and therefore static error directly. Accurately converting this static source error to an airspeed correction also requires knowledge of the test pitot tube's total pressure error (ΔPT ). This can be determined either through direct comparison with a calibrated noseboom pitot pressure or from one of the air- speed-based methods that directly yield airspeed corrections (pace, ground course, GPS). In lieu of these options, the pitot error may assumed to be zero, but this will reduce confidence in the calculated value for airspeed correction.

Page 04 - 15 SFTE Reference Handbook Third Edition 2013

4.7.4 GPS Methods

• The attraction ~ no aircraft modification required » no trailing cone or aircraft plumbing mod » no flight test boom ~ no limitation on speed or altitude » can be done down to near stall, » any altitude ~ easy data reduction » no correlation with pace aircraft, ground radar, or other references required

Various methods available, all assume steady winds and ambient temperature. You must determine wind speed and direction to calculate VT and T0 and to ensure steady winds existed during test series.

GPS accuracies are variable. Know tolerances before accepting GPS as a truth model.

If exact (± 10º ) winds are calculated inflight, you can fly one pass directly into/away from the wind VT = VG + VHeadwind

This airspeed-based Test method determines airspeed corrections directly. Accurately converting this airspeed error to a static source error requires knowledge of the test pitot tube's total pressure error (ΔPT ). This can be determined using one of the altitude-based methods that directly yield altitude corrections (tower fly-by, trailing cone or bomb). In lieu of these options, the pitot error may assumed to be zero, but this will reduce confidence in the calculated values for static pressure error and altimeter. correction

Graphs in Section 4.11 separately show the effect of measurement errors in ΔPT, air temperature, or true air- speed on ΔVC calculations . The last chart (Page 04-29) shows how each knot of accumulated ΔVC uncertainty affects the ΔHC uncertainty at various altitudes and temperatures.

Page 04 - 16 SFTE Reference Handbook Third Edition 2013

Flying four legs instead of three allows four separate calculations of wind speed & direction to confirm stable winds at that test airspeed. If several real-time calculations of winds confirm constant direction and velocity, then testing may be shortened by flying only one pass directly into or away from the wind. If this is done, an end-of-test wind calibration must be performed to confirm steady winds throughout the test series. To mini- mize temperature and wind variations, testing should be accomplished within a relatively small area.

Horseshoe Track GPS Method

• Hor seshoe track method ~ fly three legs with each perpendicular ground tracks, noting GPS ground speed on each ~ determine true airspeed by solving three equations in three unknowns •Practical problem ~ need to fly close to the ground, tracking perpendicular ground refer ences

VW c

V G3 VT

VT VG1 VT VW V VG2 b a W

1 ⎛ ⎛V 2 ⎞⎞ True airspeed: V = ⎜V 2 +V 2 +V 2 +V 2 ×⎜ 3 ⎟⎟ T ⎜ 1 2 3 1 ⎜ 2 ⎟⎟ 2 ⎝ ⎝V2 ⎠⎠

2 2 ⎛V1 −V3 ⎞ ⎛V2 −V1 ×V3 V2 ⎞ Wind velocityV W = ⎜ ⎟ + ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠

-1 ()V2 −V1 ×V3 V2 Wind directionψ W = tan ()V1 −V3

Page 04 - 17 SFTE Reference Handbook Third Edition 2013

Horseshoe Heading GPS Method

• Horseshoe Heading Method ∼ Fly three legs with perpendicular headings, noting GPS ground speed on each ∼ Determine true airspeed by solving three equations in three unknowns

W c

V VT G3

VG1 V Angle a = ψ T VG2 Angle b = 90 + ψ V W Angle c = 180 - ψ VT b a VW

2 2 2 -1 ⎡−VG1 + 2VG 2 −VG3 ⎤ Wind direction ψ = tan ⎢ 2 2 ⎥ ⎣ VG3 −VG1 ⎦ 1 2 ⎡ 2 2 2 2 ⎤ 2 ⎛ −V + 2V −V ⎞ Wind velocity V = 1 ⎢V 2 +V 2 ± ()V 2 +V 2 + /− ⎜ G1 G 2 G3 ⎟ ⎥ W 2 ⎢ G3 G1 G3 G1 ⎜ sinψ ⎟ ⎥ ⎣ ⎝ ⎠ ⎦ Whichever Works V 2 +V 2 True airspeed V = G3 G1 −V 2 T 2 W

The “Windbox” method consists of flying four legs instead of three. The extra leg provides a fourfold increase in wind calculations to improve result confidence. The “Orbis” method extends this advantage by collecting data at every heading throughout a level turn.

Page 04 - 18 SFTE Reference Handbook Third Edition 2013

Cloverleaf Method (Microsoft Excel spreadsheet adapted from Doug Gray, NSW Australia)

Fly three legs with approximately 90-120 degree difference between headings. ~ Can be accomplished in a broad turn as with the horseshoe method, or ~ Directly over a single point (cloverleaf maneuver). Accurate results require ~ Identical values for indicated airspeed (and TAS) for all legs. ~ Constant winds throughout data collection (single W/S vector in figure). ~ Approx. 10 seconds stable ground speed, Vg, (G/S in figure) during each leg. Aircraft heading results for each leg entail an airborne compass swing.

Inputs for each 3-leg data set Vg1 Vg2 Vg3 Trk1 Trk2 Trk3

Intermediate calculations X1 = Vg1*SIN(PI()*(360-Trk1)/180) Y1 = Vg1*COS(PI()*(360-Trk1)/180) X2 = Vg2*SIN(PI()*(360-Trk2)/180) Y2 = Vg2*COS(PI()*(360-Trk2)/180) X3 = Vg3*SIN(PI()*(360-Trk3)/180) Y3 = Vg3*COS(PI()*(360-Trk3)/180) M1 = -(X2 -X1)/(Y 2-Y1) M2 = -(X3 -X1)/(Y3 -Y1) B1 = (Y1 +Y2)/2 -M1*(X1 +X2)/2 B2 = (Y1 +Y3)/2 -M2*(X1 +X3)/2 VWx = (B1 -B2)/(M2 -M1) VWy = M1* VWx +B1

Results 2 2 0.5 Aircraft true airspeed = VT = [(X1 - VWx) +(Y1 - VWy) ] 2 2 0.5 Total wind speed = Vw = [(VWx + VWy )] Wind direction = ψw = Psiw = MOD(540-(180/PI()*ATAN2(VWy, VWx)),360) st 1 leg a/c heading = ψ1 = Psi1 = MOD(540-(180/PI()*ATAN2(VWy -Y1, VWx -X1)),360) nd 2 leg a/c heading = ψ2 = Psi2 = MOD(540-(180/PI()*ATAN2(VWy –Y2, VWx -X2)),360) rd 3 leg a/c heading = ψ3 = Psi3 = MOD(540-(180/PI()*ATAN2(VWy -Y3, VWx -X3)),360)

Page 04 - 19 SFTE Reference Handbook Third Edition 2013

Section 4.8 Position Error FAR 23/25.1323 and .1325/JAR Certification Requirements

100

50

30 V ( kts ) ic

H 0 Δ pc 100 200 300 -30

-50

1.3 V S 0 -100 1.8 V S 1

Maximum error at sea level must be less than ± 30 ft/100 kts between 1.3 VSO and 1.8 VSI VSO = Full flap, gear down, power off, stall speed VSI = Stall speed in a specific configuration

12 10 8 6 4 V FE 2 Δ V V ( kts ) pc 0 ic ( kts ) -2 100 200 300 400 -4 V / M -6 MO MO or V -8 NE 1.3 V S 0 -10 -12 166.7 kts

Errors must be equal to or less than +3%

of Vc or +5kts whichever is greater

Page 04 - 20 SFTE Reference Handbook Third Edition 2013

Mil Spec P-26292 C (USAF)

Landing configuration: ΔHpc ± 30 ft.

.03

.02 Air data computer .01 ΔP Mach number S 0 q .3 .4 .5 .6 .7 .8 .9 1 -.01 Compensation req. -.02

Page 04 - 21 SFTE Reference Handbook Third Edition 2013 4.9 PEC Correction Process Flow Chart a pc c ic P c e V s ic i G T V V

P σ V V V Δ V V V headwind Δ Δ − V − + + + s ÷ p − Airspeed P P ic ≡ ≡ errors V cic s − ⎤ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎦ q c 1 1 P 1 − − − V If using known known using If known using If Δ 5 5 5 . . . = 3 3 3 ; ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ; 1 1 1 pc + + + V T 2 2 2 ic c T Δ P 7 7 7 = total & static static & total = V V V = instrument-corrected pitot press. press. pitot instrument-corrected = = true total pressure, = pressure, total true = true ambient pressure, ambient true = , = instrument-corrected static press. press. static = instrument-corrected a o o a o o a s − a T s P P P P ρ ρ ρ P P P Common definitions:Common P P P ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ − P Δ ⎡ ⎢ ⎢ ⎣ reference. ⎡ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎣ , o o a T T P P T P P P P P = = = ≡ ≡ Δ c c cic c q q q T q P Δ M ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ and 1 1 1 c − − − 7 7 V 7 2 2 2 ] ] ] ] 1 1 1 1

δ o + + c − e + P q c c V 7 ο a o c a 2 e P P q q

a δ P q = ) V [ [ [ 1 T ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ with ⎛ ⎜ ⎜ ⎝ + P where a a a θ o ο

a ο c a c Δ T P P P ρ ρ P P q ρ V o − ≈ -V SUPERSONIC EQUATIONS SUPERSONIC 7 7 a [( e 7 s 5 P c Mach = = V Δ V = often 0 for fixed-wing fixed-wing for 0 A/C normal in flight. often = = c T Can replace replace Can with replace or with replace T = Δ a + c T V V e P ) Δ V V Δ 1 V cic ≡ ,written by Al Lawless 1 q < − ( M 5 . pc s 7 M 2 Subsonic from scale altitude compressibility) chart,(a.k.a. correction or from V ⎤ ⎥ ⎦ ) P Δ Exact solution requires multiple tests or noseboom with with noseboom or tests multiple requires solution Exact 1

δ Δ − e 2 and and V o ⎤ ⎥ ⎦ s R g pc a are errors to P a s ⎡ ⎢ ⎣ δ H 7 m/s Δ P e

. ρ Δ Δ V 92

o δ θ = . a = 518 ⎡ ⎢ ⎣ and and pc a o 7 ] = T 166 340.3 ( H P ρ ρ Δ Δ K = Pa Sign Convention = 25 088 . = ,

5 c a ] P q 15 c

36 σ . are corrections to be added. − mph H Note that SFTE sign convention convention SFTE sign that Note c be subtracted while stipulates × H 288 pc 6 ic [ 101325

2 3 − c = ic i = H H H C Δ H 10 Δ H 761.14 o × + + Pitot-Static Calibration Flow Chart Flow Calibration Pitot-Static kg/m = m/sec Altitude 15 in.Hg = 00004806 876 . .

ο − 225 6 . T 92 <36,088 ft ft m) (<11,000 <36,088 >36,088 ft (>11,000 m) ft (>11,000 >36,088 e . ic 1 − KTAS 80665 1 = H 29 . [ 3 9 − = = 661 c = 2 30386 a ο . K 2 = P P a H x .3325 x T slg/ft .223358 2 = = lb/ft = ft/s

pc δ δ ft/sec 22 H . Δ Slope=.2 17 . . 2 If usingIf knownpressure alt. 0023689 = geometric (tapeline) altitude . = test day ambient (abs.) temp day test = 2116 = std temp at test altitude altitude (abs.) test at temp std = 1116.45 32 ) for & outputs. ) for all inputs . G M x 47.88 = lb/in = 47.88 x s a 2 = = = T T H 2 = ο ο ο StdLevel Sea Conditions . g K P a ρ s a θ. T T . G knots = .54 x Km/hr = mph x .869 mph = x Km/hr .54 = knots Palb/ft = m/s = knots ft/s= x .51444 = knots m/s x H ) when ) when ft/s = knots x 1.68781 = mph x 1.4666 mph = x 1.68781 = knots ft/s T a R Calculations require consistent units units consistent require Calculations Δ T T 2 (e.g. ft/s, lb/ft ft/s, (e.g. ≈ 5 9 or ] 15 32 c 67 . K calculating calculating . T KM ( + Note: Must use MustNote: H 32 T 2 . Δ C − 273 absolute temperatures absolute 459 o a o + + F T ic T + 1 T Ti o 5 9 T [ T = T C = F Δ = o = o a θ + = F = T C Temperature o o K R

Page 04 - 22 SFTE Reference Handbook Third Edition 2013

4.10 Airspeed/Altitude/Mach Graphic Relation 700 800 500 600 650 750 850 900 950 1000 450 550 400 550 600 650 700 750 800 850 900 950 1000 350 500 450 300 400 250 350 300 Mach Number Mach 200 250 Airspeed: and KCAS KEAS 150 200 150 KEAS=100 KCAS=100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 0

5000

60000 55000 50000 45000 40000 35000 30000 25000 20000 15000 10000 Altitude (Feet) Altitude

Page 04 - 23 SFTE Reference Handbook Third Edition 2013 ) c . 50 KTAS 100 KTAS 150 KTAS 200 KTAS 250 KTAS 300 KTAS 350 KTAS 400 KTAS 450 KTAS 500 KTAS 550 KTAS 600 KTAS 650 KTAS ) c (1802.3 psf q psf (1802.3 600 KEAS =1218.8 psf =1218.8 psf KEAS 600 ) sf c (1484.5 psf q p 550 KEAS = 1024.1 psf KEAS = 1024.1 550 . Lawless A 500 KEAS = 846.4 psf q psf = 846.4 500 KEAS ) (1208.8 psf psf q (1208.8 c Pressure conversions conversions Pressure = psi 144.0 * psf = Pa 47.88 * psf Hg in. = .1414 * psf q 450 KEAS=685.6 sf ) (970.5 psf q psf (970.5 p c 400 KEAS=541.7 psf q 400 KEAS=541.7 ) (765.7 psf q psf (765.7 c Velocity conversions = ft/sec 1.68781 kts* = Km/hr 1.852 * kts = mph 1.15078 * kts White fairings denote const. KEAS & dynamic pressure (q) & dynamic pressure KEAS const. denote White fairings Dark fairings denote constant Mach number Dashed coloreddenote fairings constant KTAS @ ISA 350 KEAS=414.7 ) c 300 KEAS=304.7 psf q 300 KEAS=304.7 psf (443 psf q ) c 250 KEAS=211.6 psf q 250 KEAS=211.6 psf (319.5 psf q psf (319.5 ) c Calibrated Airspeed (KCAS) Airspeed Calibrated sf sf p 200 KEAS=135.4 psf q psf KEAS=135.4 200 ) c Subsonic Airspeed and Mach Number Relations 1.0M (138 psf q psf (138 ) q psf (218.5 c 150 KEAS=76.2 150 KEAS=76.2 0.8M (76.9 psf q psf (76.9 )q psf (590.8 c 0.6M 100 KEAS=33.9 psf 100 KEAS=33.9 (33.9 psf (33.9 q ) 0.4M c 50 KEAS=8.5 psf q 50 KEAS=8.5 0.2M (8.4 psf q psf (8.4 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 0

5,000

60,000 55,000 50,000 45,000 40,000 35,000 30,000 25,000 20,000 15,000 10,000 Hc, Pressure Altitude (ft) Altitude Pressure Hc,

Page 04 - 24 SFTE Reference Handbook Third Edition 2013

4.11 Effect of Errors on Calibrated Airspeed and Altitude <3psf T P Δ Calibrated Airspeed, KCAS effects are essentially if linear subsonic at all alid airspeeds T V P Δ *{ -[2.715E-13]*Vc^5 [5.75E-10]*Vc^4 + - [4.75E-07]*Vc^3 [ 1.934E-04]*Vc^2 + - [0.04047]*Vc 4.01} + T Calibrated Airspeed Correction per Total PSF Pressure Increase P Δ (kts) = = (kts) Vc Δ

50 150 250 350 450 550 650

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

Δ Δ P [+1 @ (Vc = Vc ] - Vc) - ] T

Page 04 - 25 SFTE Reference Handbook Third Edition 2013 Hc 54K' 45K' 36,089' 27K' 18K' 9K' 0 Press. Altitude Vc)

Δ Vc) Δ True Airspeed, VT above ISA] Stratosphere Troposphere 0.001973}*VT]*[#degISA] Cabove and Vc, accurate to +/- .025 and Vc, accurate to +/- .02 Calibrated Airspeed Correction per Degree Above ISA Above Degree per Correction Airspeed Calibrated Stratosphere Approximation Troposphere Approximation = [{1.7909E-16*Hc^2 - 8.543E-12*Hc + 6.6136E-7}*VT^2 + {2.3388E-8*Hc + Vc VT*[8.357E-9*Hc(ft) -0.00174]*[#deg C above ISA] Δ [{2.0287E-16*Hc^2 - 2.779E-11*Hc}*VT^2 +[{2.0287E-16*Hc^2 {1.6641E-8*Hc + 0.00172897}*VT]*[#deg C = = Vc= VT* [2.649E-8*Hc(ft) VT* Vc= -0.00239]*[#deg C above ISA] Vc Δ Δ Vc Δ (any consistent units for VT (any consistent units for VT 0 50 100 150 200 250 300 350 400 450 500

0.0

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1.0

Δ Vc = Vc @ [ISA +1] - Vc @ ISA ISA @ Vc - +1] [ISA @ Vc = Vc

Page 04 - 26 SFTE Reference Handbook Third Edition 2013 0 9K' 18K' 27K' 36,089' 45K' 54K' Pressure Altitude True Airspeed, KTAS VT*{[-2.061E-21*Hc^3 1.676E-11*Hc]*VT^2 + - 7.645E-17*Hc^2 Δ Calibrated Airspeed Correction per Increase KTAS (kts) = = (kts) Vc + [1.01E-18*Hc^3 - 1.16E-13*Hc^2 + 3.358E-9*Hc]*VT 3.358E-9*Hc]*VT + - 1.16E-13*Hc^2 [1.01E-18*Hc^3 + 1]} + -1.891E-6*Hc [3.608E-7*Hc^2 + Δ 50 100 150 200 250 300 350 400 450 500 550

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Δ Vc = (Vc @[VT+1] - Vc @VT) Vc - @[VT+1] (Vc = Vc

Page 04 - 27 SFTE Reference Handbook Third Edition 2013 54000 36089 27000 18000 45000 9000 0 Pressure Altitude (ft) Hpc @ [ISA + 20 deg C] + 20 Hpc @ [ISA Δ Hpc @ ISA conditions ISA @ Hpc Δ 2 + (-5.019E-06)*Hc - 0.06421]*Vc } - 0.06421]*Vc 2 + (-5.019E-06)*Hc Solid fairings denote fairings Solid Dashed fairings denote *Hc^2 + (-1.533E-09)*Hc - 0.000105]*Vc^2 - 0.000105]*Vc^2 + (-1.533E-09)*Hc *Hc^2 Knot Airpseed Correction KCAS Valid at all subsonic airspeeds Valid Hpc]*[1+{3.1059E-08*Hc + 0.00344}*(deg C above ISA)] C above + 0.00344}*(deg Hpc]*[1+{3.1059E-08*Hc Hpc]*[1+0.004616*(deg C above ISA)] Δ Δ (-5.034E-15)*Hc^3 + (1.502E-10)*Hc^ (-5.034E-15)*Hc^3 =[ISA =[ISA Hpc Hpc Hpc Hpc Δ Δ Altimeter Correction per Altimeter Correction Vpc{[(-1.184E-18]*Hc^3 + (4.004E-14) Vpc{[(-1.184E-18]*Hc^3 Δ Hpc = Hpc Δ ISA + [ Non-ISA: Troposphere Stratosphere 50 150 250 350 450 550 650 0

80 60 40 20

200 180 160 140 120 100

Δ Δ Vpc, (ft/kt) Vpc, / Hpc

Page 04 - 28 SFTE Reference Handbook Third Edition 2013

Section 5 Aerodynamics

5.1 Dimensional Analysis Interpretations Dynamic Pressure, Force Coefficients

5.2 General Aerodynamic Relations Continuity Equation, Conservation of Energy Equation, Resultant Aerodynamic Force

5.3 Wing Design Effects on Lift Curve Slope Aspect Ratio, Leading Edge Flap, Boundry Layer Control & Trailing Edge Flap Effects

5.4 Elements of Drag 5.4.1 Skin Friction Drag Viscosity, Reynolds Number Effects 5.4.2 Pressure Drag 5.4.3 Interference Drag 5.4.4 Induced Drag

5.5 Aerodynamic Compressibility Relations Prandtl/Glauert, Total vs Ambient Property Relations for Adiabatic Flow, Normal Shock Rela tions 5.5.1 Oblique Shocks Oblique Shock Relations, Mach Cone Angle 5.5.2 Supersonic Isentropic Expansion Relations Prandtl-Meyer Function 5.5.3 Two-Dimensional Supersonic Airfoil Approximations

5.6 Drag Polars 5.6.1 Drag Polar Construction and Terminology Simple Drag Polar Equation Limitations 5.6.2 Complicating Effects Airflow Separation, Reynolds Number, Wing Camber or Incidence Angle, Mach Number, Propeller Slipstream, and Trim Drag Effects 5.6.3 Drag Polar Analysis Optimum Aerodynamic Flight Conditions

5.7 References

Page 05 - 1 SFTE Reference Handbook Third Edition 2013

5.0 Recurring Terminology a slope of lift curve, dCL/dα a.c. aerodynamic center, location along the chord where pitching moments about this center do not change with angle of attack (25% MAC for airfoils in subsonic flow, 50% MAC for airfoils in supersonic flow) AOA angle of attack AR aspect ratio = [wing span]2/ [reference wing area] = b2/S B wing span bt horizontal tail span C coefficient, a non-dimensional representation of an aerodynamic property c wing chord length Camber maximum curvature of an airfoil, measured at maximum distance between chord line and amber line, expressed in % of MAC.Camber line theoretical line extending from an air foil’s leading edge to the trailing edge, located halfway between the upper and lower surfaces. CD drag coefficient CDi induced drag coefficient CDo,CDpe parasitic drag coefficient cf friction coefficient Chord straight-line distance from an airfoil’s leading edge to its trailing edge. CL lift coefficient Cp pressure coefficient = Δp/q e Oswald efficiency factor l distance traveled by flow, or characteristic length of surface M Mach number MAC mean aerodynamic chord, chord length of location on wing where total aerodynamic forces can be concentrated. MGC mean geometric chord, the average chord length, derived only from a plan form view of a wing (similar to MAC if wing has no twist and constant cross section & thickness-to-chord ratio). P pressure Preq’d power required 2 2 q dynamic pressure = ½ ρa VT = ½ ρo VT R gas constant Rn,Re Reynolds number S reference wing area, includes extension of wing to fuselage centerline. St horizontal tail surface area SW wetted area of surface T temperature V true velocity Ve equivalent velocity α angle of attack αi induced angle of attack δ depth of boundary layer, or surface wedge angle μ viscosity, or wave angle ν flow turning angle θ shock wave angle ρ density

Page 05- 2 SFTE Reference Handbook Third Edition 2013

•Perfect Fluid ~ incompressible, inelastic, and non-viscous ~ used in flow outside of boundary layers at M < .7 •Incompressible, inelastic, viscous ~ used for boundary layer studies at M < .7 •Compressible, non-viscous, elastic fluid ~ used outside boundary layers up to M = 5

Page 05 - 3 SFTE Reference Handbook Third Edition 2013

5.1 Dimensional Analysis Interpretations (ref 5.2)

Aerodynamic force = F

• F = f (ρ, μ, T, V, shape, orientation, size, roughness, gravity) • For aircraft ignore R, K & hypersonic effects

F Aerodynamic Force

α l Angle of Attack ength Chord L eam str ree F V

• Initially assume similar body orientations, shapes & roughness.

• Dimensional Analysis reveals four non-dimensional (π) parame- ters:

F π = Force Coefficient 1 2 2 ρV l

ρVl Reynolds Number π = 2 μ V Mach Number π = 3 a V π = Froude Number 4 l g

Page 05- 4 SFTE Reference Handbook Third Edition 2013

A closer look at the force coefficient:

F F C = ⇒ F 2 2 1 2 ρV l 2 ρV S

1 2 1 2 where /2 ρaVT = /2 ρoVe =dynamic pressure, q dimensions of reference wing area, S are the same

A feel for q 2 • Kinetic energy of a moving object = ½ mVT 2 • Block of moving air kinetic energy = ½ ρ (volume) V T 2 • Dividing through by volume yields KE per volume of moving air = ½ ρ V T • "Dynamic pressure” or “q” = potential for converting each cubic foot of the airflow's kinetic energy into frontal stagnation pressure • Feel q by extending your hand out the window of a moving car

A feel for coefficients • C F = (F /S)/q = the ratio between the total force pressure and the flow 's dy- namic pressure • Lift is the component of the total force perpendicular to the free stream flow • Drag is the component along the flow • Break total into lift and drag coefficients:

L F

D α

V ive relat

Page 05 - 5 SFTE Reference Handbook Third Edition 2013

• Froude number is not significant in aerodynamic phenomena

• Recall that forces are aslo a function of angle of attack, shape & surface roughness, therefore

Froude number is not significant in aerodynamic phenomena

Recall that forces are also a function of angle of attack, shape & surface roughness

CL,CD= f [M, Re, α] for a given shape, roughness L

C C L max Re

M Effects are exaggerated Lift Coefficient Lift

Angle of Attack

To compare test day and standard day aircraft or to match wind tunnel CF data to actual aircraft; the shape, roughness, M, Rn and α must be equal for both aircraft

LA

LM qM qA SM

SA

LA LM =CL = qASA qM SM

Page 05- 6 SFTE Reference Handbook Third Edition 2013

5.2 General Aerodynamic Relations (refs 5.1, 5.2, 5.10)

Lift & Drag forces can be described using two approaches: 1) Change in momentum of airstream, F = d{mv]/dt 2) “Bernoulli” approach which requires the continuity and conservation of energy equations

Continuity Equation Fluid M ass in = Fluid Mass out ρ 1V 1A 1 = ρ 2V 2A 2 For subsonic (incompressible) flow ρ 1 = ρ 2 V 1A 1 = V 2A 2

Conservation of Energy (Bernoulli) Equation: Potential + Kinetic + Pressure = constant (changes in Potential energy are negligible) Energy per unit volume is pressure then Dynamic Pressure + Static Pressure = Total Pressure

1 2 2 ρV + p s = constant 1 2 2 ρV + p s = p t

• This classic approach only applies in the “potential flow” region and not in the boundary layer where energy losses occur

Page 05 - 7 SFTE Reference Handbook Third Edition 2013

• Pressures around a surface can be calculated or measured from tests and converted into pressure coefficients,

cp = (plocal-pambient)/dynamic pressure = Δp/q

• cp values can be mapped out for all surfaces

C P

-1.0 Upper surface

100% x 0 c Lower surface 1.0 1.0

• Summation of all pressures perpendicular to surface yield the pitching moments and the “Resultant Aero- dynamic Force” which is broken into lift and drag components

RAF - Resultant Lift Aerodynamic Force

Center of Pressure

Drag

Angle of attack α Freestream Velocity V • Lift & drag forces are referred to the aerodynamic center (ac) where the pitching moment is constant for rea- sonable angles of attack. • Pitching moments increase with airfoil camber, are zero if symmetric. • Aerodynamic center is located at 25% MAC for fully subsonic flow and at 50% MAC for fully supersonic flow.

Page 05- 8 SFTE Reference Handbook Third Edition 2013

5.3 Wing Design Effects on Lift Curve Slope (refs 5.1, 5.2, 5.10)

Aspect Ratio Effect • Pressure differential at wingtip causes tip vortex

Resultant Flow Low

Pressure Free stream High flow airflow

• Vortex creates flow field that reduces AOA across wingspan

Tr ex ai rt Rear view lin vo g tip ti g p in vo Upwash ail rt Tr ex Upwash

Downwash

• Local AOA reductions decrease average lift curve slope

C 2D wing = wind tunnel L g in airfoil extending to walls ao D W 2 g (infinite aspect ratio). in W CL 3D a ao = Lift curve slope for an infinite wing

Coefficient of lift a = Lift curve slope for Δα a finite wing Angle of zero lift same α2D α3D Freestream Angle of attack, α

dC a • Above relationship estimated as a= L = o 57.3ao dα 1+ πAR

Page 05 - 9 SFTE Reference Handbook Third Edition 2013

Trailing Edge Flap Effects

Basic Section

Plain Flap Split Flap

Slotted Flap Fowler Flap

3.0 r Cl le w o ed 2.5 F tt lo S it pl 2.0 S n ai Pl 1.5

on 1.0 ti ec S c

Section Lift Coefficient 0.5 i as B

-10 -5 0 5 10 15 20 Section Angle of Attack, α

Page 05- 10 SFTE Reference Handbook Third Edition 2013

Leading Edge Flap Effects

Fixed Slot Automatic Slot

Slat

3.0 Cl 2.5 Fixed Slot

2.0

1.5 Basic Section 1.0

0.5 Section Lift Coefficient Section 0 5 10 15 20 Section Angle of Attack, α

Boundary Layer Control Effects

Boundary layer control Boundary layer control by upper surface suction by flap augmentation

3.0 Cl 2.5 High Suction 2.0 Low Suction 1.5 Basic Section 1.0 No Suction

0.5 Section Lift Coefficient Lift Section 0 5 10 15 20 25 Section Angle of Attack, α

Page 05 - 11 SFTE Reference Handbook Third Edition 2013

5.4 Elements of Drag (refs 5.1, 5.2, 5.10)

TotalTotal Drag Drag

Parasite Drag Wave Drag InducedInduced Drag Drag Parasite Drag Wave Drag

InterferenceInterference Drag Drag ProfileProfile Drag Drag

SkinSkin Friction Friction Drag Drag PressurePressure Drag Drag

• Skin friction shear stress is a function of velocity profile at surface

r laye Velocity proflile through dary boun the boundary layer e of r edg Oute dv dy Surface

δ dy

⎛ dv ⎞ Shear stress ℑ w = μ ⎜ ⎟ ⎝ dy ⎠ y = 0

• Viscosity (μ) increases with temperature (ref 5.9)

1.5 n ⎛ T ⎞ ⎛ ⎞ Sutherland law: ⎜ ⎟ T + S T ⎜ ⎟ ()o Power law: μ = μo ⎜ ⎟ T ⎜ T ⎟ μ = μ ⎝ o ⎠ ⎝ o ⎠ o ()T + S

Where To = 273.15 K = 518.67 R. For air: S =110.4 K = 199 R; n=.67 For air at 273 K: μ = 1.717x10-5 [kg/m s] = 3.59x10-7 [slug/ft s] o Inserting air values (TK=Kelvin and TR=Rankin) into Sutherland law gives

1.5 1.5 −6 TK ⎡ kg ⎤ −8 TR ⎡ slg ⎤ μ =1.458x10 ⎢ ⎥ = 2.2x10 ⎢ ⎥ TK +110.4 ⎣s⋅m⎦ TR +199 ⎣s⋅ ft⎦

Page 05- 12 SFTE Reference Handbook Third Edition 2013

Reynolds Number Effects (ref 5.10)

• Laminar boundary layers have more gradual change in velocity near surface than turbulent boundary layers. Thickness • High Reynolds numbers help propagate turbulent flow.

Laminar Turbulent

⎛ dv ⎞ Shearing stress ℑw = μ⎜ ⎟ ⎝ dy ⎠ y=0 ℑ ℑ Skin friction coefficient C = w = w f 1 2 2 ρ∞V∞ q∞ 1.328 Laminar boundary layer Total C f = 1/ 2 ()ReL .455 0.074 Turbulent boundary layer Total C f = 2.58 ≈ 1/5 ()log ReL ()ReL

0.010 0.008 0.006 Cf Transition 0.004 0.003 Turbulent

0.002 ReL based on total Laminar length of flat plate 0.001 105 106 107 108 109 1010 ρ∞ V ∞ L Reynolds number, ReL μ∞

• Depth of boundary layer (δ) depends on local Reynolds number (Rex) and whether the flow is turbulent or laminar. 5.2 x δ = lam Re ρ∞V∞ x Inertia Forces x Re x = ≡ μ∞ Viscous Forces .37 x δ turb = .2 x= distance traveled to point in question Re x

Page 05 - 13 SFTE Reference Handbook Third Edition 2013

5.4.2 Pressure Drag

• Ideal frictionless flow has no losses and leads to zero pressure drag

• Real fluids have friction and energy losses along surface

• Energy losses negate total pressure recovery, lead to decreasing total pressure along surface

Airfoil Ai rfoi l leading trailing Ideal fluid edge edge Real fluid

ρ Pressure distribution greatly modified

Separation occurs near here in real fluid case 0 (a) Airfoil upper surface static presure distributions Distance along flow

• Imbalance of pressures on surfaces causes pressure drag

Equal and opposite Parallel to freestream

(a) Shoulder (b) Ideal fluid air foil (no pressure drag) Sum of horizontal pressures = 0

Net downstream force = Pressure drag

(b) Shoulder (c) Real fluid airfoil (net pressure drag) more drag pressure than thrust pressure

Page 05- 14 SFTE Reference Handbook Third Edition 2013

• Profile streamlining reduces pressure drag

Separation point Relative R =105 Flat plate (Broadside) drag force length = d V 1 CD = 2.0

Separation point R =105 Cylinder diameter = d V 1 CD = 1.2

Separation point R =105 Streamline body thickness = d V1 C = 0.12 Same D total R =104 Separation point drag dCylinder diameter = V1 10 CD = 1.2 R =107 Separation point Cylinder diameter = d Larger 100 x C = 0.6 D Skin-friction drag V1 Pressure drag

5.4.3 Interference Drag

• Occurs with multiple surfaces approximately parallel to flow

• Caused by flow’s interference with itself or by excessive adverse pressure gradient due to rapidly decreasing vehicle cross section

• Most severe with surfaces at acute angles to each other

• Effects often reduced by fillets around contracting surfaces

Page 05 - 15 SFTE Reference Handbook Third Edition 2013

5.4.4 Induced Drag

• Wingtip vortex reduces local AOA at each station along wing

• Local lift vector is perpendicular to local AOA

• Local lift vector is therefore tilted back relative to freestream lift

• Induced drag defined as rearward component of local lift vector

Induced Drag

Lift (⊥ to freestream) Local lift force (⊥ to local freestream)

Local Relativ e Wind

Local Downwash

Angle of Attack, α Chord

Freestream relative wind

Induced Drag (DLiii) == Lift( )sin αα L

Induced Drag (Di ) = L(αi ) C For elliptical lift distributions α = L i πAR

⎛ CL ⎞ ∴ Di = L⎜ ⎟ but L = qSCL ⎝ πAR ⎠ D C 2 C = i = L = induced drag coefficient Di qS πAR

Oswald efficiency factor, e, accounts for losses in excess of those predicted above (due to uneven downwash and changing interference drag effects).

