1 General Information

1.1 Unit Conversions1

(references 1.1, 1.2)

Prefix Multipliers
Exponent Prefix Abbreviation
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 \(\mu\)
10-9 nano n
10-12 pico p
10-15 femto f
10-18 atto a
Multiply by To Obtain
Angles circles 1 circumferences
circles 12 signs
circles 21,600 minutes
circles 2\(\pi\) radians
circles 360 degrees
degrees 0.011 11 quadrants
degrees 3600 seconds
degrees 60 minutes
mils (Army) 0.056 25 degrees
mils (Navy) 0.057 29 degrees
quadrants 90 degrees
radians 57.2958 degrees
revolutions 360 degrees
2 sphere 4\(\pi\) steradians
Angular Acceleration rev/min2 0.001 745 rad/sec2
Angular Velocity cycles/sec 6.2814 rads/sec
rads/sec 0.1592 rev/sec (cycles/sec)
rads/sec 9.549 rpm
rad/sec 57.296 deg/sec
rpm 0.016 67 rev/sec
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 0.092 903 04 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 0.036 13 pounds/in3
grams/cm3 62.43 pounds/ft3
kg/m3 16.018 463 pounds/ft3
slugs/ft3 515.4 kg/m3
pounds/in3 1728 pounds/ft3
slugs/ft3 1.94 grams/cm3
Electrical Quantities amperes 0.1 abamperes
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 ampere-hours
farads 10-9 abfarads
farads 106 microfarads
farads 8.986x1011 statfarads
gausses 1 maxwells/cm2
gausses 6.452 lines/in2
gilberts 0.7958 ampere turns
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 0.003 336 statvolts
Energy & Work Btu 1.055x1010 ergs
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 0.238 89 calories
Joules 1 Newton-meters
Joules 1 watt-seconds
Joules 107 ergs
kilowatt-hours 3.6x106 Joules
thermies 4.1868x106 Joules
watt-seconds 0.737 56 foot-pounds
Force3 dynes 3.597x10-5 ounces
kilograms-force 9.806 65 Newtons
kiloponds 9.806 65 Newtons
kip (kilopound-force) 4,448.221 Newtons
Newtons 0.224 808 931 pounds
Newtons 100,000 dynes
ounces 20 pennyweights
ounces (troy) 480 grains
Fuel4 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
Illumination candles 1 lumens/steradian
candles/cm2 \(\pi\) 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 0.001 496 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
Length ångströms 10-10 meters
astronomical 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
5 feet 0.3048 meters
furlongs 40 rods
hands 4 inches
inches 2.54 cm
kilometers 3281 feet
kilometers 0.539 96 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 3.280 84 feet
meters 39.370 079 inches
microns 10-6 meters
mils 10-3 inches
nautical miles 1.150 78 statute miles
6 nautical miles 1,852 meters
nautical miles 6,076.115 486 feet
paces 0.762 meters
parsec 1.9163x1013 statute miles
perch 5.0292 meters
pica (printers) 0.004 217 5176 meters
point (printers) 0.000 351 4598 meters
pole (=rod) 5.0292 meters
skein 109.728 meters
statute miles 5,280 feet
statute miles 1.609 344 kilometers
statute miles 8 furlongs
yards 3 feet
Linear Acceleration feet/sec2 1.097 28 kilometers/hr/sec
feet/sec2 0.3048 meters/sec2
feet/sec2 0.6818 mph/sec
g 32.174 049 feet/sec2
g 9.806 65 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 0.035 274 ounces
grains 6.479 891x10-5 kilograms
hundredweight (long or Imperial) 50.80 kilograms
hundredweight (short) 45.359 237 kilograms
kilograms 0.068 52 slugs
kilograms 6.024x1026 atomic mass units
kilograms 2.2046 pounds
ounces (avd) 28.349 523 125 grams
ounces (troy) 31.103 4768 grams
pounds (mass) 1 pounds (force)
pounds (mass) 0.453 592 37 kilograms
pounds (mass) 0.031 081 slugs
scruples (apoth) 0.001 295 9782 kilograms
slugs 32.174 pounds
slugs 14.594 kilograms
tons (long) 1016.047 kilograms
tons (assay) 0.029 16 kilograms
tons (metric) 1000 kilograms
tons (short) 907.1847 kilograms
Moments of Inertia gram-cm2 0.737x10-7 slug-ft2
pound-ft2 0.031 081 slug-ft2
slug-in2 0.006 9444 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
Power btu/min 0.017 58 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 0.077 12 btu/min
ft(lbs)/sec 1.356 watts
horsepower 550 ft(lb)/sec
horsepower 33,000 ft(lbs)/min
horsepower 10.69 calories (kg)/min
horsepower 745.7 watts
horsepower (metric) 735.5 watts
horsepower 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 760 mm of Hg
bars 106 dynes/cm2
bars 29.52 inches of Hg
barye 0.1 Newtons/m2
dynes/cm2 10 Newtons/m2
inches of H2O 5.202 37 pound/ft2
inches of Hg 70.726 19 pounds/ft2
inches of Hg 0.491 154 pounds/in2
inches of Hg 13.595 inches of H2O
kiloPascals 100 bars
hectoPascals 1 millibars
millibars 0.029 53 inches of Hg
mm of Hg 0.019 337 pounds/in2
mm of Hg 133.32 Newtons/m2
Pascals 1 Newton/m2
pieze 1000 Newtons/m2
pounds/ft2 0.014 14 inches of Hg
pounds/ft2 47.88 Newtons/m2
pounds/in2 2.036 inches of Hg
pounds/in2 27.681 inches of H2O
pounds/in2 6894.757 28 Pascal
torrs 133.32 Newtons/m2
Temperature Kelvin Celsius + 273.15
Rankine Fahrenheit + 459.67
Celsius (Fahrenheit - 32) * 5/9
Fahrenheit (9/5 * Celsius) + 32
Time days (solar) 24 hours
days (sidereal) 23.934 hours
days (solar) 1.0027 days (sidereal)
hours 60 minutes
minutes 60 seconds
months (sidereal) 27d + 7hr + 43min + 11.47sec
months (lunar) 29d + 12hr + 44min + 2.78sec
year 365.242 198 79 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
Velocity inches/sec 0.0254 meters/sec
km/hr 0.621 371 mph
km/hr 0.9113 feet/sec
knots 1.687 81 feet/sec
knots (kts) 1.150 78 mph
knots (kts) 1.852 km/hr
knots (kts) 0.514 44 meters/sec
meters/sec 3.281 ft/sec
meters/sec 3.6 km/hr
meters/sec 196.85 feet/min
mph 1.466 667 feet/sec
Viscosity centistokes 10-6 m2/sec
ft2/sec 0.0929 m2/sec
pound sec/ ft2 47.880 258 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 0.033 81 fluid ounces
cm3 0.061 02 in3
cord-feet 4x4x1 ft3
cords 128 ft3
cups 0.5 pints (liquid)
dram (fluid) 3.696 69x10-6 m3
ft3 0.028 3167 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.) 0.003 785 m3
gals (U.S.) 3.785 liters
gals (U.S.) 4 quarts (liquid)
gals (U.S.) 0.023 8095 barrels (U.S.)
gills 7.219 in3
hogshead 2 barrels
in3 16.39 cm3
liters 0.028 38 bushels
liters 0.9081 quarts (dry)
liters 1.057 quarts (liquid)
liters 1000 cm3
liters 61.03 in3
m3 1.308 yd3
m3 1000 liters
m3 264.2 gals (U.S.)
m3 35.314 667 ft3
mil-feet (circ.) 0.000 1545 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

