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Linearized Lateral-Directional Equations of Motion!
Robert Stengel, Aircraft Flight Dynamics MAE 331, 2016
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
Reading:!Flight Dynamics!
574-591!
Learning Objectives
1
•! 6th-order -> 4th-order -> hybrid equations•! Dynamic stability derivatives •! Dutch roll mode•! Roll and spiral modes
Review Questions!!! Why can we normally reduce the 6th-order
longitudinal equations to 4th-order without compromising accuracy?!
!! What are the longitudinal modes of motion?!!! Why are the hybrid 4th-order equations useful?!!! What are “dimensional stability and control
derivatives”?!!! Under what circumstances can the 4th-order set be
replaced by two 2nd-order sets?!!! What benefits accrue from the separation, and what
are the limitations?!!! What is “normal load factor”?!!! Would you rather have one “so-so” flying car or a
good airplane and a good car?! 2
6-Component Lateral-Directional Equations of Motion
!v = YB /m + gsin! cos" # ru + pw!yI = cos" sin$( )u + cos! cos$ + sin! sin" sin$( )v + #sin! cos$ + cos! sin" sin$( )w!p = IzzLB + IxzNB # Ixz Iyy # Ixx # Izz( ) p + Ixz
2 + Izz Izz # Iyy( )%& '(r{ }q( ) ÷ Ixx Izz # Ixz2( )
!r = IxzL B+IxxNB # Ixz Iyy # Ixx # Izz( )r + Ixz2 + Ixx Ixx # Iyy( )%& '( p{ }q( ) ÷ Ixx Izz # Ixz
2( )!! = p + qsin! + r cos!( ) tan"!$ = qsin! + r cos!( )sec"
x1x2x3x4x5x6
!
"
########
$
%
&&&&&&&&
= xLD6
vypr!"
#
$
%%%%%%%%
&
'
((((((((
=
Side VelocityCrossrange
Body-Axis Roll RateBody-Axis Yaw Rate
Roll Angle about Body x AxisYaw Angle about Inertial x Axis
#
$
%%%%%%%%
&
'
(((((((( 3
4- Component Lateral-Directional
Equations of MotionNonlinear Dynamic Equations, neglecting
crossrange and yaw angle
!v = YB /m + gsin! cos" # ru + pw
!p = IzzLB + IxzNB # Ixz Iyy # Ixx # Izz( ) p + Ixz2 + Izz Izz # Iyy( )$% &'r{ }q( ) ÷ Ixx Izz # Ixz
2( )!r = IxzLB + IxxNB # Ixz Iyy # Ixx # Izz( )r + Ixz
2 + Ixx Ixx # Iyy( )$% &' p{ }q( ) ÷ Ixx Izz # Ixz2( )
!! = p + qsin! + r cos!( ) tan"
x1x2x3x4
!
"
#####
$
%
&&&&&
= xLD4
Eurofighter Typhoon
vpr!
"
#
$$$$
%
&
''''
=
Side VelocityBody-Axis Roll RateBody-Axis Yaw Rate
Roll Angle about Body x Axis
"
#
$$$$$
%
&
''''' 4
Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight
Longitudinal variables are constant
!v = YB /m + gsin! cos"N # ruN + pwN
!p = IzzLB + IxzNB( ) Ixx Izz # Ixz2( )
!r = IxzLB + IxxNB( ) Ixx Izz # Ixz2( )
!! = p + r cos!( ) tan"N
qN = 0!N = 0"N =#N
5
Lateral-Directional Force and Moments in Steady, Level Flight
YB = CYB
12!NVN
2S
LB = ClB
12!NVN
2Sb
NB = CnB
12!NVN
2Sb
Body-Axis Side ForceBody-Axis Rolling MomentBody-Axis Yawing Moment
Dynamic pressure is constant
6
Linearized Lateral-Directional Equations of Motion in Steady,
Level Flight!
