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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
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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
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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
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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
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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( )
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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
⎡
⎣⎢⎢
⎤
⎦⎥⎥
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19
37
NTSB Simulation of American Flight 587
Flight simulation derived from digital flight data recorder (DFDR) tape
38