Aircraft Flight Dynamics Robert F. Stengel Lecture19 Advanced LateralDirectional Dynamics

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 arti

Advanced Problems of

Robert Stengel, Aircraft Flight Dynamics


MAE 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"

Δr(t)

Δ  β(t) Δp(t)

Δ  φ(t)

$

%

&

&

&

&

&

'

(

) ) ) ) )

=

Nr Nβ Np 0

−1 Yβ

VN 0

g

VN

Lr Lβ Lp 0

$

%

&

&

&

&

&

&

&

'

(

) ) ) ) ) ) )

Δr(t)

Δβ(t)

Δp(t)

Δφ(t)

$

%

&

&

&

&

&

'

(

) ) ) ) ) +

~ 0 NδR 0

0 0 YδSF

VN

LδA ~ 0 0

$

%

&

&

&

&

&

&

'

(

) ) ) ) ) )

ΔδA ΔδR ΔδSF

$

%

&

&

&

' (

) ) )

Δx1

Δx2

Δx3

Δx4

"

#

$

$

$

%

&

' ' '

=

Δr

Δβ

Δp

Δφ

"

#

$

$

$

%

&

' ' '

=

Yaw Rate Perturbation Sideslip Angle Perturbation Roll Rate Perturbation Roll Angle Perturbation

"

#

$

$

$

%

&

' ' '

Δu1

Δu2

"

#

$

$

%

&

' '=

ΔδA

ΔδR

"

#

%

&

' =

Aileron Perturbation Rudder Perturbation

"

#

$

$

%

&

' '

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

Lateral-Directional

Characteristic Equation"

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

•   Typically factors into real spiral and roll roots

and an oscillatory pair of Dutch roll roots"

ΔLD(s) = s4+ Lp+ Nr+ Yβ

VN

#

$%

&

'( s 3

+ Nβ− LrNp+ LpYβ

VN + Nr

Yβ

VN + Lp

#

$%

&

'(

*

+

- s 2

+ Yβ

VN( LrNp− LpNr) + Lβ Np− g V

N

*

+,

-./ s + g V

N( LβNr− LrNβ)

= s4

+ a3s3

+ 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

Δx S = −F −1 G Δu S

Equilibrium Response of"

Δr SS

ΔβSS

#

$

%

%

&

'

( (= −

Yβ

V N

−Nβ

1 N r

#

$

%

%

%

&

'

( ( (

Yβ

V N N r + Nβ

* +

-

N δR

0

#

$

%

%

&

'

( (ΔδRSS

ΔrS= −

Yβ

VN NδR

%

&

)

*

Yβ

VN Nr+ Nβ

%

&

)

*

Δδ RS

Δ S= − NδR

Yβ

VN Nr+ Nβ

%

&

)

*

Δδ RS

•   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= − LδA

Lp

Δ δAS

Δ φ (t)S= − LδA

L Δ δ ASdt

t

∫

Δp SS

ΔφSS

#

$

%

%

&

'

(

(= −

L p 0

#

$

%

%

&

'

( (

−1

L δ A

0

#

$

%

%

&

'

( (Δ

δA SS

Lp 0

1 0

!

"

#

#

$

%

&

&

−1

=

0 0

−1 Lp

!

"

#

#

$

%

&

&

0

but"

proportional to aileron"

rate, continually increases"

•   Equilibrium state with constant aileron"

ΔpS= −Lp−1

LδAΔδAS

taken

alone"

Equilibrium Response of

4 th -Order Model"

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

ΔrS

ΔβS

ΔpS

ΔφS

$

%

&

&

&

&

&

'

(

) ) ) ) )

= −

−1 Yβ

g

VN

$

%

&

&

&

&

&

&

&

'

(

) ) ) ) ) ) )

−1

~ 0 NδR 0

0 0 YδSF

VN

$

%

&

&

&

&

&

&

'

(

) ) ) ) ) )

ΔδAS

ΔδRS

ΔδSFS

$

%

&

&

&

'

(

) ) )

Equilibrium Response of the 4 th -Order

Lateral-Directional Model"

ΔyS = HxΔxS = −HxF−1G ΔuS •  With H x =

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 "

