Flight Mechanics: Flight Performance

Chapter 2: Flight Performance

Outcomes :

1. Description of airplane, frame of reference & equations of motion.

2. Performance in level flight. 3. Power in flight: thrust required & available. 4. Climbing performance & service ceiling. 5. Turn performance. 6. Fuel consumption. 7. Range & endurance. 8. Take-off & landing. 9. Load factor. 10. Flight envelope.

1

Description of airplane, frame of reference & equations of motion

• Airplane is considered as a rigid body on which is exerted four natural forces: lift, drag, thrust, and weight.

• Aerodynamic details will no longer be a concern but focus will be on the movement of the airplane due to the above forces.

2

• Horizontal stabilizer – elevator:

- Located at the rear of the fuselage. - To provide stability and control of up and down

(pitching) motion of the a/c. - On fighter a/c, stabilizer and elevator are

combined into one large moving surface called stabilator.

3

• Vertical stabilizer – rudder:

4

- The rudder works by changing the effective shape

of the airfoil of the vertical stabilizer. Changing the angle of deflection at the rear of an airfoil will change the amount of lift generated by the airfoil.

- With increased deflection, the lift will increase in the opposite direction. The rudder and vertical stabilizer are mounted so that they will produce forces from side to side, not up to down.

5

• Ailerons:

- The ailerons are used to bank the aircraft, i.e. to cause one wing tip to move up and the other wing tip to move down. The banking creates an unbalanced side force component of the large wing lift force which causes the aircraft's flight path to curve.

6

• Frame of reference: (1) Earth and (2) body references. • Earth reference system: - X-axis oriented South-North (N) - Z-axis oriented downward (i.e. Earth center) - Y-axis perpendicular to XZ plane (E)

7

1 2

• Body reference system:

- x-axis is longitudinal - z-axis is downward - y-axis is normal to the symmetry plane - yaw attitude (azimuth angle) ψ: a/c heading w.r.t.

North (+ve if clockwise) - pitch attitude θ: angle between the longitudinal

axis and horizontal plane - bank / roll attitude φ: angle between lateral a/c y-

axis and horizontal plane

8

• Consider an a/c in flight:

- There are four forces acting on the a/c: 1) Lift, L: perpendicular to the flight path direction. 2) Drag, D: parallel to the flight path. 3) Weight, W: vertically toward the center of earth. 4) Thrust, T: inclined at angle αT w.r.t. flight path.

9

• Applying the Newton’s law along the flight path:

• Applying the Newton’s law perpendicular to the flight path:

• The above equations are the equations of motion for an a/c in two-dimensional translational motion in accelerated flight.

10

dtdVmWDTF

dtdVmmaF

T =−−=

==

∑

∑

θα sincos

cT

c

rVmWTLF

rVmF

2

2

cossin =−+=

=

∑

∑

⊥

⊥

θα

Thrust Required for Level, Unaccelerated Flight

• The study of a/c performance under unaccelerated flight conditions is called static performance.

• It leads to calculations of maximum velocity, maximum rate of climb, maximum range, etc. which are vital parameters in a/c design and operation.

• For an a/c in steady level flight (i.e. flight path along horizontal) at a given altitude and velocity, the equations of motion of a/c can be written as

• The lift L and drag D are given as

11

WLDT

==

L

D

SCqLSCqD

∞

∞

==

• Hence, where, q∞ = dynamic pressure = S = wing area CD = drag coefficient for complete a/c CL = total lift coefficient

12

L

D

SCqWLSCqDT

∞

∞

====

221

∞∞Vρ

• The drag polar for the complete a/c is given by

where, CD, 0 = parasite drag coefficient at zero lift. = drag due to lift = CD, i . e = Oswald efficiency factor. AR = aspect ratio

13

eARCCC L

DD π

2

0, +=

eARCL

π

2

• Dividing the trust T with the weight W, yields:

• Thus, the thrust required (TR) by the a/c to fly at a given velocity in level, unaccelerated flight is

• Thrust required TR for a given a/c at a given altitude

varies with velocity V∞ and can be plotted in the thrust-required curve as shown:

14

L

D

CC

WT

=

DLW

CCWT

DLR ==

• To determine a point in the above curve, the following

steps can be used: 1. Choose a value of V∞ . 2. Calculate CL . 3. Calculate CD . 4. Form the ratio CL / CD . 5. Calculate the thrust required.

