Linearized Lateral-Directional

Equations of MotionRobert Stengel, Aircraft Flight Dynamics MAE 331, 2010

• Spiral, Dutch roll, and roll modes

• Stability derivatives

Roll-mode responseof roll angle

Spiral-moderesponse ofcrossrange

Spiral-mode responseof yaw angle

Dutch-roll-mode responseof side velocity

Dutch-roll-moderesponse of rolland yaw rates

Copyright 2010 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-DirectionalEquations of Motion

State Vector,6 components

Nonlinear Dynamic Equations

!v = Y / m + gsin! cos" # ru + pw

!yI= cos" sin$( )u + cos! cos$ + sin! sin" sin$( )v + # sin! cos$ + cos! sin" sin$( )w

!p = IzzL + I

xzN # I

xzIyy# I

xx# I

zz( ) p + Ixz

2+ I

zzIzz# I

yy( )%& '(r{ }q( ) ÷ IxxIzz# I

xz

2( )

!r = IxzL + I

xxN # I

xzIyy# I

xx# I

zz( )r + Ixz

2+ I

xxIxx# I

yy( )%& '( p{ }q( ) ÷ IxxIzz# I

xz

2( )!! = p + qsin! + r cos!( ) tan"

!$ = qsin! + r cos!( )sec"

x1

x2

x3

x4

x5

x6

!

"

########

$

%

&&&&&&&&

= xLD6=

v

y

p

r

'

(

!

"

########

$

%

&&&&&&&&

=

Side Velocity

Crossrange

Body ) Axis Roll Rate

Body ) Axis Yaw Rate

Roll Angle about Body x Axis

Yaw 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 + gsin! cos" # ru + pw

!p = IzzL + I

xzN # I

xzIyy# I

xx# I

zz( ) p + Ixz

2+ I

zzIzz# I

yy( )$% &'r{ }q( ) ÷ IxxIzz# I

xz

2( )

!r = IxzL + I

xxN # I

xzIyy# I

xx# I

zz( )r + Ixz

2+ I

xxIxx# I

yy( )$% &' p{ }q( ) ÷ IxxIzz# I

xz

2( )!! = p + qsin! + r cos!( ) tan"

x1

x2

x3

x4

!

"

#####

$

%

&&&&&

= xLD4=

v

p

r

'

!

"

#####

$

%

&&&&&

=

Side Velocity

Body ( Axis Roll Rate

Body ( 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, longitudinal flight

(constant longitudinal variables)

!v = YB/ m + gsin! cos"

N# ru

N+ pw

N

= YB/ m + gsin! cos$

N# ru

N+ pw

N

!p =IzzLB+ I

xzN

B( )IxxIzz# I

xz

2( )

!r =IxzLB+ I

xxN

B( )IxxIzz# I

xz

2( )!! = p + r cos!( ) tan"N

= p + r cos!( ) tan$N

qN= 0

!N= 0

"N= #

N

Lockheed F-117

Lateral-Directional Force

and Moments

YB = CYB

1

2!NVN

2S; Body " Axis Side Force

LB = ClB

1

2!NVN

2Sb; Body " Axis Rolling Moment

NB = CnB

1

2!NVN

2Sb; Body " Axis Yawing Moment

Linearized Equationsof Motion

Body-Axis Perturbation

Equations of Motion

!!x

LD(t) = F

LD!x

LD(t) +G

LD!u

LD(t) + L

LD!w

LD(t)

! !v(t)

!p(t)

!r(t)

!"(t)

#

$

%%%%%

&

'

(((((

=

) f1

)v) f

1

) p) f

1

)r) f

1

)"

) f2

)v) f

2

) p) f

2

)r) f

2

)"

) f3

)v) f

3

) p) f

3

)r) f

3

)"

) f4

)v) f

4

) p) f

4

)r) f

4

)"

#

$

%%%%%%%%%%%

&

'

(((((((((((

!v(t)

!p(t)

!r(t)

!"(t)

#

$

%%%%%

&

'

