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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
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)