1.2 Greek Alphabet

Uppercase LaTeX Command Lowercase LaTeX Command Name Say
\(Α\) Α \(\alpha\) \alpha Alpha æl-fə
\(Β\) Β \(\beta\) \beta Beta bei-tə
\(\Gamma\) or \(\varGamma\) \Gamma or \varGamma \(\gamma\) \gamma Gamma gæ-mə
\(\Delta\) or \(\varDelta\) \Delta or \varDelta \(\delta\) \delta Delta del-tə
\(Ε\) Ε \(\epsilon\) or \(\varepsilon\) \epsilon or \varepsilon Epsilon eps-ill-aan
\(Ζ\) Ζ \(\zeta\) \zeta Zeta zei-tə
\(Η\) Η \(\eta\) \eta Eta ei-tə
\(\Theta\) or \(\varTheta\) \Theta or \varTheta \(\theta\) or \(\vartheta\) \theta or \vartheta Theta thei-tə
\(Ι\) Ι \(\iota\) \iota Iota aai-oh-tə
\(Κ\) Κ \(\kappa\) or \(\varkappa\) \kappa or \varkappa Kappa kæ-pə
\(\Lambda\) or \(\varLambda\) \Lambda or \varLambda \(\lambda\) \lambda Lambda læm-də
\(M\) M \(\mu\) \mu Mu myoo
\(Ν\) Ν \(\nu\) \nu Nu nyoo
\(\Xi\) or \(\varXi\) \Xi or \varXi \(\xi\) \xi Xi ksaai
\(Ο\) Ο \(ο\) ο Omicron oh-mə-kraan
\(\Pi\) or \(\varPi\) \Pi or \varPi \(\pi\) or \(\varpi\) \pi or \varpi Pi paai
\(P\) P \(\rho\) or \(\varrho\) \rho or \varrho Rho roh
\(\Sigma\) or \(\varSigma\) \Sigma or \varSigma \(\sigma\) or \(\varsigma\) \sigma or \varsigma Sigma sig-mə
\(Τ\) Τ \(\tau\) \tau Tau taa’u
\(\Upsilon\) or \(\varUpsilon\) \Upsilon or \varUpsilon \(\upsilon\) \upsilon Upsilon oops-ill-on
\(\Phi\) or \(\varPhi\) \Phi or \varPhi \(\phi\) or \(\varphi\) \phi or \varphi Phi faai
\(Χ\) Χ \(\chi\) \chi Chi kaai
\(\Psi\) or \(\varPsi\) \Psi or \varPsi \(\psi\) \psi Psi psaai
\(\Omega\) or \(\varOmega\) \Omega or \varOmega \(\omega\) \omega Omega oh-meg-ə

