441 lecture 8

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Axial StabilityCoupling with sideslip

To be useful, we bend our denition of static stability even further.

Consider the rolling angle coupled withmotion in the y-direction. For a positiveφ, the Lift vector becomes

F =0

L sin φ− L cos φ

∼=0

Lφ− L

In a pseudo-static world, suppose thatV y is proportional to Lφ . Specically,

let V y ∼= KφThis creates a proportional relationship between roll angle and sideslip, angle

β ∼= V yV

hence β ∼= K V

φ

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Axial StabilityCoupling with sideslip

The horizontal motion creates aerodynamic roll moments proportional to φ.

C l = dC l

dφ φ =

dC ldβ

dβ dφ

φ = C lβK V

φ

We write the equations of motion for φ.

I φ̈ = QScC l = QScK

V C lβ φ

Since Q, S , c, K and V are all positive:

The aircraft is• Axial Stable if C lβ < 0• Axial Unstable is C lβ > 0


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Rolling MomentMoment Contributions

A horizontal velocity, V y creates two primary sources of rolling moment:

Figure: Wing Dihedral: Positive Roll, FrontView

Figure: Fuselage Interaction: Positive Roll,Front View


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Rolling MomentFuselage interaction

The wing acts as a lifting surface with Q = 12 ρV 2y and α = − φ. Consider

top-mounted wings• A negative roll creates a positive V y• The fuselage blocks the free-stream V y ,

reducing the velocity at the downstream

wing.• This reduces the dynamic pressure on the

bottom of the downstream wing, Qd ,reducing the lift as Ld = C Lα φQ d S .

• A moment differential between the wings is

createdC l = C Lα lac S (Q d − Q u )φ

Figure: Fuselage Interaction:Negative Roll, Front View

Mounting wings on the bottom will create the Opposite effect!


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Rolling MomentWing Dihedral

• A negative roll creates a positive V y• With respect to V y , upstream wing has

positive AOA, Γ; downstream wing hasnegative AOS, − Γ.

•

This creates positive lift differential onupstream wing and negative liftdownstream.

• A positive moment differential between thewings is created.

Figure: Wing Dihedral: NegativeRoll, Front View

C l 0 = 2 C Lα lac SQ Γ


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Other ContributionsWing Sweep

Wing sweep creates roll stability

• A positive roll creates a positive β .• A Positive β means a negative Yaw (ψ).• A negative ψ creates a larger V ⊥ on the

right wing.• A larger V ⊥ means more lift.• The right wing rotates up, reducing φ

A result of coupling between y, ψ, and φ


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Other ContributionsTail

The same Force as for directional stability, different moment arm

• A positive roll creates a positive β .• Sideslip creates pressure on tail

surface.• Pressure on tail creates positive

Rotation.

Figure: Roll StabilityFigure: Directional Stability


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Roll ControlAilerons

Ailerons are control surfaces attached tothe wings. They are similar to elevators,but move in opposite directions.

A δ a is dened as• A downwards deection on the left.• An upwards deection on the right.


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Roll ControlAilerons

The change in lift on the right wing is

∆ C Lα = − τ a C Lα δ a

The change in lift on the left wing ispositive. The overall momentcontribution is described as

C l = C lβ β + C lδ a δ a

C lδ a = 2C Lα τ

Sb y2

y 1cydy ∼= 2C Lα τ

ycb


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Directional StabilityExample: Cessna Skyhawk


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Roll StabilityExample: Dassault Rafale


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Directional StabilityExample: Tu-154


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Directional StabilityExample: Harrier


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Directional StabilityExample: Mig 29


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Directional StabilityExample: Frégate


l b l

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Directional StabilityExample: Mirage


Di i l S bili

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Directional StabilityExample: A10


Di i l S bili

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Directional StabilityExample: P.166


R ll I t bilit ith I ti

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Roll Instability with InertiaFalling Leaf Mode

If we truly couple the dynamics of horizontal motion with roll moment, we get

ÿ

φ̈ = K y

φ

When an eigenvalue of K has positive real part, we get motion akin to the“falling leaf” mode, which was unstable in the F/A-18.

This mode will NOT be present in static stability dynamics


F lli g L f M d

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Falling Leaf ModeExample: Elevator-Induced


Falling Leaf Mode

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Falling Leaf ModeExample: F/A-18 Cockpit Camera


Conclusion

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ConclusionWe need 6DOF

The dynamics of aircraft are described by 6 degrees of freedom:Rotational Coordinates

• Roll• Pitch•

YawTranslational Coordinates

• Nose• Right• Down

Because body-xed coordinates are dened in terms of rotation angles,equations for rotational and translational motion cannot be decoupled foraircraft.


Conclusion

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Conclusion

In this Lecture, you learned:

Roll Stability• Contributions to Roll Stability or instability.

Roll Contributions result from coupling with ψ .Not really static Stability.• Control Surfaces.• A design example.• Motivation for 6DOF analysis.


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Review: Static Stability

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Review: Static StabilityLongitudinal Stability

Moment Equation:C m = C m 0 + C mα α + C mδ e

Expressions for C mα and C mδ

C M 0 ,wf + ηV H C Lα,t (ε 0 + iwf − i t )

C mα = C Lα,wf X CG

c̄ −

X AC,wf c̄

− ηV H C Lα,t 1 −dεdα

C mδ e = − ηV H C Lα,t τ

• Stable if C mα < 0• α eq = − C m 0C mα• Neutral point when C mα = 0



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Review: Static StabilityDirectional Stability

C n = C nβ β + C nδ r δ r

whereC nδ r = − ηv V v C Lα,v τ,

• Stable if C nβ > 0• Use plots/data to nd C nβ



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Review: Static StabilityRoll Stability

C l = C lβ β + C lδ a δ a

whereC lδ a

∼= 2C Lα τ ycb

• Stable if C lβ < 0• Use plots/data to nd C lβ


441 lecture 8

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