...........
.
BENDING MODE ACCELERATION INFLUENCE
ON PILOT CONTROL OF FLEXIBLE
BOOSTER DYNAMICS
by
PHILIP S.
A.B.,
KILPATRICK
Carleton College
(1964)
S.B.,
Massachusetts Institute of Technology
(1964)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1965
Signature of Author
Depaitment rof Aeronautics and
ronautics
tember 1965
Certified by
ProfesoLaurence'
R.LfingTIesis Supervisor
Accepted by
Chairman, Departmental Graduate Committee
I
I
ii
BENDING MODE ACCELERATION INFLUENCE
ON PILOT CONTROL OF FLEXIBLE
BOOSTER DYNAMICS
by
Philip S. Kilpatrick
Submitted to the Department of Aeronautics and Astronautics,
Massachusetts Institute of Technology, on September 30, 1965,
in partial fulfillment of the requirements for the degree of
Master of Science.
ABSTRACT
This investigation is concerned with the general problem
of man's ability to directly control a large flexible launch
vehicle. Specifically, the effect of a flexible body mode
on pilot control of simulated single axis Saturn V rigid
body dynamics is studied. First bending mode amplitude
and natural frequency, and the type of simulation, fixed
or moving base, are the variables considered most intensively.
Brief studies of variations in the RMS level of the disturbance
signal and comparisons of two proposed control stick filters
and vehicle augmentation schemes are included.
The effects of the flexible mode on the pilot and his
closed loop performance are analyzed by ratios of attitude
error to disturbance signal and control stick output to
attitude error, and by computed pilot transfer functions.
Results show that pilot 's sability to generate le.ad. -compensation and to control the attitude error decreased as
the bending mode amplitude increased. Significant deterioration occurred at the lowest bending mode amplitude, 1/3
the value at the proposed location of the Saturn V attitude
gyro, under study. The pilot's gain and ability to control
the attitude error decreased during the moving base experiments.
This result is attributed to dynamics and non-
linearities associated with the simulator, a less sensitive
moving base display, and possibly vestibular uncertainty.
iii
and insensitivity concerning small deflections from the
vertical. With increasing bending mode amplitude, pilot
performance deteriorates at approximately the same rate
for both w
= 5 and 7 rad/sec.
However, for a given
amplitude,n ge 5 rad/sec bending mode generates only onehalf the acceleration of the 7 rad/sec bending mode.
Thesis Supervisor:
Laurence R. Young
Title:
Assistant Professor of Aeronautics and Astronautics
iv
ACKNOWLEDGEMENTS
The author wishes to thank Professor L. R. Young, his
thesis advisor, for stimulating initial interest in the
problem and for offering guidance throughout the research.
Professor J. L. Meiry's instructions concerning operation
of the Man-Vehicle Control Laboratory's NE-.2 Motion
Simulator and assistance with digital computer aspects of
the data analysis are gratefully acknowledged.
Mr. Ernest Silagyi prepared the computer program to transform data from analog to digital form.. This work was
completed on the Electrical Engineering Department's
TX-O Computer.
The M.I.T. Computation Center granted time on its
IBM 7094 Computer for work done as Problems M4345 and
M4347.
The author also expresses his thanks to Miss Toni Mello
for typing the thesis.
This research was supported by NASA Grant NsG-577.
V
TABLE OF CONTENTS
Page No.
Chapter IV
Introduction
1
Chapter II
Simulation and Equipment
6
Chapter III
Procedure
12
Chapter Iv
Discussion of '.Results
16
Chapter V
Conclusions
30
Appendix A
Derivation and Simplification of
Simulation Equations
33
Appendix B
Pilot Describing Fundtion Measurement Tecilnique
45
Bibliography
Fig.
Fig.
Fig.
Fig.
Fig.
2
3
4
5
42
General block diagram for the manbooster control problem.
46
Block diagram for the man-booster
control problem showing components
used in this investigation.
47
Analog computer patching program
for fixed base experiments.
48
Analog computer patching program
for moving base experiments.
49
Analog computer patching program
for a modified version of the
augmented dynamics and stick
50
filter recommended by Jex and Teper.
vi
Page No.
Fig.
6
Analog computer patching program
used to shape the spectrum of the
disturbance* signal.
51
52
Fig.
7
NE-2 motion simulator.
Fig.
8
Fixed and moving base control stick
Fig.
Fig.
Fig.
Fig.
Fig.
9
10
11
12
13
characteristics.
53
Block diagram showing control
systems signals recorded on
strip chart and tape recorders.
54
Subject position and equipment
location for fixed base experiments,.
55
Subject position, control stick,
and display location for moving base
experiments.
56
Analog computer patching program for
integral square error calculations.
57
Strip chart recordings of attitude
error
Fig.
14
Table 1
Table 2
Strip chart recordings
respohse'
Table 4
of pilot
Transfer function data for fixed base
s-!ssion, w nbd = 5 rad/sec.
59
60
Transfer function data for fixed base
session, wnbd
Table 3
58
=
7
rad/sec.
Transfer function data for moving base
session, wnbd = 7 rad/sec..
61
62
Transfer function data for moving base
session, wnbd = 5 rad/sec.
63
Fig.
15
Plot of amplitude ratio data from Table 1. 64
Fig.
16
Plot of amplitude ratio data from Table 2. 65
Fig.
17
Plot of amplitude ratio data from Table 3.. 66
Fig.
18
Plot of amplitude ratio data from Table 4, 67
Fig.
19
Plot of phase data from Table 1.
68
/
vii
Page No.
Fig. 20
Plot of phase data from Table 2
69
Fig. 21
Plot of phase data from Table 3
70
Fig. 22
Plot of phase data from Table 4
71
Table 5
Summary of fitted describing
functions
72
Attitude error and pilot response
power spectrum for fixed base
session, w nbd = 5 rad/sec.
73
Attitude error and pilot response
power spectrum for fixed base
session, w nbd = 7 rad/sec
74
Table 6
Table 7
Table 8
Table 9
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Table 10
Attitude error and pilot response
power spectrum for moving base
session, w nbd = 7 rad/sec
75
Attitude error and pilot response
power spectrum for moving base
session, w nbd = 5 rad/sec
76
Plot of control stick power from
Table 6
77
Plot of control stick power from
Table 7
78
Plot of the RMS ratio
versus bending mode amplitude for
fixed base plus simulated dynamics
experiments.
79
Individual tracking run
scores versus bending mode amplitude for each subject under fixed
base,w nbd =7
rad/sec conditions
80
Analysis of variance results for
Fig. 26
Fig. 27
81
Plot of
results versus
bending mode amplitude for each
type of simulation at w nbd=7 rad/sec 82
viii
Page No.
Fig.
