Design of Aircraft Stabilizers

This lecture is intended to provide a quick guide

for the conceptual design of aircraft stabilizers

Scope

Two major criteria are described in this document for the design of aircraft horizontal and vertical stabilizers.

Design for controlability

Design for stability

The establishiment of the area for the stabilizers shall satisfy both criteria.

As guided example, the Boeing B-757 will be taken as reference airplane

Introduction

Introduction

1. They provide static and dynamic stability.

2. They enable aircraft control.

3. They provide a state of equilibrium in each flight condition.

The stability and control surfaces have the following functions

Introduction

The figure below illustrates how the various components of an aircraft contribute to the stability of the whole vehicle. The wing alone shows a nose-up tendency when its angle of attack is increased; this is accentuated when a fuselage is added. The tailplane, on the other hand, provides a stabilizing nose-down moment, so that when wing, body and tailplane are brought together in a complete unit the result is a stable configuration.

Introduction

The effect of CG position on static longitudinal stability is easily analyzed by looking at the wing in isolation. The figure at right shows the variation of pitching moment with AOA for a range of CG positions. As the CG is moved aft, so the slope of the line Cm/CL becomes less and less steep (negatively) and approaches the neutral position, where there is no change of pitching moment with CL (i.e. dm/dCL = 0), giving neutral stability. Beyond this CG position, the slope is positive (i.e. upwards to the right), showing that as the AOA is increased so the pitching moment acts so as to further increase the angle. This is a divergent (i.e. unstable) condition. For the example above, 40% wing chord is the aftmost position for the CG. If the CG is moved beyond this into the unstable region, some way must be found to counteract the resulting out-of-balance moment, which threatens then turn the wing leading edge over trailing edge. On a conventional aircraft this is achieved by means of a tailplane. If however the CG is kept ahead of the 40% point (in this hypothetical case) then the wing itself is inherently stable: once disturbed it will tend to return to its original position unaided. This aspect of stability thus depends on the relationship of the CG and the point at which the wing lift acts. Flying wings and all but the most recent deltas have relied on careful positioning of the CG in order to avoid the need for a tailplane. The main benefit of a tailplane lies however in the fact that it allows a further aft CG limit, the principal constraint being need to keep the CG forward of where the center of lift of the whole aircraft (i.e. the neutral point) is located.

Introduction

1. They shall provide a sufficiently large contribution to static and dynamic longitudinal, directional (and sometimes lateral) stability. This determines primarily their lift gradients and requires surfaces with higher aspect ratios with low sweepback angles.

2. They shall provide sufficient control capability, which again determines their lift slope. This also requires a maximum aspect ratio and for high aspect ratios minimum sweep.

3. Control shall be possible with acceptable control forces. This requires a higher aspect ratio because control force is given by

2

2h c c

VF C S c

2

is the hinge moment coefficient

is the dynamic pressure2

is the control surface area

is the mean aerodynamic chord of the control surface

h

c

c

C

V

S

c

Introduction

4. The tail surfaces shall be able to provide a maximum force sufficiently large to balance the total tail-off forces and moments so that static equilibrium is achieved in all flight conditions. This leads to specific requirements on tail surface areas and on the maximum lift coefficient for the tail surfaces with varying degree of control surface deflection, including the effect of ice roughness.

5. The tail surfaces shall be able to cope with high tailplane angles-of attack, both for the horizontal tail (in particular at higher speeds with flaps deflected) and for the vertical tail surface (high cross-winds). In this case a low aspect ratio is required and sweep is beneficial. The requirement to be able to cope with high tailplane and fin angles-of attack is aggravated when flight in icing conditions is possible.

6. For high-speed aircraft the Mach-number at which serious flow separation occurs shall preferably lie above the design dive Mach number (MD). Serious flow separation on the stabilizer will aggravate the effect of the changes in tail-off pitching moment due to changes in the wing flow. This applies in particular to aircraft with reversible control systems.

