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AERODYNAMIC SUMMARY SHEET 
Prepared by Vladimir SMOLKO 
1 
 
 
Nomenclature Subscripts 
A Aspect Ratio n Load Factor (g’s) Γ Circulation (Ft2/sec) A Aileron S Static 
 Flow Cross Section Area (Ft2) P pressure (psf) γ Flight Path Angle (Deg) ac aerodynamic center T Total 
a Speed of Sound (Ft/Sec) Roll Rate Body Axis (Rad/sec) Specific Heat Ratio CP/CV am ambient True 
 Acceleration (Ft/Sec2) Time to Oscillate One Cycle (sec) ∆ Increment Notation C calibrated Tropopause 
 Lift Curve Slope (1/Deg) Q Pitch Rate Body Axis (Rad/sec) δ Pressure Ratio P/P0 Compressible Thrust 
b Wing Span q Dynamic Pressure (Psf) Boundary Layer Thickness (Ft) Chord t tail 
C Coefficient R Yaw Rate Body Axis (Rad/sec) Control Surface Deflection (Deg) CG Center of gravity W Wind 
c Wing Mean Aerodynamic Chord (Ft) Gas Constant ε Temperature Probe Recovery Factor D Drag Wing 
D Drag Radius (Ft) Atmospheric Eddy Viscosity DP Positive Drag Y Side Force 
e wing efficiency factor RN Reynolds Number Downwash Angle (Deg) d Damping 
F Thrust (Lbs) R/C Ratio of Climb (Ft/Min) ζ Damping Ratio E Elevator 
 Force (Lbs) S Wing Area (Ft2) θ Temperature Ratio T/T0 e Equivalent 
FC Compressibility Factor Take Off Distance (Ft) Pitch Attitude Earth Axis System f Friction 
G(s) Forward Loop Transfer Function T Thrust (Lbs) Λ Sweep Angle (Deg) G Gross 
g Acceleration of gravity (Ft/Sec2) Temperature (0R) λ Taper Ratio Ground 
H Altitude Geopotential (Ft) TSFC Thrust Specific Fuel Consumption µ Friction coefficient I Indicated 
H(s) Feedback Transfer Function t Time (sec) Viscosity (Lb·Sec/Ft2) i Incidence 
K Gain V Velocity (Ft/Sec) ν Cinematic Viscosity (Ft2/Sec) L Lift 
L Lift (Lbs) Volume (Ft3) ρ Mass Air density (slug/Ft3) l Roll 
Ln Natural logarithm W Gross Weight (Lbs) σ Density ratio ρ/ρ0 m Pitch 
L’ Rolling Moment (Ft·Lbs) WA Mass Flow Air (Lbs/Sec) Side Angle (Deg) n Yaw 
L Characteristic Length (Ft) Wf Fuel Flow(Lbs/Hr) τ Time Constant (Sec) North 
M Pitching Moment (Ft·Lbs) w Specific Weight ф Bank Angle (Earth Axis System) Natural 
M Mach Number Y Side Force (Lbs) Runway Gradient (deg) 0 Sea level 
m mass (slugs) α Angle of Attack (Deg) ψ Yaw Angle (Earth Axis System) p Pressure 
N Yawing Moment (Ft·Lbs) β Angle of Sideslip (Deg) ω Frequency (Rad/sec) R Rudder 
 
ATMOSPHERE 
Pressure Density Alt 
Ft PSF δ SLUG/FT3 σ 
TEMP 
0R 
 0 2,116.22 1.0000 .002377 1.0000 518.67 
10,000 1,455.32 .6877 .001766 .7385 483.01 
20,000 972.48 .4595 .001266 .6328 447.35 
30,000 628.43 .2970 .000689 .3741 411.69 
40,000 391.69 .1851 .000585 .2452 389.97 
50,000 242.22 .1145 .000362 .1522 389.97 
 
THERMODYNAMICS 
P=Pressure (PSF) V= Volume (Ft3/Slug) T=Temp (0R) 
RTPV = TgRTP ρρ 1716== V1=ρ 
Isentropic (reversible Adiabatic) 
γ