28
Fig. 29
Fig. 30
Fig. 31
Plot of averaged /Ve-V/
results versus bending mode
amplitude for each type of
simulation at wnbd = 5 rad/sec
83
Comparison of/f
results for wbd - 5 and 7 rad/sec
base conditions
during fixed
84
Comparison of/Ej= 5 and 7 rad/sec
results for wng
during moving base conditions
85
Comparison of V7/results for wnd~ =
5 and 7 rad/sec
during fixed base plus simulator
Fig. 32
Fig. 33
dynamics conditions
86
results versus
Plot of /Vs1/
bending mode amplitude for conditions of low RMS disturbance signal
87
Plot of /'/'results versus
bending mode amplitude for condition of low RMS disturbance signal
88
Fig. 34
results
Comparison of /i2-/X2
for the two disturbance signals with
89
Wnbd = 7 rad/sec
Fig. 35
Comparison of
v'D//-%
results for the two disturbance
signals with w nbd = 5 rad/sec
90
Effective dynamics as seen by pilot
for simplified versions of two proposed vehicle augmentation and
control stick filter schemes
91
Summary of results comparing the
two proposed vehicle augmentation
and control stick filter schemes
92
Fig. Al
Root locus 1
93
A2
Root locus 2
94
Fig. 36
Table 11
Fig.
ix
LIST OF SYMBOLS
Partial derivatives of the force on the
missile with respect to a, a, 0,
respectively.
Partial derivatives of the moment on the
missile with respect to a and a respectively.
Pitch angle of attack.
Engine gimbal angle,
Total pitch attitude angle with respect
to nominal trajectory
tot
Part of 0tot due to the rigid body mode-
$rb
$bd
Part of, 0tot due to the flexible bpdy mode,
V
Nominal vehicle velocity.
Natural frequency of the rigid body mode,
Wnrb
Natural frequency of the bending mode.
LA
z
=
3w
nbd
Location of the zeroes associated with
the first bending mode
Ki 1
K21
K3 1
Constants associated with the bending
mode.
K
Constant proportional to the amplitude
of the bending mode
Krl
Root locus gain
x
n
Generalized coordinate associated with
the first bending mode.
qi
ith
zero in transfer function.
p.
ith
pole in transfer function.
p
LaPlace operator.
ISE
Integral squared error.
d (t)
aisturbance signal into the control loop.
e (t)
Displayed attitude error.
s (t)
Operator's response.
o (t)
Output of simulated missile dynamics.
0M
RMS ratio of the attitude error to
disturbance signal for one tracking run.
The same ratio averaged over several
tracking runs.
RMS ratio of the pilot response to
attitude error for one tracking run.
n
nom.
I
The same ratio averaged over several
tracking runs.
The amplitude of the bending mode at
the proposed location of the Saturn V
attitude gyro.
CHAPTER I
INTRODUCTION
Several relatively recent studies have been conducted
to determine the feasibility of using a pilot to control
the attitude and trajectory of large flexible boosters during
the launch into orbit.
Along with other problems,
these
investigations considered the ability of the pilot to effectively control the unstable rigid body mode without exciting
the relatively low frequency and highly sensitive bending mode beyond structural or attitude limitations.
Hardy, et al, simulated the rigid and flexible body
dynamics of the Saturn V booster and report that the visual
and vestibular cues from flexible motions did not create
serious problems.1
However, they obtained these resnlts
using a second order low pass filter to attenuate the high
frequency components of the control stick output.
In a theoretical study, Teper and Jex agree that it would
be possible for the pilot to directly control the missile,
but recommend, among other things, replacing the second order
stick filter with a single integration. 7
In either case, stick filters reduce the bending mode
effect at the cost of additional phase lag in series with
an already difficult set of missile dynamics.
2
The objective of this thesis is to study more comprehensively the effect of a superimposed bending mode on the pilot's
ability to control the single axis attitude of a missile with
unstable rigid body dynamics.
Hopefully, in spite of the
restrictions to a particular set of rigid body dynamics, the
results will be applicable to other missiles and large aircraft with significant flexible mddes.
The importance of the bending mode depends on many
control system parameters and pilot characteristics.
However,
the relative natural frequencies of the two modes, the
amplitude of the bending mode, and the type of simulation,
fixed or moving base, were considered the most important,
and attention was directed primarily at these
factors.
In general, the interaction between two modes in a
feedback control system increases as the separation between
the natural frequencies decreases.
In this situation, there
will be a direct relation between the degree of excitation
of the bending mode and the frequency content of the control
stick signal.
In turn, this frequency content will be deter-
mined partly by the difficulty of the rigid body control task
assigned to the pilot.
The simulated system dynamics included the dominant
inverted pendulum rigid body mode of the Saturn V at peak
dynamic pressure
W nrb
b = .15, and the first bending mode.
Bending mode frequencies of
(rad/sec)2
were studied.
nd
= 49
(rad/sec)2 and 25
The first bending mode natural
frequency of the Saturn V at maximum dynamic pressure equals
-approximately 49
(rad/sec) 2 .A
value significantly closer
3
to the pilot control frequencies chosen for the second
frequency.
For a given natural frequency, the flexible mode acceleration sensed by the pilot is proportional to the mode amplitude.
An amplitude approximately equal to that sensed at the proposed
attitude gyro station for the Saturn V was taken as a nominal
value.
Three other amplitudes,
(0,
.33 nom.,
were studied in the fixed base experiments.
range,
(0 to 4.0 nom),
and 2.0 nom.),
A wider
were used for the moving base work.
The
decrease in effective pilot control fixed the upper limit on
the amplitude, and the onset of significant bending mode
effects determined the lower.
The pilot senses the existence of the bending mode by
visual and vestibular cues.
In order to assess the relative
importance of these two inputs, the experiments were performed
both fixed and moving base using a single axis of rotation.
Fig. 1 contains a block diagram showing the position of
the pilot, booster, displays, disturbance signal, and possible
compensations in the signal axis control loop.
Fig. 2 shows
the components used for this investigation.
For the purposes of this study, display and stick filters
were not used except for a brief comparison of two suggested
stick filters.
The control stick filter was eliminated to
find the deterioration of uncompensated pilot performance
with increasing bending mode amplitude.
From comparisons of the
resulting increases in attitude error and structural loadings
with attitude specifications and structural limitations, bending
4
mode amplitudes requiring the additional complexities of
control stick filters may be determined.
Rigid body rate compensation was added after
preliminary experiments indicated the combination of
uncompensated dynamics, noise signal, and bending mode
created a very difficult control problem,
No bending
mode rate information was included in the rate signal.
Perfect filtering of the flexible portion of the rate signal
was assumed in order to study only direct control stick
excitation of the bending mode.
A random noise disturbance signal summed with the
simulated dynamics output produced the moving base drive
signal to the simulator and the fixed base error signal.
The random noise signal replaced the wind spike distrubance of
Ref. 1 to allow pilot transfer function computations.
The
noise signal entered as an attitude angle and not an acceleration into the dynamics, once again, to restrict direct
bending mode excitation to the control stick output.
The RMS value of the disturbance signal was chosen so
that typical RMS attitude errors ranged from one to three
degrees.
