Linearized Lateral-Directional Equations of Motion
Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012"
• Spiral, Dutch roll, and roll modes"
• Stability derivatives"
Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!http://www.princeton.edu/~stengel/MAE331.html!
http://www.princeton.edu/~stengel/FlightDynamics.html!
6-Component "Lateral-Directional
Equations of Motion"
State Vector, 6 components!
Nonlinear Dynamic Equations!
v = Y / m + gsinφ cosθ − ru + pwyI = cosθ sinψ( )u + cosφ cosψ + sinφ sinθ sinψ( )v + − sinφ cosψ + cosφ sinθ sinψ( )w
p = IzzL + IxzN − Ixz Iyy − Ixx − Izz( ) p + Ixz2 + Izz Izz − Iyy( )%& '(r{ }q( ) ÷ Ixx Izz − Ixz
2( )r = IxzL + IxxN − Ixz Iyy − Ixx − Izz( )r + Ixz
2 + Ixx Ixx − Iyy( )%& '( p{ }q( ) ÷ Ixx Izz − Ixz2( )
φ = p + qsinφ + r cosφ( ) tanθψ = qsinφ + r cosφ( )secθ
x1x2x3x4x5x6
!
"
########
$
%
&&&&&&&&
= xLD6 =
vyprφ
ψ
!
"
########
$
%
&&&&&&&&
=
Side VelocityCrossrange
Body − Axis Roll RateBody − Axis Yaw Rate
Roll Angle about Body x AxisYaw Angle about Inertial x Axis
!
"
########
$
%
&&&&&&&&
Douglas A-4!
4- Component "Lateral-Directional
Equations of Motion"
State Vector, 4 components!
Nonlinear Dynamic Equations, neglecting crossrange and yaw angle!
v = Y / m + gsinφ cosθ − ru + pw
p = IzzL + IxzN − Ixz Iyy − Ixx − Izz( ) p + Ixz2 + Izz Izz − Iyy( )$% &'r{ }q( ) ÷ Ixx Izz − Ixz
2( )r = IxzL + IxxN − Ixz Iyy − Ixx − Izz( )r + Ixz
2 + Ixx Ixx − Iyy( )$% &' p{ }q( ) ÷ Ixx Izz − Ixz2( )
φ = p + qsinφ + r cosφ( ) tanθ
x1x2x3x4
!
"
#####
$
%
&&&&&
= xLD4 =
vprφ
!
"
#####
$
%
&&&&&
=
Side VelocityBody − Axis Roll RateBody − Axis Yaw Rate
Roll Angle about Body x Axis
!
"
#####
$
%
&&&&&
Eurofighter Typhoon!
Lateral-Directional Equations of Motion Assuming Steady,
Level Longitudinal Flight"Nonlinear dynamic equations, assuming steady, level,
flight (longitudinal variables are constant )!
v =YB /m+ gsinφ cosθN − ruN + pwN
=YB /m+ gsinφ cosαN − ruN + pwN
p = IzzLB + IxzNB( ) Ixx Izz − Ixz2( )
r = IxzLB + IxxNB( ) Ixx Izz − Ixz2( )
φ = p+ r cosφ( ) tanθN = p+ r cosφ( ) tanαN
qN = 0γN = 0θN =αN
Lockheed F-117!
