Mae 331 Lecture 12

Linearized Lateral-Directional Equations of Motion

Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012"

•  Spiral, Dutch roll, and roll modes"

•  Stability 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 + pwyI = cosθ sinψ( )u + cosφ cosψ + sinφ sinθ sinψ( )v + − sinφ cosψ + cosφ sinθ sinψ( )w

p = IzzL + IxzN − Ixz Iyy − Ixx − Izz( ) p + Ixz2 + Izz Izz − Iyy( )%& '(r{ }q( ) ÷ Ixx Izz − Ixz

2( )r = IxzL + IxxN − Ixz Iyy − Ixx − Izz( )r + Ixz

2 + Ixx Ixx − Iyy( )%& '( p{ }q( ) ÷ Ixx Izz − Ixz2( )

φ = p + qsinφ + r cosφ( ) tanθψ = qsinφ + r cosφ( )secθ

x1x2x3x4x5x6

!

"

########

$

%

&&&&&&&&

= xLD6 =

vyprφ

ψ

!

"

########

$

%

&&&&&&&&

=

Side VelocityCrossrange

Body − Axis Roll RateBody − Axis Yaw Rate

Roll Angle about Body x AxisYaw Angle about Inertial x Axis

!

"

########

$

%

&&&&&&&&

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 = IzzL + IxzN − Ixz Iyy − Ixx − Izz( ) p + Ixz2 + Izz Izz − Iyy( )$% &'r{ }q( ) ÷ Ixx Izz − Ixz

2( )r = IxzL + IxxN − Ixz Iyy − Ixx − Izz( )r + Ixz

2 + Ixx Ixx − Iyy( )$% &' p{ }q( ) ÷ Ixx Izz − Ixz2( )

φ = p + qsinφ + r cosφ( ) tanθ

x1x2x3x4

!

"

#####

$

%

&&&&&

= xLD4 =

vprφ

!

"

#####

$

%

&&&&&

=

Side VelocityBody − Axis Roll RateBody − Axis Yaw Rate

Roll Angle about Body x Axis

!

"

#####

$

%

&&&&&

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 = IzzLB + IxzNB( ) Ixx Izz − Ixz2( )

r = IxzLB + IxxNB( ) Ixx Izz − Ixz2( )

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

qN = 0γN = 0θN =αN

Lockheed F-117!

Lateral-Directional Force and Moments"

YB =CYB

12ρNVN

2S; Body− Axis Side Force

LB =ClB

12ρNVN

2Sb; Body− Axis Rolling Moment

NB =CnB

12ρNVN

2Sb; Body− Axis Yawing Moment

Linearized Equations of Motion�

Body-Axis Perturbation Equations of Motion"

Δ v(t)Δp(t)Δr(t)Δ φ(t)

#

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%%%%%

&

'

(((((

=

∂ 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

#

$

%%%%%%%%%%%

&

'

(((((((((((

Δv(t)Δp(t)Δr(t)Δφ(t)

#

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&

'

(((((

+ Control[ ]+ Disturbance[ ]

Body-Axis Perturbation Variables"

Δu1Δu2

"

#$$

%

&''=

ΔδAΔδR

"

#$

%

&' =

Aileron PerturbationRudder Perturbation

"

#$$

%

&''

Δw1Δw2

"

#$$

%

&''=

ΔδAΔδR

"

#$

%

&' =

SideWind PerturbationVortical Wind Perturbation

"

#$$

%

&''

ΔvΔpΔrΔφ

#

$

%%%%%

&

'

(((((

=

Side Velocity PerturbationBody − Axis Roll Rate PerturbationBody − Axis Yaw Rate Perturbation

Roll Angle about Body x Axis Perturbation

#

$

%%%%%

&

'

(((((

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 of side velocity!

~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

#

$

%%%%%

&

'

(((((

Stability Matrix!

=

Yv Yp +wN( ) Yr −uN( ) gcosθNLv Lp Lr 0

Nv Np Nr 0

0 1 tanθN 0

#

$

%%%%%%

&

'

((((((

Dimensional Stability-and-Control Derivatives"∂ f1 ∂δA ∂ f1 ∂δR∂ f2 ∂δA ∂ f2 ∂δR∂ f3 ∂δA ∂ f3 ∂δR∂ f4 ∂δA ∂ f4 ∂δR

#

$

%%%%%

&

'

(((((

=

YδA YδRLδA LδRNδA NδR

0 0

#

$

%%%%%

&

'

(((((

∂ f1 ∂vwind ∂ f1 ∂ pwind∂ f2 ∂vwind ∂ f2 ∂ pwind∂ f3 ∂vwind ∂ f3 ∂ pwind∂ f4 ∂vwind ∂ f4 ∂ pwind

"

#

$$$$$

%

&

'''''

=

Yv YpLv Lp

Nv Np

0 0

"

#

$$$$$

%

&

'''''

Control Effect Matrix!

