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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