Aircraft Flight Dynamics Robert F. Stengel Lecture2 PointMass Dynamics and Aerodynamic Thrust Forces

Point-Mass Dynamics and
Aerodynamic/Thrust Forces

Robert Stengel, Aircraft Flight Dynamics, 

MAE 331, 2012!
•  Properties of the Atmosphere"
•  Frames of reference"
•  Velocity and momentum"
•  Newtons laws"
•  Introduction to Lift, Drag, and Thrust"
•  Simplified longitudinal equations of
motion"
Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!
/> />The Atmosphere
•  Air density and pressure decay
exponentially with altitude"
•  Air temperature and speed of sound
are linear functions of altitude "
Properties of the
Lower Atmosphere!
Wind: Motion of the Atmosphere"
•  Zero wind at Earths surface = Inertially rotating air mass"
•  Wind measured with respect to Earths rotating surface "
Wind Velocity Profiles vary over Time!
Typical Jetstream Velocity!
•  Airspeed = Airplanes speed with respect to air mass"
•  Inertial velocity = Wind velocity ± Airspeed "
Air Density, Dynamic Pressure,
and Mach Number"
ρ
= Air density, functionof height
=

ρ
sealevel
e
β
z
=
ρ
sealevel
e
−
β
h
ρ
sealevel
= 1.225 kg / m
3
;
β
= 1/ 9,042 m
V
air
= v
x
2
+ v
y
2
+ v
z
2

!
"
#
$
air
1/2
= v
T
v
!
"
#
$
air
1/2
= Airspeed
Dynamic pressure = q =
1
2
ρ
h
( )
V
air
2
Mach number =
V
air
a h
( )

; a = speed of sound, m / s
•  Airspeed must increase as altitude increases
to maintain constant dynamic pressure"
Contours of Constant
Dynamic Pressure, "
Weight = Lift = C
L
1
2
ρ
V
air
2
S = C
L
qS
•  In steady, cruising flight, "
q
Equations of Motion
for a Point Mass
Newtonian Frame of Reference"
•  Newtonian (Inertial) Frame of
Reference"
–  Unaccelerated Cartesian frame
whose origin is referenced to an
inertial (non-moving) frame"
–  Right-hand rule"
–  Origin can translate at constant
linear velocity"
–  Frame cannot be rotating with

respect to inertial origin"
•  Translation changes the position of an object"
r =
x
y
z
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
•  Position: 3 dimensions"
•  What is a non-moving frame?"
Velocity and Momentum "
•  Velocity of a particle"

v =
dx
dt
=

x =

x


y

z
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
v
x
v
y
v
z
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥

⎥
⎥
•  Linear momentum of a particle"
p = mv = m
v
x
v
y
v
z
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
where m = mass of particle
Newtons Laws of Motion: 

Dynamics of a Particle "
•  First Law"
–  If no force acts on a particle, it remains at rest or
continues to move in a straight line at constant
velocity, as observed in an inertial reference
frame Momentum is conserved"

d
dt
mv
( )
= 0 ; mv
t
1
= mv
t
2
Newtons Laws of Motion: 

Dynamics of a Particle "
d
dt
mv
( )
= m
dv
dt
= F ; F =
f
x
f
y
f
z
⎡
⎣
⎢
⎢

⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
•  Second Law"
–  A particle of fixed mass acted upon by a force
changes velocity with an acceleration
proportional to and in the direction of the force,
as observed in an inertial reference frame; "
–  The ratio of force to acceleration is the mass of
the particle: F = m a"
∴
dv
dt
=
1
m
F =
1
m
I
3
F =
1 / m 0 0
0 1 / m 0
0 0 1 / m

⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
f
x
f
y
f
z
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
Newtons Laws of Motion: 

Dynamics of a Particle "

•  Third Law"
–  For every action, there is an equal and opposite reaction"
Equations of Motion for a Point Mass:
Position and Velocity "

dv
dt
=

v =

v
x

v
y

v
z
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥

⎥
=
1
m
F =
1 / m 0 0
0 1 / m 0
0 0 1 / m
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
f
x
f
y
f
z
⎡
⎣
⎢
⎢
⎢
⎢

⎤
⎦
⎥
⎥
⎥
⎥

dr
dt
=

r =

x

y

z
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= v =
v

x
v
y
v
z
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
F
I
=
f
x
f
y
f
z
⎡
⎣
⎢
⎢

⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
I
= F
gravity
+ F
aerodynamics
+ F
thrust
⎡
⎣
⎤
⎦
I
Rate of change
of position!
Rate of change
of velocity!
Vector of
combined forces!
Equations of Motion for a
Point Mass "

˙

x (t) =
dx(t)
dt
= f[x(t),F]
•  Written as a single equation"
x ≡
r
v
"
#
$
%
&
'
=
Position
Velocity
"
#
$
$
%
&
'
'
=
x
y
z
v

x
v
y
v
z
"
#
$
$
$
$
$
$
$
$
%
&
'
'
'
'
'
'
'
'
•  With"


x


y

z

v
x

v
y

v
z
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥

⎥
⎥
=
v
x
v
y
v
z
f
x
/ m
f
y
/ m
f
z
/ m
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤

⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥

⎥
⎥
⎥
⎥
⎥
x
y
z
v
x
v
y
v
z
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥

⎥
⎥
⎥
⎥
+
0 0 0
0 0 0
0 0 0
1 / m 0 0
0 1 / m 0
0 0 1 / m
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
f

x
f
y
f
z
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
Dynamic equations are linear!

˙
x (t) =
dx(t)
dt
= f[x(t),F]
Equations of Motion for a
Point Mass !
Gravitational Force:

Flat-Earth Approximation"
•  g is gravitational acceleration"
•  mg is gravitational force!

•  Independent of position!
•  z measured down"
F
gravity
( )
I
= F
gravity
( )
E
= mg
f
= m
0
0
g
o
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
•  Approximation"
–  Flat earth reference is an inertial
frame, e.g.,"

•  North, East, Down"
•  Range, Crossrange, Altitude (–)"

g
o
 9.807 m / s
2
at earth's surface
Aerodynamic Force"
F
I
=
X
Y
Z
!
"
#
#
#
$
%
&
&
&
I
=
C
X
C

Y
C
Z
!
"
#
#
#
$
%
&
&
&
I
1
2
ρ
V
air
2
S
=
C
X
C
Y
C
Z
!
"

#
#
#
$
%
&
&
&
I
q S
•  Referenced to the
Earth not the aircraft"
Inertial Frame"
Body-Axis Frame" Velocity-Axis Frame"
F
B
=
C
X
C
Y
C
Z
⎡
⎣
⎢
⎢
⎢
⎤
⎦

⎥
⎥
⎥
B
q S
F
V
=
C
D
C
Y
C
L
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
q S
•  Aligned with the
aircraft axes"
•  Aligned with and
perpendicular to
the direction of

motion"
Non-Dimensional
Aerodynamic Coefficients"
Body-Axis Frame" Velocity-Axis Frame"
C
X
C
Y
C
Z
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
B
=
axial force coefficient
side force coefficient
normal force coefficient
⎡
⎣
⎢
⎢
⎢

⎤
⎦
⎥
⎥
⎥
C
D
C
Y
C
L
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
drag coefficient
side force coefficient
lift coefficient
⎡
⎣
⎢
⎢
⎢

⎤
⎦
⎥
⎥
⎥
•  Functions of flight condition, control settings, and disturbances, e.g.,
C
L
= C
L
(δ, M, δE)"
•  Non-dimensional coefficients allow application of sub-scale model
wind tunnel data to full-scale airplane"
u(t) :axial velocity
w(t) :normal velocity
V t
( )
: velocity magnitude
α
t
( )
: angle of attack
γ
t
( )
: flight path angle
θ
(t) : pitch angle
•  along vehicle centerline!
•  perpendicular to centerline!

•  along net direction of flight!
•  angle between centerline and direction of flight!
•  angle between direction of flight and local horizontal!
•  angle between centerline and local horizontal!
Longitudinal Variables"
γ
=
θ
−
α
(with wingtips level)
Lateral-Directional Variables"
β
(t) : sideslip angle
ψ
(t) : yaw angle
ξ
t
( )
: heading angle
φ
t
( )
: roll angle
•  angle between centerline and direction of flight!
•  angle between centerline and local horizontal!
•  angle between direction of flight and compass reference
(e.g., north)!
•  angle between true vertical and body z axis!
ξ