C 2 ∴ C = L Di πARe

Page 05- 16 SFTE Reference Handbook Third Edition 2013

5.5 Aerodynamic Compressibility Relations (reference 5.8)

Prandtl/Glauert Approximation Approximates Mach effects on aerodynamics below critical Mach

1 CP = CP compressib le 1− M 2 incompress ible

Total vs Ambient Property Relations for Adiabatic Flow

T γ − 1 T = 1 + M 2 Isentropic flow not required T 2 γ P γ − 1 γ −1 T ⎡ 2 ⎤ Isentropic (shockless) flow required = ⎢1 + M ⎥ P ⎣ 2 ⎦ 1 ρ ⎡ γ − 1 ⎤ γ −1 T = 1 + M 2 Isentropic flow required ρ ⎣⎢ 2 ⎦⎥

Normal Shock Relations Assumes isentropic flow on each side of the shock Assumes flow across shock is adiabatic Property changes occur in a constant area (throat)

P 1− γ + 2γM 2 2 = 1 P1 1+ γ −1 2 ρ2 ⎡2 + ()γ −1 M1 ⎤ = ⎢ 2 ⎥ ρ1 ⎣ ()γ +1 M1 ⎦ 2 2 T2 ⎡1− γ + 2γM1 ⎤⎡2 + ()γ −1 M1 ⎤ = ⎢ ⎥⎢ 2 ⎥ T1 ⎣ 1+ γ ⎦⎣ ()1+ γ M1 ⎦ 2 M 2 + 1 γ −1 M 2 = 2 2γ M 2 −1 γ −1 1

Normal shock summary P > P ρ > ρ T = T M > M T1 T2 T1 T2 T1 T2 1 2 P < P ρ < ρ T < T s < s 1 2 1 2 1 2 1 2

Page 05 - 17 SFTE Reference Handbook Third Edition 2013

5.5.1 Oblique Shocks

Oblique Shock Description

δ = surface turning angle

θ = shock wave angle

Subscript 1 denotes upstream conditions

Subscript 2 denotes downstream conditions

Oblique Shock Relations • Calculate P2/P1, T2/T1, and ρ2/ρ1 across oblique shocks by using normal shock equations and substituting M1 sinθ in place of M1 • Calculate total pressure loss across oblique shock as • Calculate relation between Mach number and angles as

1 P ⎧ γ ⎫1−γ T2 ⎪⎡γ −1 2 ⎤ ⎡ 2γ 2 2 γ −1⎤⎪ = ⎨⎢ + 2 2 ⎥ M 1 sin θ − ⎬ P γ +1 ()γ +1 M sin θ ⎢γ +1 γ +1⎥ T2 ⎩⎪⎣ 1 ⎦ ⎣ ⎦⎭⎪

2 M 2 sin 2 θ + 1 γ − 1 M 2 sin 2 ()δ − θ = 2 2γ M 2 sin 2 θ − 1 γ − 1 1

Page 05- 18 SFTE Reference Handbook Third Edition 2013

Oblique Shock Turning Angle as a Function of Wave Angle

• Two θ solutions exist for every M1 & δ combination These represent the strong and weak shock solutions Weak shocks normally occur in nature • There is a minimum Mach number for each turning angle • The wave angle of a weak shock decreases with increased Mach • For a given Mach number, θ approaches μ as δ decreases

Mach Cone Angle

Minimum Wave Angle μ = sin−1 (1/Μ)

Page 05 - 19 SFTE Reference Handbook Third Edition 2013

5.5.2 Supersonic Isentropic Expansion Relations

• The wave angle μ determines where the lower pressure can be felt and thus where the flow can be accelerat- ed • As the flow accelerates, a new wave angle forms and the subsequent lower pressure further accelerates the flow • Results in a series of Mach waves forming a “fan” until the flow turns and accelerates so that it is parallel to the new boundary

Page 05- 20 SFTE Reference Handbook Third Edition 2013

Prandtl-Meyer Function Shows flow’s required turning angle (ν) to accelerate from one Mach number to another

γ +1 ⎡ −1 γ −1 2 ⎤ −1 2 ν M = ⎢tan ()M −1 ⎥ − tan M −1 γ −1 ⎣ γ +1 ⎦

• If upstream Mach (M1) =1, then ν1 = 0, and equation directly relates downstream Mach (M2) to surface turn- ing angle (Δν) • If M1>1, determine M2 as follows: Calculate upstream ν1 from above equation Calculate ν2 = ν1 + Δν Reverse above equation to obtain corresponding M2 • Above equation is tabulated in NACA TR 1135 and is plotted below

Example: Flow initially at M1 = 2.0 accelerates through an ex- pansion corner of 24 deg. Exit Mach number is 3.0

Page 05 - 21 SFTE Reference Handbook Third Edition 2013

5.5.3 Two-Dimensional Supersonic Airfoil Approximations

• Determine surface static pressures by calculating changes through obliques shocks and expansion fans

• Ackert approximations for thin wings are based on

Δ P 2δ C p = ≅ ± q M 2 − 1

• Double wedge airfoil approximations

t 4α C L ≅ c M 2 − 1

2 2 4α 4 ⎛ t ⎞ C D ≅ + ⎜ ⎟ M 2 − 1 M 2 − 1 ⎝ c ⎠

• Biconvex wing approximations

t c 4α C L ≅ M 2 − 1 2 4α 2 5.33 ⎛ t ⎞ C D ≅ + ⎜ ⎟ M 2 − 1 M 2 − 1 ⎝ c ⎠

Page 05- 22 SFTE Reference Handbook Third Edition 2013

5.6 Drag Polars (ref 5.2)

5.6.1 Drag Polar Construction and Terminology CL = lift coefficient CD = drag coefficient CDi = induced drag coefficient CDo = parasitic drag coefficient AR = aspect ratio e = Oswald efficiency factor l = length flow has traveled Swet = wetted area of surface S = reference wing area

Simple Drag Polar Equation Limitations • No separated flow losses • Symmetric Camber • Applies at one Mach, Altitude, cg

C 2 C = C + L = C + C D Do πA Re Do Di

CL

“Polar” form of simple drag polar

C CDo D

CD Linearized form of 1 m = simple drag polar πA Re

C Do

2 CL

Page 05 - 23 SFTE Reference Handbook Third Edition 2013

5.6.2 Complicating Factors

Actual Drag CL Airflow Separation Effects Parabolic Curve C Lbreak

CD

Drag Polar Equation Accounting for Flow Separation:

2 (C − C ) C = C + L Lmin + k ()C − C D Dmin πARe 2 L Lbreak

• Delete last term if CL

Reynolds Number Effects (refs 5.4, 5.11) • Calculate length ReL and friction coefficient (cf ) for each surface as

ρVl ⎡ δ ⎤⎡T +110⎤ Re = = 7.101×106 M K l (TK = Kelvin, L 2 ⎥⎢ ⎥ μ ⎣⎢θ ⎦⎣ 398 ⎦ l= total length, ft) ⎧ ⎫ ⎪1.328 ⎪ 2 −0.12 ⎡ .074 1700⎤ c f = ⎨ ⎬[]1+ 0.1305M laminar, or = ⎢ − ⎥transition .2 Re ⎩⎪ ReL ⎭⎪ ⎣()ReL L ⎦ −258 2 −0.65 or c f = 0.455{}logReL {}1+ 0.144M turbulent

• In general, cf decreases as Rn increases (unless transitioning from laminar to turbulent flow) • Friction drag = cf q Swet for each component (Swet = wetted area) • Correct from test day to standard day aircraft drag coefficient by summing differences of each component’s drag change

Σ(c − c )S ΔC = f s ft wet D S

Page 05- 24 SFTE Reference Handbook Third Edition 2013

Wing Camber or Incidence Angle Effects

1.25 7 6 1.00 C 5 L 4 0.75 3 2 0.50 1 = AR 0.25

Note slight increase in drag as lift decreases 0.00 towards zero

-0.25 0.00 0.05 0.10 0.15 0.20 CD

CD Linearized drag po- m = 1 lafor aircraft with πARe wing camber and/or C incidence Dmin C Dpe

(− C 2) 0 CL 2 L min CL

Revised drag polar equation accounting for wing camber or incidence

2 (C − C ) C = C + L Lmin D Dmin πARe

• Generally not necessary since most flight occurs above CLmin

Page 05 - 25 SFTE Reference Handbook Third Edition 2013

Mach Number Effects Altitude = 30,000 ft Μ ≤ 0.85 cg = 25% MAC • Aircraft with No external stores low parasitic drag coefficients and high fineness ratios pay a relatively

0.9 L small “wave drag” 0.95 C penalty. 1.0

Modern fighter-type aircraft 1.3 1.5 Lift Coefficient,

0.02 0.03 0.04 0.05 0.06 0.07 0.08

Drag Coefficient

0.7 Altitude = 30,000 ft Μ ≤ 0.6 cg = 25% MAC • With external 0.6 Multiple external stores stores, same air- 0.8 craft pays larger 0.5 Mach penalty 0.85 0.4 0.9 0.95 0.3 1.0

Lift Coefficient 1.3 0.2 1.5

0.1

0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Propeller Slipstream Effects • a.k.a “scrubbing” drag • Propwash increases flow speed over surface within slipstream • More drag is created by higher q and vorticity. • Function of prop speed and power absorbed (Cp)or thrust (CT) • Problem should be addressed in airframe or propeller models

Page 05- 26 SFTE Reference Handbook Third Edition 2013

Trim Drag Effects (reference 5.4) e = wing Oswald efficiency factor et = tail Oswald efficiency factor b = span, bt = tail span x = wing ac-to-cg distance l= wing ac-to tail ac dist.

Lw LT x

l

2 2 ⎧ ⎡ ⎤ ⎫ W ⎪ 2 1 S e ⎛ b ⎞ 2 2 ⎪ ΔC D = ⎨ []x0 − x1 + ⎢1 + ⎜ ⎟ ⎥[]x0 − x1 ⎬ trim πq 2 Sb 2e lW l 2 S e ⎜ b ⎟ ⎩⎪ ⎣⎢ t t ⎝ t ⎠ ⎦⎥ ⎭⎪

2 ⎡ ⎛ ⎞ ⎤ Trim drag change relative to ΔC D trim x x b e = ⎢ ⎜ ⎟ − 2⎥ total induced drag: ΔC l ⎢ l b e ⎥ Di ⎣ ⎝ t ⎠ t ⎦ ΔC Dtrim 2 CD 1113 i ⎛ b ⎞ e 9 0.5 ⎜ ⎟ = 13 ⎝ btt⎠ e 7 11 5 0.4

0.3 Plot of above equation 0.2 9

0.1 x 7 0 l -0.2 -0.1 0 0.1 0.2 0.3

-0.1

5 -0.2

Page 05 - 27 SFTE Reference Handbook Third Edition 2013

5.6.3 Drag Polar Analysis

⎡ C 2 ⎤ ⎡ W 2 ⎤ D = q SC = q S C + L = 1 ρ V 2 S ⎢C + ⎥ D ⎢ D o ⎥ 2 o e D o 2 πA Re 1 2 ⎣ ⎦ ⎣⎢ πA Re ()2 ρ oVe S ⎦⎥

• For a given configuration (CDo, S, AR, e)

W 2 D = kV 2 +k first term = parasitic drag, 1 e 2 2 second term = induced drag Ve

• For any given weight, D = f(equivalent airspeed) only

Total Drag Induced Drag

D

Parasitic Drag

Min drag Min drag per velocity Ve

• Minimum total drag occurs when Dinduced = Dparasitic same as speed where CDi = CDo occurs at max CL /CD ratio (same as max L/D ratio) • Minimum drag/velocity occurs at min slope of Drag vs V curve same as speed where 3CDi =CDo 1/2 occurs at max CL /CD ratio Power required = drag x true airspeed

3 2 Ve Ve W Preq = DVT = D = k1 + k2 σ σ σVe

Minimum total Preq’d occurs when Pinduced = Pparasitic • same as speed where CDi = 3CDo 3/2 • occurs at max CL /CD ratio Minimum power/velocity occurs at min slope of Preq’d vs V curve • same as speed where CDi =CDo • occurs at max CL /CD ratio

Page 05- 28 SFTE Reference Handbook Third Edition 2013

Optimum Aerodynamic Flight Conditions

Gliders/ Engine-Out Flight • Max range (minimum glide slope) occurs at max CL/CD same as condition where CDo = CDi if drag polar is parabolic 3/2 • Min sink rate (minimum power req‘d) occurs at max CL /CD ratio same as condition where 3CDo = CDi if drag polar is parabolic

Reciprocating Engine Aircraft (assuming constant BSFC & prop η) • Max range (minimum power/velocity) occurs at max CL/CD ratio same as condition where CDo = CDi if drag polar is parabolic 3/2 • Max endurance (minimum power req‘d) occurs at max CL /CD same as condition where 3CDo = CDi if drag polar is parabolic

Turbine Jet Engine Aircraft (assuming constant TSFC) • Max range at constant altitude (minimum drag/velocity) 1/2 occurs at max CL /CD ratio same as condition where CDo = 3CDi if drag polar is parabolic • Best cruise/climb range (maximum [M x L/D] ratio) 3/2 occurs at max CL/CD ratio same as condition where CDo = 2CDi if drag polar is parabolic • Best endurance (minimum drag) occurs at max CL/CD ratio same as condition where CDo = CDi if drag polar is parabolic

CL d: CDi = 3 CDo

C L c: C = C = L = Di Do C D D max max b: CDi = CDo /2 a: CDi = CDo /3

C Do CD

To calculate optimum speed V2 for config- 1 4 1 uration & weight based on optimum ⎛ CD ⎞ ⎛ W ⎞ 2 2 2 ⎜ o1 ⎟ 2 V2 = ⎜ ⎟ ⎜ ⎟ V1 speed V1 at configuration1 & weight1 CD ⎝ W1 ⎠ ⎝ o2 ⎠

Page 05 - 29 SFTE Reference Handbook Third Edition 2013

5.7 References

5.1 Roberts, Sean “Aerodynamics for Flight Testers” Chapter 3, Subsonic Aerodynamics, National Test Pilot School, Mojave, CA, 1999

5.2 Lawless, Alan R., et al, “Aerodynamics for Flight Testers” Chptr 4, Drag Polars, National Test Pilot School, Mojave ,CA, 1999

5.3 Hurt Hugh H., “Aerodynamics for Naval Aviators“, University of Southern California, Los Angeles, CA, 1959.

5.4 McCormick, Barnes W., “Aerodynamics, Aeronautics, and Flight Mechanics“, Wilet &Sons, 1979

5.5 Stinton, Darryl, “The Design of the Aeroplane“, BSP Professional Books, Oxford, 1983

5.6 Roskam, Jan Dr., “Airplane Design, Part VI“, Roskam Aviation and Engineering Corp. 1990

5.7 Anon, “Equations, Tables, and Charts for Compressible Flow” NACA Report 1135, 1953

5.8 Lewis, Gregory, “Aerodynamics for Flight Testers” Chapter 6, Supersonic Aerodynamics, National Test Pilot School, Mojave CA, 1999

5.9 White, Frank M. “Fluid Mechanics” pg 29, McGraw-Hill, 1979, ISBN 0-07-069667-5.

5.10 Anderson, John D. Jr, “Introduction to Flight” pg 142, Mcraw-Hill, 1989, ISBN 0-07-001641-0.

5.11 Twaites, Bryan, Editor, “Incompressible Aerodynamics: An Account of the steady flow of incompressible Fluid Past Aerofoils, Wings, and Other Bodies,” Dover Publications, 1960.

Page 05- 30 SFTE Reference Handbook Third Edition 2013

Section 6 Axis Systems and Transformations

6.1 Earth Axis Systems

6.2 Aircraft Axis Systems Body, Stability, Wind, Principle

6.3 Euler Angles

6.4 Flight Path Angles

6.5 Axis System Transformations Earth-to-Body, Body-to-Earth

6.6 References

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6.1 Earth Axis Systems (ref 6.6.1)

Both fixed-Earth and moving-Earth axis systems keep constant orientation with respect to the Earth. The Z- axis point towards the center of the Earth. • The origin of a fixed-Earth system does not move relative to the Earth. (such as a ground radar site) • The origin of a moving Earth system does not move relative to its host (such as an aircraft inertial reference unit) .

X Moving XY plane is horizontal

Earth Y Axes

Y Z Fixed X Earth Axes . Z

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6.2 Aircraft Axis Systems (ref 6.6.2)

• The body axis system originates at the aircraft’s reference center of gravity. The +xb direction is towards the front, the +yb direction is towards the right wing tip, and the +zb direction is towards the bottom of the aircraft.

xb p u

xis x a

y a xis q

s i

x

a v z y r b

w

z b

• The stability axis system is similar to the body axis system except that it is rotated about the y-axis through the angle of attack (α)

xb RW xs α

zb ys, yb zs

Forces, velocities or accelerations along the stability axes are related to the body axes as follows

xb = xs cos α − zs sin α zb = zs cos α + xs sin α yb = ys

For cases where the z axis is defined positive upward (typical for normal-axis accelerometers)

xb = xs cos α + zs sin α zb = zs cos α - xs sin α

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• The wind axis system is similar to the stability axis system except it is rotated about the zs axis through the angle of sideslip (β).

The term “wind” refers to the fact that the freestream relative wind approaches the aircraft directly along the xw axis. This dictates that the true airspeed also lies along the xw axis.

xs Forces, velocities or accelerations xw along the wind axes are related to the stability axes as follows β xs = xw cos β − yw sin β ys = yw cos β + xw sin β z = z s w ys

y x w b x s RW u α V xw The geometric relations be- T β tween body, stability and wind axis velocities are illustrated v here. yb w

zb ω V ω V if β is small, then sinα = , β = sin α = , sinβ = VT VT VT cosβ VT ω if α is small, then α = VT

Most aircraft sideslip vanes do not measure β directly. They measure the flanking angle, which is the projection of the relative wind into the aircraft’s x-y plane. The difference between these two angles increases with angle of attack. Ignoring upwash, boom bending, and body axis rate corrections, calculate true sideslip as a function of vane α and β as follows: -1 βtrue = tan [ tan(βvane ) cosα ]

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• Wind-Body Axis Transformations (ref 6.6.1) Combining the two previous transformations, forces, velocities or accelerations along the wind axes are related to the body axes as follows

⎡xb ⎤ ⎡cosα 0 − sinα⎤ ⎡cosβ − sin β 0⎤ ⎡xW ⎤ ⎢y ⎥ = ⎢ 0 1 0 ⎥ ⎢sin β cosβ 0⎥ ⎢y ⎥ ⎢ b ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ W ⎥

⎣⎢zb ⎦⎥ ⎣⎢sinα 0 cosα ⎦⎥ ⎣⎢ 0 0 1⎦⎥ ⎣⎢zW ⎦⎥

After expansion, xb = cosα (xwcosβ – ywsinβ) - zwsinα yb = xwsinβ + ywcosβ zb = sinα (xwcosβ – ywcosβ) + zwcosα

The inverse transform, converting from the body to the wind axis system is

⎡xW ⎤ ⎡ cos β sin β 0⎤ ⎡ cosα 0 sinα ⎤ ⎡xb ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ yW = − sin β cos β 0 0 1 0 yb ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢z ⎥ ⎢ 0 0 1⎥ ⎢− sinα 0 cosα⎥ ⎢z ⎥ ⎣ W ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ b ⎦

After expansion, xw = cosβ (xbcosα + zbsinα) + ybsinβ yw = -sinβ (xbcosα+zbsinα)+ ybcosβ zw = -xbsinα + zbcosα

Note that these equations apply to the sign convention with z+ down. If sign convention (and instrumentation calibration) use z+ upward, then the above equations become: xw = cosβ(xbcosα - zbsinα) + ybsinβ yw = –sinβ(xbcosα + zbsinα) + ybcosβ zw = xbsinα + zbcosα

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• The Principle axes are those about which the products of inertia are zero. They can be equated to the axis of “dumbells” which represent concentrated mass elements. Neglecting aerodynamic and gyroscopic ef- fects, an aircraft rotating about one of its principle axes will not tend to cross-couple into motion about any other axis.

x

y

x y

z z

Wind to Body Axes Matrix Transformation

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6.3 Euler Angles (ref 6.6.1)

Euler angles are expressed as yaw, pitch, and roll. The sequence: first yaw, then pitch, then roll; must be main- tained to arrive at the proper orientation angles. The Euler angles are defined as follows:

ψ ≡ Yaw Angle: The angle between the projection of the vehicle xb- axis onto the horizontal reference plane and some initial reference position of the Earth x-axis. Yaw angle equals the vehicle heading only if the initial reference is North.

θ ≡ Pitch angle (in vertical plane) between xb and horizon.

φ ≡ Bank angle, the angle (measured in the y-z plane of the body-axis system) between the y-axis and the horizontal reference plane. Also known at the roll angle, it is a measure of the rotation (about the x-axis) to return the aircraft to a wings level condition.

x

+ θ

− ψ

+ φ y z

Earth Plane

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6.4 Flight Path Angles (ref 6.6.3) Just as the Euler angles define the attitude of the aircraft with respect to the Earth, three flightpath angles describe the vehicle's cg trajectory relative to the Earth (not the air mass).

σ = Flight path heading angle; also known as ground track heading, is the horizontal angle between some ref- erence direction (usually North) and the projection of the velocity vector on the horizontal plane. Positive rotation is from North to East. γ = Flightpath elevation angle; the vertical angle between the flightpath and the horizontal plane. Positive rota- tion is up. During a descent, this parameter is commonly known as glide path angle. μ = Flightpath bank angle; the angle between the plane formed by the velocity vector and the lift vector and the vertical plane containing the velocity vector. Positive rotation is clockwise about the velocity vector, looking forward.

The first two parameters above are easily measured using ground-based radar, or onboard GPS or inertial refer- ence systems. If only α, β, and the Euler angles are available, then assuming zero winds, the flightpath angles can be calculated as

γ =sin−1[(sinθ cosα −cosθ cosφsinα)cosβ −cosθ sinφsinβ cosφ sin β − sinφ sinα cos β σ = sin −1[ ]+ψ cosγ cosθ sinφ cos β + (sinθ cosα − cosθ cosφ sinα )sin β μ = sin −1[ ] cosγ

Technically, the above equations describe the velocity vector (angles relative to the air mass). If the air mass is moving relative to the Earth, as is usually the case, the above equations do not describe the flight path.

Editor’s note: not knowing the difference between flightpath and velocity vector angles can cause considerable confusion when analyzing data from different sources.

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6.5 Axis System Transformations (ref 6.6.2)

Transformation matrix for converting forces, velocities or accelerations from inertial (X, Y, Z) to body (x, y, z) axes

⎡ x⎤ ⎡ 001 ⎤ ⎡ θ − sin0cos θ ⎤ ⎡ ψψ 0sincos ⎤ ⎡ X ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ y⎥ = ⎢ sincos0 φφ ⎥ ⎢ 010 ⎥ ⎢− ψψ 0cossin ⎥ ⎢Y ⎥ ⎣⎢ z ⎦⎥ ⎣⎢ − cossin0 φφ ⎦⎥ ⎣⎢ θ cos0sin θ ⎦⎥ ⎣⎢ 100 ⎦⎥ ⎣⎢ Z ⎦⎥ Expanding gives: ⎡x⎤ ⎡ coscos ψθ sincos ψθ −sinθ ⎤ ⎡ X ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ Y ⎥ ⎢y⎥ = ⎢ cosφ sinψ +− sinφ sinθ cosψ cosφ cosψ +sinφ sinθ sinψ sinφ cos θ⎥ ⎢ ⎥ ⎣⎢z⎦⎥ ⎣⎢ + +− coscossinsincoscossincossincossinsin θφψθφψφψθφψφ ⎦⎥ ⎣⎢ Z ⎦⎥

The inverse of the above transform matrix converts from the body axis to the inertial axis coordinate system

⎡X⎤ ⎡ +− + cossincossinsincossinsinsincoscoscos ψθφψφψθφψφψθ ⎤ ⎡x⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢Y ⎥ = ⎢ + +− sinsincoscossinsinsinsincoscossincos ψθφψφψθφψφψθ ⎥ ⎢y⎥ ⎣⎢Z⎦⎥ ⎣⎢ −sinθ cossin θφ coscos θφ ⎦⎥ ⎣⎢z⎦⎥

Acceleration Transformations • Convert body-axis angular rates & linear accelerations into total accelerations along body axes. • Convert element (m1) location & rates into specific angular momentum

−+= x ua & qw rv −+= y va & ru pw −+= z wa & qupv H []ω rr ⇒××= m 1 `1

y ω ⎡ H ⎤ a 22 m1 ()()−−+= ()xzrxyqzyp ⎣⎢ m ⎦⎥ l n r ⎡ H ⎤ 22 1 ⇒ ⎢ ⎥ ()()−−+= ()xypyzrzxq ⎣ m ⎦ y ⎡ H ⎤ n ⇒ ()22 ()−−+= ()yzqxzpyxr ⎣⎢ m ⎦⎥ k x z

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Transformations between body axis rates and Euler angle rates

p = φ& − ψ& sin θ q = θ& cos φ + ψ& cos θ sin φ r = ψ& cos θ cos φ − θ& sin φ φ& = p + tan θ ( q sin φ + r cos φ ) θ& = q cos φ − r sin φ q sin φ + r cos φ ψ& = cos θ

Transformations from Euler & aerodynamic angles to the aircraft stability and wind axis angular rates. Subscripts b, s, and w denote the body, stability and relative wind axis systems.

( ps , qs , rs , pw , qw , rw ) = f ( pb , qb , rb ,α , β )

ps = pb cosα + rb sinα pw = ps cos β + qs sin β

qs = qb qw = qs cos β − ps sin β

rs = rb cosα − pb sinα rw = rs

Transformations from Euler angles to the three aircraft axis angular accelerations (ref 6.6.3)

( p& b , p& s , p& w ,q&b , q&s , q&w , r&b , r&s , r&w ) = f (θ ,θ&,θ&&,φ,φ&,φ&&,ψ ,ψ&,ψ&&,α,α&, β , β&)

p& b = φ&&−ψ&&sinθ −ψ&θ&cosθ

q&b = θ&&cosφ −θ&φ&sinφ +ϕ&&cosθ sinφ −ψ&θ&sinθ sinφ +ψ&φ&cosθ cosφ

r&b =ψ&& cosθ cosφ −ψ&θ&sinθ cosφ −ψ&φ&cosθ sinφ −θ&&sinφ −θ&φ&cosφ

p& s = p& b cosα +α&pb sinα + r&b sinα +α&rb cosα

q&s = q&b

r&s = r&b cosα −α&rb sinα − p& b sinα −α&pb cosα

p& w = p& s cos β − ps β& sin β + q&s sin β + qs β& cos β

q&w = q&s cos β − qs β& sin β − p& s sin β − ps β& cos β

r&w = r&s

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6.6 References

6.6.1 Lawless, Alan R., Math and Physics for Flight Testers “Chapter 7, Axis Systems and Transformations”, National Test Pilot School, Mojave CA, 1998. 6.6.2 Anon., Aircraft Flying Qualities, Chapter 4, Equations of Motion, USAF TestPilot School notes, AFFTC Edwards AFB CA, March 1991. 6.6.3 Kalviste, Juri, Flight Dynamics Reference Handbook, Northrop Corp. Aircraft Division, April 1988.

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NOTES

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Section 7 Mass Properties

7.1 Abbreviations and Terminology

7.2 Longitudinal & Lateral Center of Gravity Measurement

7.3 Vertical Center of Gravity Measurement

7.4 Moment & Product of Inertia Measurement 7.4.1 Radius of Gyration Aircraft moment of inertia summary 7.4.2 Parallel Axis Theorem Applications determine modified moment of inertia, determine modified products of inertia, correct the moment of inertia to the actual cg axis 7.4.3 Measuring Roll Inertia, Ixb 7.4.4 Measuring Pitch Inertia, Iyb 7.4.5 Measuring Yaw Inertia, Izb 7.4.6 Measuring Axis of Inclination and Ixz 7.4.7 Guidelines for Spring Oscillation Method 7.4.8 Swing Method

7.5 References

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7.1 Abbreviations and Terminology Abbreviations a perpendicular distance from the spring line of action to the oscillation axis (ft). cg center of gravity f measured frequency of oscillation (cycles/sec) g Earth’s acceleration due to gravity (g = 32.172 ft/sec2) h vertical component of the perpendicular distance from the cg to the oscillation axis. Icg moment of inertia any axis passing through the cg Ixb moment of inertia about aircraft body x-axis Iyb moment of inertia about aircraft body y-axis Izb moment of inertia about aircraft body z-axis Ixz product of inertia in aircraft body x-z plane Io moment of inertia about oscillation axis Ko component of spring stiffness perpendicular to vehicle motion. k spring constant (lb/ft) Ktot total radius of gyration (includes effect of offset pivot) L rolling moment leff effective pendulum length MAC mean aerodynamic chord METO maximum power (except for takeoff) m mass N yawing moment T period of oscillation p roll rate r yaw rate ΔIte correction for test equipment mass (slug-ft2) ΔIam equivalent moment of inertia contribution of the air (slug-ft2) δ tilt of spring assembly (measured positive if tilted nose-down relative to local horizontal). ε inclination of principle axis (positive if tilted down). φ angle between aircraft y-axis and line connecting aircraft cg with spring attach points. ωd damped frequency of oscillation (rad/sec) =2πf ωn natural frequency of oscillation (rad/sec)

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Terminology

Allowable cg range Documented on Type Certificate Data Sheet. May be different for takeoff vs landing. Forward limit usually determined by control power limitations, aft limit usually deter mined by stability requirements. datum The manufacturer defined reference plane used for distance calculations. empty weight Basic aircraft weight with only equipment on board. (without crew, passengers, or fuel). This weight may or may not inlcude oil weight, depending on civil certification date. empty weight cg range The allowable cg locations for an empty aircraft. This is defined by the manufacturer to help assure that a normally loaded aircraft will have an acceptable total cg location. lateral Along the aircraft y-axis. longitudinal Along the aircraft x-axis maximum weight Maximum allowable weight. Usually implies takeoff weight, but may apply to landing or in-flight weight. conditions after aerial refueling. minimum fuel A calculated value that represents the minimum amount of fuel any airplane should have while retaining appropriate flight reserves. Calculated as min fuel [lbs] = 0.5METO [hp] moment arm Distance between datum plane and cg of object. moment Product of moment arm and force (weight) tare The bias in weight scales due to test equipment weight or due to scale calibration errors. useful load Maximum takeoff weight minus empty weight. weighing point Location where aircraft is supported during the cg measurement.

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7.2 Longitudinal & Lateral Center of Gravity Measurement

Test set-up procedures for empty weight cg determination 1. Clean aircraft including any mud or water seepage. 2. Ensure all aircraft equipment in place. 3. Drain to only residual oil (for older aircraft that include only residual oil as part of empty weight calculations) or fill to full oil (newer aircraft include full oil for cal- culations). 4. Drain to residual fuel only. 5. Rig all equipment in a closed building to eliminate wind effects. 6 Calibrate scales. 7. Record tare (bias in scale readings, may be due to wheel chocks, jack stands, or other test equipment) 8. Place aircraft on scales and level according to manufacturer’s specifications. 9. Measure longitudinal and lateral distances between scale centers and datum. NOTE: distances behind datum are positive, distances ahead of datum are negative. 10. Record each scale weight

Calculating longitudinal center of gravity 11. Subtract the tare from each scale reading to get the correct weight. 12. Multiply each scale’s corrected weight by its distance to the datum. This product is the moment for each scale. 13. Sum the moments in step 12. 14. Sum the corrected weights. 15. Divide the total moment by the total weight [step 13/step 14] to obtain the distance from the datum to the cg. Some aircraft use this distance for all cg references (typically presented in inches). 16. Other aircraft refer cg location to a percent of the mean aerodynamic chord (MAC). In this case, subtract the distance between the datum and the leading edge of the MAC from the distance in step 15. 17. Divide the distance in step 16 by the length of the MAC. This number is the fraction of the cg location along the MAC. 18 To present the above fraction in terms of % MAC, multiply by 100.

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Example cg calculations Datum #2 Datum #1 Datum ? in ? ? in

40 in 75 in 115 in

320 lbs 816 lbs right 810 lbs left

Example Using Datum at Main Landing Gear ItemWeight × Arm = Moment Right wheel 816 lb × 00= Left wheel 810 lb × 00= Nose wheel 320 lb × -75= -24,000 inlb Total 1,946 lb × = -24,000 inlb 24,000 =12.33 in (fwd of MWCL) 1,946 Example Using Datum at Prop Spinner ItemWeight × Arm = Moment Right wheel 816 lb × 115= 93,840 Left wheel 810 lb × 115= 93,150 Nose wheel 320 lb × 40= 12,800 Total 1,946 lb × = 199,790 199,790 = 102.67in 1,946 Example Lateral cg Calculation ItemWeight × Lat. Arm = Moment Right wheel 816 lb × +70= 57,120 Left wheel 810 lb × -70= -56,700 Nose wheel 320 lb × 0 = 0 Total weight 1,946 lb × = 420 420 then = = .216 in 1,946

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Correcting empty weight cg for changes in fuel, passengers, equipment or stores.

1. Note aircraft empty weight & empty weight cg. Multiply these values to obtain the empty weight moment. 2. Note the weight and moment arm for each item added to or subtracted from the aircraft (items subtracted are listed as negative weights). 3. Multiply each item’s weight and arm to determine its moment. 4. Sum each item’s moment in step 3 with the aircraft empty weight moment. 5. Sum each item’s weight with the aircraft empty weight. 6. Calculate the new cg as [step 4/step 5].

Example cg corrections: Given aircraft with empty weight = 1,075 lbs and cg @ 84 inches. Add pilot (170 lbs, @85.5”), fuel (75 lbs @94”), and oil (15 lbs @ 31.7”).

Item Wt × Arm = Moment Airplane (empty) 1,075 × 84 = 90,300.0 Pilot 170 × 85.5 = 14,535.0 Fuel 75 × 94 = 7,050.0 Oil 15 × 31.7 = 475.5 Total 1,335 × = 112,360.5 112 ,360 .5 then = = 84 .16 in 1,335

Given aircraft with empty weight = 1,220 lbs and cg @ 25 inches. Add radio (15 lbs, @ 65”), and replace 11 lb generator with 14 lb generator at same 21.5” location (in front of datum-located on firewall).

Item Wt × Arm= Moment Airplane (empty) 1,220 × 25.0= 30,500.0 Radio 15 × 65.0= 975.0 Generator (removed) -11 × -21.5= 236.5 Generator (installed) +14 × -21.5= -294.0 New empty weight 1,238 × = 31,417.5 31,. 417 5 = 25. 38 in the new cg (empty) 1, 238

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7.3 Vertical Center of Gravity Measurement

1. Drain or block landing gear struts to keep distances G, J, and V constant. 2. Level the fuselage and measure the weight on the nose wheel (Fo). 3. Tilt the aircraft at various (θ) measure nose wheel weight (F).

C

B +

G J V

WL F SCALE FLOOR

The change in nose wheel force can be written as

⎡ BG⎤ ⎛ W ⎞ ()FO − F = V + ⎜ ⎟ ⎣⎢ C ⎦⎥ ⎝ C cotθ + G ⎠

400 F -F 4. Plot (Fo-F) vs the term in o parenthesis. 300 5. Slope of line equals term in brackets. BG 6. Solve for V after measuring B, C, G, ⎡ ⎤ Slope= ⎢V + ⎥ 200 and the slope. ⎣ C ⎦ • This method 100 applies to “gear down” cg . -16 -12 -8 -4 48 12 • For “gear up” add the manufacturer's W/(C cot θ+G) [lb/in] prediction of the cg shift to this result. -100 -200

-300

-400

-500

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7.4 Moment & Product of Inertia Measurement

The moment of inertia about any axis of a body is the summation of the product of every element’s mass and distance squared. Moments of inertia represent the resistance to rotational momentum changes.

d 2 2 2 I ≡ ()y 2 + z 2 dm d = x + z xb ∫ x I ≡ x2 + z 2 dm yb ∫ () 2 2 I zb ≡ ()x + y dm ∫ m1 z

Products of inertia are also calculated about any body axes. They represent the symmetry of mass dis- tribution (comparing opposing quadrants). x I = I ≡ xydm xy yx ∫ (−) (+) y I yz = I zy ≡ yzdm (+) (−) ∫ m1 I = I ≡ xzdm Ixy = (+) xz zx ∫

• Aircraft moments and products of inertia are calculated using body axes as the reference system.

