7/30/2019 Mae 331 Lecture 14

1/8

Linearized Lateral-Directional

Equations of MotionRobert Stengel, Aircraft Flight Dynamics MAE

331, 2008

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 2008 by Robert Stengel. All rights reserved. For educati onal 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 + 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 + Ixz2 + Izz Izz $Iyy( )[ ]r{ }q( ) IxxIzz $Ixz2( )r = I

xzL + I

xxN$ I

xzIyy$I

xx$I

zz( )r + Ixz2 + Ixx Ixx $Iyy( )[ ]p{ }q( ) IxxIzz $Ixz2( )"= p + qsin"+ rcos"( )tan#%= qsin"+ rcos"( )sec#

x1

x2

x3

x4

x5

x6

"

#

$$$$$$$

%

&

'''''''

= xLD6

=

v

y

p

r

(

)

"

#

$$$$$$$

%

&

'''''''

=

SideVelocity

Crossrange

Body * Axis Roll Rate

Body * AxisYaw Rate

Roll Angle about Body x Axis

Yaw Angle about Inertial x Axis

"

#

$$$$$$$

%

&

'''''''

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+ Ixz2+ I

zzIzz$I

yy( )[ ]r{ }q( ) IxxIzz $Ixz2( )r = I

xzL + I

xxN$ I

xzIyy$I

xx$I

zz( )r + Ixz2+ I

xxIxx$I

yy( )[ ]p{ }q( ) IxxIzz $Ixz2( )"= p+ qsin"+ rcos"( ) tan#

x1

x2

x3

x4

"

#

$$$$

%

&

''''

= xLD4

=

v

p

r

(

"

#

$$$$

%

&

''''

=

Side Velocity

Body ) Axis Roll Rate

Body ) AxisYaw Rate

Roll Angle about Body x Axis

"

#

$$$$

%

&

''''

4- Component

Lateral-Directional

Equations of Motion

State Vector, 4components

Nonlinear Dynamic Equations, assuming steady, level, longitudinal flight

(constant longitudinal variables)

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

=YB /m + gsin"cos%N $ ruN + pwN

p =IzzLB + IxzNB( )IxxIzz $Ixz

2( )

r =IxzLB + IxxNB( )IxxIzz $Ixz

2( )"= p+ rcos"( )tan#N = p+ rcos"( )tan%N

x1

x2

x3x

4

"

#

$$

$

$

%

&

''

'

'

= xLD4=

v

p

r(

"

#

$$

$

$

%

&

''

'

'

=

SideVelocity

Body ) Axis Roll Rate

Body ) AxisYaw RateRoll Angle about Body x Axis

"

#

$$

$

$

%

&

''

'

'

qN = 0

"N = 0

#N =

$N

7/30/2019 Mae 331 Lecture 14

2/8

Lateral-Directional

Force and Moments

YB = CYB1

2"NVN

2S; Body # Axis Side Force

LB =

ClB

1

2"NVN

2Sb; Body # Axis Rolling Moment

NB =

CnB

1

2"NVN

2Sb; Body # AxisYawing Moment

Body-Axis Perturbation

Equations of Motion

"xLD

(t) = FLD"x

LD(t) +G

LD"u

LD(t) + L

LD"w

LD(t)

dim("x) = 4 #1

dim("u) = 2#1

dim("w) = 2#1

"v(t)

"p(t)

"r(t)

"#(t)

$

%

&&&&

'

(

))))

=

*f1

*v

*f1

*p

*f1

*r

*f1

*#*f2*v

*f2*p

*f2*r

*f2*#

*f3*v

*f3*p

*f3*r

*f3*#

*f4*v

*f4*p

*f4*r

*f4*#

$

%

&&&&&&&&

'

(

))))))))

"v(t)

"p(t)

"r(t)

"#(t)

$

%

&&&&

'

(

))))