(((((

+

) f1

)*A) f

1

)*R) f

2

)*A) f

2

)*R) f

3

)*A) f

3

)*R) f

4

)*A) f

4

)*R

#

$

%%%%%%%%%%

&

'

((((((((((

!*A(t)

!*R(t)

#

$%%

&

'((+

) f1

)vwind

) f1

) pwind) f

2

)vwind

) f2

) pwind) f

3

)vwind

) f3

) pwind) f

4

)vwind

) f4

) pwind

#

$

%%%%%%%%%%%

&

'

(((((((((((

!vwind!pwind

#

$%%

&

'((

!u1

!u2

"

#

$$

%

&

''=

!(A

!(R

"

#$

%

&' =

Aileron Perturbation

Rudder Perturbation

"

#

$$

%

&

''

!w1

!w2

"

#

$$

%

&

''=

!(A

!(R

"

#$

%

&' =

SideWind Perturbation

Vortical Wind Perturbation

"

#

$$

%

&

''

• Perturbations from wings-level flight vN= 0; p

N= 0; r

N= 0; !

N= 0

!v!p

!r!"

#

$

%%%%%

&

'

(((((

=

Side Velocity Perturbation

Body ) Axis Roll Rate Perturbation

Body ) Axis Yaw Rate Perturbation

Roll Angle about Body x Axis Perturbation

#

$

%%%%%

&

'

(((((

Typical

Lateral-Directional Initial-Condition

Linearized Response

Roll-mode responseof roll angle

Spiral-moderesponse ofcrossrange

Spiral-moderesponse of yawangle

Dutch-roll-mode responseof side velocity

Dutch-roll-moderesponse of rolland yaw rates

Dimensional Stability-

and-Control Derivatives! f

1

!v! f

1

! p! f

1

!r! f

1

!"

! f2

!v! f

2

! p! f

2

!r! f

2

!"

! f3

!v! f

3

! p! f

3

!r! f

3

!"

! f4

!v! f

4

! p! f

4

!r! f

4

!"

#

$

%%%%%%%%%%%

&

'

(((((((((((

=

Yv Yp + wN( ) Yr ) uN( ) gcos*N

Lv Lp Lr 0

Nv Np Nr 0

0 1 tan*N 0

#

$

%%%%%%

&

'

((((((

! f1

!"A

! f1

!"R

! f2

!"A

! f2

!"R

! f3

!"A

! f3

!"R

! f4

!"A

! f4

!"R

#

$

%%%%%%%%%%

&

'

((((((((((

=

Y"A Y"R

L"A L"R

N"A N"R

0 0

#

$

%%%%%

&

'

(((((

! f1

!vwind

! f1

! pwind

! f2

!vwind

! f2

! pwind

! f3

!vwind

! f3

! pwind

! f4

!vwind

! f4

! pwind

"

#

$$$$$$$$$$$

%

&

'''''''''''

=

Yv Yp

Lv Lp

Nv Np

0 0

"

#

$$$$$

%

&

'''''

StabilityMatrix

ControlEffect Matrix

DisturbanceEffect Matrix

v

p

r

!

"

#

$

$

$

$

%

&

'

'

'

'

=

Side Velocity

Body ( Axis Roll Rate

Body ( Axis Yaw Rate

Roll Angle about Body x Axis

"

#

$

$

$

$

%

&

'

'

'

'

! !v(t)

!!p(t)

!!r(t)

! !"(t)

#

$

%%%%%

&

'

(((((

=

Yv Yp + wN( ) Yr ) uN( ) gcos*N

Lv Lp Lr 0

Nv Np Nr 0

0 1 tan*N 0

#

$

%%%%%%

&

'

((((((

!v(t)

!p(t)

!r(t)

!"(t)

#

$

%%%%%

&

'

(((((

+

Y+A Y+R

L+A L+R

N+A N+R

0 0

#

$

%%%%%

&

'

(((((

!+A(t)

!+R(t)

#

$%%

&

'((+

Yv Yp

Lv Lp

Nv Np

0 0

#

$

%%%%%

&

'

(((((

!vwind!pwind

#

$%%

&

'((

• Rolling and yawing motions

Body-Axis Perturbation

Equations of Motion

Stability Axes

Stability Axes• Alternative set of body axes

• Nominal x axis is offset from the body centerline bythe nominal angle of attack, !N

