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Root Locus Analysis of Parameter
Variations and Feedback Control!
Robert Stengel, Aircraft Flight Dynamics!
MAE 331, 2014"
•! Effects of system parameter variations on
modes of motion"
•! Root locus analysis"
–! Evanss rules for construction"
–! Application to longitudinal dynamic models"
Reading:!
Flight Dynamics!
357-361, 465-467, 488-490, 509-514!
Copyright 2014 by Robert Stengel. All rights reserved. For educational use only.!
http://www.princeton.edu/~stengel/MAE331.html!
http://www.princeton.edu/~stengel/FlightDynamics.html!
1!
2!
Characteristic Equation: A Critical
Component of the Response’s
Laplace Transform"
!x(s) = [ sI " F ]
"1
[ sI ! F]
!1
[ !x(0) + G !u(s) + L!w(s)]
Adj ( sI ! F ) CT ( s )
=
=
sI ! F
sI ! F
(n " n)
(1"1)
•! Characteristic equation defines the modes of motion!
sI ! F = "(s) = s n + an !1s n !1 + ... + a1s + a0
= ( s ! #1 ) ( s ! #2 ) (...) ( s ! #n ) = 0
•! Recall: s is a complex variable!
s = ! + j"
3!
Real Roots of the Dynamic System"
•! Roots are solutions of the
characteristic equation"
!(s) = ( s " #1 ) ( s " #2 ) (...) ( s " #n ) = 0
•! Real roots"
–! are confined to the real axis"
–! represent convergent or divergent
time response"
–! time constant, ! = –1/" = –1/µ, sec#
s Plane = (! + j" ) Plane
!i = µi (Real number)
x ( t ) = x ( 0 ) eµ t
4!
Complex Roots of the Dynamic System"
•! Complex roots"
–! occur only in complex-conjugate pairs"
–! represent oscillatory modes"
–! natural frequency, $n, and damping ratio,
%, as shown"
!1 = µ1 + j"1 = #$% n + j% n 1# $ 2
" = cos#1 $
!2 = µ 2 + j" 2 = µ1 # j"1 = !
*
1
= #$% n # j% n 1# $ 2
Stable"
Unstable"
– time constant = –1/! = 1/#$n"
– decay of exponential time-"
response envelope"
s Plane = (! + j" ) Plane
5!
Complex Roots, Damping Ratio,
and Damped Natural Frequency"
( s ! "1 ) ( s ! "1* ) = $%s ! (µ1 + j#1 )&'$%s ! (µ1 ! j#1 )&'
= s 2 ! $%(µ1 ! j#1 ) + (µ1 + j#1 )&' s + (µ1 ! j#1 ) (µ1 + j#1 )
= s 2 ! 2µ1s + (µ12 + #12 ) ! s 2 + 2() n s + ) n2
µ1 = !"# n = !1 Time constant
$1 = # n 1! " 2 ! # ndamped = Damped natural frequency
6!
Corresponding 2nd-Order
Initial Condition Response"
Stable, Lightly Damped"
2nd-order system"
Identical exponentially
decaying envelopes for
both displacement and
rate"
General form of response"
x1 (t ) = Ae!"#nt sin %&# n 1! " 2 t + $ '(
x2 (t ) = Ae!"#nt %&# n 1! " 2 '( cos %&# n 1! " 2 t + $ '(
7!
Multi-Modal LTI Responses Superpose
Individual Modal Responses"
•! With distinct roots,
(n = 4) for example,
partial fraction
expansion for each
state element is"
!xi ( s ) =
d1i
+
d2i
+
d2i
+
d2i
( s " #1 ) ( s " #2 ) ( s " #3 ) ( s " #4 )
Corresponding 4th-order time response is"
!xi (t ) = d1i e"1t + d2i e"2t + d3i e"3t + d4i e"4t
8!
Evanss Rules for
Root Locus Analysis!
9!
Root Locus Example:"
4th-Order Longitudinal
Characteristic Equation"
! Lon (s) = s 4 + a3 s 3 + a2 s 2 + a1s + a0
(
= s + DV +
4
L"
)
VN # M q s
(
with Lq = Dq = 0
3
)
L
L
L
$
'
+ &( g # D" ) V V + DV " V # M q # M q " V # M " ) s 2
N
N
N
%
(
L
L
+ M q $( D" # g ) V V # DV " V ' + D" M V # DV M " s
)
N
N(
%&
{
(
+ g MV
L"
VN # M "
LV
)
}
VN = 0
Typically factors into oscillatory phugoid and short-period modes"
! Lon (s) = s 2 + 2"# n s + # n2
(
) (s
Ph
2
+ 2"# n s + # n2
)
SP
10!