+

*f1

*+A

*f1

*+R*f2*+A

*f2*+R

*f3*+A

*f3*+R

*f4

*+A

*f4

*+R

$

%

&&&&&&&&

'

(

))))))))

"+A(t)

"+R(t)

$

%&

'

()+

*f1*vwind

*f1*pwind

*f2*vwind

*f2*pwind

*f3*vwind

*f3*pwind

*f4*vwind

*f4*pwind

$

%

&&&&&&&&&

'

(

)))))))))

"vwind"pwind

$

%&

'

()

"u1

"u2

#

$%

&

'(=

")A

")R

#

$%

&

'(=

Aileron Perturbation

Rudder Perturbation

#

$%

&

'(

"w1

"w2

#

$%

&

'(=

")A

")R

#

$%

&

'(=

SideWind Perturbation

VorticalWind Perturbation

#

$%

&

'(

Perturbations from wings-level flight

vN = 0; pN = 0; rN = 0; "N = 0

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

"f1"v

"f1"p

"f1"r

"f1"#

"f2

"v

"f2

"p

"f2

"r

"f2

"#

"f3

"v

"f3

"p

"f3

"r

"f3

"#

"f4

"v

"f4

"p

"f4

"r

"f4

"#

$

%

&&&&&&&&

'

(

))))))))

=

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

Nv Np Nr 0

0 1 tan+N 0

$

%

&&&&

'

(

))))

"f1

"#A

"f1

"#R"f

2

"#A

"f2

"#R"f

3

"#A

"f3

"#R"f

4

"#A

"f4

"#R

$

%

&&&&&&&

&

'

(

)))))))

)

=

Y#A

Y#R

L#A L#R

N#A N#R

0 0

$

%

&&&&

'

(

))))

"f1

"vwind

"f1

"pwind"f

2

"vwind

"f2

"pwind"f

3

"vwind

"f3

"pwind"f

4

"vwind

"f4

"pwind

#

$

%%%%%%%%

%

&

'

((((((((

(

=

Yv Yp

Lv Lp

Nv Np

0 0

#

$

%%%%

&

'

((((

StabilityMatrix

ControlEffect Matrix

DisturbanceEffect Matrix

v

p

r

"

#

$

%%%%

&

'

((((

=

SideVelocity

Body ) Axis Roll Rate

Body ) AxisYaw Rate

Roll Angle about Body x Axis

#

$

%%%%

&

'

((((

7/30/2019 Mae 331 Lecture 14

3/8

"v(t)

"p(t)

"r(t)

"#(t)

$

%

&&&&

'

(

))))

=

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

$

%&

'

()

"#A

"#R

$

%&

'

()=

Aileron Perturbation

Rudder Perturbation

$

%&

'

()

"#A

"#R

$

%&

'

()=

Side Wind Perturbation

VorticalWind Perturbation

$

%&

'

()

Rolling and yawing motions

"v

"p

"r

"#

$

%

&&&&

'

(

))))

=

SideVelocity Perturbation

Body * Axis Roll Rate Perturbation

Body * AxisYaw Rate Perturbation

Roll Angle about Body x Axis Perturbation

$

%

&&&&

'

(

))))

Body-Axis Perturbation

Equations of MotionStability Axes

Alternative set of body axes

Nominal xaxis is offset from the body centerline bythe nominal angle of attack, N

Transformation from

Original Body Axes to

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

= HBS

"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

+

",(t)

"p(t)

"r(t)

"#(t)

$

%

&&&&

'

(

))))Stability*Axis

+

"r(t)

",(t)

"p(t)

"#(t)

$

%

&&&&

'

(

))))Stability*Axis

7/30/2019 Mae 331 Lecture 14

4/8

Stability-Axis Lateral-

Directional Equations

"r(t)

"#(t)"p(t)

"$(t)

%

&

''''

(

)

****S

=

Nr N# Np 0

Yr

VN+1

,

-.