7
Body-Axis Perturbation Equations of Motion
! !v(t)!!p(t)!!r(t)! !"(t)
#
$
%%%%%
&
'
(((((
=
F11 F12 F13 F14F21 F22 F23 F24F31 F32 F33 F34F41 F42 F43 F44
#
$
%%%%%
&
'
(((((
!v(t)!p(t)!r(t)!"(t)
#
$
%%%%%
&
'
(((((
+ Control[ ]+ Disturbance[ ]
8
Body-Axis Perturbation Variables
!u1!u2
"
#$$
%
&''= !(A
!(R"
#$
%
&' =
Aileron PerturbationRudder Perturbation
"
#$$
%
&''
!w1!w2
"
#$$
%
&''=
!vwind!pwind
"
#$$
%
&''=
Side Wind PerturbationVortical Wind Perturbation
"
#$$
%
&''
!v!p!r!"
#
$
%%%%
&
'
((((
=
Side Velocity PerturbationBody-Axis Roll Rate PerturbationBody-Axis Yaw Rate Perturbation
Roll Angle about Body x Axis Perturbation
#
$
%%%%%
&
'
(((((
9
Vortical Wind PerturbationsWind Rotor Wake Vortex
TangentialVelocity,
ft/s
Radius, ft
Wake Vortex
Wind Rotor10
Side Velocity DynamicsNonlinear equation
First row of linearized dynamic equation
! !v(t) = F11!v(t)+ F12!p(t)+ F13!r(t)+ F14!"(t)[ ]+ G11!#A(t)+G12!#R(t)[ ]+ L11!vwind + L12!pwind[ ]
11
!v = YB /m + gsin! cos"N # ruN + pwN
F11 =
1m
CYv
!NVN2
2S"
#$%&'! Yv
Coefficients in first row of F
Side Velocity Sensitivity to State Perturbations
12
F12 =
1m
CYp
!NVN2
2S"
#$%&'+wN ! Yp +wN
F14 = gcos! cos"N # g
!v = YB /m + gsin! cos"N # ruN + pwN
F13 =
1m
CYr
!NVN2
2S"
#$%&'( uN ! Yr ( uN
Side Velocity Sensitivity to Control and Disturbance Perturbations
G11 =1m
CY!A
"NVN2
2S#
$%
&
'( ! Y!A
G12 =1m
CY!R
"NVN2
2S#
$%
&
'( ! Y!R
Coefficients in first rows of G and L
L11 = !F11L12 = !F12
13
Roll Rate DynamicsNonlinear equation
Second row of linearized dynamic equation
14
!p = IzzLB + IxzNB( ) Ixx Izz ! Ixz
2( )
!!p(t) = F21!v(t)+ F22!p(t)+ F23!r(t)+ F24!"(t)[ ]+ G21!#A(t)+G22!#R(t)[ ]+ L21!vwind + L22!pwind[ ]
F21 = Izz
!LB!v
+ Ixz!NB
!v"#$
%&' Ixx Izz ( Ixz
2( ) ! Lv
Coefficients in second row of F
Roll Rate Sensitivity to State Perturbations
15
F24 = 0
!p = IzzLB + IxzNB( ) Ixx Izz ! Ixz
2( )
F22 = Izz
!LB!p
+ Ixz!NB
!p"#$
%&'
Ixx Izz ( Ixz2( ) ! Lp
F23 = Izz
!LB!r
+ Ixz!NB
!r"#$
%&' Ixx Izz ( Ixz
2( ) ! Lr
Yaw Rate DynamicsNonlinear equation
Third row of linearized dynamic equation
16
!r = IxzLB + IxxNB( ) Ixx Izz ! Ixz
2( )
!!r(t) = F31!v(t)+ F32!p(t)+ F33!r(t)+ F34!"(t)[ ]+ G31!#A(t)+G32!#R(t)[ ]+ L31!vwind + L32!pwind[ ]
F31 = Ixz
!LB!v
+ Ixx!NB
!v"#$
%&' Ixx Izz ( Ixz
2( ) ! Nv
Coefficients in third row of F
Yaw Rate Sensitivity to State Perturbations
17F34 = 0
!r = IxzLB + IxxNB( ) Ixx Izz ! Ixz
2( )
F32 = Ixz
!LB!p