Root Locus Analysis of Parametric
Effects on Aircraft Dynamics"
•! Parametric variations alter
eigenvalues of F"
•! Graphical technique for
finding the roots with
varying parameter values"
Short Period"
Root"
Phugoid "
Roots"
Short Period"
Root"
Locus: the set of all points whose
location is determined by stated
conditions
"
s Plane!
11!
Example: How do the roots vary when
we change pitch-rate damping, Mq?"
(
! Lon (s) = s 4 + DV +
L"
)
3
VN # M q s
(
)
L
L
L
$
'
+ &( g # D" ) V V + DV " V # M q # M q " V # M " ) s 2
N
N
N
%
(
L
L
+ M q $( D" # g ) V V # DV " V ' + D" M V # DV M " s
&%
N
N)
(
L
L
+ g MV " V # M" V V = 0
N
N
{
(
)
}
•! Mq could be changed by"
–! Variation in aircraft aerodynamic configuration"
–! Effect of feedback control, i.e., control of
pitching moment (via elevator) that is
proportional to pitch rate"
12!
Effect of Parameter
Variations on Root
Location "
Walter R. Evans"
(1920-1999)"
! Lon (s) = s 4 + a3s 3 + a2 s 2 + a1s + a0
= ( s " #1 ) ( s " #2 ) ( s " # 3 ) ( s " # 4 )
(
= ( s " #1 ) ( s " #1* ) ( s " # 3 ) ( s " # 3* )
)(
)
= s 2 + 2$ P% nP s + % n2P s 2 + 2$ SP% nSP s + % n2SP = 0
•! Let root locus gain
= k = ai (just a notation change)"
–! Option 1: Vary k and calculate roots for each new value"
–! Option 2: Apply Evanss Rules of Root Locus Construction"
13!
Effect of a0 Variation on
Longitudinal Root Location"
•! Example: Root locus gain, k = a0!
! Lon (s) = "#s 4 + a3s 3 + a2 s 2 + a1s$% + [ k ] & d(s)+ kn(s)
= ( s ' (1 ) ( s ' (2 ) ( s ' (3 ) ( s ' (4 ) = 0
d ( s ) : Polynomial in s, degree = n
n ( s ) : Polynomial in s, degree = q
where
d(s) = s 4 + a3 s 3 + a2 s 2 + a1s
= ( s ! " '1 ) ( s ! " '2 ) ( s ! " '3 ) ( s ! " '4 )
n(s) = 1
Degree?"
14!
Effect of a1 Variation on
Longitudinal Root Location"
•! Example: Root locus gain, k = a1!
! Lon (s) = s 4 + a3s 3 + a2 s 2 + ks + a0 " d(s)+ kn(s)
= ( s # $1 ) ( s # $2 ) ( s # $3 ) ( s # $4 ) = 0
where
d(s) = s 4 + a3 s 3 + a2 s 2 + a0
= ( s ! " '1 ) ( s ! " '2 ) ( s ! " '3 ) ( s ! " '4 )
Degree?"
n(s) = s
15!
Three Equivalent Equations
for Defining Roots"
d(s) + k n(s) = 0
1+ k
k
n(s)
=0
d(s)
n(s)
= !1 = (1)e! j" (rad ) = (1)e! j180(deg)
d(s)
16!
Longitudinal Equation Example"
Original 4th-order polynomial!
! Lon (s) = s 4 + a3s 3 + a2 s 2 + a1s + a0
(
= ( s " #1 ) ( s " #1* ) ( s " # 3 ) ( s " # 3* )
)(
)
= s 2 + 2$ P% nP s + % n2P s 2 + 2$ SP% nSP s + % n2SP = 0
Typical flight condition!
! Lon (s) = s 4 + 2.57s 3 + 9.68s 2 + 0.202s + 0.145
2
2
= "# s 2 + 2 ( 0.0678 ) 0.124s + ( 0.124 ) $% "# s 2 + 2 ( 0.411) 3.1s + ( 3.1) $% = 0
Phugoid"
Short Period"
17!
Example: Effect of a0 Variation"
Original 4th-order polynomial!
! Lon (s) = s 4 + 2.57s 3 + 9.68s 2 + 0.202s + 0.145 = 0
Example: k = a0!