Δr S

ΔβS

Δp S

ΔφS

$

%

&

&

&

&

&

'

(

)

)

)

)

)

=

g

V N L δA Nβ −g

V N LβN δR 0

g

V N L δA N r g

V N L r N δR 0

Nβ+ N r Yβ

V N

,

-/ 0

1 L δA − Lβ+ L r Yβ

V N

,

-/ 0

1 N δR (L r Nβ− LβN r)Y δSF

V N

$

%

&

&

&

&

&

&

&

&

&

'

(

) ) ) ) ) ) ) ) )

g

V N(LβN r − L r Nβ)

ΔδAS

ΔδRS ΔδSFS

$

%

&

&

&

'

(

) ) )

Stability and Transient Response

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

N β Effect on 4 th -Order

Roots!

numerator"

ΔLD(s) = d(s) + Nβn(s) = 0

kn(s)

d(s) = −1 =

Nβ( s − z1) ( s − z2)

s − λ1

( ) ( s − λ2) s2

+ 2 ζωns + ωn

N! > 0"

N! < 0"

–  Increases Dutch roll natural frequency "

–  Damping ratio decreases but remains

stable"

–  Spiral mode drawn toward origin"

–  Roll mode unchanged "

Root Locus Gain = Directional Stability!

Roll! Spiral!

Dutch Roll!

Dutch Roll!

Zero!

Zero!

Roots"

ΔLD(s) = d(s) + Nrn(s) = 0 kn(s)

d(s) = −1 =

Nr( s − z1) s2

+ 2 µνns + νn

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"

Root Locus Gain = Yaw Damping!

Roll! Spiral!

Zero!

Dutch Roll!

Dutch Roll!

Zero!

Zero!

Roots"

ΔLD(s) = d(s) + Lpn(s) = 0

kn(s)

d(s) = −1 =

Lps s2 + 2µνns + νn

s − λ1

( ) ( s − λ2) s2

+ 2ζωns + ωn

•   Negative Lp "

–  Decreases roll mode time constant"

–  Draws spiral and roll modes together

drawn toward origin "

•   Positive Lp destabilizes roll mode"

•   Lphas negligible effect on spiral mode"

become positive at high angle of

attack"

Root Locus Gain = Roll Damping!

Roll! Zero! Spiral!

Dutch Roll & Zero!

Dutch Roll & Zero!

Coupling Stability Derivatives

and Their Effects

Dihedral Effect : Roll Acceleration

Lβ ≈ Clβ ρV

2

2Ixx

$

%

& ' (

)Sb

Clβ ≈ C ( )lβ Wing+ C ( )lβ Wing − Fuselage+ C ( )lβ Vertical Tail

principal contributors"

Typically < 0 for stability!

Dihedral Effect : Roll Acceleration

ρV2

2Ixx

$

%&

' () Sb

•   Dihedral and sweep effect"

β

Λ

' ()

* +,

•   Tapered, trapezoidal, swept wing"

•   High/low wing effect"

C lβ

b2

C lβ

( )Vertical Tail≈z vt

b( )C Yβ Vertical Tail

•   Vertical tail effect"

•   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) + Lβ g V

N− Np

n(s) d(s) = −1 =

Lβ g V

N− Np

( ) ( s − z1)

s − λS

( ) ( s − λR) s2

+ 2ζωn DRs + ωn2DR

< 0"

L! > 0"

Bizjet Example!

ΔLD(s) =

s − 0.00883

( ) ( s + 1.2 ) s2

+ 2 0.08 ( ) ( 1.39 ) s + 1.392

Roll! Zero! Spiral!

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?!

Solar Impulse!