• Careful x axis U F Documentation can yield P x G inertial x predictions

within about G y 1-5% of actual. Q

F G y z V R y axis F z z axis W

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7.4.1 Radius of Gyration

• Dimensions of all I terms are [slugs-ft2]. • Aircraft moments & products of inertia are generally assumed to be constant. • For moments of inertia, mathematically replace I with the product of total mass times some constant with the dimensions of square feet. • This constant is called the “radius of gyration” (k) • If all the mass were concentrated at this radius, it would have the same moment of inertia as the actual body.

2 2 2 I x ≡ (y + z ) dm = mkyz ∫ Aircraft moment of inertia summary Izb is always the largest value. Iyb> Ixb for fuselage-loaded aircraft Ixb> Iyb for wing-loaded aircraft.

x

y

x y

z z

• Vehicle mass distribution can be represented with concentrated “dumbbell” masses. • The lines connecting the opposing dumbbells are the principle axes. • When the principle axis lies along some line different from the body axis, the products of inertia are non- zero. • If the orientation of the principle axes and the moments of inertia about theses axes are known, then the mo- ments of inertia about any other axis system can be calculated.

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7.4.2 Parallel Axis Theorem Applications

Shows how to determine modified moment of inertia after some component alterations. Example: Correct original aircraft roll inertia, Ixb, to a modified value that accounts for the addition of wingtip fuel tanks. I = I + ΔI xbmod xb orig x comp

• ΔIx comp is composed of two components

2 ΔI x comp = I x comp + mcomp r

• Ix comp is the moment of inertia of just the new component about an axis which is parallel to the aircraft axis in question (this axis should run through the component’s cg). • Ix comp can be determined analytically by summing the inertias of every mass element throughout the compo- nent (documented for simple shapes in various engineering handbooks). • Ix comp can be determined experimentally by “swinging,” (sect’n 7.4.8). 2 • Ix comp is usually much smaller than mcomp r and can often be omitted. • mcomp is the component’s mass • r is the distance from the axis in question to the component’s cg.

Similarly, the parallel axis theorem shows how to determine modified products of inertia after some compo- nent alterations

I xz mod = I xz orig + [I xz comp + mcomp xz] x and z are the distances from the component cg to the reference axes.

Moments of inertia are calculated about a set of reference axes which all intersect at the reference cg. In general, however, the actual cg does not lie exactly at this reference cg. The parallel axis theorem shows how to correct the moment of inertia to the actual cg axis. As an example, the rolling moment of inertia about the actual cg (Ixcg) is calculated from the reference Ixb as follows.

2 2 I x cg = I xb − m[ y + z ] where m = total mass y = lateral distance between cg and aircraft x-axis. z = vertical distance between cg and aircraft x-axis.

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7.4.3 Measuring Roll Inertia, Ixb (spring oscillation method)

1. Balance aircraft on prism-like “knife edges” that support wooden cradles that conform to aircraft shape. 2. Attach springs (tension springs illustrated here) so they are parallel to line a as shown.

a O h 90 SPRING A+B

CG

WL OSCILLATION AXIS CRADLE KNIFE EDGE AB

3. Allow aircraft to oscillate freely in roll after a small disturbance. 4. Use automatic recording to determine period of oscillations (T). 5. Calculate damped frequency as 2π ωd = T 6. Record peak magnitude of each oscillation. 7. Calculate ζ using transient peak ratio method (see “Motion Analy sis,” Section 8 of this handbook). 8. Calculate natural frequency as

ωd ωn = 1−ζ 2

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Measuring Roll Inertia (continued)

9. Measure the spring’s stiffness (Ko)

Ko 10. Calculate inertia about oscillation axis (Io) as Io = 2 ωn

11. Use parallel axis theorem to correct Io to a parallel axis passing through the cg.

12. Springs, cradles and attachments hardware will change the moment of inertia. Sum their total into a com- bined “test equipment inertia” (ΔIte) and subtract this from the above result.

Combining steps 10-12 gives the complete moment of inertia equation

K a2 − mgh I = o − md 2 − ΔI cg ω 2 te n where h is the vertical distance between the cg and axis of oscillation and d is the total distance between the cg and the axis of oscillation (d = h in the illustrated roll inertia test setup).

If automatic recording is not available... Accomplish steps 1-3 as described for automatic recording case. 4. Use a stopwatch to time several oscillations and calculate ωd as before 5. Measure the spring’s rotational stiffness (Ko) 6. Approximate inertia using the damped frequency using

Ko 7. Use parallel axis theorem to correct Io to a parallel axis Io ≈ 2 passing through the cg. ωd

8. Correct for effects of test hardware moment of inertia (ΔIte). 9. To correct for errors in the Io approximation, apply an "additional mass correction" (ΔIam) which equates the flate plate area damping effect to additional inertia. This correction is detailed in ref. 7.5.3.

K a2 − mgh Combining steps 6-9 gives o 2 Icg = 2 − md − ()ΔIte + ΔIam ωd

Above methods can also be used to determine Iyb and Izb

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7.4.4 Measuring Pitch Inertia, Iyb (spring oscillation method)

1. Balance aircraft on knife edges as shown.

2. Attach spring perpendicular to line a. Only one spring is required since the aircraft cg is off-center. Spring must be stiff enough to hold the aircraft in equilibrium as well as provide a restoring moment during oscilla- tions.

CG h

WL a d O 90 SPRING

OSCILLATION AXIS KNIFE EDGE A B

Repeat steps 3-12 and apply the moment of inertia equation to determine Iyb

2 K o a − mgh 2 I cg = 2 − md − ΔI te ωn

or, for an uninstrumented aircraft,

2 Koa − mgh 2 Icg = 2 − md − ()ΔIte + ΔIam ωd

Page 07 - 13 SFTE Reference Handbook Third Edition 2013

7.4.5 Measuring Yaw Inertia, Izb (spring oscillation method)

1. Suspend aircraft as shown: fuselage reference line parallel with floor, z-body axis coincides with the oscillation axis (h = d = 0). For Izb test, set spring assembly tilt angle parallel to floor (δ = 0).

CRASH RECOVERY RESTORING MOMENT SLING N

WL δ L A+B

N

O I

T

A

L

L

I

S

I C

S X O A

2. Ideal case is where the wing attach points are in line with the aircraft cg as shown. Springs are perpendicu- lar to the AB line and are parallel.

L

W

CG

S A B I

X

9 Y A 0 O

Page 07 - 14 SFTE Reference Handbook Third Edition 2013

Measuring Yaw Inertia (continued)

If not possible to rig wing attach points in line with the cg, then keep the springs parallel to each other and measure the angle φ.

• Calculate Ko = k cos φ where k is the actual spring constant

A

CG WL

O 90 B

Y AXIS

Repeat steps 3-12 and apply the moment of inertia equation to determine Izb. Note that distances h and d are zero for a properly suspended aircraft.

2 K o a I cg = − ΔI te ω 2 n

or, for an uninstrumented aircraft,

2 K o a I cg = 2 − ()ΔI te + ΔI am ω d

Page 07 - 15 SFTE Reference Handbook Third Edition 2013

7.4.6 Measuring Axis of Inclination and Ixz

• Inclination of principle axis (ε) is positive if it lies in the +xz plane as shown.

• Ixz positive if ε is positive.

-Z

CG +X +ε WL

+X- FUSELAGE REFERENCE LINE PRINCIPAL LONGITUDINAL AXIS

+Z AXIS ORIENTATION

• From equations of motion

and L = I x p& − I xz (r& + pq) + (I z − I y )qr N = I zr& − I xz ( p& − qr) + (I y − I x ) pq

• When forcing small motions about only the yaw axis, pq and qr are negligible, giving

L = I p − I r and x & xz & N = I zr& − I xz p&

• If ε (and therefore Ixz ) is positive, then yawing to the right will generate a left rolling moment.

• If ε and Ixz are negative, a right yaw will generate a right moment.

Page 07 - 16 SFTE Reference Handbook Third Edition 2013

Measuring Axis of Inclination and Ixz (continued)

1. Determine Ixz and ε by repeating yaw experiment for different spring angles (δ). Use instrumentation to note for each δ.

2 Graphically determine the angle δo at which the restoring vector produced by the springs completely counteracts the roll motion.

p& Plot [tan δ] vs r& as shown.

+.02

+.01 p & r& 0 Tangent of Restoring Spring Assembly Angle 0.1 (tanδ)

-.02

-.03

-.04 -0.3 -0.2 -0.1 0 0.1 .02 .03 .04 .05 .06 .07

p 3. Determine the point where & equals zero. This occurs at tanδ . r o &

4. Calculate the product of inertia using I xz = I z tanδo

5. Calculate the inclination of the principle axis using

2I tan 2ε = xz I zb − I xb p& r& Note: Since the Ixz test objective is to interpolate to a condition where equals zero, only the ratio is necessary and the absolute magnitudes of the accelerations are not reuired. In other words, the roll acceleration sen- sitivity can be increased to allow for easier measurement of the ratio.

Page 07 - 17 SFTE Reference Handbook Third Edition 2013

7.4.7 Guidelines for Spring Oscillation Method

• Use only small magnitude oscillations

• Keep size and radius of knife edges as small as practical.

• Springs are typically linear except around zero load. Best results occur if springs are pre-loaded.

• Choose spring stiffness so oscillation frequency will be within instrumentation limits.

• If using a hand recorded stopwatch, best spring stiffness gives about one cycle per second (f = 1).

∗ Estimate the desired spring rate using

( I + md 2 ) (2πf )2 + hmg k = cg a 2

• The purpose of instrumentation is to provide a means for measuring frequency and magnitude. Any of sev- eral parameters will be sufficient, including angular displacement, rate or acceleration; or linear accelera- tion.

Page 07 - 18 SFTE Reference Handbook Third Edition 2013

7.4.8 Swing Method

1. Suspend component as shown.

2. Measure pendulum length (l), component mass (m ), and the period of oscillation (T).

l

• The observed period is a function of the effective pendulum length.

2 g 2π l eff ⎛ T ⎞ ω = and T = = 2π or leff = ⎜ ⎟ g leff ω g ⎝ 2π ⎠

• This effective length is the total radius of gyration leff = Ktot

• The total moment of inertia about the pivot point is the product of mass and radius of gyration squared. The parallel axis theorem states that this also is the sum of the component’s moment of inertia about its cg plus its mass times the distance between the pivot and the component’s cg.

2 2 I eff = mK tot = I cg + ml

3. Solving for the component’s moment of inertia about its own cg gives

2 ⎡ 2 ⎤ 2 2 2 2 ⎛ T ⎞ 2 I cg = mK tot − ml = m(K tot − l ) = m(⎢⎜ ⎟ g ⎥ − l ) ⎣⎢⎝ 2π ⎠ ⎦⎥

Page 07 - 19 SFTE Reference Handbook Third Edition 2013

7.5 References

7.5.1 Lawless, Alan R. et al, “Special Topics and Aircraft Subsystems Flight Testing,” Chapter 1, Mass Prop- erties, National Test Pilot School, Mojave, CA, 1999.

7.5.2 Bradfield, Edward N., "Experimental Determination of the Moments of Inertia, Product of Inertia, and Inclination of the Principle Axis of Conventional Aircraft by the Spring Oscillation Method" FTC-TIM- 1001, AFFTC, Edwards AFB, CA, 1971.

7.5.3 Malvestuto, S. F., et al, "Formulas for Additional Mass Corrections to the Moments of Inertia of Air planes" TN 1187, Langley Memorial Aeronautical Laboratory, Langley Field Virginia, 1947.

7.5.4 Lawless, Alan R., "Fixed Wing Flying Qualities Flight Testing" Chapter 7, Equations of Motion, Na tional Test Pilot School Mojave CA, 1998.

Additional Reading

7.5.5 Tanner H.L., "Measurement of the Moments of Inertia of an Airplane by a Simplified Method" NACA2201, Ames Aeronautical Laboratory, Moffet Field, CA, 1950.

7.5.6 Woodward, C.R., et al "Handbook of Instructions for Experimentally Determining the Moments of Iner tia of Aircraft by the Spring Oscillation Method" TB-822-F-2, ASTIA AD97104, Cornell Aeronautical Laboratory, Buffalo, New York, 1955.

Page 07 - 20 SFTE Reference Handbook Third Edition 2013

Section 8 Motion/Vibration Analysis

8.1 Recurring Abbreviations

8.2 First Order Motion 8.2.1 Elements of First Order Motion 8.2.2 First Order Motion Descriptive Parameters 8.2.3 Determining Descriptive Parameter τ

8.3 Second Order Motion 8.3.1 Elements of Second Order Motion 8.3.2 Second Order Motion Descriptive Parameters 8.3.3 Determining Descriptive Parameters

8.4 Complex Plane

8.5 Parameter Conversions

8.6 Vibration Nomograph

8.7 References

Page 08 - 1 SFTE Reference Handbook Third Edition 2013

8.1 Recurring Abbreviations

C 1/x number of cycles to achieve 1/x amplitude D damping D1,D2 peak-to-peak displacement (subsequent) FV final value F(t) forcing function f frequency, cycles/sec = ω/(2π) HCAR half cycle amplitude ratio (i.e., x2/x1, x3/x2, etc.) Im imaginary axis M mass MP peak overshoot Re real axis rms root mean square s1, s2 equation roots of second order T period = 1/f = 2π/ωd (seconds) Td delay time (i.e., time to 50% of FV) Tr rise time (i.e., time from 10% to 90% of FV) Tp time to peak amplitude TPR transient peak ratio Ts settling time (time to settle within x% of FV) T 1/2 time to achieve 1/x amplitude x displacement x1,x2 peak displacements (subsequent) v velocity vo peak velocity 2 .5 ε =ζωn /ωd = ζ/[1‐ζ ] φ phase lag (radians) ζ damping coefficient (non-dimensional) σ damping rate =ζωn = 1/τ τ time constant =1/ζωn ω frequency, radians/sec ωd damped natural frequency (rad/sec) ωn natural frequency (rad/sec)

Page 08 - 2 SFTE Reference Handbook Third Edition 2013

8.2 First Order Motion

Found in classical aircraft roll and spiral modes. Named first-order because the motions are described by mathe- matics using the first derivative of a parameter.

8.2.1 Elements of First Order Motion • Mechanical analogy contains elements of mass, damping and sometimes a forcing function. • Example: Determine the vertical velocity of a diver as she hits the water at 10 ft/s (assume constant body position & neutral buoyancy)

Summing vertical forces dV ∑ Fvert : M + DV = 0 dt dV D + dt = 0 V M dV D ∫ = − ∫ dt V M

− D − D Since D & M are constant lnV = t +C = t + lnc M M −D t Vc = ce M (complementary solution)

V t = 0 =10 ft s ⇒ Apply initial condition ( ) 10 = C

−D t M VT =10e Plot response over time V

10 D M

t

• Exponential rate of decay described by D/M ratio

Page 08 - 3 SFTE Reference Handbook Third Edition 2013

• Example 2: Diver with 20 lb submerged weight releases from zero velocity at top of pool (quiescent condi- tion). Solve using Laplace analysis methods: MV& + DV = 20 (step input) 20 M[]sV()s −V ()0 + DV()s = s D 20 1 sV()s −V ()0 + V()s = M M s ⎛ D ⎞ 1 20 1 20 +V()0 ⎧ 20 ⎫ V()0 V()s s + = +V()0 V ()s = s M = M + ⎜ ⎟ D ⎨ D ⎬ D ⎝ M ⎠ s M s + M ⎩s()s + M ⎭ s + M ⎧ 20 ⎫ A B ⎧ 20 ⎫ A(s+ D ) Bs A(s+ D )+ Bs use partial fraction= M = + M = M + = M ⎨ D ⎬ D ⎨ D ⎬ D D D ⎩s()s+ M ⎭ s s+ M ⎩s ()s+ M ⎭ s()s+ M s()s+ M s()s+ M −D 20 −D −20 20 D 20 let s = ; = B ⇒B = , let s = 0: = A ⇒ A= M M M M M M D ⎧ 20 − 20 ⎫ V V()s = D + D + ()0 , ⎨ D ⎬ D ⎩S s+ M ⎭ s+ M

− D t − D t − D t ⎧20 20 ()M ⎫ ()M 20⎡ ()M ⎤ v()t = ⎨ − e ⎬+ V()0 e since V()0 = 0 then v ()t = 1−e ⎩ D D ⎭ D ⎣⎢ ⎦⎥

−D ⎡ t ⎤ 20 M Note that is the steadystate value i.e. v()t =Vss ⎢1−e ⎥ D ⎣ ⎦

This “force/damping” Vss ratio is merely a scaling D large V M factor for the steady D small state. M t

• Several methods can be used to describe the quickness of convergence toward steady state (i.e., time to 99.999 % of Vss, time to 1/2 Vss). • By convention, we use a % that directly reflects the exponent. • Establish a time constant τ based on D/M.

D 1 − D t − t ≡ so e M = e τ M τ − t −τ when t = τ , then e τ = e τ = e−1 = 0.36788

D −1 so, after τ []()M seconds have elapsed, V = Vss []1− 0.36788 = 63.212%Vss τ = time constant (time for parameter to reach 63% of its steady state value)

Page 08 - 4 SFTE Reference Handbook Third Edition 2013

8.2.2 First Order Motion Descriptive Parameters

x 1.0

t .37 − x()t = e−ζωnt = e τ .05 .02 t τ 3τ 4τ

when t = 0.6931τ: x = e−.6931 = 0.5 (time to half amplitude) when t = τ: x = e−1 = 0.37 when t = 3τ x = e−3 = 0.05 when t = 4τ: x = e−4 = 0.02

• If exponent > 0, then motion is divergent. x(t) = cemt x where m > 0

t

• τ again describes the exponential rate of divergence. • By convention, the “time to double amplitude” (t2) is usually applied as the evaluation metric. m0 • x(t2) = 2x(0) where x(0) = ce

Therefore x2 = 2c 2c = ce mt 2 2 = e mt 2

ln 2 = mt 2 0.6931 ∴Time to double amplitude = t = 0.693τ m 2

Page 08 - 5 SFTE Reference Handbook Third Edition 2013

8.2.3 Determining τ from Step Input Time History

Method #1 x xss τ = time to reach 0.632 xss

0.632 xss

t τ

Method #2

1. Define xss

2. Measure x1 at some time t1 3. Calculate x = 0 .632 ()x +.58x 2 ss ()t1

4. Find t2 corresponding to x2

5. t2 −t1 =τ xss

0 .632 ( x SS + .58 x ( t1) ) = x q

x1

t t1 t2

Method #3 .t2 x xss 1. Pick any time t1. 2. Draw tangent at t1. Slope method 3. Note t2 where tangent intercepts xss. m x(t ) 4. τ = t2-t1 1 t t1

Page 08 - 6 SFTE Reference Handbook Third Edition 2013

Method #4 When X is unknown use SS Δ t τ = ⎛ A1 ⎞ ln ⎜ ⎟ ⎝ A2 ⎠

XSS X3 A2 X2

A1 X1

t2- t1= t3- t2 =Δt

time t1 t2 t3

− Δt Method #5 When Xss is known, use τ = ⎡ X − X ⎤ ln⎢ SS 1 ⎥ ⎣XSS − X 2 ⎦

Linearity check: XSS X Note parameter change be- 3 A X 2 tween even time increments. 2 A X 1 Plot parameter changes vs 1 t - t = t -t =Δt elapsed time on semi-log 2 1 3 2 scale time t1 t2 t3

Slope of line equals τ

Δ p1

Δ p2

Δp Δp 3

Δp4

t0 t1 t2 t3 t4 t5

Page 08 - 7 SFTE Reference Handbook Third Edition 2013

8.3 Second Order Motion

Found in classical aircraft phugoid, Dutch roll and short period modes as well as noise filter and vibration test- ing. Named second-order because the motions are described by mathematics using the second derivative of a parameter.

8.3.1 Elements of Second Order Motion

Mechanical systems have elements of spring, mass, and usually damping. Forcing functions can be included (see illustration).

K = spring stiffness (F/x) x = displacement from equilibrium M = mass K F(t) = forcing function D = damping x M

D F : Mx + Dx + Kx = F (t) F ∑ vert && & (t)

Natural character is observed when system is allowed to move with no external input [F(t) = 0]

∑ Fvert : M&x&+ Dx& + Kx = 0

st st 2 st Apply operator technique: let x = e , x& = se , &x& = s e

⇒ est Ms2 + Ds + K = 0 () Divide out est, since it never equals zero, the characteristic equation remains: D K Ms 2 + Ds + K = 0 or s 2 + s + = 0 M M

The values of s that satisfy this equation are called the roots

s1t s2t [x = c1e + c2e ]

Solve for the roots using the quadratic equation

Page 08 - 8 SFTE Reference Handbook Third Edition 2013

8.3.2 Second Order Motion Descriptive Parameters

Solution (x) calculated as 2

s1t s2t − D K ⎛ D ⎞ x = c1e + c2e where s1, s2 = ± j − ⎜ ⎟ 2M M ⎝ 2M ⎠

Apply Euler’s identity for complex conjugate roots −D ⎛ 2 ⎞ t K ⎛ D ⎞ x = Ae 2M sin⎜ − ⎜ ⎟ t +φ ⎟ ⎜ M 2M ⎟ ⎝ ⎝ ⎠ ⎠

• φ defines the phase shift. • A defines the initial amplitude. • The real part of the root [D/2M] defines the envelope of the motion.

• The imaginary part of the root identifies the damped frequency 2 K ⎛ D ⎞ −⎜ ⎟ of the oscillations, ωd (rad/sec). M ⎝ 2M ⎠

ω = K − D 2 d M ()2M

• If damping is reduced to D = 0 then only [K/M]1/2 remains. This is the undamped or “natural” frequency (ωη). ω ≡ K n M

If D 2 K then D is conisidered to be critical [just enough to prevent oscillations] • ()2M = M

K D = 2M = 2 KM crit M

• For oscillatory motion, actual system damping is typically expressed as a fraction critical damping. Define damping ratio as D D ζ ≡ = Dcrit 2 KM

Page 08 - 9 SFTE Reference Handbook Third Edition 2013

2 K ⎛ D ⎞ Combining ωd ≡ −⎜ ⎟ M ⎝ 2M ⎠ K D with ω ≡ and ζ = n M 2 KM D gives ω ≡ ω 1−ζ 2 and = ζω d n 2M n The values can be substituted to give

D ⎛ 2 ⎞ − t K ⎛ D ⎞ x = Ae 2M sin ⎜ −⎜ ⎟ +φ ⎟ ⎜ M 2M ⎟ ⎝ ⎝ ⎠ ⎠

−ζωnt = Ae sin ()ωd +φ

−ζωnt 2 = Ae sin (ωn 1−ζ +φ) ω ω 1−ζ 1−ζ where φ = tan−1 d = tan−1 n = tan−1 ζωn ωn ζ

−ζωnt 2 x = Ae sin (ωn 1−ζ t +φ ) d φ ω Shift Α Phase Initial Damped Envelope of Motion Amplitude Frequency, x = Ae−ζωnt

x −ζωnt 2 x = Ae sin (ωn 1−ζ t +φ)

t x = Ae−ζωnt

Τ −Α 2π = ωd

Page 08 - 10 SFTE Reference Handbook Third Edition 2013

2 − D K ⎛ D ⎞ 2 Roots s1,2 = ± j − ⎜ ⎟ = −ζωn ± jωn 1−ζ 2M M ⎝ 2M ⎠

Possible Solutions:

D 2 then roots are and system response is if ()2M

k Non-oscillatory x > t m Real and unequal ~ convergent if D > 0 x ~ divergent if D < 0 t

k Non-oscillatory x = m Real and equal ~ convergent if D > 0 ~ divergent if D < 0 t Non-oscillatory k Complex ~ convergent if D > 0 < (purely imaginary if D = 0) m ~ neutral if D = 0 ~ divergent if D < 0

• The various combination of K, M, and D and their effects on system response can be related to damping ratio ζ as follows:

ζ > 1 Real & unequal roots −ω ⎜⎛ζ + ζ 2 −1⎟⎞t −ω ⎜⎛ζ − ζ 2 −1⎟⎞t n ⎝ ⎠ n ⎝ ⎠ exponential, convergent x()t = c1e +c2e

ζ = ± 1 Real & equal roots −ζωnt −ζωnt exponential, conv or div x()t = c1e +c2te

0 < ζ < 1 Complex pair roots −ζω t x()t = Ae n sin (ω t +φ ) sinusoidal, convergent d

ζ = 0 Imaginary pair roots x()t = Asin (ωnt +φ ) sinusoidal, neutral = Acosω t n −ζωnt −1 < ζ < 0 Complex pair roots x()t = Ae sin (ωdt +φ ) sinusoidal, divergent

−ω ⎜⎛ζ + ζ 2 −1⎟⎞t −ω ⎜⎛ζ − ζ 2 −1⎟⎞t ζ < − 1 Real & unequal roots n ⎝ ⎠ n ⎝ ⎠ x()t = c1e +c2e exponential, divergent

Page 08 - 11 SFTE Reference Handbook Third Edition 2013

Damping ratio effect on second order system

Convergent Sinusoid 1.0 (x)t = e−0.3t sin (2.99t), ζ = 0.1, ω = 3 Response of various n second order systems 0.5 to an impulse input. X 0

Second-order sys- -0.5 tems are oscillatory if -1.0 -1 > ζ > 1. 0 5 101520 Time (sec)

Motion typically described by Neutrally Damped Sinusoid 2 ωn and ζ (x)t = cos 3t , ζ = 0

1 T, ωd, ωn and ζ are linked such that X 0 knowledge of any two will yield -1 the other two. -2 0 2 4 6 8 10 Time (sec)

Divergent Sinusoid 400 (x)t = e+0.2t sin t, ζ < 0

200

X 0

-200

-400 0102030 Time (sec)

Page 08 - 12 SFTE Reference Handbook Third Edition 2013

2nd order system response to unit step input for underdamped systems

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9 x 0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 2 3 4 5 6 7 8 9 101112

Normalized Time, ωn t (0 > ζ > 1)

Page 08 - 13 SFTE Reference Handbook Third Edition 2013

Decay rates (for 0 < ζ <1) and Useful Insights

ln x1 ln x1 x2 x2 • Time to decay ζωn = ⇒ Δt = t2 −t1 ζωn 2 1 ln 1 0.6931 Time to amplitude ≡ Δt 1 = = 2 2 ζωn ζωn 10 1 ln 1 2.3026 Time to amplitude ≡ Δt 1 = = 10 10 ζωn ζωn

1 ln x Time to x amplitude ≡ Δt 1 = x ζωn Δt Δtω • # cycles to 1 amplitude # cycles = = d 2 T 2π

1 0.6931ωd 0.6931 ωd 0.1103 # cycles to 2 amplitude ≡ C1 = = = 2 ζωn 2π 2π ζωn ε

1 2.3026 ωd 2.3026 ωd 0.3665 # cycles to 10 amplitude ≡ C 1 = = = 10 ζωn 2π 2π ζωn ε

1 ln x ωd # cycles to amplitude ≡ C1 = x x 2π ζωn • Same analysis applies for exponential growth x x x For same Δt 1 = 2 = 3 etc x2 x3 x4 2 x1 x1 x2 ⎛ x1 ⎞ Therefore = = ⎜ ⎟ ⇒ HCAR = full cycle amp ratio x3 x2 x3 ⎝ x2 ⎠

T ζω 2 π x ζω ()t −t ζω ()Δt ζω () n()2ω HCAR ≡ 1 = e n 2 1 = e n = e n 2 = e d x2

⎛ π ⎞ πζ ζωn ⎜ ⎟ ω 1−ζ2 −ζ2 Note = e ⎝ n ⎠ = e 1 HCAR= f (only) This can be applied in x 2 (ln 1 ) graphical form x1 πζ x2 ⇒ ln = 2 ⇒ ζ = (transient peak ratio 1−ζ 2 x 2 x2 π +()ln 1 method) x2

Page 08 - 14 SFTE Reference Handbook Third Edition 2013

8.3.3 Determining Descriptive Parameters

Time domain metrics

Mp

1.2 x(t) 1.0 0.9 0.8

0.6

0.4

0.2

0.1 0.0 T d Tp Ts time T r

Peak Value, MP: largest value

Final Value, FV: steady state value

Delay Time, Td: 50% of final value

Rise Time, Tr: 10% - 90% of FV

Peak Time, Tp: time to MP

Settling Time, Ts: time to reach some defined % of final value MP−1 % Overshoot, PO: 1 ×100% target value = unity

Page 08 - 15 SFTE Reference Handbook Third Edition 2013

Method #1 Basic Analysis

Note −ζωnt x()t = Ae sin (ωd t + 0 ) []ζω and []ω describe a time history n d

−ζωnt x e x 1 t2 t t1 x2

2π T = ωd

x1 1 2π ln x f = ω = ζω = 2 d d n T T ()t2 − t1

Easily measured values: can use any points on envelope

ln x1 x2 x1 ζω ()t −t T ln x Define ε ≡ n = 2 1 = 2 2π ωd T 2π ()t2 −t1 ln x1 T ζωn x2 where ()t2 −t1 = ⇒ = = ε = ln[]x1 / x2 /π 2 ωd π 2 recall ωd = ωn 1−ζ ζω ζ ζ 2 ε = n = ⇒ ε 2 = ⇒ ε 2 ()1−ζ 2 = ζ 2 ⇒ ε 2 = ζ 2 []1+ε 2 2 2 1−ζ 2 ωn 1−ζ 1−ζ

2 ζω ω ε ω = n = d ζ = n 2 1+ε 2 ζ 1−ζ

x1 ζω ()− ≡ Half - Cycle Amplitude Ratio (HCAR) = e n t2 t1 x2

Page 08 - 16 SFTE Reference Handbook Third Edition 2013

Method #2 Transient Peak Ratio Analysis

1) Measure either D or x distances as shown. 120

110 x D 1 D D 2 Example3 CalculationD D5 x 100 4 1 D1 x5 t 90 x2 D2 80 0 50 100 150 Time (sec)

2) Note ratio of adjacent peak values (transient peak ratios). D2 117 − 86 31 3) Average several TPRs. First TPR : = = = 0.8378 D1 117 − 80 37 4) Use equation to find ζ: D 112 − 86 26 Second TPR : 3 = = = 0.8387 2 D2 117 − 86 31 x1 ()ln x D 112 − 90 22 ζ = 2 Third TPR : 4 = = = 0.8462 2 x 2 D3 112 − 86 26 π +()ln 1 x2 Average TPR = 0.8409

4a) Can use 1.0 D1/D2 or x1/x2 ratios in above equation. 0.9

0.8 4b) In lieu of equation, use adjacent look-up curve to find ζ. 0.7

4c) Time ratio method works better 0.6 with heavy damping. 0.5

0.4

0.3

0.2

0.1

0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00 Damping Ratio, ζ

Page 08 - 17 SFTE Reference Handbook Third Edition 2013

Method #3 Multiple TPR Analysis

0.30

0.28

0.26 0.24 2 m = 1

0.22 3 4 0.20 5 0.18 Damping ratio,6 ζ 0.16 7

0.14 8 9 0.12 10 11 0.10 12 14 13 0.08 18 16 15 19 17 0.06 20 0.04

0.02

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00

Subsidence ratio, x m /x 0

To determine damping ratio

Use the m = 1 line when comparing the next ratio. Δx 2.5 ~ 1 = = 0.56 ζ = 0.16

Δx0 4.5 Δx 1.5 2 = = 0.60 ζ = 0.14 Δx1 2.5

Δx 1.5 ~ Use the m = 2 line for comparing every other 2 = = 0.33 ζ = 0.16 Δx 4.5 peak ratio. 0 Δx 1.0 3 = = ζ = 0.40 0.14 Δx1 2.5

Page 08 - 18 SFTE Reference Handbook Third Edition 2013

Method #4 Time Ratio Analysis

• If the damping ratio is between 0.5 and 1.0 (two or less overshoots), then the time ratio method can be used to determine frequency and damping ratio. Select a peak where the response if free.

• Note times for amplitude to reduce to 73.6%, 40.9%, and 19.9% of the peak value.

1.000 0.736 0.409 199 α time t 1 t 2 t 3

Input Response

• Form the time ratios t2/t1, t3/t1, and [t3-t2]/[t2-t1]

• Enter the next figure at the time ratio side to find ζ for each time ratio.

• This technique is valid if the system transfer function has no zeros.

• If recorded measurements are not available and if the number of overshoots is between 2 and 6, then

7−#overshoots ζ ≈ 10

Page 08 - 19 SFTE Reference Handbook Third Edition 2013

Time Ratio, Δt3 ,,Δt2 Δt3 − Δt2 Δt1 Δt1 Δt2 − Δt1 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.9 0.8 0.7 0.6 2 1 2 1 t t 3 t t Δ Δ t 3 1 2 1 t t t − t − Δ 2 3 Δ Δ Δ Δ Δ Δ t t n n n Δ Δ ω ω ω

ζ Damping Ratio, Damping 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.9 0.8 0.7 0.6

Frequency Time Products (ωnΔt3), (ωnΔt2), (ωnΔt1)

Page 08 - 20 SFTE Reference Handbook Third Edition 2013

Method #5 Maximum Slope Analysis

Maximum Slope Tanget Line Δx 1 Peak

Δx Peak

t Δt

4.8

2.0 4.4

1.8 4.0

1.6 t 3.6 Δ

n 1 .4 ω 3.2 , ζ 1 . io 2 at R 2.8 g 1. n 0 pi 2.4 am 0 . 8 D

0 . 6 0 0 . 2.0 0 4 . . 2 0 0.08 0.12 0.16 0.20 0.24 0.28 0.32 Δx1/Δx

• Calcu- lation of ζ somewhat sensitive to Δx1 measurement

ωnΔt • ωn = Δ t not too sensitive to Δx1

• Best if 0.5 ≤ ζ ≤ 1.4

X pK • Initial overshoot approximation: let (step inputs only) K≡ −1 X ss 1 2 ⎡ ⎛ −1 ⎞ ⎤ 2 ⎢ ⎜ lnK ⎟ ⎥ π ζ = ⎢ ⎝ ⎠ ⎥ 2 ⎢ ⎛ −1 ⎞ ⎥ ⎢1+⎜ lnK ⎟ ⎥ ⎣⎢ ⎝ π ⎠ ⎦⎥

Page 08 - 21 SFTE Reference Handbook Third Edition 2013

Method #6 Separated Real Root Analysis (when ζ>1)

1) Determine several steady state ΔX(t) values from time history

X(t) ΔX(t)

t t Response o

2) Plot ΔX vs t on semi-log scale

ΔX(t) ' ΔX0 ΔX ΔX 0 0 slope = 1/τ Scale) (Log ΔX Semilog1 Plot 1 ΔX2

t t t t 0 1 2

3) After the faster root has decayed, the semi-log plot will be a straight line whose slope determines the slower root (1/τ1)

t −t τ = 1 2 1 ⎛ Δχ ⎞ ln⎜ 1 ⎟ ⎜ Δχ ⎟ ⎝ 2 ⎠

4) Determine by extrapolating the straight line portion of the response to establish the values

⎛ 1 ⎞ Δ χ & Δ χ ' ⎜ ⎟ 0 0 ⎝τ 2 ⎠

⎛ 1 1 ⎞ ⎛ 1 1 ⎞ - ⎜ + ⎟ - ⎜ + ⎟ ⎛ Δχ0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 1 ⎝τ1 τ 2 ⎠ ⎝τ1 τ 2 ⎠ τ 2= τ1 ' ⎜ Δχ ⎟ ωn = and ζ = = ⎝ 0 ⎠ τ τ 2ω 1 1 1 2 n 2 τ1 τ 2

Page 08 - 22 SFTE Reference Handbook Third Edition 2013

Method #7 Modified Separated Real Root Analysis

• Method #6 is sensitive to errors in determining steady state values • Alternate method is to avoid need for steady state value • Define ΔX(t) ≡ [ x(t+ ΔT) -x(t)] where ΔT is a time increment 1) From time history, measure ΔX values according to definition

X

ΔX (t3){ ΔT = (t1-to) = (t2-t1), etc. ΔX (t2){ ΔX (t ){ 1 to t1 t2 t3 t4 t5 t6 t

2) Plot ΔX (t) vs time on semi-log scale

ΔX(t0) ' ΔX ΔX0 0

ΔX(t3) ΔX(t )

X(t) 4

Δ ΔX(t5) ΔX(t6) Log

t0 t1 t2 t3 t4 t5 t6 t

3) Use previous method to determine roots and characteristics

• Gross error will result if ζ is actually <1

• If ζ is near 1, check results using time ratio or slope method

Page 08 - 23 SFTE Reference Handbook Third Edition 2013 Method # 8 Frequency Sweep Analysis

Determine ωn and ζ using sinusoidal inputs. • This “forced response” method most useful when damping is heavy. • For a second order system, output/input amplitude ratio and phase shift are a function of input frequency.

