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Directional Stability

AE 430 - Stability and Control of Aerospace Vehicles

� In an equilibrium condition (figure (a)), an airplane flies so that the yaw angle is zero. To have static directional stability, the appropriate positive or negative yawing moment should be generated to compensate for a negative or positive sideslip angle excursion

Static directional stability

� Static directional stability is a measure of the aircraft's resistance to slipping. The greater the static directional stability the quicker the aircraft will turn into a relative wind which is not aligned with the longitudinal axis.

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Directional (Weathercock) stability

� The main contributor to the static directional stability is the fin. Both the size and arm of the fin determine the directional stability of the aircraft. The further the vertical fin is behind the center of gravity the more static directional stability the aircraft will have. (This is often called the weather veining effect, because it works the same way as a weather vein.)

� As mentioned previously all rotational motions of the aircraft occur around the center of gravity. Directional stability refers to motions around the normal axis.

Stable/unstable aircraft

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This figure shows the variation of yawing-moment coefficient with sideslip angle. This positively sloping line indicates a directionally stable case.

Wing contribution to directional stability

� A wing produces two effects that give a yawing moment with sideslip. The important one is due to sweep-back angle, and the other minor effect is due to geometric dihedral.

Directional and lateral effects of wing sweep due to sideslip

The second effect, due to dihedral, results from a tilt of the lift vectorwith sideslip.

(Both effects are stabilizing)

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Lateral Effects

� Wing Dihedral– Dihedral effects

due to sideslip– Sideslip produces

two important effects other than those mentioned directional effects:

� rolling moment

� side force

� Wing Sweep� Fuselage

Contribution to directional stability

� Fuselage and engine nacelles (in general are destabilizing)

wf

fs fn n RL

w

S lC k k

S bβ= −

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Contribution to directional stability

Wing-body interference factor

Reynolds numbercorrection factor

Vertical tail contribution

vv L v v vY C Q S

αα= −

vα β σ= +

Sidewash due to wing vortices

vY

vα

βσ

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Moment produced by a side force

( )v v

v v v v L v v v v L v vN l Y l C Q S l C Q Sα α

α β σ= = = +

( ) ( )v v

v v vn v L v v L

w w

N Q SC l C V C

Q Sb Q Sb α αβ σ η β σ= = + = +

v vv

S lV

Sb=

vv

w

QQ

η =

Vertical tail volume ratio

Dynamic pressure ratio

Contribution vertical tail to directional stability

1v v

n v v Ld

C V Cdβ α

σηβ

� �= +� �

� �

41 0.724 3.06 0.4 0.009

1 cosw

v wv w

c

S S zdAR

d dσηβ

� �+ = + + +� � + Λ� �

USAF Stability and Control Datcom:

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Some comments� The moment associated with yawing and rolling are cross-coupled,

i.e., the angular velocity in yaw produces rolling moments and vice versa. If a pilot steps on a rudder pedal causing the aircraft to yaw one wing will advance and the other will retreat. The faster moving wing produce more lift than the other which will cause a roll in the same direction as the yaw. This will be exaggerated by wing dihedral.

� At a normal flight, i.e., steady rectilinear symmetric motion, all the lateral motion and force variables are zeroes.

� There is no fundamental trimming problem: control surfaces (ailerons and rudder) would normally undeflected.

� Lateral control provides secondary trimming functions in the case of asymmetry.

� Effects of CG movement are negligible on lateral and directionalstability

� Due to cross-coupling effect, (e.g., the rolling motion will cause sideslip), we investigate the directional and lateral effects ofsideslip.

Directional Control

� Rudder

(+)

Positive rudder deflection, producesa positive side force, that will produce a negativeyawing moment

v vN l Y= −

vv L v vY C Q S=

vLv vn v r

w w r

dCQ SNC l

Q Sb Q Sb dδ

δ= = −

vLn v v r

r

dCC V

dη δ

δ= −

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Requirements for Directional Control

� Table from R. Nelson book– Adverse yaw– Crosswind landings– Asymmetric power condition– Spin recovery

Rudder control effectivenessv

r r

Ln n r n v v

r

dCC C C V

dδ δδ η

δ= � = −

v v

v

L L vL

r v r

dC dC dC

d d d α

α τδ α δ

= =

A 747 lands in a very strong cross-wind

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Adverse Yaw

� Roll-Yaw Coupling� Asymmetric aileron deployment produces asymmetric drag

Asymmetric drag produces adverse yaw� Rudders required for coordinated turn

Static Roll Stability

� The roll moment created on an airplane when it start to slip depend on:

– Wing dihedral angle Γ– Wing sweep Λ– Position of wing on the

fuselage– Vertical tail

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� Figure (a) shows a head-on view of an airplane that has dihedral where the wings are turned up at some dihedral angle to the horizontal. If a disturbance causes one wing to drop relative to the other (figure (b)), the lift vector rotates and there is a component of the weight acting inward which causes the airplane to move sideways in this direction. When wings have dihedral, the wing toward the free-stream velocity, hence the lower wing, will experience a greater angle of attack than the raised wing and hence greater lift. There results a net force and moment tending to reduce the bank angle (figure (c)).

nvu

α∆ = sinnv v= Γ

Dihedral Effect

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Dihedral Effect

v

vu

β ≈

α β∆ ≅ Γ

α β∆ ≅ − Γ Up-moving wing

Down-moving wing

Approximation for the sideslip

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Effect of wing placement on lateral stability -Fuselage contribution to dihedral effect

Wing sweep effect on roll stability

� The windward wing (less effective sweep) will experience more lift than the trailing wing. The result is that the sweepback adds to the dihedral effect

� On the other hand, sweep forward will decrease the effective dihedral effect

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Roll moment due to vertical tail

Roll Control

� By differential deflection of ailerons or by spoilers

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Roll Control

� By differential deflection of ailerons or by spoilers

( )LiftL y∆ = ∆

Ll

C Qcydy C cydyC

QSb QSb Sb∆∆ = = =� �

a aa

dC C C

dα α

α δ τδδ

= =� � �

2

1

2w

yL al y

CC cydy

Sbα

τδ= � 2

1

2w

a

yLl y

CC cydy

Sbα

δ

τ= �

Control power

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2rc c ybλ � �−= +� �� �

� �� � �

Tapered wing