Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Horizontal tail – static stability

References

T. C. Corke „Design of Aircraft”

D.P. Raymer „Aircraft Design, a Conceptual Approach”

J. Roskam „Airplane Design”

D. Stinton „The Design of the Aeroplane”

Why we need a horizontal tail?

• to satisfy trim (equilibrium)

conditions

• to satisfy stability• to satisfy stability

Equilibrium problem

• Classical configuration

• Tailless configuration

Equilibrium problem

• Classical

configuration

• Canard

configuration

Equilibrium problem

0)( =−−+=∑ SCHZHSCZYSC xxPxPMM

Equilibrium and stability problem

∂C

0)(2

1

2

1

2

1 222 =−−+=∑ SCHZHHHSCZmbHaSC xxCVSxCSVCCSVM ρρρ

02

12

22 =

−−+=∑

a

SCHZH

HH

a

SCZmbHaSC

C

xxC

V

V

S

S

C

xCCCSVM ρ

)( 0αα −= aCZ where: α∂

∂= ZC

a

KHHZH aaaC δδα 321 ++=

0110111 )1()()( εααε

εααε

αεαα aaaaa H −∂∂

−=−∂∂

−=−=

constconst KH == δδ ,

Equilibrium and stability problem

)1(1

αε

α

α∂∂

−=

∂∂∂∂

=∂∂

a

a

C

C

C

C

Z

ZH

Z

ZH

0)1(1

2

2

=−

∂∂

−−+∂∂

=∂∂ SCHHHSCmbHmSC

C

xx

a

a

V

V

S

S

C

x

C

C

C

C

αε

0)1(2

=∂

−−+∂

=∂ aaZZ CaVSCCC α

0)1()1(1 1

2

2

1

2

2

=∂∂

−−

∂∂

−++∂∂

=∂∂

αε

αε

a

a

V

V

C

x

S

S

a

a

V

V

S

S

C

x

C

C

C

C H

a

HHHH

a

SC

Z

mbH

Z

mSC

Equilibrium and stability problem

)1(1

)1(

1

2

2

1

2

2

αε

αε

∂∂

−+

∂∂

−+∂∂

−==

a

a

V

V

S

S

a

a

V

V

C

x

S

S

C

C

C

XX

HH

H

a

HH

Z

mbH

a

��

Equilibrium and stability problem

Assumption: free stick

0321 =++ KHH bbb δδαThe equilibrium equation for hinge moments:

thus:

3

21

b

bb HHK

δαδ

−−=

or:

2

31

b

bb KHH

δαδ

−−=

Equilibrium and stability problem

if incidence angle is sufficient, the angle of

trimming/balancing tab is equal to:

0=+ bb δα

0=Kδthen we can write:

021 =+ HH bb δαthen the elevator deflection is equal to:

2

1

b

b HH

αδ

−=

Equilibrium and stability problem

The pitching moments equation is the same:

using:

0)( =−−+=∑ SCHZHSCZYSC xxPxPMM

and:

HHZH aaC δα 21 +=

HHH

HZH aab

aba

b

baaC αα

αα '

1

12

211

2

121 )1( =−=−

+=

Equilibrium and stability problem

The neutral point position in free stick case:

)1(1

)1(

'

1

2

'

1

2

2

εαε

∂−+

∂∂

−+∂∂

−==

aVS

a

a

V

V

C

x

S

S

C

C

C

XX

HH

H

a

HH

Z

mbH

a

��

)1(21

121

'

1ba

baaa −=

)1(1 1

2 α∂−+

aVSHHa

where:

How to improve the stability?

• to increase a1 and decrease a2

– comment: it is not realistic, because decreasing of

a2 , because the problem with trimming could

occuroccur

• to increase b2 and decrease b1

– comment: usually increasing of b2

increases b1

How looks stability in canard

configuration

Equilibrium and stability problem

0)( =−+−=∑ SCHZHSCZYSC xxPxPMM

0)(2

1

2

1

2

1 222 =−+−=∑ SCHZHHSCZpmbHapSC xxCVSxCSVCCSVM ρρρ

122 −∑ xxSxVV

02

12

2

2

2

2 =

−+−=∑

a

SCHZH

H

a

SCZ

p

mbH

p

aSCC

xxC

S

S

C

xC

V

VC

V

VCSVM ρ

αααε

1)1( aCaC ZHZ =∂∂

−=

Equilibrium and stability problem

2 − ∂∂ xxaSxCVC

)1(

1

αε

α

α

∂∂

−=

∂∂∂∂

=∂∂

a

a

C

C

C

C

Z

ZH

Z

ZH

0

)1(

1

2

2

=−

∂∂

−+

−

∂∂

=∂∂

a

SCHH

a

SC

Z

mbHp

Z

mSC

C

xx

a

a

S

S

C

x

C

C

V

V

C

C

αε

)1()1(

1

2

2

1

2

2

αε

αε

∂∂

−+

∂∂

=

∂∂

−+

a

a

C

x

S

S

C

C

V

V

a

a

S

S

V

V

C

x

a

HH

Z

mbHpHp

a

�

Equilibrium and stability problem

)1(

)1(

1

2

2

1

2

2

αε

αε

∂∂

−+

∂∂

−+

∂∂

==

a

a

S

S

V

V

a

a

C

x

S

S

C

C

V

V

C

XX

Hp

a

HH

Z

mbHp

a

��

Important remark

• canard configuration is more stable in

free stick case

• it is opposite to classical configuration

Downwash angle

Downwash angle behind wing ε (in the symmetry plane).

