Creating Constraint Diagrams 04A_constraint-diagram.ppt
D. Edberg
2013/Oct/08
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Overview • • • • • • •
What are constraint diagrams? Constraints for takeoff Constraints for cruise Landing constraints Acceleration The constraint diagram Embraer ERJ-145 Regional Jet Concluding remarks
2013/Oct/08
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What are Constraint Diagrams? • Display what an airplane can and cannot do • Used for design optimization • Choose a design point based on
T SL
and
W TO
W TO S
• Design point must lie within the constraint boundaries • Designs are often “optimum” near the constraint lines
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Example Constraint Diagram
Courtesy W. Mason, Va. Tech 2013/Oct/08
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Constraint Diagram
1.00 0.90
Design Point
0.80 0.70 0.60 W / T
0.50 0.40 Cruise Out Combat Turn Ma=0.9 Combat Turn Ma=1.2 Max Mach Heavy=2.2 Landing 4k Takeoff 4k Loiter SL Mach 1.2 SL
0.30 0.20 0.10 0.00 0
20
40
60
80
100
W/S 2013/Oct/08
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JSF Constraint Diagram Courtesy Paul Bevilaqua, Lockheed Martin 1.4 1.2 Mach 1.4 1.5 1.6
1.0
Thrust / Weight Ratio T /W
0.8 Sustained G
6 5 4
0.6 0.4 0.2 0.0 20
Instantaneous G
30
40
50
9
60
Wing Loading 2013/Oct/08
8
70
7
80
90
100
W / S (psf) #$%& *
The Governing Equation • Derived from the equation for specific excess power (§ 5.15, Introduction to Aeronautics: A Design Perspective , Brandt, Stiles, Bertin, and Whitford, AIAA. PDF file in notes.) • Thrust-to-weight vs. wing loading: 0 * 2 T SL" = # 21 q ,, C D W TO " 2 # ,W TO 23 +
2
$ n# ' $ W + k & ) &% S % q ( S
TO
1
4 '/ 1 $ dh ' 1 $ dv '22 )/ + & )+ & )5 (/ V % dt ( g % dt (22 . 6
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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) 2 C = k C + k C +C 1 2 D L L D h = altitude n = load factor ! # V " 2 n = + 1 q = !# V = dynamic pressure $ % g & ' v = velocity ! = turn rate (rad/s)
O
2
0
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Values for Constants
K 1 & C D0
(Mattingly et al, Aircraft Engine Design)
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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
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Example Constraints for Takeoff (“High” Thrust, Neglect Runway Friction) Governing Equation:
T SL
2
1.44 #
W TO
=
W TO
!" C L MAX gsTO S
Name
Sample Value
" (fully fueled)
1
# (sea level)
0.002378 slugs/ft 3
! (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
2013/Oct/08
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Resulting Takeoff Constraint Equation W TO
2
S
T SL =
.004
W TO
2013/Oct/08
& lb $$ % ft
# !! "
T SL W TO
20
0.82
40
0.16
W TO
60
0.25
S
80
0.33
100
0.41
120
0.49
140
0.57
160
0.65
180
0.74
200
0.82 #$%& !'
Constraints for Cruise ' T 4 ! ! q + C D Governing Eqn: SL = & + W W TO 5 ! 4 + TO ! % , S o
+
3 n4 0 .. 2 q /
k 1 11
2
$ * ( 1 dh 1 dV ! ! 3 W TO 0 + 1 .( + # 2 S /( V dt g dt ! ! ) "
Name
Value
" (fuel lost during climb)
0.818
! (thrust at cruise speed)
0.93
q
200 lb/ft2
C L
0.575
k 1
0.03
C D 0
0.03
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Final Cruise Constraint Equation W TO S
T SL W TO
& $ 6.46 W = $ + .003 W S $ % S
TO
TO
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# ! ! ! "
& lb $$ % ft
2
# !! "
T SL W TO
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 #$%& !$
Example Constraints for Landing Governing Equation:
W TO
=
S L " C L MAX g µ 1.69 !
S
Name
Value
#
0.00238 slugs/ft 3 2.6 (Schaufele)
C L
MAX
µ (friction coefficient)
0.3 (www.asft.se)
"
0.65
S L (landing distance)
3000 ft
2013/Oct/08
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Final Landing Constraint Equation W TO
W TO =
S
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163
lb f ft
2
S
" lb f % $ 2' # ft &
T SL W TO
163
0.1
163
0.2
163
0.3
…
…
163
1.0
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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
s L
=
1.44W TO
=
" SC L
max
g0 [T # D # µ (W TO # L)] 1.69 W L
" SC L
max
2
0.7 V TO
2
g0 [ D + µ (W L # L)]
0.7 V L
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Acceleration Constraint (Level, Unbanked Flight) !
The governing equation is 0 * 2 T SL" = # 21 q ,, W TO " 2 # , 23 +
!
-
4
2 $ ' C # $ W '/ 1 $ dv '22 + k )/ + & )5 & ) &% (W S ) % q ( S (/. g % dt (226 D
TO
1
TO
What is used for q? Start, finish, or mean?