Fourth-Order Frequency Response

Responses of Business Jet"

to Rudder"

•  Yawing response to aileron is not negligible"

•  Yaw rate response is poorly characterized by the 2 nd -order model below the

Dutch roll natural frequency "

•  Sideslip angle response is adequately characterized by the 2 nd -order model"

Aileron and Rudder"

Δr j( )ω

ΔδA j( )ω

Δβ( )jω

ΔδA j( )ω

Δr j( )ω

ΔδR j( )ω

Δβ( )jω

ΔδR j( )ω

Δr jω( )

Δδ R jω( )

Δβ jω( )

Δδ R jω( )

to Aileron"

•  Roll response to rudder is not negligible"

•  Roll rate response is marginally well characterized by the 2 nd -order model"

•  Roll angle response is poorly characterized at low frequency by the 2 nd -order model"

Δp jω( )

Δδ R jω( )

Δφ jω( )

Δδ R jω( )

Δp jω( )

Δδ A jω( )

Δφ jω( )

Δδ A jω( )

Δp jω( )

Δδ A jω( )

Δφ jω( )

Δδ A jω( )

Aileron and Rudder"

Frequency and Step

• 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 j( )ω

Δ δA j( )ω

Δβ jω( )

Δδ A jω( )

Δp j( )ω

Δ δA j( )ω

Δφ jω( )

Δδ A 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"

Dutch roll natural frequency"

Δr jω( )

Δδ R jω ( )

Δβ jω ( )

Δδ R jω ( )

Δp jω( )

Δδ R jω ( )

Δφ jω ( )

Δδ R 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 "

Δxfast Δxslow

"

#

$

$

%

&

' ' =

Ffast Fslow fast

Ffast slow Fslow

"

#

$

$

%

&

' '

Δxfast

Δxslow

"

#

$

$

%

&

' ' +

Gfast

Gslow

"

#

$

$

%

&

' ' Δu

Δxf = FfΔxf + Fs fΔxs+ GfΔu Δxs = Ff sΔxf + 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"

Δx f ≈ 0 ≈ F f Δx f + F s f Δx s + G f Δu

Δx s = F f s Δx f + F s Δx s + G s Δu

•   Algebraic solution for Δxfast"

0 ≈ FfΔxf + Fs fΔxs+ GfΔu

FfΔxf = −Fs fΔxs− GfΔu

Δxf = −Ff−1 Fs f

Δxs+ GfΔu

•   Substitute quasi-steady Δxfastin differential equation for Δxslow "

Δxs = −Ff s Ff−1 Fs f

Δxs+ GfΔu

& + FsΔxs+ GsΔu

= Fs− Ff sFf−1Fs f

#$ %&Δxs+ G #$ s− Ff sFf−1Gf%&Δu

•   Residualized equation for Δxslow "

F's = Fs − Ff

s

Ff−1

Fs f

G's = Gs− Ff

s

Ff−1

Gf

where!

Residualization"

Residualized Roll-Spiral Mode"

•   Assume that the Dutch roll mode is stable

and faster than the roll mode"

roll on the roll and spiral modes"

ΔxDR ΔxRS

"

#

$

$

%

&

'

0

ΔxRS

"

#

$

$

%

&

'

' =

"

#

$

$

%

&

' '

"

#

$

$

%

&

' ' +

"

#

$

$

%

&

' '

ΔδA ΔδR

"

#

&

'

"

#

$

$

%

&

' '

Residualized Roll-Spiral Mode"

faster than the roll mode"

on the roll and spiral modes"

ΔxDR= −FDR

−1

FRS DR

ΔxRS+ GDR

Δδ A

Δδ R

$

%

( )

* + , -,

/ , 0,

ΔxRS= FRSΔxRS− FDR RSFDR−1 FRS DR

ΔxRS+ GDR Δδ A

Δδ R

$

%

( )

* + , -,

/ , 0,

+ GRS Δδ A

Δδ R

$

%

( )

= F'RSΔxRS+ G'RS Δδ A

Δδ R

$

%

( )

Model of the Residualized

Roll-Spiral Mode"

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

Δp

#

$

% &

'

( = L p 0

1 0

#

$

% &

'

( Δp

Δφ

#

$

% &

' ( − L r Lβ

0 0

#

$

% &

' (

N r Nβ

−1 Yβ

V N

#

$

%

%

&

'

( (

−1

0 g

V N

#

$

%

%

&

'

( (

Δp

Δφ

#

$

% &

' ( +

NδA NδR

0 YδR

V N

#

$

%

%

&

'

( (

Δδ A

Δδ R

#

$

&

'

, -

/

0 1

2

+ LδA LδR

0 0

#

$

&

'