!(s) = s 4 + a3s 3 + a2 s 2 + a1s + a0
= ( s 4 + a3s 3 + a2 s 2 + a1s ) + k
= s ( s 3 + a3s 2 + a2 s + a1 ) + k
= s ( s + 0.21) "#s 2 + 2.55s + 9.62$% + k
Rearrange:!
k
= %1
2
s ( s + 0.21) !" s + 2.55s + 9.62 #$
18!
Example: Effect of a1 Variation"
Example: k = a1!
!(s) = s 4 + a3s 3 + a2 s 2 + a1s + a0
= s 4 + a3s 3 + a2 s 2 + ks + a0
= ( s 4 + a3s 3 + a2 s 2 + a0 ) + ks
= #$s 2 " 0.00041s + 0.015%&#$s 2 + 2.57s + 9.67%& + ks
Rearrange:!
ks
= !1
"# s 2 ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
19!
Locations of Roots for Large and Small k"
Origins of the roots are the n poles of d(s)"
!(s) = d(s) + kn(s) #k"0
##
" d(s)
Two destinations for the roots as k becomes large"
1) q roots go to the zeros of n(s)"
d(s) + kn(s) d(s)
=
+ n(s) #k!±"
##
! n(s) = ( s $ z1 ) ( s $ z2 )!
k
k
2) (n – q) roots go to infinity"
! sn
$
! d(s) + kn(s) $ ! d(s)
$
=#
+ k & )k'±
)))
' # q + k & ' s ( n*q ) ± R ' ±(
R'±(
#
&
n(s)
"
% " n(s)
%
"s
%
20!
Origins of the roots are the n poles of d(s)"
k
= %1
s ( s + 0.21) !" s + 2.55s + 9.62 #$
2
21!
Origins of the roots are the n poles of d(s)"
ks
= !1
"# s 2 ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
22!
Destination infinity for the roots as k
becomes large"
k
= %1
s ( s + 0.21) !" s + 2.55s + 9.62 #$
2
No zeros when k = a0"
R!
23!
Destinations for the roots as k becomes large"
ks
= !1
"# s 2 ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
One zero at origin when k = a1"
R!
24!
Asymptotes of the root loci are described by"
R(+)"
s ( n!q ) = R e! j180° " # or
R(–)"
R e! j 360° " !#
s = R1 ( n!q ) e! j180° ( n!q ) " #
or
1 ( n!q )
R
e
! j 360° ( n!q )
" !#
Magnitudes of roots are the same for given k"
Angles from the origin are different"
http://www.wolframalpha.com"
25!
Asymptotes of Roots (for k -> ±%)"
4 roots to infinite radius"
Asymptotes = ±45°, ±135°"
3 roots to infinite radius"
Asymptotes = ±60°, –180°"
26!
(n – q) Roots Approach Asymptotes
as k –> ±%"
Asymptote angles for positive k"
! (rad) =
" + 2m"
, m = 0,1,...,(n # q) # 1
n#q
Asymptote angles for negative k"
! (rad) =
2m"
, m = 0,1,...,(n # q) # 1
n#q
27!
Origin of Asymptotes =
Center of Gravity"
q
n
"c.g." =
$!
i =1
"i
# $! zj
j =1
n#q
28!
Root Locus on Real Axis"
•! Locus on real axis"
–! k > 0: Any segment with odd number of
poles and zeros to the right on the axis"
–! k < 0: Any segment with even number of
poles and zeros to the right on the axis"
29!
First Example: Positive and
Negative Variations of k = a0"
k
= %1
s ( s + 0.21) !" s 2 + 2.55s + 9.62 #$
30!
Second Example: Positive and
Negative Variations of k = a1"
ks
= !1
"# s ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
2
31!
Summary of Root Locus Concepts"
Destinations "
of Roots"
Origins "
of Roots"
Center "
of Gravity"
Locus on "
Real Axis"
32!
Wind Shear Encounter"
•! Inertial Frames"
–! Earth-Relative"
–! Wind-Relative (Constant Wind)"
Pitch Angle, #
Angle of Attack, !
•! Non-Inertial Frames"
Flight Path Angle, "
–! Body-Relative"
–! Wind-Relative (Varying Wind)"
Earth-Relative Velocity
Wind Velocity
Air-Relative Velocity
33!
Pitch Angle and Normal Velocity
Frequency Response to Axial Wind"
•! Pitch angle resonance at phugoid natural frequency"
•! Normal velocity (~ angle of attack) resonance at phugoid
and short period natural frequencies"
!!