Introduction

Furthermore the following shall be kept in mind:

• A high aspect ratio has an adverse (although relatively small) effect on weight. Also, in particular for T-tails the flutter analysis requires extra care. A few degrees anhedral (negative dihedral) has a very beneficial effect.

• Excessive taper ratio may lead to premature tip stall. This risk is higher when sweep

is applied although the stall is then more gradual with less loss in lift. On the other hand tapering leads to lower weight.

Introduction

Introduction

Placement of Stability & Control Surfaces

Introduction

Introduction

A Taste of Aerodynamics

Stall Types A Taste of Aerodynamics

• Spin Recovery The primary control for spin recovery in most airplanes is the rudder. The

rudder must be powerful enough to oppose the spin rotation.

Trapezoidal wing Stall seen inboard; tail blanket; aileron control still available

Swept wing Stall outboard; tail clean; aileron control may be not available

Dorsal Fin A Taste of Aerodynamics

Fokker 100

P-3 Orion

ERJ 145

Dorsal Fin A Taste of Aerodynamics

Instead of a lightly-swept leading edge in combination with a dorsal fin also a fully-swept-back leading edge on a vertical tail surface may produce favorable sideslip characteristics. This can be demonstrated in Figures at bottom of this page which show test results of an investigation on three tail configurations with these differences in leading edge geometry, performed during the development of the Fokker F-28. Yawing moment vs. sideslip angle is presented for two aircraft angles-of-attack for the three configurations tested. For the linear regime, the three curves practically coincide. At higher side-slip angles above β = 150, it appears that for zero angle-of-attack applying fin sweep or adding a dorsal fin has nearly the same favorable effect on the yawing moment curve.

Source: Obert, E., Aerodynamic design of transport aircraft, Delft University

Horizontal Stabilizer Design for Controllability and Static Stability

Aircraft Trim Requirements

(1)

(2)

0

0

F

M


(3)0

0 (4)

x

cg

F

M

The aircraft trim must be maintained about three axes (x, y, and z): 1. lateral axis (x), 2. longitudinal axis (y), and 3. directional axis (z). When the summation of all forces in x direction (such as drag and thrust) is zero; and the summation of all moments including aerodynamic pitching moment about y axis is zero, the aircraft is said to have the longitudinal trim.


0 (5)

0 (6)

y

cg

F

N

The horizontal tail is responsible to maintain longitudinal trim and make the summations to be zero, by generating a necessary horizontal tail lift and contributing in the summation of moments about y axis. Horizontal tail can installed behind the fuselage or close to the fuselage nose. The first one is called conventional tail or aft tail, while the second one is referred to as the first tail, foreplane or canard. When the summation of all forces in y direction (such as side force) is zero; and the summation of all moments including aerodynamic yawing moment about z axis is zero, the aircraft is said to have the directional trim.


0 (7)

0 (8)

z

cg

F

L

The Vertical tail is responsible to maintain directional trim and make the summations to be zero, by generating a necessary vertical tail lift and contributing in the summation of moments about y axis. When the summation of all forces in z direction (such as lift and weight) is zero; and the summation of all moments including aerodynamic rolling moment about x axis is zero, the aircraft is said to have the directional trim.

The Vertical tail is responsible to maintain directional trim and make the summation of moment to be zero, by generating a necessary vertical tail lift and contributing in the summation of moments about z axis.

Longitudinal Trim

0 0 0 (9)0 wf h e wf hL LM M M M M

(7)0 cgM

Longitudinal Trim For the horizontal tail design process, we need to develop a few equations; hence the longitudinal trim will be described in more details. Consider the side view of a conventional aircraft (i.e. with aft tail) in figure below that is in longitudinal trim. In his figure, the aircraft is depicted when the aircraft center of gravity is forward of the wing-fuselage aerodynamic center. There are several moments about y axis (cg) that must be balanced by the horizontal tail’s lift; two of which are: 1. wing-fuselage aerodynamic pitching moment, 2. the moment of lift about aircraft center of gravity. Other source of moments about cg could be engine thrust, wing drag, landing gear drag, and store drag.