Airfoil USA 45; taper ratio 2, Cz = 1,35; λ = 6

Downwash angle behind wing

Downwash behind wing ε in distance from aerodynamic center (0,25c) equal to 1,3 (left) and 3,4 MAC (right)

Downwash behind the canard and decreasing of the

lift coefficient of main wing

Lift distribution in canard configuration (SAAB Viggen)

Stinton

Stinton

Controllability neutral point

• definition:

0=∂Cm

0=∂

∂

n

C�m

Controllability neutral point

V

qCCCSVxxCSVCCSVM a�

mqa�MZ�maM

22

,

2

2

1)(

2

1

2

1ρρρ +−+=∑

V

qCCCSVxxPCCSVM a�

mqa�MZ�maM

2

,

2

2

1)(

2

1ρρ +−+=∑

Controllability neutral point – cont.

2 −

derivative Cmq depends mainly on horizontal tail:

12

2

, aV

V

S

S

C

xxC

A

HH

a

�H�

Hmq

−−=

Controllability neutral point – cont.

the forces equation on “z” direction is:

zy PRmmgnQ =+= 2ωthus:

g

R1n

2

yω=−

Controllability neutral point – cont.

as we know:

RV yω=we obtain:we obtain:

V

ngq y

)1( −==ω

Controllability neutral point – cont.

2

2

,

2 )1(

2

1)(

2

1

V

CngCCSVxxnQCCSV a�

mqa�M�ma

−+−+ ρρ

after differentiation with respect to n

02

1)(

2 =+− �

mqa�M CgSCxxQ ρ

the position of neutral point of controllability M is equal to:

Q

CgSC

C

xx�

mqa

a

�M

2

ρ−=

−

Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1

Attitude of an aircraft

Ψ

Y0

X0

YawX1

First rotation:

Yaw

(heading)Y0

Y1

−=

0

0

0

0

1

1

100

0cossin

0sincos

k

j

i

k

j

i

ψψψψψψψψ

ψψψψψψψψ

Transformation matrix ->

Second

rotation:

pitch

X2

X1

Pitch

Transformation matrix ->

θ

Z1 Z0

−

=

0

1

1

1

1

2

cos0sin

010

sin0cos

k

j

i

k

j

i

θθθθθθθθ

θθθθθθθθ

Third rotation:

roll

φ

Y2

Y1

Roll roll

Transformation matrix ->

Z2

Z1

−

=

1

1

2

2

2

2

cossin0

sincos0

001

k

j

i

k

j

i

φφφφφφφφ

φφφφφφφφ

Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1

Dynamic Dynamic Dynamic Dynamic stabilitystabilitystabilitystability

Modes of motion – Short Period oscillations

Modes of motion – Short Period oscillations Modes of motion – Short Period oscillations

Simplified mathematical model of the Short Period

Assumption:

0,0 VUuu === & 0,0 VUuu === &

Equation:

+

+=

−

−

q

w

MVSM

ZmVZ

q

w

JMS

SZm

qxw

qw

ywx

xw

0

0

&

&

&

&

Modes of motion – Phugoid

simplified model

assumption: 0,0,0 VUqw eee ===== ααααθθθθ&&

−=

−− θθθθθθθθ

u

Z

mgXu

ZmV

m

u

u

q 00

0

0&

&

Modes of motion – Phugoid

Modes of motion – RollModes of motion – Roll

Roll – simplified model

assumptions: 0===== rqwvu assumptions: 0===== rqwvu

equation: pLpJ px =&

Modes of motion – SpiralModes of motion – Dutch roll

Modes of motion – Dutch roll Modes of motion – Dutch roll

Dutch roll – simplified model

assumptions:

0,,,0 Vvrpp ββββββββψψψψψψψψφφφφφφφφ ======= &&&

equation: )( 0VS%v%rJ xrvz ++=&

Lateral stability diagram – TS-11 Iskra

Nv

0.20

0.40

0.60 Lv

-0.10 0.00 0.10 0.20 0.30

Nv

-0.60

-0.40

-0.20

0.00

Boundaries of stability

duch roll

spiral

What say the regulations?

CS 23.181 Dynamic stability

(a) Any short period oscillation not including combined lateral-

directional oscillations occurring between the stalling speed and the

maximum allowable speed appropriate to the configuration

of the aeroplane must be heavily damped with the primary controls –

(1) Free; and (2) In a fixed position, except when compliance with

���� SAS

���� SAS

(2) In a fixed position, except when compliance with

CS 23.672 is shown.

(b) Any combined lateral–directional oscillations (“Dutch roll”)

occurring between the stalling speed and the maximum allowable speed

appropriate to the configuration of the aeroplane must be damped to 10

amplitude in 7 cycles with the primary controls –

(1) Free; and (2) In a fixed position, except when compliance with

CS 3.672 is shown.

