mae 331 lecture 14
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7/30/2019 Mae 331 Lecture 14
1/8
Linearized Lateral-Directional
Equations of MotionRobert Stengel, Aircraft Flight Dynamics MAE
331, 2008
Spiral, Dutch roll, and roll modes
Stability derivatives
Roll-mode responseof roll angle
Spiral-moderesponse ofcrossrange
Spiral-mode responseof yaw angle
Dutch-roll-mode responseof side velocity
Dutch-roll-moderesponse of rolland yaw rates
Copyright 2008 by Robert Stengel. All rights reserved. For educati onal 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 + gsin"cos#$ ru+ pw
yI = cos#sin%( )u+ cos"cos%+ sin"sin#sin%( )v + $sin"cos%+ cos"sin#sin%( )w
p = IzzL + I
xzN$ I
xzIyy$I
xx$I
zz( )p + Ixz2 + Izz Izz $Iyy( )[ ]r{ }q( ) IxxIzz $Ixz2( )r = I
xzL + I
xxN$ I
xzIyy$I
xx$I
zz( )r + Ixz2 + Ixx Ixx $Iyy( )[ ]p{ }q( ) IxxIzz $Ixz2( )"= p + qsin"+ rcos"( )tan#%= qsin"+ rcos"( )sec#
x1
x2
x3
x4
x5
x6
"
#
$$$$$$$
%
&
'''''''
= xLD6
=
v
y
p
r
(
)
"
#
$$$$$$$
%
&
'''''''
=
SideVelocity
Crossrange
Body * Axis Roll Rate
Body * AxisYaw Rate
Roll Angle about Body x Axis
Yaw Angle about Inertial x Axis
"
#
$$$$$$$
%
&
'''''''
4- Component
Lateral-Directional
Equations of Motion
State Vector,
4 components
Nonlinear Dynamic Equations, neglecting crossrange and yaw angle
v =Y/m + gsin"cos#$ ru+ pw
p = IzzL + I
xzN$ I
xzIyy$I
xx$I
zz( )p+ Ixz2+ I
zzIzz$I
yy( )[ ]r{ }q( ) IxxIzz $Ixz2( )r = I
xzL + I
xxN$ I
xzIyy$I
xx$I
zz( )r + Ixz2+ I
xxIxx$I
yy( )[ ]p{ }q( ) IxxIzz $Ixz2( )"= p+ qsin"+ rcos"( ) tan#
x1
x2
x3
x4
"
#
$$$$
%
&
''''
= xLD4
=
v
p
r
(
"
#
$$$$
%
&
''''
=
Side Velocity
Body ) Axis Roll Rate
Body ) AxisYaw Rate
Roll Angle about Body x Axis
"
#
$$$$
%
&
''''
4- Component
Lateral-Directional
Equations of Motion
State Vector, 4components
Nonlinear Dynamic Equations, assuming steady, level, longitudinal flight
(constant longitudinal variables)
v =YB /m + gsin"cos#N $ ruN+ pwN
=YB /m + gsin"cos%N $ ruN + pwN
p =IzzLB + IxzNB( )IxxIzz $Ixz
2( )
r =IxzLB + IxxNB( )IxxIzz $Ixz
2( )"= p+ rcos"( )tan#N = p+ rcos"( )tan%N
x1
x2
x3x
4
"
#
$$
$
$
%
&
''
'
'
= xLD4=
v
p
r(
"
#
$$
$
$
%
&
''
'
'
=
SideVelocity
Body ) Axis Roll Rate
Body ) AxisYaw RateRoll Angle about Body x Axis
"
#
$$
$
$
%
&
''
'
'
qN = 0
"N = 0
#N =
$N
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7/30/2019 Mae 331 Lecture 14
2/8
Lateral-Directional