180° 0.05 0.15 ζ = 0 φ 0.375 3 0.05 ζ = 1.0

x 90°

K 0.10 A

F 0.15 0.25 Angle, Phase 2 0 1 2 3 4 5 0.375 Frequency Ratio ω 0.50 ω n

1

Factor Amplification

0 12345 ω Frequency Ratio ωn

• Amplitude ratio peaks at “resonant” frequency, ωr.

• Resonant peaks increase as ζ decreases below 0.707. • Peak amplitude ratio “rolls off” as ζ increases above 0.707. • Resonant frequency approaches natural fre quency as damping decreases: 2 .5 ωr = ωn[1 − 2ζ ]

• Phase shift = 90° if excited at ωn, regardless of damping ratio.

ζ = 0.5(ω2 – ω1)/ωn

Page 08 - 24 SFTE Reference Handbook Third Edition 2013

Frequency Sweep Analysis (continued)

1. Using sinusoidal inputs excite system @ ω near ωn

output 2. Measure phase lag (φ) of input

Input

Output 45°

3. Excite system @ another ω near ωn

4. Again Measure phase lag φ

Input 120°

Output

5. Plot φ vs input frequency

ω 0° n ω φ -45°

-90° -135°

-180°

6. ωn occurs at φ = 90°

Page 08 - 25 SFTE Reference Handbook Third Edition 2013

Linearity Check /Accuracy Improvement

1) On semi-log scale, plot ratio of initial amplitude to subsequent peak amplitudes at each half cycle (points a-e).

2) Fair straight line (f) through these points.

10

a m=0 b m=1

c

m=2 d m=3

1 e m=4 TPR f Double or Half Amplitudes Half or Double g

T/2 T 3T/2 2T

3) Draw line (g) parallel (f) intercepting the ordinate at TPR=1

4) Average TPR occurs at T/2 on line g

Page 08 - 26 SFTE Reference Handbook Third Edition 2013

8.4 Complex Plane

Begin with sum of forces in spring-mass-damper example

M&x&+ Dx& + Kx = f (t) let x = est , find transient solution

Ms2 + Ds + K est = 0 () Apply quadratic equation to −D D 2 K solve for roots s1,2 = ± ()− 2M 2M M

Recall previous analogy 2 s1,2 = −ζωn ± jωn 1−ζ

m1 Imaginary 1 τ = ζω ζ = cos θ n ω 1− ζ2 = ω − ζωn n d Real 1 2 − τ − ω 1− ζ = −ω 1 n d − ± jωd ω τ n m2

Location of Roots on Complex Plane 1. Line of constant damping ratio ζ − varying C1/n and ωn 2. Line of constant ωn − varying ζ 3. Line of constant ωd and period (T) 4. Line of constant real part (ζωn) and time to damp (T1/n)

1 4

3 Imaginary

σ = ζωn = 1/τ = damping rate

2 Real

Page 08 - 27 SFTE Reference Handbook Third Edition 2013

Sample second order root plots and corresponding time histories

(time histories represent trends only)

Examples of “two real roots”

Im x

Re t m1t m2t x()t = c1e + c2 e overdamped

Im x

Re t

m1t m2t x()t = c1e + c2e

Im x

Re t −ζωnt −ζωnt x()t = c1e + c2te critically damped

Im x

Re t mt mt x()t = c1e + c2 e

Page 08 - 28 SFTE Reference Handbook Third Edition 2013

More sample second order root plots and corresponding time histories

Examples of “imaginary roots”

x Im

Re t

x()t = A sin (ω n t + φ ) undamped

Im x

Re t

− ζω nt x()t = Ae sin (ωd t + φ ) lightly damped Im x

Re t

− ζω nt x()t = Ae sin (ωd t + φ ) moderately damped Im x

Re t

ζω nt x()t = Ae sin (ω d t + φ ) negatively damped

Page 08 - 29 SFTE Reference Handbook Third Edition 2013

8.5 Parameter Conversions

For conversion of accelerometer measurements. • For magnitude conversion substitute 2πf for jω. • Assumes linear spectra. • Conversion factor should be squared for power spectra.

Acceleration to velocity to convert from to multiply by ft/s2 rms ft/s rms 1/jω ft/s2 rms in/s rms 12/jω ft/s2 rms in/s peak 16.97/jω g rms in/s rms 386/jω g rms in/s peak 545.8/jω m/s2 rms mm/s rms 1000/jω m/s2 rms mm/s peak 1414/jω g rms mm/s rms 9806/jω g rms mm/s peak 13865.7/jω

Acceleration to Displacement to convert from to multiply by ft/s2 rms in rms 12/(jω)2 ft/s2 rms in p-p 33.9/(jω)2 ft/s2 rms mil p-p 33.9 E 03/(jω)2 g rms in rms 386/(jω)2 g rms in p-p 1091.6 E 03/(jω)2 g rms mil p-p 1091.6 E 03/(jω)2 m/s2 rms mm rms 1000/(jω)2 m/s2 rms mm p-p 2828/(jω)2 m/s2 rms micron p-p 2828 E 03/(jω)2

E= engineering exponent (x10 __) g = 32.174 ft/sec2 in= inches mil = thousandths of an inch mm = milimeters p-p = peak-to-peak rms = root mean square

Page 08 - 30 SFTE Reference Handbook Third Edition 2013

8.6 Vibration Nomograph

Page 08 - 31 SFTE Reference Handbook Third Edition 2013

8.7 References

8.7.1 Lawless, Alan R., Math and Physics for Flight Testers, “Chapter 9, Motion Analysis,” National Test Pilot School, Mojave CA, 1999.

8.7.2 Ward, Don, Introduction to Flight Testing, Texas A&M, Elsevier, 1993.

8.7.3 Lang, George F., Understanding Vibration Measurements, Application Note 9, Rockland Scientific Corporation, Rockleigh, New Jersey, December 1978.

8.7.4 The Fundamentals of Modal Testing, Application Note 243-3, Hewlett-Packard Company,

Additional Reading

Hartog, J.P. Den, Mechanical Vibrations, Dover Publications, New York, New York, 1984.

Jacobsen, Ludik S. and Ayre, Robert S., Engineering Vibrations, McGraw-Hill Book Company, New York, New York, 1958.

Meirovitch, Leonard, Elements of Vibration Analysis, McGraw-Hill Book Company, New York, New York, 1986.

Meirovitch, Leonard, Analytical Methods in Vibrations, Macmillan Publishing Company, New York, New York, 1967.

Myklestad, N.O., Vibration Analysis, McGraw-Hill Book Company, New York, New York, 1944.

Page 08 - 32 SFTE Reference Handbook Third Edition 2013

Section 9 Material Strength

9.1 Terminology

9.2 Material Stress and Strain

9.3 V-n Diagram

9.4 Strain Gauges

9.5 References

Page 09 - 1 SFTE Reference Handbook Third Edition 2013

Section 9 Abbreviations

A cross-sectional area (ft2) DLL design load limit E modulus of elasticity or Young’s Modulus (lb/ft2) e strain (non-dimensional) EK gage factor GW gross weight KU effective gust velocity (ft/sec) L lift force L length (ft) Nzb normal load factor, along aircraft z-axis P applied load (lb) R unstrained resistance ΔR change in resistance due to load S wing area (ft2) V flight speed Vs stall speed Ve equivalent airspeed W aircraft weight W/S wing loading ν Poisson’s ratio σ stress (lb/ft2) σ air density (slugs/ft3)

Page 09 - 2 SFTE Reference Handbook Third Edition 2013

9.1 Loads Terminology

Annealing A heat treatment that eliminates the effects of cold working.

Brittleness Measure of a material's lack of ductility (by one definition breakage at five percent or less strain implies brittleness.

Creep rate The rate at which a material continues to stretch when stress is applied at high temperature.

Cold Working Deformation of a metal below its recrystallization temp., thereby strengthening and reshaping it.

Design Load Maximum loads expected in normal service. Limit

Ductility Ability of a material to deform without breaking.

Durability Ability to resist cracking, corrosion, thermal degradation, delamination, wear, and the effects of foreign object damage over time.

Elastic Deformation of the material that is recovered when the applied load is removed. Deformation

Elasticity Ability of a material to return to its undeformed shape after all loads have been removed.

Endurance Limit The stress below which a material will not fail in a fatigue test.

Factor of Safety Ratio of the predicted failure stress to the maximum stress anticipated in normal operation (DLL). For aircraft, the Factor of Safety is typically 1.5 DLL.

Fatigue The failure of a material when subjected to repeated loads less than the ultimate sustainable load. This effect is presented in an S-N diagram such as

60

40 Stress (S) 20

0 1 10 102 103 104 105 106 107 108 Cycles to Failure (N)

Page 09 - 3 SFTE Reference Handbook Third Edition 2013

Fatigue life The number of cycles at a particular stress before a material fails by fatigue.

Hardness Resistance to plastic deformation resulting from impact loads.

Impact Energy The energy required to fracture a specimen when the load is suddenly applied.

Limit Stress The maximum stress where the Modulus of Elasticity remains constant (proportional limit).

Margin of Safety Any load-bearing capability greater that the ultimate load, calculated as failure load as a factor of DLL - 1 1.5 DLL

Notch Sensitivity Measure the effect of a notch on impact energy

Plastic Permanent deformation of a material applied load. Plasticity Material deformation charac Deformation teristics beyond its elastic limit.

Resilience A measure of the amount of energy a material can absorb elastically in a unit volume of the material. Rupture time The time required for a specimen to fail by creep at a particular temperature and stress.

Stiffness A qualitative of the elastic deformation produced.

Strain (e) The deformation of a material under an applied load.

Strength Ability to withstand external loads without failure.

Stress (σ) The ability of a material to react a force distributed over some area.

Thermal stress Stress resulting from expansion (strain) of a material subjected to heating.

Tempering A low-temp. heat treatment which reduces hardness.

Tensile strength The stress that corresponds to the maximum load in a tensile test.

Toughness Total energy absorbed before failure occurs (area under the stress-strain curve).

Transition Temperature The temperature below which a material behaves in a brittle manner in an impact test.

True Strain The actual strain produces when a load is applied.

Ultimate Stress The stress point at which additional load cannot be reacted.

Wing Loading Aircraft weight per wing area, W/S, a ready measure of air loads for steady level flight.

Yield Stress The stress applied to a material that just causes permanent plastic deformation.

Page 09 - 4 SFTE Reference Handbook Third Edition 2013

9.2 Material Stress & Strain

Stress (σ) is the ability of a material to react a force distributed over some area. In the simple axial load case this can be presented as σ = P/A where P = the applied axial load A = cross-sectional area over which the load is applied

Strain (e) is the deformation of a material under an applied load. In the basic form this can be presented as

e = ΔL/L where ΔL is the change in dimension due to some load, and L is the original dimension

The stress-strain relationship is linear (proportional) for a large percentage of the applied load to the maximum, as expressed by the Modulus of Elasticity (Young’s Modulus)

E = σ/e

A typical stress & strain relationship for a material is illustrated as

ultimate stress fracture stress ) ~ psi) ~ Plastic Range yield stress σ proportional limit

Stress ( Stress E

Elastic Range maximum strain

Strain (e) ~ inches/inches

Page 09 - 5 SFTE Reference Handbook Third Edition 2013

9.3 V-n Diagram

Flight Path Normal Load Factor (Nzw) can be expressed during level flight, as

Nzw= 1/cosφ = L/W

where CL = lift coefficient Fn= net thrust L = lift force = wing lift + thrust lift = CLqS +FnsinαF q = dynamic pressure S = wing area W = gross weight αF = incidence angle between thrust line and relative wind φ = angle of bank

Body Axis Normal Load Factor (Nzb) is calculated as

Nzb = [Nzw-Nxbsinα]/cosα

where Nzb = load factor along aircraft body x-axis α = angle of attack

For the simplified case of negligible thrust lift, the maximum achievable Nzb at any flight speed can be calculated as

2 Nzb = (V/Vs)

where both speed must be the same units (i.e., true, equivalent, calibrated) V = flight airspeed Vs = stall speed

Page 09 - 6 SFTE Reference Handbook Third Edition 2013

A general normal load flight envelope (V-n diagram) would appear as • The envelope typically varies with: asymmetric loading; aircraft configuration; for air loads other than along the normal axis; and other structural, system, and safety considerations.

STRUCTURAL FAILURE AREA 12 POSITIVE ULTIMATE 11 11.25 STALL REGION 10 STRUCTURAL DAMAGE AREA 9 8 POSITIVE LIMIT 7.5 7 UNAVAILABLE LIFT AREA 6 AREA OF

5 STRUCTURAL VH MAXIMUM THE DAMAGE 4

z V OR FAILURE POSITIVE LIFT L 3 CAPABILITY ENVELOPE

2 LIMIT AIRSPEED 1 Indicated Airspeed ~ knots 575 KNOTS 0 n ~ Factor Load 200 300 400 500 600 -1 -2 UNAVAILABLE LIFT AREA NEGATIVE LIMIT -3 -3.0 MAXIMUM STRUCTURAL DAMAGE AREA -4 NEGATIVE LIFT NEGATIVE ULTIMATE -4.5 -5 CAPABILITY STRUCTURAL FAILURE AREA STALL REGION

• It is frequently desirable to correct measured (test) Nzb data to a standard weight or design gross weight (GW) using the relationship

Nzb= (test Nzb)(Wt/Ws)

where Wt = test weight Ws = standard weight

• The increase in load factor due to a vertical gust (Δn) is calculated as

Δn = 0.115mVe(KU)/(W/S)

where m = slope of CLα curve Ve = equivalent airspeed (knots) KU = effective gust velocity (fps) W/S = wing loading (psf)

Page 09 - 7 SFTE Reference Handbook Third Edition 2013

9.4 Strain Gages

The three strain gage configurations most commonly used are

Unixial GageBi-axial Gage Rosette Gage

Strain (e) is measured using the electrical resistance measured via the strain gage in a material subject to load. For the uniaxial gauge

K = (ΔR/R)/e

where K = gage factor (provided by manufacturer) R = unstrained resistance ΔR = change in resistance due to load (+ΔR for tension)

• For the bi-axial gage oriented coincident with the principal axes (maximum strain), each leg of the gage is analyzed as a uniaxial gage using the above equation for the principle strains. The associated stresses are

2 σmax = E(emax + υemin)/(1 -υ ) 2 σmin = E(emin + υe max)/(1 - υ ) where emax and emin are the measured principal strains in the appropriate legs of the bi-axial gage, E is the Young’s Modulus of the material, ν is Poisson’s ratio for the material. (ratio of compression and tension strains)

Page 09 - 8 SFTE Reference Handbook Third Edition 2013

• For the Rosette gage, the principal strains and stresses are derived as

2 2 emax, min = 0.5(e a + ec) ± 0.5 (e a − ec) + (2e b − ea − ec)

2 2 σ m ax, min = E/2 [(e a + e c)/(1 − u) ] ± (e a − e c) + (2e b − e a − e c) /(1 − u) where e’s denote the strains in each of the three legs of the Rosette (+ is used for the maximum and - for the minimum).

To accurately measure the very small resistance changes in a strain gage, a Wheatstone Bridge is typically used

A B

G A = active or strain-measuring gage B = temperature compensating (dummy) gage D C C = D = internal resistance in instrument G = galvanometer

Page 09 - 9 SFTE Reference Handbook Third Edition 2013

9.5 References

9.1 Dole, Charles E., Fundamentals of Aircraft Material Factors, University of Southern California, Los Angeles, California, 1987.

9.2 Norton, William J., Structures Flight Test Handbook, AFFTC-TIH-90-001, Air Force Flight Test Cen ter, Edwards AFB, California, November 1990.

Additional Reading

Military Specification Airplane Strength and Rigidity - General Specification, MIL-A-8860.

Military Specification Airplane Strength and Rigidity, Sonic Fatigue, MIL-A-008893.

Page 09 - 10 SFTE Reference Handbook Third Edition 2013

Section 10 Reciprocating Engines

10.1 Recurring Abbreviations and Terminology

10.2 Reciprocating Engine Modeling Graphic Power Model Analytic Power Model Fuel Flow Model

10.3 Reciprocating Engine Power Standardization Partial Throttle Standardization Full Throttle Standardization

10.4 FAA Approved Engine Temperature Corrections

10.5 References

Page 10 - 1 SFTE Reference Handbook Third Edition 2013

10.1 Recurring Abbreviations and Terminology (references 10.5.1-10.5.5) Abbreviations BHP brake horsepower (measured at engine crankshaft) BHPs brake horsepower at standard conditions BHPt brake horsepower at test conditions BHP alt brake horsepower at maximum altitude where test RPM and MP can be maintained. BHPslmax maximum brake horsepower at standard sea level conditions (for any given RPM) BHPc chart brake horsepower BSFC brake specific fuel consumption (fuel flow/horsepower/ hour) C manifold pressure correction factor HP horsepower (= 550ft-lb/sec) M freestream Mach number MP manifold pressure, also MAP SHP shaft horsepower (measured at propeller shaft) P power output [ft-lb/sec or HP] Pa ambient pressure Pts standard day total (ram) pressure Ptt test day total (ram) pressure Q torque [ft-lbs] q dynamic pressure RPM revolutions per minute Tct test day carburetor temperature (absolute) Tcs standard day carburetor temperature (absolute) Tas standard day ambient temperature (absolute) Tat test day ambient temperature (absolute) VT freestream true velocity ΔBHPcat change in brake horsepower due to carb. temp. change ΔBHPmp change in brake horsepower due to manifold pressure change ΔMPΔt change in manifold pressure due to temperature change Δt change between test and standard temperature (Tat-Tas) ηr carburetor ram inlet efficiency ρa ambient density In Hg inches of mercury (manifold pressure)

Page 10 - 2 SFTE Reference Handbook Third Edition 2013

Terminology Bore Piston diameter Critical Altitude The altitude at which a supercharged (or turbocharged) engine can no longer: a) maintain sea level manifold pressure, or b) maximum allowable horsepower. Detonation An operating condition where combinations of excessive temperature, high manifold pressure, and l ow RPM cause explosive fuel burn, large internal pressure pulses, and subsequent engine damage. Displacement Total volume swept by all cylinders, measured in either cubic inches or liters. Manifold Pressure Pressure of fuel-air mixture passing through intake manifold, typically measured in absolute gauge pressure (inches of mercury or lb/in2). Mixture Ratio Ratio of [fuel weight/air weight] passing through the intake manifold. • This ratio must be between .05 and .125 to burn. • Best power typically occurs at mixture ratio of 0.075 to 0.08. • Best economy typically occurs at a ratio of .0625 • To provide sufficient cooling, the mixture ratio is usually greatly increased from best economy when operating at very high or very low power settings (a.k.a. auto rich). Reduction gear Gearing between the engine crankshaft and propeller shaft that reduces the rotation speed going to the propeller. Stroke Linear distance traveled by piston. Supercharger A mechanically driven compressor that boosts the ambient air pressure to provide the engine with higher power output. Turbocharger Also known as a turbo supercharger, it is similar to a super charger except that the compressor is driven by engine exhaust pressure.

Page 10 - 3 SFTE Reference Handbook Third Edition 2013

10.2 Reciprocating Engine Modeling (ref 10.5.3)

For any given engine design, power output is primarily a function of the product of engine speed (RPM) and in- take manifold pressure (MP). Smaller but significant effects are due to the fuel/air density (ρfa) and exhaust back pressure (which is essentially ambient pressure, Pa). Less significant effects are due to the condition of the engine itself and include such factors as ignition quality & timing, piston ring leakage, fuel grade, and oil viscos- ity.

Engine models have various levels of sophistication which can account for the four most significant factors listed above. Theses models can be presented graphically or analytically. Figure 10.2a shows a typical graphic power model for determining reciprocating engine BHP.

1) The left-hand chart shows the fundamental relation between BHP and the product of RPM and MP. En- ter with MP and RPM to obtain point “B” and the associated “base brake horse power” (BHPB) at sea level standard day pressure and temperature.

1 2) Transfer this BHPB value to point “B ” on the ordinate of the right hand chart. 3) Enter the right hand chart with the same MP & RPM to obtain point “A” and the associated brake horse- power at altitude (BHPA) .

4) Connect points B1 and A with a straight line.

5) Enter the abscissa at the test pressure altitude, locate point “C”, and read the corresponding “chart horse- power” (BHPC). BHPC. is the sea level power corrected to the reduced back pressure con ditions at altitude. It does not account for non-standard temperatures.

0 6) To correct for non-standard air temperature, subtract 1% from BHPC for each 6 C warmer than test altitude 0 standard temperature. Conversely, add 1% to BHPC for each 6 C cooler than standard. For convenience, the lower right hand chart of Figure 10.2a illustrates standard temperature as a function of pressure altitude.

Page 10 - 4

SFTE Reference Handbook Third Edition 2013

34000

32000

30000

28000

26000

24000

22000

20000

18000

14

16000

16 14000 12000

18 10000

20 8000 Pressure Altitude, (ft) M

ltitude Performance ltitude P

R

22 0 0 0 0 6000 0 0 0 0 0 0 0 0 0 8 2 8 2 6 4 2 2 2 2 1

24 A 4000

C 2000 26 Sea Level Sea B 28 0

40

s ) ) T STD ALT Temp °F ( °F Temp -40 ALT STD 220 200 180 160 140 120 100 80

220 200 180 160 140 120 100

0

0

M

0 (in. Hg.)

0 0

0

P

0

0 0 0

8

R 2

4

0 2

1

2 2 6 0

2 0

8

2

e

s

B i

ru C Sea Level Performance A

18 20 22 24 26 28 30

ABS, Dry Manifold Pressure Dry ManifoldABS, Pressure Brake Horsepower Brake

Figure 10.2a Engine Performance Chart for Continental IO360D

Page 10 - 5 SFTE Reference Handbook Third Edition 2013

An analytic power model of a reciprocating engine should match the graphic model. Its principle application is in automating the power determination process rather than manually performing graphic lookups.

1) The left hand chart shows that BHPB is a linear function of MP, but it is not necessarily a linear function of RPM. Extrapolating the RPM curves towards low manifold pressure illustrates their convergence to a common point. This left hand chart can be summarized with the equation

2 BHP B = [a(RPM ) + b(RPM ) + c][ MP − e] + d a through e are determined from the graph or from separate engine tests.

2) In a similar fashion, calculate 2 BHP A = [a(RPM ) + b(RPM ) + c][ MP ]

3) Based on the direct relation between available power and density, calculate chart horsepower as

BHP = BHP [BHP − BHP ][(1− σ ) /(1− σ )] C B A B A D where σD is the standard atmosphere density ratio at the operating pressure altitude (Hc). For convenience, this calculation is presented below for flight in the tropopause

−6 4.2558 σ D = [1 − 6.876 x10 H c ] (Hc in feet)

σA is the density ratio corresponding to point A and is calculated as

BHP A σ A = .117 + BHP sl max where BHP is the full-throttle sea level power at the RPM in question. This value is located towards the right side of the BHPB chart.

4) The final step in determining test day power (BHPt) is to correct for non-standard ambient absolute tempera- ture (Ta) T BHP = BHP as t C T at where Tas is the standard absolute ambient temperature at the test altitude, and, below the tropopause, is calculat- ed as Tas = 288.15-.0019812Hc .

Page 10 - 6 SFTE Reference Handbook Third Edition 2013

The fuel flow model is centered around the brake specific fuel consumption (BSFC) defined as

fuel flow ⎛ lb/ hr ⎞ BSFC≡ ⎜ ⎟ power ⎝ BHP ⎠

Figure 10.2b shows the basic effect of RPM & BHP on BSFC (ref 10.5.4). • At any given BHP, operating at lower RPMs reduces mechanical friction and therefore BSFC. • At any given RPM, operating at very low BHP increases the percentage of piston work overcoming friction and therefore increases BSFC. • Operation at high BHP also increases BSFC, but this is due to the fuel enrichment required to prevent deto- nation at high loads.

1.05

0.90 2700 RPM

0.75

0.60 2200

0.45 1800

0.30

(lbs/BHP-hr)BSFC, 0.15

0 100 500 1000 1500 2000

Brake Horsepower, BHP

Figure 10.2b Effect of RPM and BHP on BSFC

The same effects can be modeled as shown in Figure 10.2c (ref 10.5.5).

• Not shown on these figures are the possible altitude and temperature effects. Flight testers rarely need to validate these models throughout the engine’s working range. Instead, testers typically evaluate BSFC only at the combinations of RPM and MP recommended by the manufacturer to give the desired power output.

Page 10 - 7

SFTE Reference Handbook Third Edition 2013

(supercharged engine) (load) engine) (supercharged Percent Normal Rated Power Rated Normal Percent

0 8 6 90 80 70 60 50 40 30 . 100 0 0 6 6 . 0 0 4 6 . 0 0 2 6 .

0 0

0.690 BSFC 0 6 . 0 0 8 5 . 0 0 6 5 . 0 0 4 5 .

0

0

2

5

.

0

0

0

5

.

0

r

e

w

o

p

e e

l

ab

l

i

a PistonMean Speed, (hundreds of feet per minute)

0

av 8

4 0

.

6

m

0

4

0 u

.

5

0

m 4

. i 0

0 4

4 ax

.

0 M 14 16 18 20 22 24 26 13 15 17 19 21 23 25

Figure 10.2c BSFC Curves for a Typical Supercharged Airplane Engine

Page 10 - 8 SFTE Reference Handbook Third Edition 2013

10.3 Reciprocating Engine Power Standardization (ref 10.5.4)

Correcting from test day to standard day power available uses one of several methods, depending on the test conditions. Some useful insights are summarized below. • In all cases, test day RPM must equal standard day RPM. There are no corrections if this requirement is vio- lated. • The effect of density on power output at wide open throttle has been empirically shown to be (ref. 10.5.6) BHP = BHP (1.1324σ − .1324 ) alt sl max

• The above correction is not typically applied to test data since test and standard day pressure altitudes (Hc) are usually the same. Instead, most standardization requirements center around correcting to standard tem- perature. • Engine power is actually related to the carburetor air temperature. The difference between test and standard day carburetor temperature equals the difference between test and standard day ambient temperature (Tct-Tcs = Tat-Tas). • With proper design, most of the freestream dynamic pressure (q) is converted into additional pressure at the carburetor and is known as “ram” effect. Above the critical altitude (where full throttle operation does not generate maximum manifold pressure), ram effect can be used to increase manifold pressure and therefore power output.

The different power standardization methods are described below.

Partial Throttle Standardization. If the test engine is set at some partial throttle setting to achieve a specific manifold pressure, then the same MP should be achievable on standard day with a slightly different throttle set- ting. Calculate standard day power (BHPs) as

n ⎛ T ⎞ BHPs = BHP ⎜ ct ⎟ t ⎜ T ⎟ ⎝ cs ⎠

Tcs is the standard day carburetor absolute temperature = Tas-Tat +Tct n is the power exponent, usually = 0.5

This correction does not address changes in back pressure, so Hc test = Hc std std

Page 10 - 9 SFTE Reference Handbook Third Edition 2013

Full Throttle Standardization applies only if test and standard pressure altitude are equal. If the engine is op- erated full throttle on a test day, then the change in ambient temperature would generate a power change due to carburetor air temperature (ΔBHPcat) and manifold pressure changes (ΔBHPmp). Calculate standard day power (BHPs) as BHP = BHP+ΔBHP +ΔBHP s t cat mp

The first correction is another form of the previous constant MP correction

n ⎡⎛ T ⎞ ⎤ ⎢⎜ ct ⎟ ⎥ ΔBHPcat = BHPt ⎜ ⎟ −1 ⎢ T ⎥ ⎣⎝ cs ⎠ ⎦

The second (manifold pressure) correction accounts for two effects: 1) For supercharged engines, correct for the change in pressure ratio of the supercharger due to inlet temperature changes. 2) For all engines operating below maximum MP, correct for the change in air inlet ram pressure ratio due to flight Mach number changes. ⎛ MP ⎞ ΔBHP = BHP⎜ s −1⎟ mp t ⎜ MP ⎟ ⎝ t ⎠

MPt is the test manifold pressure. MPs is the manifold pressure corrected to standard temperature and flight Mach number:

P ts MPs = MPΔt Ptt

MPΔt is the correction of manifold pressure due to changes in ambient temperature and is approximated as

ΔMPΔt = MPt CΔt

C is a constant depending upon the pressure ratio (P2/P1), carburetor air inlet temperature, and whether or not the fuel is vaporized during process. • If only the air is compressed, or if the inlet temperature is measured after fuel vaporization, then determine C using Figure 10.3a . • If the fuel is injected after the temperature is taken but before the charge is compressed, then determine C using Figure 10.3b. • By use of Figures 10.3a and 10.3b, any combination of induction processes for air only or for a fuel air mix- ture may be evaluated.

Page 10 - 10 SFTE Reference Handbook Third Edition 2013

Δt is the difference between test and standard day carburetor air temperature and was previously described as the change in ambient air temperature Δt = Tat-Tas

Pts/Ptt is the ratio between standard and test day total (ram) inlet pressures at the standard and test Mach num- bers. The first step in determining this ratio is to recognize

Pts P P ts = a P Ptt tt Pa

Pa is the pressure altitude and must be the same for test and standard days. Calculate Ptt/Pa using test Mach number and the equation

P 3.5 t 2 = η r [(1+ .2M ) −1]+1 Pa

ηr is the carburetor inlet ram efficiency and is usually between 0.7 and 0.75. A more exact value may be calcu- lated as

Pt (actual ) − Pa η r = Pt (theoretica l) − Pa

Calculate Pts /Pa using the same equations and standard Mach number. • This last calculation may be iterative because standard Mach number cannot be exactly determined from the drag polar until power output is known. • This correction is not normally made unless the flight Mach number is above 0.6 and the power change causes a speed change of more than 3 knots. • To get a feel for the dynamic pressure change (and therefore ram effect change) due to Mach number change, recall

⎡ lb ⎤ 1 2 2 q⎢ 2 ⎥ = ρ aVT = 1481δM ⎣ ft ⎦ 2

The final standard day power curves are presented in a form similar to that shown in Figure 10.3c

Page 10 - 11 SFTE Reference Handbook Third Edition 2013

0.007 1.43

Carburetor Air Temperature 0.006 1.67 -50°

0.005 0° 2.00

50° 0.004 2.50

0.003 3.33 Correction Factor, C Factor, Correction °C For 1% Correction°C 0.002 5.00 Note: This chart may be used where air only is compressed or 0.001 where inlet temperature is 10.00 measured after fuel vaporization.

0 1.02.03.04.05.06.07.08.0

Pressure Ratio, (P2/P1) Figure 10.3a - Manifold Pressure Correction When Temperature is Measured After Fuel Vaporization

0.005 2.00 Carburetor Air Temperature -50°C 0.004 2.50 0°C

0.003 50°C 3.33

0.002 5.00 Correction Factor, C Factor, Correction

Note: This chart may be used °C for 1% Correction 0.001 only where fuel is injected after 10.00 temperature is taken and before charge is compressed 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Pressure Ratio, P2/P1 Figure 10.3b—Manifold Pressure Correction When Temperature is Measured Before Fuel Injection

Page 10 - 12 SFTE Reference Handbook Third Edition 2013

30

20

10 PM R 0 0 M 0

(°C) 5 2 P cs , R t g 0 ”H 00 0 2 -10 4 g, H 0” 3 -20

-30 40”Hg, 2500 RPM Military Power Low Blower 40

(“Hg) 30 s 30”Hg, 2000 RPM MP Cruising Power 20 Low Blower 10

1000 40”Hg, 2500 RPM

s 30”Hg, 2000 RPM

BHP 600

200 36 32 28 24 20 16 12 8 4 0 Pressure Altitude, Thousands of feet Figure 10.3c Example Standard Day Supercharged Engine Performance Data

Page 10 - 13 SFTE Reference Handbook Third Edition 2013

10.4 FAA Approved Engine Temperature Corrections

The intent is to ensure that the critical engine parts, (i.e., cylinder head and cylinder barrel) do not exceed the engine manufacturer’s specified limits during worst-case climb operating conditions on a 100 oF hot day.

Test procedures are detailed in AC 23-A. The basic idea is best illustrated with the single-engine airplane proce- dure:

1) Trim in level flight at the lowest practical altitude with at least 75% maximum continuous power. Allow 2) temperatures to stabilize. 2) Increase engine power to takeoff rating and climb at a speed not greater than Vy (best climb speed). Maintain takeoff power for one minute. 3) At the end of one minute, reduce engine power to maximum continuous and continue climb for at least five minutes after temperatures peak or the maximum operating altitude is reached. Flight manual lean ing procedures should be used.

Correct the peak test day cylinder barrel temperature (Tbt) to hot day conditions (Tbh) as follows

Tbh = Tbt+0.7[100-0.0036Hc-Tat]

Correct the peak test day cylinder head or other temperature (Tht) to hot day conditions (Thh) as follows

Thh = Tht+100-0.0036Hc-Tat

Hc is the pressure altitude in feet Tat is the outside air temperature in degrees F

This method is known to be quite conservative More satisfactory temperatures may be achieved by actually test- ing during hot weather.

Page 10 - 14 SFTE Reference Handbook Third Edition 2013

10.5 References

10.5.1 Roberts, Sean C. “Light Aircraft Performance,” Flight Research Inc. Mojave, CA, 1982.

10.5.2 Lawless, Alan R, “Fixed Wing Aircraft Performance Testing” Chapters 3 and 4, National Test Pilot School, Mojave CA, 1996.

10.5.3 Baughn, J. W., “A Method for computing Brake Horsepower from the Performance Charts of Recipro- cating Aircraft Engines,” AIAA- 94-2148-CP, from 7th Biennial AIAA Flight Test Conf., 1994.

