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AERODYNAMICS AND

FLIGHT MECHANICS

Thrust and Power Requirements

2

EXAMPLE: BEECHCRAFT QUEEN AIR The results we have developed so far for lift and drag for a finite wing may also be applied to a

complete airplane. In such relations:

CD is drag coefficient for complete airplane

CD,0 is parasitic drag coefficient, which contains not only profile drag of wing (cd) but also

friction and pressure drag of tail surfaces, fuselage, engine nacelles, landing gear and any

other components of airplane exposed to air flow

CL is total lift coefficient, including small contributions from horizontal tail and fuselage

Span efficiency for finite wing replaced with Oswald efficiency factor for entire airplane

Example: To see how this works, assume the aerodynamicists have provided all the information

needed about the complete airplane shown below

Beechcraft Queen Air Aircraft Data

W = 38,220 N

S = 27.3 m2

AR = 7.5

e (complete airplane) = 0.9

CD,0 (complete airplane) = 0.03

What thrust and power levels are required of

engines to cruise at 220 MPH at sea-level?How would these results change at 15,000 ft

3

OVERALL AIRPLANE DRAG

No longer concerned with aerodynamic details

Drag for complete airplane, not just wing

eAR

CCC LDD

2

0, eAR

CCC LdD

2

Wing or airfoil Entire Airplane

Landing Gear

Engine NacellesTail Surfaces

4

DRAG POLAR

CD,0 is parasite drag coefficient at zero lift (L=0) CD,i drag coefficient due to lift (induced drag)

Oswald efficiency factor, e, includes all effects from airplane

CD,0 and e are known aerodynamics quantities of airplane

iDDL

DD CCeAR

CCC ,0,

2

0,

Example ofDrag Polarfor complete airplane

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4 FORCES ACTING ON AIRPLANE Model airplane as rigid body with four natural forces acting on it

1. Lift, L

Acts perpendicular to flight path (always perpendicular to relative wind)

2. Drag, D

Acts parallel to flight path direction (parallel to incoming relative wind)

3. Propulsive Thrust, T

For most airplanes propulsive thrust acts in flight path direction

May be inclined with respect to flight path angle, T, usually small angle4. Weight, W

Always acts vertically toward center of earth

Inclined at angle, , with respect to lift direction

Apply Newtons Second Law (F=ma) to curvilinear flight path

Force balance in direction parallel to flight path

Force balance in direction perpendicular to flight path

6

GENERAL EQUATIONS OF MOTION (6.2)

cTlarperpendicu

Tparallel

r

V

mWTLF

dt

dVmWDTF

2

cossin

sincos

Apply Newtons 2nd Parallel to flight path:

Apply Newtons 2nd Perpendicular to flight path:

Free Body Diagram

7

LEVEL, UNACCELERATED FLIGHT

JSF is flying at constant speed (no accelerations)

Sum of forces = 0 in two perpendicular directions Entire weight of airplane is perfectly balanced by lift (L = W)

Engines produce just enough thrust to balance total drag at this airspeed (T = D)

For most conventional airplanesT is small enough such that cos(T) ~ 1

T D

L

W

8

LEVEL, UNACCELERATED FLIGHT

L

D

L

D

C

C

W

T

SCVWL

SCVDT

WL

DT

2

2

2

1

2

1

DL

W

CC

WT

D

L

R

TR is thrust required to fly at a given velocity in

level, unaccelerated flight

Notice that minimum TRis when airplane is at

maximum L/D

L/D is an important aero-perfo rmance quantity

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THRUST REQUIREMENT (6.3) TRfor airplane at given altitude varies with velocity

Thrust required curve: TRvs. V

10

PROCEDURE: THRUST REQUIREMENT1. Select a flight speed, V

2. Calculate CL

SV

WCL

2

2

1

eAR

CCC LDD

2

0,

D

L

R

C

C

WT

3. Calculate CD

4. Calculate CL/CD

5. Calculate TR

This is how much thrust engine

mustproduce to fly at selected V

Recall Homework Problem #5.6, find (L/D)max for NACA 2412 airfoil

Minimum TRwhen airplane

flying at (L/D)max

11

THRUST REQUIREMENT (6.3) Different points on TRcurve correspond to different angles of attack

eAR

CCSqSCqD

SCqSCVWL

LDD

LL

2

0,

2

2

1

At a:

Large q

Small CL and D large

At b:

Small q

Large CL (or CL2) and to support W

D large

12

THRUST REQUIRED VS. FLIGHT VELOCITY

eAR

CSqSCqT

CCSqSCqDT

LDR

iDDDR

2

0,

,0,

Zero-Lift TR(Parasitic Drag)