Force and Moments
YB = CYB1
2"NVN
2S; Body # Axis Side Force
LB =
ClB
1
2"NVN
2Sb; Body # Axis Rolling Moment
NB =
CnB
1
2"NVN
2Sb; Body # AxisYawing Moment
Body-Axis Perturbation
Equations of Motion
"xLD
(t) = FLD"x
LD(t) +G
LD"u
LD(t) + L
LD"w
LD(t)
dim("x) = 4 #1
dim("u) = 2#1
dim("w) = 2#1
"v(t)
"p(t)
"r(t)
"#(t)
$
%
&&&&
'
(
))))
=
*f1
*v
*f1
*p
*f1
*r
*f1
*#*f2*v
*f2*p
*f2*r
*f2*#
*f3*v
*f3*p
*f3*r
*f3*#
*f4*v
*f4*p
*f4*r
*f4*#
$
%
&&&&&&&&
'
(
))))))))
"v(t)
"p(t)
"r(t)
"#(t)
$
%
&&&&
'
(
))))
+
*f1
*+A
*f1
*+R*f2*+A
*f2*+R
*f3*+A
*f3*+R
*f4
*+A
*f4
*+R
$
%
&&&&&&&&
'
(
))))))))
"+A(t)
"+R(t)
$
%&
'
()+
*f1*vwind
*f1*pwind
*f2*vwind
*f2*pwind
*f3*vwind
*f3*pwind
*f4*vwind
*f4*pwind
$
%
&&&&&&&&&
'
(
)))))))))
"vwind"pwind
$
%&
'
()
"u1
"u2
#
$%
&
'(=
")A
")R
#
$%
&
'(=
Aileron Perturbation
Rudder Perturbation
#
$%
&
'(
"w1
"w2
#
$%
&
'(=
")A
")R
#
$%
&
'(=
SideWind Perturbation
VorticalWind Perturbation
#
$%
&
'(
Perturbations from wings-level flight
vN = 0; pN = 0; rN = 0; "N = 0
Typical
Lateral-Directional Initial-Condition
Linearized Response
Roll-mode responseof roll angle
Spiral-moderesponse ofcrossrange
Spiral-moderesponse of yawangle
Dutch-roll-mode responseof side velocity
Dutch-roll-moderesponse of rolland yaw rates
Dimensional Stability-and-
Control Derivatives
"f1"v
"f1"p
"f1"r
"f1"#
"f2
"v
"f2
"p
"f2
"r
"f2
"#
"f3
"v
"f3
"p
"f3
"r
"f3
"#
"f4
"v
"f4
"p
"f4
"r
"f4
"#
$
%
&&&&&&&&
'
(
))))))))
=
Yv Yp + wN( ) Yr * uN( ) gcos+NLv Lp Lr 0
Nv Np Nr 0
0 1 tan+N 0
$
%
&&&&
'
(
))))
"f1
"#A
"f1
"#R"f
2
"#A
"f2
"#R"f
3
"#A
"f3
"#R"f
4
"#A
"f4
"#R
$
%
&&&&&&&
&
'
(
)))))))
)
=
Y#A
Y#R
L#A L#R
N#A N#R
0 0
$
%
&&&&
'
(
))))
"f1
"vwind
"f1
"pwind"f
2
"vwind
"f2
"pwind"f
3
"vwind
"f3
"pwind"f
4
"vwind
"f4
"pwind
#
$
%%%%%%%%
%
&
'
((((((((
(
=
Yv Yp
Lv Lp
Nv Np
0 0
#
$
%%%%
&
'
((((
StabilityMatrix
ControlEffect Matrix
DisturbanceEffect Matrix
v
p
r
"
#
$
%%%%
&
'
((((
=
SideVelocity
Body ) Axis Roll Rate
Body ) AxisYaw Rate
Roll Angle about Body x Axis
#
$
%%%%
&
'
((((
-
7/30/2019 Mae 331 Lecture 14
3/8
"v(t)
"p(t)
"r(t)
"#(t)
$
%
&&&&
'
(
))))
=
Yv Yp + wN( ) Yr * uN( ) gcos+NLv Lp Lr 0
Nv Np Nr 0
0 1 tan+N 0
$
%
&&&&
'
(
))))
"v(t)
"p(t)
"r(t)
"#(t)
$
%
&&&&
'
(
))))
+
Y,A Y,R
L,A L,R
N,A N,R
0 0
$
%
&&&&
'
(
))))
",A(t)
",R(t)
$
%&
'
()+
Yv Yp
Lv Lp
Nv Np
0 0
$
%
&&&&
'
(
))))
"vwind"pwind
$
%&
'
()
"#A
"#R
$
%&
'
()=