15

Thrust-required curve

• From the thrust required relation and curve, minimum

thrust required will be obtained when the a/c is flying at a velocity where L / D is maximum.

• L / D is a function of angle of attack, as shown below

• For most conventional subsonic a/c, (L / D)max is

reached usually on the order of 2o < α < 5o . • Hence, a relationship can be established between the

angle of attack and the thrust required.

16

L / D vs. angle of attack curve

• From the above figure, moving from right to left of the

thrust-required curve, α increases. • An interesting relation between CD,0 and CD,i can be

established from the thrust required study, where

at minimum TR .

17

Thrust required curve with associated angle of attack.

iDD CC ,0, =

Comparison of induced and parasite thrust required.

Example 1 (Home Assignment) Two types of a/c will be considered: a) A light single engine, propeller driven, private a/c

known as the Cessna Skylane (i.e. CP-1):

18

b) A jet-powered executive a/c known as the Cessna

Citation 3 (i.e. CJ-1): Calculate the TR curves at sea level for both the CP-1 and CJ-1 a/c.

19

Example 2 Consider an a/c patterned after the twin-engine Beechcraft Queen Air executive transport. The airplane weight is 38220 N, wing area is 27.3 m2, aspect ratio is 7.5, Oswald efficiency factor is 0.9 and parasite drag coefficient is 0.03. Calculate the thrust required to fly at a velocity of 350 km/h at a) standard sea level, and b) an altitude of 4.5 km.

20

Thrust Available and Maximum Velocity

• Thrust required TR is dictated by the aerodynamics and weight of the a/c.

• Thrust available TA is associated with the engine of the a/c: propulsive thrust provided by the a/c engine.

• Commonly used power plants for a/c are (a) reciprocating engine propeller and (b) turbojet engine.

21

TA curves for piston engine & turbojet engine.

• The maximum velocity Vmax that an a/c can flies at a

given altitude is obtained when the TR curve and the maximum TA curve intersect each others:

• Vmax is an important aspect of the a/c design process.

22

( )AR TTVV ==max

Maximum velocity point.

Example 3 Calculate the maximum velocity of the CJ-1 at sea level.

23

Power Required for Level, Unaccelerated Flight

• Power exerted on a moving object (i.e. block) at constant velocity V under the influence of constant force F as shown below

is given as

24

t1 t2 d

V V F F

FVtt

dFPower =

−

=×

=12time

distanceforce

• For an a/c in level, unaccelerated flight at a given

altitude and with velocity V∞. The corresponding power required PR is therefore

• The effect of the a/c aerodynamics on PR is provided by

25

∞= VTP RR

DLL

DR CCSC

CWP 2/33

23 12∝=

∞ρ

Power required curve.

• At the velocity for minimum power required, the a/c is

flying at the angle of attack which corresponds to a maximum CL

3/2 / CD. • Similarly, at minimum power required, the following

aerodynamic condition prevails:

26

iDD CC ,31

0, =

Comparison of induced, parasite and net power required.

• Point 1 in the previous figure corresponds to

minimum TR, where CD,0 = CD,i. • V∞ for minimum PR is less than V∞ for minimum TR. • The point on the PR curve that corresponds to the

minimum TR is easily obtained by drawing a line through the origin and tangent to the PR curve.

27

Tangent line to the power required curve.

Example 4 Calculate the power required curves for the a) CP-1 at sea level, and b) CJ-1 at an altitude of 6.5 km.

28

Power Available and Maximum Velocity

• Power required, PR characteristic of aerodynamic design and weight of the a/c.

• Power available, PA characteristic of the power plant of the a/c.

Reciprocating Engine-Propeller • Piston engine generates power by burning fuel in

cylinders to move pistons, which in turn deliver power to the rotating crankshaft.

29

Relation between shaft brake power and power available.

• The power deliver via the crankshaft to the propeller

is defined as the shaft brake power P. • Power available PA to drive the airplane is related to

the shaft brake power P via where, η is the propeller efficiency, η < 1. • All piston engines are rated in terms of horsepower

instead of watts (SI), hence 1 horsepower = 746 W

• It is also common to use shaft brake power bhp in place of P, and horsepower available hpA in place of PA. 30

PPA η=

( )bhphpA η=

• The power available curve for a typical piston engine

can be sketched as

Jet Engine • For jet engine, the thrust is generated by combusting

incoming stream of air and then exhausting this hot air at high velocities through a nozzle.

