takeoff & landing constraints c- 2: glides, climbs, range

AE317 Aircraft Flight Mechanics & Performance

UNIT C: Performance

ROAD MAP . . .

C-1: Equation of Motion

C-2: Glides, Climbs, Range, & Endurance

C-3: Takeoff, Landing, & Turn

C-4: V-n Diagram & Constraint Analysis

C-5: Performance Analysis Examples

Brandt, et.al., Introduction to Aeronautics: A Design Perspective

Chapter 5: Performance

5.13 V-n Diagrams5.14 Energy Height and Specific Excess Power5.16 More Details – Constraint Analysis

Unit C-4: List of Subjects

V-n Diagram

Energy Height & Specific Excess Power

Ps Comparison & Maneuverability

Constraint Analysis

Takeoff & Landing Constraints

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V-n Diagrams

• Limitations that the aircraft might have on its ability to generate the lift or sustain the structural

loading needed to perform a specific turn can be summarized on V-n diagrams.

• The maximum positive and negative load factors that the aircraft structure can sustain are shown as

horizontal lines on the chart (these structural limits are not function of velocity).

• At low speeds, the maximum load factor is limited by the maximum lift generated by aircraft:

max max

2

max max

1

2L LL n W C qS C V S= = = => max 2

max2

LC Sn V

W

= (5.71)

• The maximum lift boundary is also known as the stall boundary. Eq(5.71) actually leads to a more

general form of the stall speed:

max

stall

2

L

nWV

SC= (5.72)

• The vertical line labeled "q limit" indicates the maximum structural airspeed of the aircraft. For

high-speed aircraft, this is often set by the aircraft's critical Mach number ( )critM , because flying

above this flight speed will cause control difficulties due to shock-induced separation. For low-

speed aircraft, the limit is set by the structural strength required by the wings, windscreen, fuselage,

etc., to resist the high dynamic pressures and high stagnation point pressure. For some high-speed

aircraft, the maximum speed is set for, actually, a temperature limit. For F-104, the q limit is set by

maximum engine compressor inlet temperature.

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Corner Velocity

• The velocity labeled "corner velocity" is the velocity at which the stall limit and structural limit

make a "corner" on the graph. This velocity satisfies the conditions for quickest, tightest turn

because a faster velocity would not see an increase in n as a result of the structural limit, and slower

velocity would see n limited to less than its maximum value by the stall.

• The term "corner velocity" might also have been chosen to reflect the fact that the aircraft makes its

sharpest corner at that speed. An expression for corner velocity can be obtained by substituting the

aircraft's maximum load factor into eq(5.72):

max

* max2

L

n WV

SC= (5.73)

Solution (5.13)

Example C-4-1(V-n Diagram)

Example 5.13

An aircraft with a wing loading (W/S) of 70 lb/ft2 and has a maximum structural load limit of 9.

What is its corner velocity at sea level?

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Table 1.2 Typical minimum design requirements/constraints for a multirole jet fighter

Item Requirement

Combat mission radius 400 miles

Weapons payload 2 AIM-120, 4 2,000-lb MK-84, 600 20-mm ammunition

Takeoff distance 2,000 ft

Landing distance 2,000 ft

Mach number M = 1.8 at optimum altitude at Wmana

Instantaneous turn rate 18 deg/s at M = 0.9, 20,000 ft MSLb at Wman

Specific excess power 4 g at M = 1.2, 20,000 ft MSL at Wman

Sustained load factor 9 g at M = 0.9, 5,000 ft MSL at Wman a The maneuver weight Wman is the aircraft weight with 50% internal fuel, 2 AIM-120 AMRAAM (Advanced Medium Range Air-to-Air) missiles, and full-

cannon ammunition, but no air-to-ground weapons. b The abbreviation MSL signifies altitude above mean sea level, the average elevation of the Earth's oceans.

• Specific excess power is a measure of an aircraft's ability to increase its specific energy ( )eH :

2 2. . . . 0.5

2e

P E K E mgh mV VH h

W W g

+ += = = + (5.74)

• Specific energy is also called energy height because it has units of height. The aircraft's power

available that is in excess of its power required is its excess power. It is the portion of aircraft's

power that can be used to increase its potential and/or kinetic energy (specific excess power).