Aileron Perturbation
Rudder Perturbation
$
%&
'
()
"#A
"#R
$
%&
'
()=
Side Wind Perturbation
VorticalWind Perturbation
$
%&
'
()
Rolling and yawing motions
"v
"p
"r
"#
$
%
&&&&
'
(
))))
=
SideVelocity Perturbation
Body * Axis Roll Rate Perturbation
Body * AxisYaw Rate Perturbation
Roll Angle about Body x Axis Perturbation
$
%
&&&&
'
(
))))
Body-Axis Perturbation
Equations of MotionStability Axes
Alternative set of body axes
Nominal xaxis is offset from the body centerline bythe nominal angle of attack, N
Transformation from
Original Body Axes to
Stability Axes
HB
S=
cos"N
0 sin"N
0 1 0
#sin"N
cos"N
$
%
&&&
'
(
)))
"u
"v
"w
#
$
%%%
&
'
(((S
=HB
S
"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
Stability-Axis Lateral-
Directional Equations
Rotate body axes to stability axes
Replace side velocity by sideslip angle Revise state order
"#$"v
VN
"v(t)
"p(t)
"r(t)
"#(t)
$
%
&&&&
'
(
))))Body*Axis
+
"v(t)
"p(t)
"r(t)
"#(t)
$
%
&&&&
'
(
))))Stability*Axis
+
",(t)
"p(t)
"r(t)
"#(t)
$
%
&&&&
'
(
))))Stability*Axis
+
"r(t)
",(t)
"p(t)
"#(t)
$
%
&&&&
'
(
))))Stability*Axis
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7/30/2019 Mae 331 Lecture 14
4/8
Stability-Axis Lateral-
Directional Equations
"r(t)
"#(t)"p(t)
"$(t)
%
&
''''
(
)
****S
=
Nr N# Np 0
Yr
VN+1
,
-.
/
01
Y#
VN
Yp
VN
gcos2NVN
Lr L# Lp 0
tan2N 0 1 0
%
&
'''''
(
)
*****S
"r(t)
"#(t)
"p(t)
"$(t)
%
&
''''
(
)
****S
+
N3A N3RY3A
VN
Y3R
VNL3A L3R
0 0
%
&
'''''
(
)
*****S
"3A(t)
"3R(t)
%
&'
(
)*+
N# NpY#
VN
Yp
VNL# Lp
0 0
%
&
'''''
(
)
*****S
"#wind"pwind
%
&'
(
)*
"x1
"x2
"x3
"x4
#
$
%%%%
&
'
((((
=
"r
")
"p
"*
#
$
%%%%
&
'
((((
=
Stability+ AxisYaw Rate Perturbation
Sideslip Angle Perturbation
Stability+ Axis Roll Rate Perturbation
Stability+ Axis Roll Angle Perturbation
#
$
%%%%
&
'
((((
Why Modify the Equations?
Dutch-roll mode is primarily described by stability-axis yawrate and sideslip angle
Roll and spiral mode are primarily described by stability-axisroll 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
FDR
RSFRS
"
#$
%
&'=
FDR
small
small FRS
"
#$
%
&'(
FDR
~ 0
~ 0 FRS
"
#$
%
&'
Effects of Dutch roll perturbations
on Dutch roll motion
Effects of Dutch roll perturbations
on roll-spiral motion
Effects of roll-spiral perturbationson Dutch roll motion
Effects of roll-spiral perturbations on
roll-spiral motion
... but are the off-diagonal blocks really small?
