Lecture 1-constraint diagrams - MAELabs UCSD

Copyright © 2005-06 by Don Edberg. All rights reserved.
Overview
Creating Constraint
Diagrams
• What are constraint diagrams?
• Constraints for takeoff
• Constraints for cruise
• Landing constraints
• The constraint diagram
• Concluding remarks
Embraer ERJ-145 Regional Jet
1

Aircraft Weight (in 1000 lb)
What are Constraint Diagrams?
• Diagrams that show what airplanes can and cannot do.
TSL
• Choose a design point based on and
W
• Constraint diagrams are used for optimization
• The design point will lie within the constraint boundaries
• Design points are usually optimum near the constraint
lines
TO
WTO S
Example Constraint Diagram
2

!
The Governing Equation
• Derived from the equation for specific
excess power (Brandt et al, § 5.15)
• Thrust-to-weight vs. wing loading
0
2
*
,
T SL"
W TO
= # 2 ,
q C
D
$ n# '
2
2 ,
$ WTO '
1 , +k
" # $ W ' 1&
) & )
2 ,
& TO % q ( % S
2
(
,
%
S )
2 , (
3
+
- 4
/ 2
/ 2
/ 2
/ 5
/ 2
/ 2
/ 2
. 6
+ 1
V
dh'
)
)
dt (
+ 1 $ dv &
&
g%
dt
Variables in Constraint Equation
α = T/T TO = ratio of actual thrust to takeoff thrust
β = W/W TO = weight fraction
k 1 = induced drag term
h = altitude
n = load factor
q = ρv 2 /2 = dynamic pressure
V = velocity
$
&
&
%
'
)
)
(
3

Flight Path Considerations
If level flight, dh/dt = 0
No turns or loads means n = 1
If non-accelerating, dV/dt = 0
Takeoffs are done at 1.2 × stall speed
(factor of 1.44 affects max lift coefficient)
Landings are done at 1.3 × stall speed
(factor of 1.69 affects C Lmax)
Example Constraints for Takeoff
Governing Equation:
Name
β
ρ
α
g
S TO
W
CLMAX
TSL TO
TO
#
!" C gs
2
1.
44
=
L
MAX
TO
Value
1
0.00238 slugs/ft 3
0.84
2.2
W
S
32.2 ft/s 2
2500 ft
4

TSL TO = . 004
W
Resulting Takeoff Constraint Equation
TO
Governing Eqn:
W
S
Constraints for Cruise
W
' -
4 ! q + CD
= & +
5 ! 4 W
+
! % , S
Name
β
α
q
C L
k 1
WTO S
20
40
60
80
100
120
140
160
180
200
&
$
%
lb
2
ft
#
!
"
3 n4
0 3W
+ k11
. 1
2 q / 2 S
Value
0.818
0.93
200 lb/ft 2
0.575
0.03
0.03
T
W
SL
TO
0.82
0.16
0.25
0.33
0.41
0.49
0.57
0.65
0.74
0.82
*
$
0(
1 dh dV !
. ( + + #
/ ( V dt g dt !
)
! "
2
TSL o TO
1
TO
CDo
TO
5

W
Final Cruise Constraint Equation
&
$ 6.
46
= $ + .
W
$ TO
% S
TSL TO 003
TO
W
S
#
!
!
!
"
W TO
S
&
$
%
20
40
60
80
100
120
140
160
180
200
lb
2
ft
Example Constraints for Landing
Governing Equation:
Name
ρ
CLMAX
µ (friction coefficient)
β
S L (landing distance)
#
!
"
S " C gµ
TO L L
=
S 1.
69!
W MAX
Value
0.65
3000 ft
T
W
0.00238 slugs/ft 3
2.6 (Schaufele)
0.3 (www.asft.se)
SL
TO
0.38
0.28
0.29
0.32
0.36
0.41
0.46
0.52
0.57
0.63
6

Final Landing Constraint Equation
WTO lb
= 163
S ft
2
WTO S
163
163
163
…
163
&
$
%
lb
2
ft
Construction of Constraint
Diagram
Plot all curves on a single graph
Wing loading horizontal
Thrust-to-weight vertical
#
!
"
T
W
SL
TO
0.1
0.2
0.3
…
1.0
Identify which side of each curve is OK
Make sure the constraint curves make
sense!
Choose and identify design point
7

Tsl/Wto
Sample Constraint Diagram Diagram for Regional (Regional Jet Mission Jet)
1.2
1
0.8
0.6
0.4
0.2
0
Design point
(W/S = 50 psf,
T/W = 0.33)
0 50 100 150 200 250
Wing Loading (lb/ft^2)
Solution Space
Example Constraint Diagram
Takeoff
Cruise
Landing
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Comments on Design Point
Must fit within all constraint curves
Allow some margin (“wiggle room”) so
design changes don’t move it out
Typically, the lightest weight aircraft that
meets constraints is cheapest (often the
lowest point on the plot)
Less thrust = less engine required = less
engine cost
Concluding Remarks
• Selected design point
• Must satisfy all constraint curves
• Must fit all constraints and/or missions
(for multiple-mission aircraft)
• Go back and re-do constraint diagram
when parameters change
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References
1. Introduction to Aeronautics: A Design Perspective,
Brandt, Stiles, Bertin, and Whitford, AIAA, 1997.
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