Advanced Problems of Lateral-Directional Dynamics

Robert Stengel, Aircraft Flight DynamicsMAE 331, 2012"

•  Fourth-order dynamics"–  Steady-state response to control"–  Transfer functions"–  Frequency response"–  Root locus analysis of parameter

variations"•  Residualization"•  Roll-spiral oscillation"

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!

Stability-Axis Lateral-Directional Equations"

Δr(t)Δ β(t)Δp(t)Δ φ(t)

$

%

&&&&&

'

(

)))))

=

Nr Nβ Np 0

−1YβVN

0 gVN

Lr Lβ Lp 0

0 0 1 0

$

%

&&&&&&&

'

(

)))))))

Δr(t)Δβ(t)Δp(t)Δφ(t)

$

%

&&&&&

'

(

)))))

+

~ 0 NδR 0

0 0 YδSFVN

LδA ~ 0 00 0 0

$

%

&&&&&&

'

(

))))))

ΔδAΔδRΔδSF

$

%

&&&

'

(

)))

Δx1Δx2Δx3Δx4

"

#

$$$$$

%

&

'''''

=

ΔrΔβ

ΔpΔφ

"

#

$$$$$

%

&

'''''

=

Yaw Rate PerturbationSideslip Angle PerturbationRoll Rate PerturbationRoll Angle Perturbation

"

#

$$$$$

%

&

'''''

Δu1Δu2

"

#$$

%

&''=

ΔδAΔδR

"

#$

%

&' =

Aileron PerturbationRudder Perturbation

"

#$$

%

&''

• With idealized aileron and rudder effects (i.e., NδA = LδR = 0)"

Lateral-Directional Characteristic Equation"

ΔLD (s) = s − λS( ) s − λR( ) s2 + 2ζωns +ωn2( )DR

•  Typically factors into real spiral and roll roots and an oscillatory pair of Dutch roll roots"

ΔLD (s) = s4 + Lp + Nr +

YβVN

#$%

&'(s3

+ Nβ − LrNp + LpYβVN

+ NrYβVN

+ Lp#$%

&'(

*

+,

-

./ s

2

+YβVN

LrNp − LpNr( ) + Lβ Np −gVN( )*

+,-./s

+ gVN

LβNr − LrNβ( )= s4 + a3s

3 + a2s2 + a1s + a0 = 0

Business Jet Example of Lateral-Directional

Characteristic Equation"ΔLD (s) = s − 0.00883( ) s +1.2( ) s2 + 2 0.08( ) 1.39( )s +1.392#$ %&

Slightly unstable Spiral!

Stable Roll!

Lightly damped Dutch roll!

Dutch roll!

Spiral!

Dutch roll!

Roll!

Steady-State Response�ΔxS = −F

−1GΔuS

Equilibrium Response of"2nd-Order Dutch Roll Model"

ΔrSSΔβSS

#

$%%

&

'((= −

YβVN

−Nβ

1 Nr

#

$

%%%

&

'

(((

YβVN

Nr +Nβ

*

+,

-

./

NδR

0

#

$%%

&

'((ΔδRSS

ΔrS = −

YβVN

NδR

%

&'

(

)*

YβVN

Nr +Nβ

%

&'

(

)*

ΔδRS

ΔβS = −NδR

YβVN

Nr +Nβ

%

&'

(

)*

ΔδRS

•  Equilibrium response to constant rudder"

•  Steady yaw rate and sideslip angle are not zero"

•  What is the corresponding ground track of the aircraft [y(t) vs. x(t)]?"

Equilibrium Response of Roll-Spiral Model"

ΔpS = −LδALp

ΔδAS

Δφ(t)S = −LδALp

ΔδAS dt0

t

∫

ΔpSSΔφSS

#

$%%

&

'((= −

Lp 0

1 0

#

$%%

&

'((

−1LδA0

#

$%%

&

'((ΔδASS

Lp 0

1 0

!

"##

$

%&&

−1

=

0 0−1 Lp

!

