Advanced Problems of Lateral-Directional Dynamics !

Robert Stengel, Aircraft Flight Dynamics!MAE 331, 2016

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

•! Residualization

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

Flight Dynamics!595-627!

1

Learning Objectives

Stability-Axis Lateral-Directional Equations

!!r(t)

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

$

%

&&&&&

'

(

)))))

=

Nr N" Np 0

*1Y"

VN0 g

VNLr L" Lp 0

0 0 1 0

$

%

&&&&&&&

'

(

)))))))

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

$

%

&&&&&

'

(

)))))

+

~ 0 N+R

0 0L+A ~ 00 0

$

%

&&&&&

'

(

)))))

!+A!+R

$

%&

'

()

!x1!x2!x3!x4

"

#

$$$$$

%

&

'''''

=

!r!(

!p!)

"

#

$$$$$

%

&

'''''

=

Yaw Rate PerturbationSideslip Angle PerturbationRoll Rate PerturbationRoll Angle Perturbation

"

#

$$$$$

%

&

'''''

!u1!u2

"

#$$

%

&''=

!(A!(R

"

#$

%

&' =

Aileron PerturbationRudder Perturbation

"

#$$

%

&''

With idealized aileron and rudder effects (i.e., N!!A = L!!R = 0)

2

Lateral-Directional Characteristic Equation

!LD (s) = s " #S( ) s " #R( ) s2 + 2$%ns +%n2( )DR

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

!LD (s) = s4 + Lp + Nr +

Y"VN

#$%

&'( s

3

+ N" ) LrNp + LpY"VN

+ NrY"VN

+ Lp#$%

&'(

*+,

-./s2

+ Y"VN

LrNp ) LpNr( ) + L" Np )gVN( )*

+,-./s

+ gVN

L"Nr ) LrN"( )= s4 + a3s

3 + a2s2 + a1s + a0 = 0

3

Business Jet Example of Lateral-Directional

Characteristic Equation!LD (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

4

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

5

Approximate Roll and Spiral Modes

!!p! !"

#

$%%

&

'((=

Lp 0

1 0

#

$%%

&

'((

!p!"

#

$%%

&

'((+

L)A0

#

$%%

&

'((!)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

Neutral stability

Generally < 0

6

Approximate Dutch Roll Mode

!!r! !"

#

$%%

&

'((=

Nr N"

YrVN

)1*

+,

-

./

Y"VN

#

$

%%%%

&

'

((((

!r!"

#

$%%

&

'((+

N0R

Y0RVN

#

$

%%%

&

'

(((!0R

!DR(s) = s2 " Nr +

Y#VN

$%&

'() s + N# 1"

YrVN( ) + Nr

Y#VN

*+,

-./

0 nDR= N# 1"

YrVN( ) + Nr

Y#VN

1 DR = " Nr +Y#VN

$%&

'() 2 N# 1"

YrVN( ) + Nr

Y#VN

!nDR= N" +Nr

Y"VN

#DR = $ Nr +Y"VN

%

&'

(

)* 2 N" +Nr

Y"VN

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

•! Characteristic polynomial, natural frequency, and damping ratio

7

Effects of Variation in Primary Stability

Derivatives!

8

N"" Effect on 4th-Order Roots

•! Group !!(s) terms multiplied by N"" to form numerator

•! Denominator formed from remaining terms of !!(s)

!LD (s) = d(s)+ N"n(s) = 0

k n(s)d(s)

= #1=N" s # z1( ) s # z2( )

s # $1( ) s # $2( ) s2 + 2%& ns +& n2( )

N" > 0

N" < 0•! Positive N!

–! Increases Dutch roll natural frequency –! Damping ratio decreases but remains

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

•! Negative N"" destabilizes Dutch roll mode

Root Locus Gain = Directional Stability

Roll Spiral

Dutch Roll

Dutch Roll

Zero

Zero

9

Nr Effect on 4th-Order Roots

!LD (s) = d(s)+ Nrn(s) = 0

k n(s)d(s)

= "1=Nr s " z1( ) s2 + 2µ#ns +#n

2( )s " $1( ) s " $2( ) s2 + 2%& ns +& n

2( )•!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

Lp Effect on 4th-Order Roots

!LD (s) = d(s)+ Lpn(s) = 0

k n(s)d(s)

= "1=Lps s

2 + 2µ#ns +#n2( )

s " $1( ) s " $2( ) s2 + 2%& ns +& 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

Dihedral (L"") Effect on 4th-Order Roots

•!Negative L"" –! Stabilizes spiral and roll modes but ...–! Destabilizes Dutch roll mode

•!Positive L"" does the opposite

Root Locus Gain = Dihedral Effect

!LD (s) = d(s)+ L"gVN

# Np( )n(s) = 0k n(s)d(s)

= #1=L"

gVN

# Np( ) s # z1( )s # $S( ) s # $R( ) s2 + 2%& nDR

s +& nDR2( )

L"< 0 L" > 0

Bizjet Example

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

How can L"" and Nr be adjusted artificially , i.e., by closed-loop control?