These errors could be controlled with the maximum
control torque of 4.40/sec2 .
This value was inadvertantly
made 15% higher than that recommended in Ref. 1.
The disturbance consisted of a white Guassian signal
shaped by two first order filters with break frequencies
at 1 rad/sec.
the signal.
Two factors determined the frequency content of
First, the noise signal had to contain sufficient
5
high frequency power to permit computation of a transfer
function valid to .8 cps.
Secondly, the signal could not
vary so rapidly that the subject was unable to control the
rigid body portion of the error signal.
Bending mode influence on pilot performance was
measured by error to disturbance and pilot control stick
output to error signal ratios, and by pilot transfer functions.
The experimental part of the thesis consisted of both
fixed and moving base studies of variations in bending mode
amplitude and natural frequency, and extended fixed base
studies including simulator dynamics, disturbance signals
with lower RMS values, and stick filters.
6
CHAPTER II
SIMULATION AND EQUIPMENT
This chapter describes the simulated missile dynamics,
and the equipment needed to instrument the control loop
of Fig. 2.
The equations of motion and parameter values describing
the pitch axis dynamics of the Saturn V booster at maximum
The linearized
dynamic pressure were taken from Ref. 1.
rigid body equations for small perturbations from the
nominal trajectory are:
x = - F a - F
rb
= M a
a=
rb
-
6
F
(2.1)
(2.2)
-M
rb +
57.3
(2.3)
x
In Appendix 1, these equations are reduced to the
following relation between $rb and
(p + M
F
M V
rb =
M
5
(P3 + 57.3 F
(p+V
57.3
57.3 + F
P12
- M
a Ma
:
p + 57.3
+
M F
(2.4)
V
After replacing the parameters with their values at
maximum dynamic pressure and factoring:
(p + .0197)
$rb
(
-
5(p
4.1.15
-
.34)
(p + .40)
(p -. 0425)
(2.5)
7
This expression is simplified in Appendix 1 to:
7.67(.15)
rb
(p
(2.6)
.15)
-
Finally, when the rigid body rate compensation recommended
in Ref. 1 is added, results in Appendix 1 show:
$
7.67(.15)
(p +
1.00)(p -
.15)
(2.7)
The elastic body mode may be represented by:
K1+ K' )
2
bd
K 3 -1 K 2 1 (p
a
(p
2
2
+ 2E WnbdP + w nbd)
(2.8)
From Appendix 1, after certain assumptions and parameter
value substitutions, the equation becomes:
K(p2 + 212)
$bd
2 f2
for
nbd
=
7 rad/sec
+ .01(7)p + 7 )
(p
(2.9)
2
K(p
$bd
-
=
2
+ 15)
2
(p + .01(5)p + 5
The total expression for
2
for wnbd = 5 rad/sec
(2.10)
tot/a in terms of K, a number
proportional to the bending mode amplitude, and wnbd, the
natural frequency, becomes:
t
S
-l.15 (p +. OlwnbdP+ nbd)+K (p 2+(3wnbd)2p-.15)(p+1.00)
(p-.15) (p+l.00) (p2+.OlWnbdp+
2 bd)
(2.11)
8
The root locus technique is used in Appendix 1 to factor
the numerator of
(2.11) for the two values of wnbd
rad/sec) and four values of K (.0022,
.0066, .0132,
(5
and 7
.0264).
The results are listed below:
=
Wnbd
K =
7 rad/sec,
.0022
(p -
K =
=
K =
.0066
4
(p -
P
.
t1
.0132
.15)
0
.15)
4.5)(p -
.15)
=
.0022
(2.14)
(19.5)2)
(2.15)
r3
.0066
.0132
.0264
2
+
.01(7)p
+ 7 2)
22.7) (p -
22.7) (p 2 +
(5.8)2)
(2.16)
-tot
(p -
tot:
.15) (p + 1.00) (p2 +
.0066 (p + 9.4) (p -
-7-
(p -
.15)(p +
1
.0
---
$tot
0
.01(5)p + 52)
8.8) (p2 +
)(p
L
4.3) (p
+
(2.17)
(8.8)2)
(.01)5 p + 52)
2
+
(2.18)
(11.5) 2)
~
(p -
K =
(18.6) )
rad/sec
.0132 (p J- 5) (p K =
2
2
p
2.4) (p2 +
(p + 1.00) (p
.0022(p +
K
.01(7)p + 72
.0264
Wnbd
=
(2.13)
(15.9)2)
.15) (p + 1.00) (p2 + .01(7)p + 7,)
(p -
K
( 2.12)
+ 72)
.01(7)p
+
3.8)
.0264 (p + 3.2) (p -
$ tot
+ 102)
6.5) (p 2 +
(p +1.00) (p2
132(p'+
-
K =
(p + 1.00) (p 2 +
.0066 (p + 7.1) (p -
tot
(nom.)
18.6) (p2
+ 19) (p -
.0022(p
tot
.15) (p + 1.00) (P2
.0264 (p + 3.3) (p '
kp -
-.
-'
-
-
2.6) (p
- -L-')) kP -t- I. 00) (p
(.01)5 p + 52)
+
2
+
(2.19)
(13.2)2)
+- (.01) 5 p +- 5
2
2
9
Figs. 3 and 4 show the analog computer program for the
simulation of these missile dynamics.
An EAI TR-48 analog
computer was used for the fixed base investigations and
Philbrick amplifiers for the moving base experiments.
For the case of the experiments concerning the stick
filter proposed in Ref. 2, the dynamics associated with
Teper and Jex recommend, in addition
rigid body change.
to the single integration stick filter, feeding
back position as well as rate information.
Appendix 1 shows that with these modifications and
certain simplifications the effective rigid body dynamics
become:
crb
-
2
p
85(4)
(2.20)
-
+ 2(.56)2 p + 2
With this change in rigid body dynamics the relation
between
tot and 6 becomes:
2
tot
-3.4(p +.01(7)p+7
2
2
2
2
)+.0066(p +2(.56)2p+22) (p +21
2
(p'+2(.56)2p+2 2)(p2+(.01)7p+7 )2.21)
The root locus technique is applied to factor the numerator
and the results appear below for K = .0066 and wnbd = 7 rad/sec.
10
4tot
S
.0066 (p
(p
11) (p
w
15.5) (p
+ (10.5)
+ 2(.56)2p + 2 ) (p' +(.01)7p
(2.22)
+ 7 )
for these equations
The modified analog computer program
appears in Fig. 5.
Fig. 6 contains the analog computer program required to
unbias, amplify, and filter the random noise signal.
The
signal on the tape recorder had a frequency spectrum flat
to 1000 cps.
A motion simulator capable of rotation about two axes
was used for the moving base experiments
(see Fig. 7).
Because of superior roll frequency response, the experiments
wexe performed about the roll and not pitch axis.
The
freqctency, response of the simulator with subject was measured
and found to be second order with a natural frequency of
1.6 cps and a damping ratio,,
equal to .4.
shift at .8 cps was 30P and 550 at 1.2 cps.