Lateral-Directional Force and Moments"
YB =CYB
12ρNVN
2S; Body− Axis Side Force
LB =ClB
12ρNVN
2Sb; Body− Axis Rolling Moment
NB =CnB
12ρNVN
2Sb; Body− Axis Yawing Moment
Linearized Equations of Motion�
Body-Axis Perturbation Equations of Motion"
Δ v(t)Δp(t)Δr(t)Δ φ(t)
#
$
%%%%%
&
'
(((((
=
∂ f1∂v
∂ f1∂ p
∂ f1∂r
∂ f1∂φ
∂ f2∂v
∂ f2∂ p
∂ f2∂r
∂ f2∂φ
∂ f3∂v
∂ f3∂ p
∂ f3∂r
∂ f3∂φ
∂ f4∂v
∂ f4∂ p
∂ f4∂r
∂ f4∂φ
#
$
%%%%%%%%%%%
&
'
(((((((((((
Δv(t)Δp(t)Δr(t)Δφ(t)
#
$
%%%%%
&
'
(((((
+ Control[ ]+ Disturbance[ ]
Body-Axis Perturbation Variables"
Δu1Δu2
"
#$$
%
&''=
ΔδAΔδR
"
#$
%
&' =
Aileron PerturbationRudder Perturbation
"
#$$
%
&''
Δw1Δw2
"
#$$
%
&''=
ΔδAΔδR
"
#$
%
&' =
SideWind PerturbationVortical Wind Perturbation
"
#$$
%
&''
ΔvΔpΔrΔφ
#
$
%%%%%
&
'
(((((
=
Side Velocity PerturbationBody − Axis Roll Rate PerturbationBody − Axis Yaw Rate Perturbation
Roll Angle about Body x Axis Perturbation
#
$
%%%%%
&
'
(((((
Linearized Lateral-Directional Response to Yaw Rate Initial
Condition"
~Roll-mode response of roll angle!
~Spiral-mode response of crossrange!
~Spiral-mode response of yaw angle!
~Dutch-roll-mode response of side velocity!
~Dutch-roll-mode response of roll and yaw rates!
Dimensional Stability-and-Control Derivatives"
∂ f1 ∂v ∂ f1 ∂ p ∂ f1 ∂r ∂ f1 ∂φ∂ f2 ∂v ∂ f2 ∂ p ∂ f2 ∂r ∂ f2 ∂φ∂ f3 ∂v ∂ f3 ∂ p ∂ f3 ∂r ∂ f3 ∂φ∂ f4 ∂v ∂ f4 ∂ p ∂ f4 ∂r ∂ f4 ∂φ
#
$
%%%%%
&
'
(((((
Stability Matrix!
=
Yv Yp +wN( ) Yr −uN( ) gcosθNLv Lp Lr 0
Nv Np Nr 0
0 1 tanθN 0
#
$
%%%%%%
&
'
((((((
Dimensional Stability-and-Control Derivatives"∂ f1 ∂δA ∂ f1 ∂δR∂ f2 ∂δA ∂ f2 ∂δR∂ f3 ∂δA ∂ f3 ∂δR∂ f4 ∂δA ∂ f4 ∂δR
#
$
%%%%%
&
'
(((((
=
YδA YδRLδA LδRNδA NδR
0 0
#
$
%%%%%
&
'
(((((
∂ f1 ∂vwind ∂ f1 ∂ pwind∂ f2 ∂vwind ∂ f2 ∂ pwind∂ f3 ∂vwind ∂ f3 ∂ pwind∂ f4 ∂vwind ∂ f4 ∂ pwind
"
#
$$$$$
%
&
'''''
=
Yv YpLv Lp
Nv Np
0 0
"
#
$$$$$
%
&
'''''
Control Effect Matrix!
Disturbance Effect Matrix!
Stability Axes�
Stability Axes"• Alternative set of body axes"• Nominal x axis is offset from the body centerline by
the nominal angle of attack, αN "
Transformation from Original Body Axes to Stability Axes"
HBS =
cosαN 0 sinαN
0 1 0−sinαN 0 cosαN
#
$
%%%
&
'
(((
ΔuΔvΔw
"
#
$$$
%
&
'''S
= HBS
ΔuΔvΔw
"
#
$$$
%
&
'''B
ΔpΔqΔr
"
#
$$$
%
&
'''S
= HBS
ΔpΔqΔr
"
#
$$$
%
&
'''B
• Side velocity (Δv) and pitch rate (Δq) are unchanged by the transformation "
Stability-Axis State "• Rotate body axes to stability axes"
Δv(t)Δp(t)Δr(t)Δφ(t)
#
$
%%%%%
&
'
(((((Body−Axis
⇒αN ⇒
Δv(t)Δp(t)Δr(t)Δφ(t)
#
$
%%%%%
&
'
(((((Stability−Axis
Stability-Axis State"
Δv(t)Δp(t)Δr(t)Δφ(t)
#
$
%%%%%
&
'
(((((Stability−Axis
⇒Δβ ≈ΔvVN
⇒
Δβ(t)Δp(t)Δr(t)Δφ(t)
#
$
%%%%%
&
'
(((((Stability−Axis
• Replace side velocity by sideslip angle"
Stability-Axis State"
Δβ(t)Δp(t)Δr(t)Δφ(t)
$
%
&&&&&
'
(
)))))Stability−Axis
⇒
Δr(t)Δβ(t)Δp(t)Δφ(t)
$
%
&&&&&
'
(
)))))Stability−Axis
=
Stability− Axis Yaw Rate PerturbationSideslip Angle Perturbation
Stability− Axis Roll Rate PerturbationStability− Axis Roll Angle Perturbation
$
%
&&&&&
'
(
)))))
• Revise state order"
Stability-Axis Lateral-Directional Equations"
Δr(t)Δ β(t)Δp(t)Δ φ(t)
$
%
&&&&&
'
(
)))))S
=
Nr Nβ Np 0
YrVN
−1+
,-
.