Transformation fromOriginal Body Axesto Stability Axes

HB

S=

cos!N

0 sin!N

0 1 0

" sin!N

cos!N

#

$

%%%

&

'

(((

!u

!v

!w

"

#

$$$

%

&

'''S

= HB

S

!u

!v

!w

"

#

$$$

%

&

'''B

!p

!q

!r

"

#

$$$

%

&

'''S

= HB

S

!p

!q

!r

"

#

$$$

%

&

'''B

• Side velocity (!v) and pitch rate (!q) are unchanged by the transformation

Stability-Axis Lateral-Directional Equations

• Rotate body axes to stability axes

• Replace side velocity by sideslip angle

• Revise state order

!" #!v

VN

!v(t)

!p(t)

!r(t)

!"(t)

#

$

%%%%%

&

'

(((((Body)Axis

*

!v(t)

!p(t)

!r(t)

!"(t)

#

$

%%%%%

&

'

(((((Stability)Axis

!v(t)

!p(t)

!r(t)

!"(t)

#

$

%%%%%

&

'

(((((Stability)Axis

*

!+(t)

!p(t)

!r(t)

!"(t)

#

$

%%%%%

&

'

(((((Stability)Axis

!"(t)

!p(t)

!r(t)

!#(t)

$

%

&&&&&

'

(

)))))Stability*Axis

+

!r(t)

!"(t)

!p(t)

!#(t)

$

%

&&&&&

'

(

)))))Stability*Axis

Stability-Axis Lateral-Directional Equations

!!r(t)

! !"(t)!!p(t)

! !#(t)

$

%

&&&&&

'

(

)))))S

=

Nr N" Np 0

Yr

VN*1

+

,-.

/0Y"

VN

Yp

VN

gcos1 N

VN

Lr L" Lp 0

tan1 N 0 1 0

$

%

&&&&&&&

'

(

)))))))S

!r(t)!"(t)!p(t)!#(t)

$

%

&&&&&

'

(

)))))S

+

N2A N2R

Y2A

VN

Y2R

VN

L2A L2R

0 0

$

%

&&&&&&

'

(

))))))S

!2A(t)!2R(t)

$

%&&

'

())+

N" Np

Y"

VN

Yp

VN

L" Lp

0 0

$

%

&&&&&&&

'

(

)))))))S

!"wind

!pwind

$

%&&

'

())

!x1

!x2

!x3

!x4

"

#

$$$$$

%

&

'''''

=

!r!(

!p

!)

"

#

$$$$$

%

&

'''''

=

Stability * Axis Yaw Rate Perturbation

Sideslip Angle Perturbation

Stability * Axis Roll Rate Perturbation

Stability * Axis Roll Angle Perturbation

"

#

$$$$$

%

&

'''''

Why Modify the Equations?

• Dutch-roll mode is primarily described by stability-axis yawrate and sideslip angle

• Roll and spiral mode are primarily described by stability-axisroll 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 Modifythe Equations?

FLD

=FDR

FRS

DR

FDR

RSFRS

!

"

##

$

%

&&=

FDR

small

small FRS

!

"

##

$

%

&&'

FDR

~ 0

~ 0 FRS

!

"

##

$

%

&&

Effects of Dutch roll perturbations

on Dutch roll motion

Effects of Dutch roll perturbations

on roll-spiral motion

Effects of roll-spiral perturbationson Dutch roll motion

Effects of roll-spiral perturbations on

roll-spiral motion

... but are the off-diagonal blocks really small?

Dassault Rafale

Stability, Control, andDisturbance Matrices

FLD =FDR FRS

DR

FDRRS

FRS

!

"

##

$

%

&&=

Nr N' Np 0

Yr

VN(1

)

*+,

-.Y'

VN

Yp

VN

gcos/ N

VN

Lr L' Lp 0

tan/ N 0 1 0

!

"

#######

$

%

&&&&&&&

GLD

=

N!A N!R

Y!A

VN

Y!R

VN

L!A L!R

0 0

"

#

$$$$$$

%

&

''''''

LLD

=

N! Np

Y!