1.3 Greek Symbols Used for Aircraft

Symbol Used For
\(\alpha\) angle of attack (degrees or radians)
\(\alpha_{\tau}\) tail angle of attack
\(\beta\) angle of sideslip (degrees)
\(\gamma\) flight path angle relative to horizontal
\(\gamma\) specific heat ratio (1.4 for air)
\(\delta\) relative pressure ratio ( \(\frac{P_a}{P_0}\))
\(\delta_a\) aileron deflection angle
\(\delta_r\) rudder deflection angle
\(\delta_e\) elevator deflection angle
\(\varepsilon\) downwash angle at tail (degrees)
\(\zeta\) damping ratio
\(\eta\) efficiency
\(\theta\) body axis/pitch angle
\(\theta\) relative temperature ratio, \(T_a / T_0\)
\(\iota\) angle of incidence
\(\iota_F\) thrust angle of incidence
\(\iota_T\)* horizontal tail angle of incidence
\(\lambda\) pressure lag constant
\(\Lambda\) wing sweep angle
\(\mu\) coefficient of absolute viscosity =\(\rho \nu\)
\(\mu\) Mach cone angle
\(\nu\) kinematic viscosity =\(\mu / g\)
\(\pi\) nondimensional parameter
\(\rho\) density
\(\rho_a\) ambient air density
\(\rho_0\) standard atmospheric density (slugs/ft^3 )
\(\sigma\) air density ratio \((\rho_{\alpha} / \rho_0)\)
\(\sigma_{\mathrm{cr}}\) critical density
\(\tau\) shear stress (pounds per square inch) psi
\(\tau_R\) Roll Mode Time Constant (sec)
\(\phi\) bank angle (degrees)
\(\psi\) aircraft heading (degrees)
\(\omega\) frequency
\(\omega\) rotational velocity (radians per second)
\(\omega_d\) damped natural frequency
\(\omega_n\) natural undamped frequency