+ Ixx!NB
!p"#$
%&'
Ixx Izz ( Ixz2( ) ! Np
F33 = Ixz
!LB!r
+ Ixx!NB
!r"#$
%&' Ixx Izz ( Ixz
2( ) ! Nr
Roll Angle DynamicsNonlinear equation
Fourth row of linearized dynamic equation
18
!! = p + r cos!( ) tan"N
! !"(t) = F41!v(t)+ F42!p(t)+ F43!r(t)+ F44!"(t)[ ]+ G41!#A(t)+G42!#R(t)[ ]+ L41!vwind + L42!pwind[ ]
F41 = 0
Coefficients in fourth row of F
Roll Angle Sensitivity to State Perturbations
19
!! = p + r cos!( ) tan"N
F42 = 1
F43 = cos!N tan"N # tan"N
F44 = rN sin!N tan"N = 0
Linearized Lateral-Directional Response to Initial Yaw Rate
~Roll response of roll angle
~Spiral response of crossrange
~Spiral response of yaw angle
~Dutch-roll response of side velocity
~Dutch-roll response of roll
and yaw rates
20
Dimensional Stability Derivatives
F11 F12 F13 F14F21 F22 F23 F24F31 F32 F33 F34F41 F42 F43 F44
!
"
#####
$
%
&&&&&
Stability Matrix
=
Yv Yp +wN( ) Yr !uN( ) gcos"NLv Lp Lr 0
Nv Np Nr 0
0 1 tan"N 0
#
$
%%%%%%
&
'
((((((
21
Dimensional Control- and Disturbance-Effect Derivatives
G11 G12G21 G22
G31 G32
G41 G42
!
"
#####
$
%
&&&&&
=
Y'A Y'RL'A L'R
N'A N'R
0 0
!
"
#####
$
%
&&&&&
L11 L12L21 L22L31 L32L41 L42
!
"
#####
$
%
&&&&&
=
Yv YpLv Lp
Nv Np
0 0
!
"
#####
$
%
&&&&&
Control Effect Matrix
DisturbanceEffect Matrix
22
! !v(t)!!p(t)!!r(t)! !"(t)
#
$
%%%%%
&
'
(((((
=
Yv Yp +wN( ) Yr ) uN( ) gcos*N
Lv Lp Lr 0
Nv Np Nr 0
0 1 tan*N 0
#
$
%%%%%%
&
'
((((((
!v(t)!p(t)!r(t)!"(t)
#
$
%%%%%
&
'
(((((
+
Y+A Y+RL+A L+R
N+A N+R
0 0
#
$
%%%%%
&
'
(((((
!+A(t)!+R(t)
#
$%%
&
'((+
Yv YpLv Lp
Nv Np
0 0
#
$
%%%%%
&
'
(((((
!vwind!pwind
#
$%%
&
'((
Rolling and yawing motions
LTI Body-Axis Perturbation Equations of Motion
23
Asymmetrical Aircraft: DC-2-1/2
24
DC-3 with DC-2 right wingQuick fix to fly aircraft out of harm s way during WWII
Asymmetric Aircraft - WWIIBlohm und Voss, BV 141 B + V 141 derivatives
B + V P.202
Recent Asymmetric AircraftScaled Composites Ares
Scaled Composites Boomerang
Oblique Wing Concepts•! High-speed benefits of wing sweep without the
heavy structure and complex mechanism required for symmetric sweep
•! Blohm und Voss, R. T. Jones , Handley-Page concepts
•! Improved supersonic L/D by reduction of shock-wave interference and elimination of the fuselage in flying-wing version
27
Handley-Page Oblique Wing Concepts•! Advantages
–! 10-20% higher L/D @ supersonic speed (compared to delta planform)
–! Flying wing: no fuselage•! Issues
–! Which way do the passengers face?