• The power available PA from jet engine is expressed as

31

∞= VTP AA

Power available curve for piston engine.

• The power available curve for a typical jet engine can

be sketched as

• The maximum flight velocity Vmax is determined by the

intersection of the maximum PA and the PR curves.

32

Power available curve for jet engine.

33

Determination of maximum flight velocity Vmax from the PA and PR curves.

Piston engine

Jet engine

Example 5 Calculate the maximum flight velocity for the CP-1 at sea level and CJ-1 at altitude of 6.5 km.

34

Altitude Effects on Power Required & Available

• Power required curve at varying altitudes can be obtained easily via these simple ratio relations:

• Geometrically, these equations allow us to plot a point on the PR curve at altitude from a given point on the sea level curve.

35

2/1

00,,

2/1

00

=

=

altRaltR

altalt

PP

VV

ρρ

ρρ

Corresponding points on sea level & altitude power required curve.

• Using this method, the complete PR curve at altitude

can be obtained from the sea level curve:

• With regard to PA, the lower air density at altitude

causes a reduction in power for both the reciprocating and jet engines.

• Hence, PA and TA are assumed to be proportional to ambient density:

36

Effect of altitude on power required.

00,

,

00,

, and,ρρ

ρρ alt

A

altAalt

A

altA

PP

TT

==

• Vmax varies with altitude, i.e. decreasing. • The minimum velocity of an airplane is dictated by the

stall velocity Vstall or the low speed intersection between the power required & power available curves.

37

Effect of altitude on power available.

• Hence, Vmin = Vstall , if Vstall > Vintersection. Vmin = Vintersection , if Vintersection > Vstall.

38

Minimum velocity at high altitude.

Example 6 Plot the power required curve for the CJ-1 airplane at 6.5 km. Then, using the method in this section, obtain the CJ-1 power required curve at sea level. Compare the maximum velocities at both altitudes.

39

Rate of Climb • When an airplane climbs, it is important to know how

fast and how long does it take to reach a certain altitude?

• Consider an airplane in steady, unaccelerated, climbing flight as shown below:

• Under this flight condition, the thrust T is not only working to overcome the drag, but also to support a component of weight.

40

A/c in climbing flight.

• Summing forces parallel to the flight path:

• Forces perpendicular to the flight path:

• The above equations are the equations of motion for

steady, unaccelerated, climbing flight. • Multiply the thrust equation by V∞ and rearranging:

where,

41

θsinWDT +=

θcosWL =

θsin∞∞∞ =− WVDVTV

availablepower=∞TVo20 if required,power <=∞ θDV

powerexcesssin =∞ θWV

• The vertical velocity of the airplane is called the rate

of climb R/C:

42

WDVTV

WVCR ∞∞ −===

power excesssin/ θ

Excess power for (a) propeller a/c, (b) jet a/c.

• The excess power is different at different values of V∞ . • There will be a point in which the excess power will

be maximum and at this point, the R/C will be maximum:

• The maximum R/C can be determined via graphical method by plotting R/C vs. V∞ :

43

WCR max

maxpower) (excess)/( =

Determination of maximum R/C.

• Another approach to determine the maximum R/C is

the hodograph diagram, which is a plot of the a/c vertical velocity Vv vs. horizontal velocity Vh.

horizontal tangent to the hodograph defines the

point of maximum R/C. line through the origin and intersecting the

hodograph defines the climb angle θ. tangent line to the hodograph and through the

origin defines the maximum climb angle θmax. 44

Determination of maximum R/C via hodograph.

Example 7 Calculate the rate of climb at sea level for (a) CP-1: At velocity 46 m/s. (b) CJ-1: At velocity 152 m/s.

45

Gliding Flight • Consider an a/c flying at steady, unaccelerated,

descending flight with no power, i.e. gliding flight:

• Summing forces along flight path:

• Summing forces perpendicular to flight path:

46

A/c in gliding flight

θsinWD =

θcosWL =

• The equilibrium glide angle is

Example 8 The maximum lift to drag ration for CP-1 is 13.6. Calculate the minimum glide angle and the maximum range measured along the ground covered by the CP-1 in a power-off glide that starts at an altitude of 3 km.