=
2
1
12
V
V
PP
1
1
2
12
−






=
γ
γ
T
T
PP
 
γ
1
2
1
12 





=
P
P
VV
1
1
2
1
12
−






=
γ
T
T
VV
 
γ
γ 1
1
2
12
−






=
P
P
TT
1
2
1
12
−






=
γ
V
V
TT
 
For Air Atmospheric Temperature, below 50,000Ft 
FLbBTUCP
0/240. ⋅= FLbBTUC
V
0/1715. ⋅=
 40.1==
VP
CCγ 






⋅





⋅=
T
T
P
P 0
0
002327.0ρ
 
w=ρg Lbs/Ft3 
 
Speed of sound, Standard Day sea level a=1116.45Ft/sec 
)()(04.29)/()(02.49
00
KnotsRTSecFtRTa ⋅=⋅=
 
 
BOUNDARY LAYER 
 
Flat Plate 
 
Laminar Flow 
NR
l2.5
=δ
 
 
Turbulent Flow 
2.0
37.0
NR
l
=δ
 
νµ
ρ lVlV
R TTN ==
 
72.198
102697.2
8
+
⋅
=
−
T
TT
µ
 
Standard Day sea level µ=3.7372·10-5 Lb·Sec/Ft2 ν=µ/ρ 
 
 
AERODYNAMICS BASIC LAWS 
Continuity Equation Momentum Equation 
 
tConsAV tan=ρ VmF ∆=
⋅
 
Bernoullis Equation 
Incompressible Compressible 
2
2V
PPT
ρ
+=
 
211
2VPP
T
T +⋅
−
=⋅
− ργ
γ
ργ
γ
 
 
AERODYNAMIC COEFFICIENTS 
qS
L
CL =
 
qS
D
CD =
 
cqS
M
CM =
 
q
P
CP
∆
=
 
qSb
L
Cl
'
=
 
qSb
N
CN =
 
qS
Y
CY =
 
qS
T
CT =
 
 
 
GEOMETRY 
For Straight Tapered Surface 
 
Aspect Ratio 
S
b
A
2
=
 






+
++
⋅⋅=
1
1
3
2 2
λ
λλ
tCc
 
 
Taper Ratio 
r
t
C
C
=λ
 






+
+
⋅=
λ
λ
1
21
6
b
v
 
 
Area 
( )
2
rt
CCb
S
+⋅
=
 
 
 
 
 
 
 
 
 
Altitude (Ft) Sea Level->36,089 A1=6.87536·10
-6 
Pressure ratio δ=(1-A1H)
5.2563 H=145,448(1-δ0.19026) 
Density ratio σ=(1-A1H)
4.2563 H=145,448(1-σ0.23496) 
Temperature ratio θ=(1-A1H) H=145,448(1-θ) 
Altitude (Ft) 36,089->65,617 
A2=4.80634 10
-5 
65,617->104,987; 
A3=3.17176 10
-5 
Pressure ratio δ=0.22336·e-A2·(H-36,089) δ=A3θ
-34.1632 
Density ratio σ=0.29708·e-A2·(H-36,089) σ=A3θ
-34.1632 
Temperature ratio θ=0.75187 θ=0.68246+1.05778·10-6 
 
Airspeed/Mach/Dynamic Pressure/Total Conditions 
 
Dynamic Pressure )(
295
)(
1481
2
2
2
2
PSF
KtsV
M
V
q eT =⋅⋅== δ
ρ 
 
Impact Pressure 
(Pitot-Static Measure) 