/0
YβVN
YpVN
gcosγNVN
Lr Lβ Lp 0
tanγN 0 1 0
$
%
&&&&&&&
'
(
)))))))S
Δr(t)Δβ(t)Δp(t)Δφ(t)
$
%
&&&&&
'
(
)))))S
+
NδA NδR
YδAVN
YδRVN
LδA LδR0 0
$
%
&&&&&&
'
(
))))))S
ΔδA(t)ΔδR(t)
$
%&&
'
())+
Nβ Np
YβVN
YpVN
Lβ Lp
0 0
$
%
&&&&&&&
'
(
)))))))S
ΔβwindΔpwind
$
%&&
'
())
Why Modify the Equations?"• Dutch-roll mode is primarily described by stability-axis yaw
rate and sideslip angle"
• Roll and spiral mode are primarily described by stability-axis roll rate and roll angle"
• Linearized equations allow the three modes to be studied"
Stable Spiral!
Unstable Spiral!Roll!
Dutch Roll, top! Dutch Roll, front!
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?!
Dassault Rafale!
Stability, Control, and Disturbance Matrices"
FLD =FDR FRS
DR
FDRRS FRS
!
"
##
$
%
&&=
Nr Nβ Np 0
YrVN
−1)
*+
,
-.
YβVN
YpVN
gcosγNVN
Lr Lβ Lp 0
tanγN 0 1 0
!
"
#######
$
%
&&&&&&&
GLD =
NδA NδR
YδAVN
YδRVN
LδA LδR0 0
"
#
$$$$$$
%
&
''''''
LLD =
Nβ Np
YβVN
YpVN
Lβ Lp
0 0
"
#
$$$$$$$
%
&
'''''''
Δx1Δx2Δx3Δx4
"
#
$$$$$
%
&
'''''
=
ΔrΔβ
ΔpΔφ
"
#
$$$$$
%
&
'''''
Δu1Δu2
"
#$$
%
&''=
ΔδAΔδR
"
#$
%
&'
Δw1Δw2
"
#$$
%
&''=
ΔδAΔδR
"
#$
%
&'
Lateral-Directional Stability Derivatives�
2nd-Order Approximate Modes of Lateral-Directional Motion�
2nd-Order Approximations in System Matrices"
FLD =FDR 00 FRS
!
"##
$
%&&=
Nr Nβ 0 0
YrVN
−1)
*+
,
-.
YβVN
0 0
0 0 Lp 0
0 0 1 0
!
"
#######
$
%
&&&&&&&
GLD =
NδA 0YδAVN
0
0 LδR0 0
"
#
$$$$$$
%
&
''''''
LLD =
Nβ 0
YβVN
0
0 Lp
0 0
"
#
$$$$$$$
%
&
'''''''
Second-Order Models of Lateral-Directional Motion"
• Approximate Spiral-Roll Equation"
• Approximate Dutch Roll Equation"
ΔxDR =ΔrΔ β
#
$%%
&
'((≈
Nr Nβ
YrVN
−1+
,-
.
