Linearized Lateral-Directional Equations of Motionstengel/MAE331Lecture13.pdf ·...
Transcript of Linearized Lateral-Directional Equations of Motionstengel/MAE331Lecture13.pdf ·...
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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
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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!
"
#
$$$$
%
&
''''
=
Side VelocityBody-Axis Roll RateBody-Axis Yaw Rate
Roll Angle about Body x Axis
"
#
$$$$$
%
&
''''' 4
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Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight
Longitudinal variables are constant
!v = YB /m + gsin! cos"N # ruN + pwN
!p = IzzLB + IxzNB( ) Ixx Izz # Ixz2( )
!r = IxzLB + IxxNB( ) Ixx Izz # Ixz2( )
!! = p + r cos!( ) tan"N
qN = 0!N = 0"N =#N
5
Lateral-Directional Force and Moments in Steady, Level Flight
YB = CYB
12!NVN
2S
LB = ClB
12!NVN
2Sb
NB = CnB
12!NVN
2Sb
Body-Axis Side ForceBody-Axis Rolling MomentBody-Axis Yawing Moment
Dynamic pressure is constant
6
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Linearized Lateral-Directional Equations of Motion in Steady,
Level Flight!
7
Body-Axis Perturbation Equations of Motion
! !v(t)!!p(t)!!r(t)! !"(t)
#
$
%%%%%
&
'
(((((
=
F11 F12 F13 F14F21 F22 F23 F24F31 F32 F33 F34F41 F42 F43 F44
#
$
%%%%%
&
'
(((((
!v(t)!p(t)!r(t)!"(t)
#
$
%%%%%
&
'
(((((
+ Control[ ]+ Disturbance[ ]
8
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Body-Axis Perturbation Variables
!u1!u2
"
#$$
%
&''= !(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
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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
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Side Velocity Sensitivity to Control and Disturbance Perturbations
G11 =1m
CY!A
"NVN2
2S#
$%
&
'( ! Y!A
G12 =1m
CY!R
"NVN2
2S#
$%
&
'( ! Y!R
Coefficients in first rows of G and L
L11 = !F11L12 = !F12
13
Roll Rate DynamicsNonlinear equation
Second row of linearized dynamic equation
14
!p = IzzLB + IxzNB( ) Ixx Izz ! Ixz
2( )
!!p(t) = F21!v(t)+ F22!p(t)+ F23!r(t)+ F24!"(t)[ ]+ G21!#A(t)+G22!#R(t)[ ]+ L21!vwind + L22!pwind[ ]
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F21 = Izz
!LB!v
+ Ixz!NB
!v"#$
%&' 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[ ]
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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[ ]
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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
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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
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! !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
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Asymmetric Aircraft - WWIIBlohm und Voss, BV 141 B + V 141 derivatives
B + V P.202
Recent Asymmetric AircraftScaled Composites Ares
Scaled Composites Boomerang
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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
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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
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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
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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
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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
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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
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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
"
#$$
%
&''=
!(A!(R
"
#$
%
&'
39
2nd-Order Approximate Modes of Lateral-Directional
Motion!
40
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2nd-Order Approximations in System Matrices
FLD =FDR 00 FRS
!
"##
$
%&&=
Nr N' 0 0
YrVN
(1)*+
,-.
Y'
VN0 0
0 0 Lp 0
0 0 1 0
!
"
#######
$
%
&&&&&&&
GLD =
N!R 0Y!RVN
0
0 L!A
0 0
"
#
$$$$$$
%
&
''''''
LLD =
N! 0
Y!
VN0
0 Lp
0 0
"
#
$$$$$$$
%
&
''''''' 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 +
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1 0
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42
![Page 22: Linearized Lateral-Directional Equations of Motionstengel/MAE331Lecture13.pdf · Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight Longitudinal variables](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f21537f8b9a84178c67c0/html5/thumbnails/22.jpg)
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)]
![Page 23: Linearized Lateral-Directional Equations of Motionstengel/MAE331Lecture13.pdf · Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight Longitudinal variables](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f21537f8b9a84178c67c0/html5/thumbnails/23.jpg)
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
![Page 24: Linearized Lateral-Directional Equations of Motionstengel/MAE331Lecture13.pdf · Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight Longitudinal variables](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f21537f8b9a84178c67c0/html5/thumbnails/24.jpg)
Primary Lateral-Directional Control Derivatives
L!A = Cl!A
"NVN2
2Ixx
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N!R = Cn!R
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47
Initial Condition Response of Approximate Dutch Roll Mode
!r t( ) !" t( )
48
![Page 25: Linearized Lateral-Directional Equations of Motionstengel/MAE331Lecture13.pdf · Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight Longitudinal variables](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f21537f8b9a84178c67c0/html5/thumbnails/25.jpg)
Dutch roll VideoCalspan Variable-Stability Learjet Dutch roll Oscillation
http://www.youtube.com/watch?v=1_TW9oz99NQ
49