10.5.4 Herrington, R. M. Major, USAF, et al, “Flight Test Engineering Handbook,” USAF Technical report No. 6273, AFFTC, Edwards AFB, CA, May 1951.

10.5.5 Chatfield, C. H., et al, “The Airplane and its Engine,” McGraw Hill, 1949.

10.5.6 Gagg, R.F., and Farrar, E.V., “Altitude Performance of Aircraft Engines Equipped with Gear-Driven Superchargers,” SAE Transcripts, Vol 29, pg 217-223, 1934.

10.5.7 anon., “Flight Test Guide for Certification of Part 23 Air planes,“ U.S. Department of Transportation Advisory Circu- lar 23-8A, 1989.

Additional Reading and Second Generation References

10.5.8 Smith, H. C., and Dreier M. E., “A computer Technique for the Determination of Brake Horsepower Output of Normally Aspirated Reciprocating Aircraft Engines,” SAE paper No. 770465, March 1977.

10.5.9 Hamlin, B., “Flight Testing Conventional and Jet-Propelled Airplanes,” The Macmillan Co., New York, NY, 1946.

10.5.10 Operators Manual for Series IO-360 Aircraft Engines, form No. X-30032, FAA Approved March 1979.

Page 10 - 15 SFTE Reference Handbook Third Edition 2013

NOTES

Page 10 - 16 SFTE Reference Handbook Third Edition 2013

Section 11 Propellers

11.1 Abbreviations and Terminology

11.2 Propeller Geometry

11.3 Propeller Coefficients

11.4 Propeller Efficiency and States

11.5 Propeller Theory

11.6 Propeller Modeling Examples Static Thrust Chart, In-Flight Charts, Blocking Correction Factor Charts, Tip Compressibility Factor Charts

11.7 Propeller Flight Test

11.8 References

Page 11 - 1 SFTE Reference Handbook Third Edition 2013

11.1 Abbreviations and Terminology

Abbreviations 100 ,000 1 .0⎛ b ⎞ ⎜ ⎟ x 3 dx AF activity factor = 16 ∫.15 ⎝ D ⎠

B number of blades b blade section width (feet) BHP brake horsepower (measured at engine crankshaft) CLD blade section design lift coefficient

1 .0 3 CLi integrated design lift coefficient = 4 ()C LD x dx ∫.15 CP power (absorbed) coefficient CT thrust coefficient D propeller diameter (feet) fc ratio of speed of sound at standard day sea level to speed of sound at operating condition 1 f = HP horsepower (1 HP = 550 ft-lb/sec) c θ G.R. gear ratio, propeller speed/engine speed J Propeller advance ratio =VT/nD (nondimensional) M aircraft Mach number N propeller speed, revolutions per minute (RPM) n propeller speed, revolutions per second Ne engine speed, RPM Pa ambient pressure P power output (ft-lb/sec) Q torque (ft-lb) q dynamic pressure T thrust Ta absolute ambient temperature R blade radius at propeller tip (feet) r radius at blade element (feet) SHP shaft horsepower (measured at propeller shaft) T propeller thrust (pounds) VT freestream velocity (ft/sec) VK freestream velocity (knots) x fraction of propeller tip radius, r/R Vtan tangential velocity VR resultant velocity Vtip tip speed α local angle of attack β local blade twist angle, measured between chord and plane of rotation, same as θ (degrees). ΔM Mach number adjustment for effect of blade camber φ propeller disk angle of attack η isolated propeller efficiency. ηcomp composite prop efficiency (includes tip and blockage corrections) θ 3/4 propeller blade twist angle at x =3/4 (degrees), same as β 3/4 σ ratio of operating density to sea level standard density = ρa/ρo. ω propeller rotation speed (radians/second)

Page 11 - 2 SFTE Reference Handbook Third Edition 2013

Terminology blade aspect ratio measured as [R / max blade width]. effective pitch actual advance per revolution. experimental pitch necessary advance to generate zero thrust. geometric pitch (p) advance per revolution if blade element moves according to β (i.e., with no slip). reduction gear gearing between the engine crankshaft and prop shaft that reduces the propeller rotation speed . right-handed moves clockwise (viewed from the slipstream). solidity fraction of prop disk covered by blade area = 2πR/Bb. total width ratio (TWR) measured as [WR x B] thickness ratio (TR) blade thicknessmeasured locally or at .75R to represent entire prop. width ratio (WR) calculated as {b @ .75R } / D

Page 11 - 3 SFTE Reference Handbook Third Edition 2013

11.2 Propeller Geometry

δr • δr is the width of any element along blade radius.

r R

• x = r/R, the local fraction of prop tip radius

• Prop blade chord extends from leading edge to trailing edge. • Blade twist angle θ, measured between rotation plane and local chord. • Relative wind is the resultant velocity (VR), comprised of aircraft forward speed and tangential speed at radial location along blade.

2 2 V R = VT + ()2πrn

-1 VT -1 VT -1 VT -1 1 VT φ = tan = tan = tan : φ tip = tan rω r 2πn xD πn π nD

• Angle φ is measured between plane of rotation and local VR

1 V 1 V J α x = θ x −φ = θ x - tan-1 T = θ x - tan-1 T = θ x − tan-1 π r2n π xDn πx

• Αdvance ratio (J) is defined as J = VT/nD. • Local angle of attack at any fraction of radius(αx) is measured between the local chord line and relative wind • Lift and drag are perpendicular and parallel to VR, respectively • Thrust (T) and torque (Q) are perpendicular and parallel to the plane of rotation, respectively.

Page 11 - 4 SFTE Reference Handbook Third Edition 2013

11.3 Propeller Coefficients

Integrating lift and drag along a blade gives the thrust (T) and torque (Q). Multiply by number of blades (B) to determine total T and Q.

R2 c T = qB 2 ()C L cos φ − C D sin φ dr ∫R1 sin φ

R2 cr Q = qB ()C L sin φ + C D cos φ dr ∫R 2 1 sin φ

T Thrust Coefficient, CT ≡ ρ n 2 D 4

Q Torque Coefficient, C ≡ Q ρ n 2 D 5

P Q × ω Q × 2πn Q ≡ = = = 2π = 2πC Power Coefficient, CP ρn 3 D 5 ρn 3 D 5 ρn 3 D 5 ρn 2 D 5 Q

CQ CT

.008 .08 Typical effect of advance ratio on thrust and torque coefficients .006 .06

CQ

.004 .04 CT

.002 .02

0 0 .2 .4 .6 .8 1.0 Advance Ratio

Page 11 - 5 SFTE Reference Handbook Third Edition 2013

11.4 Propeller Efficiency and States

Propeller efficiency (η)

P Thrust×V C ρn2 D4 ×V 1 C V C η ≡ out = T = T T = T T = T J 2 5 Pin Q×ω CQ ρn D ×2πn 2π CQ nD CP

η Cp

0.8 .04 η

0.6 .03

C p 0.4 .02

0.2 .01 V N D 0 0 0 0.2 0.4 0.6 0.8 Windmill Propeller State State Brake State

Propeller state: positive thrust & efficiency, power supplied by engine. Brake state: negative thrust & efficiency, power supplied by engine. Windmill state: negative thrust & η, power supplied by freestream.

Efficiency of constant speed prop η

20º 25º 15º 10º Fixed-pitch blade twist at .75R Efficiency,

Propeller Advance Ratio, J

Page 11 - 6 SFTE Reference Handbook Third Edition 2013

11.5 Propeller Theory

Simple momentum theory describes pressure jump (ΔP) across propeller disk . • The downstream velocity increment (v1) is twice the velocity increment at the disk (v) . • Thrust (ΔP) = ΔP x disk area TV V η ≡ T = T • Froude’s momentum theory: efficiency = T()V +v V +v T T

V V + v V + v1

p p' p' + Δ p

Δp Atmospheric Pressure

Free stream Velocity v1 v

Blade element theory tends to be more complex and may include corrections for tip swirl losses, Mach effects, nacelle blockage, etc. Below is a comparison of typical calculated and measured thrust distribution.

Thrust

Calculated

Measured

Radius

Page 11 - 7 SFTE Reference Handbook Third Edition 2013

11.6 Propeller Modeling

• For a specified propeller geometry; CT , CP, J, and blade angle (θ) are interrelated such that knowledge of any two defines the other two. • Calculate propeller efficiency as η = JCT /CP . • Models assume isolated conditions, i.e., without nacelle blockage. • Models assume negligible Mach effects at propeller tips. • Different models required for static and “in-flight” conditions.

Determine static CT and CP using “Static Thrust Chart” (ref 11.2)

3.2 2 blades / 80 AF

2.8

2.4 Integrated design C L 0.3 2.0 Optimum CT/CP 0.5 1.6 0.7

1.2 C T C P .8

.4 Advance Ratio, J 0 0 .04 .08 .12.16 .20 .24

∼ Separate charts exist for each combination of AF and # of blades (B). ∼ Enter chart at appropriate J & Cli C SHP Static Thrust = T = T 33,000 where N = Propeller RPM Determine isolated propeller in-flight effi static C ND P ciency (η) from the appropriate “Flight Charts.” They are typically presented in one of two forms.

Page 11 - 8 SFTE Reference Handbook Third Edition 2013 2.6 2

1 . 0 8 06 .0 0. 0 = p C 2.2 SHP 3

n

ρ V

n 0.40875

0.62 =

p Mass density Mass of air =

SHP = 0.00381 = x (n)SHP x Torque (lb - ft) True airspeed,V knots % speed, rotational Propeller n ρ J C 1.8 Advance RatioAdvance (J) 1.4

1.0

0

3

.

C

=

p

C 8

0

2

6 .

2

8 4 2 . 0

.

1 2

0

. .

0

2

6

6 0 0

2 4

1 .

0

.

.

1

0

.

0

0

12

0

0

= 8

0.

1

.

0

p

. 0

C 0 0.6 0.4

0.9 1.0 0.5 0.8 1.7

0.6

p P r o p l e e l r E

f f c i i e n c 7 ) y (

Above example for AiResearch TPE331-3U-303G engines and Hart- zell T10282HDB-4R 3-blade, constant speed, feathering propellers.

Page 11 - 9 SFTE Reference Handbook Third Edition 2013

The other “in-flight η” format also requires calculation of CP and J. Below is a typical flight chart published by Hamilton Standard (Ref 11.2). This applies to a propeller with 2 blades, AF= 80, and CLi =0.5

5 3.0 5

7

. 0 8

.

8

3

. =

8

. η

2.8

0

9

.

1 9

.

o

5

.

8 2

9

.

=

η

2.4 2.6

o 1

5 9

.

2

5

0 9

9

8

o 8

0 8

. 7

0 8

5 6

8 2.0 2.2

5 8

0

8 J

.

Li

=

C η 2.0

1.8

o

5

.

7 4 Advance Ratio,Advance

o

0

.

5

4 1.4 1.6

o 2 Blades/ 80AF/0.5

5

. o

2

0 .

4

0

4 =

4 o /

5

3 .

o

7

0 θ

3 .

5 0

3 7

5 6

0 6

5

5

0

5

.

x

η

5 4

0 0.6 0.8 1.2 4

5 3

0 3 0.4

4 / 3 0.2 θ o o 5 o o . 0 . @ o 2 5 5 o . o . o o o 0 0 3 5 7 2 . 0 3 0 . 0 5 . . . . 2 2 0 7 5 5 2 0 2 1 2 1 1 1 0 0 .28 .12 .32 .16 .04 .36 .24 .20 .08 P C

Page 11 - 10 SFTE Reference Handbook Third Edition 2013

A body correction factor (FBC) should be applied to account for reduced efficiency due to body flow blockage immediately behind the propeller. Two examples follow.

Society of British Aircraft Constructors, Ltd. S/A = [body diameter/Prop dia]2 Use body diameter @ 3/4 prop diameter behind disk

1.00

.98

.96 S/A = .141

S/A = .146 F BC S/A = .153

.94 S/A = .160

S/A = .167

.92

S/A = .175

.90 S/A = .184

S/A = .193

S/A = .199 .88

0 .5 1.0 1.5 2.0 2.5 3.0

Advance Ratio

Page 11 - 11 SFTE Reference Handbook Third Edition 2013

Hamilton Standard also publishes a generalized nacelle blocking correction for typical scoop and annual inlet nacelles used on typical turboprops. 2.4 2.8 2.4 ANNULAR ANNULAR SCOOP J Advance ratio, 0 0.4 0.8 1.2 1.6 2.0 2.0 1.6 1.2 0.8 0.4 0 1.03 1.02 1.01 1.00 1.04 BC F

Page 11 - 12 SFTE Reference Handbook Third Edition 2013

To determine if tip compressibility corrections are appropriate, find the maximum integrated design lift coeffi- cient, CLimax from the graph below. ∼ Enter at flight Mach number, and move across at appropriate NDfc.

1 f = c θ

∼ If CLimax is below calculated Cli , then corrections are required. L Approximate integratedApproximate design C losses compressibility avoid to 0 4,00 = 1 x f C ND 0 ,00 15 00 ,0 16 00 ,0 17 0 00 8, 0 Number Mach Aircraft 1 00 9, 1 = f C x 0 D 0 ,5 N 9 1

0 0 ,0 0 2 0 5 ,2 0 2 =

f C x D N 0.1 0.2 0.3 0.4 0.5 0.5 0.4 0.3 0.2 0.1 1.0 0.9 0.8 0.7 0.6 3

0.

0.6 0.1 0.7 0.5 0.4 0.2

L L

C C

n n g g i i s s e e D D

d d e e t t a a r r g g e e t t n n I I

m m u u m m i i x x a a M M

Page 11 - 13 SFTE Reference Handbook Third Edition 2013

If tip compressibility corrections are necessary, then the first step is to • Determine the Mach number adjustment for the effect of blade camber (ΔM) from the figure below.

0.06

0.04

0.02

ΔM

0

-0.02

-0.04

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Integrated Design CL , CLi

• Next, add ΔM from above to flight Mach number to get Meff. • Enter adjacent generalized compressibility correction chart to determine propeller efficiency tip factor (Ft) • Calculate composite propeller efficiency as

ηcomp = η x Ft x FBC

• Calculate in-flight thrust as η comp SHP 326η comp SHP T = = VT KTAS

Page 11 - 14 SFTE Reference Handbook Third Edition 2013 3.6 3.2 8 2. .70 4 2. .65 0 J 2. .60 .55 1.6 M Δ .50 + Advance Ratio, 2 M 1. .45 = eff M .40 = airplane Mach number airplane Mach = = adjustment for blade camber for adjustment = 0.8 .35 M M Δ .30 .4 0 0 9 8 7 0 6 0. 0. 0. 1. 0. t F

Page 11 - 15 SFTE Reference Handbook Third Edition 2013

11.7 Propeller Flight Test The best method for determining ηcomp is to instrument the prop shaft and/or engine mounts to measure thrust and torque.

Slip Rings

Calculate efficiency as

T V T η comp = Q ω

Page 11 - 16 SFTE Reference Handbook Third Edition 2013

As an alternate, the incremental drag method requires an accurate engine power model, a load cell and a small drag device.

• Trim the aircraft at test RPM, VT , & altitude. Note SHP required. • Repeat above test with drag device and load cell attached. Note the power requirement change (ΔSHP ) and load cell drag (ΔD).

V η . BHP = T D p AC 550

VT η p . BHPAC + ΔD = ()D + ΔD 550

Load Cell

• Calculate aircraft drag and prop efficiency as

ΔD()SHP V T D D = η = ΔSHP 550 ()SHP

• This technique assumes the same η for both tests and is valid if J is constant and the CP change is small. The drag device must therefore be small enough to not violate this assumption, yet large enough for the change in SHP to be measurable on engine instruments.

Page 11 - 17 SFTE Reference Handbook Third Edition 2013

11.8 References

11.8.1 Roberts, Sean, “Light Aircraft Performance for Test Pilots and Flight Test Engineers,” Flight Re search Inc., Mojave CA, 1982.

11.8.2 anon., Hamilton Standard Propeller Efficiency Charts (a.k.a.Redbook), PDB 6101.

11.8.3 Von Mises, Richard, “Theory of Flight,” McGraw-Hill, 1945.

Page 11 - 18 SFTE Reference Handbook Third Edition 2013

Section 12 Fixed-Wing Performance Standardization

12.1 Recurring Abbreviations 12.2 Standardization Techniques Difference Method, Ratio Method 12.3 Takeoff Distance 12.3.1 Empirical Standardization Method 12.3.2 Takeoff Distance Prediction direct approximation of takeoff ground roll, total takeoff distance 12.4 Landing Distance 12.4.1 Empirical Standardization Method 12.4.2 Landing Distance Prediction direct approximation of landing ground roll, direct approximation of the landing air distance, exact prediction of landing performance, fixed time increment approach 12.5 Climb/Descent Acceleration . 12.5.1 Climb/Descent/Acceleration Prediction 12.5.2 Correcting to Standard Climb Rate 12.5.3 WAT Limits 12.6 Level Turn Performance 12.6.1 Sustained Level Turn Performance Prediction 12.6.2 Sustained Level Turn Performance Correction 12.6.3 Level Limit Turn Performance Correction 12.7 Reciprocating Engine Cruise Performance 12.7.1 Power Standardization 12.7.2 Fuel Flow Standardization 12.7.3 Endurance Optimization and Prediction 12.7.4 Range Optimization and Prediction 12.8 Jet Aircraft Cruise Performance 12.8.1 Thrust Standardization 12.8.2 Fuel Flow Standardization 12.8.3 Endurance Optimization and Prediction 12.8.4 Range Optimization and Prediction 12.9 References

Page 12 - 1 SFTE Reference Handbook Third Edition 2013

12.1 Recurring Abbreviations (references 10.5.1-10.5.5) a acceleration BHP brake horsepower BSFC brake specific fuel consumption (fuel flow per horsepower per hour) CD drag coefficient CDiOGE induced drag coefficient out of ground effect CDiIGE induced drag coefficient in ground effect cg center of gravity CL lift coefficient CLIGE lift coefficient in ground effect CLOGE lift coefficient out of ground effect D drag Fn net thrust Fg gross thrust Fe ram thrust Fex excess thrust F/δ corrected thrust g reference acceleration due to gravity (32.174 ft/sec²) GECL ground effect correction factor for lift coefficient GECD ground effect correction factor for drag coefficient H, h geopotential altitude Hc pressure altitude L lift LW lift of the wing M Mach number m mass Nxw longitudinal load factor along flight path (wind axis) Nzw, nz load factor normal to flight path P power output Pa ambient pressure Po std ambient pressure sea level (2116.22 lb/ft² = 29.921 in Hg) Piw standard day sea-level power required Pm mission-day power required Ps specific excess power Ps standard power required Pt test-day power required q dynamic pressure

Page 12 - 2 SFTE Reference Handbook Third Edition 2013

R range R/C rate of climb Rn Reynolds number RF range factor S reference wing area Sa horizontal air distance Sg ground roll SLD total landing distance SR specific range STO total takeoff distance T ambient air temperature (absolute) T thrust To ambient temperature sea level standard (288.15 °K=15.0 °C) V inertial speed Vc calibrated airspeed Ve equivalent airspeed Viw standard day sea-level true airspeed VT true airspeed VTt test true airspeed VTs test true airspeed Wt test weight Ws weight standard W& f Fuel Flow W& standard day sea-level fuel flow f iw

α angle of attack β sideslip angle δ ambient air pressure ratio σ ambient air density ratio ιT thrust incidence angle γ flightpath angle φ bank angle μ rolling coefficient of friction ω turn rate (rad/sec)

Page 12 - 3 SFTE Reference Handbook Third Edition 2013

12.2 Standardization Techniques (ref 12.5) Performance data is usually corrected to “standard” conditions which are specified values of weight, alti- tude, cg and Mach number. The process also corrects data to some standard ambient air temperature, usually defined by the 1976 U.S. Standard Atmosphere. In some cases the data is corrected to “standard hot” day or “standard cold” day conditions which are a specified increment relative to the true standard conditions. The standardization process usually relies on models of drag, thrust (or power), fuel flow, and propeller effi- ciency if appropriate. The overall principle is to collect test data as near as practical to standard conditions (+/-10%) and correct the results to standard using the models. Even with a 10% modeling error, correcting test data that is 10% from standard leads to only 1% total error in the standardized results. The most common of the two standardization methods is the difference method which adds a correction to the test day parameter. This correction is the difference between the model predictions for standard and test conditions:

Ps = Pt + (Ps’ - Pt’) (Eq’n 12.1) where Ps = standardized parameter Pt = test day parameter Ps’ = standard day parameter predicted by models Pt’ = test day parameter predicted by models

The parameter of interest can be one of the basic modeling values such as thrust, drag, power, or fuel flow. The parameter can also be the end result of the predictive process, and may include values like takeoff/landing dis- tance, climb/sustained turning capability, or cruise range. The ratio method is the other standardization process. It corrects to standard conditions by multiplying the test values by a correction factor. This factor is the ratio of the model predictions for standard and test conditions. Ps = Pt (Ps’ / Pt’) (Eq’n 12.2)

The preferred approach is whichever gives the lowest total error. If the prediction models are in error by approx- imately a constant percentage, then the ratio method yields the least error. If the models are in error by approxi- mately a constant magnitude, then the increment method yields the least error. Less exact empirical methods can also be used.

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12.3 Takeoff Distance (refs 12.1, 12.2, 12.5) The total takeoff distance is the sum of the ground roll distance (Sg, from brake release to main wheel lift- off) and the horizontal component of the air distance (Sa, from liftoff to main gear reaching either 35 or 50 feet altitude-depending on the requirements).

STO = Sg + Sa

Both Sg and Sa can be standardized using the increment or ratio method, or by empirical relations. The empiri- cal methods are useful when detailed aircraft models are not available. The more exact process of predicting takeoff distance using models is described in section 12.3.2.

12.3.1 Empirical Standardization Method First correct for the effects of the test day wind. Define headwind velocity as Vw , liftoff true airspeed as VLO, and test day ground roll as Sgw. With a typical variation of thrust per headwind, estimate the test day zero- wind ground roll (Sgzw) using the following empirical equation:

1.85 ⎛V ⎞ S = S ⎜ LO ⎟ (Eq’n 12.3) gzw gw ⎜ ⎟ ⎝ Vg ⎠

If the average thrust is not appreciably affected by velocity, then the exponent should be 2.0 in lieu of 1.85. The zero-wind air distance (Sat) correction is

S = S + V t a t a w w (Eq’n 12.4)

where t is the time from liftoff to 35 (or 50) feet altitude. The second correction is for the effect of runway slope (θ, positive uphill) and therefore applies only to the ground roll. Correct the above zero-wind distance (Sgzw) to the test day zero-slope distance (Sgt) as follows:

⎡ W sin θ ⎤ S = S ⎢1 − ⎥ (Eq’n 12.5) g t g zw F ⎣⎢ []ex avg ⎦⎥

If the average excess thrust is not known, then approximate Fex as that at 70% of the liftoff airspeed or from the zero-wind ground roll distance:

mV 2 []F ≈ LO (Eq’n 12.6) ex avg 2S g zw

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After correcting the test day distance to zero wind and slope, use the following empirical equations to correct for non-standard weight, density, and temperature. Any desired values can be treated as the "standard" conditions.

Aircraft Propulsion Type Standard Day Distance

2.4 −2.4 0.5 Fixed pitch propellers ⎛Ws ⎞ ⎛σ s ⎞ ⎛ Tas ⎞ S = S ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ gs gt ⎜ ⎟ ⎜ ⎟ ⎝ Wt ⎠ ⎝ σ t ⎠ ⎝ Tat ⎠ (neglect temp correction for 2.2 −2.2 0.6 constant rpm evaluation) ⎛W ⎞ ⎛σ ⎞ ⎛ T ⎞ S = S ⎜ s ⎟ ⎜ s ⎟ ⎜ as ⎟ as at ⎜ W ⎟ ⎜ σ ⎟ ⎜ T ⎟ ⎝ t ⎠ ⎝ t ⎠ ⎝ at ⎠

2.6 −1.7 −0.7 −0.9 Turbo-propeller aircraft ⎛ W ⎞ ⎛ σ ⎞ ⎛ N ⎞ ⎛ P ⎞ S = S ⎜ s ⎟ ⎜ s ⎟ ⎜ s ⎟ ⎜ s ⎟ gs gt ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ Wt ⎠ ⎝ σ t ⎠ ⎝ N t ⎠ ⎝ Pt ⎠ (for heavy aircraft, replace 2.3 & -1.2 2.3 −1.2 −0.8 −1.1 with 2.6 & -1.5 respectively) ⎛ W ⎞ ⎛ σ ⎞ ⎛ N ⎞ ⎛ P ⎞ S = S ⎜ s ⎟ ⎜ s ⎟ ⎜ s ⎟ ⎜ s ⎟ as at ⎜ W ⎟ ⎜ σ ⎟ ⎜ N ⎟ ⎜ P ⎟ ⎝ t ⎠ ⎝ t ⎠ ⎝ t ⎠ ⎝ t ⎠

2.3 1.3 Large jet aircraft ⎛ W ⎞ ⎛ σ ⎞⎛ F ⎞ S = S ⎜ s ⎟ ⎜ t ⎟⎜ nt ⎟ gs gt ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ Wt ⎠ ⎝ σ s ⎠⎝ Fns ⎠ (for lights jets, replace 2.3 & 0.7 2.3 0.7 1.6 exponents with 2.6 & 1 respectively) ⎛ W ⎞ ⎛ σ ⎞ ⎛ F ⎞ S = S ⎜ s ⎟ ⎜ t ⎟ ⎜ nt ⎟ as at ⎜ W ⎟ ⎜ σ ⎟ ⎜ F ⎟ ⎝ t ⎠ ⎝ s ⎠ ⎝ ns ⎠ where Pt = Test day brake power at the propeller Fnt = Tot = Avg. Test net thrust (approx .94 x static thrust @ test conditions)

Ps = Standard day brake power at the propeller Fns = Tos = Avg. Standard net thrust (approx .94 x static thrust @ std.conditions)

Nt = Test day propeller RPM

Ns = Standard day propeller RPM

These empirical corrections are valid only for small (<10%) changes. (above equations from ref 12.3)

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12.3.2 Takeoff Distance Prediction (refs 12.1, 12.5)

Estimating takeoff ground roll without numerical methods

Define V = airspeed VTO = liftoff airspeed Sg = ground roll distance Sw = reference wing area Fex = excess thrust Fn = net thrust CLIGE = lift coeffient in ground effect CDIGE = drag coefficient in ground effect W = aircraft weight m = rolling friction g = acceleration due to gravity A = acceleration = g*Fex/

Assuming zero initial speed for takeoff run, ground roll distance

2 VTO ⋅W Sg = ⎛ 1 2 ⎞ (Eq’n 12.7) 2⋅ g ⋅⎜Tavg − μ ⋅W + ⋅ ρ ⋅ Sw ⋅VTO ⋅ ()μ ⋅CLIGE − CDIGE ⎟ ⎝ 6 ⎠

Where Fnavg may be estimated as the average of static thrust and net thrust at liftoff airspeed. If the initial air- speed is non-zero, its value should be inserted into Equation 10.3 in place of VL/O and the answer subtracted from the zero-wind case.

where μ is the rolling friction coefficient (typically between 0.015 and 0.025 for hard dry runways), and CLIGE is the lift coefficient in ground effect while at ground roll attitude. Estimate CLIGE by determining the out-of-ground-effect lift coefficient (CLOGE) at the ground roll angle of at- tack and correcting it as follows: n (Eq’ 12.8) CLIGE = CLOGE GECL

where the ground effect factor, GECL = [0.8609 –0.6282 log10(h/b)] and h is the wing height above the surface and b is the wingspan. The above correction is not used above the height that predicts

GECL < 1

CDIGE is the induced drag coefficient while in ground effect. Estimate this by determining the out-of-ground- effect drag coefficient (CDiOGE ) at the appropriate angle of attack and correcting it as fol- lows: (Eq’n12.9)

CDiIGE = CDiOGE GECD

where the ground effect factor, GECD = [0.2412 ln(h/b)+1.0829] The above correction is not used above the height that predicts GECD > 1

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A direct approximation of takeoff air distance requires the desired speeds at liftoff and at 50 feet (typically 1.1Vs and 1.2Vs, respectively). It also requires an estimate of the average excess thrust as the aircraft climbs out of ground effect.

W ⎡ V 2 − V 2 ⎤ ( 50 ) ( LO ) (Eq’n 12.10) Sa = ⎢ +50⎥ ()T − D avg ⎣ 2g ⎦

A direct approximation of the total takeoff distance (STO) can be calculated as the sum of the ground and air distances or can be estimated by multiplying the ground roll distance by a “planform factor” (Fpl).

STO = SgFpl (Eq’n 12.11)

Fpl combines the effects of wing type, thrust-to-weight ratio, and pilot technique. The following values charac- terize the typical aircraft.

straight wing: Fpl = 1.15 swept wing: Fpl = 1.36 delta wing: Fpl = 1.58

A more exact prediction of takeoff performance (ref 12.5) requires accurate thrust and drag models and an integration of the aircraft’s velocity over the takeoff time. This is equivalent to a double integration of the air- craft’s acceleration or its specific excess thrust.

F 1 ex (Eq’n 12.12) STO = VT dt = adt = dt = Fexdt ∫ ∫∫ ∫∫ m m ∫∫

This double integration can be performed numerically or graphically. Alternately, use planar kinematics and sum the distances required to accelerate between incremental true airspeeds from brake release (V0) to the true airspeed when the aircraft reaches the takeoff altitude (V50‘).

V50 ' 2 2 m V 2 − V1 S TO = ∑ (Eq’n 12.13) 2 V0 Fex

Both methods above are typically split into pre-rotation ground roll, rotation/post-rotation ground roll, and airborne segments. Both methods require calculation of the excess thrust, addressed below.

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Solving for the excess net thrust during the ground roll for either takeoff or landing cases requires a simultane- ous solution of the three equations of motion along the aircraft’s longitudinal & vertical axes and about the pitch axis. These equations (in the above order) are as follows:

Fex + μnwRnw + μmRm = [FgcosιT + Fe – Dwb – Dt – Wsinθrw] (Eq’n 12.14)

(Eq’n 12.15) Rnw+Rm = [Wcosθrw – Lw – Lt]

(X1 + X2)Rnw = [WcosθrwX2 + Wsinθrw Z1 + {Fgcos(θ+ιT)-Fe}Z1 + Lt(X3 + X4 – X2) - Lw(X2 – X3) - Dt (Z1 + Z2 )] (Eq’n 12.16) where Fex = excess net thrust μnw = nose wheel coefficient of friction (about 0.02 for takeoff, 0.5 for maximum dry runway braking) Rnw = reaction force (weight) on nose wheel (positive) μm = main wheel coefficient of friction (positive) Rm = reaction force (weight) on main wheel (positive) Fg = gross engine thrust (positive, aligned with engine axis) ιT = thrust incidence angle (positive denotes thrust that generates lift) Fe = ram thrust (or drag) due to momentum change of the air outside the engine, measured along drag axis- aligned with relative wind (typically negative at low speed, positive at high speed) Dwb = aerodynamic drag of wing and body (excludes horizontal tail drag) Dt = aerodynamic drag of horizontal tail (positive aft) W = aircraft weight (positive) θrw = runway slope (positive denotes uphill) Lw = main wing lift (positive denotes up) Lt = horizontal tail (positive denotes up) X1 = distance from nose gear to aircraft cg (positive) X2 = distance from the main gear to aircraft cg (positive) Z1 = distance from the ground plane to the aircraft body axis (positive) θ = aircraft pitch attitude (positive denotes nose up) X3 = horiz. dist. from wing’s aerodynamic center to aircraft cg (positive) X4 = horizontal distance from the wing’s aerodynamic center to the horizontal tail’s aerodynamic center (positive) Z2 = vertical distance from the horizontal tail’s aerodynamic center to the aircraft body axis (positive)

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The previous equations were arranged so that the right hand side of each can be abbreviated as A1, A2, and A3 respectively. This step allows for a compact matrix form of the equations using a 3x3 matrix

⎡1 μnw μm ⎤⎡ Fex ⎤ ⎡ A1 ⎤

⎢0 1 1 ⎥⎢R ⎥ = ⎢A ⎥ ⎢ ⎥⎢ nw ⎥ ⎢ 2 ⎥ ⎣⎢0 X 1 + X 2 0 ⎦⎥⎣⎢ Rm ⎦⎥ ⎣⎢ A3 ⎦⎥

Solve for Fex by pre-multiplying both sides by the inverse of the first matrix

−1 ⎡ Fex ⎤ ⎡1 μnw μm ⎤ ⎡ A1 ⎤

⎢R ⎥ = ⎢0 1 1 ⎥ ⎢A ⎥ ⎢ nw ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎣⎢ Rm ⎦⎥ ⎣⎢0 X 1 + X 2 0 ⎦⎥ ⎣⎢ A3 ⎦⎥

Although wheel reaction forces are not required for takeoff distance prediction, they are useful for accurate calculation of rotation capability and for braking effectiveness during landing ground rolls. For takeoff calculations, several simplifying assumptions can be made such as:

μnw = μm = 0.02 Fg >> Fe Dt = 0

This above approach can be repeated for the segment between rotation and liftoff. This is slightly more complicated because the changing angle of attack alters drag and acceleration.

Precise predictions of the takeoff air distance can be made by applying Equation 12.10 in small incre- ments using accurate models that describe thrust as a function of airspeed and the lift & drag changes due to climbing out of ground effect.

Along with the incremental Sg and Sa calculations, the time to accelerate between the corresponding incre- mental velocities can be calculated as

V −V Δt = m 2 1 Fex

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12.4 Landing Distance (refs 12.1, 12.2)

The total landing distance (SLD) is the sum of the ground roll distance (Sg, from touchdown to full stop) and the horizontal component of the air distance (Sa, from the screen height to touchdown). The screen height can be either 35 or 50 feet above the surface, depending on the requirements

SLD = Sg + Sa

Both Sg and Sa can be standardized according to the increment or ratio methods described by equations 12.1 and 12.2, or by empirical relations. The empirical methods are useful when detailed aircraft models are not availa- ble. The more exact process of predicting landing distance using models is described in section 12.4.2.

12.4.1 Empirical Standardization Method First correct for the effects of the test day wind. Define headwind velocity as Vw , touchdown true airspeed as VTD, and test day ground roll as Sgw. With a typical variation of thrust per headwind, estimate the test day zero- wind ground roll (Sgzw) using the approach applied to takeoff ground roll:

1.85 ⎛ V +V ⎞ S = S ⎜ TD w ⎟ g zw g w ⎜ ⎟ (Eq’n 12.17) ⎝ VTD ⎠

If the average thrust is not appreciably affected by velocity, then the exponent should be 2.0 in lieu of 1.85.

Apply Equation 12.4 to correct to the zero-wind air distance (where t is the time to descend from the screen height to touchdown).

To correct to a zero-slope runway, apply Equation 12.5 to the zero-wind ground roll distance (note that Fex is negative). If the average excess thrust is not known, then approximate [Fex ]avg as that at 70% of the touchdown airspeed. Alternately, approximate [Fex ]avg from the zero-wind ground roll distance using

2 mV TD []F ≈ (Eq’n 12.18) ex avg 2 S g zw

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After correcting the test day distance to zero wind and slope, use the following empirical equations to cor- rect the ground roll to standard weight and air density.