( )( ). . . .

A R

d P E K EP P V T D

dt

+− = − = => dividing both sides by weight =>

( ) 2. . . .

2

eA Rs

V T D dHP P d P E K E d V dh V dVP h

W W dt W dt dt g dt g dt

− − + = = = = = + = +

(5.75)


(5.74) (5.75)

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Ps Diagrams

• Fig. 5.42 is a typical Ps diagram for a multi-role fighter aircraft. Ps values are indicated by contour

lines (lines connecting points with equal values of Ps). Lines of constant energy height are also

plotted on the diagram (energy height = altitude, when velocity is zero).

• In essence, the Ps diagram functions as a 3-dimensional (with altitude as the third dimension) power-

available / power-required curve. Ps = 0 contour is the aircraft's operating envelope. For all

combinations of altitude and Mach number inside this envelope, the aircraft has sufficient thrust to

sustain level flight. Where Ps > 0, the aircraft can climb and/or accelerate.

• The aircraft's absolute ceiling would be the highest point on the Ps = 0 contour.

• Likewise, its service ceiling would be the highest point on its Ps = 100 ft/min (not ft/s given in fig.

5.42) contour.

• The aircraft's absolute maximum speed in level flight occurs at the altitude and Mach number where

Ps = 0 contour reaches farthest to the right.

Zoom Climb

• A zoom climb occurs when an aircraft climbs, so as to convert airspeed into altitude. If an aircraft is

flown to the edge of its operating envelope (T = D) and then forced to climb, it will move along a

constant energy height line on the Ps diagram.

• If the aircraft whose Ps diagram is shown in fig. 5.42 were at its absolute ceiling (h = 57,000 ft and V

= 630 kts) this would correspond to an energy height of 74,000 ft. If it entered a zoom climb from

this condition (and thrust maintained equal to drag), it would move along the He = 74,000 ft line,

decelerating until it reaches zero velocity at an altitude of 74,000 ft.

Minimum Time to Climb

• The maximum rate of climb at any given altitude is achieved at the speed where Ps is maximum.

However (because the aircraft can be zoomed at the end of its climb to get to a particular altitude

faster), the minimum time to climb is achieved by changing energy height (not just height) as fast as

possible.

• The aircraft increases energy height the fastest when it moves perpendicular to constanteH = lines

at the point where Ps is maximum on each line.

o Transonic regime: the constant-energy-height decent/acceleration moves the aircraft quickly

through the transonic regime to an altitude and supersonic speed, where Ps is maximum along

the higher constanteH = lines.

Maneuvering Ps

• Fig. 5.43 is a Ps diagram for the same aircraft as in fig. 5.42 but at a load factor n = 5. Note that Ps

has decreased everywhere on the diagram. This is because more induced drag and therefore more

power required from the five times greater lift required to generate a load factor n = 5.

Energy Height & Specific Excess Power (Continued)

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Solution (5.14)

Example C-4-2(Energy Height & Specific Excess Power)

For the aircraft, whose Ps diagrams are depicted in Figs. 5.42 and 5.43, determine the followings:1. What is the aircraft’s maximum 1-g level-flight speed and at what altitude does it occur?2. What is the aircraft’s maximum zoom altitude?3. What is the aircraft’s best rate of climb at sea level, and at what velocity does it occur?4. What is this aircraft’s minimum level-flight speed at sea level? What causes this limit?5. What is this aircraft’s maximum level-flight speed at sea level? What causes this limit?6. What is the maximum altitude at which this aircraft can sustain a 5-g turn, and at what speed does it

occur?7. What is the maximum speed at which this aircraft can sustain a 5-g turn, and at what altitude does it

occur?8. What is the minimum speed at which this aircraft can sustain a 5-g turn, at which altitude and airspeed

does it occur, and what causes this limit?

Example 5.14

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Aircraft Ps Comparisons

• Fig. 5.44 illustrates one of the most important uses of Ps diagrams: two different fighter aircraft (A &

B) are compared.

