Aircraft Flight Dynamics Robert F. Stengel Lecture12 LateralDirectional Dynamics

Linearized Lateral-Directional and roll modes" derivatives" Copyright 2012 by Robert Stengel.. All rights reserved... but are the off-diagonal blocks really small?!. Dassault Rafale!.

Linearized Lateral-Directional

and roll modes"

derivatives"

Copyright 2012 by Robert Stengel All rights reserved For educational use only.!

http://www.princeton.edu/~stengel/MAE331.html !

http://www.princeton.edu/~stengel/FlightDynamics.html !

6-Component " Lateral-Directional Equations of Motion"

State Vector, 6 components!

Nonlinear Dynamic Equations!

v = Y / m + gsinφ cosθ − ru + pw

y I= cosθ sinψ ( )u + cosφ cosψ + sinφ sinθ sinψ( )v + − sinφ cosψ + cosφ sinθ sinψ( )w

p = I zz L + I xz N − I xz(I yy − I xx − I zz)p + I xz

2

+ I zz(I zz − I yy)

2

r = I xz L + I xx N − I xz(I yy − I xx − I zz)r + I xz

2

+ I xx(I xx − I yy)

2

φ = p + qsinφ + r cosφ( ) tanθ



ψ = q sinφ + r cosφ( ) secθ

x1

x2

x3

x4

x5

x6

!

"

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= xLD6 =

v y p r

φ ψ

!

"

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=

Side Velocity Crossrange Body − Axis Roll Rate Body − Axis Yaw Rate Roll Angle about Body x Axis Yaw Angle about Inertial x Axis

!

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Douglas A-4!

4- Component "

Lateral-Directional Equations of Motion"

State Vector, 4 components!

Nonlinear Dynamic Equations, neglecting crossrange and yaw angle!

v = Y / m + gsinφ cosθ − ru + pw

p = I zz L + I xz N − I xz(I yy − I xx − I zz)p + I xz

2

+ I zz(I zz − I yy)

2

r = I xz L + I xx N − I xz(I yy − I xx − I zz)r + I xz

2

+ I xx(I xx − I yy)

2

φ = p + qsinφ + r cosφ( )tanθ

x1

x2

x3

x4

!

"

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&

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= xLD4 =

v p r

φ

!

"

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$

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=

Side Velocity Body − Axis Roll Rate Body − Axis Yaw Rate Roll Angle about Body x Axis

!

"

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Eurofighter Typhoon!

Lateral-Directional Equations

of Motion Assuming Steady, Level Longitudinal Flight"

Nonlinear dynamic equations, assuming steady, level, flight (longitudinal variables are constant )!

v = YB / m + gsinφ cosθN − ruN + pwN

= YB / m + gsinφ cosαN − ruN+ pwN

p = I ( zzLB+ IxzNB) ( IxxIzz − Ixz2 )

r = I ( xzLB+ IxxNB) ( IxxIzz− Ixz2)

φ = p+ r cosφ ( ) tanθN = p + r cos ( φ ) tanαN

q N= 0

γN= 0

θN =αN

Lockheed F-117!

Lateral-Directional Force and

Moments"

1

2 ρNVN

2

S; Body − Axis Side Force

2 ρNVN

2

Sb; Body − Axis Rolling Moment

2 ρNVN

2

Sb; Body − Axis Yawing Moment

Linearized Equations

of Motion

Body-Axis Perturbation

Equations of Motion"

Δ v(t)

Δp(t)

Δr(t)

Δ  φ(t)

#

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&

'

(

(

(

(

(

=

∂ f1

∂ v

∂ f1

∂ f1

∂ r

∂ f1

∂φ

∂ f2

∂ v

∂ f2

∂ f2

∂ r

∂ f2

∂φ

∂ f3

∂ v

∂ f3

∂ f3

∂ r

∂ f3

∂φ

∂ f4

∂ v

∂ f4

∂ f4

∂ r

∂ f4

∂φ

#

$

%

%

%

%

%

%

%

%

%

%

%

&

'

( ( ( ( ( ( ( ( ( ( (

Δv(t) Δp(t) Δr(t)

Δφ(t)

#

$

%

%

%

%

%

&

'

( ( ( ( (

+ Control [ ] + Disturbance [ ]

Body-Axis Perturbation

Variables"