"##

$

%&&

0

but"

•  Steady roll rate proportional to aileron"

•  Roll angle, integral of roll rate, continually increases"

•  Equilibrium state with constant aileron"

ΔpS = −Lp−1LδAΔδAS

taken alone"

Equilibrium Response of 4th-Order Model"

•  Equilibrium state with constant aileron, rudder, and side-force panel deflection"

ΔrSΔβSΔpSΔφS

$

%

&&&&&

'

(

)))))

= −

Nr Nβ Np 0

−1YβVN

0 gVN

Lr Lβ Lp 0

0 0 1 0

$

%

&&&&&&&

'

(

)))))))

−1

~ 0 NδR 0

0 0 YδSFVN

LδA ~ 0 00 0 0

$

%

&&&&&&

'

(

))))))

ΔδASΔδRSΔδSFS

$

%

&&&

'

(

)))

Equilibrium Response of the 4th-Order Lateral-Directional Model"

ΔyS =HxΔxS = −HxF−1GΔuS

•  With Hx = Identity matrix"

•  Observations"–  Steady-state roll rate is zero"–  Aileron and rudder commands produce steady-state yaw rate, sideslip angle, and roll

angle"–  Side force command produces steady-state roll angle but has no effect on steady-state

yaw rate or sideslip angle"

ΔrSΔβSΔpSΔφS

$

%

&&&&&

'

(

)))))

=

gVN

LδANβ −gVN

LβNδR 0

gVN

LδANrgVN

LrNδR 0

0 0 0

Nβ +NrYβVN

,

-.

/

01LδA − Lβ + Lr

YβVN

,

-.

/

01NδR LrNβ − LβNr( )YδSFVN

$

%

&&&&&&&&&

'

(

)))))))))

gVN

LβNr − LrNβ( )

ΔδASΔδRSΔδSFS

$

%

&&&

'

(

)))

Stability and Transient Response�

4th-Order Initial-Condition Responses of Business Jet"

•  Initial roll angle and rate have little effect on yaw rate and sideslip angle responses"

•  Initial yaw rate and sideslip angle have large effect on roll rate and roll angle responses"

Initial !yaw rate!

Initial !sideslip angle!

Initial !roll rate!

Initial !roll angle!

Effects of Variation in Primary Stability

Derivatives�

Nβ Effect on 4th-Order Roots!

•  Group Δ(s) terms multiplied by Nβ to form numerator"

•  Denominator formed from remaining terms of Δ(s)"

ΔLD (s) = d(s) + Nβn(s) = 0

kn(s)d(s)

= −1 =Nβ s − z1( ) s − z2( )

s − λ1( ) s − λ2( ) s2 + 2ζωns +ωn2( )

N! > 0"

N! < 0"

•  Positive NΒ "–  Increases Dutch roll natural frequency "–  Damping ratio decreases but remains

stable"–  Spiral mode drawn toward origin"–  Roll mode unchanged"

•  Negative Nβ destabilizes Dutch roll mode"

Root Locus Gain = Directional Stability!

Roll! Spiral!

Dutch Roll!

Dutch Roll!

Zero!

Zero!

Nr Effect on 4th-Order Roots"

ΔLD (s) = d(s) + Nrn(s) = 0

kn(s)d(s)

= −1 =Nr s − z1( ) s2 + 2µνns + νn2( )

s − λ1( ) s − λ2( ) s2 + 2ζωns +ωn2( )

• Negative Nr "–  Increases Dutch roll damping "–  Draws spiral and roll modes together

drawn toward origin"• Positive Nr destabilizes Dutch roll mode"

Nr < 0" Nr > 0"Root Locus Gain = Yaw Damping!

Roll! Spiral!

Zero!

Dutch Roll!

Dutch Roll!

Zero!

Zero!

Lp Effect on 4th-Order Roots"

ΔLD (s) = d(s) + Lpn(s) = 0

kn(s)d(s)

= −1 =Lps s

2 + 2µνns + νn2( )

s − λ1( ) s − λ2( ) s2 + 2ζωns +ωn2( )

Lp < 0" Lp > 0"

•  Negative Lp "–  Decreases roll mode time constant"–  Draws spiral and roll modes together

drawn toward origin"•  Positive Lp destabilizes roll mode"•  Lp has negligible effect on spiral mode"•  Normally negative; however, can

become positive at high angle of attack"

Root Locus Gain = Roll Damping!

Roll! Spiral!Zero!

Dutch Roll & Zero!

Dutch Roll & Zero!

Coupling Stability Derivatives and Their Effects�

�Dihedral Effect�: Roll Acceleration Sensitivity to Sideslip Angle, Lβ!

Lβ ≈ Clβ

ρV 2

2Ixx

$

%&

'

()Sb

Clβ≈ Clβ( )

Wing+ Clβ( )

Wing−Fuselage+ Clβ( )

Vertical Tail

•  Wing, wing-fuselage interference, and vertical tail are principal contributors"

Typically < 0 for stability!

�Dihedral Effect�: Roll Acceleration Sensitivity to Sideslip Angle, Lβ!