++++=⋅=−= ...
80404
1;;
842 MMM
FFqqPPq CCCSTC
 
( )[ ] SC PMq ⋅−⋅+= 12.01 5.32 
 
 
Mach No 










−










+








−














⋅+⋅= 111
5.661
)(
2.015
286.0
5.3
2
0 KtsV
P
P
M C
 
Indicated Airspeed VI = VC + ∆Vpitot Static Source Error 
 
Calibrated Airspeed )(1
_
11.1479
28571.0
0
Knots
P
PP
V ST
c
−





+⋅=
 
 
Equivalent Airspeed )(11174.32
28571.0
Knots
P
PP
PV
S
ST
Se








−




 −
+⋅=
 
 
True Airspeed 
a
V
M
V
V TeT ==
σ
 
Total condition M < 1.0 γ=1.4 
( )22 2.01
2
1
1 MTMTTT ⋅+⋅=





⋅
−
+⋅= εε
γ
 
 
( ) 5.22
1
1
2
2.01
2
1
1 MMT ⋅+⋅=





⋅
−
+⋅=
−
ρ
γ
ρρ
γ
 
 
( ) 5.32
1
2
2.01
2
1
1 MPMPPT ⋅+⋅=





⋅
−
+⋅=
−γ
γ
γ
 
Total condition M = 1.0 γ=1.4 
528.0634.0833.0 ===
TTT P
P
T
T
ρ
ρ
 
 
Conversion Factors 
bars x 75.006 =CM Hg 00C Ft/sec x0.6818 =MPH 
BTU x 778.169 =Ft·Lbs Ft/sec x1.0973 =Km/Hr 
BTU/sec x1054.118x =Watts Ft·Lbs/Sec x1.1818·10-3 =H.P. 
CM Hg x 27.845 =Lbs/Ft2 Fluid Oz x29.6 =Cu CM 
Ft3 x 28.317 =Liter Gals Imp x1.201 =U.S. Gal 
dynes x2.248·10-6 =Lbs Kilogram x2.205 =Lbs 
Ergs x7.376·10-8 =Ft·Lbs Knots x1.688 =Ft/sec 
Ft x0.3048 =Meters Liters x0.2642 =U.S. Gals 
Ft/sec x0.5925 =Kts Rad x57.29 =Deg 
 
Constants Weigths Lbs/Gal At 0C 
g=32.174 Ft/Sec2 JP-4 6.55 20 
R= 63.35 Ft·Lbs/LB0R Water 8.345 4 
γ=1.4 
 
AERODYNAMIC SUMMARY SHEET 
Prepared by Vladimir SMOLKO 
2 
LONGITUDINAL STABILITY LATERAL DIRECTIONAL STABILITY 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
AXIS TRANSFORMATION 
 
 
 
 
 
 
 
 
 
 
 
Component Along Earth Axes X
Y
Z
 XE YE ZE 
XB ψθ coscos ⋅ ψθ sin⋅cos θsin− 
YB ψφψθφ sinsinsin ⋅−⋅⋅ coscos
 ψφψθφ coscos ⋅+⋅⋅ sinsinsin θφ cos⋅sin 
Body 
Axes 
Component 
ZB ψφψθφ sinsinsin ⋅+⋅⋅ coscos
 ψφψθφ coscos ⋅−⋅⋅ sinsinsin θφ coscos ⋅ 
 
Angular Velocity relations 
θψφ sinP ⋅−= && θφθφψ seccossec ⋅⋅+⋅⋅= RsinQ&
 
θφψφθ coscos ⋅⋅+⋅= sinQ & φφθ sinRQ ⋅−⋅= cos& 
θφψφθ coscos ⋅⋅+⋅−= && sinR θφθφφ tancostan ⋅⋅+⋅⋅+= RsinQP& 
 
Lateral –Directional Axis System 
Small perturbation Assumption 
 
Lateral-Directional Equation of motion 
RArp RA
YYrYpYYgrUv δδβθφ δδβ ++++∆+⋅=+ 00 cos&
 Ft/Sec2 
 
( ) RArpxxxz RA LLrLpLLrIIp δδβ δδβ ++++∆=⋅− &&
 Rad/Sec2 
 
( ) RArpzzxz RA NNrNpNNpIIr δδβ δδβ ++++∆=⋅− &&
 Rad/Sec2 
 
Lateral-Directional Dimensional Stability Derivatives 
 
Side force due to sideslip Roll moment due to aileron deflection 
)sec(
2−⋅= Ft
m
qSC
Y
Yβ
β
 
)(
12 −− ⋅= DegSec
I
qSC
L
xx
l
A
A
δ
δ
 
Side force due to roll rate Roll moment due to rudder deflection 
)sec(
2
1
0
−⋅= Ft
mU
qSbC
Y
pY
p
 
)(
12 −− ⋅= DegSec
I
qSC
L
xx
l
R
R
δ
δSide force due to rate of yaw Aircraft’s directional stability 
)sec(
2
1
0
−⋅= Ft
mU
qSbC
Y r
Y
r
 