The phase
A dead-zone of
approximately 1/30 existed.
The characteristics of the control sticks for the fixed
and moving base experiments appear in Fig. 8.
The fixed
base control stick was restrained by a stronger spring.
Attitude error, fixed base, and cab position, moving
base, was the only information displayed to the subject.
In both cases, the error angle was represented on an oscilloscope
11
by the horizontal distance from the center of the screen to
a generated vertical line.
The scope sensitivity was 1 cm/deg
for fixed base work and .6 cm/deg for moving base experiments.
Four signals, distrubance, attitude error, stick output,
and simulated missile dynamics output were recorded on strip
chart recorders and a four channel F-M tape recorder,
(see Fig. 9).
Figs. 10 and 11 show the subjects' position, display,
and control stick for the fixed and moving base investigations respectively.
12
CHAPTER III
PROCEDURE
Three students served as subjects in the experiments.
The subjects were screened by testing their ability
to control unstable dynamics in a compensatory tracking task
and to balance on one foot without visual uses.
After selection, the subjects practiced controlling
the Saturn V dynamics until no. further improvement in
performance could be detected.
Typically, sessions lasted two hours and included twenty
to twenty-five tracking runs of 90 or 120 seconds
by three minute rest'
separated
periods.
During a session, only the bending mode amplitude changed.
Usually,
the session was divided into six runs at each of four
amplitudes.
The bending mode amplitude increased as the
session progressed.
Before each session, the subjects were informed of the
bending mode natural frequency.
In addition, they were told
of changes in bending mode amplitude.
The subjects were instructed to use the control stick as
necessary to minimize the displayed error until the distraction
of bending mode oscillations forced a reduction of control
efforts.
The subjects learned fairly quickly by trial and
error how much control to use at each bending mode amplitude.
13
The chart below shows the content and sequence of the
sessions and the participating subject.
St40LATION
3MgrA9L-S
A
Sue 3e.1
BENJCT4- MoME WAT'JUE L.RDUOk
sesie
AK)J AAP .wIEc RMAE
#8/64=~ Z-1 4jAj=-7 #7 w,,,= 5 *9, WAA
K= 0 -ro 2(OM)
IED 8OfA SE Ka _'_ = K=Om4(NM) K=OTo
43OWnO=S73 45 = ) ~ L-Se=7 #9, A6--Z IOJ1~~A=5
EIX
7O ro
9 =C0 T
4
K-- ) TO ToamZ)
) ILA =b&
Es O
K--or Q o
I
COWD171ONS
FlyD
4i)
IIAJnAV1 IJ
K'A TO 2(NJOM KQ
A~r
f 4=0 oMTo
AIXCD F#11)
BA SET
p
BAW~ %Wgpl #2
I
~SME
OF 4tDNEr' AL.
F
(E~ ?STKJZ
TE=
FO1D
*LS
26)
rez
20
WAOM
_
_
_
__
fi LemOnt
K: 0o"
#29
ahsE
--10 TUACN1NO
AMPL1UVPC CJ3
INs__
arc
e z-O
To (00
---
K:= WoM
0 rA-Kk
u%
PT
13CWO1W~r MODE
I-.
=0 ro 200 ) KC6 Tz Z(OM
IL TCAr(V
-1
po 4
o
K=O re L(4om)
K- MdOA
VCLE ADMWEWhing
STfCK RLTER
40 To 4)
5 #2Oj4 5'
4-4=7 #2Zaw5 K44 A0=5
COV~rTIOI45
Fl-E
)
ro46i_
OhW=7i
1O 5L.A
aoKOiL:)O
--- 1
--
kll,4
s5MULATO WOPI.
C MOVW6
rl:E
-STICK I0
) KK= VI,4OM
A)
T.()
MOVN46- BASE V . *O'o2Mb4
K=-OT>ZLbM)
f
'=IAJ~~
"130A
*L
K=Ob2(Ov)
14
The experiments
part
in Series C vere conctucted
to find what
of the difference Detween fixed and moving base results
should be attributed to the combination of simulator dynamics,
decreased display sensitivity,
and lightly restrained control
stick.
The interaction between disturbance signal amplitude and
performance deterioration with increasing bending mode amplitude was studied in Series D.
Series E and F reflect curiosity about the effect of
essentially changing the dynamics the pilot must control
from fourth to either fifth or sixth order by the addition
of a stick
filter.
To confirm that deterioration in performance with
increasing bending mode amplitude did not occur because
of the order of presentation or knowledge of amplitude, a
subject, without this information, was tested in Series G.
Fatigue was checked as a possible factor by testing the
subjects' ability to control the rigid body mode alone at
various times during the two hour sessions.
Integral square error values of the noise, error, and
stick
outputsijna1s
were calculated
by amplifying,
attenuating and integrating these signals
squaring,
for each run.
Fig. 12
contains the analog computer patching program that performed
these operations.
These scores, pilot transfer functions, and strip chart
recordings provided the means
to analyze the effect of
variations in dynamics and experimental conditions.
15
The error to disturbance ratio,
measured the pilot's ability to control the attitude of
the simulated missile.
The control output to error ratio,
and pilot transfer functions show more directly the results
of changing conditions on the pilot.
This second ratio represents an average gain for the
pilot for the run, but ignores the well-known dynamics
associated with the human operator.
The pilot transfer functions were computed by a spectral
analysis method described in Appendix 2.
The approximate levels of accelerations due to bending
mode oscillations were determined by anaiysis of the individual
strip chart recordings.
These recordings also show clearly
changes in pilot control actions as a function of bending
more amplitude.
16
CHAPTER IV
DISCUSSION OF RESULTS
The effect of variations in bending mode parameters
and simulation conditions will be discussed from two
standpoints:
1.
Changes in pilot control characteristics.
2.
Changes in closed loop attitude error performance.
Strip chart recordings of the displayed attitude
error and control stick movement are presented in Figs. 13
and 14.
These are taken from a fixed base session with
Wnbd = 7 rad/sec.
These recordings show qualitatively
the reduction of effective pilot control and the increase
in attitude error as the bending mode amplitude increases.
With the bending mode removed entirely, the pilot
used all available control power and behaved very nonlinearily.
In this case, only the amount of control
power as set by the Saturn V recommendations of Ref. 1
restricted the pilot, and the control stick output contained
a significant amount of high frequency power.
A revision
in pilot control strategy became necessary with non-zero
bending mode amplitudes.
If the subject failed to restrain
17
his control action, intolerable bending mode oscillations
developed at even the lowest amplitude
study.
(1/3 nom.)
under
In the process of attempting to alleviate this
problem, the subject began to lose effective control
of the rigid body portion of the attitude error.
To analyze these effects more quantitatively, several
criteria were employed.
Pilot describing functions were
measured for four sessions by a power spectral technique
5,6
explained in detail in Appendix B and associated references.'
Power spectral estimates of the attitude error and control
stick signals are available from the describing function
computation.