1.4 Common Subscripts

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

1.5 Common Abbreviations

Abbreviation Meaning
\(a\) 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
\(a_x\) longitudinal acceleration
\(a_y\) lateral acceleration
AZ azimuth
\(b\) span of wing (feet)
B/N bombardier/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
°C degrees centigrade (see T)
\(c\) brake specific fuel consumption (BSFC)
\(c\) 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
\(C/N_0\) carrier to noise ratio
CADC central air data computer
CARD cost analysis requirement document
\(C_D\) coefficient of drag
\(C_{D_i}\) induced drag coefficient
\(C_{D_0}\) 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
\(C_f\) coefficient of friction
CFE contractor furnished equipment
CFT conformal fuel tank
cg center of gravity (normally in % MAC)
\(C_H\) hinge moment coefficient
cine cinetheodolite
\(C_l\) rolling moment coefficient, airfoil section lift coefficient
\(C_L\) lift coefficient
CLHQ closed loop handling qualities
\(C_{\mathrm{lp}}\) roll damping coefficient
\(C_{\mathrm{lr}}\) roll moment due to yaw rate coefficient
\(C_m\) pitching moment coefficient
\(C_M\) moment coefficient
cm centimeters
cos cosine
cot cotangent
\(C_{l_{\beta}}\) (dihedral) rolling moment due to sideslip
\(C_{l_{\delta_a}}\) aileron power coefficient
\(C_{m_q}\) pitch damping coefficient
\(C{m_{\alpha}}\) longitudinal static stability coefficient
\(C{m_{\delta e}}\) elevator power coefficient
\(C_n\) yawing moment coefficient
\(C_{n_r}\) yaw damping coefficient
cnst constant
\(C_{n_{\beta}}\) directional stability coefficient
\(C_{n_{\delta a}}\) adverse yaw coefficient
\(C_{n_{\delta r}}\) rudder power coefficient
COTS commercial, off–the-shelf
CP center of pressure
\(C_P\) propeller power coefficient
CPU central processing unit
\(c_r\) wing root chord
CRM crew resource management
\(c_t\) wing tip chord
CTF combined test force
CY calendar year
\(C_Y\) side force coefficient
\(C_{Y_{\beta}}\) side force due to sideslip coefficient
\(C_{Y_{\delta r}}\) 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
\(\mathrm{e}\) natural mathematical constant = 2.718 281 828 459
E energy
E lift-to-drag ratio ( \(C_L / C_D\) , \(L/D\) )
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
\(E_{\mathrm{max}}\) maximum lift-to-drag ratio
EMC electromagnetic compatibility
EMI electromagnetic interference
EMP electromagnetic pulse
EO electro optical
EOM equations of motion
EPR engine pressure ratio
EPROM electrically programmable read only memory
\(E_s\) 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
°F degrees Fahrenheit
\(f\) frequency…hertz (originally cycles per second)
F.S. fuselage station
\(F_a\) aileron force
FAA Federal Aviation Administration
FAR Federal Aviation Regulation
FCF functional check flight
FDC flight data computer
\(F_e\) elevator force
\(F_{\mathrm{ex}}\) excess thrust
\(F_g\) gross thrust
FL flight level
FLIP flight information publication
FLIR forward-looking infrared
FM frequency modulation
FMC fully mission capable
FMS flight management system
FMS foreign military sales
\(F_n\) net thrust
\(F_n / \delta\) 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
\(F_r\) 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...
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
\(g_0\) 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
\(H_c\) calibrated altitude (assumed to be pressure altitude in flight test)
\(H_D\) density altitude
HDDR high density digital recorder
HDOP horizontal dilution of precision
HF high frequency
Hg mercury
\(H_i\) indicated altitude
\(h_m\) stick-fixed maneuver point (%MAC)
\(h_{'m}\) stick-free maneuver point (%MAC)
\(h_n\) 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
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
\(I_x \text{, } I_x\text{, } I_z\) moments of inertia
\(I_{xy}\text{, }I_{xz}\text{, } I_{yz}\) 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
\(K\text{, }k\) 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 \(\mathrm{e}\)
LO low observables
Log common log, to the base 10
LOS line of sight
\(l_t\) distance from \(cg\) to tail’s aerodynamic cent
\(L_{\delta a}\) 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
mb millibar
MCA minimum crossing altitude
\(M_{\mathrm{cr}}\) critical Mach number
\(M_d\) drag divergence Mach number
\(M_{\mathrm{ac}}\) mean aerodynamic cord
\(M_{\mathrm{GC}}\) mean geometric chord
MHz megahertz
mHZ millihertz
\(M_{\mathrm{ic}}\) instrument-corrected Mach number
MilSpec military specification
MIL-STD military standard (publication)
min minute (time)
mm millimeter
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
\(N_1\) low pressure compressor speed
\(N_2\) 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
\(N_x\) longitudinal load factor (g's)
\(N_y\) lateral load factor (g's)
\(N_z\) 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)
\(P\) pressure (N/m2 ,pounds per square inch)
\(P_a\) 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
\(P_{\mathrm{iw}}\) total thrust horsepower required
Pk probability of kill
PLF power for level flight
\(P_0\) standard atmospheric pressure (2116.22 lb/ft2 )
POC point of contact
\(P_p\) pitot pressure
ppm parts per million
Prop propeller
\(P_s\) specific power
\(P_s\) static pressure
PS pulse search
psf pounds per square foot
psi pounds per square inch
\(P_T\) total pressure
PW pulse width
\(Q\) or \(q\) dynamic pressure = \(0.5 \rho V^2\)
q aircraft pitch rate
Q engine torque
\(q_c\) impact pressure (\(P_t − P_a\))
°R degrees Rankine = °F + \(459.67\)
R perfect gas constant = \(8314.34 \left[ \text{J/kmol K} \right]\)
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)
R/T receiver/transmitter
RTO Rejected/refused takeoff
RTO responsible test organization
\(S\) wing area (ft2 or m2)
\(S_a\) 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
\(S_g\) 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
\(S_T\) tail area
std standard
\(S_T\) total takeoff or landing distance \(S_a + S_g\))
STOL short takeoff and landing
STOVL short takeoff and vertical landing
\(T\) period of oscillation
\(T\) temperature
\(t\) thickness
\(T\text{, }t\) time (sec)
t/c thickness-to-chord ratio
\(T_a\) ambient temperature
TACAN tactical air navigation
tan tangent
\(T_{\mathrm{as}}\) 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
TF terrain following
\({\mathrm{THP}}\) Thrust Horsepower
\({\mathrm{THP}}_{\mathrm{alt}}\) horsepower available at altitude
\({\mathrm{THP}}_{\mathrm{max}}\) maximum horsepower available
\({\mathrm{THP}}_{\mathrm{min}}\) minimum horsepower required
\({\mathrm{THP}}_{\mathrm{SL}}\) horsepower required at sea level
TIT turbine inlet temperature
TM telemetry
TMN true Mach number
T/O takeoff
\(T_0\) 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
\(T_t\) 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
\(V_H\) horizontal tail volume coefficient
\(V_V\) vertical tail volume coefficient
\(V_1\) takeoff decision speed
\(V_2\) takeoff safety speed
\(V_A\) design maneuvering speed
VAC volts AC
\(V_b\) buffet airspeed
\(V_B\) design speed for max gust intensity
\(V_{\mathrm{br}}\) velocity for best range
\(V_c\) calibrated airspeed
\(V_D\) design diving speed
VDC volts DC
VDOP vertical dilution of precision
\(V_e\) equivalent velocity
\(V_{\mathrm{FE}}\) maximum flap extended speed
VFR visual flight rules
\(V_g\) ground speed
VHF very high frequency
\(V_i\) indicated airspeed
\(V_{\mathrm{ic}}\) indicated airspeed corrected for instrument error
\(V_{\mathrm{iw}}\) velocity at sea level std day and std weight
VLE max speed with landing gear extended
\(V_{\mathrm{LO}}\) max speed while operating landing gear
\(V_{\mathrm{LOF}}\) lift off speed
VLSIC very large scale integrated circuit
\(V_{\mathrm{mc}}\) minimum directional control speed
VMC visual meteorological conditions
\(V_{\mathrm{mca}}\) minimum directional control speed in the air
\(V_{\mathrm{mcg}}\) minimum directional control speed on the ground
\(V_mo/M_mo\) maximum operating limit speed
\(V_{\mathrm{mu}}\) minimum unstick speed
\(V_{\mathrm{NE}}\) never exceed velocity
\(V_{\mathrm{no}}\) max structural cruising speed
\(V_{\mathrm{opt}}\) optimum velocity for endurance flight
VOR VHF omni-directional range
VORTAC VOR + TACAN
\(V_{P_{\mathrm{min}}}\) velocity for minimum power
\(V_{P_{\mathrm{min}_{\mathrm{SL}}}}\) velocity for minimum power at sea level
\(V_R\) rotation speed
\(V_S\) stall speed
\(V_{S_0}\) stall speed in landing configuration
\(V_{S_1}\) stall speed in some defined configuration
VSTOL vertical/short takeoff and landing
\(V_T\) true airspeed
VTOL vertical takeoff & landing
VVI vertical velocity indicator
\(V_W\) wind velocity
\(V_X\) speed for best angle of climb
\(V_Y\) speed for best rate of climb
\(W\) weight
\(w\) component of velocity along aircraft's Z-axis
WDL weapon data link
\(W / \delta\) weight-to-pressure ratio
\(W_f\) fuel weight
WGS-84 World Geodetic System, 1984
WI watch item
WIT watch item
WOD word of day
WOW weight on wheels
WPT waypoint
wrt with respect to
\(\frac{\dot{W_f}}{\delta \sqrt{\theta}}\) corrected fuel flow parameter
W/S wing loading
\(W_f\) fuel flow (lb/hr)
\(x\) aircraft longitudinal axis, a line running through the nose & tail
\(X_{\mathrm{ac}}\) distance from leading edge to aerodynamic center
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 lateral axes
\(\Delta H_{\mathrm{ic}}\) altimeter instrument correction
\(\Delta H_{\mathrm{pc}}\) altimeter position error correction
\(\Delta P_{p}\) pitot pressure error
\(\Delta P_{s}\) static pressure error
\(\Delta V_{c}\) scale attitude correction to airspeed
\(\Delta V_{\mathrm{ic}}\) instrument correction to airspeed indicator
\(\Delta V_{\mathrm{pc}}\) correction for airspeed position error
\(\infty\) infinity, or freestream conditions