)(
2−= Sec
I
qSC
N
zz
nβ
β
 
Side force due to aileron deflection 
)sec(
12 −− ⋅⋅= DegFt
m
qSC
Y A
A
δ
δ
 
)(
2−= Sec
I
qSC
N
zz
n
T
Tβ
β
 
Side force due to rudder deflection Yaw moment due to roll rate 
)sec(
12 −− ⋅⋅= DegFt
m
qSC
Y R
R
δ
δ
 
)(
2
1
0
2
−= Sec
UI
CqSb
N
zz
n
p
p
 
‘Dihedral effect’, lateral stability ‘Yaw damping’ 
)(
2−= Sec
I
qSbC
L
xx
lβ
β
 
)(
2
1
0
2
−= Sec
UI
CqSb
N
zz
n
r
r
 
‘Roll damping’ Yaw moment due to aileron deflection 
)(
2
1
0
2
−= Sec
UI
CqSb
L
xx
l
p
p
 
)(
12 −− ⋅= DegSec
I
qSbC
N
zz
n
A
A
δ
δ
 
Roll moment due to yaw rate Yaw moment due to rudder deflection 
)(
2
1
0
2
−= Sec
UI
CqSb
L
xx
l
r
r
 
)(
12 −− ⋅= DegSec
I
qSbC
N
zz
n
R
R
δ
δ
 
 
Perturbation Equation 
 
vVV += 0
 
0Uv=∆β
 
pPP += 0
 
0Uv&
& =∆β 
rRR += 0
 ψψψ ∆+= 0
 
 φφ ∆= 
 
XE, YE - Earth Axes 
XSo, YSo – Initial Stability Axes 
XS YS – Disturbed Stability Axes 
βo=0 
Longitudinal Axis System 
Small perturbation Assumption 
 
Longitudinal Equation of motion 
ETu Eu
XXuXuXgu δαθθ δα +∆+++⋅−= 0cos&
 Ft/Sec2 
 
Equ E
ZqZZZuZsingqUw δααθθ δαα ++∆+∆++⋅−=− && &00
 Ft/Sec2 
 
EqTTu Eu
MqMMMMuMuMq δααα δαα α ++∆+∆+∆++= && &
 Rad/Sec2 
 
Longitudinal Dimensional Stability Derivatives 
 
‘Speed damping’ derivative Z-force due to Elevator 
)(sec
)2(
1
0
0 −
+−
=
mU
CCqS
X
DD
u
u
 
)(
12 −− ⋅⋅
−
= DegSecFt
m
qSC
Z E
E
Lδ
δ
 
Thrust changes due to speed ‘Speed effect stability’ 
)(sec
)2(
1
0
−
+
=
mU
CCqS
X
XuX
u
TT
T
 ( )
)(
2
11
0
0 −− ⋅
+−
= SecFt
UI
CCcqS
M
YY
mm
u
u
 
X-force due to AoA M due to Thrust / Speed changes 
)sec(
)(
20 −⋅
−−
= Ft
m
CCqS
X
LDα
α
 ( )
)(
2
11
0
−− ⋅
+−
= SecFt
UI
CCcqS
M
YY
mm
T
TuT
u
 
‘Trim drag’ ‘AoA stability’ 
)sec(
12 −− ⋅⋅
−
= DegFt
m
qSC
X E
E
Dδ
δ
 
)(
2−= Sec
I
CcqS
M
YY
mα
α
 
Z-force due to forward speed M due to Thrust / AoA changes 
)(sec
)2(
1
0
0 −
+−
=
mU
CCqS
Z
LL
u
u
 
)(
2−= Sec
I
CcqS
M
YY
m
T
Tα
α
 
Z-force due to AoA ‘Downwash lag’ derivative 
)sec(
)(
20 −⋅
+−
= Ft
m
CCqS
Z
DLα
α
 
)(
2
1
0
2
−= Sec
UI
CcqS
M
YY
mα
α
&
&
 
Z-force due to AoA rate ‘Pitch damping’ 
)sec(
2
1
0
−⋅
−
= Ft
mU
qSC
Z
L εα
α
&
&
 
