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C H A P T E R

21

Performance e

Descent

O U T L I N E

21.1 Introduction 925

21.1.1 The Content of this Chapter 926

21.2 Fundamental Relations for the DescentManeuver 926

21.2.1 General Two-dimensional Free-body

Diagram for an Aircraft 92621.2.2 Planar Equations of Motion (Assumes No

Rotation about Y-axis) 927

21.3 General Descent Analysis Methods 927

21.3.1 General Angle-of-descent 927Derivation of Equation (21-9) 927

21.3.2 General Rate-of-descent 927Derivation of Equation (21-10) 928

21.3.3 Equilibrium Glide Speed 929

Derivation of Equation (21-11) 930

21.3.4 Sink Rate 930Derivation of Equations (21-12) and (21-13) 930

21.3.5 Airspeed of Minimum Sink Rate, VBA 931Derivation of Equation (21-14) 931

21.3.6 Minimum Angle-of-descent 931Derivation of Equation (21-15) 931

21.3.7 Best Glide Speed, VBG 931Derivation of Equation (21-16) 932

21.3.8 Glide Distance 932Derivation of Equation (21-17) 933

Variables 933

References 934

21.1 INTRODUCTION

For powered aircraft, the analysis of gliding flight isessential from the standpoint of safety. For unpoweredflight, glide performance is what sets one sailplaneapart from another. Analysis of descent provides avery important insight into how efficient an airplaneis. It can even expose possible handling problems. Forinstance, a powered airplane with a high glide ratio(L/D ratio) will find this feature very favorable if it

experiences engine failure and must rely on this prop-erty to get to the nearest emergency airport. However,if the L/D ratio is very high, this will actually make itharder to land as it would have to fly at a very shallowapproach path angle to keep the airspeed low. Theshallow angle would not only make it more difficult toassess where it will touch-down, but would also tendto make the airplane float once it enters ground effect,compounding the difficulty. Figure 21-1 shows an

organizational map displaying the descent among othermembers of the performance theory.

This section will present the formulation of and thesolution of the equation of motion for the descent,and present practical, as well as numerical solutionmethodologies, that can be used both for propellerand jet-powered aircraft. When appropriate, eachmethod will be accompanied by an illustrative exampleusing the sample aircraft. The primary information wewant to extract from this analysis is characteristics like

minimum rate-of-descent (ROD), best (lowest) angle-of-descent (AOD), the corresponding airspeeds, theAOD for a given power setting, and unpowered glidedistance.

In general, the methods presented here are the“industry standard” and mirror those presented by avariety of authors, e.g. Perkins and Hage [1], Torenbeek[2], Nicolai [3], Roskam [4], Hale [5], Anderson [6] andmany, many others. Also note that sailplane design

925General Aviation Aircraft Design Copyright 2014 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/B978-0-12-397308-5.00021-0

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and glide analyses methods are detailed in AppendixC4, Design of Sailplanes.

21.1.1 The Content of this Chapter

• Section 21.2 develops fundamental relationshipsnecessary to evaluate the characteristics of glidingflight, most importantly the equations of motion fordescending flight.

• Section 21.3 presents an assortment of methods topredict the various descent characteristics of anairplane.

21.2 FUNDAMENTAL RELATIONS FORTHE DESCENT MANEUVER

In this section, we will derive the equation of motionfor the descent maneuver, as well as all fundamental re-lationships used to evaluate its most important charac-teristics. First, a general two-dimensional free-body

diagram will be presented to allow the formulation to be developed. Only the two-dimensional version of theequation will be determined as this is sufficient for all as-pects of aircraft design.

21.2.1 General Two-dimensional Free-bodyDiagram for an Aircraft

Thefree body forthe descending flight is developed in asimilar manner to that for the climbing flight. Figure 21-2shows a free-body diagram of an airplane moving alonga flight path. The x- and z-axes are attached to the centerof gravity (CG) of the airplane, just as for the climb andhave identical orientation with respect to the datum of the

airplane. The angle between the datum and the tangentto the flight path is the angle-of-attack and the force (orthrust) generated by the airplane’s power plant, T , may beat some angle ε with respect to the x-axis. The figure showsthatthis coordinate systemcan change its orientation withrespect to the horizon depending on the maneuver beingperformed. Then, the angle between the horizon and thex-axis, the descent angle, is denoted byq, where these threesimple rules apply: If q > 0, then the aircraft is said to beclimbing. If q¼ 0 then the aircraft issaid to be flying straightand level (cruising). If q < 0, then the aircraft is said to bedescending. This chapter only considers the final rule.