Δu1

Δu2

"

#

$

$

%

&

' '=

Δδ A

Δδ R

"

#

&

' =

Aileron Perturbation Rudder Perturbation

"

#

$

$

%

&

' '

Δw1

Δw2

"

#

$

$

%

&

' '=

Δδ A

Δδ R

"

#

&

' =

Side Wind Perturbation Vortical Wind Perturbation

"

#

$

$

%

&

' '

Δv Δp Δr

Δφ

#

$

%

%

%

%

%

&

'

( ( ( ( (

=

Side Velocity Perturbation Body − Axis Roll Rate Perturbation Body − Axis Yaw Rate Perturbation Roll Angle about Body x Axis Perturbation

#

$

%

%

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&

'

( ( ( ( (

Linearized Lateral-Directional

Response to Yaw Rate Initial

Condition"

~Roll-mode response of roll angle!

~Spiral-mode response of crossrange!

~Spiral-mode response of yaw angle!

~Dutch-roll-mode response

~Dutch-roll-mode

response of roll

and yaw rates!

Dimensional Stability-and-Control

Derivatives"

∂ f1 ∂ v ∂ f1 ∂ p ∂ f1 ∂ r ∂ f1 ∂φ

∂ f2 ∂ v ∂ f2 ∂ p ∂ f2 ∂ r ∂ f2 ∂φ

∂ f3 ∂ v ∂ f3 ∂ p ∂ f3 ∂ r ∂ f3 ∂φ

∂ f4 ∂ v ∂ f4 ∂ p ∂ f4 ∂ r ∂ f4 ∂φ

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'

( ( ( ( (

Stability Matrix!

=

Yv ( Yp+ wN) ( Yr− uN) g cos θN

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&

'

( ( ( ( ( (

Dimensional

Stability-and-Control Derivatives"

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'

( ( ( ( (

=

YδA YδR

LδA LδR

NδA NδR

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$

%

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&

'

( ( ( ( (

∂ f1 ∂ vwind ∂ f1 ∂ pwind

∂ f2 ∂ vwind ∂ f2 ∂ pwind

∂ f3 ∂ vwind ∂ f3 ∂ pwind

∂ f4 ∂ vwind ∂ f4 ∂ pwind

"

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$

%

&

' ' ' ' '

=

Yv Yp

Lv Lp

Nv Np

"

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$

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$

%

&

' ' ' ' '

Control

Effect Matrix!

Disturbance

Effect Matrix!

Stability Axes

Stability Axes"

•  Nominal x axis is offset from the body centerline by

the nominal angle of attack, αN "

Transformation from Original Body Axes to Stability Axes"

HB S =

cosαN 0 sinαN

−sinαN 0 cosαN

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'

( ( (

Δu Δv Δw

"

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' ' 'S

= HB S

Δu Δv Δw

"

#

$

$

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%

&

' ' 'B

Δp Δq Δr

"

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$

$

$

%

&

' ' '

S

=H B S

Δp Δq Δr

"

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$

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&

' ' '

B

•   Side velocity (Δv) and pitch rate (Δq) are unchanged

by the transformation "

Stability-Axis State "

Δv(t)

Δp(t)

Δr(t)

Δ φ (t)

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'

(

(

(

(

(

Body−Axis

⇒ αN ⇒

Δv(t) Δp(t) Δr(t)

Δ φ (t)

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'

( ( ( ( (

Stability−Axis

Stability-Axis State"

Δv(t) Δp(t) Δr(t)

Δ φ (t)

#

$

%

%

%

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&

'

( ( ( ( (

Stability−Axis

VN ⇒

Δ β (t)

Δp(t) Δr(t)

Δ φ (t)

#

$

%

%

%

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&

'

( ( ( ( (

Stability−Axis

•  Replace side velocity by sideslip angle"

Stability-Axis State"

Δβ(t)

Δp(t)

Δr(t)

Δφ(t)

$

%

&

&

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'

(

) ) ) ) )

Stability−Axis

⇒

Δr(t)

Δβ(t)

Δp(t)

Δφ(t)

$

%

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&

&

&

&

'

(

) ) ) ) )

Stability−Axis

=

Stability − Axis Yaw Rate Perturbation

Sideslip Angle Perturbation Stability − Axis Roll Rate Perturbation

Stability − Axis Roll Angle Perturbation

$

%

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'