/

01

Y#

VN

Yp

VN

gcos2NVN

Lr L# Lp 0

tan2N 0 1 0

%

&

'''''

(

)

*****S

"r(t)

"#(t)

"p(t)

"$(t)

%

&

''''

(

)

****S

+

N3A N3RY3A

VN

Y3R

VNL3A L3R

0 0

%

&

'''''

(

)

*****S

"3A(t)

"3R(t)

%

&'

(

)*+

N# NpY#

VN

Yp

VNL# Lp

0 0

%

&

'''''

(

)

*****S

"#wind"pwind

%

&'

(

)*

"x1

"x2

"x3

"x4

#

$

%%%%

&

'

((((

=

"r

")

"p

"*

#

$

%%%%

&

'

((((

=

Stability+ AxisYaw 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 Modify the

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?

Stability, Control, and

Disturbance Matrices

FLD =FDR

FRS

DR

FDR

RSFRS

"

#$

%

&'=

Nr

N( Np 0

Yr

VN

)1*

+, -

./ Y

(

VN

Yp

VN

gcos0NVN

Lr

L( Lp 0

tan0N 0 1 0

"

#

$$$$$

%

&

'''''

GLD

=

N"A

N"R

Y"A

VN

Y"R

VN

L"A

L"R

0 0

#

$

%%%%%

&

'

(((((

LLD

=

N" NpY"

VN

Yp

VN

L" Lp

0 0

#

$

%%%%%

&

'

(((((

"x1"x

2

"x3

"x4

#

$

%%%%

&

'

((((

=

"r")

"p

"*

#

$

%%%%

&

'

((((

"u1

"u2

#

$%

&

'(=

")A

")R

#

$%

&

'(

"w1

"w2

#

$%

&

'(=

")A

")R

#

$%

&

'(

7/30/2019 Mae 331 Lecture 14

5/8

Second-Order Models of

Lateral-Directional Motion

Approximate LTI Spiral-Roll Equation

Approximate LTI Dutch Roll Equation

"xDR

=

"r

"#$

%&

'

()*

Nr

N#

Yr

VN

+1,

-.

/

01

Y#

VN

$

%

&&&

'

(

)))

"r

"#

$

%&

'

()+

N2R

Y2R

VN

$

%

&&

'

(

))"2R +

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

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 -1.2

0.2501 -2.408 -1.1616 0 2.3106 0 -1.16e-01 + 1.39e+00j 8.32E-02 1.39E+000 0 1 0 0 0 -1.16e-01 - 1.39e+00 8.32 -02 1.39 +00

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

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

Roll-Spiral Approximation= = genva ue amp ng req. rad s

-1.1616 0 2.3106 01 0 0 -1.16

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

Approximate

Dutch Roll Mode

"r

"#$

%&

'

()=

Nr

N#

Yr

VN

*1+

,-

.

/0

Y#

VN

$

%

&&&

'

(

)))

"r

"#

$

%&

'

()+

N1R

Y1R

VN

$

%

&&

'

(

))"1R

"DR

(s) = s2 # N

r+

Y$VN

%

&'

(

)*s+ N$ 1#

Yr

VN

%&'

()*+ N

r

Y$VN

+

,-.

/0

1nDR

= N$ 1#Yr

VN

%&'

()*+ N

r

Y$

VN

2DR

= # Nr+

Y$VN

%

&'

(

)* 2 N$ 1#

Yr

VN

%&'

()*+ N

r

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

7/30/2019 Mae 331 Lecture 14

6/8

Weathercock Stability: Yaw Acceleration

Sensitivity to Sideslip Angle,N

N" #$C

n

$"

%V2

2

&

'(

)

*+Sb

Izz

=Cn"

%V2

2

&

'(

)

*+Sb

Izz

Vertical tail, fuselage, and wing are principalcontributors to directional stability 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 to

