Constraint Diagrams

7/23/2019 Constraint Diagrams

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Creating Constraint Diagrams

04A_constraint-diagram.ppt

D. Edberg

!#$%&2013/Oct/08 !

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Embraer ERJ-145 Regional Jet

Overview

• What are constraint diagrams?

• Constraints for takeoff

• Constraints for cruise

• Landing constraints

• Acceleration

•

The constraint diagram

• Concluding remarks

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• Display what an airplane can and cannot do

•

Used for design optimization• Choose a design point based on

• Design point must lie within the constraint

boundaries

• Designs are often “optimum” near the

constraint lines

T SL

W TO

andW

TO

S

What are Constraint Diagrams?

(2013/Oct/08

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Example Constraint Diagram

$

Courtesy W. Mason, Va. Tech

2013/Oct/08



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Constraint Diagram

)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 20 40 60 80 100

W/S

T / W

Cruise Out

Combat Turn Ma=0.9Combat Turn Ma=1.2

Max Mach Heavy=2.2

Landing 4k

Takeoff 4kLoiter SL

Mach 1.2 SL

Design Point

2013/Oct/08

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JSF Constraint Diagram

*

0.0 0.2 0.4 0.6

0.8 1.0 1.2 1.4

20 30 40 50 60 70 80 90 100

Wing Loading W / S (psf)

Thrust /

Weight

RatioT /W

Sustained G

5 4

Instantaneous G 9 8 7

6

Mach 1.4 1.5 1.6

Courtesy Paul Bevilaqua, Lockheed Martin

2013/Oct/08



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• Derived from the equation for specific excess

power (§ 5.15, Introduction to Aeronautics: A

Design Perspective, Brandt, Stiles, Bertin, andWhitford, AIAA. PDF file in notes.)

• Thrust-to-weight vs. wing loading:

The Governing Equation

SL" T TOW = # "

q

#

C D

W TO S

+k 1

n#

q

$

% &

'

( )

2

W TO

S

$ % &

' ( )

*

+

, , ,

-

.

/ / / +1

V

dh

dt

$

% &

'

( ) +

1

g

dv

dt

$

% &

'

( )

0

1 2 2

3 2 2

4

5 2 2

6 2 2

+2013/Oct/08

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Variables in Constraint Equation

! = T /T TO = ratio of actual thrust to takeoff thrust(accounts for thrust loss due to altitude, V )

" = W /W TO = weight fraction (fuel use, stores drop)

k 1 = induced drag term

k 2 = drag term (Brandt, p. 134)

h = altitude

n = load factor

q = !# V 2 = dynamic pressure

v = velocity

! = turn rate (rad/s)

2

1 2O

D L L DC k C k C C = + +

2

0

1 V

n g

! "#= + $ %

& '

,2013/Oct/08



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Values for Constants K 1 & C D0

(Mattingly et al, Aircraft Engine Design)

-2013/Oct/08

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Flight Path Considerations

!

If level flight, dh/dt = 0

!

If no turns or loads, n = 1

! If no acceleration, dv/dt = 0

! Usually take off at 1.2 " stall speed

(apply a factor of 1.44 to v2 with takeoff assumed

using max lift coefficient C L max)!

Landing at 1.3 " stall speed (factor of 1.69 for

landing)

!#2013/Oct/08



Governing Equation:

Name Sample Value

" (fully fueled) 1

# (sea level) 0.002378 slugs/ft3

! (from v at 0.7 liftoff speed) 0.84

C L max (estimated, similar aircraft) 2.2

g 32.2 ft/s2

S TO (requirement) 2500 ft

S

W

gsC W

T TO

TO LTO

SL

MAX !"

# 244.1=

Example Constraints for Takeoff(“High” Thrust, Neglect Runway Friction)

!!#$%&2013/Oct/08

S

W

W

T TO

TO

SL004.=

20 0.82

40 0.16

60 0.25

80 0.33

100 0.41120 0.49

140 0.57

160 0.65

180 0.74

200 0.82

S

W TO

TO

SL

W

T

!!"

#$$%

&2 ft

lb

Resulting Takeoff Constraint Equation

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Governing Eqn:

Name Value

" (fuel lost during climb) 0.818

! (thrust at cruise speed) 0.93

q 200 lb/ft2

C L 0.575

k 1 0.03

C D 0 0.03

!!"

!!#

$

!!%

!!&

'

++

(((

)

*

+++

,

-

./

012

3../

0112

3+=

dt

dV

g dt

dh

V S

W

q

nk

S

W

C q

W

T TO

TO

D

TO

SL o 11

2

1

4

4 5

4

Constraints for Cruise

!(#$%&2013/Oct/08

!!!