What say the regulations?

(c) Any long-period oscillation of the flight path (phugoid) must not be

so unstable as to cause an unacceptable increase in pilot workload or

otherwise endanger the aeroplane. When, in the conditions of CS

23.175, the longitudinal control force required to maintain speeds

differing from the trimmed speed by at least plus or minus 15% is

suddenly released, the response of the aeroplane must not exhibit any

CS 23.175 – static stability conditions

suddenly released, the response of the aeroplane must not exhibit any

dangerous characteristics nor be excessive in relation to the magnitude

of the control force released.

Other regulations:

Norm:

MIL-F-8785C

the most coherent document about

flying qualities of an aircraftflying qualities of an aircraft

New (confident) version:

MIL-STD-1797A

Aircraft class

Class Definition

I Small, light airplanes (m ≤≤≤≤ 5000kg) such as:

Light utility, Primary trainer, Light observation

II Medium (5000÷÷÷÷30000 kg) weight, low-to-medium ma-

neuverability airplanes such as:

Heavy utility/search and rescue, Light or medium trans-

port/cargo/tanker, Early warning/electronic counter-

measures/airborne command, control, or communica-measures/airborne command, control, or communica-

tions relay, Antisubmarine, Assault transport, Recon-

naissance, Tactical bomber, Heavy attack, Trainer for

Class II

III Large, heavy, low-to-medium maneuverability airplanes

such as: Heavy transport/cargo/tanker, Heavy bomber,

Patrol/early warning/electronic countermea-

sures/airborne command control, or communications re-

lay, Trainer for Class III

IV High-maneuverability airplanes such as: Fighter (inter-

ceptor), Attack, Tactical reconnaissance, Observation,

Trainer for Class IV

Flight phases

Phase definition Typical flights

A. Those nonterminal Flight Phases

that require rapid maneuvering, preci-

sion tracking, or precise flight-path

control

a. Air-to-air combat (CO)

b. Ground attack (GA)

c. Weapon delivery/launch (WD)

d. Aerial recovery (AR)

e. Reconnaissance (RC)

f. In-flight refueling (receiver) (RR)

g. Terrain following (TF)

h. Antisubmarine search (AS)

i. Close formation flying (FF).

B. Those nonterminal Flight Phases

that are normally accomplished using

a. Climb (CL)

b. Cruise (CR)

c. Loiter (LO) that are normally accomplished using

gradual maneuvers and without preci-

sion tracking, although accurate

flight-path control may be required.

c. Loiter (LO)

d. In-flight refueling (tanker) (RT)

e. Descent (D)

f. Emergency descent (ED)

g. Emergency deceleration (DE)

h. Aerial delivery (AD).

C. Terminal Flight Phases are normal-

ly accomplished using gradual ma-

neuvers and usually require accurate

flight-path control

a. Takeoff (TO)

b. Catapult takeoff (CT)

c. Approach (PA)

d. Wave-off/go-around (WO)

e. Landing (L)

Flight quality levels

Level The ability to complete the operational missions

1 Flying qualities clearly adequate for the mission

Flight Phase

2 Flying qualities adequate to accomplish the mission

Flight Phase, but some

increase in pilot workload or degradation in mission

effectiveness, or both, exists

3 Flying qualities such that the airplane can be con-3 Flying qualities such that the airplane can be con-

trolled safely, but pilot workload

is excessive or mission effectiveness is inadequate, or

both. Category A Flight

Phases can be terminated safely, and Category B and

C Flight Phases can be

completed.

Cooper-Harper rating scale

Cooper-Harper scale vs. MIL flight levels

Relation between Cooper-Harper scale and Flight levels

Pilot rating Level Definition

1 - 3½ 1 Clearly adequate for the mission

flight phase

3½ - 6½ 2 Adequate to accomplish mission

flight phase flight phase

Increase in pilot workload, or loss of

effectiveness of mission, or both

6½ - 9 3 Aircraft can be controlled

Pilot workload excessive – mission

effectiveness impaired

Category A flight phases can be

terminated safely.

Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1

Introduction to controlIntroduction to control

Definition of the control surfaces

deflection and sign conventionControls – sign convention

The general rule:

• positive stick deflection causes positive

aircraft reaction (moment)

• positive stick deflection causes negative • positive stick deflection causes negative

control surface deflection

• negative control surface deflection causes

positive moment

Controls – sign convention

General rule:

force and

stick control surfacestick control surface

deflection deflection moment

(+) � (-) � (+)

Controls – sign convention

which moment are positive:

• roll - X axis forward – right wing downward

• pitch – Y axis on the right wing – nose upward

• yaw - Z axis downward – turn right• yaw - Z axis downward – turn right

Controls – sign convention

• roll (+) � stick right (+)

� right aileron up, left aileron down (-)

� positive rolling moment

• pitch (+) � stick back (+) • pitch (+) � stick back (+)

� TE of elevator up (-)

� positive pitching moment (nose up)

• yaw (+) � right pedal forward (+)

� rudder right (-)

� positive yawing moment (on right)