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Advanced Problems of Lateral-Directional Dynamics

Robert Stengel, Aircraft Flight DynamicsMAE 331, 2018

• 4th-order dynamics– Steady-state response to control– Transfer functions– Frequency response– Root locus analysis of parameter variations

• Residualization

Copyright 2018 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

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

Flight Dynamics595-627

1

Learning Objectives

Stability-Axis Lateral-Directional Equations

Δ!r(t)

Δ !β(t)Δ!p(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 NδR0 0LδA ~ 00 0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ΔδAΔδR

⎡

⎣⎢

⎤

⎦⎥

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)

2

http://www.princeton.edu/~stengel/FlightDynamics.htmlhttp://adg.stanford.edu/aa241/AircraftDesign.html

12/4/18

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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

⎛⎝⎜

⎞⎠⎟ s

3

+ Nβ − LrNp + LpYβVN

+ NrYβVN

+ Lp⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥s2

+ Yβ VNLrNp − LpNr( ) + Lβ Np − gVN( )⎡⎣⎢ ⎤⎦⎥ s

+ gVNLβNr − LrNβ( )

= s4 + a3s3 + a2s

2 + a1s + a0 = 0

3

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 SpiralStable Roll

Lightly damped Dutch roll

Dutch roll

Spiral

Dutch roll

Roll

4

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4th-Order Initial-Condition Responses of Business Jet

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

• Roll angle is slightly divergent (spiral mode)• 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

5

ΔRS (s) = s s− Lp( )λS = 0λR = Lp

Approximate Roll and Spiral Modes

ΔpΔ φ

#

$%%

&

'((=

Lp 0

1 0

#

$%%

&

'((

ΔpΔφ

#

$%%

&

'((+

LδA0

#

$%%

&

'((ΔδA

Characteristic polynomial has real roots

• Roll rate is damped by Lp• Roll angle is a pure integral of roll rate

Neutral stability

Generally < 0

6

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Approximate Dutch Roll Mode

ΔrΔ β

#

$%%

&

'((=

Nr NβYrVN

−1*

+,

-

./

YβVN

#

$

%%%%

&

'

((((

ΔrΔβ

#

$%%

&

'((+

NδRYδRVN

#

$

%%%

&

'

(((ΔδR

ωnDR = Nβ +NrYβVN

ζDR = − Nr +YβVN

%

&'

(

)* 2 Nβ +Nr

YβVN

• With negligible side-force sensitivity to yaw rate, Yr

• Characteristic polynomial, natural frequency, and damping ratio

7

ΔDR(s) = s2 − Nr +

YβVN

⎛⎝⎜

⎞⎠⎟ s + Nβ 1−

YrVN( )+ Nr Yβ VN⎡⎣⎢ ⎤⎦⎥

ω nDR = Nβ 1−YrVN( )+ Nr Yβ VN

ζDR = − Nr +YβVN

⎛⎝⎜

⎞⎠⎟ 2 Nβ 1−

YrVN( )+ Nr Yβ VN

Effects of Variation in Primary Stability

Derivatives

8

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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

k n(s)d(s)

= −1=Nβ s − z1( ) s − z2( )

s − λ1( ) s − λ2( ) s2 + 2ζω ns +ω n2( )

Nβ > 0

Nβ < 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

9

Nr Effect on 4th-Order Roots

ΔLD (s) = d(s)+ Nrn(s) = 0

k n(s)d(s)

= −1=Nr s − z1( ) s2 + 2µνns +νn2( )

s − λ1( ) s − λ2( ) s2 + 2ζω ns +ω n2( )• Negative Nr

– Increases Dutch roll damping – Draws spiral and roll modes together

• Positive Nr destabilizes Dutch roll mode

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

Roll Spiral

Zero

Dutch Roll

Dutch Roll

Zero

Zero

10

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Lp Effect on 4th-Order Roots

ΔLD (s) = d(s)+ Lpn(s) = 0

k n(s)d(s)