=
ψ
+
β
(with wingtips level)
Introduction to
Lift and Drag
Lift and Drag are Oriented
to the Velocity Vector"
•  Drag components sum to produce total drag"
–  Skin friction"
–  Base pressure differential"
–  Shock-induced pressure differential (M > 1)"
•  Lift components sum to produce total lift"
–  Pressure differential between upper and lower surfaces"
–  Wing"
–  Fuselage"
–  Horizontal tail"
Lift = C
L
1
2
ρ
V
air
2
S ≈ C
L
0
+

∂
C
L
∂α
α
%
&
'
(
)
*
1
2
ρ
V
air
2
S
Drag = C
D
1
2
ρ
V
air
2
S ≈ C
D
0
+

ε
C
L
2
$
%
&
'
1
2
ρ
V
air
2
S
Aerodynamic Lift"
•  Fast flow over top + slow flow over bottom =
Mean flow + Circulation"
•  Speed difference proportional to angle of attack"
•  Kutta condition (stagnation points at leading and
trailing edges)"
Chord Section!
Streamlines!
Lift = C
L
1
2
ρ
V
air

2
S ≈ C
L
wing
+ C
L
fuselage
+ C
L
tail
( )
1
2
ρ
V
air
2
S ≈ C
L
0
+
∂
C
L
∂α
α
%
&
'
(

)
*
qS
2D vs. 3D Lift"
•  Inward flow over upper surface"
•  Outward flow over lower surface"
•  Bound vorticity of wing produces tip vortices"
Inward-Outward Flow!
Tip Vortices!
2D vs. 3D Lift"
Identical Chord Sections!
Infinite vs. Finite Span!
•  Finite aspect ratio reduces lift slope"
What is aspect ratio?!
Aerodynamic Drag"
•  Drag components"
–  Parasite drag (friction, interference, base pressure
differential)"
–  Induced drag (drag due to lift generation)"
–  Wave drag (shock-induced pressure differential)"
•  In steady, subsonic flight"
–  Parasite (form) drag
increases as V
2
"
–  Induced drag proportional
to 1/V
2
"
–  Total drag minimized at

one particular airspeed"
Drag = C
D
1
2
ρ
V
air
2
S ≈ C
D
p
+ C
D
i
+ C
D
w
( )
1
2
ρ
V
air
2
S ≈ C
D
0
+
ε

C
L
2
$
%
&
'
qS
2-D Equations of Motion
2-D Equations of 

Motion for a Point Mass"


x

z

v
x

v
z
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤

⎦
⎥
⎥
⎥
⎥
⎥
=
v
x
v
z
f
x
/ m
f
z
/ m
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥

⎥
=
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
x
z
v
x
v
z
⎡
⎣
⎢
⎢
⎢
⎢

⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
+
0 0
0 0
1 / m 0
0 1 / m
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
f
x
f
z
⎡

⎣
⎢
⎢
⎤
⎦
⎥
⎥
•  Restrict motions to a vertical plane
(i.e., motions in y direction = 0)"
•  Assume point mass location
coincides with center of mass"
Transform Velocity
from Cartesian to
Polar Coordinates"


x

z
⎡
⎣
⎢
⎤
⎦
⎥
=
v
x
v
z

⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
V cos
γ
−V sin
γ
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⇒
V
γ
⎡
⎣
⎢
⎢
⎤
⎦

⎥
⎥
=

x
2
+

z
2
−sin
−1

z
V
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥

⎥
⎥
⎥
=
v
x
2
+ v
z
2
−sin
−1
v
z
V
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥

⎥
⎥
⎥


V

γ
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
d
dt
v
x
2
+ v
z
2
−sin
−1
v
z
V

⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
d
dt
v
x
2
+ v
z
2
−
d
dt

sin
−1
v
z
V
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
•  Inertial axes -> wind axes and back"
•  Rates of change of velocity and flight path angle"
Longitudinal Point-Mass
Equations of Motion"


r(t) =

x(t) = v
x
= V (t)cos
γ
(t)

h(t) = −

z(t) = −v
z
= V (t)sin
γ
(t)

V (t) =
Thrust − Drag − mg h
( )
sin
γ
(t)
m
=
C
T
− C
D
( )

1
2
ρ
h
( )
V
2
(t)S − mg h
( )
sin
γ
(t)
m

γ
(t) =
Lift − mg h
( )
cos
γ
(t)
mV(t)
=
C
L
1
2
ρ
h
( )