"# ( j$ )
"Vwind ( j$ )
! VN
MacRuer, Ashkenas, and Graham, 1973"
!" ( j# )
!Vwind ( j# )
34!
Pitch Angle and Normal Velocity
Frequency Response to Vertical Wind"
•! Pitch angle resonance at phugoid and short
period natural frequencies"
•! Normal velocity (~ angle of attack) resonance
at short period natural frequency"
!
!" ( j# )
VN !$ wind ( j# )
=
!" ( j# )
!" wind ( j# )
MacRuer, Ashkenas, and Graham, 1973"
35!
Microbursts"
1/2-3-km-wide
Jetimpinges on
surface"
Ring vortex
forms in
outlow"
High-speed outflow
from jet core"
Outflow strong enough
to knock down trees"
36!
http://en.wikipedia.org/wiki/Microburst!
The Insidious Nature of
Microburst Encounter"
The wavelength of the phugoid mode and
the disturbance input are comparable"
DELTA 191 (Lockheed L-1011)!
http://www.youtube.com/watch?v=BxxxevZ0IbQ&NR=1!
Headwind!
Downdraft!
Tailwind!
Landing Approach!
http://en.wikipedia.org/wiki/Delta_Air_Lines_Flight_191!
37!
Importance of Proper Response
to Microburst Encounter"
!! Stormy evening July 2, 1994"
!! USAir Flight 1016, Douglas DC-9, Charlotte"
!! Windshear alert issued as 1016 began descent along glideslope"
!! DC-9 encountered 61-kt windshear, executed missed approach"
!! Plane continued to descend, striking trees and telephone poles
before impact"
!! Go-around procedure begun correctly -- aircraft's nose rotated up -but power was not advanced"
!! Together with increasing tailwind aircraft stalled "
!! Crew lowered nose to eliminate stall, but descent rate increased,
causing ground impact"
http://en.wikipedia.org/wiki/US_Airways_Flight_1016!
38!
Optimal Flight Path !
Through Worst JAWS Profile"
•!
•!
•!
•!
Graduate research of Mark Psiaki"
Joint Aviation Weather Study (JAWS)
measurements of microbursts (Colorado
High Plains, 1983)"
Negligible deviation from intended path
using available controllability"
Aircraft has sufficient performance
margins to stay on the flight path"
Downdraft"
Headwind"
Airspeed"
Angle of Attack"
Pitch Angle"
Throttle Setting"
39!
Optimal and 15° Pitch
Angle Recovery during!
Microburst Encounter"
Graduate Research of Sandeep Mulgund"
Altitude vs. Time"
Airspeed vs. Time"
Encountering
outflow"
Angle of Attack vs. Time"
Rapid arrest of
descent"
FAA Windshear Training Aid, 1987, addresses proper
operating procedures for suspected windshear"
40!
Tactical Airplane Maneuverability!
Chapter 10, Airplane Stability and Control,
Abzug and Larrabee!
•! What are the principal subject and scope of the
chapter?"
•! What technical ideas are needed to understand the
chapter?"
•! During what time period did the events covered in
the chapter take place?"
•! What are the three main "takeaway" points or
conclusions from the reading?"
•! What are the three most surprising or remarkable
facts that you found in the reading?"
41!
Root Locus Analysis of
Simplified Longitudinal Modes!
42!
Approximate Phugoid Model"
2nd-order equation"
!!x Ph
# T+ T
*g & #
( !V & %
(+ L
%
0 ( %$ !" (' % + T V
%$
('
N
#
# !V! & % *DV
=%
( ) % LV
%$ !"! (' %
VN
$
&
(
( !+ T
('
Characteristic polynomial"
sI ! FPh = det ( sI ! FPh ) "
#(s) = s 2 + DV s + gLV / VN
= s 2 + 2$% n s + % n 2
Parameters"
gLV / VN , DV
43!
Approximate Phugoid Roots"
Approximate Phugoid Equation (&N = 0)"
!!x Ph
#
# !V! & % *DV
=%
( ) % LV
%$ !"! '( %
VN
$
# T+ T
*g & #
( !V & %
(+ L
%
0 ( %$ !" (' % + T V
%$
('
N
&
(
( !+ T
('
Characteristic polynomial"
sI ! FPh = det ( sI ! FPh ) " #(s) = s 2 + DV s + gLV / VN
= s 2 + 2$% n s + % n 2
Natural frequency and damping ratio"
! n = gLV / VN
"=
DV
2 gLV / VN
&
LV
" NVN2
1 #
!