= −1=Lps s

2 + 2µνns +νn2( )

s − λ1( ) s − λ2( ) s2 + 2ζω ns +ω n2( )Lp < 0 Lp > 0

• Negative Lp– Roll mode time constant– Draws spiral mode toward origin

• Positive Lp destabilizes roll mode• Lp: negligible effect on Dutch roll

mode• Lp can become positive at high angle

of attack

Root Locus Gain = Roll Damping

Roll SpiralZero

Dutch Roll & Zero

Dutch Roll & Zero

11

Coupling Stability Derivatives

12

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Dihedral (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) = 0k n(s)d(s)

= −1=Lβ

gVN

− Np( ) s − z1( )s − λS( ) s − λR( ) s2 + 2ζω nDR s +ω nDR2( )

Lβ< 0 Lβ > 0

Bizjet Example

ΔLD (s) =

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

Roll SpiralZero

Dutch Roll

Dutch Roll

13

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

14

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Oscillatory Roll-Spiral ModeΔRSres = s − λS( ) s − λR( ) or s2 + 2ζω ns +ω n2( )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( )⎡

⎣⎢

⎤

⎦⎥

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4th-Order Frequency Response

17

Yaw Rate and Sideslip Angle Frequency Responses of Business Jet

2nd-Order Response to Rudder

Yawing response to aileron is not negligibleYaw 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( )

18

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Roll Rate and Roll Angle Frequency Responses of Business Jet

2nd-Order Response to Aileron

Roll response to rudder is not negligibleRoll 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

19

Frequency and Step Responses to Aileron Input

Roll rate response is relatively benignRatio 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( )

20

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Frequency and Step Responses to Rudder Input

Lightly damped yaw/sideslip responsewould 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( )

21

Reduction of Model Order by Residualization

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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 fastΔxslow

"

#

$$

%

&

''=

Ffast Fslowfast

Ffastslow Fslow

"

#

$$

%

&

''

Δx fastΔxslow

"

#

$$

%

&

''+

G fastGslow

"

#

$$

%

&

''Δu

Δx f = FfΔx f + FsfΔxs +G fΔu

Δxs = FfsΔx f + FsΔxs +GsΔu 23

Express as 2 separate equations

Approximation for Fast-Mode ResponseAssume that fast mode reaches steady state very quickly

compared to slow-mode response

Steady-state solution for Δxfast

Δx f = −Ff−1 Fs

fΔxs +G fΔu( )24

Δ!x f ≈ 0 ≈ FfΔx f + FsfΔxs +G fΔu

Δ!xs = FfsΔx f + FsΔxs +GsΔu

0 ≈ FfΔx f + FsfΔxs +G fΔu

FfΔx f = −FsfΔxs −G fΔu

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Substitute quasi-steady Δxfast in differential equation for Δxslow

Residualized differential equation for Δxslow

Δxs = F 's Δxs +G 's Δu

F 's = Fs − FfsFf

−1Fsf⎡⎣ ⎤⎦

G 's = Gs − FfsFf

−1G f⎡⎣ ⎤⎦

where

Adjust Slow-Mode Equation for Fast-Mode Steady State

25

Δ!xs = −Ffs Ff

−1 FsfΔxs +G fΔu( )⎡⎣ ⎤⎦ + FsΔxs +GsΔu

= Fs − FfsFf

−1Fsf⎡⎣ ⎤⎦Δxs + Gs − Ff

sFf−1−1G f⎡⎣ ⎤⎦Δu

Model of the ResidualizedRoll-Spiral Mode

Residualized roll/spiral equation

Δ!pΔ !φ

⎡

⎣⎢⎢

⎤

⎦⎥⎥"

Lp −Np Lr

YβVN

+ Lβ⎛⎝⎜

⎞⎠⎟

Nβ + NrYβVN

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

gVN

LrNβ − LβNr( )Nβ + Nr

YβVN

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

ΔpΔφ

⎡

⎣⎢⎢

⎤

⎦⎥⎥+#

=f11 f121 0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

ΔpΔφ

⎡

⎣⎢⎢

⎤

⎦⎥⎥+#

26

Yawing motion is assumed to be instantaneous compared to rolling motions

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ΔRSres = s2 +1.0894s − 0.0108 = 0