• Comparative Ps diagrams are useful to fighter pilots as they plan how to conduct an aerial battle

against an adversary aircraft of a particular type.

o Aircraft A would attempt to bring the fight to a lower altitudes and higher speeds.

o Aircraft B would attempt to keep the fight higher altitudes and lower speeds.

• Similar diagrams made for higher load factors are also used because most extended aerial fights

involve a great deal of turning.

Energy Maneuverability Diagrams

• Another very useful performance diagram combines the Ps and V-n diagrams, but plots them in such

a way that turn rate and turn radius can also be read from them (fig. 5.45). This is called energy

maneuverability (or simply maneuverability) diagram.

• The diagram is plotted for a fixed aircraft weight, configuration, and altitude (load factor is a

variable). The major axes of the diagram are airspeed (or Mach number) and turn rate. Contour

lines of constant load factor and turn radius are added, and then the aerodynamic limits and Ps = 0

curves are plotted.

• The diagram displays maximum instantaneous turn performance and sustainable (Ps = 0) turn

capability.

• In fig. 5.45:

o Absolute minimum turn radius is not achieved at corner velocity but at a lower speed, due to

a degradation in maxLC at speeds above critical Mach number. However, the corner velocity

V* is usually the choice for turning maneuver. For aircraft with corner velocities well below

their critical Mach number (max

constantLC = ), then:

o The maximum turn rate occurs at V* (corner velocity).

o The minimum turn radius also occurs at V* (corner velocity).

Ps Comparison & Maneuverability

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Solution (5.15)

Example C-4-3(Ps Comparison & Maneuverability)

What are the maximum instantaneous turn rate, minimum instantaneous turn radius, maximum sustained turn rate, and minimum sustained turn radius at h = 15,000 ft for the aircraft whose maneuverability diagram is depicted in Fig. 5.45?

Example 5.15

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Constraint Analysis

• There are so many interrelated variables to control and choices to make for an aircraft design.

• An analysis method called constraint analysis narrows down the choices to help designers focus on

the most promising concepts.

• Constraint analysis calculates ranges of values for an aircraft concept's takeoff wing loading

( )TOW S and takeoff thrust loading or takeoff thrust-to-weight ratio ( )SL TOT W , which will allow

the design to meet specific performance requirements.

• In many cases, constraint analysis will eliminate some aircraft concepts from further consideration.

In other instances, constraint analysis will identify two conflicting design requirements that no single

aircraft configuration can satisfy.

Constraint Analysis Master Equation

• The methodology of constraint analysis is based on a modification of the equation for specific excess

power:

( )s

V T D dh V dVP

W dt g dt

−= = + (5.76) =>

1 1T D dh dV

W W V dt g dt− = + (5.77)

• Substitute the following relations into eq(5.77):

o SLT T= , where (the thrust lapse ratio) depends on SL and M

o TOW W= , where is the weight fraction for a given constraint

o ( )0

2

1 2D D L LD C qS C k C k C qS= = + +

o ( ) ( ) ( )LC L qS nW qS= =

• This produces the master equation for constraint analysis:

0

2

SL TO TO1 2

TO TO

1 1D

T W WqS n n dh dVk k C

W W q S q S V dt g dt

= + + + +

(5.78)

Constraint Analysis

(5.76)

(5.77)(5.79)

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• For convenience, break the parasite drag coefficient into two parts: 0 xD DC C+

o 0DC : zero-lift drag of a clean aircraft

o xDC : extra drag of external stores ( 0

xDC = for a stealthy aircraft)

0

2

SL TO TO1 2

TO TO

1 1xD D

T W WqS n n dh dVk k C C


= + + + + +

(5.79)

• For each requirement (such as specified in Table 1.2), eq(5.79) is used to calculate ( )SL TOT W values

required to meet that requirement for a range of ( )TOW S values. When the results are plotted, the

line is called a constraint line because all values of ( )SL TOT W below the line are plotted on a single

set of axes, a constraint diagram (fig. 5.46) is formed. The portion of the constraint diagram that is

above all of the constraint lines is called the solution space because all combinations of ( )SL TOT W

and ( )TOW S within that portion of the diagram will satisfy all of the design requirements.