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 viewpoints 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. (\(V_w =V_T - V_g\)).

Body Axes Sign Convention

The generally accepted body axes sign convention is based on the establishment of a three-dimensional axis system 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.

Translational displacements, rates, accelerations, & forces are positive along the positive body axes directions. 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 clockwise 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 clockwise 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.

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 horizontal 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) \(\sigma_g\), is the horizontal angle between true North and the projection 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. Positive rotation is up. During a descent, this parameter is commonly known as glide path angle.

Flightpath bank angle; \(\mu\), 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 established by the design group.

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
Longitudinal acceleration \(a_x\) along +x axis
Lateral acceleration \(a_y\) along +y axis
Vertical acceleration \(a_z\) along +z axis
Longitudinal load factor \(N_x\) along +x axis
Lateral load factor \(N_y\) along +y-axis
Normal load factor \(N_z\) along -z axis
Longitudinal force \(F_x\) along the +x axis
Lateral force \(F_y\) along the +y axis
Normal force \(F_z\) 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

Summary of Generally Accepted Body Axes Sign Convention

Parameter Name Symbol Positive Direction
Angular Measurements
Bank angle \(\phi\) right wing down
Pitch angle \(\theta\) nose-up
Heading \(\psi\) 0 North, +Eastward
Angle of attack \(\alpha\) normal flight attitude
Angle of sideslip \(\beta\) “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 \(\mu\) right wing down
Flightpath elevation \(\gamma\) climb
Flightpath heading \(\sigma_g\)

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 positive. 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 difference between the two as a measure of rolling moment.

The rationale within the wind tunnel community is also logical: any control surface deflection that increases 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, specifically 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.

\(\delta_{a} = \frac{\delta_{\mathrm{a_R}} - \delta_{\mathrm{a_L}}}{2}\) or \(\delta_{a} = \frac{\delta_{\mathrm{a_L}} - \delta_{\mathrm{a_R}}}{2}\)

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:

\(\delta_{a_R} = +\mathrm{TED}\), \(\delta_{a_L} = +\mathrm{TEU}\).

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 generate positive surface deflections. Only the flight tester’s convention states that positive inputs yield positive motions and deflections. All approaches are mathematically connected to the hinge moment sign convention discussed 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) elevator deflection (positive stick force generates a dive).

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

Based on the above background, the SFTE technical council proposes the following standard convention for inceptor & 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 positive 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 moments should have negative values.

 

Conventions for Positive Control Surface Deflections

Parameter Symbol Flight Test SFTE/ Wind Tunnel
Horizontal Stabilizer \(\delta_i\) TEU TED
Elevator \(\delta_e\) TEU TED
Elev. Tab \(\delta_{et}\) TED TED
Stabilators or Rollervators, \(\delta_{eL}\), \(\delta_{eR}\) TEU TED
average: \(\delta_e\) \[\frac{\delta_{eR} + \delta_{eL}}{2}\]
differential: \(\Delta\delta_e\) \[\frac{\delta_{eR} - \delta_{eL}}{2}\]
Elevons \(\delta_{vL}\), \(\delta_{vR}\) TEU TED
average: \(\delta_v\) \[\frac{\delta_{vR} + \delta_{vL}}{2}\]
differential: \(\Delta\delta_v\) \[\frac{\delta_{vR} - \delta_{vL}}{2}\]
Flaperons or trailing edge flap \(\delta_{fL}\), \(\delta_{fR}\) TED TED
average: \(\delta_f\) \[\frac{\delta_{fR} + \delta_{fL}}{2}\]
differential: \(\Delta\delta_f\) \[-\frac{\delta_{fR} - \delta_{fL}}{2}\] \[\frac{\delta_{fR} - \delta_{fL}}{2}\]
Canards \(\delta_{cL}\), \(\delta_{cR}\) TED TED
average: \(\delta_c\) \[\frac{\delta_{cR} + \delta_{cL}}{2}\]
differential: \(\Delta\delta_c\) \[-\frac{\delta_{cR} - \delta_{cL}}{2}\] \[\frac{\delta_{cR} - \delta_{cL}}{2}\]
Leading edge flap \(\delta_{lefL}\), \(\delta_{lefR}\) Leading Edge Down Leading Edge Down
Ruddervators \(\delta_{rvL}\), \(\delta_{rvR}\) TEU TED
average: \(\delta_{rv}\) \[\frac{\delta_{rvR}+\delta_{rvL}}{2}\]
differential: \(\Delta\delta_{rv}\) \[-\frac{\delta_{rvR} - \delta_{rvL}}{2}\]
Ailerons \(\delta_{aL}\), \(\delta_{aR}\)