–! Where is the cockpit?–! How are the engines and vertical
surfaces swiveled?–! What does asymmetry do to
stability and control?
28
NASA Oblique Wing Test Vehicles•! Stability and control issues
abound: The fact that birds and insects are symmetric should give us a clue (though they use huge asymmetry for control)
–! Strong aerodynamic and inertial longitudinal-lateral-directional coupling
–! High side force at zero sideslip angle
–! Torsional divergence of the leading wing
•! Test vehicles: Various model airplanes, NASA AD-1, and NASA DFBW F-8 (below, not built)
29
Stability Axis Representation of Dynamics!
30
Stability Axes•! Stability axes are an alternative set of body axes•! Nominal x axis is offset from the body centerline by
the nominal angle of attack, !!N
31
Transformation from Original Body Axes to Stability Axes
HBS =
cos!N 0 sin!N
0 1 0"sin!N 0 cos!N
#
$
%%%
&
'
(((
!u!v!w
"
#
$$$
%
&
'''S
= HBS
!u!v!w
"
#
$$$
%
&
'''B
!p!q!r
"
#
$$$
%
&
'''S
= HBS
!p!q!r
"
#
$$$
%
&
'''B
Side velocity (!!v) and pitch rate ("q) are unchanged by the transformation 32
Stability-Axis State Vector Rotate body axes to stability axes
!v(t)!p(t)!r(t)!"(t)
#
$
%%%%%
&
'
(((((Body)Axis
*!+N *
!v(t)!p(t)!r(t)!"(t)
#
$
%%%%%
&
'
(((((Stability)Axis
33
Stability-Axis State Vector
!v(t)!p(t)!r(t)!"(t)
#
$
%%%%%
&
'
(((((Stability)Axis
*!+ ,!vVN
*
!+(t)!p(t)!r(t)!"(t)
#
$
%%%%%
&
'
(((((Stability)Axis
Replace side velocity by sideslip angle
34
Stability-Axis State Vector
!"(t)!p(t)!r(t)!#(t)
$
%
&&&&&
'
(
)))))Stability*Axis
+
!r(t)!"(t)!p(t)!#(t)
$
%
&&&&&
'
(
)))))Stability*Axis
=
Stability-Axis Yaw Rate PerturbationSideslip Angle Perturbation
Stability-Axis Roll Rate PerturbationStability-Axis Roll Angle Perturbation
$
%
&&&&&
'
(
)))))
Revise state order
35
Stability-Axis Lateral-Directional Equations
!!r(t)
! !"(t)!!p(t)! !#(t)
$
%
&&&&&
'
(
)))))S
=
Nr N" Np 0
YrVN
*1+,-
./0
Y"
VN
YpVN
gVN
Lr L" Lp 0
0 0 1 0
$
%
&&&&&&&
'
(
)))))))S
!r(t)!"(t)!p(t)!#(t)
$
%
&&&&&
'
(
)))))S
+
N1A N1R
Y1AVN
Y1RVN
L1A L1R
0 0
$
%
&&&&&&
'
(
))))))S
!1A(t)!1R(t)
$
%&&
'
())+
N" Np
Y"
VN
YpVN
L" Lp
0 0
$
%
&&&&&&&
'
(
)))))))S
!"wind
!pwind
$
%&&
'
())
36
Why Modify the Equations?Dutch-roll motion is primarily described by stability-
axis yaw rate and sideslip angle
Roll and spiral motions are primarily described by stability-axis roll rate and roll angle
Stable Spiral
Unstable SpiralRoll
Dutch Roll, top Dutch Roll, front
37
Why Modify the Equations?
FLD =FDR FRS
DR
FDRRS FRS
!
"##
$
%&&=
FDR smallsmall FRS
!
"##
$
%&&'
FDR 00 FRS
!