)(
2
1
0
2
−= Sec
UI
CcqS
M
YY
m
q
q
 
Z-force due to rate of pitch Elevator control effectiveness 
)sec(
2
1
0
−⋅
−
= Ft
mU
qSC
Z
qL
q
ε 
)(
2−= Sec
I
CcqS
M
YY
m
E
E
δ
δ
 
Perturbation Equation 
 
uUU += 0
 
0Uw=∆α
 
wWW += 0
 
0Uw&& =∆α
 
qQQ += 0
 ααα ∆+= 0
 
 θθθ ∆+= 0
 
 αθγ ∆−= 
 
XE, ZE - Earth Axes 
XSo, ZSo – Initial Stability Axes 
XS ZS – Disturbed Stability Axes 
Earth to body Axis 
Yaw-Pitch-Roll Sequence 
 
Vector Transformation Matrix 
Earth to body Axes (Yaw-Pitch-Roll Sequence) 
Component Along Wind Axes X
Y
Z
 XW YW ZW 
XB βα coscos ⋅
 βα sin⋅− cos αsin− 
YB βsin
 βcos 0 
Body 
Axes 
Component 
ZB βα cos⋅sin
 βα cos⋅− sin αcos 
 
Angular Velocity relations 
αβ sinP ⋅= & ααβ cscsec ⋅=⋅−= PR& 
α&=Q Q=α& 
αβ cos⋅−= &R αtan0 ⋅+= RP 
 
Body Axis to stability Axis 
Angular Velocity relations 
αα sinRPP
s
⋅+⋅= cos
 αα sinRPP
ss
⋅−⋅= cos
 
QQ
s
=
 
s
QQ =
 
αα cos⋅+⋅−= RsinPR
s
 αα cos⋅+⋅=
ss
RsinPR
 
 
AERODYNAMIC SUMMARY SHEET 
Prepared by Vladimir SMOLKO 
3 
LONGITUDINAL 
Static trim 
H
t
LMDLMM
V
q
q
CC
c
Z
C
c
X
CCC
tNACWBcaCG
⋅⋅−+⋅+⋅+=
..
 
For symmetrical Airfoil ( )εααα −+−⋅=
it tiWtL
aC
 
Static Stability 






−⋅⋅⋅−+⋅+⋅=
α
ε
αα ααα d
d
V
q
q
C
d
dC
c
Z
d
dC
c
X
CC H
t
L
MD
LM
H
NAC 1
 
Elevator/g 
Turn
n
A
PullUpForA
SW
gAcC
dC
dC
Cq
SW
dn
d
PMCG
PMN
m
L
m
MZ
e q
e






+=
=












⋅
+⋅
⋅
=
−
−
2
..
..
1
1
1
4
0 444 3444 21
43421
ρδ
δ
 
Neutral Point (N0) 
 
 
AirplaneL
M
CG
dC
dC
XN 





−=0
 
 
 
( )0NXCC CGLM −⋅= αα
 
 
cS
lS
V
d
d
V
q
q
a
a
dC
dC
NN HHHH
t
w
t
L
M
Nac
BODYWING ⋅
⋅
=





−⋅⋅





⋅+−=
α
ε
100
 
Manoeuvre Point (M.P.) 4444 84444 7644 844 76 Turn
M
UpPull
M
nW
qcSC
N
W
qcSC
NN
qq






+⋅
⋅
−=
⋅
−=
2000
1
1
44
ρρ 
Short Period Approximation 
Natural Freq.,Damping 
 
sp
sp
n
q
q
n
M
U
Z
M
SecRadM
U
MZ
ω
ςω
α
α
α
α
⋅






++−
=−=
2
;/
0
0
& 
Phugoid Approximation 
p
p
n
UU
n
X
SecRad
U
gZ
ω
ςω
⋅
−
=
−
=
2
;/
0
 
Hinge Moments 
αδδ
αδδ
⋅+⋅+⋅+=
⋅⋅
= HcHTABHHH
cc
H CCCCC
cSq
HM
C
cTAB0
 
Oscillatory Motion 
 
 
d
d
P
π
ω
2
=
 
 
n
ωζσ ⋅= 
 
2
1 ζωω −⋅= nd
 
 
12
2
1
TT
X
X
Ln
−






=σ
 
 
n
Amp
T
ωζ ⋅
=
693.0
2
1
 
 
 
 
 