Disturbance Effect Matrix!

Stability Axes�

Stability Axes"•  Alternative set of body 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"

HBS =

cosαN 0 sinαN

0 1 0−sinαN 0 cosαN

#

$

%%%

&

'

(((

ΔuΔvΔw

"

#

$$$

%

&

'''S

= HBS

ΔuΔvΔw

"

#

$$$

%

&

'''B

ΔpΔqΔr

"

#

$$$

%

&

'''S

= HBS

ΔpΔqΔr

"

#

$$$

%

&

'''B

•  Side velocity (Δv) and pitch rate (Δq) are unchanged by the transformation "

Stability-Axis State "•  Rotate body axes to stability axes"


#

$

%%%%%

&

'

(((((Body−Axis

⇒αN ⇒


#

$

%%%%%

&

'

(((((Stability−Axis

Stability-Axis State"


#

$

%%%%%

&

'


⇒Δβ ≈ΔvVN

⇒

Δβ(t)Δp(t)Δr(t)Δφ(t)

#

$

%%%%%

&

'


•  Replace side velocity by sideslip angle"

Stability-Axis State"

Δβ(t)Δp(t)Δr(t)Δφ(t)

$

%

&&&&&

'

(

)))))Stability−Axis

⇒

Δr(t)Δβ(t)Δp(t)Δφ(t)

$

%

&&&&&

'

(

)))))Stability−Axis

=

Stability− Axis Yaw Rate PerturbationSideslip Angle Perturbation

Stability− Axis Roll Rate PerturbationStability− Axis Roll Angle Perturbation

$

%

&&&&&

'

(

)))))

•  Revise state order"

Stability-Axis Lateral-Directional Equations"

Δr(t)Δ β(t)Δp(t)Δ φ(t)

$

%

&&&&&

'

(

)))))S

=

Nr Nβ Np 0

YrVN

−1+

,-

.

/0

YβVN

YpVN

gcosγNVN

Lr Lβ Lp 0

tanγN 0 1 0

$

%

&&&&&&&

'

(

)))))))S

Δr(t)Δβ(t)Δp(t)Δφ(t)

$

%

&&&&&

'

(

)))))S

+

NδA NδR

YδAVN

YδRVN

LδA LδR0 0

$

%

&&&&&&

'

(

))))))S

ΔδA(t)ΔδR(t)

$

%&&

'

())+

Nβ Np

YβVN

YpVN

Lβ Lp

0 0

$

%

&&&&&&&

'

(

)))))))S

ΔβwindΔpwind

$

%&&

'

())

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

FDRRS FRS

!

"

##

$

%

&&=

FDR smallsmall FRS

!

"##

$

%&&≈

FDR 00 FRS

!

"##

$

%&&

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

FDRRS FRS

!

"

##

$

%

&&=

Nr Nβ Np 0

YrVN

−1)

*+

,

-.

YβVN

YpVN

gcosγNVN

Lr Lβ Lp 0

tanγN 0 1 0

!

"

#######

$

%

&&&&&&&

GLD =

NδA NδR

YδAVN

YδRVN

LδA LδR0 0

"

#

$$$$$$

%

&

''''''

LLD =

Nβ Np

YβVN

YpVN

Lβ Lp

0 0

"

#

$$$$$$$

%

&

'''''''

Δx1Δx2Δx3Δx4

"

#

$$$$$

%

&

'''''

=

ΔrΔβ

ΔpΔφ

"

#

$$$$$

%

&

'''''

Δu1Δu2

"

#$$

%

&''=

ΔδAΔδR

"

#$

%

&'

Δw1Δw2

"

#$$

%

&''=

ΔδAΔδR

"

#$

%

&'

Lateral-Directional Stability Derivatives�

2nd-Order Approximate Modes of Lateral-Directional Motion�

2nd-Order Approximations in System Matrices"

FLD =FDR 00 FRS

!

"##

$

%&&=

Nr Nβ 0 0

YrVN

−1)

*+

,

-.

YβVN

0 0

0 0 Lp 0

0 0 1 0

!

"

#######

$

%

&&&&&&&

GLD =

NδA 0YδAVN

0

0 LδR0 0

"

#

$$$$$$

%

&

''''''

LLD =

Nβ 0

YβVN

0

0 Lp

0 0

"

#

$$$$$$$

%

&

'''''''

Second-Order Models of Lateral-Directional Motion"

•  Approximate Spiral-Roll Equation"

•  Approximate Dutch Roll Equation"

ΔxDR =ΔrΔ β

#

$%%

&

'((≈

Nr Nβ

YrVN

−1+

,-

.

