Linearized Lateral-Directional Equations of Motion!

Robert Stengel, Aircraft Flight Dynamics MAE 331, 2016

Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

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

Reading:!Flight Dynamics!

574-591!

Learning Objectives

1

•! 6th-order -> 4th-order -> hybrid equations•! Dynamic stability derivatives •! Dutch roll mode•! Roll and spiral modes

Review Questions!!! Why can we normally reduce the 6th-order

longitudinal equations to 4th-order without compromising accuracy?!

!! What are the longitudinal modes of motion?!!! Why are the hybrid 4th-order equations useful?!!! What are “dimensional stability and control

derivatives”?!!! Under what circumstances can the 4th-order set be

replaced by two 2nd-order sets?!!! What benefits accrue from the separation, and what

are the limitations?!!! What is “normal load factor”?!!! Would you rather have one “so-so” flying car or a

good airplane and a good car?! 2

6-Component Lateral-Directional Equations of Motion

!v = YB /m + gsin! cos" # ru + pw!yI = cos" sin$( )u + cos! cos$ + sin! sin" sin$( )v + #sin! cos$ + cos! sin" sin$( )w!p = IzzLB + IxzNB # Ixz Iyy # Ixx # Izz( ) p + Ixz

2 + Izz Izz # Iyy( )%& '(r{ }q( ) ÷ Ixx Izz # Ixz2( )

!r = IxzL B+IxxNB # Ixz Iyy # Ixx # Izz( )r + Ixz2 + Ixx Ixx # Iyy( )%& '( p{ }q( ) ÷ Ixx Izz # Ixz

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

x1x2x3x4x5x6

!

"

########

$

%

&&&&&&&&

= xLD6

vypr!"

#

$

%%%%%%%%

&

'

((((((((

=

Side VelocityCrossrange

Body-Axis Roll RateBody-Axis Yaw Rate

Roll Angle about Body x AxisYaw Angle about Inertial x Axis

#

$

%%%%%%%%

&

'

(((((((( 3

4- Component Lateral-Directional

Equations of MotionNonlinear Dynamic Equations, neglecting

crossrange and yaw angle

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

!p = IzzLB + IxzNB # Ixz Iyy # Ixx # Izz( ) p + Ixz2 + Izz Izz # Iyy( )$% &'r{ }q( ) ÷ Ixx Izz # Ixz

2( )!r = IxzLB + IxxNB # Ixz Iyy # Ixx # Izz( )r + Ixz

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

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

x1x2x3x4

!

"

#####

$

%

&&&&&

= xLD4

Eurofighter Typhoon

vpr!

"

#

$$$$

%

&

''''

=

Side VelocityBody-Axis Roll RateBody-Axis Yaw Rate

Roll Angle about Body x Axis

"

#

$$$$$

%

&

''''' 4

Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight

Longitudinal variables are constant

!v = YB /m + gsin! cos"N # ruN + pwN

!p = IzzLB + IxzNB( ) Ixx Izz # Ixz2( )

!r = IxzLB + IxxNB( ) Ixx Izz # Ixz2( )

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

qN = 0!N = 0"N =#N

5

Lateral-Directional Force and Moments in Steady, Level Flight

YB = CYB

12!NVN

2S

LB = ClB

12!NVN

2Sb

NB = CnB

12!NVN

2Sb

Body-Axis Side ForceBody-Axis Rolling MomentBody-Axis Yawing Moment

Dynamic pressure is constant

6

Linearized Lateral-Directional Equations of Motion in Steady,

Level Flight!

7

Body-Axis Perturbation Equations of Motion

! !v(t)!!p(t)!!r(t)! !"(t)

#

$

%%%%%

&

'

(((((

=

F11 F12 F13 F14F21 F22 F23 F24F31 F32 F33 F34F41 F42 F43 F44

#

$

%%%%%

&

'

(((((

!v(t)!p(t)!r(t)!"(t)

#

$

%%%%%

&

'

(((((

+ Control[ ]+ Disturbance[ ]

8

Body-Axis Perturbation Variables

!u1!u2

"

#$$

%

&''= !(A

!(R"

#$

%

&' =

Aileron PerturbationRudder Perturbation

"

#$$

%

&''

!w1!w2

"

#$$

%

&''=

!vwind!pwind

"

#$$

%

&''=

Side Wind PerturbationVortical Wind Perturbation

"

#$$

%

&''

!v!p!r!"