Lift-Induced TR(Induced Drag)

Zero-Lift TR~ V2

(Parasitic Drag)

Lift-Induced TR~ 1/V2

(Induced Drag)

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THRUST REQUIRED VS. FLIGHT VELOCITY

iDL

D

DR

RR

DR

CeAR

CC

eARSq

WSC

dq

dT

dq

dV

dV

dT

dq

dT

eARSq

WSCqT

,

2

0,

2

2

0,

2

0,

0

At point of minimum TR, dTR/dV=0

(or dTR/dq=0)

CD,0 = CD,i at minimum TR and maximum L/D

Zero-Lift Drag = Induced Drag at minimum TRand maximum L/D14

HOW FAST CAN YOU FLY? Maximum flight speed occurs when thrust available, TA=TR

Reduced throttle settings, TR< TA

Cannot physically achieve more thrust than TA which engine can provide

Intersection of TRcurve and maximum

TA defined maximum

flight speed of airplane

15

FURTHER IMPLICATIONS FOR DESIGN: VMAX Maximum velocity at a given altitude is important specification for new airplane

To design airplane for given Vmax, what are most important design parameters?

21

0,

0,

2

maxmax

max

2

0,

2

2

0,22

2

0,

2

0,

4

0

D

DAA

D

DD

L

LDD

C

eAR

C

W

T

S

W

S

W

W

T

V

eARS

WTqSCq

eARSq

WSCq

eARSq

WCSqT

Sq

WC

eAR

CCSqSCqTD

Steady, level flight: T = D

Steady, level flight: L = W

Substitute into drag equation

Turn this equation into a quadratic

equation (by multiplying by q)

and rearranging

Solve quadratic equation and setthrust, T, to maximum available

thrust, TA,max

16

FURTHER IMPLICATIONS FOR DESIGN: VMAX

TA,max does not appear alone, but only in ratio (TA/W)max

S does not appear alone, but only in ratio (W/S)

Vmax does not depend on thrust alone or weight alone, but rather on ratios

(TA/W)max: maximum thrust-to-weight ratio

W/S: wing loading

Vmax also depends on density (altitude), CD,0, eAR

Increase Vmax

by

Increase maximum thrust-to-weight ratio , (TA/W)max

Increasing wing loading, (W/S)

Decreasing zero-lift drag coefficient, CD,0

21

0,

0,

2

maxmax

max

4

D

DAA

C

eAR

C

W

T

S

W

S

W

W

T

V

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AIRPLANE POWER PLANTS

Two types of engines common in

aviation today

1. Reciprocating piston engine with

propeller

Average light-weight, general

aviation aircraft

Rated in terms ofPOWER

2. Jet (Turbojet, turbofan) engine

Large commercial transports

and military aircraft

Rated in terms ofTHRUST

18

THRUST VS. POWER Jets Engines (turbojets, turbofans for military and commercial applications) are

usually rate in Thrust

Thrust is a Force with units (N = kg m/s2)

For example, the PW4000-112 is rated at 98,000 lb of thrust

Piston-Driven Engines are usually rated in terms ofPower

Power is a precise term and can be expressed as:

Energy / time with units (kg m2/s2) / s = kg m2/s3 = Watts

Note that Energy is exp ressed in Joules = kg m2/s2

Force * Velocity with units (kg m/s2) * (m/s) = kg m2/s3 = Watts

Usually rated in terms of horsepower (1 hp = 550 ft lb/s = 746 W)

Example:

Airplane is level, unaccelerated flight at a given altitude with speed V

Power Required, PR=TR*V

[W] = [N] * [m/s]

19

POWER AVAILABLE (6.6)

Jet EnginePropeller Drive Engine

20

POWER AVAILABLE (6.6)

Jet EnginePropeller Drive Engine

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POWER REQUIRED (6.5)

PRvs. V qualitatively

(Resembles TRvs. V)

22

POWER REQUIRED (6.5)

D

LL

DR

L

D

LR

L

L

D

L

RR

CCSC

CWP

SC

W

CC

WP

SC

WVSCVWL

V

CC

WVTP

233

23

2

12

2

2

2

1

PRvaries inversely as CL3/2/CD

Recall: TRvaries inversely as CL/CD

23

POWER REQUIRED (6.5)

eAR

CSVqVSCqP

VCCSqVSCqDVVTP

LDR

iDDDRR

2

0,

,0,

Zero-Lift PR Lift-Induced PR

Zero-Lift PR~ V3

Lift-Induced PR~ 1/V

24

POWER REQUIRED

03

1

2

3

2

12

1

,0,

2

2

0,

3

iDDR

DR

CCSVdV

dP

eARSV

WSCVP

iDD CC ,0,3

1

At point of minimum PR, dPR/dV=0

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POWER REQUIRED V

for minimum PRis less than V for minimum TR

iDD CC ,0,

iDD CC ,0,3

1

26

WHY DO WE CARE ABOUT THIS? We will show that for a piston-engine propeller combination