 
Feedback Control system 
 
Transfer Function Block Diagram 
 
Closed Loop 






⋅+
=
)()(1
)(
)(
)(
sHsG
sG
sX
sY
 
 
Open loop 
)()( sHsG 
 
Root Locus Analysis 
)(
)(
1
)(
)(
)(
sD
sN
K
sG
sX
sY
+
=
 
 
)()()(
)()()(
)(
)(
21
21
n
m
PsPsPs
ZsZsZs
K
sD
sN
K
+⋅⋅⋅+⋅+
+⋅⋅⋅+⋅+
=
 
s ~ Laplace operator (o) Zeros~Roots of Nominator Z1, Z2,···Zm 
 (x) Poles~Roots of Denominator P1, P2,···Pn 
Table of Laplace Transforms 
Time Function Laplace Transform 
Unit Impulse 
δ(t) 
 
 
1 
Unit Step 
u(t) 
s
1 
Unit Ramp 
t 
2
1
s
 
Polinomial 
tn 
1
!
+ns
n 
 
Exponential 
 
e-at 
as +
1 
 
Sine Wave 
 
sin(ωt) 
22 ω
ω
+s
 
 
Cosine Wave 
 
cos(ωt) 
22 ω+s
s 
 
Damping Sine Wave 
 
e-atsin(ωt) 
22)( ω
ω
++ as
 
 
Damping Cosine Wave 
 
e-atcos(ωt) 
22)( ω++
+
as
as
 
 
Stability Derivative Effect 
 
 Stability 
Derivative 
Quantity Most 
Affected 
How affected 
Cmq Damping of the 
short period 
Increase Cmq to 
increase the damping 
Cmα Natural frequency 
of the short period 
Increase Cmα to 
increase the 
frequency 
CXu Damping of the 
phugoid 
Increase CXu to 
increase the damping 
 
 
 
 
Longitudinal 
Czu Natural frequency 
of phugoid 
Increase to increase 
the frequency 
Cnr Damping of the 
Dutch Roll 
Increase Cnr to 
increase the damping 
Cnβ Natural Frequency 
of the Dutch Roll 
Increase Cnβ to 
increase the natural 
frequency 
Clp Roll subsidence Increase Clp to 
increase 1/TR 
 
 
 
Lateral 
Directional 
Clβ Spiral divergence Increase Clβ for 
spiral stability 
 
LATERAL-DIRECTIONAL 
 
Static trim 0
' =+++ qSbLCCC TRlAll
RA
δδβ
δδβ
 
0=+++ qSbNCCC TRnAnn
RA
δδβ
δδβ
 
0cos =⋅⋅+++ γφδδβ
δδβ
sinCCCC LRYAYY
RA
 
 
Static Stability 
bS
lS
V
d
d
V
q
q
aC VVVV
t
Vn ⋅
⋅
=





−⋅⋅





⋅−=
β
σ
β
1
 
 
Roll Performance 
( )1)(
2
−⋅
⋅
+⋅
⋅
−= tL
P
A
P
A PAA e
L
L
t
L
L
t
δδ
φ δδ
 
P
A
ss
T
t
ss
L
L
PePtP AR
δδ ⋅
−=





−⋅=
−
1)(
 
RadAngleHelix
C
C
V
bP
A
l
l
P
A δδ ⋅=
⋅
⋅
2
 
 
Roll Time Constant 
1
2
2
0 sec
2
4
1 −














−⋅
⋅
⋅
⋅+⋅−=
pP nL
ZZ
n
l
l
XXR
CC
bm
I
C
C
C
I
bSU
T
β
βρ
 
 
Spiral Time Constant 
1
0
sec
1 −








−⋅
⋅
−=
rr
P
ln
n
l
lS
CC
C
C
CU
g
T
β
β
 
 
Dutch Roll Approximation 
( ) SecRadYNUNNY
U
rrn /
1
0
0
βββω −+=
 














⋅−⋅








⋅−+−=
pr n
ZZ
L
n
l
XX
ZZ
n
ZZ
Y
n
d C
I
mb
C
C
C
I
I
C
I
mb
C
m
SU
224
22
0
β
β
βω
ρ
ς
 