Longitudinal Trim

0 0 0 0wf h e wf hL LM M M M M

0 0 0 0wf h e wf cg ac mac h h cg ac macM M M L x c L l x c

is given as a fraction of the mean aerodynamic chordcg acx

Longitudinal Trim

2 2 2 2 2

0 0 0 02 2 2 2 2wf h ew mac h h mac w mac Lwf w cg ac mac Lh h h h cg ac mac

V V V V VS c Cm S c Cm S c Cm C S x c C S l x c

11

Longitudinal Trim

2 2

0 0 02 2wf e hw mac Lwf cg ac h h Lh h mac Lh mac cg ac

V VS c Cm Cm C x S C l c Cm C c x

0 0

0

wf e

h

Lwf cg ac mach

w h Lh h cg ac mac mac

Cm Cm C x cS

S C l x c c Cm

11

10

Longitudinal Trim

12

Another form of (11) is as follow

0 0wf e

h h h LhLwf cg ac

w mac

S l CCm Cm C x

S c

12a

For the derivation of (11a) it was considered that Cm0h is negligible

0 0

0

wf e

h

Lwf cg ac mach

w h Lh h cg ac mac mac

Cm Cm C x cS

S C l x c c Cm

with h h cg ac macl l x c

Longitudinal Trim

0 0wf e

h h h LhLwf ac cg

w mac

S l CCm Cm C x

S c

12a

The combination 𝑆ℎ𝑙´ℎ

𝑆𝑤𝑐𝑚𝑎𝑐 in equation 11a of is an important non-dimensional parameter in

the horizontal tail design, and is referred to as the “Horizontal tail Volume Coefficient”. The name originates from the fact that both numerator and denominator have the unit of volume (e.g. m3). The numerator is a function of horizontal tail parameters, while the denominator is a function of wing parameters.

h hh

w mac

S lV

S c

This non-dimensional parameter Vh has a limited range in values and also it is not a function of aircraft size or weight. From a small aircraft such as Cessna 172 to a large jet aircraft such as Boeing 747 all have similar tail volume coefficient.

13

Longitudinal Trim Table below shows typical values for tail volume coefficient for several aircraft types. The tail volume coefficient is an indication of handling quality in longitudinal stability and longitudinal control. As the Vh increases, the aircraft tends to be more longitudinally stable, and less longitudinally controllable. The fighter aircraft that are highly maneuverable tend to have a very low tail volume coefficient, namely about 0.2. On the other hand, the jet transport aircraft which must be highly safe and stable tend to have a high tail volume coefficient, namely about 1.1. This parameter is a crucial variable in horizontal tail design and must be selected at the early stages of tail plane design. Although the primary function of the horizontal tail is the longitudinal stability, but the tail volume coefficient serves as a significant parameter both in the longitudinal stability and longitudinal trim issues.

Airplane Type MTOW (kg) Wing

area (m2) Overall

length (m) HT area

(m2) Vh

Cessna 172 Light GA (piston

powered) 1,100 16.2 7.9 1.94 0.76

Piper PA-46-350P Light transport (piston

powered) 1,950 16.26 8.72 0.66

Alenia G222 Military transport

(turboprop) 28,000 82 22.7 0.85

Fokker 100 Jet airliner (R&R Tay

620) 43,090 93.5 32.5 21.72 1.07

Boeing F/A-18C Fighter 23,400 46 16.8 0.49

Pilatus PC-12 Transport (single

engine turboprop) 4,100 25.81 14.14 1.08

Airbus A340-200 Jet airliner 257,000 363.1 59.39 72.90 1.11

Boeing 747-400 Jet airliner 396,830 525 68.63 136.60 0.81

Horizontal Stabilizer • Design for static stability

Another design requirements must be examined: aircraft static and dynamic longitudinal stability. The static longitudinal stability is examined through the sign of the longitudinal stability derivative Cm or the location of the aircraft neutral point. For an aircraft with a fixed aft tail, the aircraft longitudinal stability derivative is determined as:

1 1wf h

h h hm L L h h ac cg L h

w w mac

S S ld dC C C x C

S d S c d

14

0 0

2

[ ( )]

12

m wf wf ac cg m h h h ac cg

w macm

C L x x C L l x xd

V S cdC

d d

0 1h L h L h h h

dL C C S

d

with

15

Horizontal Stabilizer • Design for static stability

When the derivative Cm is negative or when the neutral point is behind the aircraft cg, the aircraft is said to be statically longitudinally stable. The limit of the design is found when Eq. 15 is set to zero.

16

0 1 1wf h

h h hL L h h ac cg L h

w w mac

S S ld dC C x C

S d S c d

1

wfL ac cgh

w hL h h cg ac

mac

C xS

S ldC x

d c

The vertical tail tends to have two primary functions:

1. directional stability, 2. 2. directional trim. 3. Moreover, the vertical tail is a major contributor in maintaining directional

control which is the primary function of the rudder.

The primary function of the vertical tail is to maintain the aircraft directional stability. In summary, the stability derivatives Cn must be positive (to satisfy the static directional stability requirements), but the stability derivatives Cnr must be negative (to satisfy the dynamic directional stability requirements). Two major contributors to the value of these stability derivatives are vertical tail area (SV) and vertical tail moment arm (lV). If vertical tail area is large enough and vertical tail moment arm is long enough, the directional stability requirements could be easily satisfied. The directional stability analysis is performed after all aircraft components are designed and the roots () of the lateral-directional characteristic equation are calculated. A general form of the aircraft lateral-directional characteristic equation looks like the following:

4 3 2

2 2 2 2 2 0A B C D E

where coefficients A2, B2, C2, D2, and E2 are functions of the several stability derivatives such as Cnr and Cn. The reader is encouraged to consult with [Roskam, 2007] to see how to derive the aircraft lateral-directional characteristic equation. The directional stability derivatives cannot be determined unless all aircraft components including wing and fuselage have been designed. Hence, we have to resort to some other simplifying criterion that could be a base for the vertical tail preliminary design.

The second function of the vertical tail is to maintain the aircraft directional trim. As discussed in Section 6.3, the summation of all forces along the y-axis and the summation of all moments about z-axis must be zero.

0 (5)

0 (6)

y

cg

F

N

Sizing for Controllability

0

2

ee

T yN

21

2D Dwm ref eN C V S y

Nv

18

17


21

2D Dwm ref eN C V S y

2 2

2

20.0785 1

1 0.16 4

is the Mach number

is the engine inlet diameter

is the nozzle exit velocity

0.92 for high by-pass ratio engines

is the wing reference area

n ni i

Dwm

ref

i

n

n

ref

V Vd d

M V VC

S

M

d

V

V

V

S

19

Sizing for ControllabilityThe maximum available yawing moment coefficient is obtained at an equilibrium flight condition with a given bank angle () and a given maximum rudder deflection (r). The bank angle is limited to a maximum of 5° by FAR 25.149, and the aircraft is allowed to have some sideslip ().

The sideslip angle is found by summing the forces along the y-axis:

sin 0y a y r y LC a C r C C

The bank angle is found by solving the rolling moment equation:

sin 0l a l r l LC a C r C C

20

21


The rudder deflection is initially set to the given maximum allowable steady-state value, and the sideslip angle and aileron deflection for equilibrium flight are determined by Eqs. (20) and (21). The maximum allowable steady-state deflection is typically 20°-25°. This allows for an additional 5° of deflection for maneuvering. The maximum available yawing moment is found by summing the contributions due to the ailerons, rudder, and sideslip:

,n available n a n r nC C a C r C

However, we will follow here a simplified an conservative approach considering that the airplane is experiencing no bank and sideslip angle.