The free-body diagram of Figure 21-2 is balanced in

terms of inertia, mechanical, and aerodynamic forces.The lift is the component of the resultant aerodynamicforce generated by the aircraft that is perpendicular tothe flight path (along itsz-axis). The drag is the component

FIGURE 21-2 A two-dimensional free- body of the airplane in a powered glidingflight.

FIGURE 21-1 An organizational map placing performance theory among the disciplines of dynamics of flight, and highlighting the focus of this section: descent performance analysis.

21. PERFORMANCE e DESCENT926

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of the aerodynamic force that is parallel (along its x-axis).These are balanced by the weight, W , and the correspond-ing components of T . The presentation of Figure 21-2 cannow be used to derive the planar equations of motion forthe descent maneuver, which are sufficient to accuratelypredict the vast majority of descent maneuvers.

21.2.2 Planar Equations of Motion (AssumesNo Rotation about Y -axis)

The equations of motion for gliding flight can bederived using the free-body diagram in Figure 21-2:

L W cosqþ T sinε ¼ W g

dV Zdt

(21-1)

D þ W sinqþ T cosε ¼ W g

dV X dt

(21-2)

The equations of motion can be adapted for descending

flight by making the following assumptions:(1) Steady motion implies dV /dt ¼ 0.(2) The descent angle, q, is a non-zero quantity.(3) The angle-of-attack, a, is small.(4) The thrust angle, ε, is 0.

Equations of motion for a steady unpowered (T ¼ 0)descent:

L W cosq ¼ 0 0 L ¼ W cosq (21-3)

D W sinq ¼ 0 0 D ¼ W sinq (21-4)

Equations of motion for a steady powered (T > 0)descent:

L W cosq ¼ 0 0 L ¼ W cosq (21-5)

D þ W sinqþ T ¼ 0 0 D ¼ T þ W sinq (21-6)

Vertical airspeed:

V V ¼ V sinq (21-7)

AOD is also known as angle-of-glide (AOG) or glide angle.

21.3 GENERAL DESCENT ANALYSISMETHODS

21.3.1 General Angle-of-descent

The angle-of-descent is the flight path angle to thehorizontal and is computed from:

Unpowered descent:

tanq ¼ DL ¼ 1

L=Dz

D

W (21-8)

Powered descent:

sinq ¼ DW T

W z

1

L=D T

W (21-9)

The right approximations (z) are valid for low descentangles, q, and when the CG is not too far forward, as thiscan put a high load on the stabilizing surface andinvalidate the approximation L zW . Many airplanes, in

particular sailplanes, have such high glide ratios thatlanding becomes difficult. For this reason, they areequipped with speed brakes or spoilers, which arepanels that deflect from the wing surface and cause flowseparation, increasing drag and reducing lift. The sameholds for high-speed jets.

Derivation of Equation (21-9)

We get Equation (21-8) by dividing Equation (21-4) by (21-3):

D

L ¼ W sinq

W cosq ¼ tanq

We get Equation (21-9) from Equation (21-6):

D ¼ T þ W sinq 5 sinq ¼ D T W

¼ DW T

W

QED

21.3.2 General Rate-of-descent

The rate at which an aircraft reduces altitude is given below:

V V ¼ DV W ¼ V ðCL=CDÞ (21-10)

The above expression has units of ft/s or m/s. Generally,the units preferred by pilots are in terms of feet per min-ute or fpm for general aviation, commercial aviation,and military, but m/s for sailplanes and some nationsthat use the metric system. To convert Equation (21-10)into units of fpm multiply by 60. FIGURE 21-3 Airspeed components during climb.