2 ⎡W ⎤ σ S = S s t g s gt ⎢ ⎥ (Eq’n 12.19) ⎣Wt ⎦ σ s

Any desired values can be treated as the "standard" conditions. Correct the air distance to standard weight and air density using the zero-wind air distance as follows (for a 50-foot screen height) h h 2+ v v hv +50 hv +50 ⎡Ws ⎤ ⎡σ t ⎤ Sa = Sa ⎢ ⎥ ⎢ ⎥ s t W σ (Eq’n 12.20) ⎣ t ⎦ ⎣ s ⎦ where hv is the specific kinetic energy change during the air phase. For the case of a 50-foot screen height, this term is calculated as V 2 − V 2 50 ' TD h v = (Eq’n 12.21) 2 g

12.4.2 Landing Distance Prediction

With reasonably precise models available, the landing distance can be predicted through calculation. Test distances can then be standardized using either the increment or ratio method (Equations 12.1-12.2). A direct approximation of landing ground roll can be obtained by applying the same Equation 12.7 used for the takeoff case. This method requires a value for the average net thrust (Tavg) across the landing roll speed range and reasonable values for the wheel braking friction coefficient (0.35<μ<0.5 for typical dry runway max braking). The same equations for estimating ground effect also apply. A direct approximation of takeoff air distance including the flare requires the desired lift and associat- ed drag coefficients, the thrust, and the applied normal load factor during the landing flare (n = 1.15).

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W ⎛ C T ⎞ D ⎜ ⎟ 50 S ⎝ CL W ⎠ Sa = + (Eq’n 12.22) ⎛ C T ⎞ Tρ g()n −1 C ⎜ D ⎟ o L ⎜ − ⎟ ⎝ CL W ⎠

A more exact prediction of landing performance requires accurate thrust and drag models and an integration of the aircraft’s velocity across the landing time. This is equivalent to a double integration of the aircraft’s acceleration as shown in Equation 12.12. This double integra- tion can be performed numerically or graphically. Similarly, Equation 12.13 can be adapted for landing as fol- lows:

m V50 ' V 2 −V 2 S = 2 1 LD ∑ (Eq’n 12.23) 2 V Fex 0

As with the takeoff case, this equation is usually broken into the air phase and the ground roll phase. Cal- culation of excess thrust during the ground roll needs to consider the changing weight on the wheels and associ- ated braking force. This requires a simultaneous solution of the three equations of motion along the aircraft’s longitudinal & vertical axes and about the pitch axis, previously shown as Eq’ns 12.14-12.16. Precise calculation of excess thrust during the air phase must consider the change in normal and longitudi- nal load factor during the flare and the changes in lift and drag coefficients due to entering ground effect (previously described). If the desired flare technique is some schedule of flight path angle (γ) versus altitude, then the normal load factor can be calculated from γ and the rate of γ using . VT γ N z = cosγ + g

. . The longitudinal load factor can be calculated as H V T N x = + VT g

. H where sin γ = V T

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An alternate method of calculating distance is the fixed time increment approach. The following air distance example is based on a constant angle of attack landing technique (ref 12.5).

Fixed inputs Initial inputs angle of attack, α initial ground speed, Vgo wing area, S initial air distance, Sao = 0 air density, ρ initial altitude, ho weight, W initial sink rate, hodot wingspan, b initial lift coefficient, CLOGE head wind, Vw wing aspect ratio, AR net thrust, Fn (Fn = Fg cosα –Fe) wing Oswald efficiency factor, e time increment, Δt (.05 sec works well) height of wing above ground when on gear, hwing

Initial calculations initial true airspeed, VTo = Vgo + Vw -1 initial glide slope, γo =sin [hodot]/ VTo initial load factor, Nz = cosγo (assumes dγo/dt =0) 1/2 initial trim speed, VT =[2NzW/ρCLS]

Incremental calculations (values with prime symbols represent the result of the previous iteration).

1) CL = CLOGE GECL = CLOGE[0.8609 –0.6282 log10(h/b)] 2 2) L = 0.5NzWρCLSVT 3) Nz = L/W 4) γdot = g(Nz –cosγ)/VT 5) γ = [γdot]’Δt +γ ’ 6) hdot = VT sinγ 7) h = (hdot + [hdot]’)Δt/2 + h’ 2 8) CDiIGE = CDiOGE GECD = [CL /πARe][0.2412 ln(h/b)+1.0829] 9) CD = CDo +CDiIGE 2 10) drag, D = CDρSVT /2 11) Fex = Fn - D 12) Nx = Fex/W 13) Vdot = g(Nx - sinγ ) 14) Vg = Vg’ +Vdot(Δt) 15) Sa = (Vg + Vg’)Δt/2 +Sa’

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12.5 Climb/Descent/Level Acceleration (ref 12.4) Standard performance can be determined either by predicting results using (flight test validated) models or by correcting individual flight test climb/acceleration results to standard conditions Performance predictions require accurate net thrust and aerodynamic models. Net thrust is the sum of the gross thrust and ram drag, while the aero model includes the drag polar and lift curve. Corrections to individual climb/accelerations tests require models that only show the change in thrust & drag between test and standard conditions. The following sections address both the prediction and correction approaches.

12.5.1 Climb/Descent/Acceleration Prediction According to basic energy theory, an aircraft’s specific excess power (Ps) is related to the change in kinetic and potential energy as fol- lows 2 Pex ()Fn − D V dH H dW V dV V dW Ps = = = + + + W W dt W dt g dt 2Wg dt (Eq’n 12.24) where V is technically inertial speed. True airspeed and an assumption of zero wind is usually used instead of inertial speed. Since aircraft typically have negligible weight change during a maneuver, the above reduces to F − D V dH V dV ()n Ps = = + (Eq’n 12.25) W dt g dt

This shows not only how climb rate or acceleration performance can be predicted, but also shows how the climb and acceleration capabilities can be exchanged at any given specific excess power. Dividing this equation through by V shows the relation between specific excess thrust and climb angle, γ

• PS F − D H 1 dV = n = + = sin γ (Eq’n 12.26) V W V g dt . When predicting climb performance capability using this approach, iterations may be required because the re- sulting climb angle affects the normal load factor (Nz = cosγ) and the therefore the induced drag.

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12.5.2 Correcting to Standard Climb Rate The below sequence corrects results at the test q and (usually) pressure altitude (i.e., Ve test = Ve std, and Htest =Hstd). 1) If the test day vertical velocity is measured by timing pressure altitude changes, then first correct the altimeter readings for instrument error and then convert the indicated pressure altitude rate to geometric (tapeline) climb rate as follows dH ⎛ T ⎞ dH = ⎜ t ⎟ dt ⎜ T ⎟ dt t ⎝ s ⎠ indicated

2) Equation 12.25 yields the climb rate correction that accounts for the change in power (or thrust) between test and standard days (at the test weight and velocity)

• ΔP ΔF V Δ H = ΔP = = n s P W W where ΔP or ΔFn comes from engine models. For reciprocating engines without models that can predict this power change, estimate the correction using only a standard day power chart and the following equation

• 550ηBHP ⎡ T ⎤ Δ H = s ⎢1− s ⎥ P Wt ⎣ Tt ⎦

3) A changing horizontal headwind with altitude will alter climb results. If this change (dVw/dH) is known, then add the following correction to the tapeline climb rate

• V ⎛ dV ⎞⎛ dH ⎞ Δ H = hw ⎜ ⎟⎜ ⎟ g ⎝ dH ⎠⎝ dt ⎠t

Usually the exact wind shear profile is unknown. In this case, fly perpendicular to the known crosswind direc- tion and repeat each climb speed at the reciprocal heading. After completing the remaining corrections listed below, average the reciprocal results to obtain a standard climb rate.

4) If the climb is flown at constant indicated airspeed or Mach, then true airspeed will change with air density. Correct for any change in true airspeed with the following “acceleration factor” correction

⎛V −V ⎞ • ⎜ final initial ⎟ ΔH =V dV =V ⎜ ⎟ g g ⎜ ⎟ dt ⎜ time to climb ⎟ AF ⎝ ⎠

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5) Combine the previous corrections then multiply this by the “inertial correction” factor that accounts for the inertial effects of changing the weight from test to standard conditions

• W H = t I W s

6) To the above result, add a correction for the change in induced drag due to weight change.

• 2 ⎡W 2 −W 2 ⎤ ΔH = t s ⎢ ⎥ Ind πAReρaltVT S ⎣ Ws ⎦

Summary of climb rate corrections

• ⎧⎛ T ⎞dH • • • ⎫ • • H = ⎜ t ⎟ + ΔH +ΔH +ΔH H +ΔH ⎨⎜ ⎟ ⎬ std ⎝Ts ⎠ dt indicated P hw AF I Ind(Eq’n 12.27) ⎩ ⎭

Equation 12.27 can also be used to correct descents, level accelerations, and level decelerations to a stand- ard climb rate. The primary difference is that for level accelerations, the accelerations factor is the dominant term while the indicated climb rate is near zero.

12.5.3 Weight/Altitude/Temperature (WAT) Limits To ensure safety, aviation authorities specify minimum climb gradients (γreq‘d) for many aircraft opera- tions. The most straight forward way to comply with the specified gradients is to document the maximum allow- able weight at various pressure altitude/temperature combinations. Assuming the test day CL for best γ equals that for any other day, calculate the maximum allowable weight by applying the following correction to the best test γ results.

D L ⎡ C O C ⎤ ⎢sin γt + + ⎥Wt + ΔFn ⎣ CL πARe ⎦ Wmax = CD CL O sin γ req'd + + CL πARe where ΔFn (= Fnstd -Fntest ) comes from the engine model. To ensure accuracy, the test configuration (i.e., one engine inoperative) must equal the standard configuration. Level acceleration results are not an acceptable sub- stitute for actual climb data.

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12.6 Level Turn Performance (ref 12.1)

Standard level turn performance can be determined either by predicting results using (flight test validated) models or by correcting individual turn results to standard conditions. It is possible to predict turn performance using climb or level ac- celeration data, but this approach is not always accurate and should be validated with actual turn results. Performance predictions require accurate net thrust and aero models (drag polar and lift curve). Corrections to test day turn results require models that only show the change in thrust & drag between test and standard conditions. The following sections address both the prediction and correction approaches. For either approach, load factor (nzw) is usually determined first, then the corresponding turn rate (ω-radians/sec) and radius (R-ft) are cal- culated using the equations below.

2 V 2 g n 2 − 1 ⎛ ωV ⎞ R = T ω = zw n = ⎜ ⎟ + 1 (Eq’n 12.28) 2 zw ⎜ ⎟ g n −1 VT g zw ⎝ ⎠

12.6.1 Sustained Level Turn Performance Prediction

1) At the desired speed, altitude, temperature, and throttle setting use the engine model to determine the gross thrust (Fg). Sophisticated models may show this to be a function of the inlet angle of attack as well.

2) At the same conditions, use the engine and airframe models to determine the ram drag (Fe).

3) Calculate net thrust as Fn = Fg cos αF +Fe where αF = (α + ιT) and is ιT the incidence angle of the thrust line (TED positive).

4) The total lift is the sum of the wing lift and the thrust lift:

L = LW + Fg sinαF.

Since L = nzwW , then LW = nzwW - Fg sinαF

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5) For any sustained turn, the net thrust equals the drag

⎡ 2 ⎤ ⎛ LW ⎞ ⎢ ⎜ ⎟ ⎥ 2 ⎜ ⎟ ⎡ n W − F sin α ⎤ ⎢ ⎝ qS ⎠ ⎥ ()zw g F Fn = D = qSC D = qS C D o + = qS ⎢C D o + 2 ⎥ ⎢ πA Re ⎥ ()qS πA Re ⎢ ⎥ ⎣⎢ ⎦⎥

⎣⎢ ⎦⎥

Solving for load factor gives

⎛ 1/ 2 ⎞ 1 ⎜⎧⎡Fg cos()α +ιT + Fe ⎤ 2 ⎫ ⎟ nzw = ⎨ −CD ()qS πARe⎬ + Fg sin (α +ιT ) W ⎜ ⎢ qS o ⎥ ⎟ ⎝⎩⎣ ⎦ ⎭ ⎠ (Eq’n 12.29)

For any combination of weight, altitude, and airspeed, calculation of the standard sustained load factor requires knowledge of the gross thrust, ram drag, drag polar (C , e), and angle1/ 2of attack. ⎛ ⎧⎡ F D0 ⎤ ⎫ ⎞ n = W ⎜ qS n − C πA Re + F sin α ⎟ zw ⎜ ⎨⎢ D o ⎥ ⎬ n F ⎟ 6) To determine the standard angle of attack,⎩⎣ startqS with the⎦ lift curve⎭ slope model ⎝ ⎠ dC C = C + L α = C + C α L Lα =0 dα Lα =0 Lα

Rearrange to solve for α

2 ⎛ n W − F sin(α +ι )⎞ ⎜ zw g T ⎟ − C C − C ⎜ qS ⎟ Lα =0 L Lα =0 ⎝ ⎠ α = = (Eq’n 12.30) C C Lα Lα

Because α cannot be solved for explicitly, calculate it using successive iterations of Equations 12.29 and 12.30.

Fg n zw In cases where sin α F < W 10 the angle of attack can be roughly estimated without significant error to the final result.

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12.6.2 Sustained Level Turn Performance Correction The best method for obtaining standardized sustained level turn data is to correct actual level turn results to standard conditions. It is also possible to correct level acceleration or climb data to give standard level turn results. This approach may not work as well since any drag polar or engine model errors will be magnified. Additionally, inlet distortion that accompanies actual turn thrust is different during (low angle of attack) climbs and accelerations. The equation below corrects any combination of test day climb, turn, and acceleration to a load factor for a sustained turn at the same dynamic pressure but at standard conditions.

F n = g std sin α + zw std Fs Ws

1 ⎧ 2 ⎫2 ⎪⎛ W Fg ⎞ πARe qS ⎡WtV&T WH& ⎤⎪ ⎜n t − t sinα ⎟ + ⎢ t + t t + F cosα − F cosα ⎥ (Eq’n 12.31) ⎨⎜ zwt Ft ⎟ 2 gstd Fstd gt Ft ⎬ Ws Ws Ws g VT ⎩⎪⎝ ⎠ ⎣⎢ t ⎦⎥⎭⎪

Fg nzw If sin α F < then the above equation can be closely approximated as W 10

1 2 ⎡V&T Wt H& W ⎤ n = ()n W + πAReqS ⎢ t + t t + ΔF ⎥ (Eq’n 12.32) zw std W zwt t g V ex S ⎣⎢ T ⎦⎥ where ΔFex = Fgstd(cosαFstd) - Fgt(cos αFt)

The primary difference between using turn, accel, or climb test data is the dominant term in the above correc- tions. In all cases, the test and standard day thrust values come from engine models.

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12.6.3 Level Limit Turn Performance Correction A limit turn is one in which the aircraft performs a level turn beginning from maximum speed and maxi- mum load factor and continues to decelerate at the Nzb limit until reaching the maximum CL . At this point, the aircraft continues its level turning deceleration at the lift limit. This maneuver is also known as a “slow-down” turn. Test day limit turn data is corrected to a standard specific excess power (Ps) for each given combination of altitude, Mach number and load factor (or AOA) limit. The following correction accounts for changes in trim drag, weight, and atmospheric affects on thrust.

P = P +ΔP ss st s Wa ⎛ t xwt ⎞ ⎜ ⎟VT ()F V ()m a V ⎜ g ⎟ t where P = ext Tt = t xwt Tt = ⎝ ⎠ = N V = Ma θ N st xwt Tt o t xwt Wt Wt Wt and

θs θt qt θt qs θs ΔPs = Mao{()Fg cosαF +Fe W −()Fg cosαF +Fe W +SCD [ W − W ] s s s t t t o t s

N W −Fg sinα 2 N W −F sinα 2 Sqt θt ⎛ zwt t nt Ft ⎞ Sqs θs ⎛ zws s gs Fs ⎞⎫ + W ⎜m[]q S +ΔCD ⎟− W ⎜m[]q S +ΔCD ⎟⎬ t ⎝ t trimt ⎠ s ⎝ s trims ⎠⎭ where C N W − F sin α α = ι − Loα + zwt t gt Ft Ft Tt (Eq’n 12.33) a aqt S C N W − F sin α and α = ι − Loα + zws s g s Fs Fs Ts a aq S s

As with the sustained level turn case, one cannot solve explicitly for αF, so either assume an approximate value or iterate until a solution converges. In For the simplified case where δt = δs, cgt = cgstd , and sinαF = 0, then the above equation reduces to

2 2 θ N W θ N W ⎧ s ⎡ ()zws s ⎤ t ⎡ ( zwt t ) ⎤⎫ ΔPs = Mao ⎨ Fn −qSCD − − Fn −qSCD − ⎬ (Eq’n 12.34) ⎩ Ws ⎣⎢ s o qSπARe ⎦⎥ Wt ⎣⎢ t o qSπARe ⎦⎥⎭

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12.7 Reciprocating Engine Cruise Performance (ref 12.1) Cruise performance standardization consists of correcting test day range and endurance results to standard conditions. Standard conditions are typically the standard aircraft weight & cg location, the nearest 5,000 ft in- crement of pressure altitude, and standard ambient temperature at that altitude. Although not included in this section, additional corrections can be made to adjust fuel flow to a standard heating value and to adjust the thrust and fuel flow for the slight gravity effects due to changes in latitude and centrifugal relief (see section 3.2). Although any weight can be called “standard,” several are quite common. General aviation aircraft typi- cally have the test data corrected to the maximum takeoff weight. Transport aircraft often use a mid-mission weight (maximum payload and one-half fuel) as standard, and fighter/attack aircraft typically use full ordnance and half internal fuel as standard for any given configuration. Once standard cruise results are documented, mission planning can be conducted by reversing the stand- ardization equations to the desired “mission” conditions. If desired, test day results can be directly corrected to mission conditions by simply treating the mission conditions as standard. These options are shown below. It is common practice to correct test data for only minor changes in altitude & temperature conditions. Because large changes in Mach and Reynolds numbers alter drag polars and engine efficiency, it is not common practice to correct results across altitude differences of more than 5,000 feet. This leads to a series of results separated by altitude. . 12.7.1 Power Standardization If fuel flow is directly proportional to power output only, the power and optimal velocity for cruise perfor- mance can be determined from a power required curve as shown below.

Power Best Required End.

Best Range

VT

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To correct the power required curve to any standard altitude/weight condition, the usual approach is to treat the lift coefficient as the anchor (CL test = CL std). This leads to the following power and velocity standardization equations 3 1 1 ⎛ W ⎞ 2 ⎛ σ ⎞ 2 ⎛ W σ ⎞ 2 P = P ⎜ s ⎟ ⎜ t ⎟ V = V ⎜ s t ⎟ (Eq’ns 12.35) s t ⎜ ⎟ ⎜ ⎟ T s T t ⎜ ⎟ ⎝ W t ⎠ ⎝ σ s ⎠ ⎝ W t σ s ⎠

Because the drag polar of low performance propeller aircraft generally collapses (generalizes) well to a single curve, it is often acceptable to correct all power required data to a single standard altitude/weight condition. When this condition is chosen to be standard day sea level at maximum weight, the above correction simplifies to what is known as the “Piw ~Viw” values.

3 1 2 1 2 ⎛Ws ⎞ ⎛Ws ⎞ P = P = P⎜ ⎟ ()σ 2 V = V = V ⎜ ⎟ (Eq’ns 12.36) s iw t ⎜ ⎟ t T s iw et ⎜ ⎟ ⎝Wt ⎠ ⎝ Wt ⎠

Although all points along the test day power curve can be standardized, the most useful points are those for best range and endurance. When corrected to standard conditions, the performance of the test aircraft can be fairly compared to that of another aircraft which has also been corrected to the same flight conditions.

Additionally, once the standard power and velocity are known and documented, the required power and airspeed for any “mission” conditions can be predicted by reversing Equations 12.36 as follows

3 1 1 1 2 2 2 2 ⎛ W ⎞ ⎛ 1 ⎞ ⎛ Ws ⎞ ⎛ 1 ⎞ P = P ⎜ m ⎟ ⎜ ⎟ V = V ⎜ ⎟ ⎜ ⎟ (Eq’ns 12.37) m iw ⎜ ⎟ ⎜ ⎟ T m iw ⎜ W ⎟ ⎜ σ ⎟ ⎝ W s ⎠ ⎝ σ m ⎠ ⎝ t ⎠ ⎝ m ⎠

The power & optimal speed for best mission range (and mission endurance) are determined by applying the above equations to correct the points noted on the above figure. To correct directly from test conditions to mission conditions, apply Equation 12.35 and substitute mission weight and density in lieu of standard values.

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12.7.2 Fuel Flow Standardization Because reciprocating engine fuel flow is essentially proportional to power output, Equations 12.35 and 12.36 can be modified to correct the test fuel flow to standard values. For the following standardization equations to be accurate, the propeller efficiency and brake specific fuel consumption (BSFC) must be the same for test and standard days.

3 1 3 ⎛ W ⎞ 2 ⎛ σ ⎞ 2 2 1 s t ⎛ W s ⎞ W& = W& ⎜ ⎟ ⎜ ⎟ 2 f ft or W& = W& ⎜ ⎟ ()σ (Eq’ns 12.38) s ⎜ W ⎟ ⎜ σ ⎟ f iw ft ⎜ ⎟ t ⎝ t ⎠ ⎝ s ⎠ ⎝ Wt ⎠

Because BSFC is affected by engine RPM (due to friction losses), fuel flow results at one engine RPM are never corrected to another RPM. Separate tests must be performed for each engine speed of interest. BSFC may also be affected by ambient air pressure and temperature. If the relation between fuel flow and power can be repre- sented with a model as shown, then the fuel flow is a linear function of BSFC.

test Fuel Flow W& std f W& f = b + BSFC • BHP BSFC b

BHP

If the values for b and BSFC are known for both the test and standard conditions, then test fuel flow can be more exactly corrected to standard weight and density conditions as follows

3 1 η BSFC ⎛ W ⎞ 2 ⎛ σ ⎞ 2 W& = b + W& − b t s ⎜ s ⎟ ⎜ t ⎟ (Eq’n 12.39) f s s ()f t t ⎜ ⎟ ⎜ ⎟ η s BSFC t ⎝ Wt ⎠ ⎝ σ s ⎠

Note that this correction requires only a knowledge of the ratio of test and standard BSFC values. If both values have the same percent error, then the effect is self-canceling. The above equation also corrects for changes in fuel flow due to changing propeller efficiency.

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12.7.3 Endurance Optimization and Prediction To determine the optimum endurance flight profile and time aloft for any condition (at the same RPM as the test condition), plot the test day specific endurance parameter (SEP) vs the test day lift coefficient (CL)

RPM = xxx

3 2 ()n z W t SEP = t W& f t Max end.

2nz Wt CL = 2 ρ ()V S a Tt

The maximum endurance occurs at the peak of the SEP curve. The associated lift coefficient is the optimum endurance condition for the aircraft (at that same RPM). The results of this test change with engine speed. If the aircraft operates at this optimum CL or any other constant CL , then the total endurance time (t) while at con- stant altitude can be calculated from this test day data using

1 ⎛ σ ⎞ 2 ⎡ 2 2 ⎤ t = SEP ⎜ s ⎟ ⎢ − ⎥ (Eq’n 12.40) ⎜ σ ⎟ ⎝ t ⎠ ⎣⎢ W F W I ⎦⎥ where the SEP comes from the above test day curve at whatever CL is chosen. WI is the total aircraft weight at the start of the endurance segment and WF is the final weight of the endurance segment. This equation accounts for the effect of how a change in air density alters the power required and the subsequent fuel flow, but does not account for changes in propeller efficiency, BSFC, or the fuel flow intercept, b.For endurance at a constant CL and VT, use the following equation and the SEP and test weight associated with the lift coefficient at the start of the endurance segment.

SEP W t = ln I (Eq’n 12.41) W t W F

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12.7.4 Range Optimization and Prediction (ref 12.1) To determine the optimum range flight profile and distance for any condition (at the same RPM as the test condition), plot the test day range factor (RF) vs the test day lift coefficient (CL)

RPM = xxx

V W RF = Tt t W& f t Max Range

2nz Wt CL = ρ ()V 2 S a Tt

The maximum range occurs at the peak of the RF curve. The associated lift coefficient is the optimum range condition for the aircraft (at that same RPM). The results of this test change with engine speed. If the aircraft operates at this optimum CL or any other constant CL , then the range at constant altitude can be calculat- ed from this test day data using W R = RF ln I (Eq’n 12.42) W F where the RF comes from the above test day curve at whatever CL is chosen. WI is the total aircraft weight at the start of the range segment and WF is the final weight of the range segment. Although not explicitly shown in this equation, the correction does account for changes in air density, but does not account for changes in propel- ler efficiency, BSFC, or the fuel flow intercept, b. For cruise at constant airspeed and altitude, use the following equation and the RF associated with the lift coefficient at the start of the cruise segment.

⎡ W ⎤ 1− F ⎢ W ⎥ R = RF • 2 tan −1 ⎢ I ⎥ (Eq’n 12.43) W ⎢1+ F ⎥ ⎢ ⎥ ⎣ WI ⎦

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12.8 Jet Aircraft Cruise Performance (ref 12.1) Refer to section 12.7 for a general discussion of cruise performance standardization. . 12.8.1 Thrust Standardization If fuel flow is directly proportional to net thrust output only, the thrust and optimal velocity for cruise per- formance can be determined from a thrust required curve as shown below.

Thrust Best End. Required

Best Range

VT Because jet aircraft typically cruise at speeds where changes in Mach number affect the drag polar, it is custom- ary to treat both the lift coefficient and Mach numbers as anchors (CL test = CL std , M test = Mstd). In terms of Mach number, cruise (nz = 1) lift coefficient is calculated as

W C = δ (Eq’n 12.44) L 1481M 2S

From this relation, the only way to match test & standard values for both CL and M is to match test & standard values for W/δ . In this case, the test day net thrust required curve can be corrected to standard conditions as fol- lows

W θ F = F s V = V s (Eq’ns 12.45) n s nt T s T t W t θ t

Although all points along the test day thrust curve can be standardized, the most useful points are those for best range and endurance. When corrected to standard conditions, the performance of the test aircraft can be fairly compared to that of another aircraft which has also been corrected to the same flight conditions. Addition- ally, once the standard thrust and velocity are known and documented, the required power and airspeed for any “mission” conditions can be predicted by reversing Equations 12.45.

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12.8.2 Fuel Flow Standardization Both the thrust and fuel flow of a simple (fixed-geometry turbojet) turbine engine are functions of engine speed (N), Mach number (M), ambient pressure (δ), and ambient temperature (θ). Dimensional analysis and ex- perimental results show these parameters to be related approximately as illustrated in the figures below.

Corrected Fuel Flow Corrected Thrust vs vs Corrected RPM & M Corrected RPM & M

w& f F δθ δ

F E 100 M D M

ABC N ABC N θ θ

Thrust specific fuel consumption (TSFC ) is defined as the fuel flow per thrust. At any given level of cor- rected fuel flow, the above figures can be cross-plotted onto a single figure that relates corrected thrust (F/δ) to corrected fuel flow at various Mach numbers. .

F 200 150 δ F w& f E =100 D δθ

M 0.3 0.6 0.9

The slopes of the above figure exaggerate the typical case where TSFC changes with Mach number. If, at any given Mach number, steady increments of corrected fuel flow are evenly spaced vertically, then

TSFC ≈ constant at that Mach number. θ

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Standard fuel flow can be determined from these relations. If flight test data is to be corrected from test to standard conditions at the same CL and M , then the CD will also be the same for both test & standard conditions. Because thrust equals drag* during cruise, the following relations show that corrected thrust (F/δ) must be the same for test and standard conditions

D F δ δ C D = = 1481 M 2 S 1481 M 2 S

• technically Fn cos(α+ιT)+Fe = D, where Fn = net thrust, Fe = ram thrust, and ιT is the thrust incidence angle

If Mach number and F/δ are equal for both test & standard conditions, then the previous cross plot shows that corrected fuel flow must also be the same for both conditions.

W& W& f t = f s δ θ δ θ t t s s

This relation allows standard fuel flow to be calculated as

δ θ W& = W& s s (Eq’n 12.46) f s ft if CL test = CL std , M test = Mstd δ t θ t

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12.8.3 Endurance Optimization and Prediction Test day results are corrected to standard results at the same Mach, CL (and therefore the same W/δ , ac- cording to Eq‘n 12.44) as the test condition. For each W/δ ratio tested, plot the test day corrected fuel flow vs the test Mach number.

w & f δθ W δ

M

At any given W/δ , the maximum endurance occurs at the Mach corresponding to the bottom of the curve. This optimal Mach and W/δ define the optimum lift coefficient for endurance (Eq‘n 12.44). The corrected fuel flow for any desired Mach & W/δ combination can be interpolated from the above figure. Calculate the actual fuel flow using Equation 12.46.

Even with simple turbojets, experience has shown that the above curves do not generalize well if the desired standard altitudes

If the aircraft maintains flight at any combination of constant Mach & W/δ , then the corrected fuel flow will be constant. For flight at a constant CL, endurance time can be calculated using

1 θ C W (Eq’n 12.47) t = L ln I θ c C D W F where c is the thrust specific fuel consumption at sea level std conditions. WI is the total aircraft weight at the start of the endurance segment and WF is the final weight of the endurance segment. Although not explicitly shown in this equation, the correction process does account for changes in aircraft weight, and ambient pressure & temperature.

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12.8.4 Range Optimization and Prediction As with endurance analysis, test day range results are standardized at a common Mach & W/δ. Using the same corrected fuel flow vs Mach test data illustrated in sec- tion 12.8.3, create a cross plot of range factor (RF) vs Mach number for each W/δ tested.

W where range factor can be calculated as δ RF V M W RF = SR ⋅W = Tt W = t a t t W& t W& δ o ft ft t δ θ t t M (Eq’ns 12.48)

The optimum CL for range at any given W/δ occurs at the Mach corresponding to the top of the curve. The best overall W/δ is the highest. These curves do not usually generalize well if the desired standard altitudes are more than about 5,000 ft away from the test altitudes. If the aircraft cruises at any combination of constant Mach & W/δ , then the range factor will be constant, and range is calculated as W R = RF ln I (Eq’n 12.49) WF where RF comes from the above test day figure at whatever Mach & W/δ is chosen. It is often reasonable to interpolate the above test data to define a RF for the desired standard conditions. WI and WF are the total aircraft weights at the start and end of the range segment. For cruise at constant altitude, fly at a constant CL by allowing the airspeed to decrease with weight. Cal- culate range from test day results using

V Tt R = 2 Wt ( Wi − W f ) (Eq’n 12.50) W& ft

For this equation to be valid, use the VTt and fuel flow corresponding to the same CL and altitude of the desired standard conditions. Both of the above correction equations account for changes in aircraft weight and ambient temperature.

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12.9 References

12.1 Lawless, Alan R, “Fixed Wing Aircraft Performance Testing,” Volume III, Professional Textbook Series, National Test Pilot School, Mojave CA, 1998.

12.2 anon, “Performance Flight Testing Phase” notes, USAF TPS , Edwards AFB, CA, 1991.

12.3 Lush, Kenneth J “Standardization of Take-Off Performance Measuresments for Airplanes,” AFFTC Tech- nical Note R-12, USAF Air Research and Development Command, Edwards AFB, CA, circa 1955.

12.4 Roberts, S.C., Light Aircraft Performance for Test Pilots and Flight Test Engineers, NTPS publication

12.5 Olson, Wayne, “Performance Testing Handbook,” (publication number pending ) AFFTC, Edwards AFB, CA, 1999.

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Section 13 Acoustics

13.1 Abbreviations and Terminology

13.2 Velocities, Spectrum and Reference Levels

13.3 Pressure, Intensity

13.4 Weighting Curves

13.5 1/3 Octave Center Frequencies

13.6 References

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13.1 Abbreviations and Terminology

Abbreviations ANSI Acoustic National Science Institute dB decibels f frequency, cycles/sec Hz Hertz nm 10-9 meters P sound power p pressure pW 10-12 Watts x RMS value of quantity xo reference value of quantity μPa 10-6 Pascals

Terminology decade band with the upper frequency x10 that of thelower. decibels measure of a magnitude, dB = 10log10(mag). far field beyond the near field (region where sound level drops -6 dB as distance from the source doubles). Hertz frequency in cycles/second. narrow band band whose width is less than one-third octave but less than 1% of the center frequency near field range within a distance equal to the wavelength of the lowest frequency emitted or twice the greatest dimension of the subject. octave a band with the upper freq exactly twice the lower freq. (common octaves include .0375-.075, .075-.15, 15-.3, .6-1.2, 1.2-2.4, 2.4-4.8, 4.8-9.6 kHz). pink noise has equal energy in each octave from 20 to 20,000 Hz, or with an energy content inversely proportional to frequency. random noise does not have a uniform frequency spectrum and has an amplitude, as a function of time, consis tent with a Gaussian distribution curve. third-octave highest frequency =1.26 x lower frequency (ratio= 21/3) white noise has a constant spectrum level over the entire band of audible frequencies (need not be random).