In addition, RMS ratios of the control stick
to the attitude error signal,
attitude error to disturbance signal, V
,
and the
-
were measured for the individual tracking runs.
The describing function was calculated by the following
formula:
13 (W)
where:
D 1 3 (w).=the cross power spectral estimate between the
disturbance signal and the control stick signal
012(w)= the cross power spectral estimate between the
disturbance signal and the displayed error
signal.
18
The degree to which the describing function accounted
for the pilot's behavior was measured by the correlation
coefficient:
.(4.2)
where:
fiw
(~)>
= the power spectrum of the disturbance
=
signal
the power spectrum of the control stick signal
The value of 1
should be near unity if the describing
function accounts for most of the operator's characteristics.
The describing function data is presented in Tables 1
through 4 and plotted in Figures 15 through 22.
The
amplitude data and fitted amplitude ratios appear in the
first four figures.
four.
The phase data are shown in the last
Pertinent information about the associated experi-
mental condition is
listed
in the Tables.
It should be noted that the correlation coefficients
are not close to unity above 3 rad/sec for the fixed
base experiments and 2 rad/sec for the moving base work.
The recorded disturbance signal contained occasional
spikes from an extraneous source.
These spikes showed up
as an uncorrelated wide-band signal superimposed on the
spectrum of the disturbance.
On the average, the wide-band
spectrum amplitude reached 1/10 of the disturbance signal
19
power spectrum at 2.7 rad/sec fixed base, and at 1.8 rad/sec
moving base.
Above these frequencies, the reliability of the data
drops sharply because only a fraction of the disturbance
signal is uncontaminated.
However, the form of the
amplitude and phase data remains reasonable to 3.6 rad/sec.
With these factors in mind, the describing function results
are retained up to this frequency.
The describing functions are summarized in Table 5.
The important results are:
1.
The level of pilot gain decreases as the bending
mode amplitude increases.
2.
The phase lead generated by the pilot decreases
with increasing bending mode amplitude.
3.
The subjects' gain was significantly lower during
the moving base tests than in the fixed base
experiments.
The lowering of subject gain reflects his attempts to
minimize bending mode excitations.
The decrease in phase lead as the bending mode amplitude
increases seems to be best accounted for by a larger deadtime delay in the pilot describing function.
There is some
shifting of the pole-zero cqrbination, but this is not a
major factor.
The distinct reduction of subject moving base gain
compared with fixed base results was unexpected.
Simulation
and scaling factors were carefully checked for errors that
might explain the effect.
20
The power spectral estimates of the error signal
presented in Tables 6 through 9 show that the reduced
moving base gain occurs because of increased error rather
than lower control stick power.
A series of fixed base-experiments including
the second order dynamics and dead-zone associated with
the motion simulator plus the moving base, control stick
and display will be discussed in more detail later.
However, these experiments did not indicate that the increased
increased error could be attributed entirely to these
factors.
It should be pointed out, though, that not all
the motion simulator non-linearities, such as stiction
and backlash, were considered.
The effect of these factors
cannot be discounted because typical simulator movements
were within the range of + 50.
Apart from this, the increased errors may be
attributed to subject uncertainty about the location
of the vertical during the moving base simulation.
The
subject's vestibular system may provide orientation information that is in error by a degree or two.
Furthermore,
the scale on the moving base display was less sensitive
and the grid was not as well defined as the fixed base
display.
In spite of the reduced gain, the subject was able
to generate the same amount if not more phase lead.
For
these reasons, the best explanation seems to involve
neglected simulation non-linearities, a less sensitive
21
display, and perhaps vestibular confusion and insensitivity
concerning very small angular deflections about the vertical.
Tables 6 through 9 include the power spectral estimates
of the error and control stick for the four sessions where
describing functions were computed.
The control stick power from Tables 7 and 8 has been
plotted in Figures 23 and 24.
These power spectral
estimates show:
1.
In general pilot control-power at the primary
rigid body control frequencies of .45 to 1.8 rad/sec
decreases with increasing bending mode amplitudes.
There
are exceptions to this statement, however, considering the
increases in rigid body error power at these frequencies,
effective pilot control has certainly decreased.
The same
general effect appears on the amplitude plots for the
describing function.
In this case there is no considera-'
tion of linear correlation.
2.
In certain cases, there has been an attempt to
track the higher frequency bending mode error as indicated
by secondary peaks a little below the bending mode natural
frequency.
3.
Control stick power drops drastically with increas-
ing bending mode amplitude at and above the natural frequency.
4.
The subjects were remarkably adaptive in their
attempts to avoid bending mode excitation by elimination
22
of the high frequency components from their control
stick action.
The reduced gain versus increased bending mode amplitude
is shown by a slightly different criterion in Fig. 25-.
The
RMS ratio of control stick signal to error signal,
is plotted versus bending mode amplitude for the set of
fixed base experiments including simulator dynamics.
This ratio represents a pseudo-gain for the subject that
ignores dynamics and linear correlation.
Once again, pilot
gain is inversely related to bending mode amplitude.
The
decreasing gain consists of both in increasing RMS error
and decreasing RMS control power.
The ratio of the RMS value of the error to disturbance
signal,
V7
measures the effect of variations in
pilot control strategy on closed loop performance.
Before discussing the main body of
results,
several tests for spurious variables will be described.
Three subjects were used for the fixed base experiments
with the bending mode natural frequency, w nbd = 7 rad/sec.
In order to check. intersubject differences an analysis
of variance was performed on the results of this series
of experiments.
Intersession and intersubject variances
were compared for each of the four amplitudes.
Fig. 26
presents the individual tracking run scores for the two
sessions under study for each of the subjects.
shows the results of the analysis of variance.
Table 10
23
The intersubject interaction was significant at the
level for only one .mplitude
2x(nom.).
On this
bais
.05
inter-
subject interactions were ignored.
In order to test the importance of subject knowledge
of the bending mode amplitude and the fixed order of bending mode amplitude variations, ten tracking runs were
taken at an amplitude, 1.33 nom.,
unknown to the subject.
The experimental conditions were:
fixed base simulation,
and wn =
7 rad/sec.
score was
The average,
.84
compared with an expected .70 based on results that will
If such a limited amount of data is
appear in Fig. 27.
significant at all, it would indicate that subject performance
at a given amplitude would not improve if the amplitude was
varied randomly with no information being given to the
subject.
On several occasions,
scores for the condition
of no superimposed bending mode were taken towards the
end of a session.
No increase in these scores over the
ones, at the start of the session was noted indicating
no fatigue effects.
ThevJ
-T
Wnbd = 7 rad/sec
ratio is plotted against the four
bending mode amplitudes for the three
types of simulation, fixed base, moving base, and fixed
base plus simulator dynamics in Fig. 27.
The scores
plotted for each amplitude and type of simulation represents
the average of all the individual tracking runs for the
specific condition.
24
Both moving and fixed base experiments were performed
in an attempt to find the relative importance of vestibular
and visual cues.