Original Root Locus Increased |Nr|

Solar Impulse

14

Roll Acceleration Due to Yaw Rate, Lr

Lr !Clr

"VN2

2Ixx

#$%

&'(Sb

= Clr̂

b2VN

#$%

&'(

"VN2

2Ixx

#$%

&'(Sb = Clr̂

"VN4Ixx

#$%

&'(Sb2

15

!LD (s) = d(s)+ LrNpn(s) = 0

kn(s)d(s)

= "1=LrNp s " z1( ) s " z2( )

s " #1( ) s " #2( ) s2 + 2$% ns +% n2( )

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

Yaw Acceleration Due to Roll Rate, Np

Np ! Cnp

"VN2

2Izz

#

$%&

'(Sb

= Cnp̂

b2VN

#

$%&

'("VN

2

2Izz

#

$%&

'(Sb = Cnp̂

"VN4Ixx

#

$%&

'(Sb2

16

!LD (s) = d(s)+ Npn(s) = 0

kn(s)d(s)

= "1=Nps s " z1( )

s " #1( ) s " #2( ) s2 + 2$% ns +% n2( )

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

Oscillatory Roll-Spiral Mode!RSres

= s " #S( ) s " #R( ) or s2 + 2$% ns +% n2( )RS

The characteristic equation factors into real or complex roots

Real roots are roll mode and spiral mode when

L!Nr > LrN! and

Np L! + LrY! /VN( ) 2 gVN

L!Nr " LrN!( )#

$%

&

'( <1

L!Nr < LrN!

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

17

Roll-Spiral Oscillation of the M2-F2 Lifting Body Test Vehicle

18

4th-Order Frequency Response!

19

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

!r j"( )!#A j"( )

!" j#( )!$A j#( )

!r j"( )!#R j"( )

!" j#( )!$R j#( )

!r j"( )!#R j"( )

!" j#( )!$R j#( )

20

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

!p j"( )!#R j"( )

!" j#( )!$R j#( )

!p j"( )!#A j"( )

!" j#( )!$A j#( )

!p j"( )!#A j"( )

!" j#( )!$A j#( )

4th-Order Response to Aileron and Rudder

21

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

!r j"( )!#A j"( )

!" j#( )!$A j#( )

!p j"( )!#A j"( )

!" j#( )!$A j#( )

!v t( )

!y t( )

!r t( )

!p t( )

!" t( )

!" t( )

22

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

!r j"( )!#R j"( )

!" j#( )!$R j#( )

!p j"( )!#R j"( )

!" j#( )!$R j#( )

!v t( ) !y t( )

!r t( )

!p t( )!" t( )

!" t( )

23

24

Next:!Lateral-Directional

Flying Qualities!Learning Objectives

•! LCDP•! "#/"d•! #/!•! Eigenvectors•! Pilot-Vehicle Interactions•! PIO

Supplemental Material!

25

Review Questions!!! What is a flying qualities “thumbprint?”!!! What is the best Cooper-Harper pilot rating?!!! What is MIL-F-8785C?!

!! What are the Airplane Types?!!! What are the Flight Phases?!!! What are the Levels of Performance?!

!! What is the “Time to Double?”!!! What is the “Control Anticipation Parameter?”!!! What is the “C* Criterion?”!!! What is the “Gibson Dropback Criterion for Pitch

Angle Control?”!