(

) ) ) ) )

•  Revise state order"

Stability-Axis Lateral-Directional

Equations"

Δr(t)

Δ β(t) Δp(t)

Δ φ(t)

$

%

&

&

&

&

&

'

(

) ) ) ) )

S

=

Y r

V N −1

+ ,

/

0 Yβ

V N

Y p

V N

g cosγ N

V N

tanγN 0 1 0

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%

&

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'

(

) ) ) ) ) ) )

S

Δr(t)

Δβ(t)

Δp(t)

Δφ(t)

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%

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'

(

) ) ) ) )

S

+

N δA N δR

Y δA

V N

Y δR

V N

L δA L δR

$

%

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&

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&

'

(

) ) ) ) ) )

S

ΔδA(t) ΔδR(t)

$

%

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&

' (

) )+

Nβ N p

Yβ

V N

Y p

V N

Lβ L p

$

%

&

&

&

&

&

&

&

'

(

) ) ) ) ) ) )S

Δβwind

Δp wind

$

%

&

&

' (

) )

Why Modify the Equations?"

•   Dutch-roll mode is primarily described by stability-axis yaw

rate and sideslip angle"

•   Roll and spiral mode are primarily described by stability-axis

roll rate and roll angle"

•   Linearized equations allow the three modes to be studied"

Stable Spiral!

Unstable Spiral!

Roll!

Dutch Roll, top! Dutch Roll, front!

Why Modify the Equations?"

FLD= FDR FRS

DR

FDR RS FRS

!

"

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&=

FDR small small FRS

!

"

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&≈

FDR 0

0 FRS

!

"

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Effects of Dutch roll perturbations

on Dutch roll motion"

Effects of Dutch roll perturbations

on roll-spiral motion"

Effects of roll-spiral perturbations

on Dutch roll motion"

Effects of roll-spiral perturbations on roll-spiral motion"

but are the off-diagonal blocks really small?!

Dassault Rafale!

Stability, Control, and

Disturbance Matrices"

FLD= FDR FRS

DR

FDR RS FRS

!

"

#

#

$

%

&

&=

N r Nβ N p 0

Y r

V N −1

)

*

V N

Y p

V N

g cosγ N

V N

L r Lβ L p 0

!

"

#

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GLD=

NδA NδR

YδA

YδR

LδA LδR

"

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$

$

$

$

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$

%

&

' ' ' ' ' '

LLD=

Yβ

V N

Y p

V N

"

#

$

$

$

$

$

$

$

%

&

' ' ' ' ' ' '

Δx1

Δx2

Δx3

Δx4

"

#

$

$

$

$

%

&

' ' ' '

=

Δr

Δ

Δp

Δφ

"

#

$

$

$

$

%

&

' ' ' '

Δu1

Δu2

"

#

$ %

&

' = Δδ A

Δδ R

"

#

%

&

Δw1

Δw2

"

#

$ %

&

' = ΔδA ΔδR

"

#

%

&

Lateral-Directional Stability Derivatives

2 nd -Order Approximate

Modes of

Lateral-Directional Motion

2 nd -Order Approximations

in System Matrices"

FLD= FDR 0

0 FRS

!

"

#

#

$

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&

&=

N r Nβ 0 0

Y r

V N −1

)

*

!

"

#

#

#

#

#

#

#

$

%

&

&

&

&

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&

&

GLD=

NδA 0

YδA

"

#

$

$

$

$

$

$

%

&

' ' ' ' ' '

LLD=

Yβ

V N 0

"

#

$

$

$

$

$

$

$

%

&

' ' ' ' ' ' '

Second-Order Models of

Lateral-Directional Motion"

•   Approximate Spiral-Roll Equation"

•   Approximate Dutch Roll Equation"

ΔxDR= Δr

Δ  β

#

$

%

%

&

'

( (≈

Y r

V N

−1 + ,

/

0 Yβ

V N

#

$

%

%

%

%

&

'

( ( ( (

Δr

Δβ

#

$

%

%

&

'

( (+

N δR

Y δR

V N

#

$

%

%

%

&

'

( ( ( ΔδR +

Nβ

Yβ

V N

#

$

%

%

%

%

&

'

( ( ( (

Δβwind

Δ φ

#

$

%

%

&

'