Lβ ≈ Clβ

ρV 2

2Ixx

$

%&'

()Sb

•  Dihedral and sweep effect"

Clβ( )Wing

=1+ 2λ6 1+ λ( )

ΓCLαwing+

CL tanΛ1− M 2 cos2 Λ

'()

*+,

•  Tapered, trapezoidal, swept wing"

Wing and Tail Location Effects on Lβ!

•  High/low wing effect"

Clβ( )Wing−Fuselage

= 1.2 ARzWing h + w( )

b2

Clβ( )Vertical Tail

≈zvtb

CYβ( )Vertical Tail

•  Vertical tail effect"

Lβ Effect on 4th-Order Roots!

• Negative Lβ "–  Stabilizes spiral and roll modes

but ..."–  Destabilizes Dutch roll mode"

• Positive Lβ does the opposite"

Root Locus Gain = Dihedral Effect!

ΔLD (s) = d(s) + LβgVN

− Np( )n(s) = 0n(s)d(s)

= −1 =Lβ

gVN

− Np( ) s − z1( )

s − λS( ) s − λR( ) s2 + 2ζωnDRs +ωnDR

2( )L!!!

< 0"L! > 0"

Bizjet Example!

ΔLD (s) =

s − 0.00883( ) s +1.2( ) s2 + 2 0.08( ) 1.39( )s +1.392#$ %&

Roll! Spiral!Zero!

Dutch Roll!

Dutch Roll!

Stabilizing Lateral-Directional Motions"

• Provide sufficient Lβ (–) to stabilize the spiral mode"• Provide sufficient Nr (–) to damp the Dutch roll mode"

How can Lβ and Nr be adjusted �artificially�, i.e., by closed-loop control?!

Original Root Locus! Increased |Nr|!

Solar Impulse!

Fourth-Order Frequency Response�

Yaw Rate and Sideslip Angle Frequency Responses of Business Jet"

2nd-Order Response to Rudder"

•  Yawing response to aileron is not negligible"•  Yaw rate response is poorly characterized by the 2nd-order model below the

Dutch roll natural frequency "•  Sideslip angle response is adequately characterized by the 2nd-order model"

4th-Order Response to Aileron and Rudder"

Δr jω( )ΔδA jω( )

Δβ jω( )ΔδA jω( )

Δr jω( )ΔδR jω( )

Δβ jω( )ΔδR jω( )

Δr jω( )ΔδR jω( )

Δβ jω( )ΔδR jω( )

Roll Rate and Roll Angle Frequency Responses of Business Jet"

2nd-Order Response to Aileron"

•  Roll response to rudder is not negligible"•  Roll rate response is marginally well characterized by the 2nd-order model"•  Roll angle response is poorly characterized at low frequency by the 2nd-

order model"

Δp jω( )ΔδR jω( )

Δφ jω( )ΔδR jω( )

Δp jω( )ΔδA jω( )

Δφ jω( )ΔδA jω( )

Δp jω( )ΔδA jω( )

Δφ jω( )ΔδA jω( )

4th-Order Response to Aileron and Rudder"

Frequency and Step Responses to Aileron Input"

•  Roll rate response is relatively benign"•  Ratio of roll angle to sideslip response

is important to the pilot"

•  Yaw/sideslip sensitivity in the vicinity of the Dutch roll natural frequency"

Δr jω( )ΔδA jω( )

Δβ jω( )ΔδA jω( )

Δp jω( )ΔδA jω( )

Δφ jω( )ΔδA jω( )

Δv t( )

Δy t( )

Δr t( )

Δp t( )

Δψ t( )

Δφ t( )

Frequency and Step Responses to Rudder Input"

•  Lightly damped yaw/sideslip response would be hard to control precisely"

•  Yaw response variability near and below the Dutch roll natural frequency"

•  Significant roll rate response near the Dutch roll natural frequency"

Δr jω( )ΔδR jω( )

Δβ jω( )ΔδR jω( )

Δp jω( )ΔδR jω( )

Δφ jω( )ΔδR jω( )

Δv t( ) Δy t( )

Δr t( )

Δp t( )Δψ t( )

Δφ t( )

Order Reduction by Residualization�

Approximate Low-Order Response"

•  Dynamic model order can be reduced when"–  One mode is stable and well-damped,

and it and is faster than the other"–  The two modes are coupled"

Δx fast

Δxslow

"

#

$$

%

&

''=

Ffast Fslowfast

Ffastslow Fslow

"

#

$$

%

&

''

Δx fast

Δxslow

"

#

$$

%

&

''+

G fast

Gslow

"

#

$$

%

&

''Δu

Δx f = FfΔx f + FsfΔxs +G fΔu

Δxs = FfsΔx f + FsΔxs +GsΔu

or!