21.3 GENERAL DESCENT ANALYSIS METHODS 927

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Begin by multiplying Equation (21-4) by V , and thenrewrite V sin q using Equation (21-7):

D ¼ W sinq and V V ¼ V sinq 0 DV ¼ WV sinq

¼ WV V 5 V V ¼ DV W

QED

E X A M P L E 2 1 - 1

Plot the rate-of-descent for the Learjet 45XR at S-L,

15,000 ft, and 30,000 ft at a weight of 20,000 lb f (assuming

no thrust). Plot the descent rate as a function of true

airspeed in knots (KTAS).

Solution

Sample calculation for the sample aircraft gliding at 175

KCAS (LDmax) at S-L and at 20,000 lbf (no thrust). Note

that LDmax is calculated in EXAMPLE 19-5.

D ¼ W LDmax

¼ 20;00015:45

¼ 1294 lbf

At S-L the density is 0.002378 slugs/ft3. Therefore:

V V ¼ DV

W ¼ ð1294Þð175$1:688Þ

20000 ¼ 19:1 ft=s

This amounts to 1147 fpm. The rate-of-descent for other

airspeeds is plotted in Figure 21-4. Figure 21-5 shows the

corresponding glide angle and L/D for the aircraft at

S-L. Figure 21-6 shows how important performance char-

acteristics, such as the airspeed for minimum power

required and best glide ratio, can be extracted from the

rate-of-descent plot.

FIGURE 21-4 A flight polar, also known as ROD vs airspeed graph.


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21.3.3 Equilibrium Glide Speed

Equilibrium glide speed is the airspeed that must bemaintained to achieve a specific glide angle for a specific

AOA. One common use is to determine the airspeedrequired to maintain a specific flight path angle, q.

V ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cosq

rCL

W

S

s (21-11)

The lift coefficient can be determined based on the AOArequired for the airspeed using CL ¼ CLo þ CLa$a

E X A M P L E 2 1 - 1 (cont’d)

FIGURE 21-5 ROD, L/D and glide angle superimposed on the same graph (at S-L).

FIGURE 21-6 Important characteristics extracted from the flight polar (at S-L).


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From Equation (21-3) we get: L ¼ W cosq 5 1

2rV 2SCL ¼ W cosq 5 V

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cosq

rCL

W

S

s

QED

21.3.4 Sink Rate

Sink rate is the rate at which an aircraft loses altitude.This is most commonly expressed in terms of feet perminute or meters per second. If the lift and drag coeffi-

cients can be determined for a specific glide condition(e.g. from knowing the AOA), the sink rate can becomputed from:

Straight and level sink:

V V ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

2

r

C3L=C2D

W S

s ¼ CD

C3=2L

ffiffiffiffiffiffiffiffiffi2

r

W

S

s (21-12)

The above expression will return the sink rate in terms of ft/s or m/s. To convert to fpm multiply by 60. If theairplane is turning and the bank angle is given by the bankangle 4, then the sink rate increases and amounts to:

Sink rate while banking at 4:

V V ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi

2

rðCLcosfÞ3=C2D

W S

v uut ¼ CDC3=2L cos

3=2f

ffiffiffiffiffiffiffiffiffi2

r

W

S

s

(21-13)

Derivation of Equations (21-12) and (21-13)

Substitute Equation (21-11) into (21-7) to get:

V V ¼ V sinq ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cosq

rCL

W

S

s sinq (i)

Divide Equation (21-4) by (21-3) to get:

sinq

cosq ¼ D

L 5 sinq ¼ D

L cosq ¼ CD

CLcosq (ii)

Substitute Equation (ii) into (i) to get:

V V ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cosq

rCL

W

S

s sinq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cosq

rCL

W

S

s CDCL

cosq

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi

2cos3q

r

C3L=C2D

W S

s

If we assume cos q w 1 we get Equation (21-12).

E X A M P L E 2 1 - 2

During flight testing, the pilot of an SR22 wants to

maintain a 3 glide path angle at an AOA of 5.What airspeed must be maintained? Assume a test

weight of 3250 lbf , ISA at S-L conditions, CLo ¼ 0.4 andCLa ¼ 5.5 /rad.