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

Roots of the 2nd-Order Residualized Roll-Spiral Mode

For the business jet model

Slightly unstable spiral modeSimilar to 4th-order roll-spiral results

ΔLD (s) = s − 0.00883( ) s +1.2( ) s2 + 2 0.08( ) 1.39( )s +1.392#$ %& 27

sI− F 'RS = s1 00 1

⎡

⎣⎢

⎤

⎦⎥ −

f11 f121 0

⎡

⎣⎢⎢

⎤

⎦⎥⎥= ΔRSres

= s2 − Lp − NpLβ + LrYβ /VNNβ + NrYβ /VN

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥s + g

VN

LβNr − LrNβNβ + NrYβ /VN

⎛

⎝⎜⎞

⎠⎟

= s − λS( ) s − λR( ) or s2 + 2ζω ns +ω n2( )RS = 0

28

Next:Flying Qualities

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Supplemental Material

29

• 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 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 -1.20.2501 -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

Unstable

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Roll Acceleration Due to Yaw Rate, Lr

Lr ≈ClrρVN

2

2Ixx

⎛⎝⎜

⎞⎠⎟Sb

= Clr̂b2VN

⎛⎝⎜

⎞⎠⎟

ρVN2

2Ixx

⎛⎝⎜

⎞⎠⎟Sb = Clr̂

ρVN4Ixx

⎛⎝⎜

⎞⎠⎟Sb2

31

ΔLD (s) = d(s)+ LrNpn(s) = 0

kn(s)d(s)

= −1=LrNp s − z1( ) s − z2( )

s − λ1( ) s − λ2( ) s2 + 2ζω ns +ω n2( )

Root Locus Gain = Roll Due to Yaw Rate Lr < 0 Lr > 0

Yaw Acceleration Due to Roll Rate, Np

Np ≈ CnpρVN

2

2Izz

#

$%&

'(Sb

= Cnp̂b2VN

#

$%&

'(ρVN

2

2Izz

#

$%&

'(Sb = Cnp̂

ρVN4Ixx

#

$%&

'(Sb2

32

ΔLD (s) = d(s)+ Npn(s) = 0

kn(s)d(s)

= −1=Nps s − z1( )

s − λ1( ) s − λ2( ) s2 + 2ζω ns +ω n2( )

Np > 0 Np < 0 Root Locus Gain = Yaw due to Roll Rate

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Steady-State ResponseΔxS = −F

−1GΔuS

33

Equilibrium Response of2nd-Order Dutch Roll Model

ΔrSSΔβSS

#

$%%

&

'((= −

YβVN

−Nβ

1 Nr

#

$

%%%

&

'

(((

YβVN

Nr +Nβ*

+,

-

./

NδR0

#

$%%

&

'((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

34

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Equilibrium Response of 2nd-Order Roll-Spiral Model

ΔpS = −LδALp

ΔδAS

Δφ(t)S = −LδALp

Δδ AS 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

35

Equilibrium Response of 4th-Order Model

Equilibrium state with constant aileron and rudder deflection

ΔrSΔβSΔpSΔφS

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= −

Nr Nβ Np 0

−1YβVN

0 gVN

Lr Lβ Lp 0

0 0 1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

−1

~ 0 NδR0 0LδA ~ 00 0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ΔδASΔδRS

⎡

⎣⎢⎢

⎤

⎦⎥⎥

36

ΔrSΔβSΔpSΔφS

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

gVN

LδANβ −gVN

LβNδR

gVN

LδANrgVN

LrNδR

0 0

Nβ + NrYβVN

⎛⎝⎜

⎞⎠⎟LδA − Lβ + Lr

YβVN

⎛⎝⎜

⎞⎠⎟NδR

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

gVN

LβNr − LrNβ( )ΔδASΔδRS

⎡

⎣⎢⎢

⎤

⎦⎥⎥

12/4/18

19

37

NTSB Simulation of American Flight 587

Flight simulation derived from digital flight data recorder (DFDR) tape

38