Residualization Provides an Approximation for Low-Order Dynamics"

•  Assume that fast mode reaches steady state on a time scale that is short compared to the slow mode"

Δx f ≈ 0 ≈ FfΔx f +FsfΔxs +G fΔu

Δxs = FfsΔx f +FsΔxs +GsΔu

•  Algebraic solution for Δxfast"

0 ≈ FfΔx f +FsfΔxs +G fΔu

FfΔx f = −FsfΔxs −G fΔu

Δx f = −Ff−1 Fs

fΔxs +G fΔu( )

•  Substitute quasi-steady Δxfast in differential equation for Δxslow "

Δxs = −Ffs Ff

−1 FsfΔxs +G fΔu( )#$ %& + FsΔxs +GsΔu

= Fs − FfsFf

−1Fsf#$ %&Δxs + Gs − Ff

sFf−1−1G f#$ %&Δu

•  Residualized equation for Δxslow "

Δxs = F 's Δxs +G 's Δu

F 's = Fs − FfsFf

−1Fsf"# $%

G 's = Gs − FfsFf

−1G f"# $%

where!

Residualization"

Residualized Roll-Spiral Mode"•  Assume that the Dutch roll mode is stable

and faster than the roll mode"•  Calculate effect of the quasi-steady Dutch

roll on the roll and spiral modes"

ΔxDRΔxRS

"

#$$

%

&''≈

0ΔxRS

"

#$$

%

&''=

FDR FRSDR

FDRRS FRS

"

#

$$

%

&

''

ΔxDRΔxRS

"

#$$

%

&''+GDR

GRS

"

#$$

%

&''

ΔδAΔδR

"

#$

%

&'

"

#$$

%

&''

Residualized Roll-Spiral Mode"•  Assume that the Dutch roll mode is stable and

faster than the roll mode"•  Calculate effect of the quasi-steady Dutch roll

on the roll and spiral modes"

ΔxDR = −FDR−1 FRS

DRΔxRS +GDRΔδAΔδR

$

%&

'

()

*+,

-,

./,

0,

ΔxRS = FRSΔxRS − FDRRSFDR

−1 FRSDRΔxRS +GDR

ΔδAΔδR

$

%&

'

()

*+,

-,

./,

0,+GRS

ΔδAΔδR

$

%&

'

()

= F 'RS ΔxRS +G 'RSΔδAΔδR

$

%&

'

()

Model of the Residualized Roll-Spiral Mode"

•  2nd-order approximation for roll and spiral modes"

ΔpΔ φ

#

$%%

&

'((=

Lp 0

1 0

#

$%%

&

'((

ΔpΔφ

#

$%%

&

'((−

Lr Lβ0 0

#

$%%

&

'((

Nr Nβ

−1YβVN

#

$

%%%%

&

'

((((

−1Np 0

0 gVN

#

$

%%%%

&

'

((((

ΔpΔφ

#

$%%

&

'((+

NδA NδR

0 YδRVN

#

$

%%%

&

'

(((

ΔδAΔδR

#

$%

&

'(

,

-..

/..

0

1..

2..

+LδA LδR0 0

#

$%%

&

'((

ΔδAΔδR

#

$%

&

'(

ΔpΔ φ

#

$%%

&

'((=

Lp −

Np LrYβVN

+ Lβ+

,-

.

/0

Nβ +NrYβVN

+

,-

.

/0

#

$

%%%%

&

'

((((

gVN

LrNβ − LβNr( )

Nβ +NrYβVN

+

,-

.

/0

#

$

%%%%

&

'

((((

1 0

#

$

%%%%%%

&

'

((((((

ΔpΔφ

#

$%%

&

'((+

=f11 f121 0

#

$%%

&

'((

ΔpΔφ

#

$%%

&

'((+

Roots of the Residualized Roll-Spiral Mode"

sI − F 'RS = s 1 00 1

"

#$

%

&' −

f11 f121 0

"

#$$

%

&''= ΔRSres

= s2 − Lp − Np

Lβ + LrYβ /VNNβ + NrYβ /VN

*

+,

-

./

"

#$$

%

&''s +

gVN

LβNr − LrNβ

Nβ + NrYβ /VN

*

+,

-

./

= s − λS( ) s − λR( ) or s2 + 2ζωns +ωn2( )RS = 0

•  For the business jet model"