SolutionLift coefficient:

CL ¼ CL0 þ CLa,a ¼ 0:4þ 5:55 p180

¼ 0:8800

Knowing that the wing area is 144.9 ft2, we can now

compute the airspeed necessary to maintain the said glide

path angle:


rCL

W

S

s

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2cosð5Þð0:002378Þð0:8800Þ

3250

144:9

s ¼ 146 ft=s ¼ 86:6 KTAS


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To get Equation (21-13) we refer to Figure 19-26and see that W ¼ L$cos f ¼ qS$CL$cos f. When level(f ¼ 0) the same relationship is W ¼ L ¼ qS$CL. Thisshows that the lift really depends on the product CL$cos

f. Therefore, it is more accurate to replace the lift coeffi-cient in the above formulation with the product.

QED

21.3.5 Airspeed of Minimum Sink Rate, V BA

Just like the rate-of-climb, the magnitude of the sinkrate varies with airspeed. This implies it has a mini-mum value that would be of interest to the operatorof the vehicle as the kinetic energy of the vertical speedis then also at a minimum and, thus, may have animpact on survivability in an unpowered glide (asimpact energy is a function of the square of the speed).It turns out that if the simplified drag model applies, theminimum sink speed can be calculated directly as fol-lows. Note that this expression only holds in air massthat is neither rising nor sinking (see Appendix C4,Design of Sailplanes).

V BA ¼ V Emax ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

r

W

S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik

3$CDmin

s v uut (21-14)


Inspection of Equation (21-12) reveals that whenCL1.5/CD is maximum, V V is minimum. The airspeed at

which this takes place has already been derived asEquation (19-14).

QED

21.3.6 Minimum Angle-of-descent

This angle results in a maximum glide distance from agiven altitude and is of great importance to both gliderpilots and pilots of powered aircraft.

tanqmin ¼ 1

LDmax ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi4$k $CDminp (21-15)Note that the drag model yields a qmin which is indepen-dent of altitude.


Equation (19-18) gives the maximum lift-to-drag ra-tio for the simplified drag model (repeated below forconvenience) and is inserted into Equation (21-8):

LDmax ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi4$CDmin$k p QED

21.3.7 Best Glide Speed, V BG

The best glide speed is the airspeed at which theairplane will achieve maximum range in glide. It is amatter of life and death for the occupants of an aircraft,as is evident from its inclusion in 14 CFR 23.1587(c)(6),Performance Information. A part of pilot training requires

this airspeed to be remembered in case of an engine fail-ure. It can be calculated using Equation (21-16) below.Note that this expression only holds in air mass that isneither rising nor sinking (see Appendix C4, Design of Sailplanes).

V BG ¼ V LDmax ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi2

r

ffiffiffiffiffiffiffiffiffiffiffiffiffik

CDmin

s W

S

v uut (21-16)

E X A M P L E 2 1 - 3

Determine the minimum angle-of-descent for the

sample aircraft flying at S-L at a weight of 20,000 lbf .Solution

tanqmin ¼ 1LDmax

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4$k $CDmin

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4$0:05236$0:020

p

¼ 0:0647 rad

This amounts to 3.7.


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Using Equation (21-11) and the assumption that at the best glide angle cos q z 1, we get:


rCL

W

S

s 0 V ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

rCL

W

S

s

It was demonstrated in the derivation for Equation(19-18) that at LDmax the lift coefficient CL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCDmin=k

p .

Inserting this into the above expression and manipulatingleads to:

V ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

rCL

W

S

s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi2

r ffiffiffiffiffiffiffiffiffiCDmink q W

S

v uut

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi2

r


CDmin

s W

S

v uut

QED

E X A M P L E 2 1 - 4

Determine the airspeed of minimum angle-of-descent

for the sample aircraft flying at 30,000 ft and S-L at a

weight of 20,000 lbf .

Solution

At 30,000 ft the density is 0.0008897 slugs/ft3.

Therefore:

V LDmax ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

r


CDmin

s W

S

v uut

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

0:0008897

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:05236

0:020

r 20000

311:6

s

¼ 483:2 ft=s ð286 KTASÞ

At S-L the density is 0.002378 slugs/ft3. Therefore:

V LDmax ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

r


CDmin

s W

S

v uut

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

0:002378

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:052360:020

r 20000

311:6

s ¼ 295:9 ft=s ð175 KTASÞ

This amounts to 175 KCAS in both cases.