\(\delta_{aR}\) TEU & \(\delta_{aL}\) TED

or

\(\delta_{aR}\) & \(\delta_{aL}\) TED

 \(\delta_{aR}\) & \(\delta_{aL}\) TED
Aileron Tab \(\delta_{at}\) \[\frac{\delta_{aR}+\delta_{aL}}{2}\] \(\delta_{at}\) TED
average: \(\delta_{a}\) \[-\frac{\delta_{aR} - \delta_{aL}}{2}\] \[\frac{\delta_{aR} - \delta_{aL}}{2}\]
Spoilers \(\delta_{sL}\), \(\delta_{sR}\) Extended Extended
average: \(\delta_{s}\) \[\frac{\delta_{sR}+\delta_{sL}}{2}\]
differential: \(\Delta\delta_{s}\) \[\frac{\delta_{sR} - \delta_{sL}}{2}\] \[-\frac{\delta_{sR} - \delta_{sL}}{2}\]
Rudders \(\delta_{rL}\), \(\delta_{rR}\) TER TEL
average: \(\delta_{r}\) \[\frac{\delta_{rR}+\delta_{rL}}{2}\]
Rudder tab \(\delta_{rt}\) TEL
Speed brake \(\delta_{sb}\) Extended

Conventions for Positive Inputs and Hinge Moments

Parameter Symbol Flight Test SFTE/ Wind Tunnel

Stick/Wheel

Long Force

 \(F_e\) Pull

Stick/Wheel

Lateral Force

 \(F_a\) Right
Pedal Force \(F_r\) Right pedal push

Stick/Wheel

Longitudinal deflection

 \(\delta_{s_e}\) Aft

Stick/wheel

Lateral deflection

 \(\delta_{s_a}\) Right
Pedal deflection \(\delta_{pR}\), \(\delta_{pL}\) Right pedal push
Aerodynamic Hinge Moments

\(C_{h_\delta}\)

\(C_{h_\alpha}\)

\(C_{h_{\delta_o}}\)

\(C_{h_{\delta_\text{tab}}}\)

positive moments

generate

positive deflections

Inceptor

Hinge Moments

\(C_{h_{F_e}}\)

\(C_{h_{F_a}}\)

\(C_{h_{F_r}}\)

+ moments generate

+ deflections

+ moments generate

- deflections

*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, Australian, 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. Although SFTE does not expect all organizations to adopt this standard, it still provides a cornerstone for reference purposes

1.7 Thermodynamics Relations

(references 1.3, 1.4, 1.5, 1.6)

1.7.1 Thermodynamic Definitions

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. heated air moving from one room to another room).

Radiation is the energy transmission through space.

1.7.2 Thermodynamic Symbols

Symbol Use
\(A\) area
\(C\) compressibility factor
\(c\) speed of sound
\(E = u\) specific internal energy (e.g. Btu/lb)
\(H\) specific enthalpy (e.g. Btu/lb)
\(J\) Joule
\(Q\) 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)
\(\overline{V}\) velocity
\(\Delta\) change ( final - initial value)
\(Z\) altitude
\(S\) specific entropy
\(R\) gas constant for each gas (for air = 287 J/kg/K = 53.35 ft-lb/lbmR)
\(\overline{R}\) 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
\(\rho\) density

1.7.3 Thermodynamic Laws

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): \(W + Q = (E_2 - E_1)\)

The Second Law of Thermodynamics states that entropy increases during any irreversible process: \(S_2 > S_1\)

Ideal Gas Equation of State (a.k.a. Perfect gas law):

\[\begin{align} PV&=RT\\ P &= \rho RT\\ PV &= mRT\\ PV &= nRT\\ \end{align}\]

\[\delta = \sigma \theta\]

where

\[\begin{align} \delta &= \frac{P_a}{P_0}\\ \sigma &= \frac{\rho_a}{\rho_0}\\ \theta &= \frac{T_a}{T_0} \end{align}\]

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

\[P_1 V_1 = P_2 V_2\]

where \[T_1 = T_2\]

Charles’ 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.

\[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]

where \[V_1 = V_2\]

Real Gas Relation:

\[PV = CRT\]

For reversible processes:

\[\begin{align} W &= −\int_{}^{}{PdV}\\ Q &= \int_{}^{}{TdS} \end{align}\]

For reversible adiabatic processes:

\[\begin{align} \frac{P_1}{P_2} &= \Bigg[ \frac{V_2}{V_1} \Bigg]^{\gamma} \\ \frac{T_1}{T_2} &= \Bigg[ \frac{V_2}{V_1} \Bigg]^{\gamma - 1} \\ \frac{T_1}{T_2} &= \Bigg[ \frac{P_1}{P_2} \Bigg]^{\frac{\gamma - 1}{\gamma}} \\ \frac{P_1}{P_2} &= \Bigg[ \frac{\rho_1}{\rho_2} \Bigg]^{\gamma} \\ \end{align}\]