47

DL1

cossintan ==

θθθ

Absolute and Service Ceiling • The effects of altitude on PA and PR were discussed

earlier. • Similarly, altitude also has an effect on the excess

power:

i.e. altitude increases maximum R/C decreases. • Such effect is illustrated by

48

Effect of altitude on excess power.

• At altitude high enough, the PA curve becomes tangent

to the PR curve:

49

Altitude vs. maximum R/C.

PA curve tangent to PR curve.

• When this occurs: Excess power = 0 W Max. R/C = 0 m/s • The altitude at which the maximum R/C is zero is

defined as the absolute ceiling. • A more useful quantity is the service ceiling, defined

as the altitude where the maximum R/C = 100 ft/min (30.48 m/min).

• Hence, the service ceiling represents the practical upper limit of steady, level flight.

50

• The absolute and service ceilings can be determined

as follows: 1. calculate values of maximum R/C for a number of

different altitudes. 2. Plot altitude vs. maximum R/C. 3. Extrapolate the curve to 100 ft/min and 0 ft/min to

find the service and absolute ceilings, respectively.

51

Example 9 Calculate the absolute and service ceilings for CP-1 & CJ-1 using the steps explained beforehand.

52

Time to Climb • The time for an a/c to climb to a given altitude is a n

important design consideration: fighter a/c: to engage the advancing enemy a/c. commercial a/c: to minimize discomfort, risks & air traffic. • R/C is defined as time rate of change altitude, or

• Rearranging in terms of time differential dt and integrating from one altitude h1 to another altitude h2:

53

dtdhCR =/

∫=2

1 /h

h CRdht

• The time to climb t, is usually considered from sea

level, where h1 = 0, hence:

• The above result shows that if curve of (R/C)-1 vs. h is plotted, the time to climb is equal to the area under the curve from h = 0 to h = h2.

54

∫=2

0 /h

CRdht

Area under the curve to determine time to climb.

Range and Endurance

• Range : the total distance (measured w.r.t. the ground) traversed by the a/c on a tank of fuel.

• Endurance : the total time that an a/c stays in the air on a tank of fuel.

• The parameters that maximize range are different from those which maximize endurance.

• They are also different for propeller- and jet-powered a/c.

55

Propeller-powered airplane : • One of the critical factors influencing range and

endurance is the specific fuel consumption (SFC). • SFC : weight of fuel consumed per unit power per unit

time. • Maximum endurance for a propeller-driven a/c occurs

when the a/c is flying at minimum power required. • Maximum range for a propeller-driven a/c occurs when the

a/c is flying at minimum thrust required.

56

Conditions of maximum range and endurance.

• Mathematical formulation: Assume: c = SFC (N/W/s) P = engine power (W) dt = small increment in time (s) Hence, cPdt = differential change in the weight due to fuel consumption over the period dt (N) cPdt = - dW If, W0 = gross weight of a/c (empty weight + full fuel

+ payload) W1 = weight of a/c without fuel

57

Integrating between time t = 0, where W = W0, and

time t = E, where W = W1, yields : • In the above equation, E is the endurance in seconds. • To obtain an analogous expression for range R, the

relation for endurance is multiplied with V∞ :

58

∫−=1

0

W

W cPdWE

∫ ∞∞ −== 1

0

W

W cPdWVREV

Area under the curve to determine endurance & range.

• A simpler but approximate analytic expressions for R

and E are provided by Breguet formulas : Assumptions, level unaccelerated flight PR = D V∞ to maintain steady flight PA = PR = D V∞ P = shaft brake power of propeller engine PA = η P hence, • Substituting the above result into the relation for

range yields :

59

ηη∞==

DVPP A

• Assuming that η, L / D and c are constant throughout

the flight. The above relation can be integrated to yield

• Similar formula can be obtained for endurance by

applying the same approximations used to get the range relation. Hence,

• Since L = W = , then

60

∫−=1

0

W

W WdW

DL

cR η

1

0lnWW

CC

cR

D

Lη=

∫∞

−= 1

0

W

W WdW

DVL

cE η

LSCV 221

∞∞ρ LSCWV ∞∞ = ρ/2

• Thus,

• Similarly, assuming that CL, CD, η, c, and ρ∞ are all constant. Integrating the above equation yields :

• From the Breguet relations, it is observed that endurance depends on altitude, whereas range is independent of altitude.