,n available n rC C r or 21

2ref n rEngine torque V S C r b

23

24


In order to proceed with the calculation of Cnr we can write

cos sinv v

n r y r

l zC C

b

,

,

, ,

L v CL L vy r b f L theory

l v L theory refCl

C C SC K K C

C C S

• Kb is the span flap factor. • Kf is an empyrical correction factor for large control deflections. It is can be estimated by

interpolation in curves on Roskam Vol. VI.

Force coef. Moment coef.

Arm

Reference length

25

26

,

is found from Figure 8.15L

L theory

C

C

is found from Figure 8.53CL

Cl


Thus, we can write

2

, 1

cos sin1 +

2

v vL theory b f v

l zEngine drag torque V C K K K S r b

b

Finally

2

,

2 ( )

cos sinv

L theory b f v v

Engine drag torqueS

V C K K K r l z

27

with

,

, ,

L v CL L

l v L theoryCl

C CK

C C

Sizing for Stability

cg

Moment caused by fuselage Now we will handle another condition for the design of the vertical stabilizer: the balancing of the yaw moment when a sideslip angle is present


Resulting VT wing fuselageM M M M

2 2 2 2

,Re , , ,

1 1 1 1

2 2 2 2w N sulting w N VT w N wing w N fuselageV b C V b C V b C V b C

,Re , , ,N sulting N VT N wing N fuselageC C C C

28

29


,Re , , ,n sulting n VT n wing n fuselageC C C C

, 0n wingC

,

180fbsn fuselage N Rl

ref w

lSC K K

S b

, ,

cos sinv v

n VT y VT

w

l zC C

b

is an empyrical factor 0.0011NK

, , ,

Re

1 VTy VT Cy VT L VT v

f

SdC k C

d S

S fuselage side area 0.83bs f fl d

10 6

Re0.46 log 1

1 10RlK

Fig. 10.29 [Roskam 73]

30

31 32


, , ,

Re

1 VTy VT Cy VT L VT v

f

SdC k C

d S

,

,

,

, ,

,

,

0.75 2

1 5 2 3.5

6 12

1 3.5

vCy VT

f v

v vCy VT

f v f v

vCy VT

f v

bk

d

b bk

d d

bk

d

33



14

1 0.724 3.06 0.4 0.091 cos

v

w wv

SS Zd

Ad d

Zw is the vertical distance from the wing root quarter-chord line to the fuselage centerline, positive downward. CLα,VT can be calculated by using the following expression

22 2

12

2 2

2

tan2 1 4

eff

L

eff

ARC

AR

34

35

AReff is the effective aspect ratio

21 M

2

lC



( ) ( )

( )

1 1V B V HB

eff V H

V V B

A AAR A k

A A


[Roskam 73]

Horizontal and Vertical Stabilizer Boeing 757-200 Example

Design for Controlability

Design for Static Stability

max,

2

0

2.81

185.25

325,653 (sea level, )

L landing

w

C

S m

T N ISA

2

0

25

1.56

0.35

50.35

0.211

30

0.33

4.59

v

v

h

w

h

h

h

AR

S m

AR

115680 kg

M 0.86mo

MTOW

0

25

0

50

0

50

25

25.826

28.5

7.82

w

h

v

wAR

3.8

47.32

f

f

d m

l m

Horizontal Stabilizer Boeing 757-200

• Design for controllability

This criterium derives a equation where the static margin plays an important role. A critical condition for the horizontal surface sizing is the landing, where the flaps are deflected at its maximum deflection angle. The center of pressure moves rearwards and a negative picth moment is present. During the approach and landing phases the engines were set for flight idle but they contribute with a pitching moment as well.