• Performing a constraint analysis allows an aircraft designer to make much more intelligent choices

about aircraft configuration. These choices involve choosing a design point, specific values of

( )SL TOT W and ( )TOW S from within the solution space that the aircraft concept will be designed to

achieve.

• In most cases some constraints will be more constraining than others. Those constraints that form

the boundaries of the solution space and drive the choice of the design point are called driving

constraints.

• Constraint analysis occasionally reveals two design requirements that conflict so completely with

each other that their constraint lines do not permit a solution space, or they only have a solution for

unreasonably high values of ( )SL TOT W . When this happens, it is time to negotiate which constraint

can be relaxed or what kind of compromise can be made to allow a solution.

Constraint Analysis (Continued)

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Takeoff Constraints

• For the takeoff constraint, the takeoff distance equation, eq(5.57), is rewritten in terms of ( )SL TOT W

and ( )TOW S using the very conservative and commonly true assumption that lift is approximately

zero prior to rotation:

( )max

TO

2

TOTO

TO 0.7

1.44

L V

Ws

SC g T D W L =

− − − (5.57) =>

( )TO

max

TO0.7 TO

TO TO

1.44V

L

T D W W

W C gs S

− − =

TO

max

0.7SL TO

TO TO TO

1.44 V

L

DT W

W C gs S W

= + +

=>

( )

0TO

max

20.7SL TO

TO TO TO

1.44 DV

L

C qT W

W C gs S W S

= + +

(5.80)

• The takeoff performance calculation is based on drag at TO0.7V . First of all, TOV is:

max max

TO TOTO

2 21.2 1.2

L L

W WV

SC S C = =

• The dynamic pressure at TO0.7V is:

( ) ( )( )max max

2

2 TO TOTO

2 0.70.5 0.7 0.5 0.7 1.2

L L

W Wq V

S C S C

= = =

• Therefore, eq(5.80) becomes:

0

max max

2

SL TO

TO TO

0.71.44 D

L L

CT W

W C gs S C

= + +

(5.81)

Landing Constraints

• The landing constraint is slightly different because SLT is not present in eq(5.59), and so the

constraint equation is written only in terms of ( )TOW S :

( )max

2

0.7

1.69

L

LL

L L V

Ws

SC g D W L =

+ − (5.59) =>

( )max 0.7TO

1.69

LL L L VL

L

s C g D W LWW

S S W

+ − = =

( )max 0

2

TO0.7TO

TO1.69

b bL

L L D L LV

s C g qS C kC C WW

S W

+ − +

= (5.82)

bLC : Lift coefficient maintained during braking

Takeoff & Landing Constraints

(5.57)

(5.80)

(5.81)

(5.59)

(5.83)

(5.84)

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• As with takeoff, the effect of drag on landing can be reasonably approximated by calculating it using

the dynamic pressure and drag polar at 0.7 LV :

max max

TO2 21.3 1.3L

L

L L

WWV

SC S C

= =

• The dynamic pressure at 0.7 LV is:

( ) ( )( )max max

2

2 TO TO2 0.830.5 0.7 0.5 0.7 1.3L

L L

W Wq V

S C S C

= = =

• Therefore, eq(5.82) becomes:

( )max 0

max

TO

TOTO

2

0.083

1.69

L L D

L

W Ss C g C

C S WW

S

+ =

( )max max 0TO

2

0.083

1.69

L L L Ds C g C CW

S

+= (5.83)

• With typical values of 0.5 = for braking on dry pavement and typical values of 0

0.05DC = unless

a deceleration parachute is used, aerodynamic drag in this condition is typically much less than the

deceleration force available from the wheel brakes, especially at low speeds. For this common

situation and conservative assumption, eq(5.83) simplifies to:

maxTO

21.69

L Ls C gW

S

= (5.84)