"

#

$$$

%

&

+=

S

W

S

W W

T TO

TOTO

SL003.

46.6

20 0.38

40 0.28

60 0.29

80 0.32

100 0.36

120 0.41

140 0.46

160 0.52

180 0.57

200 0.63

S

W TO

TO

SL

W

T !!"#

$$%&

2 ft lb

Final Cruise Constraint Equation

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Governing Equation:

Name Value

# 0.00238 slugs/ft3

2.6 (Schaufele)

µ (friction coefficient) 0.3 (www.asft.se)

" 0.65S L (landing distance) 3000 ft

!

µ "

69.1

g C S

S

W MAX L LTO

=

MAX LC

Example Constraints for Landing

!)#$%&2013/Oct/08

W TO

S =163

lbf

ft2

163 0.1

163 0.2

163 0.3

… …

163 1.0

S

W TO

TO

SL

W

T

lbf

ft2

"

#

$ %

&

'

Final Landing Constraint Equation

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Non-Fighters Must Consider Runway

Friction! Use “effective” acceleration at 70% of takeoff or

landing speed

Landing assumes no thrust

! = runway friction

W TO, W L = takeoff and landing weights

sTO =1.44W TO

2

" SC Lmax

g0 T # D#µ W TO # L( )[ ]

0.7V TO

s L =

1.69W L

2

" SC Lmaxg0 D+ µ W L # L( )[ ]

0.7V L

!+2013/Oct/08

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Acceleration Constraint(Level, Unbanked Flight)

! The governing equation is

!

What is used for q? Start, finish, or mean?

!,

SL" T TOW = # "

q

#

C D

W TO S ( )

+k 1

#

q

$

% &

'

( )

2

W TO

S

$ % &

' ( )

*

+

, , ,

-

.

/ / / +1

g

dv

dt

$

% &

'

( )

0

1 2 2

3 2 2

4

5 2 2

6 2 2

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Construction of Constraint Diagram

! Plot all curves on a single graph

! Wing loading horizontal

!

Thrust-to-weight vertical

! Identify which side of each curve is OK

! Make sure the constraint curves make sense!

! Choose and identify design point

!-2013/Oct/08

Constraint Diagram for Regional Jet Mission

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

Wing Loading (lb/ft^2)

T

s l / W t o

Takeoff

Cruise

Landing

Sample Constraint Diagram —Regional

JetDesign point

(W /S = 50 psf,

T /W = 0.33)

Solution Space

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Example Constraint Diagram

(Black lines, Cartoon box, Design point)

2013/Oct/08

0.0

0.5

1.0

1.5

2.0

2.5

3.0

20 40 60 80 100 120

T / W

W / S (lbf /ft2)

Fighter Constraint Diagram

Turn

Horiz Accel

Takeoff

Braking

Design Point

W / S = 38 psf

T /W = 1.7

Solution Space

'!

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Design Point

104 ft2, 45 hp

0

10

20

30

40

50

60

70

80

20 40 60 80 100 120 140 160

Wing Area (ft2)

E n g i n e H o r s e p o w e r

Ceres UAV Constraint Diagram

F e r r y

L a n d

i n g

T ak eof f

Courtesy Nathan Olson, CPP ‘08

M a n

u f a c t u

r i n g C

o s t

''2013/Oct/08



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Comments on Design Point

•

Must fit in allowable areas of all constraints

•

Allow some margin (“wiggle room”) so

design changes don’t move it out•

Lowest-weight aircraft meeting constraints isoften cheapest

•

Less thrust = less engine required = lessengine cost, usually

•

Existing engine thrust may not match what’sneeded

•

Constraint diagram does not know how manyengines. (Multi-engines provide nT of thrust)

'(2013/Oct/08

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Summary of Constraint Diagrams• Select design point• Must satisfy all constraint curves• Must fit all constraints and/or missions (for

multiple-mission aircraft)• SHOW SIMILAR AIRCRAFT on your plot!• Be sure to include off-nominal conditions i.e.

• Phoenix Sky Harbor Airport @ 120° F

•

Denver @ 100°F • Go back and re-do constraint diagram when

parameters or design changes• Do not put tables of constraint curve data in your

slides! I WILL take off points.