22
1
1
Rd
XXn
ZZl
d
T
IC
IC
ω
β
φ
β
β
+
−=
 
 
AERODYNAMIC SUMMARY SHEET 
Prepared by Vladimir SMOLKO 
4 
 
Jet Airplane Performance 
Ground 
 
( )[ ]φµµ sinWqSCCWT
W
g
a
LDTx
⋅−⋅−−−⋅=
 
For average Values TT, W, CD, CL, µ, ф (Runway Gradient) At 0.707 VT.O. 
)(
0444.0
tan
2
..
2
Feet
BVA
BVA
Ln
B
W
ceDis
OT
W






−
−
⋅
⋅
=
 
)(
1
11
1026.0
..
.. Sec
VAB
VAB
VAB
VAB
Ln
BA
W
Time
W
W
OT
OT
















⋅+
⋅−








⋅−
⋅+
⋅
⋅
⋅
=
 
( ) ( )KtsVKTASV
WOT
→→..
 ( )φµ sinWWTA
T
⋅−−= ( )
LD
CCSB µσ −⋅⋅⋅= 00339.0 
 
In Flight 
 
 
 
AxisXAccelLongu
S
→&
 
 
 
AxisZAccelVertw
S
→&
 
 
 
 
 
Wing level ф=0 
( )γsinWDaT
W
g
u ⋅−−⋅⋅= cos&
 
( )αγ sinTWL
W
g
w ⋅−⋅−⋅= cos&
 





 ⋅
−−=
g
V
g
w
nZ
θ&&
1
 
 
Energy Equation T.E. =P.E.+K.E.=W·h+1/2mVT
2 
 
Specific Energy 
( )
W
VDT
dt
dV
g
V
dt
dh
dt
dE
g
V
hE hh
⋅−
=⋅+=+=
2
2 
T.E. – Total Energy, P.E. – Potential Energy, K.E. – Kinetic Energy 
 
 
Navigation 
 Two points A and B )( WestDegLatitudeLat +→ )( NothernDegLongitudeLong +→ 
Straight Line Navigation (Rhumb line Navigation)- Loxodromia 
[ ]
( )
( )( ) ( )( )[ ]









⋅+−⋅+⋅
−⋅
= −
A
0
B
00
BA1
NorthDeg
Lat5.045tanLnLat5.045tanLn180
LongLong
tan
π
Ψ
 
[ ]
( )
44444 844444 76 0cosif
AB
AB
.M.N
)Latcos(LongLong60
cos
LatLat
60S
=
⋅−⋅=
−
⋅=
Ψ
Ψ
 
Great Circle Navigation - Orthdomia 
[ ]
[ ])LongLongcos()Latcos()Latcos()Lat(sin)Lat(sincos60 ABBABA
1
.M.N
S −⋅⋅+⋅⋅=
−
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Jet Airplane Performance 
Climb__ 
Descent 
min/Ft
dH
dV
g
V
1
)Kts(V
W
DT
28.101CR
T


















⋅+
⋅




 −
⋅=
 






⋅
dH
dV
g
V
 For 
constant 
 
 
VC 
 
 
Ve 
 
 
M 
Above 36089 --- 0.7�M2 0 
Below 36089 -0.014055�M+0.65667�M2-0.214204�M3 0.567�M2 -0.133�M2 
Range 






⋅=





⋅





⋅
⋅⋅
=
2
1
2
10
.)M.N( W
W
LnW
W
W
Ln
D
L
TSFC
Ma
R
θ
 
Lb
NAM
W
 Range 
Factor 
Endurance 






=





⋅





⋅=
2
1
2
1
)Hours( W
W
Ln
V
W
W
W
Ln
D
L
TSFC
1
T
 
a0–Speed Sound knots at Sea level 
 
Lb
NAM 
Nautical 
Air 
Miles 
Per Lb 
Fuel 
Parabolic Drag Polar 
 
( )21
D
21
L
.)M.N(
MAX WW
C
C
STSFC
40.34
R
RANGE
MAX
−





⋅
⋅⋅
=
σ
 
eA
C
CC
2
L
DD 0 π
+=
 
 
3
eAC
C 0
RANGE
MAX
D
L
π
=
 
0
RANGE
MAX
RANGE
MAX
D
L
D
21
L
C4
C
3
C
C
=





 
 