Static margin

0 0

0

wf e

h

Lwf cg ac mach

w h Lh h mac

Cm Cm C x cS

S C l c Cm

Since the horizontal tail surface is usually designed with a uncambrede airfoil (or with a very small cambered) airfoil, the pitch moment coefficient, Cm0h, of the horizontal tail is usually negligible. Eq. (11) then becomes

.1II

0 0wf e Lwf cg ac mach

w h Lh h

Cm Cm C x cS

S C l

.2II



For starting designing the Boeing 757-200 horizontal tail, the following parameters were estimated

.1II0 0

( )

wf e Lwf ac cg mach

w h Lh h ac cg mac

Cm Cm C x cS

S C l x c

.2II

max 2.81

-0.50

0.90

18.03

5.66

Lwf L

Lh

h

h

mac

C C

C

l m

c m

Eq. II.1 can be now written as

We consider Eq. II.1 relating the area ratio Sw/Sh to the static margin

0 012.57835.344

18.03 5.66 18.03 5.66

wf ecg ach

w cg ac cg ac

Cm CmxS

S x x



.3II

The pitch moment coefficient of the wing can be estimated by

2

25 00,

25

cos

2cos

mmw m flapped t

t

AR CC C

AR

The pitch moment coefficient for deflected flaps is estimated using II.4

0, 0 ,

ref cp

m flapped m L flap

mac

x x cC C C

c c c

.4II

• 𝐶𝑚0, the pitch moment coefficient at retracted flap condition is estimated to be -0.07

• The increase in maximum lift coeficient due to the flap system is 2.81-1.5 = 1.31

• 0.25

ref aft cg

mac mac

x x

c c and 00.25 cos 45

1.1768c c

cc c

Results 0, 0.4208m flappedC

[Roskam VI, Eq. 8.70]

Is found from Figure 8.91 [Roskam VI]



=0.5

Estimative of ∆𝐶𝑚0𝜖𝑡

=1.0

=0.0

For the Boeing 757-200 we obtain

10 0.065 degm

t

C



Estimative of 𝑥𝑐𝑝

𝑐′

Figure 8.91 – Location of center of pressure incremental due to flaps.



Considering that t = -30, AR = 7.82, and 25= 250 we obtain

2

25 00,

25

cos0.300

2cos

mmw m flapped t

t

AR CC C

AR

Now the pitching moment coefficient originated by the engine thrust must be calculated

eme

w mac

T zC

q S c

0

2

1.3

0.3 97965,7

64

5.66

185.25

e

app

mac

w

z m

T T N

mVs

c m

S m

0.0482meC

.5II



Equation II.2 can now be fully derived

0 012.535.344 35.344 4.68648

18.03 5.66 18.03 5.66 18.03 5.66 18.03 5.66

wf ecg ac cg ach

w cg ac cg ac cg ac cg ac

Cm Cmx xS

S x x x x


• Design for static stability

Left. Geometric parameters for horizontal tail location

z

HT

Wing

h

Right. Definition of dimensional and non-dimensional Aerodynami center location



Now we have to consider the following equation

1

wf

h

L ac cg mach

wL h h

C x cS

dSC l

d

2 2 2

50

2

2 1 tan 4wL

ARC

AR M

The lift slope of the wing cab be estimated using Eq.

.6II

For the Boeing 757-200 at Mach=0.86 we obtain

1 13.2482 0.0567degwLC rad

14

1 14.93 0.0086deghLC rad



The downwash gradient is calculated as the following manner

1.19

25

0

4.44 cosL M

A h w

L M

Cdk k k

d C

.7II

1.7

3

1 10.09846

1

10 31.32957

7

1

0.97752

A

w w

w

h

h

h

kAR AR

k

zb

kl

b

Where

1

, 02.29 L w M

C rad

From Eq. II.6

0.7466d

d

5.3614

3.1855

cg ach

w cg ac

xS

S x


• Obtaining the HT area

The two previously derived linear equations can now be combined into a single graph. It should be observed that the aft center of gravity must be positioned at a safe distance to the natural stability limit. For a jet transport aircraft Roskam establishes this value as 5% of MAC. However, according to Raymer, this value can be further reduced by 3% of MAC.

The permitted areas of focus are now between the limit lines of controllability and that of the stability requirement. Between these lines now, the required cg range can thus be fitted to find out the smallest horizontal tail surface area. Thus, we choose for the cg range the value of 0.32. This result an area ratio of 0.25. Considering the wing reference area of 185.25 m2 , the horizontal tail area is then 46.31 m2. This is a pretty much good result, since the actual value is 50.35 m2.