#

$

%%%%

&

'

((((

=

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

Roll Angle about Body x Axis Perturbation

#

$

%%%%%

&

'

(((((

9

Vortical Wind PerturbationsWind Rotor Wake Vortex

TangentialVelocity,

ft/s

Radius, ft

Wake Vortex

Wind Rotor10

Side Velocity DynamicsNonlinear equation

First row of linearized dynamic equation

! !v(t) = F11!v(t)+ F12!p(t)+ F13!r(t)+ F14!"(t)[ ]+ G11!#A(t)+G12!#R(t)[ ]+ L11!vwind + L12!pwind[ ]

11

!v = YB /m + gsin! cos"N # ruN + pwN

F11 =

1m

CYv

!NVN2

2S"

#$%&'! Yv

Coefficients in first row of F

Side Velocity Sensitivity to State Perturbations

12

F12 =

1m

CYp

!NVN2

2S"

#$%&'+wN ! Yp +wN

F14 = gcos! cos"N # g

!v = YB /m + gsin! cos"N # ruN + pwN

F13 =

1m

CYr

!NVN2

2S"

#$%&'( uN ! Yr ( uN

Side Velocity Sensitivity to Control and Disturbance Perturbations

G11 =1m

CY!A

"NVN2

2S#

$%

&

'( ! Y!A

G12 =1m

CY!R

"NVN2

2S#

$%

&

'( ! Y!R

Coefficients in first rows of G and L

L11 = !F11L12 = !F12

13

Roll Rate DynamicsNonlinear equation

Second row of linearized dynamic equation

14

!p = IzzLB + IxzNB( ) Ixx Izz ! Ixz

2( )

!!p(t) = F21!v(t)+ F22!p(t)+ F23!r(t)+ F24!"(t)[ ]+ G21!#A(t)+G22!#R(t)[ ]+ L21!vwind + L22!pwind[ ]

F21 = Izz

!LB!v

+ Ixz!NB

!v"#$

%&' Ixx Izz ( Ixz

2( ) ! Lv

Coefficients in second row of F

Roll Rate Sensitivity to State Perturbations

15

F24 = 0

!p = IzzLB + IxzNB( ) Ixx Izz ! Ixz

2( )

F22 = Izz

!LB!p

+ Ixz!NB

!p"#$

%&'

Ixx Izz ( Ixz2( ) ! Lp

F23 = Izz

!LB!r

+ Ixz!NB

!r"#$

%&' Ixx Izz ( Ixz

2( ) ! Lr

Yaw Rate DynamicsNonlinear equation

Third row of linearized dynamic equation

16

!r = IxzLB + IxxNB( ) Ixx Izz ! Ixz

2( )

!!r(t) = F31!v(t)+ F32!p(t)+ F33!r(t)+ F34!"(t)[ ]+ G31!#A(t)+G32!#R(t)[ ]+ L31!vwind + L32!pwind[ ]

F31 = Ixz

!LB!v

+ Ixx!NB

!v"#$

%&' Ixx Izz ( Ixz

2( ) ! Nv

Coefficients in third row of F

Yaw Rate Sensitivity to State Perturbations

17F34 = 0

!r = IxzLB + IxxNB( ) Ixx Izz ! Ixz

2( )

F32 = Ixz

!LB!p

+ Ixx!NB

!p"#$

%&'

Ixx Izz ( Ixz2( ) ! Np

F33 = Ixz

!LB!r

+ Ixx!NB

!r"#$

%&' Ixx Izz ( Ixz

2( ) ! Nr

Roll Angle DynamicsNonlinear equation

Fourth row of linearized dynamic equation

18

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

! !"(t) = F41!v(t)+ F42!p(t)+ F43!r(t)+ F44!"(t)[ ]+ G41!#A(t)+G42!#R(t)[ ]+ L41!vwind + L42!pwind[ ]

F41 = 0

Coefficients in fourth row of F

Roll Angle Sensitivity to State Perturbations

19

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

F42 = 1

F43 = cos!N tan"N # tan"N

F44 = rN sin!N tan"N = 0

Linearized Lateral-Directional Response to Initial Yaw Rate

~Roll response of roll angle

~Spiral response of crossrange

~Spiral response of yaw angle

~Dutch-roll response of side velocity

~Dutch-roll response of roll

and yaw rates

20

Dimensional Stability Derivatives

F11 F12 F13 F14F21 F22 F23 F24F31 F32 F33 F34F41 F42 F43 F44

!