Stability, Control, and
Disturbance Matrices
FLD =FDR
FRS
DR
FDR
RSFRS
"
#$
%
&'=
Nr
N( Np 0
Yr
VN
)1*
+, -
./ Y
(
VN
Yp
VN
gcos0NVN
Lr
L( Lp 0
tan0N 0 1 0
"
#
$$$$$
%
&
'''''
GLD
=
N"A
N"R
Y"A
VN
Y"R
VN
L"A
L"R
0 0
#
$
%%%%%
&
'
(((((
LLD
=
N" NpY"
VN
Yp
VN
L" Lp
0 0
#
$
%%%%%
&
'
(((((
"x1"x
2
"x3
"x4
#
$
%%%%
&
'
((((
=
"r")
"p
"*
#
$
%%%%
&
'
((((
"u1
"u2
#
$%
&
'(=
")A
")R
#
$%
&
'(
"w1
"w2
#
$%
&
'(=
")A
")R
#
$%
&
'(
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7/30/2019 Mae 331 Lecture 14
5/8
Second-Order Models of
Lateral-Directional Motion
Approximate LTI Spiral-Roll Equation
Approximate LTI Dutch Roll Equation
"xDR
=
"r
"#$
%&
'
()*
Nr
N#
Yr
VN
+1,
-.
/
01
Y#
VN
$
%
&&&
'
(
)))
"r
"#
$
%&
'
()+
N2R
Y2R
VN
$
%
&&
'
(
))"2R +
N#
Y#
VN
$
%
&&
'
(
))"#
wind
"xRS ="p
"#$
%&
'
()*
Lp 0
1 0
$
%&
'
()"p
"#
$
%&
'
()+
L+A
0
$
%&
'
()"+A +
Lp
0
$
%&
'
()"pwind
2nd-order-model eigenvalues are close to those of the 4th-order model
Eigenvalue magnitudes of Dutch roll and roll roots are similar
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 -1.2
0.2501 -2.408 -1.1616 0 2.3106 0 -1.16e-01 + 1.39e+00j 8.32E-02 1.39E+000 0 1 0 0 0 -1.16e-01 - 1.39e+00 8.32 -02 1.39 +00
Dutch Roll ApproximationF = G = Eigenvalue Damping Freq. (rad/s)
-0.1079 1.9011 -1.1196 -1.32e-01 + 1.38e+00 9.55 -02 1.38 +00-1 -0.1567 0 -1.32e-01 - 1.38e+00 9.55 -02 1.38 +00
Roll-Spiral Approximation= = genva ue amp ng req. rad s
-1.1616 0 2.3106 01 0 0 -1.16
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
Approximate
Dutch Roll Mode
"r
"#$
%&
'
()=
Nr
N#
Yr
VN
*1+
,-
.
/0
Y#
VN
$
%
&&&
'
(
)))
"r
"#
$
%&
'
()+
N1R
Y1R
VN
$
%
&&
'
(
))"1R
"DR
(s) = s2 # N
r+
Y$VN
%
&'
(
)*s+ N$ 1#
Yr
VN
%&'
()*+ N
r
Y$VN
+
,-.
/0
1nDR
= N$ 1#Yr
VN
%&'
()*+ N
r
Y$
VN
2DR
= # Nr+
Y$VN
%
&'
(
)* 2 N$ 1#
Yr
VN
%&'
()*+ N
r
Y$VN
"nDR
= N# + Nr
Y#
VN
$DR
= % Nr+
Y#
VN
&
'(
)
*+ 2 N# + Nr
Y#
VN
With negligible Yr
Characteristicpolynomial,natural frequency,and damping ratio
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7/30/2019 Mae 331 Lecture 14
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Weathercock Stability: Yaw Acceleration
Sensitivity to Sideslip Angle,N
N" #$C
n
$"
%V2
2
&
'(
)
*+Sb
Izz
=Cn"
%V2
2
&
'(
)
*+Sb
Izz
Vertical tail, fuselage, and wing are principalcontributors to directional stability Side force contributions times respective moment arms
Non-dimensional stability derivative
> 0 for stability
Dimensional stability derivative
Cn" # Cn"( )Vertical Tail + Cn"( )Fuselage + Cn"( )Wing + Cn"( )Propeller
Contributions to
Directional Stability,N
Cn"( )Fuselage =
#2K VolumeFuselage
Sb
K= 1"dmax
Lengthfuselage
#
$%
&
'(
1.3
Cn"( )Vertical Tail # aFin$vt
SFin
lvt
Sb= a
Fin$
vtV
vt
Vertical tail
Vvt=
SFin
lvt
Sb=Vertical TailVolume Ratio