C
S + C LN " NVN S (
LV
%
VN mVN $
2
'
&
" NVN2
1#
DV ! %C DV
S + C DN " NVN S (
m$
2
'
44!
Effect of Airspeed on Approximate Phugoid
Natural Frequency and Period"
Neglecting compressibility effects"
g
2g
=
mVN2
LV g
! #$C LN " N S %&
VN m
1
2g
2g 2
#
2 %
'$C LN 2 " NVN S (& = mV 2 [ mg ] = V 2
N
N
13.87
!n " 2 gV "
(m / s)
N
VN
Period, T = 2! / " n
# 0.45VN sec
45!
Effect of L/D on Approximate
Phugoid Damping Ratio"
Neglecting compressibility effects"
DV !
!=
DV
2 gLV / VN
!
C DN " NVN S m
2 2 g VN
Natural
Velocity Frequency
m/s
rad/s
50
0.28
100
0.14
200
0.07
400
0.035
1
#C D " NVN S %&
m$ N
=
C DN " NVN2 S 2
2mg
Period
sec
23
45
90
180
=
1 # C DN &
2 %$ C LN ('
!"
1
2 ( L / D )N
Damping
L/D Ratio
5
10
20
40
0.14
0.07
0.035
0.018
46!
Effect of LV/VN Variation on
Approximate Phugoid Roots"
k = gLV/VN"
!(s) = ( s 2 + DV s ) + k
= s ( s + DV ) + k
Change in damped
natural frequency"
! ndamped ! ! n 1" # 2
47!
Effect of DV Variation on
Approximate Phugoid Roots"
k = DV"
(
!(s) = ( s 2 + gLV / VN ) + ks
= s + j gLV / VN
)( s " j
)
gLV / VN + ks
Change in
damping ratio"
!
48!
Approximate Short-Period Model"
Approximate Short-Period Equation (Lq = 0)"
!!x SP
M"
# Mq
# !q! & %
=%
()%
!
"
!
%$
'( % 1
$
*
L"
VN
&
# M+ E
( # !q & %
( % !" ( + % *L+ E
(' %
VN
(' %$
$
&
(
( !+ E
('
Characteristic polynomial"
$L
'
$
L '
!(s) = s 2 + & " # M q ) s # & M " + M q " )
VN (
% VN
(
%
= s 2 + 2*+ n s + + n 2
Parameters"
M! , M q ,
L!
VN
49!
Approximate Short-Period Roots"
Approximate Short-Period Equation (Lq = 0)"
!!x SP
# Mq
# !q! & %
=%
()%
%$ !"! (' % 1
$
M"
*
L"
VN
&
# M+E
( # !q & %
( % !" ( + % *L+ E
(' %
VN
( %$
$
'
Characteristic polynomial"
$L
' $
L '
!(s) = s + & " # M q ) s # & M " + M q " )
VN (
% VN
( %
2
= s 2 + 2*+ n s + + n2
&
(
( !+ E
('
Generally,
L! > 0
M! < 0
Mq < 0
Natural frequency and damping ratio"
$
L '
! n = " & M # + M q # ); * =
VN (
%
$ L#
'
& " Mq )
% VN
(
$
L '
2 "& M # + M q # )
VN (
%
50!
Effect of M' on Approximate
Short-Period Roots"
k = M'"
$L
'
$
L '
!(s) = s 2 + & " # M q ) s # & M q " ) # k = 0
VN (
% VN
(
%
$
L '
= & s + " ) s # Mq # k = 0
VN (
%
(
)
Change in damped
natural frequency"
51!
Effect of Mq on Approximate
Short-Period Roots"
k = Mq"
Change primarily in
damping ratio"
!(s) = s 2 +
$
L"
L '
s # M" # k & s + " )
VN
VN (
%
2
2
0 *
-4 0 *
-4 $
$ L# '
$ L# '
L '
2 , L#
2 2 , L#
2
/
!(s) = 1 s "
+ &
+ M # 5 1s "
" &
+ M# /5 " k & s + # ) = 0
)
)
VN (
% 2VN (
% 2VN (
/ 2 2 , 2VN
/2 %
23 ,+ 2VN
.6 3 +
.6
52!