• Obtaining the HT area

Controllability

Stability

252.98 hS m

Vertical Stabilizer Boeing 757-200


2

,

max, 0 max,

1.2 2 64.28 /

2.42

MC stall

stall takeoff

L takeoff w L takeoff

V V MTOWV m s

C S C

0 325653 6.48 1055115.72

2 2

ee

T y N mN N m

77.14 /MCV m s

2

,

2 ( )

cos sinv

MC b f l theory v v

Engine torque Drag torqueS

V K K K C r l z



2 2

2

20.0785 1

1 0.16 40.0037

is the Mach number

is the engine inlet diameter

is the nozzle exit velocity

0.92 for high by-pass ratio engines

is the wing reference

n ni i

Dwm

ref

i

n

n

ref

V Vd d

M V VC

S

M

d

V

V

V

S

area

2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z



2 21S y =0.5 1.225 (77.14) 0.0037 185.25 6.48 16161 N m

2D Dwm ref eN V C

1071300 e DTotal torque N N N m

2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z



1

2 2 2

50

22.0485

2 1 tan 4L v

ARC rad

AR M

0

18.97 (aft cg)

=25

vl m

r

, 6.245l vC

,

,

0.3280L v

l v

C

C

2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z

kf

Vertical Stabilizer Sizing for Controllability

Boeing 757-200

Kf can be determined from Fig. 8.3 of Roskam’s Vol. VI. Considering a chord ratio of 0.30 and a flap deflection of 250, kf = 0.684.


Fig. 8.13 [Roskam VI]

2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z


Boeing 757-200

Kb can be determined from Fig. 8.52 of Roskam’s Vol. VI that is reproduced above. Considering =0.85, we find kb = 0.93.

Graph below outlines the procedure for kb estimation



2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z


Boeing 757-200

Now we need calculate K, which is comprised of three terms

,

,

is found from Figure 8.53



l

l theory

l theory

CL

Cl

C

C

C

,

, ,

L v CL l

l v l theoryCl

C CK

C C

,

,

C is found from page 386

is found from Section 8.1.1.2

L v

l vC

2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z


Boeing 757-200

1.18CL

Cl

0.30fc

c

Effect of aspect ratio and flap chord ratio on three-dimensional flap effectiveness ratio.


2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z


Boeing 757-200

,

0.985l

l theory

C

C

Correction factor for plain flap lift 0.3

fc

c

2

,

2

cos sinv

b f L theory v v

Engine torqueS

V K K K C r l z

,

, ,

L v CL l

l v l theoryCl

C CK

C C



Boeing 757-200

,

, ,

0.3280 1.18 0.985 0.38123L v CL l

l v l theoryCl

C CK

C C

2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z


Boeing 757-200

, 4.6l theoryC

Lift effectiveness for a plain flap 0.3

fc

c

2

,

2

cos sinv

b f l theory v v

Engine torqueS

V K K K C r l z


Boeing 757-200

2

2 1071300

1.225 (77.14) 0.38123 0.93 0.684 4.6 0.4363 18.97cos15 4.32sin15vS

231.06 vS m

Considering that the actual figure for the vertical stabilizer area of Boeing 757-200 is 34.37 m2, the calculated value above is an excellent estimate, indeed.

2

,

2 ( )