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13.2 Acoustic Velocities, Spectrum, and Reference Levels

Acoustic Velocity (speed of sound)

Medium Approximate Velocity Air (20o C) 343 m/s Fresh water 1,480 m/s Aluminum 5,150 m/s Concrete 3,600 m/s Glass 5,300 m/s Steel 6,000 m/s

Wavelength (λ) = acoustic velocity frequency wavelength (m) in air

frequency (Hz)

Human hearing range is approximately 20 to 20,000 Hz • Ultrasound lies above 20,000 Hz • Infrasound lies below 20 Hz

Acoustic Reference Levels Quantity Formula 2 Velocity (Lv) 20log(v/v0) vo = 10 nm/s 2 Intensity (LI) 10log(I/I0) Io= 1 pW/m Sound Power Level (LW) 10log(P/P0) Po = 1 pW Sound Pressure Level “SPL” (Lp) 20log(p/p0) 20μPa (air) Pressure Spectrum Level (PSL)* SPL – 10logΔf (dB) Pressure Band Level (PBL) PSL + 10logΔf (dB) SPL/10 Overall SPL (OASPL) 10log10 Σ10 20 μPa (air)

* the SPL contained within a band 1 Hz wide

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13.3 Acoustic Pressure and Intensity

Sound Pressure from Sound Power

Transmission Environment Lp Free Field LW + log Q - 20 log r - 10.8 dB Reflecting Plane LW + log Q - 20 log r - 7.8 dB Reverberant Room LW + log Q - 20l og R - 6.2 dB

where r = distance from source Q = directivity index of source R = room constant

Acoustic Intensity

I - Imaginary[Gyx(f)] = Im[Gyx(f)](for air) 4πρ0Δrf 16.25 Δrf

3 where ρ0 = fluid density = 1.293 kg/m for air Δr = microphone spacing (meters) f = frequency

Intensity Spectrum Level (ISL)

Intensity level of a sound contained within a band 1Hz wide

ISL = 10 log _I_ = IL – 10 logΔf (dB) IoΔf

where f = center frequency of band I = sound intensity (watts/m2) -12 2 Io = 10 watt/m reference intensity Δf = bandwidth (Hz)

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13.4 Acoustic Weighting Curves (ANSI S1.4 1983) Weighting for SPL Nominal Exact A B C Freq (Hz) Freq (Hz) (dB) (dB) (dB) 10 10.00 -70.4 -38.2 -14.3 12.5 12.59 -63.6 -33.3 -11.3 16 15.85 -56.4 -28.3 -8.4 20 19.95 -50.4 -24.2 -6.2 25 25.12 -44.8 -20.5 -4.4 31.5 31.62 -39.5 -17.1 -3.0 40 39.81 -34.5 -14.1 -2.0 50 50.12 -30.3 -11.6 -1.3 63 63.10 -26.2 -9.4 -0.8 80 79.43 -22.4 -7.3 -0.5 100 100.0 -19.1 -5.6 -0.3 125 126.9 -16.2 -4.2 -0.2 160 158.5 -13.2 -2.9 -0.1 200 199.5 -10.8 -2.0 .0 250 251.2 -8.7 -1.4 .0 315 316.2 -6.6 -0.9 .0 400 398.1 -4.8 -0.5 .0 500 501.2 -3.2 -0.3 .0 630 631.0 -1.9 -0.1 .0 800 794.3 -0.8 .0 .0 1,000 1,000 .0 .0 .0 1,250 1,259 0.6 .0 .0 1,600 1,585 1.0 .0 -0.1 2,000 1,995 1.2 -0.1 -0.2 2,500 2,512 1.3 -0.2 -0.3 3,150 3,162 1.2 -0.4 -0.5 4,000 3,981 1.0 -0.7 -0.8 5,000 5,012 0.6 -1.2 -1.3 6,300 6,310 -0.1 -1.9 -2.0 8,000 7,943 -1.1 -2.9 -3.0 10,000 10,000 -2.5 -4.3 -4.4 12,500 12,589 -4.3 -6.1 -6.2 16,000 15,849 -6.7 -8.5 -8.6 20,000 19,953 -9.3 -11.2 -11.3

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13.5 1/3 Octave Center Frequencies (ANSI S1.6 1984)

Band Nominal Exact Octave No. Center (Hz) Center (Hz) Center (Hz) 1 1.25 1.26 2 1.60 1.58 3 2.00 2.00 2.0 4 2.50 2.51 5 3.15 3.16 6 4.00 3.98 4.0 7 5.00 5.01 8 6.30 6.31 9 8.00 7.94 8.0 10 10.00 10.00 11 12.5 12.59 12 16.0 15.58 16.0 13 20.0 19.95 14 25.0 25.12 15 31.5 31.62 31.5 16 40.0 39.81 17 50.0 50.12 18 63.0 63.10 63.0 19 80.0 79.43 20 100.0 100.00 21 125.0 125.89 125.0 22 160.0 158.49 23 200.0 199.53 24 250.0 251.19 250.0 25 315.0 316.23 26 400.0 398.11 27 500.0 501.19 500.0 28 630.0 630.96 29 800.0 794.33 30 1,000 1,000.0 1,000 31 1,250 1,258.9 32 1,600 1,584.9 33 2,000 1,995.3 2,000 34 2,500 2,511.9 35 3,150 3,162.3 36 4,000 3,981.1 4,000 37 5,000 5,011.9 38 6,300 6,309.6 39 8,000 7,943.3 8,000 40 10,000 10,000.0 41 12,500 12,589.3 42 16,000 15,848.9 16,000 43 20,000 19,952.6

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13.6 References

13.1 Beranek, Leo L., Acoustic Measurements, John Wiley & Sons, New York, New York, 1956.

13.2 Peterson, Arnold P.G. and Gross, Ervin E., Jr., Handbook of Noise Measurement, GenRag Incorporated, Concord, Massachusetts, 1978.

13.3 Measuring Sound, (Pamphlet), Bruel & Kjaer, Naerum, Denmark, September 1984.

13.4 Pocket Handbook, Noise, Vibration, Light, Thermal Comfort, Bruel & Kjaer, Naerum, Denmark, 1986.

Additional Reading

Hunter, Joseph L., Acoustics, Prentice-Hall Incorporated, Englewood Cliffs, New Jersey, 1957.

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NOTES

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Section 14 Electromagnetic Compatibility

14.1 Electromagnetic Compatibility (EMC) 14.2 Abbreviations 14.3 Terms 14.4 Fundamentals 14.4.1 Electric and Magnetic Fields 14.4.2 Antennas 14.4.3 Spectra 14.4.4 Non-Ideal behavior of components 14.5 Electromagnetic Interference (EMI) 14.5.1 Interference Model 15.5.2 Conducted Emissions 15.5.3 Radiated Emissions 15.5.4 Aviation Frequency Spectrum 14.6 Testing 14.6.1 Lab Testing 14.6.2 Aircraft Ground Testing 14.6.3 Aircraft Flight Testing 14.6.4 Avionics changes and EMI testing 14.7 Lightning 14.7.1 Aircraft Lightning Zones 14.7.2 Direct Effects 14.7.3 Indirect Effects 14.7.4 Instrumentation Precaution 14.8 High Intensity Radiated Fields (HIRF) 14.9 Precipitation Static (Pstatic) 14.10 Reference Material

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14.1 Electromagnetic Compatibility (EMC) This section gives the Flight Test Engineer a basic introduction to terms and concepts used by EMC engineers and insight into good testing philosophy and appropriate practices thus improving the interaction between the EMC and Flight Test engineers.

14.2 Abbreviations A Area m2 c Speed of Light 3.0E8 m/s E Electric Field Intensity Volts/meter, V/m f Frequency Hertz, Hz H Magnetic Field Intensity Ampere/meter, A/m I Current Ampere, A L Inductance Henries Q Charge Coulomb, C V Electric Potential Volt, V XC Capacitive Impedance Ohms XL Inductive Impedance Ohms λ Wavelength meter

14.3 Terms AC Alternating Current DC Direct Current EMC Electromagnetic Compatibility EMI Electromagnetic Interference Far Field Distance beyond 10λ HIRF High Intensity Radiated Fields RF Radio Frequency DecibelLogarithmic (base 10) expression for amplitude ratios.

dB(power) = 10 Log10 (P1/P2)

dB(voltage) = 20 Log10 (V1/V2)

dB(current) = 20 Log10 (I1/I2) Commonly used decibels for EMC: dBm decibels relative to 1 milliwatt dBW decibels relative to 1 watt dBμV decibels relative to 1 microvolt dBi antenna gain relative to an isotropic antenna

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Power V or I dB Ratio Ratio 0 1 1 3 2.0 1.4 6 4 2 10 10 3.2 20 100 10 30 1000 32 Common Decibel Values The sensitivity of a radio receiver can be on the order of 1 µV/m, while RF field strengths for HIRF can be 1000V/m, a factor of a billion or 180dBµV/m.

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14.4 Fundamentals

14.4.1 Electric and Magnetic Fields A static charge Q, creates a static Electric field, ‘E’.

+ q

A common magnet produces a static magnetic field, ‘H’.

N S

Transferring charge Q, i.e. DC current on a wire creates both a constant magnitude Electric and Magnetic field.

i wire i wire

Electric Field ‘E’ Magnetic Field ‘H’ An amplitude varying charge, i.e. changing current (AC) will generate a time varying Electric and Magnetic Field in different planes.

y y

z z

x x Electric Field ‘E’ Magnetic Field ‘H’

Plane Waves that are self sustaining Electric and Magnetic Fields and combine in the far field, are commonly called an Electromagnetic Wave.

y E

z

H x

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14.4.2 Antennas Antennas can transmit and/or Receive RF equally well. Electrical length determines effectiveness.

λ / 2 is an efficient antenna element length c where: λ = f

Frequency, f Wavelength, λ 3 MHz 100m 30 MHz 10 m 150 MHz 2 m 300 MHz 1 m 3 GHz 0.10 m

Common Wavelengths

Slots that have favorable electrical lengths are effective antenna elements also, i.e. hatches, doors, avionics metal enclosure seams and ventilation holes.

Loop Area is the area encapsulated between the signal line and its return path that can be an effective antenna. The larger the loop (capture) area, the better the antenna effectiveness is.

14.4.3 Spectra are the frequency content of the electronic signals and are an important consideration in understanding EMI issues. Periodic signals contain energy at various frequencies and as such, a frequency domain approach is needed. How much energy at what frequency depends largely on the type of periodic signal, (i.e. square wave or sine wave), initial frequency and rise/fall times of the signal. The faster the rise/fall times are, the more spectral content will be developed in the signal, most of which will be unintentional and unwanted. This is mathematically demonstrated by the use of a trigonometric Fourier series.

14.4.4 Non-Ideal behavior of components can exists in discrete components such as resistors, capacitors, inductors and even wire when operated at off nominal conditions, for example temperature. Another condition is frequency. For example, a short grounding wire from a DC perspective is a dead short, neglecting the extremely small inductance. But at some frequency, this inductance gets large enough to be a factor, for example on a bonding strap for lightning protection. A 26 gauge wire, 1 inch above a ground plane will have 0.028 µH per inch of inductance (L).

Where: XL=2пfL

For f = 150 MHz; XL=26.4 Ohms per inch of wire which can be significant. To reduce this, replace the ground wire with a wide strap.

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14.5 Electromagnetic Interference (EMI) Electromagnetic Compatibility (EMC) is defined as systems that: a) don’t interfere with other systems; b) are tolerant of interference from other systems; c) don’t interfere with itself. Broadband Interference is interfering signals over a large range of frequencies. These can be associated with spark producing equipment like motors that can create signals with lots of spectral content. Narrowband Interference is interfering signals that have a limited range of frequencies, usually a single frequency along with it’s harmonics. These can be associated with digital devices that have periodic characteristics like clocks. 14.5.1 Interference Model The classic interference model is: Source Æ Path Æ Victim To reduce the interference you can: –Reduce the emissions from the Source –Disrupt the Path –Harden the Victim

Sources of interference can be clocks, switching power supplies, CPUs, data buses, network systems, relays, local oscillators, and transmitter harmonics. Coupling Paths can be signal and power lines, radiating wires, apertures or slots on LRUs, windows, door and hatch openings or antennas themselves. Front Door coupling is meant to be interference coming in the normal path to the system, i.e. through the antenna ports to the radio, and can cause interference at extremely low power levels (-100dBm). Back Door coupling is interference coming into the system with the wires leading to the system and is of relatively higher power. Capacitive coupling primarily involves electric waves in the near field and is due to voltages on wires. Inductive coupling primarily involves magnetic waves in the near field and is due to current on either wires or chassis. The aircraft fuselage is sometimes incorrectly thought of as a Faraday Cage encapsulating the RF energy inside or preventing it from entering because of its aluminum structure, but actually it is not. All of the windows, doors and hatches allow RF energy to travel through quite easily. Victims of interference can be radio receivers, VHF, HF, VOR, ILS, ADF, Display systems, Audio and Passenger Address system, smoke and fire detection circuits, fuel quantity systems. Typically, low energy systems can be susceptible. The reduction or elimination of EMI can be done in three areas; the systems end; by modifying the emissions and/or susceptibility requirements; or at the aircraft end by modifying the aircrafts wiring or structure.

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14.5.2 Conducted Emissions Current/signal on wires that are not the intended or primary signal is considered conducted emissions. This ‘extra’ current will be passed along to other systems and/or can radiate on those wires acting like antennas.

Differential Mode current is made up of the intended signal or information and/or noise that goes out on the signal wires and comes back on the return lines.

Common Mode current is usually just noise that goes out on two or more signal/return lines and returns via some other path. This is usually the most troublesome in terms of emissions and should be eliminated whenever possible.

14.5.3 Radiated Emissions RF energy emanating from the unit/LRU itself through holes, slots and apertures or from the interconnecting wires is considered radiated emissions.

14.5.4 Aviation Frequency Spectrum The table below lists the frequency spectrum of interest to the aviation community. The range is from 100kHz to 10GHz, a factor of 108 , (90dB). The primary interest is with equipment that is sensitive to RF energy, i.e. radio receivers, which are primarily intended to detect small signals (-105dBm). Emission requirements are set at a low level that will still allow proper operation of the radio receivers. For EMI purposes, emissions from equipment should stay clear of these frequencies.

Band Frequency ADF 190–1750 kHz HF 2–30 MHz Marker Beacon 75 MHz VHF Nav 108-118 MHz VHF Comm 118–138 MHz Glideslope 328-335 MHz DME, ATC, TCAS 960-1220 MHz GPS 1227, 1558, 1575 MHz Glonass 1609 MHz Radio Altitude 4.2-4.4 GHz MLS 5.0-5.25 GHz WXR 5.4, 8.8, 9.0-9.3 GHz Aviation Frequencies of Interest

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14.6 Testing

Regulations and Industry Guidance The following references are regulations and industry guidelines that address procedures and acceptable limits for interference testing. -RTCA DO160D, Chapter 21 -FARs Part 25.1353 and 25.1431 -MIL STD 461 -CISPR -FCC Part 15 -Aircraft manufacturers own standards

14.6.1 Lab Testing Lab testing of the unit using established standards and practices is the first and best means of testing. Not only is this where you will find the trouble spots (i.e. frequencies) but also is a place where some troubleshooting could alleviate potential problem areas. Contracts with LRU vendors should be written to require the equipment pass these tests, identified above, before delivery. A list of frequencies that exceed an established limit is the result.

14.6.2 Aircraft Ground Testing After lab testing, the unit should be installed in the airplane and be tested with the installed shops wiring. Testing will consist of measuring conducted emissions with current probes on wire bundles associated with the new equipment.

Radiated emissions are tested by using the aircrafts antennas hooked to test equipment to determine how much RF energy is getting into these sensitive systems. Again, a list of frequencies that exceed an established limit is the result.

14.6.3 Aircraft Flight Testing Only after both lab and ground testing is accomplished can a meaningful flight test occur. The results of the ground test should produce a list of frequencies of some exceedance or observed interference. It is usually only these frequencies that need to be cleared in flight. The appropriate systems should be tuned to those frequencies and with the equipment to be tested in its’ operating mode, determine if there is objectionable interference, (usually a pilots subjective opinion). Pilots can evaluate systems only if adequate lab/ground testing has been done beforehand. EMI issues that are found in Flight Test are very difficult and expensive to fix at this stage, and can typically only reduce or mask the problem.

14.6.4 Avionics changes and EMI testing Changes in the hardware/wiring of a piece of avionics that could affect EMI testing are: -Processor speeds -Power Supply changes -Frequency sensitive components, capacitors and inductors -Circuit card layout and repackaging changes

Software changes typically don’t affect EMI unless software controls/switches hardware related functions, i.e. speeds, options, peripherals etc.

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14.7 Lightning Lightning is a very large electrical transient that can impart thousands of Amperes of current through an aircraft structure. The structure needs to present a low impedance path for the lightning current so that no damage causing arcing and/or over-heating occurs. Additionally nearby wiring needs to be shielded to protect against the induced current produced by the ever changing magnetic fields.

14.7.1 Aircraft Lightning Zones The aircraft is divided into different areas that relate to the probability of a lightning attachment. The nose, tail, wingtips and engine nacelles (extremities) are more likely areas.

14.7.2 Direct Effects Direct effects of a lightning attachment can be in the form of heating, arcing and acoustic issues. Designing the structure to handle the current flow and providing a low impedance path for the lightning current will greatly minimize these effects.

14.7.3 Indirect Effects Indirect effects considers the current that is induced by the transient and coupled onto aircraft wiring that is parallel to the main lightning current flow. The protection is two fold. Systems are designed and tested to handle these types of transients as well as the wiring is addressed to minimize the induced transient to these systems. Shielding and good grounding with short pigtails at both ends is a good method to reduce the induced current.

14.7.4 Instrumentation Precaution Any flight test instrumentation wiring that lies outside the protective fuselage needs to be evaluated for both direct and indirect effects of a nearby lightning attachment. The sensor itself must be protected from the direct attachment and the wiring must be protected from induced current onto that wiring. This current may damage the data system equipment and/or, other aircraft systems that are also instrumented. Good shielding and grounding techniques will minimize these effects. For more information see the 10-6 Reference at the end of this handbook section.

14.8 High Intensity Radiated Fields (HIRF) Aircraft can be exposed to large RF energy produced by high powered radio transmitters or military/airport surveillance radars. These RF fields can penetrate the aircraft fuselage through windows and doors/slots which could couple with aircraft wiring and/or systems and potentially interfere. This threat is addressed by both the aircraft and systems approach.

The systems themselves are designed and tested to be immune to a particular level of RF. These levels are determined by the criticality of the systems and are specified in regulatory material. Testing is usually done in a laboratory environment.

From the aircraft side, the internal wiring for critical systems is protected with appropriate shielding and grounding. Aircraft ground testing is done at special facilities that can radiate the vehicle with large RF fields with instrumentation inside to measure the penetration and to verify correct system operation.

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14.9 Precipitation Static (Pstatic) This occurs due to a buildup of static charges that discharge by noisy arcing from/to various parts of the aircraft. The static buildup is caused by tribo-electric charging from the aircraft impacting snow/rain/ash particles in the air while flying. This charge should gracefully exit the aircraft through static wicks installed on the wingtips and empennage tips. If it doesn’t the problem shows up as broad banded noise (white noise) heard on receivers such as ADF, HF and to some extent VHF as the aircraft flies through the precipitation.

Typical causes are access panels (composite and metal), cowling and fairings that are not properly grounded. Ground straps do a good job of not isolating parts. (Note: these straps should not be used for lightning protection as they usually are not sized to handle the current).

14.10 Reference Material

10.1) Paul, C. R., “Introduction to Electromagnetic Compatibility”, John Wiley & Sons Publishing, 1992 10.2) Ott, H. W, “Noise Reduction Techniques in Electronic Systems”, John Wiley & Sons Publishing, 1988 10.3) Hrehov, D. W. and Walen, D. B., “What Flight Test Crews Need to Know About EMI/EMC”, 34th Annual SFTE Symposium Workshop, 2003 10.4) Federal Aviation Regulations, Part 25 10.5) RTCA DO160D, “Environmental Conditions and Test Procedures for Airborne Equipment”, 1997 10.6) Hrehov, D. W., “What Instrumentation Engineers Need to Know About Lightning”, 31st Annual SFTE Symposium, 2000 10.7) Fisher, F. A., Perala, F. A., and Plumer, J A., “Lightning Protection for Aircraft”, Lightning Technologies Inc., 1990

Page 14 - 10 SFTE Reference Handbook Third Edition 2013 15 Handling Qualities

15.1 Cooper-Harper Rating Related Figures

Figure 15.1-1 Elements of Closed-Loop Handling Qualities

Figure 15.1-2 Undesirable step input responses and pilot compensation to achieve desired response a) Lag compensation, b) lead-lag compensation.

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Figure 15.1-3 Optimum Short Period Frequency and Damping Based on Pilot Opinion

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Figure 15.1-4 General Pilot Ratings for Handling Qualities

Closed Loop Handling Qualities Test Requirements 1) Explicit mission definition: what the pilot must accomplish. Identify the circumstances and operating condi- tions.

2) Define mission tasks. Tasks should be repeatable, require sufficient control input frequency to stress the sys- tem, and they should be of adequate duration to differentiate transient from steady state responses.

3) Establish desirable and acceptable criteria for task performance. Criteria established should be quantifiable, recordable, and realistic. Desirable criteria specify a satisfactory level of performance. Acceptable criteria speci- fy the level of performance that is marginally adequate.

4) Test should include realistic typical distractions and disturbances.

5) Record task performance relative to the criteria established (comments, video, audio, pipper movement, etc.)

6). Measuring & record pilot workload and compensation.

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Figure 15.1-5 Cooper Harper Workload and Handling Qualities Rating Scale

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Figure 15.1-6 Pilot Induced Oscillations Rating Scale

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NOTES

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Section 16 Rotary Wing

16.1 Principal Aeroderivatives

16.2 Forward Flight Static And Dynamic Stability

16.3 References (part 1)

16.4 Helicopter Performance Parameters

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16.1 PRINCIPAL AERODERIVATIVES

Derivative Common Name Principal Contributors Typical Sign CONTROL POWER

M Pitch control power MR Thrust vector - B1 Mast bending moment Control gearing Rotor type Effective hinge offset L Roll control power MR Thrust vector - A1 Mast bending moment Control gearing Rotor type Effective hinge offset N Yaw control power TR thrust - θTR TR moment arm Control gearing Z Heave control power MR thrust - θC Control gearing STATIC STABILITY M Speed stability MR flap back + u Mast bending moment Horizontal tailplane M Static/Incidence/Angle of MR flap back w Attack stability Mast bending moment Horizontal tailplane Fuselage L Lateral static stability MR ‘flap back’ - v (dihedral effect) TR vertical moment arm Fuselage N Directional static stability TR thrust + v (weathercock effect) Vertical tailplane Fuselage DAMPING X Drag damping Rotor drag - u Fuselage drag Y Side force Rotor drag - v Fuselage drag Heave damping MR characteristics - Z w L Roll damping Main rotor - p Effective hinge offset M Pitch damping Main rotor - q Effective hinge offset Horizontal tailplane N Yaw damping Tail rotor - r Vertical tailplane Fuselage

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16.1 PRINCIPAL AERODERIVATIVES (Continued)

Derivative Common Name Principal Contributors Typical Sign CROSS COUPLING L Tail rotor roll Tail rotor vertical position + θTR M Pitch change with power Forward speed + θC Main rotor N Torque reaction Torque θC Y Tail rotor drift Tail rotor θTR

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16.2 FORWARD FLIGHT STATIC AND DYNAMIC STABILITY

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16.3 References: Padfield, G.D., (2007), Helicopter Flight Dynamics, 2nd Edition, Blackwell Publishing, UK. Cooke, A., Fitzpatrick, E., (2002), Helicopter Test and Evaluation, Wiley Blackwell, UK. Leishman, J.G., (2006), Principles of Helicopter Aerodynamics, 2nd Edition, Cambridge University Press, UK.

Page 16- 5 SFTE Reference Handbook 2017 Addendum to Third Edition 16.4 Helicopter Performance Parameters

Compiled by Mark Roots

Symbol Description units D engine dimension (diameter) m G fuel flow (volumetric, weight or mass) l/h; N/h; kg/h P shaft power from engine kW W helicopter weight (mg) N V true airspeed kn Vc rate of climb feet per min (fpm) Ω rotor speed; rpm R rotor radius; m Z wheel height; ft ρ air density; kg/m3 p atmospheric pressure N/m2 T ambient temperature K RTMN - rotor tip Mach number, √ 0 (subscript) denotes sea level ISA values -

σ relative density = ρ/ρ0 -

δ relative pressure = p/p0 -

θ relative temperature = T/T0 -

ω relative rotor speed = Ω/Ω0 - ∆ engine configuration (eg guide vane position) -

Cp power coefficient

Ct thrust coefficient

Helicopter Performance Turbine engine helicopter performance can be expressed as non-dimensional parameters known as referred or reduced parameters [1][2]. This allows correcting test measurements to conditions other than those tested. The power to maintain a flight condition can be expressed in the general form (1) = , , , Ω, R, Z, , Including ambient temperature above allows investigating Rotor Tip Mach Number (RTMN) effects if required. Non-dimensionalizing the above leads to

(2) = , , , , √ To investigate RTMN effects, testers must adjust rotor speed to maintain constant . √ For an atmosphere defined as a function of pressure and temperature 16- 6 - SFTE Reference Handbook 2017 Addendum to Third Edition

(3) √ = , , , , √ Fuel flow is required for range & engine performance testing and power. Fuel flow (G) replaces P from equation (2)

(2a) = , , , , √ General equations (2) and (3) can adapt to the specific performance test (i.e. hover, level flight, engine).

Hover performance During hover, equation (2) simplifies to

(4) = , √ , If rotor speed is fixed, then RTMN cannot be accounted for and the term can be dropped, further reducing equation (4) to √

(4a) = , Alternatively from equation 3:

(5) √ = , , √ Any rotor speed effect shows as increased power at the same referred weight for increasing . √ In the USA different notation (and units) may be used and for the hover the groups may be presented in the form

× (6) = , √ , Or more succinctly,

(7) = , √ , Alternative referred groupings can be used, but consult the literature to determine the most appropriate [3][5]. Practical guidance on the use of referred groups for flight test is readily available [5]. Figure 1 illustrates a plot of equation 4a across various heights and referred weight combinations.

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Figure 1; Referred Power vs Referred Weight for varying wheel heigh t

Level Flight Performance For LFP, equations (2) and (3) reduce to

(8) = , , √

(9) √ = , , √

Equation (8) has the rotor speed in both the weight ( and rotor speed ( parameters, thus it is not ) ) possible to make independent adjustments by changing altitude and rotor speed.√ Figure 2 presents typical results using this method.

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Figure 2; LFP data for one referred weight

Equation (9) is only used when the rotor speed can be varied which allows for a systematic assessment of RTMN. At each test point the referred weight is kept constant by increasing altitude after each test point and the rotor speed is varied to keep constant. With sufficient data this method lends itself to the production of √ a power carpet for each RTMN, ( .as shown in Figure 3. √)

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Figure 3; LFP carpet plot for one referred rotor speed

Specific Air Range and Specific Endurance Use level flight performance data and fuel flow data to determine helicopter range and endurance. Whether expressed as weight flow, mass flow, or volume flow, fuel flow (G) leads directly to specific endurance and range calculations

Specific Endurance: =

Specific Range: =

For fixed values of W, Hp and T the variation of these parameters with airspeed can be plotted as shown in Figure 4. This figure shows how the speeds for maximum range and maximum endurance are determined.

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Figure 4; Specific range and specific endurance for constant weight altitude and temperature.

Engine Performance Engine power depends on compressor speed, engine configuration ( ∆ - guide inlet position, bleed air, etc), rotor speed (e.g. power turbine speed), and atmospheric conditions. (10) = , , Δ, Ω, , ) Referred power is

(11) √ = √ , √ , Δ Referred fuel flow follows as

(12) √ = √ , √ , Δ Referred engine temperature such as power turbine inlet temperature (PTIT) or jet pipe temperature (T4) is

(13) = √ , √ , Δ

Figures 5 illustrate how test data can be presented as plots of referred engine speed versus referred √ power , referred engine temperature , and referred fuel flow . √ √

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Referred Engine Speed, (Nc ) Referred Engine Speed, (Nc ) Referred Engine Speed, (Nc )

Figure 5: Example referred Engine Performance Plots

Careful test planning may collect data across much of the flight envelope with a relatively small test program. An understanding of aircraft limits allows testing to those limits. Evidence with advanced aerofoil sections shows that a greater element of “non linearity” may be seen if results are extrapolated too far. A sensible guide may be ±5000ft or ±20 °C unless there is solid evidence to support further extrapolation. Further guidance on helicopter performance testing and planning is readily available [1][5][6].

References: 1. P A Knowles – The application of non dimensional methods to the planning of helicopter performance flight test and the analysis of results , Ministry of Technology, UK, Jan 1966. 2. ESDU 73026 –Introduction to non dimensional methods for the measurement of performance of turbine engined helicopters , Engineering Science and Data Unit, March 1977 3. ESDU 73027 – Non dimensional methods for the measurement of hover performance of turbine engined helicopters , Engineering Science and Data Unit, March 1977 4. ESDU 74042 – Non-dimensional methods for the measurement of level flight performance of turbine engine helicopters, Engineering Science and Data Unit, November 1974 5. A Cooke and E Fitzpatrick, “Helicopter Test and Evaluation“, Blackwell Science, 2002 6. Mark Roots & Richard Blake, “Level Flight Performance – an Engineers Guide”, AHS Forum 58, June 2002 Acknowledgement This summary would not have been possible without the generosity of Gerhard Jordaan, Antoine Van Gent, & Carl Ockier and Al Lawless for proof reading and offering their many suggestions to improve the content.

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Section 17 Gas Turbine Propulsion

17.1 Turbine Engine Basics 17.1.1 Turbine Engine Types 17.2.2 The Brayton Cycle 17.2.3 Component Descriptions 17.2 Propulsion System Analysis 17.2.1 Standard Day Corrections 17.2.2 Pressure and Temperature Relationships with Flight Parameters 17.2.3 Reynold’s Number Index 17.2.4 Thrust Calculations 17.3 Turbine Engine Operation 17.3.1 Compressor operation during accelerations 17.3.2 Compressor operation during decelerations 17.3.3 Bleed are impacts during engine starts 17.3.4 Stator vane cambering effects 17.3.5 Compressor impacts due to nozzle area for turbojet engines 17.3.6 Compressor impacts due to nozzle area for turbofan engines 17.3.7 Combustion stability 17.4 Additional Information 17.4.1 Engine Stations 17.4.2 Key Propulsion Terminology 17.4.3 Common Aircraft and Associated Engines 17.4.4 Additional Propulsion Resources

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17.1 Turbine Engine Basics

17.1.1 Turbine Engine Types

Three basic types of aircraft turbine include turbojets, turbofans, and Turboshafts.

Turbojet Turboprop

Low Bypass Turbofan High Bypass Turbofan

The primary advantage of turbojets is their efficiency at high speed/altitude and small diameter. Turbojets are less efficient at low speed and are currently only used in older aircraft (B-52, T-38, Boeing 707). A turbofan engine is essentially a turbojet to which a second compression system has been added. Turbofans are further divided into low bypass and high bypass engines, where bypass describes the amount of air used by the fan that bypasses the compressor. In low bypass (0-1 bypass ratio) engines, most of the thrust comes from the nozzle. In high bypass (1-11+ bypass ratio) engines, most the thrust comes from the fan. Turbofans are the most popular engine for new medium and large aircraft due to the range of bypass ratios available, allowing optimization for most flight regimes. Turbofan advantages include: high thrust and low fuel consumption at low airspeed (subsonic), lower engine noise compared to turbojets, and generally lower operating temperatures allowing the use of lower cost materials. Turbofan disadvantages include large engine diameters that can increase aircraft drag and cause ground clearance issues and slower engine response compared to turbojets. Turboprops generate the majority of their thrust by driving an external propeller. They generally operate at slightly higher altitudes and faster airspeeds than conventional piston driven aircraft, but performance at higher mach numbers is limited due to compressibility effects at the propeller tip. The primary advantages of turboprops compared to reciprocating engines are fuel consumption improvements and increased reliability.

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Engine Type Operating Envelopes Some applications have unique flight envelopes and are required to support large horsepower extractions to power on-board sensors (e.g. Global Hawk).

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17.1.2 The Brayton Cycle

Aircraft turbine engines generally operate on the Brayton thermodynamic cycle. A simplistic explanation is provided using pressure-volume (P-V) and temperature-entropy (T-S) diagrams.

Ideal thermodynamic cycles (dotted) are those where: 1) The inlet and compressor (engine stations 0 to 3) isentropically compress the air. 2) The combustor (engine stations 3 to 4) provides isobaric heating. 3) The turbine and nozzle (engine stations 4 to 9) isentropically expand the air to free stream jet. 4) Free stream exhaust jet is at a higher velocity and temperature (and entropy) than the inlet.

In the real case (solid/dashed line), the inlet and compressor induce increased entropy (friction losses), the combustor has pressure losses, the turbine and exhaust nozzle do not perfectly expand the air to free stream pressure, and the exhaust jet is still at a higher velocity and temperature than the inlet. All of these factors decrease the efficiency of real turbines.

Turbojet, Real (solid/dashed) & Ideal (dotted) A more complex case is the two spool turbofan with afterburning. Although following the same trends, the additional “reheat” from the afterburner (stations 5-7) provides a significant increase in free stream exhaust jet temperature and velocity. Because the increase in temperature is never recovered by a turbine, actual efficiency is lower.

Turbofan With Afterburning, Real (solid/dashed) & Ideal (dotted)

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17.1.3 Component Descriptions (Example Axial Dual-spool Augmented Turbofan)

Inlet (Station 0 to 2) Inlets usually provide laminar, subsonic flow with minimal total pressure loss across a variety of Mach numbers and angles of attack. Subsonic inlets are typically simple with fixed geometry and supersonic inlets range from simple to complex using variable bleeds and bypasses depending on the operating conditions.

0 2 Complex Supersonic Inlet Simple Subsonic Turbofan Inlet

Compressor (Fan/Core) (Stations 2 to 3) The low pressure compressor, or fan, provides increased thrust and efficiency by accelerating a larger mass flow of air (compared to the high pressure compressor). It consists of stators (S) and rotor blades (R). A splitter in the duct following the fan separates airflow from the bypass duct. Below example, the fan is coupled via an inner shaft to the low pressure turbine.

Typical Low Pressure Compressor Typical High Pressure Compressor

The high pressure compressor provides airflow to the combustor and turbines. It can have many stages, each stage consisting of a rotor and a stator. The rotors impart kinetic energy into the airflow, while the stators convert the kinetic energy to a pressure rise. For improved operation, stators can also have variable geometry. Overall compression ratios can be 10 to 40 times ambient and the temperature rise more than 600 deg F. Bleed air from later compression stages can also be extracted to cool the turbine blades and provide airflow for auxiliary power or ice protection. Shaft power is also extracted through an engine mounted gearbox attached to the high pressure spool to power electrical and hydraulic systems.

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Combustor (Station 3 to 4) Fuel is injected, ignited, and burned in the combustor. Modern combustors are annular, while older designs consisted of multiple cans surrounding the shaft. Combustors slow the airflow entering the chamber to allow fuel-air mixture and prevent combustion outside the chamber. If air velocities are too great in the combustor, combustion stability is affected.

Igniter

Fuel Line & Injector

3 Dilution

Diffuser Primary Burn 4

Liner Compressor Exit Swirler Dome

Annular Combustor Combustor Cross-Section Combustors induce turbulence into the air to achieve proper fuel-air mixing and even burning. Approximately half of the air entering the combustor is used for the combustion process; the remainder cools the combustor panels. The exhaust gas temperature can exceed 3,100 deg F, leading to significant material challenges within the combustor and turbine sections.

Turbine (Station 4) Turbines extract energy from the combustor exhaust to drive the compressors. Extreme blade and vane temperatures drive special materials and/or active cooling requirements. State-of-the- art turbines may include single crystal nickel based alloys with thermal barrier coatings, internal cooling passages, and external film air cooling. The high pressure turbine powers the high pressure compressor, and the low pressure turbine powers the low pressure compressor.

Augmentor (Station 6) Afterburning burns fuel between the turbine and the exhaust nozzle to reheat the airflow. This reheat increases flow velocity and thus thrust, but because the temperature increase is not recovered as in the turbine, afterburners are very inefficient. While an increase in thrust can be obtained from a larger engine, the commensurate increase in weight and drag is not economical for short period requirements (e.g. aircraft takeoff).

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Exhaust Nozzle (Station 7 to 9) Air from the augmentor exits through the exhaust nozzle to provide the airplane with thrust. Efficient exhaust nozzles also reduce aircraft drag by matching exhaust pressure and ambient pressure. They can provide thrust vectoring to enhance aircraft stability, thrust reversing to improve aircraft braking, and noise suppression. Large commerical jets usually have fixed nozzles that are optimized for one cruise condition. Fighter aircraft often have variable nozzles to increase performance at all flight conditions.

Variable Converging-Diverging Crossection

Accessories Turbine engines require a variety of accessories to support engine and aircraft functions. Engine control can be managed by hydro-mechanical, analog, digital, or a combination of the control types. Modern engines use a full authority digital engine control (FADEC) to schedule engine operation throughout its operating range. Engine or aircraft sensors (e.g. T t2) provide operating conditions to the engine controller. An anti-ice valve can supply bleed air to the engine face struts to prevent ice build-up. A gearbox also extracts power from the high pressure compressor shaft to run electrical generators and aircraft hydraulic systems.

17.2 Propulsion System Analysis

Typically, a propulsion system’s operation is segregated into the five sub-categories or disciplines.

1. Overall: Integrated System Utility (Does it meet the users’ needs?). Topics include adequate engine bay ventilation, anti-ice, gun or gas ingestion, and inlet compatibility.