"

#####

$

%

&&&&&

Stability Matrix

=

Yv Yp +wN( ) Yr !uN( ) gcos"NLv Lp Lr 0

Nv Np Nr 0

0 1 tan"N 0

#

$

%%%%%%

&

'

((((((

21

Dimensional Control- and Disturbance-Effect Derivatives

G11 G12G21 G22

G31 G32

G41 G42

!

"

#####

$

%

&&&&&

=

Y'A Y'RL'A L'R

N'A N'R

0 0

!

"

#####

$

%

&&&&&

L11 L12L21 L22L31 L32L41 L42

!

"

#####

$

%

&&&&&

=

Yv YpLv Lp

Nv Np

0 0

!

"

#####

$

%

&&&&&

Control Effect Matrix

DisturbanceEffect Matrix

22

! !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+RL+A L+R

N+A N+R

0 0

#

$

%%%%%

&

'

(((((

!+A(t)!+R(t)

#

$%%

&

'((+

Yv YpLv Lp

Nv Np

0 0

#

$

%%%%%

&

'

(((((

!vwind!pwind

#

$%%

&

'((

Rolling and yawing motions

LTI Body-Axis Perturbation Equations of Motion

23

Asymmetrical Aircraft: DC-2-1/2

24

DC-3 with DC-2 right wingQuick fix to fly aircraft out of harm s way during WWII

Asymmetric Aircraft - WWIIBlohm und Voss, BV 141 B + V 141 derivatives

B + V P.202

Recent Asymmetric AircraftScaled Composites Ares

Scaled Composites Boomerang

Oblique Wing Concepts•! High-speed benefits of wing sweep without the

heavy structure and complex mechanism required for symmetric sweep

•! Blohm und Voss, R. T. Jones , Handley-Page concepts

•! Improved supersonic L/D by reduction of shock-wave interference and elimination of the fuselage in flying-wing version

27

Handley-Page Oblique Wing Concepts•! Advantages

–! 10-20% higher L/D @ supersonic speed (compared to delta planform)

–! Flying wing: no fuselage•! Issues

–! Which way do the passengers face?

–! Where is the cockpit?–! How are the engines and vertical

surfaces swiveled?–! What does asymmetry do to

stability and control?

28

NASA Oblique Wing Test Vehicles•! Stability and control issues

abound: The fact that birds and insects are symmetric should give us a clue (though they use huge asymmetry for control)

–! Strong aerodynamic and inertial longitudinal-lateral-directional coupling

–! High side force at zero sideslip angle

–! Torsional divergence of the leading wing

•! Test vehicles: Various model airplanes, NASA AD-1, and NASA DFBW F-8 (below, not built)

29

Stability Axis Representation of Dynamics!

30

Stability Axes•! Stability axes are an alternative set of body axes•! Nominal x axis is offset from the body centerline by

the nominal angle of attack, !!N

31

Transformation from Original Body Axes to Stability Axes

HBS =

cos!N 0 sin!N

0 1 0"sin!N 0 cos!N

#

$

%%%

&

'

(((

!u!v!w

"

#

$$$

%

&

'''S

= HBS

!u!v!w

"

#

$$$

%

&

'''B

!p!q!r

"

#

$$$

%

&

'''S

= HBS

!p!q!r

"

#

$$$

%

&

'''B

Side velocity (!!v) and pitch rate ("q) are unchanged by the transformation 32

Stability-Axis State Vector Rotate body axes to stability axes

!v(t)!p(t)!r(t)!"(t)

#

$

%%%%%

&

'

(((((Body)Axis

*!+N *

!v(t)!p(t)!r(t)!"(t)

#

$

%%%%%

&

'

(((((Stability)Axis

33

Stability-Axis State Vector

!v(t)!p(t)!r(t)!"(t)

#

$

%%%%%

&

'

(((((Stability)Axis

*!+ ,!vVN

*

!+(t)!p(t)!r(t)!"(t)

#

$

%%%%%

&

'

(((((Stability)Axis

Replace side velocity by sideslip angle

34

Stability-Axis State Vector

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

$

%

&&&&&

'

(

)))))Stability*Axis

+

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

$

%

&&&&&

'

(

)))))Stability*Axis

=

Stability-Axis Yaw Rate PerturbationSideslip Angle Perturbation

Stability-Axis Roll Rate PerturbationStability-Axis Roll Angle Perturbation

$

%

&&&&&

'

(

)))))

Revise state order

35

Stability-Axis Lateral-Directional Equations

!!r(t)

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

$

%

&&&&&

'

(

)))))S

=

Nr N" Np 0

YrVN

*1+,-

./0

Y"