To fly longest distance (maximum range) we fly airplane at speed

corresponding to maximum L/D

To stay aloft longest (maximum endurance) we fly the airplane at minimum

PRor fly at a velocity where CL3/2/CD is a maximum

Power will also provide information on maximum rate of climb and altitude

27

POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)

Propeller Drive

Engine

1 hp = 550 ft lb/s = 746 W

PR

PA

28

POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)

Jet Engine

PR

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ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)

Recall PR= f()Subscript 0 denotes seal-level conditions

21

00,,

21

00

RALTR

ALT

PP

VV

30

ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)Propeller-Driven Airplane

Vmax,ALT < Vmax,sea-level

31

RATE OF CLIMB (6.8)

Boeing 777: Lift-Off Speed ~ 180 MPH

How fast can it climb to a cruising altitude of 30,000 ft?

32

RATE OF CLIMB (6.8)

cos

sin

WLWDT

Governing Equations:

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RATE OF CLIMB (6.8)

sin/

sin

sin

sin

VCR

VW

DVTV

WVDVTV

WDT

Vertical velocity

Rate of Climb:

TV

is power available

DV

is level-flight power required (for small neglect W)TV

- DV

is excess power

34

RATE OF CLIMB (6.8)

Jet EnginePropeller Drive Engine

Maximum R/C Occurs when Maximum Excess Power

35

EXAMPLE: F-15 K Weapon launched from an F-15 fighter by a small two stage rocket, carries a heat-

seeking Miniature Homing Vehicle (MHV) which destroys target by direct impact

at high speed (kinetic energy weapon)

F-15 can bring ALMV under the ground track of its target, as opposed to a ground-

based system, which must wait for a target satellite to overfly i ts launch site.

36

GLIDING FLIGHT (6.9)

DL

L

D

1tancos

sin

cos

sin

0

WL

WD

T

To maximize range, smallest

occurs at (L/D)max

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EXAMPLE: HIGH ASPECT RATIO GLIDER

To maximize range, smallest occurs at (L/D)maxA modern sailplane may have a glide ratio as high as 60:1

So = tan-1(1/60) ~ 1

RANGE AND ENDURANCE

How far can we fly?

How long can we stay aloft?

How do answers vary for propeller-driven vs. jet-engine?

39

RANGE AND ENDURANCE

Range: Total distance (measured with respect to the ground) traversed by airplane

on a single tank of fuel

Endurance: Total time that airplane stays in air on a single tank of fuel

1. Parameters to maximize range are different from those that maximize endurance

2. Parameters are different forpropeller-poweredandjet-powered aircraft

Fuel Consumption Definitions

Propeller-Powered:

Specific Fuel Consumption (SFC)

Definition: Weight of fuel consumed per unit powerper unit time

Jet-Powered:

Thrust Specific Fuel Consumption (TSFC)

Definition: Weight of fuel consumed per unit thrustper unit time

40

PROPELLER-DRIVEN: RANGE AND ENDURANCE

SFC: Weight of fuel consumed per unit power per unit time

hourHPfueloflb

SFC

ENDURANCE: To stay in air for longest amount of time, use minimum

number of pounds of fuel per hour

HPSFC

hour

fueloflb

Minimum lb of fuel per hour obtained with minimum HP

Maximum endurance for a propeller-driven airplane occurs whenairplane is flying at minimum power required

Maximum endurance for a propeller-driven airplane occurs when

airplane is flying at a velocity such that CL3/2/CD is a maximized

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PROPELLER-DRIVEN: RANGE AND ENDURANCE SFC: Weight of fuel consumed per unit power per unit time

hourHPfueloflb

SFC

RANGE: To cover longest distance use minimum pounds of fuel per mile

V

HPSFC

mile

fueloflb

Minimum lb of fuel per hour obtained with minimum HP/V

Maximum range for a propeller-driven airplane occurs when airplane

is flying at a velocity such that CL/CD is a maximum

42

PROPELLER-DRIVEN: RANGE BREGUET FORMULA

To maximize range:

Largest propeller effici ency,

Lowest possible SFC

Highest ratio of Winitial to Wfinal, which is obtained with the largest fuel weight

