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Level Flight Performance

Equations of Motion

Thrust and Power Required

Minimum Drag

MAE 155A

Oblique Wing Research Aircraft (NASA Image)

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2MAE 155A

Geodetic and Local Coordinates

North

East

Up

Z

X (Prime Meridian)

Y

(X, Y, Z) is an Earth-Centered,Earth-Fixed Coodinate System

Latitude

Longitude

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Velocity Vector Orientation

The flight path angleis positive when the

airplane is climbing.

The flight path headingis positive clockwisefrom true north.

VelocityVector

Down

North

North

East

Flight PathHeading

Flight PathAngle

VelocityVector

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4MAE 155A

Forces Acting on the Airplane

Lift Up

Gravity

Velocity

Vector

ThrustDrag

Lift Up

Gravity

VelocityVector

Thrust

Drag

The bank angle is positive when the

lift vector is tilted to cause a rightturn (heading rate is positive).

The thrust vector isaligned in the lift-dragplane, but is rotatedabove the velocity vector.

ThrustAngle

BankAngle

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Non-Rotating Spherical Earth

dV

dt=g

[

TcosD

W

sin

]ddt=

g

V {[n TW sin ]cos[ V2

g R1]cos}

d

dt=

g

V

{[n

T

W

sin

]sin

cos

V2

g Rcossin tan

}dWdt=CT

d

dt=

1

R VcoscosVWN

d

dt=

1

R cos VcossinVWE

dR

dt=Vsin V

WU

g = gravity

T = thrustD = dragW = weightC = thrust specific fuel consumptionV = airspeedL = liftP = powern = L/W = load factorR = radial distance from Earth center

= flight path

=thrust angle

=bank angle

= flight path heading

= latitude

= longitude

(VWN

, VWE

, VWU

) = wind speed in (North, East, Up) direction

dW

dt=C

P

or

=propulsive efficiency

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Low Speed Aircraft

dV

dt=g

[TDW sin

]ddt=

g

V n coscos

d

dt

=g

V

nsin

cos

dW

dt=CT

d

dt=

1

R VcoscosVWN

d

dt=

1

R cos VcossinVWE

dh

dt=VsinV

WU

Assumptions:

h = altitude = R RE

RE

= Earth radius

V2

g Ris small

is small

dW

dt=C

P

or

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Steady Level Flight

Steady (unaccelerated) level flight performance equations make the followingassumptions and simplifications:

0=dV

dt=g[ TDW sin ]

0=d

dt = gV n coscos

0=d

dt=

g

V

n sin

cos

0=dh

dt=Vsin

constant airspeed

constant flight path

constant heading

constant altitudeand zero winds

=0

=0

n=cos

cos= LW=1

T=DWsin=D

L=W

T=D

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Drag Polar

The drag and lift forces are typically normalized by dynamic pressure and (wing)reference area.

The resulting coefficients are non-dimensional.

CD

=CD ,

0K C

L

2 K1

e AR

CD=

D

q SC

L=

L

q S

The drag polar provides the basic aerodynamic information needed to predict levelflight performance.

q=1

2V2

AR=b2

S

q = dynamic pressureV = airspeedAR = aspect ratiob = wingspanS = reference area

=atmospheric density

e = wing planform efficiency factorCD,0

= zero-lift drag coefficient

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Thrust Required

Turbojet engines are usually rated in terms of thrust.

Thrust required refers to the engine thrust needed to maintain steady, level flight.

TR=D=q S C

DC

D=C

D ,0K C

L2

TR=D=q S CD ,0q S K CL2 W=L=q S C

L

TR=q S C

D ,0

K W2

q Sq=

1

2V2

TR=

1

2V2 S C

D ,0

2K W2

V2 S TR

= thrust required

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Thrust Available

Thrust available from a turbojet engine is approximately constant relative to airspeed.

Available thrust generally varies with altitude:

TR

V

Thrust requiredat sea level

Thrust availableat sea level Thrust required

at altitude

Thrust available

at altitude

Max airspeedat sea level

Max airspeedat altitude

TAT

SL

SL T

A= thrust available at altitude

TSL

= thrust available at sea level

=density at altitude

SL=density at sea level

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Minimum Thrust Required

What conditions lead to minimum drag (and therefore minimum thrust required)?

W=L=q S CL

T=D=q S CD

TR=

W

CL/C

D

T

W=

D

L=

1

L /D=

1

CL/C

D

Minimum drag occurs when CL/C

Dis maximized.

CL

CD

=C

L

CD ,0

K CL2

C

L CLC

D=CD ,0K CL

2

CD ,0

K CL2=0 C

L=CD ,0K

CL

CD

=1

2CD ,0KV=

2W

S KCD ,0Minimum Drag

Minimum DragMinimum Drag

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Power Required and Available

PR=V TR=V D=1

2 V3

S CD ,02K W

2

V S

PR

VAirspeed forminimum power

Minimumpower required

CL=3CD ,0K

Minimum

power requiredoccurs at:

Power available frompiston engines isapproximately constantwith airspeed butvaries with alttiude

PAP

SL SL

Power available (piston)

Power available (jet)