VN

YpVN

gVN

Lr L" Lp 0

0 0 1 0

$

%

&&&&&&&

'

(

)))))))S

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

$

%

&&&&&

'

(

)))))S

+

N1A N1R

Y1AVN

Y1RVN

L1A L1R

0 0

$

%

&&&&&&

'

(

))))))S

!1A(t)!1R(t)

$

%&&

'

())+

N" Np

Y"

VN

YpVN

L" Lp

0 0

$

%

&&&&&&&

'

(

)))))))S

!"wind

!pwind

$

%&&

'

())

36

Why Modify the Equations?Dutch-roll motion is primarily described by stability-

axis yaw rate and sideslip angle

Roll and spiral motions are primarily described by stability-axis roll rate and roll angle

Stable Spiral

Unstable SpiralRoll

Dutch Roll, top Dutch Roll, front

37

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?

38

Stability, Control, and Disturbance Matrices

FLD =FDR FRS

DR

FDRRS FRS

!

"##

$

%&&=

Nr N' Np 0

YrVN

(1)*+

,-.

Y'

VN

YpVN

gVN

Lr L' Lp 0

0 0 1 0

!

"

#######

$

%

&&&&&&&

GLD =

N!A N!R

Y!AVN

Y!RVN

L!A L!R

0 0

"

#

$$$$$$

%

&

''''''

LLD =

N! Np

Y!

VN

YpVN

L! Lp

0 0

"

#

$$$$$$$

%

&

'''''''

!x1!x2!x3!x4

"

#

$$$$$

%

&

'''''

=

!r!(

!p!)

"

#

$$$$$

%

&

'''''

!u1!u2

"

#$$

%

&''=

!(A!(R

"

#$

%

&'

!w1!w2

"

#$$

%

&''=

!(A!(R

"

#$

%

&'

39

2nd-Order Approximate Modes of Lateral-Directional

Motion!

40

2nd-Order Approximations in System Matrices

FLD =FDR 00 FRS

!

"##

$

%&&=

Nr N' 0 0

YrVN

(1)*+

,-.

Y'

VN0 0

0 0 Lp 0

0 0 1 0

!

"

#######

$

%

&&&&&&&

GLD =

N!R 0Y!RVN

0

0 L!A

0 0

"

#

$$$$$$

%

&

''''''

LLD =

N! 0

Y!

VN0

0 Lp

0 0

"

#

$$$$$$$

%

&

''''''' 41

2nd-Order Models of Lateral-Directional Motion

Approximate Spiral-Roll Equation

Approximate Dutch Roll Equation

!!xDR =!!r! !"

#

$%%

&

'(()

Nr N"

YrVN

*1+,-

./0

Y"

VN

#

$

%%%%

&

'

((((

!r!"

#

$%%

&

'((+

N1R

Y1RVN

#

$

%%%

&

'

(((!1R +

N"

Y"

VN

#

$

%%%

&

'

(((!"wind

!!xRS =!!p! !"

#

$%%

&

'(()

Lp 0

1 0

#

$%%

&

'((

!p!"

#

$%%

&

'((+

L*A0

#

$%%

&

'((!*A +

Lp

0

#

$%%

&

'((!pwind

42

Comparison of 4th- and 2nd-Order Dynamic Models!

43

Comparison of 2nd- and 4th-Order Initial-Condition Responses of Business Jet

Fourth-Order Response Second-Order Response

Speed and damping of responses is adequately portrayed by 2nd-order modelsRoll-spiral modes have little effect on yaw rate and sideslip angle responses

BUT Dutch roll mode has large effect on roll rate and roll angle responses44

4 initial conditions [r(0), !(0), p(0), !(0)]

Next Time:!Analysis of Time Response!

!

Reading:!Flight Dynamics!298-313, 338-342!

45

•! Methods of time-domain analysis–! Continuous- and discrete-time models–! Transient response to initial conditions and inputs–! Steady-state (equilibrium) response–! Phase-plane plots–! Response to sinusoidal input

Learning Objectives

Supplemental Material

46

Primary Lateral-Directional Control Derivatives

L!A = Cl!A

"NVN2

2Ixx

#$%

&'(Sb

N!R = Cn!R

"NVN2

2Izz

#$%

&'(Sb

47

Initial Condition Response of Approximate Dutch Roll Mode

!r t( ) !" t( )

48

Dutch roll VideoCalspan Variable-Stability Learjet Dutch roll Oscillation

http://www.youtube.com/watch?v=1_TW9oz99NQ

49