61

∫ ∞−= 1

02/32

W

WL

D

L

WdWSC

CC

cE ρη

( ) ( )2/10

2/11

2/12/3

2 −−∞ −= WWS

CC

cE

D

L ρη

Example 10 Estimate the maximum range and maximum endurance for the CP-1. Given, (CL / CD)max = 13.62, (CL

3/2 / CD)max = 12.81, weight of aviation gasoline = 25.1 N/gallon, specific fuel consumption = 0.00268 N/W/hr, and fuel tank capacity = 65 gallons.

62

Jet-powered airplane : • For jet a/c, the specific fuel consumption (SFC) is

defined as the weight of fuel consumed per unit thrust per unit time.

• It differs from the propeller a/c where thrust is used for the jet a/c instead of brake power.

• The maximum endurance for jet a/c occurs when the a/c is flying at minimum thrust required.

• The maximum range for jet a/c occurs when the a/c is flying at a velocity such that CL

½ / CD is maximum.

63

Points of maximum range and endurance.

• Mathematical formulations : Assumptions, ct = thrust specific fuel consumption (N / N /s) dt = small increment in time (s) TA = thrust available of a/c (N) Hence, ct TA dt = elemental change in weight of the a/c due to fuel consumption ct TA dt = - dW If, W0 = gross weight of a/c (full fuel + payload) W1 = weight of a/c without fuel 64

• Integrating between time t = 0, where W = W0, and t = E, where W = W1 :

• Applying the approximate relations under Breguet : level unaccelerated flight TR = D L = W to maintain steady flight TA = TR = D • Assuming constant ct and CL / CD = L / D : 65

∫−=1

0

W

WAtTc

dWE

∫−=1

0

1W

Wt W

dWDL

cE

1

0ln1WW

CC

cE

D

L

t

=

• The corresponding relation for range can be obtained by multiplying the endurance relation with velocity :

• Substituting the approximate relations by Breguet and integrating, yields :

66

∫ ∞∞ −== 1

0

W

WAtTc

dWVEVR

( )2/11

2/10

2/118 WWCC

cSR

D

L

t

−=∞ρ

Example 11 Calculate the maximum range and endurance for the CJ-1 with the following information : fuel tank capacity = 1119 gallons of kerosene weight of aviation kerosene = 29.7 N/gallon, thrust specific fuel consumption = 0.6 N of fuel/(N of fuel)/hr, and cruising altitude = 7000 m. (CL / CD)max = 16.9 (CL

1/2 / CD)max = 23.4

67

Relations between CD,0 & CD,i

• Various aspects of the performance of a/c depend on the aerodynamic ratios (CL

/ CD), (CL ½ / CD), and

(CL 3/2 / CD).

• Where, - for , - for , - for ,

68

max

D

LC

CiDD CC ,0, =

max

2/1

D

LC

CiDD CC ,0, 3=

max

2/3

D

LC

CiDD CC ,3

10, =

• The maximum value for the various aerodynamics ratios can be calculated as follows :

- - -

69

( )0,

2/10,

max 2 D

D

D

L

CeARC

CC π

=

( )0,3

4

4/10,3

1

max

2/1

D

D

D

L

CeARC

CC π

=

( )0,

4/30,

max

2/3

43

D

D

D

L

CeARC

CC π

=

Calculate the aerodynamics ratio for the following airplanes: - For CP-1: - For CJ-1:

70

max

D

LC

C

max

2/1

D

LC

C

max

2/3

D

LC

C

max

D

LC

C

Takeoff Performance • Past discussions of a/c performance assumed that

acceleration is zero. • For the remainder of this chapter, several aspects of

a/c performance that involve finite acceleration, i.e. takeoff, landing, and turning are considered.

• An important parameter related to takeoff flight is the running length along the ground by a/c to lift from ground.

• This length is defined as the ground roll or lift-off distance, SLO.

• Consider the following body :

71

• From Newton’s second law :

• Integrating the above relation for time t : • The incremental distance covered during an

incremental time dt is

• Integrating the above relation yields : • Substituting the result for time t :

72

dtdVmmaF ==

FVmt =

tdtmFVdtds ==

2

2tmFs =

FmVs

2

2

=

• This equation gives the distance required for a body of mass m to accelerate to velocity V under the action of a constant force F.