"##
$
%&&
Effects of Dutch roll perturbations on Dutch roll motion
Effects of Dutch roll perturbations on roll-spiral motion
Effects of roll-spiral perturbations on Dutch roll motion
Effects of roll-spiral perturbations on roll-spiral motion
... but are the off-diagonal blocks really small?
38
Stability, Control, and Disturbance Matrices
FLD =FDR FRS
DR
FDRRS FRS
!
"##
$
%&&=
Nr N' Np 0
YrVN
(1)*+
,-.
Y'
VN
YpVN
gVN
Lr L' Lp 0
0 0 1 0
!
"
#######
$
%
&&&&&&&
GLD =
N!A N!R
Y!AVN
Y!RVN
L!A L!R
0 0
"
#
$$$$$$
%
&
''''''
LLD =
N! Np
Y!
VN
YpVN
L! Lp
0 0
"
#
$$$$$$$
%
&
'''''''
!x1!x2!x3!x4
"
#
$$$$$
%
&
'''''
=
!r!(
!p!)
"
#
$$$$$
%
&
'''''
!u1!u2
"
#$$
%
&''=
!(A!(R
"
#$
%
&'
!w1!w2
"
#$$
%
&''=
!(A!(R
"
#$
%
&'
39
2nd-Order Approximate Modes of Lateral-Directional
Motion!
40
2nd-Order Approximations in System Matrices
FLD =FDR 00 FRS
!
"##
$
%&&=
Nr N' 0 0
YrVN
(1)*+
,-.
Y'
VN0 0
0 0 Lp 0
0 0 1 0
!
"
#######
$
%
&&&&&&&
GLD =
N!R 0Y!RVN
0
0 L!A
0 0
"
#
$$$$$$
%
&
''''''
LLD =
N! 0
Y!
VN0
0 Lp
0 0
"
#
$$$$$$$
%
&
''''''' 41
2nd-Order Models of Lateral-Directional Motion
Approximate Spiral-Roll Equation
Approximate Dutch Roll Equation
!!xDR =!!r! !"
#
$%%
&
'(()
Nr N"
YrVN
*1+,-
./0
Y"
VN
#
$
%%%%
&
'
((((
!r!"
#
$%%
&
'((+
N1R
Y1RVN
#
$
%%%
&
'
(((!1R +
N"
Y"
VN
#
$
%%%
&
'
(((!"wind
!!xRS =!!p! !"
#
$%%
&
'(()
Lp 0
1 0
#
$%%
&
'((
!p!"
#
$%%
&
'((+
L*A0
#
$%%
&
'((!*A +
Lp
0
#
$%%
&
'((!pwind
42
Comparison of 4th- and 2nd-Order Dynamic Models!
43
Comparison of 2nd- and 4th-Order Initial-Condition Responses of Business Jet
Fourth-Order Response Second-Order Response
Speed and damping of responses is adequately portrayed by 2nd-order modelsRoll-spiral modes have little effect on yaw rate and sideslip angle responses
BUT Dutch roll mode has large effect on roll rate and roll angle responses44
4 initial conditions [r(0), !(0), p(0), !(0)]
Next Time:!Analysis of Time Response!
!
Reading:!Flight Dynamics!298-313, 338-342!
45
•! Methods of time-domain analysis–! Continuous- and discrete-time models–! Transient response to initial conditions and inputs–! Steady-state (equilibrium) response–! Phase-plane plots–! Response to sinusoidal input
Learning Objectives
Supplemental Material
46
Primary Lateral-Directional Control Derivatives
L!A = Cl!A
"NVN2
2Ixx
#$%
&'(Sb
N!R = Cn!R
"NVN2
2Izz
#$%
&'(Sb
47
Initial Condition Response of Approximate Dutch Roll Mode
!r t( ) !" t( )
48
Dutch roll VideoCalspan Variable-Stability Learjet Dutch roll Oscillation
http://www.youtube.com/watch?v=1_TW9oz99NQ
49