( (≈

L p 0

#

$

%

%

&

'

( (

Δp

Δφ

#

$

%

%

&

'

( (+

L δ A

0

#

$

%

%

&

'

( (Δδ A +

L p

0

#

$

%

%

&

'

( (Δp wind

Approximate Roll and Spiral Modes"

Δp

Δ  φ

#

$

%

%

&

'

( ( =

Lp 0

1 0

#

$

%

%

&

'

( (

Δp

Δ φ

#

$

%

%

&

'

( ( +

Lδ A

0

#

$

%

%

& '

( ( Δ δ

A

ΔRS(s) = s s − L ( p)

λS = 0

λR = Lp

•   Characteristic polynomial has real roots"

•  Roll rate is damped by L p"

•  Roll angle is a pure integral of roll rate"

Δp t( ) Δφ t( )

•   Initial condition response"

Neutral stability!

Generally < 0!

Roll Damping Due to Roll Rate, Lp!

L p ≈ C l p

ρV N

2

#

$

'

b

#

$

'

2

#

$

'

(Sb

= C l ˆp

ρV N

#

$

'

C l

ˆp

( )Wing=∂ ΔC( l)Wing

C L

α 12

1 + 3λ

1 + λ

&

'(

)

*+

C l ˆp

( )Wing = −π AR

32

C l

ˆp ≈ C( )l ˆp Vertical Tail + C( )l ˆp Horizontal Tail + C( )l ˆp Wing

principal contributors"

< 0 for stability!

NACA-TR-1098, 1952!

NACA-TR-1052, 1951 !

Roll Damping Due

•  Tapered vertical tail"

•  Tapered horizontal tail"

ˆp = pb

2V N

C l ˆp

( )ht =∂ ΔC( l)ht

∂ ˆp = −

C L

αht

12

S ht S

%

&

' ( )

* 1+ 3λ 1+ λ

%

&

)

*

•  pb/2V Ndescribes helix

C l ˆp

( )vt =∂(ΔC l)vt

∂ˆp = −

C Y

βvt

12

S vt S

%

&

)

* 1+ 3λ 1+λ

%

&

)

*

Approximate Dutch

Roll Mode"

Δr

Δ  β

#

$

%

%

&

'

( (=

Y r

V N

−1

* +

- / Yβ

V N

#

$

%

%

%

%

&

'

( ( ( (

Δr

Δ β

#

$

%

%

&

'

( (+

N δR

Y δR

V N

#

$

%

%

%

&

'

( ( (

Δ δR

ΔDR (s) = s2

− N r+ β

V N

$

%

' (

)s + Nβ 1−Y r

V N

( )+ N r

Yβ

V N

* +,

-./

ωn DR = Nβ 1−Y r

V N

( )+ N r Yβ

V N

ζDR = − N r+ β

V N

$

%

' (

) 2 Nβ 1−Y r

V N

( )+ N r

Yβ

V N

Yβ

V N

ζDR = − N r+ β

V N

%

&

( )

* 2 Nβ+ N r

Yβ

V N

•   With negligible

side-force

sensitivity to

yaw rate, Y r"

•   Characteristic

polynomial,

natural

frequency, and

damping ratio"

Initial Condition Response of Approximate Dutch Roll Mode"

Y ≈ ∂CY

∂β qS • β = CYβqS • β

CY

β ≈ C ( )Yβ Fuselage+ C ( )Yβ Vertical Tail+ C ( )Yβ Wing

C Yβ

( )Vertical Tail≈ ∂C Y

∂β

$

%

& '

(

)

vt

ηvt S Vertical Tail

S

C Y

β

( )Fuselage≈ −2S Base

πd Base2

4

C Yβ

( )Wing ≈ −C D Parasite, Wing − kΓ2

ηvt= Vertical tail efficiency

k = π AR

Γ = Wing dihedral angle, rad

•  Fuselage, vertical tail, and wing are main contributors"

N ≈ ∂C n

∂β

ρV2

2

%

&

)

*Sb • β = C nβ

ρV2

2

%

&

)

*Sb • β

!  Side force contributions times respective moment arms"

– Non-dimensional stability derivative"