• For an a/c during its ground roll, the F.B.D. is given as

• Summing forces parallel to the ground: • Summing forces normal to the ground:

73

dtdVmNDTRDTFF r =−−=−−== µ//

0=+−=⊥ NWLF

• Combining these two relations yields:

• From the above relation: W is constant T is reasonably constant (for jet engine) μr is coefficient of rolling friction, from 0.02 to 0.1 L & D varies with velocity: where, φ accounts for the reduced drag in the

presence of ground effect. 74

dtdVmLWDTF r =−−−= )(µ

+=

=

∞∞

∞∞

eARCCSVD

SCVL

LD

L

πφρ

ρ2

0,2

21

221

• An approximate relation for determining φ is given by McCormick:

where, h = height of wing above ground, and b = wingspan.

• Hence, the variation of these forces with distance along the ground during takeoff can be sketched as follows:

75

( )( )2

2

/161/16

bhbh

+=φ

• A simple but approximate expression for the lift-off distance SLO can be obtained as follows:

assume T is constant assume average value for drag & resistance force summation: • Due to these assumptions:

• This result is fairly reasonable as shown by the difference between the thrust curve and the dashed line in the Force vs. SLO figure.

• Using the above result, the lift-off distance SLO can be determined from this relation:

76

[ ]aver LWD )( −+ µ

[ ] constLWDTF aver =−+−= )(µ

FmVs

2

2

=

by replacing, s = SLO, V = VLO = lift-off velocity, m = W/g, and Which yields, • To ensure margin of safety during takeoff,

• Substituting this relation into SLO relation yields:

77

[ ]aver LWDTF )( −+−= µ

( )( )( )[ ]{ }aver

LOLO LWDT

gWVs−+−

=µ2

/2

( )max,

22.12.1%20L

stallstallLO SCWVVV

∞

==>=ρ

( )[ ]{ }averLLO LWDTSCg

Ws−+−

=∞ µρ max,

244.1

• Shevell suggests that the lift L and drag D forces appearing in the SLO relation is to be calculated at a velocity

• Further simplification: Assume T >> D and T >> R during takeoff Ignore D and R in the SLO relation Hence, • Physical interpretations: 1. SLO α W2 if W doubled, then SLO is quadrupled. 2. SLO α 1 / ρ∞ 2 (because T α ρ∞)

78

LOVV 7.0=∞

TSCgWsL

LOmax,

244.1

∞

=ρ

3. Increase wing area S, CL, max, or T then SLO will decrease and vice-versa. • The total takeoff distance as defined by FAR is the

sum of SLO and the distance (along the ground) to clear a 35 ft (civilian jet transports a/c) or 50 ft (other a/c) height obstacle.

79

Example 12 Estimate the lift-off distance for the CJ-1 at sea level. Assume paved runway (μr = 0.02). Also, during the ground roll, the angle of attack of the airplane is restricted by the requirement that the tail not drag the ground, and therefore assume that CL,max during ground roll is limited to 1.0. Also, when the airplane is on the ground, the wings are 1.83 m above the ground.

80

Landing Performance • When an a/c is landing and has touched the ground,

the F.B.D. during the ground roll is exactly the same as takeoff.

• However, to minimize the distance required to come to a complete stop. T is assumed to be zero, i.e. T = 0.

• The resulting equation of motion is

• Variations of the forces of a/c during landing is given as

81

( )dtdVmLWDF r =−−−= µ

where, SL = ground roll distance between touchdown at velocity VT & complete stop • From Newton’s 2nd law of motion, the ground roll

distance for landing is given by

82 FmVSL 2

2

−=

• To develop an approximate expression for SL, the following assumptions are made:

V = VT F = constant deceleration force given by the a/c’s equation of motion Lift L and drag D are evaluated at V∞ = 0.7 VT m = W / g • Upon substitution of the above assumptions, SL

becomes

83

( )( )[ ]aver

TL LWD

gWVS−+

=µ2

/2

( )[ ]aver LWDF −+−= µ

• For safety reason:

• Finally,

• During landing, brakes are applied which leads rolling friction coefficient μr = 0.4 for paved surface.

