mae 331 lecture 2
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Point-Mass Dynamics andAerodynamic/Thrust ForcesRobert Stengel, Aircraft Flight Dynamics,
MAE 331, 2010• 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 2010 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
The Atmosphere
Air Density, Dynamic
Pressure, and Mach Number! = Air density, functionof height
= !sea levele"z
!sea level = 1.225 kg / m3; " = 1 / 9,042m
V = vx2+ vy
2+ vz
2#$ %&1/2
= vTv#$ %&
1/2
= Airspeed
Dynamic pressure = q =1
2!V 2
Mach number =V
a; a = speed of sound, m / s
Properties of the Lower Atmosphere
• Air density and pressure decay exponentially with altitude
• Air temperature and speed of sound are linear functions of altitude
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 + Airplane velocity
• Airspeed must increase as altitude increasesto maintain constant dynamic pressure
Contours of Constant
Dynamic Pressure,
Weight = Lift = CL
1
2!V 2
S = CLqS• In steady, cruising flight,
q
Equations of Motionfor a Point Mass
Newtonian Frame of
Reference
• Newtonian (Inertial) Frame ofReference
– Unaccelerated Cartesian framewhose origin is referenced to aninertial (non-moving) frame
– Right-hand rule
– Origin can translate at constantlinear velocity
– Frame cannot be rotating withrespect 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
!
"
###
$
%
&&&
=
vx
vy
vz
!
"
####
$
%
&&&&
• Linear momentum of a particle
p = mv = m
vx
vy
vz
!
"
####
$
%
&&&&
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
dtmv( ) = 0 ; mv
t1= mv
t2
Newton!s Laws of Motion:Dynamics of a Particle
d
dtmv( ) = m
dv
dt= F ; F =
fx
fy
fz
!
"
####
$
%
&&&&
• Second Law– A particle of fixed mass acted upon by a force
changes velocity with an accelerationproportional to and in the direction of the force,as observed in an inertial reference frame;
– The ratio of force to acceleration is the mass ofthe particle: F = m a
!dv
dt=1
mF =
1
mI3F =
1 / m 0 0
0 1 / m 0
0 0 1 / m
"
#
$$$
%
&
'''
fx
fy
fz
"
#
$$$$
%
&
''''
Newton!s Laws of Motion:Dynamics of a Particle
• Third Law– For every action, there is an equal and opposite reaction
Equations of Motion for a PointMass: Position and Velocity
dv
dt= !v =
!vx
!vy
!vz
!
"
####
$
%
&&&&
=1
mF =
1 / m 0 0
0 1 / m 0
0 0 1 / m
!
"
###
$
%
&&&
fx
fy
fz
!
"
####
$
%
&&&&
dr
dt= !r =
!x
!y
!z
!
"
###
$
%
&&&
= v =
vx
vy
vz
!
"
####
$
%
&&&&
FI =
fx
fy
fz
!
"
####
$
%
&&&&I
= Fgravity + Faerodynamics + Fthrust!" $% I
Rate of changeof position
Rate of change
of velocity
Vector of
combined forces
Equations of Motion for aPoint Mass
˙ x (t) =dx(t)
dt= f[x(t),F]
• Written as a single equation
x !r
v
"
#$
%
&' =
Position
Velocity
"
#$$
%
&''=
x
y
z
vx
vy
vz
"
#
$$$$$$$$
%
&
''''''''
• With
Scalar Dynamic Equations fora Point Mass
!x
!y
!z
!vx
!vy
!vz
!
"
########
$
%
&&&&&&&&
=
vx
vy
vz
fx / m
fy / m
fz / 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
vx
vy
vz
!
"
########
$
%
&&&&&&&&
+
0 0 0
0 0 0
0 0 0
1 / m 0 0
0 1 / m 0
0 0 1 / m
!
"
#######
$
%
&&&&&&&
fx
fy
fz
!
"
####
$
%
&&&&
Dynamic equations are linear
Gravitational Force:Flat-Earth Approximation
• g is gravitational acceleration
• mg is gravitational force
• Independent of position
• z measured down
Fgravity( )I= Fgravity( )
E= mg f = m
0
0
go
!