V
2
(t)S − mg h
( )
cos
γ
(t)
mV(t)
r = range
h = height (altitude)
V = velocity
γ
= flight path angle
•  Equations of motion, assuming mass is fixed, thrust is
aligned with the velocity vector, and windspeed = 0"
•  In steady, level flight"
•  Thrust = Drag"
•  Lift = Weight"
Introduction to
Propulsion
Reciprocating (Internal Combustion)
Engine (1860s)"
•  Linear motion of pistons converted
to rotary motion to drive propeller"
Single Cylinder"
Turbo-Charger (1920s)"
•  Increases pressure of incoming air"
•  Thrust
produced
directly by

exhaust gas"
Axial-flow Turbojet (von Ohain, Germany)!
Centrifugal-flow Turbojet (Whittle, UK)!
Turbojet
Engines
(1930s)"
Turboprop
Engines
(1940s)"
•  Exhaust gas drives a
propeller to produce
thrust"
•  Typically uses a
centrifugal-flow
compressor"
Turbojet + Afterburner (1950s)"
•  Fuel added to exhaust"
•  Additional air may be introduced"
•  Dual rotation rates, N1 and N2, typical"
Turbofan Engine (1960s)"
•  Dual or triple rotation rates"
High Bypass Ratio Turbofan"
Propfan Engine!
Aft-fan Engine!
Ramjet and Scramjet"
Ramjet (1940s)"
Scramjet (1950s)"
Talos!
X-43!
Hyper-X!

Thrust and Thrust Coefficient"
Thrust ≡ C
T
1
2
ρ
V
2
S
•  Non-dimensional thrust coefficient,
C
T!
–  C
T
is a function of power/throttle
setting, fuel flow rate, blade angle,
Mach number, "
•  Reference area, S, may be aircraft
wing area, propeller disk area, or
jet exhaust area"

I
sp
=
Thrust

m g
o
 Specific Impulse, Units =
m/s

m/s
2
= sec

m ≡ Mass flow rate of on − board propellant
g
o
≡ Gravitational acceleration at earth's surface
Thrust and Specific Impulse"
Sensitivity of Thrust to Airspeed"
Nominal Thrust = T
N
≡ C
T
N
1
2
ρ
V
N
2
S
•  If thrust is independent of velocity (= constant)"
∂
T
∂
V
= 0 =
∂
C

T
∂
V
1
2
ρ
V
N
2
S + C
T
N
ρ
V
N
S
.
( )
N
= Nominal or reference
( )
value
•  Turbojet thrust is independent of airspeed over a
wide range"
∂
C
T
∂
V
= −C

T
N
/ V
N
Power"
•  Assuming thrust is aligned with airspeed vector"
Power = P = Thrust × Velocity ≡ C
T
1
2
ρ
V
3
S
•  If power is independent of velocity (= constant)"
∂
P
∂
V
= 0 =
∂
C
T
∂
V
1
2
ρ
V
N

3
S +
3
2
C
T
N
ρ
V
N
2
S
•  Velocity-independent power is typical of propeller-
driven propulsion (reciprocating or turbine engine,
with constant RPM or variable-pitch prop)"
∂
C
T
∂
V
= −3C
T
N
/ V
N
Next Time:
Aviation History

Reading
Airplane Stability and Control, Ch. 1!

Virtual Textbook, Part 3

Supplementary
Material
Early Reciprocating Engines"
•  Rotary Engine:"
–  Air-cooled"
–  Crankshaft fixed"
–  Cylinders turn with propeller"
–  On/off control: No throttle"
Sopwith Triplane!
SPAD S.VII!
•  V-8 Engine:"
–  Water-cooled"
–  Crankshaft turns with propeller"
Reciprocating Engines"
•  Rotary"
•  In-Line"
•  V-12"
•  Opposed"
•  Radial"
Turbo-compound
Reciprocating Engine"
•  Exhaust gas drives the turbo-compressor"
•  Napier Nomad II shown (1949)"
Jet Engine Nacelles"
Pulsejet"
Flapper-valved motor (1940s)"
Dynajet Red Head (1950s)"
V-1 Motor!

Pulse Detonation Engine"
on Long EZ (1981)"
/>Fighter Aircraft and Engines"
Lockheed P-38!
Allison V-1710!
Turbocharged Reciprocating Engine!
Convair/GD F-102!
P&W J57!
Axial-Flow Turbojet Engine!
MD F/A-18!
GE F404!
Afterburning Turbofan Engine!
SR-71: P&W J58
Variable-Cycle
Engine (Late 1950s)"
Hybrid Turbojet/
Ramjet"