21.3.8 Glide DistanceFor a powered airplane, knowing how far one can

glide in case of an emergency is not just a matter of safety, but of survivability. Such information is required infor-mation for the operation of GA aircraft by 14 CFR

Part 23, x23.1587(d)(10), Performance Information, andmust be determined per x23.71, Glide: Single-engine

Airplanes (see below) and presented to the operator of the aircraft. This is typically done in the form of a glidechart, which shows clearly how far the airplane willglide for every 1000 ft lost in altitude.

During the design phase, this distance can be calcu-lated from the following expression (which is shownschematically in Figure 21-7). Note that this expressiononly holds in air mass that is neither rising nor sinking(see Appendix C4, Design of Sailplanes).

R glide ¼ h$

L

D

¼ h$

CLCD

(21-17)

x 2 3 . 7 1 G L I D E : S I N G L E - E N G I N E A I R P L A N E SThe maximum horizontal distance traveled in still air,

in nautical miles, per 1000 feet of altitude lost in a glide,

and the speed necessary to achieve this must be

determined with the engine inoperative, its propeller in

the minimum drag position, and landing gear and wing

flaps in the most favorable available position.


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First we note the following relation between thespeed and distance:

V V V H

¼ h=DtR glide=Dt

¼ hR glide

Assuming that q is small, we can say thatV HzV . There-

fore, using Equation (21-5) we get:

V V ¼ DV W 0

V V V ¼ h

R glide¼ D

W

Using Equation (21-3) and the assumption that for smallangles cos q z 1, we get:

L ¼ W cosq 0 hR glide

¼ DW ¼ D

L=cosq 0

h

R glide

¼ D

LcosqzD

LQED

VARIABLES

Symbol Description Units (UK and SI)

CLa 3D lift curve slope /deg or /rad

CD Drag coefficient

CDmin Minimum dragcoefficient

CL Lift coefficient

CL0 Lift coefficient at zero AOA


D Drag lbf or N

g Acceleration due togravity

ft/s2 or m/s2

h Altitude ft or m

k Coefficient for lift-induced drag

L Lift lbf or N

LDmax Maximum lift-to-dragratio

Rglide Glide distance ft or m

h

Rglide

θ

FIGURE 21-7 Distance covered during glide can be estimated using the L/D ratio.

E X A M P L E 2 1 - 5

Determine the maximum glide distance for the sample

aircraft flying at 30,000 ft.

Solution

Using the maximum LD calculated in EXAMPLE 19-7

we get (LDmax ¼ 15.45):

R glide ¼ h$

L

D

max

¼ 30;000

ð15:45

Þ¼ 463;500 ft ð76:3 nmÞ

VARIABLES 933

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S Wing area ft2 or m2

T Thrust lbf or N

V Airspeed ft/s or m/s

V BA Airspeed of minimumsink rate

ft/s or m/s

V BG Best glide airspeed ft/s or m/s

V Emax Airspeed of maximumendurance

ft/s or m/s

V H Horizontal airspeed ft/s or m/s

V LDmax Velocity of maximumlift-to-drag ratio

ft/s or m/s

V V Rate-of-descent ft/s or m/s

V X Horizontal velocity ft/s or m/s

V Z Vertical velocity ft/s or m/s

W Weight lbf or N

Dt Change in time sec

a Angle-of-attack deg or rad

ε Thrust angle deg or rad


f Banking angle deg or rad

q Descent angle deg or rad

qmin Minimum angle-of-descent

deg or rad

r Density slugs/ft3 or kg/m3

References

[1] Perkins CD, Hage RE. Airplane Performance, Stability, andControl. John Wiley & Sons; 1949.

[2] Torenbeek E. Synthesis of Subsonic Aircraft Design. 3rd ed. DelftUniversity Press; 1986.

[3] Nicolai L. Fundamentals of Aircraft Design. 2nd ed. 1984.[4] Roskam J, Lan Chuan-Tau Edward. Airplane Aerodynamics and

Performance. DARcorporation; 1997.[5] Hale FJ. Aircraft Performance, Selection, and Design. John Wiley

& Sons; 1984. pp. 137e138.[6] Anderson Jr JD. Aircraft Performance & Design. 1st ed. McGraw-

Hill; 1998.[7] Raymer D. Aircraft Design: A Conceptual Approach. AIAA Ed-

ucation Series; 1996.

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