"
###
$
%
&&&
• Approximation
– Flat earth reference is an inertialframe, e.g.,
• North, East, Down
• Range, Crossrange, Altitude (–)
go! 9.807 m / s
2 at earth's surface
Aerodynamic
Force
FI =
X
Y
Z
!
"
###
$
%
&&&I
=
CX
CY
CZ
!
"
###
$
%
&&&I
1
2'V 2
S
=
CX
CY
CZ
!
"
###
$
%
&&&I
q S
• Referenced to the
Earth not the aircraft
Inertial Frame Body-Axis Frame Velocity-Axis Frame
FB =
CX
CY
CZ
!
"
###
$
%
&&&B
q S FV =
CD
CY
CL
!
"
###
$
%
&&&q S
• Aligned with the
aircraft axes
• Aligned with and
perpendicular to
the direction of
motion
Non-Dimensional
Aerodynamic CoefficientsBody-Axis Frame Velocity-Axis Frame
CX
CY
CZ
!
"
###
$
%
&&&B
=
axial force coefficient
side force coefficient
normal force coefficient
!
"
###
$
%
&&&
CD
CY
CL
!
"
###
$
%
&&&=
drag coefficient
side force coefficient
lift coefficient
!
"
###
$
%
&&&
• Functions of flight condition, control settings, and disturbances, e.g.,CL = CL(!, M, "E)
• Non-dimensional coefficients allow application of sub-scale modelwind tunnel data to full-scale airplane
u(t) :axial velocity
w(t) :normal velocity
V t( ) : velocity magnitude
! t( ) :angle of attack
" t( ) : flight path angle
#(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 localhorizontal
• angle between centerline and local horizontal
Longitudinal Variables
! = " #$ (with wingtips level)
Lateral-Directional Variables
! =" + # (with wingtips level)
!(t) : sideslip angle
" (t) : yaw angle
# t( ) :heading angle
$ t( ) : roll angle
• angle between centerline and direction of flight
• angle between centerline and local horizontal
• angle between direction of flight and compassreference (e.g., north)
• angle between true vertical and body z axis
Introduction toLift 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 = CL
1
2!V 2
S " CL0
+#CL
#$$%
&'()*1
2!V 2
S
Drag = CD
1
2!V 2
S " CD0
+ #CL
2$% &'1
2!V 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 andtrailing edges)
Chord Section
Streamlines
Lift = CL
1
2!V 2
S " CLw+ CLf
+ CLht( )1
2!V 2
S " CL0
+#CL
#$$%
&'()*1
2!V 2
S
2D vs. 3D Lift
• Inward flow over upper surface
• Outward flow over lower surface
• Bound vorticity of wing produces tipvortices
Inward-Outward Flow
Tip Vortices
Identical Chord Sections
Infinite vs. Finite Span
Aerodynamic Drag
• Drag components
– Parasite drag (friction, interference, base pressuredifferential)
– Induced drag (drag due to lift generation)
– Wave drag (shock-induced pressure differential)
• In steady, subsonic flight
– Parasite (form) drag increasesas V2
– Induced drag proportional to1/V2
– Total drag minimized at oneparticular airspeed
Drag = CD
1
2!V 2
S " CDp+ CDi
+ CDw( )1
2!V 2
S " CD0
+ #CL
2$% &'1
2!V 2
S
2-D Equations of Motion
2-D Equations ofMotion for a Point Mass
!x
!z
!vx
!vz
!
"
#####
$
%
&&&&&
=
vx
vz
fx / m
fz / m
!
"
#####
$
%
&&&&&
=
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
!
"
####
$
%
&&&&
x
z
vx
vz
!
"
#####
$
%
&&&&&
+
0 0
0 0
1 / m 0
0 1 / m
!
"
####
$
%
&&&&
fx
fz
!
"
##
$
%
&&
• Restrict motions to a vertical plane (i.e., motions in
y direction = 0)
Transform Velocity
from Cartesian to
Polar Coordinates
!x
!z
!
"#
$
%& =
vx
vz
!
"##
$
%&&=
V cos'(V sin'
!
"##
$
%&&
)V
'!
"##
$
%&&=
!x2+ !z
2
( sin(1 !z
V
*+,
-./
!
"
####
$
%
&&&&
=
vx
2+ v
z
2
( sin(1 vz
V
*+,
-./
!