The fixed base experindents with the second
order dynamics and dead-zone of the motion sumulator plus
moving base display and control stick were conducted to
find the significance of these factors.
The results show that:
1.
The
ratio increases significantly with
increasing bending mode amplitude.
2.
The attitude error is much larger for moving
base experiments compared with the fixed base.
3.
This difference cannot be completely accounted
for by any conditions tested in the fixed base plus
simulated dynamics series.
4.
The attitude error increases less rapidly for
the moving base and fixed base plus simulated dynamics
than for the fixed base experiments.
The first result reflects decreased subject gain and
phase lead generation.
The second has been discussed earlier in the chapter.
Since the rate of attitude error increase is similar
for the moving base and fixed base plus simulator dynamics,
the difference between the fixed and moving base rates
cannot be necessarily attributed to vestibular effects.
25
Unfortunately, the dead-zone associated with the motion
simulator suppresses bending mode oscillations, and makes
fixed and moving base comparisons difficult.
This emphasizes the fact that moving and fixed base
differences can be the result of vestibular effects or
simulator characteristics.
Furthermore, the simulator
dynamics and non-linearities must be located at a-very bad
place, between the actual position and the displayed position, in this control loop.
For this reason, the fixed
base results are probably more realistic and reliable.
The same results are plotted in Fig. 28 for the bending mode of natural frequency w n = 5 rad/sec.
trends are very similar.
The
The general
ratio has been plotted
versus amplitude and not effective acceleration.
For the
same bending mode amplitude, the effective acceleration
for wnbd
5 rad/sec is only 1/2 of the -acceleration for
the case of wnbd = 7 rad/sec.
Comparisons of the i/l
ratio for wnbd = 5 and
7 rad/sec are plotted for each type of simulation in
Figures 29 through 31 versus amplitude and not effective
acceleration.
The
ratio increased more rapidly
for w n = 5 during fixed base simulation and for w
during fixed base plus simulator dynamics.
= 7
On the other
hand, there was little difference during the moving base
simulation.
The only certain conclusion is that one-half
the acceleration at w nbd = 5 rad/sec compared with
26
Wnbd =
7 rad/sec caused approximately the same performance
deterioration.
The results from the fixed base experiments using a
disturbance signal with one-half the RMS value of the
previous experiments appear in Figures 32 through 35.
The averaged
scores for the two sessions at
both w nbd = 5 and 7 rad/sec are presented in Fig. 32.
Consistent with the earlier fixed base results, the attitude
error for a given amplitude is greater for w nbd = 5 rad/sec.
Fig. 33 contains the
conditions.
results for the same
Note, that at each amplitude, the subject gain
is lower for ()nbd = 5 rad/sec curve and the attitude error
is greater.
This same correlation between gain and error
holds at each amplitude for the fixed base plus simulated
dynamics experiments except that the gain is lower and
attitude error greater for
27,
nbd = 7 rad/sec, see Fig. 25,
and 28.
With this correlation in mind, the interaction between
the RMS value of the disturbance signal and the rate of
increases with respect to bending mode
amplitude will be examined.
Comparisons of the attitude
error performance for the two disturbance signals at each
bending mode natural frequency appears in Figs. 34 and 35.
Somewhat surprizingly, for each non-zero bending mode
amplitude and both frequencies, the subject's
score was better for the high RMS disturbance signal.
In
27
addition, for the one high RMS fixed base session where the
ratio was measured, the subject operates
with a higher gain for the low RMS disturbance signal.
This
result is not consistent with the previous high gain ratiolow error ratio correlation just discussed.
The final series of experiments studied two vehicle
augmentation and control stick filtering schemes.
Both
approaches have been simplified, and the simulated dynamics
are only first approximations to the actual control systems.
For all previous experiments, the augmented missile
dynamics have been similar to those proposed by Hardy,
et al,in"Ref.1. To approximate the entire system recommended
there, a second order stick filter was added in series with
these dynamics.
The effective dynamics as seen by the
subject appear in Fig. 36a.
The simplification of the missile augmentation proposed
by Teper and Jex in Ref. 7 has been discussed in Appendix A.
The single integration-gain stick filter was placed in
series with this set of simplified dynamics to form the
complete system, as shown in Fig. 36b.
The bending mode amplitude was fixed at the nominal
value and the natural frequency at 7 rad/sec for all
experiments in this series.
Twenty tracking runs divided between two sessions were
taken for each system.
28
Before discussing the results, the choice of one parameter must be explained.
There was some uncertainty
about the correct choice of maximum control troque for the
single integration system.
torque has been 1.150/
Up to the present, the maximum
sec 2 /S with amaK = 3.870.
Preliminary
experiments with both systems indicated that the second
order filter removed about 10% of the pilot response signal,
and the single integration-attenuation filter output was
1/3 of the pilot's response.
For this reason, the maximum
control was increased by a factor of three for the latter
system.
For these experiments, the
ratio was measured
for both the control stick output and the stick filter
output signal.
by.
the
As usual, the attitude control was measured
ratio.
The results are summarized in Table 11 and show that:
1.
The average
ratio for the single integra-
tion-stabilized dynamics version was .88, compared with
1.20 for the second order-rate augmented dynamics system
2.
Both systems reduced bending mode excitation to an
occasional oscillation or two at an amplitude of less than
one-half degree.
3.
As mentioned earlier, the single integration filter
removed a much greater portion of the operator's response
than the second order filter.
29
4.
The effective pilot gain, as measured after
the filter, is higher for the case of stable r gid poles
and single integration stick filter.
Because of this last reason, the improved performance
with the approximation to Jex and Teper's proposal may be
due to the arbitrarily increased control torque.
However,
the complete stabilization of the rigid poles seems like a
very reasonable suggestion, and should contribute to
improved performance.
On the other-hand, the single
integration-attetuation filter, suppreSses a large percentage of the operator's response.
30
CHAPTER V
CONCLUSIONS
From observation of the strip chart recordings, attitude
error increases and pilot control strategy changes markedly
with increasing bending mode amplitude.
The operator switches
from a relay-like non-linear response to a combination of
lower amplitude of pulsing and at times smooth tracking.
Furthermore,,.significant bending mode excitation can develop
at the lowest amplitude, 1/3 nom.,
under study.
The pilot describing functions show that as the bending mode amplitude increases, the pilot gain and phase
lead compensation decrease.
The decreased phase lead is
best accounted for by greater subject dead-time delay.
The pilot attempts to track the oscillations that occur
slightly below the bending mode natural frequency, however
pilot response power decreases sharply at and above the
natural frequency.
The RMS ratio of attitude error to disturbance signal
substantiates quantitatively the increase of attitude
error with respect to bending mode amplitude.
The subject performance is much poorer moving base than
fixed base for all values of bending mode amplitude.
The
pilot exerts approximately the same control power in both
cases.
This results in a lower moving base gain.
31
Fixed base experiments including second order simulator
dynamics and dead-zone plus the moving base display and
control did not account for a major portion of the difference.