Steady Flow Energy Equation

\[ Q + H_1 + \frac{\overline{V}_1^2}{2g} + Z_1 = W + H_2 + \frac{\overline{V}_2^2}{2g} + Z_2 \] 

Bernoulli Equation:

\[ \frac{\Delta P}{\rho g} + \frac{\overline{V}_2^2 - \overline{V}_1^2}{2g} + \Delta Z = 0 \]

Flow per Unit Area:

\[ \frac{W}{A} = \sqrt{\frac{\gamma}{R} \frac{P}{\sqrt{T}} \frac{M}{ \Big( 1 + \frac{\gamma - 1}{2} M^2 \Big) \frac{\gamma + 1}{2 (\gamma - 1)} }} \]

Velocity of sound in a perfect gas:

\[ c = \sqrt{\gamma g R T} \]

Development of Specific Heat Relations:

Specific heat at constant pressure (for air = 1004.76 J/kg/K) \[ c_p \equiv \frac{\partial H}{\partial T} \Bigg\rvert_{P} \]

Specific heat at constant volume (for air = 717.986 J/kg/K) \[ c_v \equiv \frac{\partial u}{\partial T} \Bigg\rvert_{V} \]

Ratio of specific heats \[ \kappa = \gamma \equiv \frac{c_p}{c_v} \]

Enthalpy equation in differential form is: \[dH = du + d(PV)\]

Substituting definitions and ideal gas law gives

\[\begin{align} c_p \, dT &= c_v \, dT + R \, dt\\ &\;\;\mathrm{or} \\ c_p &= c_v + R \end{align}\]

Rearranging gives

\[\begin{align} c_p &= R \frac{\kappa}{\kappa - 1}\\ &\;\;\mathrm{and} \\ c_v &= R \frac{1}{\kappa - 1} \end{align}\]

Development of Poisson’s Equation:

1) From the 1st law: \[W+Q = E_2-E_1\]

2) Substitution for each term gives: \[T\,dS - P\,dV = du\]

3) Divide through by T: \[dS = \frac{du}{T} + \frac{P\,dV}{T}\]

4) Recall: \[du = c_v\,dT\] and \[PV = RT\]

5) Substitution gives: \[dS = c_v\frac{dT}{T} + R \frac{dV}{V}\]

6) Assume constant specific heat and integrate: \[s_2 - s_1 = c_v\,\ln \frac{T_2}{T_1} + R\,\ln \frac{V_2}{V_1}\]

7) Assuming a reversible adiabatic process: \[c_v\,\ln \frac{T_2}{T_1} = - R\,\ln \frac{V_2}{V_1}\]

8) Substitute \[c_v = R\frac{1}{\kappa - 1}\] to get: \[\frac{T_2}{T_1} = \Big( \frac{V_1}{V_2} \Big)^{\kappa -1}\]

9) Differentiate H: \[dH = du + P\,dV + V\,dP\]

10) Substitution into step #2: \[T\,dS = dH - V\,dP\]

11) Integrate: \[s_2 - s_1 = c_p\,\ln \frac{T_2}{T_1} + R\,\ln \frac{P_2}{P_1}\]

12) Assuming a reversible adiabatic process: \[c_p\,\ln \frac{T_2}{T_1} = - R\,\ln \frac{P_2}{P_1}\]

13) Substitute \[c_v = R\frac{\kappa}{\kappa - 1}\] to get: \[\frac{T_2}{T_1} = \Big( \frac{P_2}{P_1} \Big)^{\frac{\kappa -1}{\kappa}}\]

14) Combine steps #8, #13 to get: \[\frac{P_2}{P_1}=\Big( \frac{V_1}{V_2} \Big)^{\kappa}\] or \[(PV)^{\kappa} = \mathrm{const}\]

1.8 Mechanics Relations

1.8.1 Mechanics Symbols

Symbol Use
\(a\) linear acceleration
\(a_r\) centripetal (radial) acceleration
\(a_T\) tangential acceleration
\(F\) force
\(g\) acceleration due to gravity
\(G\) moment
\(H\) angular momentum
\(H\) height
\(Hp\) horsepower
\(I\) rotational moment of inertia (see section 10)
\(J\) impulse = change in momentum
\(k\) radius of gyration
\(m\) mass
\(N_r\) radial load factor
\(P\) power
\(L\) linear momentum
\(Q\) moment (a.k.a. torque)
\(r\) radius
\(S\) distance, displacement
\(s\) seconds
\(t\) time
\(V\) true inertial velocity
\(V_0\) initial inertial velocity
\(W\) work
\(q\) angular displacement
\(\mathrm{Vol}\) volume
\(\omega\) angular velocity (radians/second)
\(\dot{\omega}\) angular acceleration

1.8.2 Newton’s 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, \(m\,V\) ).

\[ dF = \frac{d(mV)}{dt} \]

For constant mass in rectilinear motion: \[F = ma\]

For constant mass distribution in curvilinear motion: \[G = \dot{\omega} I\]

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)”