/0
YβVN
#
$
%%%%
&
'
((((
ΔrΔβ
#
$%%
&
'((+
NδR
YδRVN
#
$
%%%
&
'
(((ΔδR+
Nβ
YβVN
#
$
%%%%
&
'
((((
Δβwind
ΔxRS =ΔpΔ φ
#
$%%
&
'((≈
Lp 0
1 0
#
$%%
&
'((
ΔpΔφ
#
$%%
&
'((+
LδA0
#
$%%
&
'((ΔδA +
Lp
0
#
$%%
&
'((Δpwind
Approximate Roll and Spiral Modes"
ΔpΔ φ
#
$%%
&
'((=
Lp 0
1 0
#
$%%
&
'((
ΔpΔφ
#
$%%
&
'((+
LδA0
#
$%%
&
'((ΔδA
ΔRS (s) = s s − Lp( )λS = 0λR = Lp
• Characteristic polynomial has real roots"
• Roll rate is damped by Lp"• Roll angle is a pure integral of roll rate"
Δp t( ) Δφ t( )
• Initial condition response"
Neutral stability!
Generally < 0!
Roll Damping Due to Roll Rate, Lp!Lp ≈ Clp
ρVN2
2Ixx
#
$%
&
'(Sb =Clp̂
b2VN
#
$%
&
'(ρVN
2
2Ixx
#
$%
&
'(Sb
=Clp̂
ρVN4Ixx
#
$%
&
'(Sb2
Clp̂( )Wing
=∂ ΔCl( )Wing
∂ p̂= −
CLα
121+ 3λ1+ λ
&'(
)*+
• Wing with taper"
• Thin triangular wing"Clp̂( )
Wing= −
πAR32
Clp̂≈ Clp̂( )
Vertical Tail+ Clp̂( )
Horizontal Tail+ Clp̂( )
Wing
• Vertical tail, horizontal tail, and wing are principal contributors"
< 0 for stability!
NACA-TR-1098, 1952!NACA-TR-1052, 1951 !
Roll Damping Due to Roll Rate, Lp!
• Tapered vertical tail"
• Tapered horizontal tail"
p̂ = pb2VN
Clp̂( )ht =∂ ΔCl( )ht∂ p̂
= −CLαht
12ShtS
%
&'
(
)*1+ 3λ1+λ
%
&'
(
)*
• pb/2VN describes helix angle for a steady roll"
Clp̂( )vt =∂ ΔCl( )vt∂ p̂
= −CYβvt
12SvtS
%
&'
(
)*1+ 3λ1+λ
%
&'
(
)*
Approximate Dutch Roll Mode"
ΔrΔ β
#
$%%
&
'((=
Nr Nβ
YrVN
−1*
+,
-
./
YβVN
#
$
%%%%
&
'
((((
ΔrΔβ
#
$%%
&
'((+
NδR
YδRVN
#
$
%%%
&
'
(((ΔδR
ΔDR(s)= s2 − Nr +
YβVN
$
%&
'
()s+ Nβ 1−
YrVN( )+Nr
YβVN
*
+,-
./
ωnDR= Nβ 1−
YrVN( )+Nr
YβVN
ζDR = − Nr +YβVN
$
%&
'
() 2 Nβ 1−
YrVN( )+Nr
YβVN
ωnDR= Nβ +Nr
YβVN
ζDR = − Nr +YβVN
%
&'
(
)* 2 Nβ +Nr
YβVN
• With negligible side-force sensitivity to yaw rate, Yr"
• Characteristic polynomial, natural frequency, and damping ratio"
Initial Condition Response of Approximate Dutch Roll Mode"
Δr t( ) Δβ t( )
Side Force due to Sideslip Angle"
Y ≈∂CY
∂βqS•β =CYβ
qS•β
CYβ≈ CYβ( )Fuselage + CYβ( )Vertical Tail + CYβ( )Wing
CYβ( )Vertical Tail ≈∂CY
∂β
$
%&
'
()vt
ηvt
SVerticalTailS
CYβ( )Fuselage ≈ −2SBaseS; SB =
πdBase2
4
CYβ( )Wing ≈ −CDParasite,Wing− kΓ2
ηvt = Vertical tail efficiency
k = πAR1+ 1+ AR2
Γ = Wing dihedral angle, rad
• Fuselage, vertical tail, and wing are main contributors"
Yawing Moment due to Sideslip Angle!