'$2013/Oct/08



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Comments on

Thrust-to-Weight (T /W )

and Wing Loading (W /S )

(Based on Chapter 5 of Raymer’s

Aircraft Design: A Conceptual Approach)

')2013/Oct/08

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Thrust-To-Weight RatioT

/W

(Raymer Tables 5.1, 5.3)

Typical Installed T /W Jet trainer 0.4

Jet fighter (dogfighter) 0.9

Jet fighter (other) 0.6

Military cargo/bomber 0.25

Jet transport 0.25 – 0.4

Statistical T /W o Estimation (vs. max Mach M max)T /W o = aM max

c a c

Jet trainer 0.488 0.728

Jet fighter (dogfighter) 0.648 0.594

Jet fighter (other) 0.514 0.141

Military cargo/bomber 0.244 0.341

Jet transport 0.267 0.363

'*2013/Oct/08



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Power-To-Weight Ratio P /W (Raymer Tables 5.2, 5.4)

Typical Installed W / P Powered Sailplane 25

Homebuilt 12

GA-Single engine 14

GA-Twin engine 6Agricultural 11

Twin turboprop 5

Flying boat 10

Statistical P /W o Estimation (vs. vmax in kt) P /W o = avmaxc

a c

Powered Sailplane 0.043 0.0

Homebuilt 0.005 0.57

GA-Single engine 0.004 0.57

GA-Twin engine 0.025 0.22Agricultural 0.009 0.50

Twin turboprop 0.013 0.50

Flying boat 0.030 0.23'+2013/Oct/08

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Power Loading &Horsepower-to-Weight Ratio

! Propeller-Powered Aircraft:

! T = % p P /v = 550 % pHP/v

! so, T /W = (% p/v)( P /W )

= (550 % p/v)(HP/W ) using fps units. (R5.1)

! Define “Power Loading” W o/HP = 1/(HP/Wo)

! Note: reversed meaning compared to T /W !

! C = C pv/(% p) = C bhpV /(550% p)

',2013/Oct/08



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Power-to-Weight Ratio (Raymer Tables 5.2, 5.4)

TYPICAL

INSTALLED

P /W :

STATISTICAL

P /W

ESTIMATION

(vs. vmax)

'-2013/Oct/08

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Thrust Matching(T & W are actual values, NOT takeoff values)

In cruise: T = D, L = W , so:

(T /W )cruise = ( D/ L)cruise = 1/( L/ D)cruise

In climb: T = D + W sin & , L = W cos & , so:

(T /W )climb = 1/( L/ D)climb + sin &

= 1/( L/ D)climb + vvert /vhoriz (R5.4)

Must ratio results back to takeoff values for comparison

T

W

"

# $

%

& ' takeoff

=

T

W

"

# $

%

& ' cruise

W cruise

W takeoff

"

# $ $

%

& ' ' T takeoff

T cruise

"

# $

%

& '

(#2013/Oct/08



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Wing Loading (W /S ) Comments

Higher W /S

Smaller Wing

! Higher stall speed

! Longer takeoff and landing distances

! Poorer maneuvering performance

But benefits are:! Reduced friction drag and weight

(!2013/Oct/08

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Stall Speed

W = L = qstallSC Lmax = 1/2 # v stall 2SC Lmax

W /S = q stall C Lmax = 1/2 # v stall 2C Lmax

# = 0.002378 slugs/ft3 @ sea level

# = 0.00189 slugs/ft3 @ 5000 ft, hot day (Denver)

v stall defined by FAR, MIL SPEC, or design reqts

v stall

= 61 kt (FAR-23: Single engine, W o< 12,500 lb)

v stall may be set by vapproach

Civil: vapproach = 1.3 v stall

Military: vapproach = 1.2 v stall

Carrier-Based: vapproach = 1.15 v stall

('2013/Oct/08



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Maximum Lift Coefficient(Raymer Fig. 5.3)

WINGS OF MODERATE ASPECT RATIO (4-8)

C Lmax

Quarter-Chord Sweep((2013/Oct/08

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Takeoff Distance Estimation (Raymer Fig. 5.4)

Takeoff

Distance

(x 1000 ft)

NUMBEROF JET

ENGINES

BALANCED

FIELD

LENGTH

Jet

Prop

TAKEOFF PARAMETER: W /S or W /S

#C LTOT /W #C LTO

HP/W ($2013/Oct/08



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Introduction to Aeronautics: A Design

Perspective, Brandt, Stiles, Bertin, and

Whitford, AIAA. PDF file in notes. Aircraft Engine Design, Mattingly, Heiser, and

Daley, AIAA

Aviation Week & Space Technology Source Book,

information on currently available engines and

aircraft:

www.avweek.com/aw/sourcebook/index.jsp

References

()2013/Oct/08

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FARs (Federal Aviation Regulations)& Other Regulation Information

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Constraint Diagrams

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