• For modern jet transports, thrust reversal TRV is utilized during landing. Hence, SL is modified according to

84

( )[ ]averLL LWDSCg

WS−+

=∞ µρ max,

269.1

max,

23.13.1L

stallT SCWVV

∞

==ρ

( )[ ]{ }averRVLL LWDTSCg

WS−++

=∞ µρ max,

269.1

• Another method to decrease SL is to destroy the lift generated by the wing via spoilers.

• The total landing distance as defined by FAR is equal to the sum of the ground roll distance SL plus the distance to achieve touchdown in a glide condition from a 50 ft height.

85

Example 13 Estimate the landing ground roll distance at sea level for the CJ-1. No thrust reversal is used; however, spoilers are employed such that L = 0. The spoilers increase the parasite drag coefficient by 10%. The fuel tanks are essentially empty, so neglect the weight of fuel carried by the a/c. The maximum lift coefficient, with flaps fully employed at touchdown is 2.5.

86

Turning Performance & V-n Diagram

• So far, a/c performance considerations are limited to rectilinear (translational) motion only.

• Flight cases which involve curved (curvilinear) flight path are

i) level turn ii) pull up iii) pull down

87

Level Turn • A typical a/c in level turn condition is shown below

• The wings are banked through angle φ, causing the lift to incline at angle φ to vertical. 88

• From the force diagram: L cos φ = W Resultant force, Fr = (L2 - W2)1/2 • The resultant force is perpendicular to the flight path

and cause the a/c to turn in a circular path with a radius R.

• Introducing a new term, i.e. load factor n:

• Load factor is usually quoted in terms of “g”. • Rewriting the resultant force Fr, yields

89

WLn =

12 −= nWFr

• The radial acceleration for the a/c moving at V∞ is

• From Newton’s 2nd law:

• Thus, the radius of curvature R can be expressed as

• The angular velocity or the turn rate of the a/c is given as

• For the maneuvering performance of a/c, it is desirable to have the smallest R and the largest ω as possible.

90

RV 2∞

RV

gW

RVmFr

22∞∞ ==

12

2

−= ∞

ngVR

∞

∞ −===

Vng

RV

dtd 12θω

Pull Up • An a/c in pull up maneuver is shown as follows

• The flight path is curved in the vertical plane, with a turn rate ω = dθ / dt.

• From the force diagram, the resultant force Fr is 91

)1( −=−= nWWLFr

• From Newton’s 2nd law:

• Combining both yields

• The turning rate or angular velocity is

92

RV

gW

RVmFr

22∞∞ ==

)1(

2

−= ∞

ngVR

∞

∞ −===

Vng

RV

dtd )1(θω

Pull Down • An a/c in pull down maneuver is shown as follows

• Similarly, the flight path is curved in the vertical plane,

with a turn rate ω = dθ / dt. • Using similar analysis as pull up maneuver, the

following results are obtained: and

93

)1(

2

+= ∞

ngVR

∞

∞ +===

Vng

RV

dtd )1(θω

V-n Diagram • High performance fighter a/c is designed to operate at

high load factors, typically from 3 to 10. • Reason lower R and higher ω. • If n is large, then n+1 ≈ n and n-1 ≈ n, hence

• Substituting the following relations to the above relations

94

gnVR

2∞=

∞

=Vgnω

WLnand

SCLV

L

==∞

∞ ρ22

yields where the factor W / S is known as wing loading. • The above results show the following conditions for

getting smaller turn radii and larger turn rate: small wing loading CL = CL,max n = nmax • At low speeds, nmax is a function of CL,max because

95

)/(22

SWnCgand

SW

gCR L

L

∞

∞

=

=

ρωρ

( )SWC

Vn L

/max,2

21

max ∞∞= ρ

• At high speeds, nmax is limited by the structural design of the a/c, i.e. V-n diagram.

point 1: CL < CL,max, hence n < nmax point 2: CL = CL,max, hence, n = nmax point 3: unattainable because of stall V* = corner velocity

96

For V∞ < V*: nmax is limited by the CL,max for V∞ > V*: nmax is limited by structural limit, i.e. positive & negative load factor of a/c. CD: high speed limit point B: known as maneuver point where both CL & n are at maximum value for a given a/c. • The corner velocity is given by

97

SW

CnV

L max,

max2*∞

=ρ