The best explanation seems to involve a combination
of neglected simulator non-linearities, a poorly marked
and less sensitive moving base display grid, and perhaps
vestibular confusion and insensitivity to very small
deflections.
The rate of increase of the attitude error with respect
to bending mode amplitude was less rapid moving base
compared to fixed base.
The results were attributed to
the simulator dead-zone and not to vestibular effects
because the fixed base plus simulator dynamics results
show a rate equal to that for moving base experiments.
For a given amplitude the bending mode with natural
frequency of 5 rad/sec creates one-half the acceleration
Equal amplitudes for the two cause
of the 7 rad/sec mode.
an approximately equal performance deterioration.
Possible spurious effects due to intersubject variance,
subject knowledge of bending mode amplitude and order of
occurance, and fatigue were checked and not considered
important.
For a given natural frequency and non-zero amplitude,
the RMS ratios,/
.
were typically 10% higher for the
case of the low RMS disturbance signal compared with the
32
high RMS signal.
The high RMS value was twice that of
the low.
Highly simplified versions of two proposed vehicle
augmentation and control stick filter schemes were
studied experimentally.
The results favor the one with
stable missile rigid poles and single .integration stick
filter judged on the basis of attitude error performance.
Both eliminated bending mode oscillations.
Because the single integration filtered out much more
of the subjects response, the maximum available control
The
power was arbitrarily increased by a factor of three.
attitude error difference may be due to this change.
A relatively low frequency first bending mode decreases
pilot lead compensation and closed loop attitude performance
in addition to creating structural problems.
Significant
effects develop at bending mode amplitudes equal to onethird that sensed at the Saturn V attitude station.
Stick
filters apparently solve the structual problem, but add
phase lag in series with typically difficult dynamics.
This requires effective stabilization of the vehicle
dynamics to alleviate the pilot control problems in the
presence of disturbance signals.
33
APPENDIX A
DERIVATION AND SIMPLIFICATION OF SIMULATION EQUATIONS
The equations of motion, parameters values, and
following- diagram are taken from Ref. 1.
The equations of
motion are linearized and valid for small perturbations about
the booster's nominal trajectory.
The equations are written
with respect to a coordinate system moving at the booster
velocity along the trajectory.
VEuIcL
NOMIANA . To hTt CTW
CL
AWrLA4 F03
'
NOMIlNAL AMITIt-a
The rigid body equations are listed
x=-
F a - F
rb
$rb =M a
=
Y rb =-F a
(A 1.0)
rb
(A 2.0)
-M
$rb +
below:
57.3
(
(A 3.0)
Vx
- FSS -
..
57.3
$rb
(A 4.0)
Differentiating and rearranging equation(A 3.0)gives:
rb 57.3
(A 5.0)
34
After substitution into (A 1.0):
V
(cx- $rb 57.3 =
FF
~
-F
-F
(A 6.0)
Frb
x
Differentiating and rearranging equation
1
9
"'
(A 6.0) results in:
Substitution into
1
*
M
0A
(A 7.0)
($rb + M6)
cx=
m
(A 2.0):
rb + Ra-
6
$rb) 57.3
F
M
Rearranging
F,.
+,,
rb
V
rb + F $rb
rb ~
57.3
M
M
=-
M
-F66
M 57.3
8.0)
(A 8. 0):
v
57.3 M
(A
rb +rb
(A
9.0)
(A
10.0)
Fa
Using LaPlace operator notation:
m
rb _ _
M
vm
Ot
57.3 p + F9 + F
F
V
(57.3 M
(p + M
p3 +
012
)
-
m
57.3 + F
5
57.3
3
p
+
573
57.3 M
(p3 + 57.3 Fa P2 - M p +, .-.)
V
ax
V
F
(A 10.1)
35
At maximum dynamic pressure the parameters have the
following values:
2
Ma
=
.141/sec
F
= .36 meters/sec /deg
M
= 1.15/sec
F
= .13 meters/sec 2 /deg
V
=
F
=
486
m/sec
.30 meters/sec
After substitution of these values
rb
-- =-
1.15
(p +
1.15
(A 10.1) becomes:
.-
.I 4 p +
.0060)
(A
10.2)
.04)
(A
10.3)
(p + .02)
(p -
Considering only (
1 and
.34)
.4)
(p +
1.15
(pr34) (p +
.40)
(p -
for p = jw
. (p + .02)
For simulation purposes
rb
/deg
.02)
(p 3 + .0153P
= -
2
-
L(p
>
-
.4 rad/sec:
.04)
00
(A 10.3) was simplified accordingly:
.40)
(A
11.0)
(A
11.1.)
1.15
p
-
.15
If'rate compensation is added, the feedback polarity must
be as shown to decrease the instability:
36
4
-1.157
tip.
t
Then:
1.15
4
rb
.75p(l.15)
_
p -. 15
p 2 -. 15
(A
rb
12.0)
After rearrangement:
2
I b (p
+ .86p -
.15)
=
-
1.156
(A 13.0)
1-.15
rb
(p
2
+
.
8
6p -
(A 14.0)
.15)
1.15
(p + 1.00) (p
-
.15)
(A
14.1)
The following equations describe the flexible body mode:
0
De
2
+
2
= Kjj
Ewnbd1 + W nbdn
Obd = K 3 1 n
+ K21a
(A 15.0)
(A 16.0)
Combining and rearranging:
2
K 31K 2
+bd
_
_
=
(p
1
(P. +
K
)
+ 2 w nbdP + w nbd)
(A 17.0)
37
The values for these parameters at maximum Kynamic
pressure are:
K1
=
.46/deg-sec
K2 1
=
.00077/deg
K
=
8.6 deg
3 1
.005
=
Wnbd =-7.33*rad/sec
After substitution:
+
(P2
K(2 +W2)
+ (24.4) )
.0066(p
bd
.01(7.33)p
+
2
(7.33)2)
P
P
For convenience, wz was made equal to 3w
values of w
(A 18.0)
+wz)
p+LO 2)
(p +.0l
-K(p
for all
under study.
The total attitude error,
ktotal' equals the sul of the
rigid body and flexible body contributions, therefore:
f3
K(p2+(3wnbd
1.15
o-
=
(p+1.00) (p-.15)
-
1.15(p2+
nbd
numerator.
locus
19.0)
nbd) + K(g +( 3 wbd) ) (p+1.00) (p-.15)
(p+1.00) (p-.15) (p +.Olw
The root
(A
2
2
(p +.Olnbdptwnbd
technique was
nbd
p+
b
nbd
(A
used to factor
20.0)
the
A separate root locus plot appears for both
bending mode frequencies,
(see Fig. Al and A2).
The
numerator must be written in the following form to apply
the technique:
l. 15 (p
1
-
2
+
2
.OlnbdP + wnbd
=
K (p2
+
(3wnbd) )(P
-
.15)
(p + 1.00)
0
(A
21.0)
38
From inspection, the 0
criterion must be used, and
the root locus gain equals 1.15/K.