Fuselage
"vt="
elas1+ #$
#%&'( )
*+V
Tail
2
VN
2
&
'(
)
*+
aFin= () Lift slope of isolated finreferenced to fin area, SFin
VTail= Airspeed at vertical tail;scrubbing lowers VTail,propeller slipstreamincreases VTail
, Sidewash: Deflection of flow
from freestream direction dueto wing-fuselage interference
Contributions to
Directional Stability,N
Cn"( )Wing = 0.75CLN#+ fcn $,AR,%( )CLN2
Cn"( )Propeller Propeller fin effect Function of blade geometry,
number of blades, thrustcoefficient, and advance ratio
J= V/nd
n= rpm
d= propeller diameter
" = Sweep angleAR= Aspect ratio$=Taper ratio%= Dihedral angle
Wing Differential drag
Interference effects
see NACA-TR-819
Side Acceleration Sensitivity
to Sideslip Angle, Y
Side-force sensitivity tosideslip angle, Y
Y" # CY"$V2
2m
%&'
()*S
CY" # CY"( )Fuselage + CY"( )Vertical Tail + CY"( )Wing
CY"( )Vertical Tail # aFin$vtS
Fin
S
CY"( )Fuselage # %2
SBaseS; SB =
&dBase2
4
CY"( )Wing =%CDParasite, Wing % k'2
k="AR
1+ 1+ AR2
Fuselage, vertical tail, and wing are principal contributors
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7/30/2019 Mae 331 Lecture 14
7/8
Yaw Damping Due to
Yaw Rate, Nr
Cn r " Cn r( )Vertical Tail + Cn r( )Wing
Cn r
( )Wing
= k0C
L
2
+ k1CDParasite,Wing
k0and k1 are functions of aspectratio and sweep angle
Wing contribution
Vertical tail contribution
" Cn( )Vertical Tail =# Cn$( )Vertical Tail
rlvt
V
%&'
()*=# C
n$( )Vertical Taillvt
b
%
&'
(
)*
b
V
%
&'
(
)*r
Cn
r( )
vt=
+" Cn( )Vertical Tail
+ rb2V( )
=
+" Cn( )Vertical Tail+r
=#2 Cn$( )
Vertical Tail
lvt
b
%
&'
(
)*
r =rb
2VN
Yaw Damping Due to Yaw Rate, Nr
Dimensional stability derivative
Nr "Cnr#V
N
2
2Izz
$
%&
'
()Sb
=
Cn r
b
2VN
$
%&
'
()
#VN
2
2Izz
$
%&
'
()Sb
=Cnr
#VN
4Izz
$
%&
'
()Sb
2 < 0 for stability
High wing-sweep anglecan lead to Nr> 0
Approximate Roll
and Spiral Modes
"p
"#$
%&
'
()=
Lp 0
1 0
$
%&
'
()"p
"#
$
%&
'
()+
L*A
0
$
%&
'
()"*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
Roll Damping Due to Roll Rate,Lp
Lp " Clp#VN
2
2Ixx
$
%&
'
()Sb = Cl
p
b
2VN
$
%&
'
()
#VN2
2Ixx
$
%&
'
()Sb = Cl
p
#VN
4Ixx
$
%&
'
()Sb
2
Clp
( )Wing
=
" #Cl( )Wing"p
= $CL%
12
1+ 3&
1+ &
'
()
*
+,
Wing with taper
Thin triangular wing
Clp
( )Wing
= "#AR
32
Clp" Cl
p( )
VerticalTail+ Cl
p( )
HorizontalTail+ Cl
p( )
Wing
Vertical tail, horizontal tail, and wing areprincipal contributors
< 0 for stability
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7/30/2019 Mae 331 Lecture 14
8/8
Roll Damping Due to
Roll Rate,Lp
Vertical tail
Tapered horizontal tail
p =pb
2VN
Clp
( )ht=
" #Cl( )ht"p
= $CL%ht
12
Sht
S
&
'(
)
*+1+ 3,
1+ ,
&
'(
)
*+
pb/2VNdescribes helix anglefor a steady roll
Cl p( )vt=
"#Cl( )vt"p
=
$
CY%vt
12
Svt
S
&
'(
)
*+
1+ 3,
1+ ,
&
'(
)
*+
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