Δδ A

Δδ R

#

$

&

'

Δp

Δ  φ

#

$

%

%

&

'

(

( =

Lp−

Np LrYβ

VN+ Lβ

+ ,

/ 0

Nβ+ NrYβ

VN

+ ,

/ 0

#

$

%

%

%

%

&

'

( ( ( (

g

VN( LrNβ− LβNr)

Nβ+ NrYβ

VN

+ ,

/ 0

#

$

%

%

%

%

&

'

( ( ( (

#

$

%

%

%

%

%

%

&

'

( ( ( ( ( (

Δp

Δφ

#

$

%

%

&

'

( ( +

= f11 f12

#

$

%

%

&

'

( (

Δp

Δφ

#

$

%

%

&

'

( ( +

Roots of the Residualized

Roll-Spiral Mode"

sI − F'RS = s 1 0

0 1

"

#

&

' − f11 f12

"

#

$

$

%

&

' '

= ΔRS res

= s2

− Lp− Np Lβ+ LrYβ/ VN

Nβ+ NrYβ/ VN

* +

-.

"

#

$

$

%

&

' '

s + g

VN

LβNr− LrNβ

Nβ+ NrYβ/ VN

* +

-.

= ( s − λS ) ( s − λR ) or s2

+ 2ζωns + ωn

•   For the business jet model"

ΔRS res = 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"

ΔRS res = 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

#

$

' ( < 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

Supplemental Material

Equilibrium Response of

4 th -Order Model"

•   Equilibrium state with constant aileron and

spiral wind perturbations"

ΔrSS

ΔβSS

ΔpSS

ΔφSS

$

%

&

&

&

&

&

'

(

) ) ) ) )

=

$

%

&

&

&

&

'

(

) ) ) )

Δδ ASS

Δδ RSS

ΔδSFSS

$

%

&

&

&

'

(

) ) )

•   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

L r ≈ C l r

ρV N

2

2I xx

#

$

&

'

(Sb

= C l ˆr

b

2V N

#

$

&

'

ρV N2

2I xx

#

$

&

'

(Sb = C l ˆr

ρV N 4I xx

#

$

&

'

(Sb2

•   Wing is the principal contributor"

–  Differential lift induced by yaw rate "

Cl ˆr

( )Wing= ∂ ΔC ( l)Wing

∂ ˆr = −

CL

α

12

1+ 3λ 1+ λ

&

'

)

*

M2 cos2

Λ − 2

M2 cos2

Λ −1

&

'

* +

•   Thin triangular wing"

Cl ˆr

( )Wing = π αN

9AR

•   Vertical tail"

Cl ˆr

( )Vertical Tail = zvt

lvt

Cn ˆr

( )Vertical Tail

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

(negative here)"

•   Similar to Nβ effect on the Dutch

the spiral mode"

Root Locus Gain = Roll Due to Yaw Rate!

Yaw Acceleration

N p ≈ C n p

ρV N

2

2I zz

#

$%

&

'(Sb

= C n ˆp

b

2V N

#

$%

&

'(

ρV N

2

2I zz

#

$%

&

'(Sb = C n ˆp

ρV N

4I xx

#

$%

&

'(Sb

2

–   Differential yaw moment induced by roll rate "

Cn ˆp

( )Wing= ∂ ( ΔCn)Wing

∂ ˆp =

1 12

1 + 3λ

1 + λ

$

%&

' ()

∂ CD Parasite, Wing

∂α ± CL

$

%

' (

•   Thin triangular wing"

Cn ˆp

( )Wing= − π αN

9AR

•   Vertical tail"

Cn ˆp

( )Vertical Tail = −2αN

lvt b

#

$%

&

'( ( ) Cnβ Vertical Tail

(–): Subsonic!

(+): Supersonic!

Roots"

ΔLD (s) = d(s) + N p n(s) = 0 kn(s)

d(s)= −1 =

N p s s − z( 1)

s −λ1

( ) (s −λ2 )s2

+ 2ζωn s +ωn

sub- and supersonic flight"

•   Effect is analogous to Lβ effect"

Root Locus Gain = Yaw due to Roll Rate!