Takeoff & Landing Constraints (Continued)

of 14 Unit C-4

1. First, let's calculate the constraint of subsonic combat turn from the given information:

• Altitude: 5,000 fth =

=> Speed of Sound (5,000 ft): 1,096.9 ft sa = , Density: 30.002048 slug ft =

• Mach number: 0.9M =

=> Airspeed: ( )1,096.9 ft s 0.9 987.2 ft sV = = , Dynamic Pressure: 2 20.5 997.9 lb ftq V= =

• Thrust Lapse Ratio: SL 1.4T T = = , Weight Fraction:

TO 0.8W W = = (from manW of table 1.2)

• Load factor (9-g sustained turn): 9n =

• Aerodynamic Properties: 0

0.0243DC = , 1 0.121k = ,

2 0.0094k = − (from unit B-3)

Using the master equation for constraint analysis:

0

2

SL TO TO1 2

TO TO

1 1xD D

T W WqS n n dh dVk k C C


= + + + + +

( )

( )( )0

2

TO1 2

TO

DC n Wk kqn

W S q S

= + +

( )

( ) ( )( )

22

TO

2

TO

9 0.8 0.0094997.9 lb ft 0.0243 0.1219 0.8

1.4 1.4 997.9 lb ft 1.4

W

W S S

− = + +

=> Table 5.2

2. Second, let's calculate the constraint of supersonic combat turn from the given information:

• Altitude: 20,000 fth =

=> Speed of Sound (20,000 ft): 1,036.9 ft sa = , Density: 30.001267 slug ft =

• Mach number: 1.2M =

=> Airspeed: ( )1,036.9 ft s 1.2 1,244 ft sV = = , Dynamic Pressure: 2 20.5 980.8 lb ftq V= =

• Thrust Lapse Ratio: SL 0.98T T = = , Weight Fraction:

TO 0.8W W = = (from manW of table 1.2)

• Load factor (4-g sustained turn): 4n =


0.0412DC = , 1 0.169k = (from unit B-3)

Example C-4-4 (1)(Constraint Analysis Example)

Consider the multi-role fighter design requirements from Table 1.2 and the aerodynamic model for the F-16 developed in Chapter 4. Table 1.2 specifies the following performance requirements:1. Combat turn (maximum AB): 9.0-g sustained at 5,000 ft / M = 0.9.2. Combat turn (maximum AB): 4.0-g sustained at 20,000 ft / M = 1.2.3. Takeoff and breaking distance: ft.Calculate and plot the constraint lines for these requirements.

Constraint Analysis (Example)

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( )

( )( )0

2

SL TO1 2

TO TO

DC nT Wk kqn

W W S q S

= + +

( )

( )2

2

TO

2

TO

4 0.8980.8 lb ft 0.0412 0.196

0.98 0.98 980.8 lb ft

W

W S S

= +

=> Table 5.3

3. Third, let's calculate for takeoff constraint from the given information:

• Altitude: 0h = , max TO

1.2LC = (from unit B-3)

• Thrust Lapse Ratio: SL 1.105T T = = (estimated: with AB), Weight Fraction:

TO 1W W = =

• Rolling Friction Coefficient: 0.03 =


0.0519DC = (typical value is 0.05: takeoff configuration, landing gear

down + flap/flaperon full down)

0

max max

2

SL TO

TO TO

0.71.44 D

L L

CT W

W C gs S C

= + +

( )( )( )( )( )

( )TO

23

0.7 0.051.440.03

1.21.105 0.002377 slug ft 1.2 32.2 ft s 2,000 ft

W

S

= + +

=> Table 5.4

Lastly, we need to calculate the landing constraint, which we will plot on the constraint diagram as a

vertical line:

• Altitude: 0h = , max L

1.37LC = (from unit B-3)

• Thrust Lapse Ratio: SL 1T T = = , Weight Fraction:

TO 1W W = =

• Rolling Friction Coefficient: 0.5 =

( )( )( )( )( )max

3 2

2TO

2

2,000 ft 0.002377 slug ft 1.37 32.2 ft s 0.562 lb ft

1.69 1.69

L Ls C gW

S

= = =

Example C-4-4 (2)(Constraint Analysis Example)

takeoff & landing constraints c- 2: glides, climbs, range

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