/0

YβVN

#

$

%%%%

&

'

((((

ΔrΔβ

#

$%%

&

'((+

NδR

YδRVN

#

$

%%%

&

'

(((ΔδR+

Nβ

YβVN

#

$

%%%%

&

'

((((

Δβwind

ΔxRS =ΔpΔ φ

#

$%%

&

'((≈

Lp 0

1 0

#

$%%

&

'((

ΔpΔφ

#

$%%

&

'((+

LδA0

#

$%%

&

'((ΔδA +

Lp

0

#

$%%

&

'((Δpwind

Approximate Roll and Spiral Modes"

ΔpΔ φ

#

$%%

&

'((=

Lp 0

1 0

#

$%%

&

'((

ΔpΔφ

#

$%%

&

'((+

LδA0

#

$%%

&

'((ΔδA

ΔRS (s) = s s − Lp( )λS = 0λR = Lp

•  Characteristic polynomial has real roots"

•  Roll rate is damped by Lp"•  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!Lp ≈ Clp

ρVN2

2Ixx

#

$%

&

'(Sb =Clp̂

b2VN

#

$%

&

'(ρVN

2

2Ixx

#

$%

&

'(Sb

=Clp̂

ρVN4Ixx

#

$%

&

'(Sb2

Clp̂( )Wing

=∂ ΔCl( )Wing

∂ p̂= −

CLα

121+ 3λ1+ λ

&'(

)*+

•  Wing with taper"

•  Thin triangular wing"Clp̂( )

Wing= −

πAR32

Clp̂≈ Clp̂( )

Vertical Tail+ Clp̂( )

Horizontal Tail+ Clp̂( )

Wing

•  Vertical tail, horizontal tail, and wing are principal contributors"

< 0 for stability!

NACA-TR-1098, 1952!NACA-TR-1052, 1951 !

Roll Damping Due to Roll Rate, Lp!

•  Tapered vertical tail"

•  Tapered horizontal tail"

p̂ = pb2VN

Clp̂( )ht =∂ ΔCl( )ht∂ p̂

= −CLαht

12ShtS

%

&'

(

)*1+ 3λ1+λ

%

&'

(

)*

•  pb/2VN describes helix angle for a steady roll"

Clp̂( )vt =∂ ΔCl( )vt∂ p̂

= −CYβvt

12SvtS

%

&'

(

)*1+ 3λ1+λ

%

&'

(

)*

Approximate Dutch Roll Mode"

ΔrΔ β

#

$%%

&

'((=

Nr Nβ

YrVN

−1*

+,

-

./

YβVN

#

$

%%%%

&

'

((((

ΔrΔβ

#

$%%

&

'((+

NδR

YδRVN

#

$

%%%

&

'

(((ΔδR

ΔDR(s)= s2 − Nr +

YβVN

$

%&

'

()s+ Nβ 1−

YrVN( )+Nr

YβVN

*

+,-

./

ωnDR= Nβ 1−

YrVN( )+Nr

YβVN

ζDR = − Nr +YβVN

$

%&

'

() 2 Nβ 1−

YrVN( )+Nr

YβVN

ωnDR= Nβ +Nr

YβVN

ζDR = − Nr +YβVN

%

&'

(

)* 2 Nβ +Nr

YβVN

•  With negligible side-force sensitivity to yaw rate, Yr"

•  Characteristic polynomial, natural frequency, and damping ratio"

Initial Condition Response of Approximate Dutch Roll Mode"

Δr t( ) Δβ t( )

Side Force due to Sideslip Angle"

Y ≈∂CY

∂βqS•β =CYβ

qS•β

CYβ≈ CYβ( )Fuselage + CYβ( )Vertical Tail + CYβ( )Wing

CYβ( )Vertical Tail ≈∂CY

∂β

$

%&

'

()vt

ηvt

SVerticalTailS

CYβ( )Fuselage ≈ −2SBaseS; SB =

πdBase2

4

CYβ( )Wing ≈ −CDParasite,Wing− kΓ2

ηvt = Vertical tail efficiency

k = πAR1+ 1+ AR2

Γ = Wing dihedral angle, rad

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

Yawing Moment due to Sideslip Angle!