⋅





⋅=
2
1
D
L
)Hours(
MAX
W
W
Ln
C
C
TSFC
1
T
RANGE
MAX
 
 
0
ENDURANCE
MAX
DL AeCC π=
 
 
 
Range Correction Factor 




+⋅





+
=



+
TSFC
TSFC
1
C
C
1
1
R
R
1
D
D ∆∆
∆
 
 
Manoeuvres 
 
Steady State Pull Up 
22
PP XRRZ −−=
 
( ))cosn(29.11VR 2
)Kts()Ft(
P γ−⋅=
 
V
)cosn(g
Q
)Sec/Rad(
γ
γ
−⋅
== &
 
 
 
Steady State Turn 
φcos
1n =
 






⇒
⋅
=
Sec
Rad
V
tang φ
Ψ&
 
( )φtang5.2132VR 2
)Kts()Ft(
T ⋅⋅=
 
 
φΨ sinQ ⋅= & φΨ cosR ⋅= & 
Stability Axis 






−⋅=
n
1
n
V
g
Q
)Sec/Rad(
 
1n
Vn
g
R 2
)Sec/Rad(
−⋅
⋅
=
 
)utes(mintanV0055.0360TurnToTime
Kts
0 φψ ⋅=
 
TRAJECTORY 
 






+⋅−=
W
SqC
sing
dt
dV Dγ
 
 
 
V
cosg
dt
d γγ ⋅−
=
 
 
γcosVV TTx =
 
γsinVV TTz =
 
 
 
Drag Coefficient Data 
 
One side 
Flat Plate 
 
Laminar Flow 
 
n
f
R
328.1
C =
 
Turbulent 
Flow 
 
( ) 58.2n10
f
Rlog
455.0
C =
 
 
CD Rn 10
4 To 106 
 
Based on Frontal Area 
Wetted 
Area 
Direction 
 
Of Flow 
 
� 
 
 
 
 
 
� 
 
� 
 
⊳600 
 
| 
Rn=9�10
6 
 
t/c=12% 
2-Dim 1.17 1.20 2.30 2.05 1.55 1.55 1.98 0.0058 
3-Dim .47 0.38 1.42 1.05 0.80 0.50 1.17 0.0030 
 
JET ENGINE PERFORMANCE 
Gross Thrust 
 
P&W Engine Station 
 (J-57 and TF-33) 
( )0999aG PPAV
g
W
F −⋅−⋅=
 
( ) ( )09909an PPAVV
g
W
F −⋅−−⋅=
 
 
444444 3444444 21
4444444 34444444 21
444 3444 21
44 344 21
DragRam
am
t
t
t
t
ta
t
ta
tr
ThrustGross
imaryPr
c
h
cgp0
Fan
fff0
am
n
P
P
P
PW
W
F
A
A
ACPACP
F
1
1
2
2
2
2
2
2






⋅








⋅








⋅














−





⋅⋅⋅⋅+⋅⋅⋅=
δ
θ
δ
θ
δ
ΨΨ
δ
 










−





⋅
−
=
−
1
P
P
1
2
1
am
t
unchoked
γ
γ
γ
γ
Ψ
 
1
1
2
P
P
2
1
1
am
t
choked −





+
⋅⋅=
−γ
γ
Ψ
 
fC
 Fan Gross Thrust Coefficient )PP(f amT3
 
gC
 Primary Gross Thrust Coefficient )PP(f amT7
 
cf A,A
 Exhaust Exit Area Fan, Primary 
ch
AA
 Ratio A9 hot to cold )PT(f amT7
 
2
2
t
taW
δ
θ 
is Corrected engine airflow Lbs/sec )
N
,PP(f
2
2
t
amT
θ
 
2
2
2
t
ta
tr
W
F
δ
θ
δ
 
 
 
is Ram Drag Parameter Lbs/(Lbs/Sec) 
2
0
M
2
1
1
M
g
a
−
+
⋅
γ
 
1
2
t
t
P
P
 
is inlet pressure recovery ratio, 
dimensionless 







M,
W
f
2
2
t
ta
δ
θ
 
am
t
P
P
1
 
is ram pressure ratio, dimensionless 
1
2
M
2
1
1
−





 −
+
γ
γ
γ