26

•! 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

27

Steady-State Response!!xS = "F

"1G!uS

28

Equilibrium Response of2nd-Order Dutch Roll Model

!rSS!"SS

#

$%%

&

'((= )

Y"VN

)N"

1 Nr

#

$

%%%

&

'

(((

Y"VN

Nr +N"

*

+,

-

./

N0R

0

#

$%%

&

'((!0RSS

!rS = "

Y#VN

N$R

%

&'

(

)*

Y#VN

Nr +N#

%

&'

(

)*

!$RS

!#S = "N$R

Y#VN

Nr +N#

%

&'

(

)*

!$RS

Equilibrium response to constant rudder

Steady yaw rate and sideslip angle are not zero

29

Equilibrium Response of 2nd-Order Roll-Spiral Model

!pS = " L#A

Lp

!#AS

!$(t)S = " L#A

Lp

!# AS dt0

t

%

!pSS!"SS

#

$%%

&

'((= )

Lp 0

1 0

#

$%%

&

'((

)1L*A0

#

$%%

&

'((!*ASS

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

!pS = "Lp"1L#A!#AS

taken alone

30

Equilibrium Response of 4th-Order Model

Equilibrium state with constant aileron and rudder deflection

!rS!"S

!pS!#S

$

%

&&&&&

'

(

)))))

= *

Nr N" Np 0

*1Y"

VN0 g

VNLr L" Lp 0

0 0 1 0

$

%

&&&&&&&

'

(

)))))))

*1

~ 0 N+R

0 0L+A ~ 00 0

$

%

&&&&&

'

(

)))))

!+AS!+RS

$

%&&

'

())

31

!rS!"S

!pS!#S

$

%

&&&&&

'

(

)))))

=

gVN

L*AN" + gVN

L"N*R

gVN

L*ANrgVN

LrN*R

0 0

N" + Nr

Y"

VN,-.

/01L*A + L" + Lr

Y"

VN,-.

/01N*R

$

%

&&&&&&&&&

'

(

)))))))))

gVN

L"Nr + LrN"( )!*AS!*RS

$

%&&

'

())

32

NTSB Simulation of American Flight 587

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

33

Reduction of Model Order by Residualization!

34

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

!!x fast

!!xslow

"

#

$$

%

&

''=

Ffast Fslowfast

Ffastslow Fslow

"

#

$$

%

&

''

!x fast

!xslow

"

#

$$

%

&

''+

G fast

Gslow

"

#

$$

%

&

''!u

!!x f = Ff!x f + Fsf!xs +G f!u

!!xs = Ffs!x f + Fs!xs +Gs!u 35

Express as 2 separate equations

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

compared to slow-mode response

!!x f " 0 " Ff!x f + Fsf!xs +G f!u

!!xs = Ffs!x f + Fs!xs +Gs!u

Steady-state solution for !!xfast

0 ! Ff"x f + Fsf"xs +G f"u

Ff"x f = #Fsf"xs #G f"u

!x f = "Ff"1 Fs

f!xs +G f!u( )36

Substitute quasi-steady !!xfast in differential equation for !!xslow

!!xs = "Ffs Ff

"1 Fsf!xs +G f!u( )#$ %& + Fs!xs +Gs!u

= Fs " FfsFf

"1Fsf#$ %&!xs + Gs " Ff

sFf"1"1G f#$ %&!u

Residualized differential equation for !!xslow

!!xs = F 's !xs +G 's !u

F 's = Fs ! FfsFf

!1Fsf"# $%

G 's = Gs ! FfsFf

!1G f"# $%

where

Adjust Slow-Mode Equation for Fast-Mode Steady State

37

Model of the Residualized Roll-Spiral Mode

Residualized roll/spiral equation

!!p! !"

#

$%%

&

'(("

Lp )Np Lr

Y*

VN+ L*

+,-

./0

N* + Nr

Y*

VN+,-

./0

#

$

%%%%

&

'

((((

gVN

LrN* ) L*Nr( )N* + Nr

Y*

VN+,-

./0

#

$

%%%%

&

'

((((

1 0

#

$

%%%%%%

&

'

((((((

!p!"

#

$%%

&

'((+#

=f11 f121 0

#

$%%

&

'((

!p!"

#

$%%

&

'((+#

38

Yawing motion is assumed to be instantaneous compared to rolling motions

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

sI! F 'RS = s 1 00 1

"

#$

%

&' !

f11 f121 0

"

#$$

%

&''= (RSres

= s2 ! Lp ! Np

L) + LrY) /VNN) + NrY) /VN

*

+,-

./"

#$$

%

&''s + g

VN

L)Nr ! LrN)

N) + NrY) /VN

*

+,-

./

= s ! 0S( ) s ! 0R( ) or s2 + 212 ns +2 n2( )RS = 0

For the business jet model

!RSres= s2 +1.0894s " 0.0108 = 0

= s " 0.0098( ) s +1.1( ) = s " #S( ) s " #R( )Slightly unstable spiral mode

Similar to 4th-order roll-spiral results

!LD (s) = s " 0.00883( ) s +1.2( ) s2 + 2 0.08( ) 1.39( )s +1.392#$ %& 39