Directional Stability,N

Cn"( )Fuselage =

#2K VolumeFuselage

Sb

K= 1"dmax

Lengthfuselage

#

$%

&

'(

1.3

Cn"( )Vertical Tail # aFin$vt

SFin

lvt

Sb= a

Fin$

vtV

vt

Vertical tail

Vvt=

SFin

lvt

Sb=Vertical TailVolume Ratio

Fuselage

"vt="

elas1+ #$

#%&'( )

*+V

Tail

2

VN

2

&

'(

)

*+

aFin= () Lift slope of isolated finreferenced to fin area, SFin

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

, Sidewash: Deflection of flow

from freestream direction dueto wing-fuselage interference

Contributions to

Directional Stability,N

Cn"( )Wing = 0.75CLN#+ fcn $,AR,%( )CLN2

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 angleAR= Aspect ratio$=Taper ratio%= Dihedral angle

Wing Differential drag

Interference effects

see NACA-TR-819

Side Acceleration Sensitivity

to Sideslip Angle, Y

Side-force sensitivity tosideslip angle, Y

Y" # CY"$V2

2m

%&'

()*S

CY" # CY"( )Fuselage + CY"( )Vertical Tail + CY"( )Wing

CY"( )Vertical Tail # aFin$vtS

Fin

S

CY"( )Fuselage # %2

SBaseS; SB =

&dBase2

4

CY"( )Wing =%CDParasite, Wing % k'2

k="AR

1+ 1+ AR2

Fuselage, vertical tail, and wing are principal contributors

7/30/2019 Mae 331 Lecture 14

7/8

Yaw Damping Due to

Yaw Rate, Nr

Cn r " Cn r( )Vertical Tail + Cn r( )Wing

Cn r

( )Wing

= k0C

L

2

+ k1CDParasite,Wing

k0and k1 are functions of aspectratio and sweep angle

Wing contribution

Vertical tail contribution

" Cn( )Vertical Tail =# Cn$( )Vertical Tail

rlvt

V

%&'

()*=# C

n$( )Vertical Taillvt

b

%

&'

(

)*

b

V

%

&'

(

)*r

Cn

r( )

vt=

+" Cn( )Vertical Tail

+ rb2V( )

=

+" Cn( )Vertical Tail+r

=#2 Cn$( )

Vertical Tail

lvt

b

%

&'

(

)*

r =rb

2VN

Yaw Damping Due to Yaw Rate, Nr

Dimensional stability derivative

Nr "Cnr#V

N

2

2Izz

$

%&

'

()Sb

=

Cn r

b

2VN

$

%&

'

()

#VN

2

2Izz

$

%&

'

()Sb

=Cnr

#VN

4Izz

$

%&

'

()Sb

2 < 0 for stability

High wing-sweep anglecan lead to Nr> 0

Approximate Roll

and 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#VN

2

2Ixx

$

%&

'

()Sb = Cl

p

b

2VN

$

%&

'

()

#VN2

2Ixx

$

%&

'

()Sb = Cl

p

#VN

4Ixx

$

%&

'

()Sb

2

Clp

( )Wing

=

" #Cl( )Wing"p

= $CL%

12

1+ 3&

1+ &

'

()

*

+,

Wing with taper

Thin triangular wing

Clp

( )Wing

= "#AR

32

Clp" Cl

p( )

VerticalTail+ Cl

p( )

HorizontalTail+ Cl

p( )

Wing

Vertical tail, horizontal tail, and wing areprincipal contributors

< 0 for stability

7/30/2019 Mae 331 Lecture 14

8/8

Roll Damping Due to

Roll Rate,Lp

Vertical tail

Tapered horizontal tail

p =pb

2VN

Clp

( )ht=

" #Cl( )ht"p

= $CL%ht

12

Sht

S

&

'(

)

*+1+ 3,

1+ ,

&

'(

)

*+

pb/2VNdescribes helix anglefor a steady roll

Cl p( )vt=

"#Cl( )vt"p

=

$

CY%vt

12

Svt

S

&

'(

)

*+

1+ 3,

1+ ,

&

'(

)

*+