VN

Yp

VN

L! Lp

0 0

"

#

$$$$$$$

%

&

'''''''

!x1

!x2

!x3

!x4

"

#

$$$$$

%

&

'''''

=

!r!(

!p

!)

"

#

$$$$$

%

&

'''''

!u1

!u2

"

#

$$

%

&

''=

!(A

!(R

"

#$

%

&'

!w1

!w2

"

#

$$

%

&

''=

!(A

!(R

"

#$

%

&'

Lateral-DirectionalStability Derivatives

“Weathercock Stability”: Yaw AccelerationSensitivity to Sideslip Angle, N"

N! "#C

n

#!$V 2

2

%&'

()*Sb

Izz

= Cn!

$V 2

2

%&'

()*Sb

Izz

• Vertical tail, fuselage, and wing areprincipal contributors to directionalstability– Side force contributions times

respective moment arms

– Non-dimensional stability derivative

> 0 for stability

• Dimensional stability derivative

Cn!" Cn!( )

Vertical Tail+ Cn!( )

Fuselage+ Cn!( )

Wing+ Cn!( )

Propeller

Contributions toDirectional Stability, N"

Cn!( )Fuselage

="2K VolumeFuselage

SbK = 1! dmax

Lengthfuselage

"#$

%&'1.3

Cn!( )

Vertical Tail

" aFin#vt

SFinlvt

Sb= a

Fin#vtVvt

• Vertical tail

Vvt=SFinlvt

Sb= Vertical Tail Volume Ratio

• Fuselage

!vt= !

elas1+ "# "$( ) V

Tail

2

VN

2

%

&'(

)*

aFin = (–) Lift slope of isolated fin

referenced to fin area, SFin

VTail = Airspeed at vertical tail;“scrubbing” lowers VTail,propeller slipstreamincreases VTail

#, Sidewash: Deflection of flowfrom freestream direction dueto wing-fuselage interference

Contributions toDirectional Stability, N"

Cn!( )Wing

= 0.75CLN" + fcn #,AR,$( )CLN

2

Cn!( )Propeller• Propeller fin effect

– Function of blade geometry,number of blades, thrustcoefficient, and advance ratio

– J = V/nd

– n = rpm

– d = propeller diameter

" = Sweep angle

AR = Aspect ratio

$ = Taper ratio

% = Dihedral angle

• Wing– Differential drag

– Interference effects

see NACA-TR-819

Side Acceleration Sensitivityto Sideslip Angle, Y"

• Dimensional stability derivative

Y! " CY!

#V 2

2m

$%&

'()S

CY!" CY!( )

Fuselage+ CY!( )

Vertical Tail+ CY!( )

Wing

CY!( )Vertical Tail

" aFin#vt

SFinS

CY!( )Fuselage

" $2SBase

S; SB =

%dBase2

4

CY!( )Wing

= $CDParasite,Wing$ k& 2

k =!AR

1+ 1+ AR2

• Fuselage, vertical tail, and wing are principal contributors

Yaw Damping Due to Yaw Rate, Nr

• Dimensional stability derivative

Nr! C

nr

"VN

2

2Izz

#

$

%

&

'

( Sb = Cn

ˆ r

b

2VN

#

$ %

&

' ( "V

N

2

2Izz

#

$

%

&

'

( Sb

= Cn

ˆ r

"VN

4Izz

#

$

%

&

'

( Sb2 < 0 for stability

• High wing-sweep anglecan lead to Nr > 0

Martin Marietta X-24B

Yaw Damping Due toYaw Rate, Nr

Cnr̂! Cnr̂( )

Vertical Tail+ Cnr̂( )

Wing

Cnr̂( )

Wing= k

0CL

2+ k

1CDParasite,Wing

k0 and k1 are functions of

aspect ratio and sweep angle

• Wing contribution

• Vertical tail contribution

! Cn( )

Vertical Tail= " C

n#( )Vertical Tail

rlvt

V( ) = " Cn#( )

Vertical Tail

lvt

b

$%&

'()

b

V

$%&

'()r

Cnr̂

( )vt

=*! C

n( )Vertical Tail

* rb2V( )

=*! C

n( )Vertical Tail

* r̂= "2 C

n#( )Vertical Tail

lvt

b

$%&

'()

r̂ =rb

2VN

Approximate Rolland Spiral Modes

!!p

! !"