2. Performance: The ability to produce thrust at a prescribed level with a specified fuel flow. Usually prescribed over the life of the engine and is modeled with a propulsion system simulation. If an inlet rake is used for testing, we can calculate inlet recovery, which is an integral part of engine/aircraft performance.

3. Operability: The ability to resist or recover from an engine instability. These instabilities primarily refer to compressor stall or surge, which are aggravated by inlet temperature and pressure distortions. However, operability can include several other aspects, such as flameout, overspeed, overtemp, engine starting, and afterburner lighting and stability.

4. Response: The ability to change thrust conditions within a prescribed time in response to a commanded change.

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5. Life/Durability: The ability to withstand extended operating conditions (pressure, temperature, and rpm) over a prescribed lifetime (usually described in terms of engine operating hours or Total Accumulated Cycles) at a specified level of performance and operability.

A number of Aerospace Recommended Practices (ARP) exist to aid in standardization of gas turbine design, testing, and analysis. Aerospace Information Reports (AIR) also provide similar guidance. This handbook scope does not include the theory and concepts of these practices, however, some of the practices most relevant to propulsion system analysis are presented in the Additional Propulsion Resources section.

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17.2.1 Standard Day Corrections

Due to varying atmospheric conditions, engine tests are rarely conducted at the same flight conditions. Therefore, to compare tests results, data must be standardized to a common flight condition. By applying standard day corrections, the effects of changes in temperature and pressure can be removed from test results. Quantity Normal Corrected

Air mass flow rate Fuel flow rate 22

Rotational speed Thrust 0

θ δ i=total temperature ratio ( ) and i=total pressure ratio ( ) [Values can also be found in the standard atmosphere table, Section 3-14. Reference is sea-level, standard day. These values are also corrected for Mach number in Figure x-11.]

17.2.2 Pressure and Temperature Relationships with Flight Parameters

Two of the main inputs to a propulsion system are engine face pressure and temperature (P t2 and Tt2 respectively), yet the flight envelope is defined in altitude, airspeed, and Mach number. Consequently, it is helpful to be able to translate from one to the other. Unfortunately, because many engine inlets are variable, and therefore introduce varying pressure and temperature losses, the handbook assumes Pt2=P t0 and T t2= Tt0. The following chart is useful when describing the engine operating conditions relative to flight conditions.

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17.2.3 Reynolds Number Index

Assuming the characteristic length of an engine is constant in differing operating environments, another way of describing the pressure and temperature relationship is with the Reynolds number index. Similar to standard day corrections, the Reynolds Number Index provides a method of comparing engine operation across varying inlet-pressure loses, inlet temperatures, flight speeds, and altitudes. The Reynolds Number Index is:

where / 718.2 , , 199.5 2116 518.7

Here, P t2 is in psf, and T t2 is in degrees Rankin.

Pt2 and T t2 are assumed to be equal to aircraft total pressure and temperature. The actual in-flight Reynolds Number Index depends on inlet recovery losses, since these are the conditions in which the engine is operating.

100 KCAS 200 KCAS 300 KCAS 400 KCAS 50,000 Reynolds Number Index 500 KCAS 0.20

45,000

600 KCAS 40,000 0.30

0.40 35,000 700 KCAS

30,000 0.50 800 KCAS

0.60 25,000 0.70 Altitude (ft)

0.80 20,000 0.90

1.00 15,000 1.10

1.20 10,000

5,000

0 0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25 2.5 Mach Number

Reynolds Number Index versus Altitude and Mach Number

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17.2.4 Thrust Calculations

Various formulae exist to calculate gross engine thrust (Fg). However, modern engines are too complex for standard textbook formulas to accurately predict thrust; this is normally left to complex computer algorithms. The current standard for new models is the Numerical Propulsion System Simulation (NPSS). Despite this, a control volume approach relying on conservation of momentum will provide gross engine thrust if the required parameters are known.

(ri29 + mfuel)v9 - ~OVO Thrust = + (P9 - P0)A9 9c Where subscript 9 represents the nozzle exit and subscript 0 represents the freestream conditions. m, V, and, P represent mass flow rate, velocity, and static pressure respectively.

Variations and simplifications of this formula exist:

Turbojet and Low Bypass Turbofan mo(v9 - Thrust = (Mixed Streams) 9c High Bypass Turbofan mf an mcme Thrust = -(vf - Vo) + -(Vg - Vo) (Separate Streams) 9c 9c Atmospheric temperature, Mach, and altitude also sigmficantly affect engine thrust and efficiency.

r Takeoff Thrust Partial Throttle Performance

hfaximum climb rating Maximum cruise ra~ing

I n...... 40M) 60a, 8Wl lO.@XJ l2.ooD 14.000 16MXI Uninstalled Thrust, F (lb,)

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17.3 Turbine Engine Operation

Compressor maps allow engine operation evaluation, showing corrected engine airflow (fan or core) versus pressure ratio. It is critical to know where an engine is operating within the compressor map in order to avoid stalls and flameouts. The following sections review compressor operation during several maneuvers and geometry changes.

17.3.1 Compressor operation during accelerations Surge Prevention logic: • Pressure ratio limits • fuel to air ratio limits Surge Region (stall line lowered by: inlet distortion, • Engine Acceleration Rate pressure, temperature, or engine deterioration)

Pressure Ratio Transient Operating Line

Surge margin Mil/Max Constant remaining Corrected RPM, N/ √θθθ Steady-State Op-Line •Raised by power extraction Idle • Lowered by airbleed

Airflow (pps) * √θθθ/

Above is a typical engine acceleration compressor map. As the engine accelerates from idle to maximum power, the engine follows the transient operating line scheduled by the engine controller. This usually includes surge prevention logic (e.g. pressure ratio limits as a function of airflow). Engine surge susceptibility is generally determined by either component bench tests or from altitude development tests of the full scale engine. Once the surge region is determined for the baseline engine, it can be further reduced by inlet distortion, power extraction, manufacturing tolerances, deterioration, or thermal transients, which affect compressor tip clearances. These affects are considered when determining the transient acceleration schedule needed to provide sufficient surge margin in the most demanding situations.

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17.3.2 Compressor operation during decelerations

Steady-State Operating Line Mil/Max Pressure Ratio Blowout prevention: Idle • Fuel to air ratio minimum Deceleration Transient • Engine Deceleration Rate

Operating Line

Blowout Region

Airflow (pps) * √θθθ/ Above is a typical engine deceleration compressor map. As the engine decelerates to idle power, the engine follows the transient operating line. To protect from combustor blowout during the deceleration, transient operation schedules usually include blowout prevention logic that schedules a minimum fuel to air ratio within the combustor as a function of engine airflow. This blowout region is usually determined by combustor bench tests or from altitude development tests of the full scale engine. Once the blowout region is determined, the engine schedules are set to include margin to account for engine-to-engine variability.

17.3.3 Bleed air impacts during engine starts

Open Start Bleed Position Closed

N/ √θ Idle Speed Stall “Hot Start” Region Start bleed Idle Pressure Closed Ratio Constant Corrected Off Start bleed RPM open

Additional Stall Margin Blowout Region

Airflow (pps) * √θθθ/ Above is a typical engine start compressor map. Turbine engines can be challenging to accelerate from off to idle power due to little stall margin at low airflow conditions. As a means to increase engine surge margin by reducing compressor back pressure and allowing quicker accelerations, engine bleed air can be removed from the engine core via a bleed valve. As the engine approaches idle power, the bleed air valve closes and the engine accelerates the remaining way to idle.

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17.3.4 Variable Stator vane cambering effects

α W β1 1 α 1 1 Cambered throat area C C1 Ca 1 1 Vane Axial Cw 1 Stator • • • rω Vane Cambered

Minimum Axial throat area W2’ β2 α2’ β2’ α2’ α2 i2’ W2 α2 i2 W2’ C2’ W2 C2

Rotor Cambering stator vanes • Reduces airflow passage area • Reduces airflow and increases surge margin

• Reduces relative velocities (W 2 vs. W2’ )

• Decreases incidence to rotor blades ( i2 vs. i2’ ) ` Above are typical engine compressor stator and rotor velocity diagrams. Stator vanes control engine surge margin and thrust; as they are cambered closed, the throat area between the vanes is reduced, thereby decreasing airflow. Cambering the vanes also decreases the incidence angle of the rotor blade. These lower airflows and reduced incidence angles increase engine surge margin.

Stators Cambered Pressure Ratio Stators Mil/Max Axial Additional Stall Margin

N√√√θ√θθθ Idle

Airflow (pps) * √θθθ/

Above is a compressor map illustrating how variable geometry and stator vane cambering can be used to affect engine operation and performance. The solid lines show how the compressor would operate if the stator vanes remained fixed in the axial position. The dashed lines show how the compressor would operate if the stator vanes remained fixed in the cambered closed position. When the stator vanes are cambered closed, additional surge margin is provided, and when the stator vanes are axial, additional airflow capability (or thrust) is provided. As a result, engines typically camber closed stators at low airflow to increase surge margin and acceleration capability, then camber the vanes axial open to maximize performance at higher engine rpms.

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17.3.5 Compressor impacts due to nozzle area for turbojet engines

• Nozzle closed shows insufficient stall margin • Opening nozzle moves op-line away from surge Nozzle Stall line closed Pressure Ratio Insufficient stall Nozzle margin w/ open smaller nozzle area

N√√√θθθ

Airflow (pps) * √θθθ/

Above is a compressor map illustrating impacts to stability margin from opening and closing the engine exhaust nozzle for a single-spool engine (e.g. J85 engine in T-38 aircraft). In this example, it is shown that stall margin is insufficient at lower airflows with the smaller nozzle area. Opening the nozzle for this single spool configuration moves the engine away from surge. As a result, a typical engine might run nozzle open at idle power to reduce thrust and keep the nozzle open until engine stability was no longer a concern.

17.3.6 Compressor impacts due to nozzle area for turbofan engines (mixed flow, proximate splitter)

• Closing nozzle moves fan away from surge . • Opposite impact compared to single-spool engine • High pressure compressor operating line not impacted by nozzle position - if low pressure turbine is choked • Variable nozzle allows ability to set airflow and thrust independently

Nozzle Stall Line open Stall Line Compressor Fan Pressure Nozzle closed Pressure Ratio Ratio Nozzle Nozzle open closed

θ Airflow (pps) * √θ/ Airflow (pps) * √ / Above is a compressor map illustrating stability margin impacts from opening and closing the engine exhaust nozzle for a turbofan engine (e.g. F100 engine in F-16 aircraft). Closing the nozzle moves the fan away from surge, which is opposite from the turbojet application. The high pressure compressor (HPC) is not impacted by the nozzle opening or closing (assuming the low pressure turbine [LPT] is choked). The fan and compressor’s independent reactions to nozzle movement allow the ability to set airflow and thrust independently. This is an important feature for flutter vibration or stability issues at particular rpm ranges.

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17.3.7 Combustion Stability

Match Point Fuel Flow Variations Variations • Engine operates closer to blowout region at high altitude and low Mach • Variations in fuel flow or match point Air Mass Blowout can result in blowout Region Flow Stable • Combustion instabilities cause rough (V/PT) Region running and setup vibrations and reduce part life • Rumble - low freq 20-200Hz Lean limit Rich limit (too rich in ULHC) • Screech - high freq ~ 3000Hz (too lean & H/W issue in LRHC

Fuel to Air Ratio

Above is a combustion stability plot showing engine airflow versus fuel-to-air-ratio. These plots are generally developed during component bench testing and are then used by engine designers to schedule combustor or augmentor fuel flow. The plot shows regions of stable and unstable combustion. Also, the engine operates closer to a blowout region at low air mass flow (high altitude and low Mach number [ULHC]). Combustion instabilities can have various effects on engine operation, including blowout, running rough, and vibrations that can reduce part life.

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17.4 Additional Information

17.4.1 Engine Stations (Reference SAE ARP-755 Aircraft Propulsion System Performance Station Designation and Nomenclature for additional details)

Engine station designations provide a consistent definition of the process the gas undergoes, regardless of the type of engine cycle The six main processes specifically isolated are: a. kinetic compression (inlet/diffuser) b. mechanical compression/work addition/fluidic compression (compressor/propeller) c. heat addition or exchange (combustor/augmentor/heat exchanger) d. mechanical expansion/work extraction (turbine) e. kinetic expansion (nozzle) f. mixing (mixer/ejector/eductor).

Dual Spool Turbofan With Afterburning

0 – Free stream air conditions 1 – First station of interest to the engine manufacturer. Inlet or aerodynamic interface plane (AIP). 2 – First compressor or fan front face 3 – Last compressor discharge or combustor entrance 4 – Combustor discharge or first turbine entrance 5 – Last turbine discharge 6 – Mixer or afterburner entrance 7 – Exhaust nozzle entrance 8 – Exhaust nozzle throat 9 – Exhaust nozzle discharge

Notes: 1. Incremental (or sub) stations may be indicated with suffix nomenclature (e.g. 2.5 to indicate fan discharge on a dual spool compression system). 2. There are a multitude of variations on this theme. SAE ARP-755 includes descriptions for most turbine engine configurations.

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17.4.2 Key Propulsion Terminology

Aerodynamic Interface Plane - A defined plane of intersection between the inlet and the engine. Afterburner - Any type of auxiliary (post turbomachine) combustion to enhance propulsion system thrust. Also known as the augmentor or reheat. Compressor Loading -The general ratio of work across the compressor stages. Forward compressor loading indicates the forward stages are more loaded (higher pressure ratio) than the aft stages. Compressor Map - A compressor’s total pressure ratio defined by corrected airflow and corrected rotation speed. Corrected - An adjustment for standard day temperature and/or pressure (at an engine station) to an engine parameter (like rotational speed, air flow or fuel flow). Also see referred.

Delta - Pressure ratio, , where i is the reference station. Flame-out - Can be synonymous with blow-out; however, it is more typically used in reference to the main combustor flame extinguishing. Gross Thrust - The momentum change at the nozzle exit or aft side of the propeller. The first term in the thrust equation. Horsepower Extraction - Any form of removal of power (bleed or mechanical) from a turbomachine other than for the generation of thrust. Inlet Compatibility - A type of test used to determine if the combined effects of inlet distortion and engine stability are compatible (e.g.; no stalls occur). Inlet Distortion - The measurement of variation in pressure, temperature, or vector at the aerodynamic interface plane. Inlet Recovery - The average total pressure at the Aerodynamic Interface Plane divided by the free stream total pressure. Instability - Can be used in many contexts. The two main contexts are in combustion stability and compressor stability. The former refers to a flame’s (either combustor or augmentor) ability to stay lit and the later to compression system flow disturbance. Operability - The sub-discipline of propulsion related to a turbine engine’s characteristic operational limits. This includes but is not limited to the regions of the flight envelope where stalls or flame-outs may occur, where augmentation is limited, or where airstarts can be accomplished. Recycle - A full no-light or blowout and relight sequence where the engine control continues to try to light the combustor or augmentor. Most typically refers to the augmentor. Referred - An adjustment for standard day temperature and/or pressure (at an engine station) to an engine parameter (like rotational speed, air flow or fuel flow). See also corrected. Reynolds Number Index - Ratio of actual Reynolds Number to standard atmosphere Reynolds Numbers assuming a constant length scale.

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Rotating Stall - A cyclic disruption of airflow (surge) across one or more fan or core compressor blades. May or may not be noticeable by the operator, but can produce cycle fatigue damage to the compressor blades. Rumble - A low frequency augmentor induced vibration. Screech - A combustion induced acoustic vibration in the augmentor. Usually in the several hundred Hertz frequency range Stagnation - A series of stalls that have become non-recoverable (no response to engine control inputs— requires the disruption of fuel flow to clear). The series of stalls has disrupted the airflow through the compressor so severely that ram flow will not recover the engine. Characterized by no engine core response and increasing exhaust gas temperature. Stall - A disruption of airflow across one or more fan or core compressor blades. Also known as surge. Stage - A blade (or rotor) and stator pair. Station - Defined locations within a propulsion system. See Section 20.2.1 Stator - The non-rotating blades of a stage within a turbomachine compressor or turbine. Swirl - Non-axial vector of inlet airflow. Temperature Profile - Usually used in reference to the span wise temperature distribution across the turbine inlet guide vanes.

Theta -Temperature ratio, , where i is the reference station Thrust Specific Fuel Consumption - The amount of fuel required to produce a unit of thrust, W TFSC = f Fn Total Accumulated Cycles - A conglomerate measurement (based on an empirical relationship) of the number of cycles an engine has experienced. It is used as a measure of engine health or life. Upper Left Hand Corner - An area of the flight envelope chart (Mach Number on the x-axis and altitude on the y-axis) characterized by areas of low speed and high altitude. Windmill - The free rotation of the rotational components of the engine driven solely by ram airflow.

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17.4.3 Aircraft and Associated Engines

Military Aircraft Designation Name Engine A-10 Thunderbolt II TF34-GE-100/-100A AC-130 Gunship T56-A-15 AH-1 Cobra T400-CP-400, T53-L-703 AH-64 Apache T700-GE-701C AV-8 Harrier F402-RR-401/402, F402-RR-406A/408 B-1 Lancer F101-GE-102 B-2 Spirit F118-GE-100 B-52 Stratofortress J57-PW-43WB, TF33-PW-3/103 C-12 Huron T74 C-130 Hercules T56-A-15/7/7B/9D, RR-AE2100D3 C-135 Stratolifter J57-PW-59W, TF33-PW-5 C-141 Starlifter TF33-PW-7/7A C-17 Globemaster III F117-PW-100 C-20 Gulfstream III F113-RR-100, F126-RR-100 C-23 Sherpa T101-CP-100 C-37 Gulfstream V RR-BR710A1 C-5 Galaxy TF39-GE-1A/1C, F138-GE-100 CH-3 Jolly Green Giant T58-GE-1/3/100 CH-47 Chinook T55-L-5/7/11/712/714 CH-53 Sea Stallion T64-GE-412 CV-22 Osprey T406-AD-400 E-3 Sentry TF33-PW-100A E-8 Joint Stars TF33-PW-102C F-14 Tomcat TF30-PW-412A, F110-GE-400 F-15 Eagle F100-PW-100/220/220E/229/229A F-16 Fighting Falcon F100-PW-200/220/220E/229/229A, F110-GE-100/129/132 F-18 Hornet F404-GE-400, F414-GE-400 F-22 Raptor F119-PW-100 F-35 Lightning II F135-PW-100, F136-GE-100 F-4 Phantom II J79-GE-2/8/10/15/17, F103-GE-100 F-5 Tiger/Freedom Fighter J85-GE-13/21, F404-GE-400 KC-10 Extender F103-GE-101 KC-135 Stratotanker J57-PW-43WB/-59W, TF33-PW-102, F108-CF-100 MQ-9 Reaper Honeywell TPE331-10 MQM-107 Streaker J402-CA-700/702 MQM-74 Chukar J400-WR-400/401 RQ-3 DarkStar F129-WR-100 RQ-4 Global Hawk F137-AD-100 SR-71 Blackbird J58-PW-4 T-1 Jayhawk PW-JT15D T-2 Buckeye J85-GE-4 T-33 Shooting Star J33-A-5 T-37 Tweet J69-T-25A T-38 Talon J85-GE-5/H/J/L/R/S T-6 Texan II PW-PT6A-68 U-2 Dragon Lady F118-GE-101 X-31 F404-GE-400 X-47 Pegasus JT15D-5C Civilian Aircraft Boeing 737 PW-JT8D, CFM-56 Boeing 747 PW-JT9D, GE-CF6, RR-RB211, GEnx Boeing 757 RR-RB211, PW-2000 Boeing 767 PW-JT9D, PW-4000, GE-CF6, RR-RB211, RR-800 Boeing 777 GE-90, PW-4000 Boeing 787 GEnx, RR-1000 Airbus A300 GE-CF6, PW-JT9D, PW-4000 Airbus A310 GE-CF6, PW-JT9D, PW-4000 Airbus A320 CFM-56, PW-6000, IAE-V2500 Airbus A330 GE-CF6, PW-4000, RR-700 Airbus A340 CFM-56, RR-500 Airbus A380 RR-900, GP-7000

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17.4.4 Additional Propulsion Resources

Jack Mattingly’s Engine Design Site http://www.aircraftenginedesign.com/

NASA EngineSim http://www.grc.nasa.gov/WWW/K-12/airplane/ngnsim.html

AeroFiles http://www.aerofiles.com/home.html

NASA Smithsonian http://www.nasm.si.edu/

National Museum of the US Air Force http://www.nationalmuseum.af.mil/

Aerospace Recommended Practices

ARP1420 Gas Turbine Engine Inlet Flow Distortion Guidelines AIR1419 Inlet Total-Pressure-Distortion Considerations for Gas-Turbine Engines ARP4990 Turbine Flowmeter Fuel Flow Calculations

Additional standards for Emissions, Test Cell Correlation, Noise, Temperature Measurement, and Health Management are also available through SAE International’s website: http://standards.sae.org/power-propulsion/engines/gas-turbines/standards/

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Section 18 - Telemetry Control Room and Radio Communications “SFTE Tech Council “Best Practice Guidelines”

In accordance with experienced FTE feedback from across industry, below are suggested best practices for communications between flight and ground crews. This guideline uses the following terminology • “Ground station” is a generic term for any station providing radio support, whether it is a simple radio, fixed‐base TM control room, mobile TM station, or other. • “R/O” is a generic term for the ground station radio operator. Different organizations may use different job titles for the person commanding the ground station (i.e., TM Chief, Test Director, Test Conductor). The radio call sign for this role is typically “TM” or “Control” • “Flight crew” to refer to whichever test aircraft crew member (pilot‐in‐command, copilot, or other) is designated to communicate with the test ground station.

Overall Guidance Unless otherwise established during the preflight briefing, radio operations between the test aircraft and ground station should use some standard phraseology established by the testing organization. The following are presented as radio transmission terminology between designated participants. The same phrases equally apply to communications within the telemetry (TM) room during a test. This phraseology is not necessarily applied to any communication with air traffic control (ATC).

“Stand by” ‐ Instruction issued by any participant to direct others to stay on established flight conditions and remain prepared for continuing, but to not execute the next action. Used when everything is OK, but the caller requires more time for some reason (e.g. ATC conversations, data analysis, airspace maneuvering).

“Cleared for ___” ‐ Transmitted by the R/O to authorize proceeding to the specified condition, typically the next planned test point, but sometimes a out‐of‐sequence test point, airspeed, or flight condition. The flight crew will typically “parrot back” the same phrase to acknowledge clearance. When in doubt, the flight crew requests “Confirm clear for ___.” Routine operations generally do not require clearance; these transmissions are typically used only when the flight crew requires clearance for saefty of flight (SOF) or technical reasons – as established during the preflight briefing.

“On Condition” – Normally transmitted by test pilot to announce the aircraft is properly trimmed or otherwise set up to beginf the test.

“Hold” – Instruction by R/O to maintain current flight condition to either extend data collection period or emphasize critical parameter (e.g “Hold condition” or “Hold altitude”).

"3,2,1, HACK" – countdown to event.

Silent countdown “ Five, four, three ______” ‐ an intentional silence at the end of a countdown to allow other participants a chance to call for an abort. Most common with ordinance employment.

"Condition Complete" or “Test Complete” – Call from test conductor or pilot to announce a normal completion.

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"Continue" – transmitted by R/O when there is a series of steps in the maneuver and flight crew is cleared to continue to the next one.

"Recover" called by either participant at test point completion or non‐urgent cessation of a test point. Instructs the flight crew to smoothly return to a normal flying condition.

“Copy” – Routine shorthand reply meaning “I understood your previous transmission.” This single statement is generally preferred. This can be replaced by a non‐verbal radio double‐click (zipper) to acknowledge simple statements. Some organizations may also aknowledge a transmission using “Roger.” Flight testers do not use CB radio slang such as “10‐4”

“Stop Test” or “Knock it Off”‐ Instruction transmitted by flight crew or R/O to stop test underway using normal means. There is no implied requirement to change flight conditions. Used when testing does not present SOF concerns but continuing will not provide useful data. The R/O would typically make this call if witnessing incorrect execution or experiencing critical data dropouts or other technical difficulties. The R/O would also make this call on behalf of a participating engineer who does not have radio access. The flight crew would make this call if testing is invalidated by ATC or weather interference or in the case of an improperly executed procedure. The caller should state the reason for stopping the test (e.g. reaching __ limit).

“ABORT, ABORT, ABORT” ‐ Emergency instruction transmitted by flight crew, chase aircraft, or R/O to stop test immediately and recover to safe flight conditions as soon as possible. Used when continued testing presents SOF concerns. The R/O may make this call if witnessing a dangerous situation or if unable to confirm SOF because of critical data dropouts or other technical difficulties. The flight crew would make this call to advise the R/O of actions they are taking. In certain cases pre‐arranged at the prebriefing, a ground station participant other than the R/O may be authorized and physically positioned to make radio abort calls. The caller should state the reason for stopping the test (e.g. reaching __ limit).

“BAIL OUT, BAIL OUT, BAIL OUT” ‐ Emergency instruction to or between the flight crew to leave the aircraft. Nominally used when out‐of‐control aircraft descends through a pre‐set minimum or other briefed altitude. In circumstance leading up to any potential bail out call, the R/O will transmit “__ altitude” callouts every thousand feet during attempted recovery of an out‐of‐control aircraft.

“Terminate” is the instruction to destroy or otherwise incapacitate a flight vehicle. Applies to vehicles with a flight termination system designed to quickly end its flight and limit possible damage to the surroundings.

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Call sign designation. Depending on prior agreements, ATC may communicate to an aircraft by referencing its registration (a.k.a. tail) number or by an accepted call sign such as “Zoom 86.” Call signs potentially have the advantage of being easier to say & understand and bringing familiarity between operators. The R/O may use this same call sign if so arranged prior to the flight, but will otherwise refer to the test aircraft as “Test” and a chase aircraft as “Chase.” The call sign for the ground station is “Ground” and is “[Company] Weather” for the mobile ground weather station. The call sign for any other mobile radio‐equipped ground crew is “[Company] Mobile” unless otherwise specified during the pre‐flight briefing. The dispatcher’s radio station, is “[Company] Dispatch.” Security personnel employed who control the gate at the taxiway and are called “[Company] Security.”

Call sign use. When initiating radio communications at the beginning of a flight or after a period of silence, the caller will first state the receiver’s call sign then his own (e.g. R/O transmits “Test – Ground”). This is an abbreviated version of the formal “Calling Zoom 86, this is Ground, Over” which is not useful unless radio transmission quality is poor. After two‐way communication has been initiated, each transmission need only to begin with the recipient’s call sign.

Thumbs Up. A convenient non‐verbal communication between control room participants is the “thumbs‐up” hand signal. During active testing, this signal is preferred for routinely acknowledging a message or for indicating readiness. Not only does this signal eliminate unnecessary discussion, but allows all ground station participants to signal simultaneously and continuously if needed.

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NOTES

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Section 19 - The Electromagnetic Spectrum

19.1 Electromagnetic Waves

All energy in the universe radiates in waves. Figure 1 depicts an electromagnetic wave, characterized by an electric field vector (E) and a magnetic field vector (H) oscillating orthogonal to each other. The electromagnetic wave propagation direction is determined by the righth -hand rule and crossing the electric field into the magnetic field as illustrated. The electromagnetic wavelength (λ) is the distance between two consecutive electric field peaks. The electromagnetic wave frequency (f or ν) is inversely proportional to its wavelength. The product of any electromagnetic radiation’s frequency and wavelength equals the speed of light (C=λν).

.

Figure 1 - The Electromagnetic Spectrum

19.2 The Electromagnetic Spectrum

Figure 2 illustrates several concepts related to the electromagnetic spectrum

• Size reference for various wavelengths. • Numerical relation between wavelength (meters between peaks), wave number (peaks per cm), energy (electron volts), and frequency (Hz). • Separate frequency bands identified within the overall spectrum. • Common sources & uses within frequency bands.

Many aircraft and spacecraft systems operate within limited regions of the electromagnetic spectrum. Common examples include radar, electro-optical sensors, radios, data linkss, electronic warfare, and navigation systems.

Page 19-1 SFTE Reference Handbook Third Edition 2013 Figure 2 - The Electromagnetic Spectrum

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19.3 Radio Frequency Electromagnetic Radiation

One of the most heavily used regions of the electromagnetic spectrum is the radio frequency or RF spectrum, from ~3 kilohertz to ~ 300 gigahertz. The RF Spectrum encompasses an array of telecommunications devices including radios, television, satellite communications, data links, radio-navigation aids, and radar. Table 1 shows how the RF Spectrum is subdivided into frequency bands.

Band Designation Label Frequency Spread Extremely Low Frequency ELF 3 - 30 Hz Super Low Frequency SLF 30 - 300 Hz Ultra Low (Voice) Frequency ULF or VF 300 Hz - 3 KHz Very Low Frequency VLF 3 - 30 KHz Low Frequency LF 30 - 300 KHz Medium Frequency MF 300 KHz - 3 MHz High Frequency HF 3 - 30 MHz Very High Frequency VHF 30 - 300 MHz Ultra High Frequency UHF 300 MHz - 3 GHz Super High Frequency SHF 3 - 30 GHz Extremely High Frequency EHF 30 - 300 GHz

Table 1 - Radio Frequency Band Designations

Depending on the type of RF system, additional subdivisions of the bands also exist. Table 2 applies to radar systems.

IEEE US Origin Frequency Wavelength NATO, US ECM (Old RADAR Range (New RADAR Designation) Designation) W W follows V in alphabet 75-111 GHz 400 mm-270 mm M (60-100 GHz) V Very Short 40-75 GHz 700 mm – 400 mm L (40-60 GHz)

KA Kurtz (above) 26-40 GHz 1.1 cm -.7 cm K (20-40 GHz)

K Kurtz 18-26 GHz 1.6 cm – 1.1 cm J (10-20 GHz) KU Kurtz (under) 12.4-18 GHz 2.5 cm – 1.6 cm

X WWII fire control - as an 8-12.4 GHz 3.7 cm -2.5 cm I (8-10 GHz) “X” for crosshairs C Compromise between S 4-8 GHz 7.5 cm -3.7 cm H (6-8 GHz) and X G (4-6 GHz)

S Short Wave 2-4 GHz 15 cm – 7.5 cm F (3-4 GHz) E (2-3 GHz) L Long Wave 1-2 GHz 30 cm – 15 cm D (1-2 GHz) UHF .3-1 GHz <1 m – 30 cm C (.5-1 GHz)

Table 2 - Radar Frequency Band Designations

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Electromagnetic wave propagation does not stop at national boundaries. Most governments regulate radio frequency band use via frequency or spectrum allocation. For technical and economic reasons, governments try to harmonize and standardize RF band allocation. A number of forums and standards bodies address frequency allocation. The International Telecommunication Union (ITU) is the United Nations agency for information and communication technologies. The ITU allocates global radio spectrum and satellite orbits and develops technical standards that ensure networks and technologies seamlessly interconnect. There are numerous users and spectrum allocation is complicated to implement and regulate. Figure 3 illustrates the United States frequency allocations.

[Editor’s Note: Figure 3 is highly detailed and not legible in either 8.5 x 11” or 11 x 17” format. Posters may be purchased via http://bookstore.gpo.gov/products/sku/003-000-00694-8. A high resolution online version is available at http://www.ntia.doc.gov/files/ntia/publications/spectrum_wall_chart_aug2011.pdf ]

Page 19-4 SFTE Reference Handbook Third Edition 2013 Figure 3 - U.S. Radio Frequency Allocations Figure 3 - U.S. Radio Frequency (2011)

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19.4 Optical Frequency Electromagnetic Radiation

Optical frequency electromagnetic radiation includes ultraviolet (UV), visible, and infrared (IR) light ranging from about 0.01 microns (µmµ ) out to about 1000 µm wavelength. These comprise the optical spectrum as shown in Figure 4.

ÅOptical SpectrumÆ

Figure 4 - The Optical Spectrum and the Visible Spectrum

Just as the RF spectrum can be subdivided into various special bands, Table 3 shows bands in the optical spectrum. The visible, near infrared (NIR), mid wave IR (MWIR), and long wave IR (LWIRR) bands are most commonly used for airborne electro-optical sensor systems. All bands can be useful, depending on the specific mission and operational requirements. [Editor’s Note: Some bandd nomenclature and boundaries depend on author and text. SFTE has not established standards on this subject]

The most familiar example of electromagnetic radiation is the lighht spectrum huumans see. Different colors of visible light have different wavelengths, ranging from violet at the shorter wavelengths (0.4 µm) to red at the longer wavelengths (0.7 µm). Aerospace vehicles employ many visible light seensor systems. Visible light sensors provide useful and highly recognizable images, but logically enough, do not woork well in poor visibility or at night without some sort of artificial illumination or image intensification.

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Table 3 - Optical Spectrum Bands

Infrared radiation refers to the optical spectrum between ~0.7 µm and ~1000 µm. Infrared radiation is popularly known as "heat or thermal radiation", butu light and electromagnetic waves of any frequency will heat surfaces that absorb them. Infrared light from the sun accounts for ~49% of earth heating, the rest is by visible light that is absorbed then re-radiated at longer wavelengths. Objects at room temperature emit radiation mostly concentrated in the 8 to 25 µm region.

Infrared sensors are useful in numerous civil and military remote sensing appliications. At wavelengths beyond about 14 microns, infrared radiation is not useful for most airbornne remote sensing applications due to the earth’s atmosphere attenuation. This is especially true in hot and humid atmospheric conditions with extreme infrared radiation attenuation.

19.5 Atmospheric Transmission Windows

The sun is the earth’s major source of natural energy across the eleectromagnetiic spectrum and its radiation bombards the atmosphere constantly. The earth's atmosphere protects its life from excessive exposure to a range of higher energy waves such as Gamma rays, x-rays, and some ultraviolet waves. These are "ionizing" radiation because they have sufficiently high energy to knock electrons out of atoms, alter atoms and molecules, and damage organic cells. Figure 5 shows atmospheric opacity (blocking) across a wide wavelength spectrum.

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Figure 5 - Atmospheric Opacity vs. Wavelength

The opposite of opacity is transmission. The earth’s atmosphere absorbs, refleccts, refracts, or allows electromagnetic radiation transmission. Suspended particles suchh as dust and raindrops cann reflect and refract radiation. Figure 6 shows electromagnetic radiation transmission through the atmosphere across the visible and infrared wavelength spectrumu (subset of the Figure 5 spectrum). Some radiation bands, including visible light and some infrared pass through the earth’s atmosphere with little to no attenuation. Such “atmospheric windows” or “transmission windows” allow infrared remote sensing from standoff distancces. The bottom scale of Figure 6 shows which molecules are primary absorbers at various infrared radiation wavelengths. The most important absorbers are water vapor (H2O), carbon dioxide (CO2), and ozone (O3). Flight testers should understand wavelengths, absorption, refraction and atmospheric windows when designing tests to evaluate sensors.

Figure 6 - Infrared Atmospheric Transmission Windows

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SFTE Reference Handbook

Quick Index

Tab Section______

1 General Information 2 Mathematics 3 Earth and Atmosphere 4 Pitot Statics 5 Aerodynamics 6 Axis Systems and Transformations 7 Mass Properties 8 Motion/Vibration Analysis 9 Material Strength (Loads) 10 Reciprocating Engines 11 Propellers 12 Fixed-Wing Performance Standardization 13 Acoustics 14 Electromagnetic Compatibility 15 Handling Qualities 16 Rotary Wing 17 Gas Turbine Propulsion 18 Radio Communications 19 The Electromagnetic Spectrum