"
####
$
%
&&&&
!V
!!
"
#$$
%
&''=d
dt
vx
2+ v
z
2
( sin(1 vz
V
)*+
,-.
"
#
$$$$
%
&
''''
=
d
dtvx
2+ v
z
2
(d
dtsin
(1 vz
V
)*+
,-.
"
#
$$$$$
%
&
'''''
Longitudinal Point-MassEquations of Motion
!V (t) =Thrust ! Drag ! mg sin" (t)
m=
CT ! CD( )1
2#V (t)2S ! mg sin" (t)
m
!" (t) =Lift ! mgcos" (t)
mV (t)=
CL
1
2#V (t)2S ! mgcos" (t)
mV (t)
!h(t) = ! !z(t) = !vz = V (t)sin" (t)
!r(t) = !x(t) = vx = V (t)cos" (t)
V = velocity
! = flight path angle
h = height (altitude)
r = range
• Assuming thrust is aligned with the velocity vector
• When airplane is in steady, level flight,CT = CD
Flight of aPaper Airplane
Flight of a Paper AirplaneExample 1.3-1, Flight Dynamics
• Red: Equilibriumflight path
• Black: Initial flightpath angle = 0
• Blue: plusincreased initialairspeed
• Green: loop
• Equations ofmotionintegratednumerically toestimate theflight path
Flight of a Paper AirplaneExample 1.3-1, Flight Dynamics
• Red: Equilibriumflight path
• Black: Initial flightpath angle = 0
• Blue: plusincreased initialairspeed
• Green: loop
Introduction toPropulsion
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
Reciprocating Engines• Rotary
• In-Line
• V-12 • Opposed
• Radial
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
Turbo-compound
Reciprocating Engine• Exhaust gas drives the turbo-compressor
• Napier Nomad II shown (1949)
• Thrustproduceddirectly byexhaust gas
Axial-flow Turbojet (von Ohain, Germany)
Centrifugal-flow Turbojet (Whittle, UK)
TurbojetEngines(1930s)
Turboprop
Engines
(1940s)
• Exhaust gas drives apropeller to producethrust
• Typically uses acentrifugal-flowcompressor
Turbojet + Afterburner (1950s)
• Fuel added to exhaust
• Additional air may be introduced
• Dual rotation rates, N1 and N2, typical
Ramjet and Scramjet
Ramjet (1940s) Scramjet (in progress)
Turbofan Engine (1960s)
• Dual or triple rotation rates
Fighter Aircraft and Engines fromQuick Quiz #1
Lockheed P-38
Allison V-1710Turbocharged Reciprocating Engine
Convair/GD F-102
P&W J57Axial-Flow Turbojet Engine
MD F/A-18
GE F404Afterburning Turbofan Engine
SR-71 : P&W J58
Variable-CycleEngine (Late 1950s)
Hybrid
Turbojet/Ramjet
Thrust and Thrust Coefficient
Thrust ! CT
1
2"V 2
S
• Non-dimensional thrustcoefficient, CT
– CT is a function of power/throttlesetting, fuel flow rate, blade angle,Mach number, ...
• Reference area, S, may be aircraftwing area, propeller disk area, orjet exhaust area
Isp =Thrust
!m go, Units =
m/s
m/s2
= sec
!m ! Mass flow rate of on " board propellant
go ! Gravitational acceleration at earth 's surface
Thrust and Specific Impulse
Sensitivity of Thrust to Airspeed
Nominal Thrust = TN! C
TN
1
2"V
N
2S
• If thrust is independent of velocity (= constant)
!T
!V= 0 =
!CT
!V
1
2"V
N
2S + C
TN"V
NS
!CT
!V= #C
TN/V
N
.( )N= Nominal or reference( ) value
• Turbojet thrust is independent of airspeed over awide range
Power• Assuming thrust is aligned with airspeed vector
Power = P = Thrust !Velocity " CT
1
2#V 3
S
• If power is independent of velocity (= constant)
!P
!V= 0 =
!CT
!V
1
2"V
N
3S +
3
2CTN"V
N
2S
!CT
!V= #3C
TN/V
N
• Velocity-independent power is typical of propeller-driven propulsion (reciprocating or turbine engine,with constant RPM or variable-pitch prop)
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