#

$%%

&

'((=

Lp 0

1 0

#

$%%

&

'((

!p

!"

#

$%%

&

'((+

L)A

0

#

$%%

&

'((!)A

!RS (s) = s s " Lp( )#S = 0

#R = Lp

• Characteristic polynomial has real roots

• Roll rate is damped by Lp

• Roll angle is a pure integral of roll rate

Roll Damping Due to Roll Rate, Lp

Lp ! Clp

"VN2

2Ixx

#

$%&

'(Sb = Clp̂

b

2VN

#

$%&

'("VN

2

2Ixx

#

$%&

'(Sb

= Clp̂

"VN4Ixx

#

$%&

'(Sb

2

Clp̂( )Wing

=! "Cl( )

Wing

! p̂= #

CL$

12

1+ 3%1+ %

&'(

)*+

• Wing with taper

• Thin triangular wingClp̂( )

Wing= !

"AR

32

Clp̂! Clp̂( )

Vertical Tail+ Clp̂( )

Horizontal Tail+ Clp̂( )

Wing

• Vertical tail, horizontal tail, and wing areprincipal contributors

< 0 for stability

Roll Damping Dueto Roll Rate, Lp

• Vertical tail

• Tapered horizontal tail p̂ =pb

2VN

Clp̂( )ht=! "Cl( )

ht

! p̂= #

CL$ht

12

Sht

S

%&'

()*1+ 3+1+ +

%&'

()*

• pb/2VN describes helix anglefor a steady roll

Clp̂( )vt=! "Cl( )

vt

! p̂= #

CY$vt

12

Svt

S

%&'

()*1+ 3+1+ +

%&'

()*

Primary Lateral-DirectionalControl Derivatives

N!R = Cn!R

"VN

2

2Izz

#

$%&

'(Sb

L!A = Cl!A

"VN

2

2Ixx

#

$%&

'(Sb

Comparison of Fourth-and Second-OrderDynamic Models

Second-Order Models of

Lateral-Directional Motion

• Approximate Spiral-Roll Equation

• Approximate Dutch Roll Equation

!!xDR

=!!r! !"

#

$%%

&

'(()

Nr

N"

Yr

VN

*1+

,-.

/0Y"

VN

#

$

%%%%

&

'

((((

!r!"

#

$%%

&

'((+

N1R

Y1R

VN

#

$

%%%

&

'

(((!1R +

N"

Y"

VN

#

$

%%%%

&

'

((((

!"wind

!!xRS =!!p

! !"

#

$%%

&

'(()

Lp 0

1 0

#

$%%

&

'((

!p

!"

#

$%%

&

'((+

L*A

0

#

$%%

&

'((!*A +

Lp

0

#

$%%

&

'((!pwind

• 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 Modelsand 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

Comparison of Second- and Fourth-OrderInitial-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

Approximate DutchRoll Mode

!!r! !"

#

$%%

&

'((=

Nr

N"

Yr

VN

)1*

+,-

./Y"

VN

#

$

%%%%

&

'

((((

!r!"

#

$%%

&

'((+

N0R

Y0R

VN

#

$

%%%

&

'

(((!0R

!DR(s) = s

2 " Nr+Y#VN

$%&

'()s + N# 1"

Yr

VN

( ) + Nr

Y#VN

*+,

-./

0nDR

= N# 1"Yr

VN

( ) + Nr

Y#VN

1DR

= " Nr+Y#VN

$%&

'()

2 N# 1"Yr

VN

( ) + Nr

Y#VN

!nDR

= N" + Nr

Y"VN

#DR

= $ Nr+Y"VN

%&'

()*

2 N" + Nr

Y"VN

• With negligible Yr

• Characteristicpolynomial,natural frequency,and damping ratio

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