Effects of Airspeed, Altitude, Mass, and
Moment of Inertia on Fighter Aircraft
Short Period"
Airspeed variation at constant altitude"
Airspeed
m/s
91
152
213
274
Dynamic
Pressure
P
2540
7040
13790
22790
Angle of
Attack
deg
14.6
5.8
3.2
2.2
Natural
Frequency
rad/s
1.34
2.3
3.21
3.84
Period
sec
4.7
2.74
1.96
1.64
Damping
Ratio
0.3
0.31
0.3
0.3
Altitude variation with constant dynamic pressure"
Airspeed
m/s
122
152
213
274
Altitude
m
2235
6095
11915
16260
Natural
Frequency
rad/s
2.36
2.3
2.24
2.18
Period
sec
2.67
2.74
2.8
2.88
Damping
Ratio
0.39
0.31
0.23
0.18
Mass variation at constant altitude"
Mass
Variation
%
-50
0
50
Natural
Frequency
rad/s
2.4
2.3
2.26
Period
sec
2.62
2.74
2.78
Damping
Ratio
0.44
0.31
0.26
Moment of inertia variation at constant altitude"
Moment of
Inertia
Variation
%
-50
0
50
Natural
Frequency
rad/s
3.25
2.3
1.87
Period
sec
1.94
2.74
3.35
Damping
Ratio
0.33
0.31
0.31
53!
Flight Control Systems!
SAS = Stability Augmentation System!
54!
Effect of Scalar Feedback Control
on Roots of the System"
K
H (s) =
kn(s)
d(s)
Block diagram algebra"
!y(s) = H (s)!u(s) =
kn(s)
kn(s)
!u(s) =
K !" (s)
d(s)
d(s)
= KH (s) [ !yc (s) " !y(s)]
!y(s) = KH (s)!yc (s) " KH (s)!y(s)
55!
Scalar Closed-Loop
Transfer Function"
K
H (s) =
kn(s)
d(s)
[1+ KH (s)] !y(s) = KH (s)!yc (s)
KH (s)
!y(s)
=
!yc (s) [1+ KH (s)]
56!
Roots of the Closed-Loop
Control System"
kn(s)
Kkn(s)
Kkn(s)
!y(s)
d(s)
=
=
=
kn(s) % [ d(s)+ Kkn(s)] ! closed ( s )
!yc (s) "
1+
K
$
'
loop
d(s) &
#
K
Closed-loop roots are solutions to"
! closed (s) = d(s) + Kkn(s) = 0
loop
or"
kn(s)
K
= !1
d(s)
57!
Root Locus Analysis of Pitch Rate Feedback
to Elevator (2nd-Order Approximation)"
k q ( s # zq )
!q(s)
KH ( s ) = K
=K 2
= #1
2
!" E(s)
s + 2$ SP% nSP s + % nSP
!! # of roots = 2"
!! Angles of asymptotes, (, for
!! # of zeros = 1"
the roots going to %"
!! K -> +%: –180 deg"
!! Destinations of roots (for k =
±%):"
!! K -> –%: 0 deg"
!! 1 root goes to zero of n(s)"
!! 1 root goes to infinite radius"
58!
Root Locus Analysis of Pitch Rate Feedback
to Elevator (2nd-Order Approximation)"
•! Center of gravity on real
axis"
•! Locus on real axis"
–! K > 0: Segment to the left of
the zero"
–! K < 0: Segment to the right of
the zero"
Feedback effect is analogous
to changing Mq"
59!
Asymmetrical Aircraft: DC-2-1/2"
DC-3 with DC-2 right wing"
s way during WWII"
Quick fix to fly aircraft out of harm
60!
Asymmetric Aircraft - WWII"
Blohm und Voss, BV 141"
B + V 141 derivatives"
B + V P.202"
Recent Asymmetric Aircraft"
Scaled Composites Boomerang"
NASA AD-1"
Scaled Composites Ares"
Next Time:!
Advanced Longitudinal
Dynamics!
!
63!
Supplemental Material!
!
64!
Effect of L#/VN on Approximate
Short-Period Roots"
k = L'/VN"
•! Change primarily
in damping ratio"
(
!(s) = s 2 " M q s " M # + k s " M q
)
2
2
0 *M
-4 0 * M
-4
$ Mq '
$ Mq '
2 , q
2
2
2
q
= 1s +
" &
+ M # / 5 1s + ,
" &
+ M# /5 + k s " M q = 0
)
)
% 2 (
% 2 (
/2 2 , 2
/2
23 ,+ 2
.6 3 +
.6
65!
(
)
Root Locus Criterion"
•! All points on the locus of roots must satisfy the
equation k[n(s)/d(s)] = –1"
•! i.e., all points on the root locus must have a
phase angle(–1) = ±180 deg"
Spirule"
(Invented by Walter Evans)"
66!
1/--pages
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