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.!http://www.princeton.edu/~stengel/MAE331.html!

http://www.princeton.edu/~stengel/FlightDynamics.html!

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= ρsealevele

βz = ρsealevele−βh

ρsealevel =1.225 kg /m3; β =1/ 9,042m

Vair = vx2 + vy

2 + vz2!" #$air1/2= vTv!" #$air

1/2= Airspeed

Dynamic pressure = q = 12ρ h( )Vair2

Mach number = Vaira 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 =CL12ρVair

2 S =CLqS•  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 =xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  Position: 3 dimensions"•  What is a non-moving frame?"

Velocity and Momentum "•  Velocity of a particle"

v = dxdt

= x =xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

vxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

•  Linear momentum of a particle"

p = mv = mvxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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"

ddt

mv( ) = 0 ; mv t1= mv t2

Newton�s Laws of Motion: Dynamics of a Particle "

ddt

mv( ) = m dvdt

= F ; F =fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

•  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"

∴dvdt

=1mF =

1mI3F =

1 / m 0 00 1 / m 00 0 1 / m

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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 "

dvdt

= v =vxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=1mF =

1 / m 0 00 1 / m 00 0 1 / m

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

drdt

= r =xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= v =

vxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

FI =fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥ I

= Fgravity + Faerodynamics + Fthrust⎡⎣ ⎤⎦ 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 ≡ rv

"

#$

%

&'=

PositionVelocity

"

#$$

%

&''=

xyzvxvyvz

"

#

$$$$$$$$

%

&

''''''''

•  With"

xyzvxvyvz

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

vxvyvzfx / mfy / m

fz / m

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

=

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

xyzvxvyvz

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

+

0 0 00 0 00 0 0

1 / m 0 00 1 / m 00 0 1 / m

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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"

Fgravity( )I = Fgravity( )E = mg f = m00go

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  Approximation"–  Flat earth reference is an inertial

frame, e.g.,"•  North, East, Down"•  Range, Crossrange, Altitude (–)"

go 9.807 m / s2 at earth's surface

Aerodynamic Force"

FI =XYZ

!

"

###

$

%

&&&I

=

CX

CY

CZ

!

"

###

$

%

&&&I

12ρVair

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 Coefficients"

Body-Axis Frame" Velocity-Axis Frame"

CX

CY

CZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥B

=axial force coefficientside force coefficient

normal force coefficient

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

CD

CY

CL

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

drag coefficientside force coefficient

lift coefficient

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  Functions of flight condition, control settings, and disturbances, e.g., CL = CL(δ, M, δE)"

•  Non-dimensional coefficients allow application of sub-scale model wind tunnel data to full-scale airplane"

u(t) :axial velocityw(t) :normal velocityV t( ) : velocity magnitudeα t( ) :angle of attackγ t( ) : flight path angleθ(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 angleψ (t) : yaw angleξ t( ) :heading angleφ 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 =CL12ρVair

2 S ≈ CL0+∂CL

∂αα

%

&'(

)*12ρVair

2 S

Drag =CD12ρVair

2 S ≈ CD0+εCL

2$% &'12ρVair

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 =CL12ρVair

2 S ≈ CLwing+CLfuselage

+CLtail( ) 12 ρVair2 S ≈ CL0

+∂CL

∂αα

%

&'(

)*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 V2"–  Induced drag proportional

to 1/V2"–  Total drag minimized at

one particular airspeed"

Drag =CD12ρVair

2 S ≈ CDp+CDi

+CDw( ) 12 ρVair2 S ≈ CD0

+εCL2$% &'qS

2-D Equations of Motion�

2-D Equations of Motion for a Point Mass"

xzvxvz

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

vxvzfx / mfz / m

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

0 0 1 00 0 0 10 0 0 00 0 0 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xzvxvz

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

+

0 00 0

1 / m 00 1 / m

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

fxfz

⎡

⎣⎢⎢

⎤

⎦⎥⎥

•  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"

xz

⎡

⎣⎢

⎤

⎦⎥ =

vxvz

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

V cosγ−V sinγ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⇒Vγ

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

x2 + z2

− sin−1 zV

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=vx2 + vz

2

− sin−1 vzV

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Vγ

⎡

⎣⎢⎢

⎤

⎦⎥⎥=ddt

vx2 + vz

2

− sin−1 vzV

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

ddt

vx2 + vz

2

−ddtsin−1 vz

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)= vx =V (t)cosγ (t)h(t)= − z(t)= −vz =V (t)sinγ (t)

V (t)=Thrust −Drag−mg h( ) sinγ (t)

m=CT −CD( ) 1

2ρ h( )V 2 (t)S −mg h( ) sinγ (t)

m

γ (t)=Lift −mg h( )cosγ (t)

mV (t)=CL12ρ h( )V 2 (t)S −mg h( )cosγ (t)

mV (t)

r = rangeh = height (altitude)V = velocityγ = 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 ≡ CT12ρV 2S

•  Non-dimensional thrust coefficient, CT!–  CT 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"

Isp =Thrustm go

Specific Impulse, Units = m/sm/s2 = sec

m ≡Mass flow rate of on−board propellantgo ≡Gravitational acceleration at earth 's surface

Thrust and Specific Impulse"

Sensitivity of Thrust to Airspeed"Nominal Thrust = TN ≡ CTN

12ρVN

2S

•  If thrust is independent of velocity (= constant)"

∂T∂V

= 0 = ∂CT

∂V12ρVN

2S+CTNρVNS

.( )N = Nominal or reference( ) value

•  Turbojet thrust is independent of airspeed over a wide range"

∂CT

∂V= −CTN

/VN

Power"•  Assuming thrust is aligned with airspeed vector"

Power = P = Thrust ×Velocity ≡ CT12ρV 3S

•  If power is independent of velocity (= constant)"∂P∂V

= 0 = ∂CT

∂V12ρVN

3S+ 32CTN

ρVN2S

•  Velocity-independent power is typical of propeller-driven propulsion (reciprocating or turbine engine, with constant RPM or variable-pitch prop)"

∂CT

∂V= −3CTN

/VN

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)"

http://airplanesandrockets.com/motors/dynajet-engine.htm!

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"