ΔRSres= s2 +1.0894s − 0.0108 = 0

= s − 0.0098( ) s +1.1( ) = s − λS( ) s − λR( )

•  Slightly unstable spiral mode"•  Similar to 4th-order roll-spiral results"

ΔLD (s) = s − 0.00883( ) s +1.2( ) s2 + 2 0.08( ) 1.39( )s +1.392#$ %&

Oscillatory Roll-Spiral Mode"ΔRSres

= s − λS( ) s − λR( ) or s2 + 2ζωns +ωn2( )RS

•  The characteristic equation factors into real or complex roots"–  Real roots are roll mode and spiral mode when"

LβNr > LrNβ and

Np Lβ + LrYβ /VN( ) 2 gVN

LβNr − LrNβ( )#

$%

&

'(<1

LβNr < LrNβ

–  Complex roots produce �roll-spiral oscillation� or �lateral phugoid mode� when!

Roll-Spiral Oscillation of a Lifting Reentry Vehicle"

Next Time:�Flying Qualities Criteria�

�

Reading�Aircraft Stability and Control, Ch. 21 �

Virtual Textbook, Part 20 �

Supplemental Material�

Equilibrium Response of 4th-Order Model"

•  Equilibrium state with constant aileron and spiral wind perturbations"

ΔrSSΔβSSΔpSSΔφSS

$

%

&&&&&

'

(

)))))

=

a b 0c d 00 0 0e f g

$

%

&&&&

'

(

))))

ΔδASSΔδRSSΔδSFSS

$

%

&&&

'

(

)))

•  Observations"–  Aileron command"–  Rudder command"–  Side-force panel command"–  Steady-state roll rate is zero"–  Steady-state roll angle is bounded!

Effects of Variation in Secondary

Stability Derivatives�

Roll Acceleration Due to Yaw Rate, Lr!Lr ≈ Clr

ρVN2

2Ixx

#

$%

&

'(Sb

=Clr̂

b2VN

#

$%

&

'(ρVN

2

2Ixx

#

$%

&

'(Sb =Clr̂

ρVN4Ixx

#

$%

&

'(Sb2

•  Wing is the principal contributor"–  Differential lift induced by yaw rate"

Clr̂( )Wing =∂ ΔCl( )Wing

∂ r̂= −

CLα

121+ 3λ1+λ

&

'(

)

*+M 2 cos2Λ−2M 2 cos2Λ−1

&

'(

)

*+

•  Thin triangular wing"

Clr̂( )Wing

=παN

9AR

•  Vertical tail"

Clr̂( )Vertical Tail

=zvtlvt

Cnr̂( )Vertical Tail

Lr Effect on 4th-Order Roots"

ΔLD (s) = d(s) + LrNpn(s) = 0

kn(s)d(s)

= −1 =LrNp s − z1( ) s − z2( )

s − λ1( ) s − λ2( ) s2 + 2ζωns +ωn2( )

Lr < 0" Lr > 0"

• Effect depends on the sign of Np (negative here)"

• Similar to Nβ effect on the Dutch roll, but opposite to its effect on the spiral mode"

Root Locus Gain = Roll Due to Yaw Rate!

Yaw Acceleration Due to Roll Rate, Np!Np ≈ Cnp

ρVN2

2Izz

#

$%&

'(Sb

= Cnp̂

b2VN

#

$%&

'(ρVN

2

2Izz

#

$%&

'(Sb = Cnp̂

ρVN4Ixx

#

$%&

'(Sb2

•  Wing is the principal contributor"–  Differential yaw moment induced by roll rate"

Cnp̂( )Wing

=∂ ΔCn( )Wing

∂ p̂=112

1+ 3λ1+ λ

$%&

'()∂CDParasite,Wing

∂α± CL

$

%&

'

()

•  Thin triangular wing"

Cnp̂( )Wing

= −παN

9AR

•  Vertical tail"

Cnp̂( )Vertical Tail

= −2αNlvtb

#$%

&'(Cnβ( )

Vertical Tail

(–): Subsonic!(+): Supersonic!

Np Effect on 4th-Order Roots"

ΔLD (s) = d(s) + Npn(s) = 0

kn(s)d(s)

= −1 =Nps s − z1( )

s − λ1( ) s − λ2( ) s2 + 2ζωns +ωn2( )

Np > 0" Np < 0"

•  Tends to have opposite signs in sub- and supersonic flight"

•  Effect is analogous to Lβ effect"

Root Locus Gain = Yaw due to Roll Rate!