N ≈∂Cn
∂βρV 2
2%
&'
(
)*Sb•β =Cnβ
ρV 2
2%
&'
(
)*Sb•β
! Side force contributions times respective moment arms"– Non-dimensional stability
derivative"
Cnβ≈ Cnβ( )
Vertical Tail+ Cnβ( )
Fuselage+ Cnβ( )
Wing+ Cnβ( )
Propeller
Cnβ( )
Vertical Tail≈ −CYβvt
ηvtSvtlvtSb −CYβvt
ηvtVVT
Vertical tail contribution"
VVT =
SvtlvtSb
=Vertical Tail Volume Ratio
ηvt =ηelas 1+∂σ ∂β( ) Vvt2
VN2
%
&'
(
)*
Yawing Moment due to Sideslip Angle!
lvt Vertical tail length (+)= distance from center of mass to tail center of pressure= xcm − xcpvt [x is positive forward; both are negative numbers]
Cnβ( )Fuselage =−2K VolumeFuselage
Sb
K = 1−dmax Lengthfuselage"
#$
%
&'1.3
Fuselage contribution"
Cnβ( )Wing = 0.75CLNΓ+ fcn Λ,AR,λ( )CLN
2
Wing (differential lift and induced drag) contribution"
Yawing Moment due to Sideslip Angle!
Yaw Damping Due to Yaw Rate, Nr!• Dimensional stability derivative"
Nr ≈ Cnr
ρVN2
2Izz
#
$%
&
'(Sb =Cnr̂
b2VN
#
$%
&
'(ρVN
2
2Izz
#
$%
&
'(Sb
=Cnr̂
ρVN4Izz
#
$%
&
'(Sb2 < 0 for stability!
• High wing-sweep angle can lead to Nr > 0"
Martin Marietta X-24B!
Yaw Damping Due to Yaw Rate, Nr!Cnr̂
≈ Cnr̂( )Vertical Tail
+ Cnr̂( )Wing
Cnr̂( )Wing
= k0CL2 + k1CDParasite,Wing
k0 and k1 are functions of aspect ratio and sweep angle"
• Wing contribution"
• Vertical tail contribution"Δ Cn( )Vertical Tail = − Cnβ( )Vertical Tail
rlvtVN( ) = − Cnβ( )Vertical Tail
lvtb
$
%&
'
()bVN
$
%&
'
()r
r̂ = rb2VN
Cnr̂( )vt =∂Δ Cn( )Vertical Tail∂ rb
2VN( )=∂Δ Cn( )Vertical Tail
∂ r̂= −2 Cnβ( )Vertical Tail
lvtb
%
&'
(
)*
NACA-TR-1098, 1952!NACA-TR-1052, 1951 !
Comparison of Fourth- and Second-Order
Dynamic Models�
• 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!
Comparison of Second- and Fourth-Order Initial-Condition Responses of Business Jet"
Fourth-Order Response! Second-Order Response!
• Speed and damping of responses is adequately portrayed by 2nd-order models"• Roll-spiral modes have little effect on yaw rate and sideslip angle responses"• Dutch roll mode has large effect on roll rate and roll angle responses"
Primary Lateral-Directional Control Derivatives"
LδA =ClδA
ρVN2
2Ixx
#
$%
&
'(Sb
NδR =CnδR
ρVN2
2Izz
#
$%
&
'(Sb
Next Time:�Analysis of Time Response�
�Reading�
Flight Dynamics, 298-314, 338-342 �
Virtual Textbook, Part 13 �
Supplemental Material"
Δ v(t)Δp(t)Δr(t)Δ φ(t)
#
$
%%%%%
&
'
(((((
=
Yv Yp +wN( ) Yr −uN( ) gcosθNLv Lp Lr 0
Nv Np Nr 0
0 1 tanθN 0
#
$
%%%%%%
&
'
((((((
Δv(t)Δp(t)Δr(t)Δφ(t)
#
$
%%%%%
&
'
(((((
+
YδA YδRLδA LδRNδA NδR
0 0
#
$
%%%%%
&
'
(((((
ΔδA(t)ΔδR(t)
#
$%%
&
'((+
Yv YpLv Lp
Nv Np
0 0
#
$
%%%%%
&
'
(((((
ΔvwindΔpwind
#
$%%
&
'((
• Rolling and yawing motions"
Body-Axis Perturbation Equations of Motion"