Cnβ ≈ C( )nβ Vertical Tail + C( )nβ Fuselage+ C( )nβ Wing + C( )nβ Propeller

( )C nβ Vertical Tail ≈ −C Y βvtηvt

S vt l vt

Sb  −C Y βvtηvtVVT

Vertical tail contribution"

VVT =S vt l vt

ηvt=ηelas(1+∂σ∂β) V vt

2

V N

2

%

&

)

*

l vt Vertical tail length (+)

= distance from center of mass to tail center of pressure

= x cm − x cp vt [x is positive forward; both are negative numbers]

C nβ

( )Fuselage=−2K Volume Fuselage

Sb

K = 1− d max

Length fuselage

"

#

%

&

1.3

Fuselage contribution"

C nβ

( )Wing = 0.75C L N Γ + fcn Λ, AR,λ( )C L2N

Wing (differential lift and induced drag) contribution"

•  Dimensional stability derivative "

Nr ≈ Cn

r

ρVN

2

2Izz

#

$

'

(Sb = Cn ˆr

b

2 VN

#

$

'

2

2Izz

#

$

'

(Sb

= Cn ˆr

ρVN

4Izz

#

$

'

(Sb2 < 0 for stability!

•  High

wing-sweep angle

can lead to

N r > 0"

Martin Marietta X-24B!

Yaw Damping Due

to Yaw Rate, Nr!

C n ˆr ≈ C( )n ˆr Vertical Tail + C( )n ˆr Wing

Cn ˆr

Parasite, Wing

k 0 and k 1 are functions of

•   Wing contribution"

•   Vertical tail contribution"

Δ C( )n Vertical Tail = − C( )nβ Vertical Tail

rl vt

V N

( )= − C( )nβ Vertical Tail

l vt b

$

%

' (

b

V N

$

%

& ' (

)r

ˆr = rb

2V N

C n ˆr

( )vt=∂Δ C( )n Vertical Tail

N

∂Δ C( )n Vertical Tail

l vt b

%

&

( )

NACA-TR-1098, 1952!

Comparison of Fourth-

and Second-Order

Dynamic Models

•   2 nd -order-model eigenvalues are close to those of the 4 th -order model"

•   Eigenvalue magnitudes of Dutch roll and roll roots are similar"

Bizjet Fourth- and Second-Order Models and Eigenvalues "

Fourth-Order Model

-0.1079 1.9011 0.0566 0 0 -1.1196 0.00883

-1 -0.1567 0 0.0958 0 0 -1.2 0.2501 -2.408 -1.1616 0 2.3106 0 -1.16e-01 + 1.39e+00j 8.32E-02 1.39E+00

0 0 1 0 0 0 -1.16e-01 - 1.39e+00j 8.32E-02 1.39E+00

Dutch Roll Approximation

-0.1079 1.9011 -1.1196 -1.32e-01 + 1.38e+00j 9.55E-02 1.38E+00 -1 -0.1567 0 -1.32e-01 - 1.38e+00j 9.55E-02 1.38E+00

Roll-Spiral Approximation

Unstable!

Comparison of Second- and Fourth-Order

Initial-Condition Responses of Business Jet"

Fourth-Order Response! Second-Order Response!

Primary Lateral-Directional Control Derivatives"

LδA = Cl δA ρ VN

2

2Ixx

#

$

'

(Sb

NδR = Cn δR ρ VN

2

2Izz

#

$

'

(Sb

Next Time:

Analysis of Time Response

Reading

338-342

Supplemental Material"

Δ v(t)

Δp(t)

Δr(t)

Δ φ(t)

#

$

%

%

%

%

%

&

'

(

(

(

(

(

=

Y v (Y p+w N) (Y r−u N) g cosθN

#

$

%

%

%

%

%

%

&

'

( ( ( ( ( (

Δv(t) Δp(t) Δr(t)

Δφ(t)

#

$

%

%

%

%

%

&

'

( ( ( ( (

+

Y δA Y δR

L δA L δR

N δA N δR

#

$

%

%

%

%

%

&

'

( ( ( ( (

ΔδA(t)

ΔδR(t)

#

$

%

%

&

'

( (+

Y v Y p

L v L p

N v N p

#

$

%

%

%

%

%

&

'

( ( ( ( (

Δv wind

Δp wind

#

$

%

%

&

'

( (

•  Rolling and yawing motions"

Body-Axis Perturbation

Equations of Motion"