Roots have been located for the appropriate values of
The results are listed in Chapter II
K on the two plots.
in transfer function form.
The root locus gains correspond-
ing to bending amplitudes are listed below:
K
K Root Locus
.0022
522.7
.0066(nom.)
174.2
.0132
87.1
.0264
43.6
The root locus
gain is related to the spirule reading
accordingly:
=
K
(pirule:
reading)
1P1 P 2 P 3 P
S
(A
22.0)
where S is scale factor equal to the number of radians/sec
represented by 5" on the plot and pi and q
are the loca-
tions of the poles and zeroes.
For
J
w
F r=
I(Q(-5
)5,
.
1(-.15) (1.00) (15
1P1P2P3P4
) (-15 J)
49
1(7 i ) (-7 i) I
1
'
1pip2
(-.15) (1.00) ( 2 1 ) (-21 )1
3p4
In this case S =
20, so
25
33.75
66.15
*74
the spirule readings and the
bending mode amplitudes have the following relationship:
(spirule reading)
=
IP PP
3
2 p.
I 1q 21
1
S p-q
1.15
K
K
(A
23.0)
Finally:
Spirule Reading
K
.0022
1.77
.0066
.59
.0132
.29
.0264
.15
Jex and Teper augment the dynamics of the Saturn V
by feeding back both attitude rate and position.
The sum
of the control stick filter output and the feedback signal
passes through two first order lags with break frequencies
at 5 rad/sec.
This filtered signal provides the command
to the engine gimbal.
The complicated closed loop transfer
function including two bending modes as taken from Ref. 2
appears below:
Sattitude gyro
_
(A 24.0)
stick filter
4,900(s+.019)(s+4.5)(s-4.6)( 0.0050
12
(s+0.035) (s+.62) (s+6) (s+30) (0.56
2
where
(
0.30
30)
denotes
2
(p
0.038
7.2
00066)
21
0 018
0 30
12
30
2
+ 2(.30)30 p + 302
The dominant rigid body modes are now stable and have a
natural frequency of 2 rad/sec and a damping ratio, C * .56.
In order to reduce the complexity of this set of
dynamics to a level equal to the previous work, the two
dominant rigid body poles were placed in parallel with the
simulated first bending mode.
40
The extreme simplification not only reduces the
complexity, but also, the difficulty of the dynamics
proposed by Teper and Jex.
However, in spite of this fact, a first approximatidon
to a realizable set of stabilized Saturn V rigid body
dynamics ha s been selected.
The new set of missile dynamics has the following
form:
tot
4k
p
+
+ 2(.56)2p + 2
.0066 (p2 + 212)
(p
(A 25.0)
+ 2(.005)7p + 7
The natural frequency of the bending mode was placed
at 7 rad/sec and K = .0066 = nom. to correspond closely
with the Saturn V design conditions.
remaining parameter, k,
ments with this system.
discussed in Chapter IV.
k = .85, equation
tot
The value of the
was selected after initial experi-
The basis for the choice is
With the selected value of
(A 25.0) becomes:
3. 4 (p 2 + (.01) 7p+7 ) + .0066 (p +21
(b +2(.56)
2p+2 ) (p +(.01)7
)(p +2(.56)2p+22
p + 7 )
(A 26.0)
After placing the numerator in a form suitable for solution
by the root locus technique:
41
I
-
(p 2 +
3.4
.0066
(p
2
(.01)7
p + 72)
+ 21 ) (p2 + 2(.56)2
= 0
(A
27.0)
p + 2 )
After solution of the root locus:
.0066 (p-li) (p+15.5) (p +(10.5) 2)
$tot
6
2 +2
(p2 +2(.56)2p+22)(p2+.0l(7)p+7 .)
(A 28.0)
42
APPENDIX B
PILOT DESCRIBING FUNCTION MEASUREMENT TECHNIQUE
The pilot describing functions were computed by the
following equation:5
(13 (W)
Y p(s)
=
12 (W)
(Bl.0)
where:
D13(w)
= cross power spectral density of the input
disturbance signal and the operator's
response-
*12 (W) = cross power spectral density of the
input
disturbance signal and the displayed error.
The operator's response contains a part correlated
with the input disturbance signal and an uncorrelated
remnant.
The degree of correlation determines how well
a describing function accounts for his behavior.
This correlation is measured by the following ratio:
2
3
2
(
(B 2(W)
43
where:
011(w)
= input disturbance signal power spectral density
0 3 3 (w) = operator's response power spectral density.
If p2 is near unity the describing function is a
close approximation to the operator's behavior.
A program, written by the staff of Health Sciences
Computing Facility, UCLA, and made available to the ManVehicle Control Laboratory by Ames Research Center, NASA,
was used to compute the necessary power spectrums.
The
program was modified in the Man-Vehicle Control Laboratory
to compute the describing function by (Bl.0).6
This method computes the power spectral estimates of
an analog signal T seconds
long by sampling every AT
seconds.
A total of M = T/AT points are available for computation.
The correlation function
O(T)
of the sampled signal
is computed for m lags of AT.
Under these conditions:
1.
The sampled data will have no spectral power
above a frequency whigh where:
Whigh
2.
(B3.)
=
The spectral density will be computed at m
equally spaced frequencies between 0 and whigh.
the frequency resolution will be:
AW
=
m1 A
Hence
44
3.
The probable error of the computed spectral
error will be:
If N independent spectral densities are saveraged, the
probable error is reduced to:
For this work the following values were chosen:
T = 70 sec
AT = 0.1 sec
M = 70 0
M = typically 5
For these values:
Whigh
hiW
-
Aw
5 cps
.07 cps
M = 700 points
E =
5(700)'
.14
45
BIBLIOGRAPHY
1.
Hardy, G. H., West, J. V., and Gunderson, R. W.,
"Evaluation of Pilot's Ability to Stabilize a Flexible
Launch Vehicle During First-Stage Boost," NASA
TN D-2807, May 1965.
2.
Jex, H. R. and Teper, G. L.,
with author..
3.
Lukens,
D.
R.,
Schmitt, A.
Personal communication
F.,
and Broucek, G.
T.,
"Approximate Transfer Functions for Flexible-Boosterand-Autopilot Analysis," WADC TR-61-93, April 1961.
4.
McNemar, Q., Psychological Statistics, John Wiley and
Sons, Inc., 1962.
5.
D,. T. 'and Krendel, E. S. , "Dynamic Response
McRuer,
of Human Operators," WADC TR56-524, August 1957.
6.
Meiry, J. L., "The Vestibular System and Human
Dynamic Space Orientation," Doctoral Thesis, ManVehicle Control Laboratory, Massachusetts Institute
of Technology, June 1965.
7.
Teper, G. L., and Jex, H. R., "Synthesis of Manned
Booster Control Systems Using Mathematical Pilot
Models," Sixth Annual Symposium of the Professional
Group on Human Factors in Electronics, IEEE, May 1965.
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