N ≈∂Cn

∂βρV 2

2%

&'

(

)*Sb•β =Cnβ

ρV 2

2%

&'

(

)*Sb•β

!  Side force contributions times respective moment arms"–  Non-dimensional stability

derivative"

Cnβ≈ Cnβ( )

Vertical Tail+ Cnβ( )

Fuselage+ Cnβ( )

Wing+ Cnβ( )

Propeller

Cnβ( )

Vertical Tail≈ −CYβvt

ηvtSvtlvtSb −CYβvt

ηvtVVT

Vertical tail contribution"

VVT =

SvtlvtSb

=Vertical Tail Volume Ratio

ηvt =ηelas 1+∂σ ∂β( ) Vvt2

VN2

%

&'

(

)*


lvt Vertical tail length (+)= distance from center of mass to tail center of pressure= xcm − xcpvt [x is positive forward; both are negative numbers]

Cnβ( )Fuselage =−2K VolumeFuselage

Sb

K = 1−dmax Lengthfuselage"

#$

%

&'1.3

Fuselage contribution"

Cnβ( )Wing = 0.75CLNΓ+ fcn Λ,AR,λ( )CLN

2

Wing (differential lift and induced drag) contribution"


Yaw Damping Due to Yaw Rate, Nr!•  Dimensional stability derivative"

Nr ≈ Cnr

ρVN2

2Izz

#

$%

&

'(Sb =Cnr̂

b2VN

#

$%

&

'(ρVN

2

2Izz

#

$%

&

'(Sb

=Cnr̂

ρVN4Izz

#

$%

&

'(Sb2 < 0 for stability!

•  High wing-sweep angle can lead to Nr > 0"

Martin Marietta X-24B!

Yaw Damping Due to Yaw Rate, Nr!Cnr̂

≈ Cnr̂( )Vertical Tail

+ Cnr̂( )Wing

Cnr̂( )Wing

= k0CL2 + k1CDParasite,Wing

k0 and k1 are functions of aspect ratio and sweep angle"

•  Wing contribution"

•  Vertical tail contribution"Δ Cn( )Vertical Tail = − Cnβ( )Vertical Tail

rlvtVN( ) = − Cnβ( )Vertical Tail

lvtb

$

%&

'

()bVN

$

%&

'

()r

r̂ = rb2VN

Cnr̂( )vt =∂Δ Cn( )Vertical Tail∂ rb

2VN( )=∂Δ Cn( )Vertical Tail

∂ r̂= −2 Cnβ( )Vertical Tail

lvtb

%

&'

(

)*

NACA-TR-1098, 1952!NACA-TR-1052, 1951 !

Comparison of Fourth- and Second-Order

Dynamic Models�

•  2nd-order-model eigenvalues are close to those of the 4th-order model"•  Eigenvalue magnitudes of Dutch roll and roll roots are similar"

Bizjet Fourth- and Second-Order Models and Eigenvalues "

Fourth-Order ModelF = G = Eigenvalue Damping Freq. (rad/s)

-0.1079 1.9011 0.0566 0 0 -1.1196 0.00883

-1 -0.1567 0 0.0958 0 0 -1.20.2501 -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 ApproximationF = G = Eigenvalue Damping Freq. (rad/s)

-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 ApproximationF = G = Eigenvalue Damping Freq. (rad/s)

-1.1616 0 2.3106 0

1 0 0 -1.16

Unstable!

Comparison of Second- and Fourth-Order Initial-Condition Responses of Business Jet"

Fourth-Order Response! Second-Order Response!

•  Speed and damping of responses is adequately portrayed by 2nd-order models"•  Roll-spiral modes have little effect on yaw rate and sideslip angle responses"•  Dutch roll mode has large effect on roll rate and roll angle responses"

Primary Lateral-Directional Control Derivatives"

LδA =ClδA

ρVN2

2Ixx

#

$%

&

'(Sb

NδR =CnδR

ρVN2

2Izz

#

$%

&

'(Sb

Next Time:�Analysis of Time Response�

�Reading�

Flight Dynamics, 298-314, 338-342 �

Virtual Textbook, Part 13 �

Supplemental Material"

Δ v(t)Δp(t)Δr(t)Δ φ(t)

#

$

%%%%%

&

'

(((((

=

Yv Yp +wN( ) Yr −uN( ) gcosθNLv Lp Lr 0

Nv Np Nr 0

0 1 tanθN 0

#

$

%%%%%%

&

'

((((((


#

$

%%%%%

&

'

(((((

+

YδA YδRLδA LδRNδA NδR

0 0

#

$

%%%%%

&

'

(((((

ΔδA(t)ΔδR(t)

#

$%%

&

'((+

Yv YpLv Lp

Nv Np

0 0

#

$

%%%%%

&

'

(((((

ΔvwindΔpwind

#

$%%

&

'((

•  Rolling and yawing motions"

Body-Axis Perturbation Equations of Motion"

Mae 331 Lecture 12

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