cos sinv

MC b f l theory v v


V K K K C r l z

Vertical Stabilizer Sizing for Stability

Boeing 757-200

,

180fbsn fuselage N Rl

ref w

lSC K K

S b

is an empyrical factor 0.0011NK

is parameter depending on Reynolds numberRlK

S fuselage side area 0.83bs f fl d

, 6

Re0.46log 1 2.7236

10R lK

8

6

0.348331 0.86 295 47.32Re 4.23 10

9.55 10

fVl

22 3

,

0.83 1.1 10 2.7226 0.83 47.32 180180f f

n fuselage N Rl

ref w

d lC K K

S b

Bibliography Boeing 757-200

[ABBOTT] ABBOTT, I.H.; DOENHOFF, A.E.: Theory of wing sections, New York: Dover, 1959. [BANSA] Bansa F.: Flugzeugentwurf am Beispiel einer Fairchild Dornier 728 Jet, HAWHamburg, FB. Flugzeugbau, 2003. [BECHNER] Bechner, T.: Boeing 757 and 767, Crowood Press Ltd Ramsbury, 1999. [BIRTLES] Birtles P.: Boeing 757/767/777, Modern Civil Aircraft 6, Ian Allan Publishing, England, 1999 [DATCOM 78] HOAK, D.E.: USAF Stability and Control Datcom, Wright-Patterson Air Force Base, Air Force Flight Dynamics Laboratory, Flight Control Division, Ohio, 1978. – Distribution&Sales: NTIS. [DUBS 87] UBS, F.: Aerodynamik der reinen Unterschallströmung, Basel : Birkhäuser, 1987. [JAR 25] JOINT AVIATION AUTHORITIES: Joint Aviation Requirements, JAR-25,Large Aeroplanes. [HOWE] HOWE, DENIS: Aircraft Conceptual Design Synthesis, Professional Engineering Publishing, London and Bury St Edmunds, UK, 2000.

Bibliography (Cont.) Boeing 757-200

[LOFTIN 80] Loftin, L.K.: Subsonic Aircraft: Evolution and the Matching of size to Performance, NASA Reference Publication 1060, 1980. [MARCKWARDT 98A] MARCKWARDT, K: Unterlagen zur Vorlesung Flugzeugentwurf, Fachhochschule Hamburg, Fachbereich Fahrzeugtechnik, 1998. [MARCKWARDT 98B] MARCKWARDT, K: Unterlagen zur Vorlesung Flugmechanik I, Fachhochschule Hamburg, Fachbereich Fahrzeugtechnik, 1998. [OBERT 97] OBERT, E.: Aircraft Design and Aircraft Operation, Short Course Notes, Linköping Technische Hochschule, 1997. [RAYMER 89] RAYMER, D.P.: Aircraft Design: A Conceptual Approach, AIAA Education Series, Washington D.C:AIAA, 1989.

Bibliography (Cont.) Boeing 757-200

[ROSKAM I] ROSKAM, J.: Airplane Design. Bd. 1 : Preliminary Sizing of Airplanes, Ottawa, Kansas, 1990 - Distribution&Sales: DARcorporation. [ROSKAM II] ROSKAM, J.: Airplane Design. Bd. 2 : Preliminary Configuration Design and Integration of the Propulsion System, Ottawa, Kansas, 1990 - Distribution&Sales: DARcorporation. [ROSKAM III] ROSKAM, J.: Airplane Design. Bd. 3 : Layout Design of Cockpit, fuselage, Wing and Empennage: Cutaways and Inboard Profiles, Ottawa, Kansas, 1990 - Distribution&Sales: DARcorporation. [ROSKAM VI] ROSKAM, J.: Airplane Design. Bd. 6 : Preliminary Calculation of Aerodynamic, Thrust and Power Characteristics, Ottawa, Kansas, 1990 - Distribution&Sales: DARcorporation. [ROSKAM 73] ROSKAM, J.: Methods for Estimating Stability and Control Derivatives of Conventional Subsonic Airplanes, Library of Congress Card No. 77-173353, 1973. [SINGFIELD] Singfield,T.: Verkehrsflugzeuge Weltweit, NARA-Verlag, Allershausen, 1997. [SCHMITT 98] Schmitt, D.: Luftfahrttechnik, Flugzeugentwurf, Technische Universität München, Lehrstuhl für Luftfahrttechnik, Skript zur Vorlesung, 1988. [TORENBEEK 88] TORENBEEK, E.: Synthesis of Subsonic Airplane design, Delft: University Press, 1988.

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Design of Aircraft Stabilizers