1.8.3 Planar Kinetics, Work, Power and Energy

Rectilinear motion Curvilinear motion
displacement \[S\] angular displacement \[\theta\]
velocity \[V = \frac{dS}{dt}\] angular velocity \[\omega = \frac{d\theta}{dt}\]
acceleration \[a = \frac{dV}{dt}\] angular acceleration \[\dot{\omega} = \frac{d\omega}{dt}\]
inertia \[m\] rotational inertia \[I = \int_{}^{}{r^2 dm}\]
momentum \[L = m\,V\] angular momentum \[H = I \omega\]
force \[F = m\,a\] torque \[Q = I \dot{\omega}\]
work \[W = \int_{}^{}{F\,dS}\] work \[W = \int_{}^{}{Qd\theta}\]
power \[P = F\,V\] power \[P = Q \omega\]
kinetic energy \[\frac{1}{2} mV^2\] kinetic energy \[\frac{1}{2} I \omega^2\]
potential energy \[mgH\] n/a

1.8.4 Planar Kinematics at Constant Acceleration

Rectilinear motion Curvilinear motion
\[V = V_0 + at\] \[\omega = \omega_0 + \dot{\omega} t\]
\[V^2 = V_0^2 + 2aS\] \[\omega^2 = \omega_0^2 + 2 \dot{\omega}\theta\]
\[S = V_0 t + \frac{1}{2} a t^2\] \[\theta = \omega_0 t + \frac{1}{2} \dot{\omega} t^2\]
\[S = \frac{1}{2}(V + V_0) t\] \[\theta = \frac{1}{2}(\omega + \omega_0)t\]
\[S = \frac{\left( V^{2} - V_{0}^{2} \right)}{2a}\] \[\theta = \frac{\left( \omega^{2} - \omega_{0}^{2} \right)}{2\dot{\omega}}\]
\[t = \frac{- V_{0} + \sqrt{V_{0}^{2} + 2\text{aS}}}{a}\] \[t = \frac{- \omega_{0} + \sqrt{\omega_{0}^{2} - 2\dot{\omega}\theta}}{\dot{\omega}}\]
\[a = \frac{2\left( S - V_{0}t \right)}{t^{2}}\] \[\dot{\omega} = \frac{2(\theta - \omega_{0}t)}{t^{2}}\]

1.8.5 Curvilinear motion with constant acceleration and radius

\[r = \frac{V^2}{g N_r}\]
\[V = \omega r\]
\[N_r = \frac{a_r}{g}\]
\[\omega = \frac{g N_r}{V}\]
\[\dot{\omega} = \frac{\dot{V}}{R}\]
\[a_r = r \omega^2 = \frac{V^2}{r}\]
\[a_{r} = \dot{\omega}r\]

1.8.6 Aircraft in level turn

\(N_{z_w}\) = load factor normal to flight path
\(r\) = turn radius
\(\Omega\) = turn rate (rad/sec)
\[r = \frac{V^{2}}{g\sqrt{N_{\mathrm{zw}}^{2} - 1}}\]
\[\omega = \frac{g\sqrt{N_{\mathrm{zw}}^{2} - 1}}{V}\]
\[N_{z_w} = \sqrt{\frac{\omega V}{g} + 1}\]
\[V = \mathrm{inertial velocity} \]

1.8.7 Level Turn Kinematics Character

 

1.8.8 Gyroscopic Motion

(reference 1.7)

For bodies spinning about an axisymmetric axis

\(\dot{\psi}\) = spin rate
\(\dot{\phi}\) = precession rate
\(\dot{\theta}\) = nutation rate
\(I_z\) = moment of inertia about spin axis
\(I_t\) = transverse moment of inertia about the spin point (perpendicular to spin axis)
\(I_{\mathrm{cg}}\) = moment of inertia about the \(\mathrm{cg}\) (perpendicular to spin axis)
\(M_x\) = moment about spin point (acting along plane that defines \(\theta\) )

For steady precession (constant \(\dot{\theta}\), \(\dot{\phi}\) , \(\dot{\psi}\) )

\[ \sum M_{x} = - I_{t}\dot{\phi^{2}}\sin\theta\cos\theta + I_{z}\dot{\theta}\sin\theta\left( \dot{\phi}\cos\theta + \dot{\psi} \right) \]

For torque free motion (gravity is only external force)

\[\dot{\psi} = \frac{I_{\mathrm{cg}} - I_{z}}{I_{z}}\dot{\phi}\cos\theta\]

note that \(I_{\mathrm{cg}} > I_z\) yields regular precession

while \(I_{\mathrm{cg}} < I_z\) yields retrograde precession

1.9 International Phonetic Alphabet and Morse Code

Character Say 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 Tree • • • — —
4 Four • • • • —
5 Fife • • • • •
6 Six — • • • •
7 Seven — — • • •
8 Eight — — — • •
9 Niner — — — — •
0 Zee-ro — — — — —

1.10 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 Marie 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/.

  1. Common FTE conversions in boldface↩︎

  2. solid angle measurement↩︎

  3. Converting between force and mass (e.g. kg force to kg mass or pound force to pound mass) uses \(g = 32.174 \frac{ft}{s^2}\)↩︎

  4. Fuel densities are temperature dependent↩︎

  5. The foot is defined as exactly 0.3048 meters https://www.nist.gov/pml/us-surveyfoot↩︎

  6. The SI defines the nautical mile as exactly 1852 meters. https://www.bipm.org/en/publications/si-brochure/↩︎