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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
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Sample Constraint Diagram —Regional Constraint Diagram for JetRegional Jet Mission Design point (W /S = 50 psf, T /W = 0.33)
1.2 1
Solution Space
o 0.8 t W 0.6 / l s T
Takeoff Cruise Landing
0.4 0.2 0 0
50
100
150
200
250
Wing Loading (lb/ft^2) 2013/Oct/08
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Example Constraint Diagram (Black lines, Cartoon box, Design point) Fighter Constraint Diagram 3.0
Solution Space
2.5
Design Point W / S = 38 psf T /W = 1.7
2.0
Turn Horiz Accel Takeoff
W1.5 T 1.0
Braking
/
0.5 0.0 20
40
60 80 W / S (lbf /ft2)
100
120
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80
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Ceres UAV Constraint Diagram
r 70 e w o60 p e50 s r o40 H e30 n i g20 n E10
g i n d n a L
t o s C g r in c tu fa u a n F er r y M
Design Point 2
104 ft , 45 hp
T ak eof f
0 20
40
60
80
100
120
140
160
2
Wing Area (ft ) Courtesy Nathan Olson, CPP ‘08 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 is often cheapest • Less thrust = less engine required = less engine cost, usually • Existing engine thrust may not match what’s needed • Constraint diagram does not know how many engines. (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 Ratio T /W (Raymer Tables 5.1, 5.3) Typical Installed T /W Jet trainer Jet fighter (dogfighter) Jet fighter (other) Military cargo/bomber Jet transport
0.4 0.9 0.6 0.25 0.25 – 0.4
Statistical T /W o Estimation (vs. max Mach M max) T /W o = aM max
c
Jet trainer Jet fighter (dogfighter) Jet fighter (other) Military cargo/bomber Jet transport 2013/Oct/08
a 0.488 0.648 0.514 0.244 0.267
c 0.728 0.594 0.141 0.341 0.363 #$%& '*
Power-To-Weight Ratio
P /W (Raymer Tables 5.2, 5.4)
Typical Installed W / P Powered Sailplane Homebuilt GA-Single engine GA-Twin engine Agricultural Twin turboprop Flying boat
25 12 14 6 11 5 10
Statistical P /W o Estimation (vs. vmax in kt) P /W o = avmaxc Powered Sailplane Homebuilt GA-Single engine GA-Twin engine Agricultural Twin turboprop Flying boat 2013/Oct/08
a 0.043 0.005 0.004 0.025 0.009 0.013 0.030
c 0.0 0.57 0.57 0.22 0.50 0.50 0.23
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Power Loading & Horsepower-to-Weight Ratio
!
Propeller-Powered Aircraft:
!
T = % p P /v = 550 % pHP/v
!
!
so, T /W = (% p/v)( P /W ) = (550 % p/v)(HP/W ) using fps units. (R5.1) Define “Power Loading” W o/HP = 1/(HP/Wo)
Note: reversed meaning compared to T /W !
! !
C = C pv/(% p) = C bhpV /(550% p)
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Power-to-Weight Ratio
(Raymer Tables 5.2, 5.4)
TYPICAL INSTALLED P /W :
STATISTICAL P /W ESTIMATION (vs. vmax)
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Thrust Matching (T & W are actual values, NOT takeoff values)
In cruise: T = D, L = W , so: (T /W )cruise = ( D/ L)cruise = 1/( L/ D)cruise In climb: T = D + W sin & , L = W cos & , so: (T /W )climb = 1/( L/ D)climb + sin & = 1/( L/ D)climb + vvert /vhoriz
(R5.4)
" T % " T % " W cruise %" T takeoff % '$ $ ' $ ' $ ' # W & takeoff # W & cruise $# W takeoff '&# T cruise & =
Must ratio results back to takeoff values for comparison
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Wing Loading (W /S ) Comments
Higher W /S Smaller Wing !
Higher stall speed
!
Longer takeoff and landing distances
!
Poorer maneuvering performance
But benefits are: !
Reduced friction drag and weight 2013/Oct/08
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Stall Speed W = L = qstallSC Lmax = 1/2 # v stall 2SC Lmax W /S = q stall C Lmax = 1/2 # v stall 2C Lmax
# = 0.002378 slugs/ft3 @ sea level # = 0.00189 slugs/ft3 @ 5000 ft, hot day (Denver) v stall defined by FAR, MIL SPEC, or design reqts v stall = 61 kt (FAR-23: Single engine, W o < 12,500 lb) v stall may be set by vapproach Civil: vapproach = 1.3 v stall Military: vapproach = 1.2 v stall Carrier-Based: vapproach = 1.15 v stall 2013/Oct/08
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Maximum Lift Coefficient (Raymer Fig. 5.3)
C Lmax
WINGS OF MODERATE ASPECT RATIO (4-8)
Quarter-Chord Sweep 2013/Oct/08
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Takeoff Distance Estimation (Raymer Fig. 5.4) Takeoff Distance (x 1000 ft)
NUMBER OF JET ENGINES
Jet
BALANCED FIELD LENGTH
Prop
TAKEOFF PARAMETER: 2013/Oct/08
W /S
or
#C L
T /W
TO
W /S #C L
HP/W
TO
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References Introduction to Aeronautics: A Design Perspective , Brandt, Stiles, Bertin, and Whitford, AIAA. PDF file in notes. Aircraft Engine Design , Mattingly, Heiser, and Daley, AIAA Aviation Week & Space Technology Source Book, information on currently available engines and aircraft: www.avweek.com/aw/sourcebook/index.jsp
2013/Oct/08
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FARs (Federal Aviation Regulations) & Other Regulation Information !""#$%%&'()*++)',-% ./'0(+",&12+3425064+37/2869&+&1% &':;.)3<*%=+63:&+>/?@#/3:&+>/A/" !""#$%%CCC)+6&C/9)*++)',-% ./'0(+",&12+3425064+37/2869&+&1% &':;.)3<*%
2013/Oct/08
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