Fly at maximum L/D

final

initial

D

L

W

W

C

C

SFCR ln

43

PROPELLER-DRIVEN: RANGE BREGUET FORMULA

final

initial

D

L

W

W

C

C

SFCR ln

PropulsionAerodynamics

Structures and Materials

44

PROPELLER-DRIVEN: ENDURACE BREGUET FORMULA

To maximize endurance:

Largest propeller effici ency, Lowest possible SFC

Largest fuel weight

Fly at maximum CL3/2/CD

Flight at sea level

2

12

12

12

3

2 initialfinalD

L WWSC

C

SFCE

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JET-POWERED: RANGE AND ENDURANCE TSFC: Weight of fuel consumed per thrust per unit time

hourthrustoflbfueloflb

TSFC

ENDURANCE: To stay in air for longest amount of time, use minimum

number of pounds of fuel per hour

ThrustTSFC

hour

fueloflb

Minimum lb of fuel per hour obtained with minimum thrust

Maximum endurance for a jet-powered airplane occurs when

airplane is flying at minimum thrust required

Maximum endurance for a jet-powered airplane occurs when

airplane is flying at a velocity such that CL/CD is a maximum

46

JET-POWERED: RANGE AND ENDURANCE TSFC: Weight of fuel consumed per unit power per unit time

hourthrustoflbfueloflb

TSFC

RANGE: To cover longest distance use minimum pounds of fuel per mile

V

ThrustSFC

mile

fueloflb

Minimum lb of fuel per hour obtained with minimum Thrust/V

Maximum range for a jet-powered airplane occurs when airplane isflying at a velocity such that CL1/2/CD is a maximum

D

L

D

L

R

CC

CSC

WS

V

T

21

12

2

1

47

JET-POWERED: RANGE BREGUET FORMULA

To maximize range:

Minimum TSFC

Maximum fuel weight

Flight at maximum CL1/2/CD

Fly at high altitudes

21

212

1

122 finalinitial

D

L WWC

C

TSFCSR

48

JET-POWERED: ENDURACE BREGUET FORMULA

To maximize endurance:

Minimum TSFC

Maximum fuel weight

Flight at maximum L/D

final

initial

D

L

W

W

C

C

TSFCE ln

1

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SUMMARY: ENDURANCE AND RANGE Maximum Endurance

Propeller-Driven

Maximum endurance for a propeller-driven airplane occurs when airplane is

flying at minimum power required

Maximum endurance for a propeller-driven airplane occurs when airplane is

flying at a velocity such that CL3/2/CD is a maximized

Jet Engine-Driven

Maximum endurance for a jet-powered airplane occurs when airplane is

flying at minimum thrust required

Maximum endurance for a jet-powered airplane occurs when airplane is

flying at a velocity such that CL/CD is a maximum

Maximum Range

Propeller-Driven

Maximum range for a propeller-driven airplane occurs when airplane is flying

at a velocity such that CL/CD is a maximum

Jet Engine-Driven

Maximum range for a jet-powered airplane occurs when airplane is flying at avelocity such that CL1/2/CD is a maximum

EXAMPLES OF DYNAMIC

PERFORMANCE

Take-Off Distance

Turning Flight

51

TAKE-OFF AND LANDING ANALYSES (6.15)

F

mVs

Vdtds

t

m

FV

dtm

FdV

dt

dVmmaF

2

2

dtdV

mLWDTRDTF r

Rolling resistance

r= 0.02

s: lift-off distance

52

NUMERICAL SOLUTION FOR TAKE-OFF

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USEFUL APPROXIMATION (T >> D, R)

Lift-off distance very sensitive to weight, varies as W2

Depends on ambient density

Lift-off distance may be decreased:

Increasing wing area, S

Increasing CL,max

Increasing thrust, T

TSCg

Ws

L

OL

max,

2

..

44.1

sL.O.: lift-off distance

54

EXAMPLES OF GROUND EFFECT

55

TURNING FLIGHT

V

ng

R

V

dt

d

ng

VR

R

VmF

nWF

W

Ln

WLF

WL

r

r

r

1

1

1

cos

2

2

2

2

2

22

Load Factor

R: Turn Radius

: Turn Rate56

EXAMPLE: PULL-UP MANEUVER

V

ng

ng

VR

R

VmF

nWWLF

r

r

1

1

1

2

2

R: Turn Radius

: Turn Rate

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V-n DIAGRAMS

SW

CVn

W

SCV

W

Ln

L

L

max,2

max

2

2

1

2

1

58

STRUCTURAL LIMITS