THEORY OF AVIATION GYROSCOPIC INSTRUMENTS

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• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R002000020008-5
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 384
• Document Creation Date: December 27, 2016
• Document Release Date: May 16, 2013
• Sequence Number: 8
• Publication Date: January 1, 1956
• Content Type: REPORT

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AIR TECHNICAL INTELLIGENCE
TRANSLATION
(Title Unclassified)
THEORY OF AVIATION GYROSCOPIC INSTRUMENTS
(Teoriya Aviatsionnykh Giroskopicheskikh Priborov)
Author: A. S. Kozlov
Source: State Publishing House For The
Defense Industry
Moscow
1956
256 Pages
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A .S .KOZLOV
THEORY OF AVIATION GYROSCOPIC INSTRUMENTS
Approved by the Central Administration
of Higher Polytechnic and Machine?Building
Schools, Ministry of Higher Education USSR,
as a Text Book for Higher Aviation Schools
State Publishing House for the Defense Industry
Moscow 1956
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This book gives the principles of the general theory of
gyroscopes. The use of gyroscopes with displaced center of
gravity to indicate the vertical and meridian is discussed. The
principles of designing gyroscope correction systems are stated.
The behavior of corrected gyroscopes on fixed and moving bases,
and the theory of gyro horizons, gyroscopic course indicators
and rate gyroscopes is discussed. Fundamental information is
given on power gyroscope systems.
The book handles separately questions related to the prin-
ciples of constructing correction systems and the behavior of
corrected gyroscopes on fixed and moving bases, as questions
common for all gyroscopic instruments. Particular attention is
paid to questions connected with the use of gyroscopic instru-
ments in flight: errors of instruments in flight, longitudinal
accelerations, in turns and other maneuvers, as well as methods
of diminishing these'errors.
Reviewers: B.A.Ryabov, Doctor of Technical Sciences,
S.S.Tikhmenev, Doctor of Technical Sciences
Scientific Editor, M.S.Kozlov, Candidate in Technical
Sciences
Editor-in-chief,
Engineer A.I.Sokolov
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PREFACE
The study of the aviation gyroscope has developed widely during the past 10-15
years. If we speak of the main trend of this development, it consists in the effort
to increase the reliability of operation and increase the accuracy of gyroscopic
instruments and units. This trend in development has been due to the continuous in-
crease of the range, speed, and altitude of flight. As has been shown by experi-
ence, the most effective way of improving the reliability of operation and increas-
ing the accuracy of gyro instruments and units is the development of the aviation
electro-gyroscope, a fact that was well understood by our designers and production
men, who, as far back as 1936-1937, produced in series production, completely elec-
trified directional gyroscopes for use in autopilots, where the demands on the reli-
ability of operation and accuracy of reading are particularly stringent.
The development of USSR aviation gyroscopy is based on the works of the Russian
classical school of mechanicians, Soviet scientists and specialists, who have creat-
ed the theory of gyroscopic instruments, which has been formed today into an inde-
pendent scientific discipline. First of all we must point out the fundamental works
of Academician A.N.Krylov (Bib1.1) and B.V.Bulgakov (Bib1.2), corresponding member
AN SSSR. These works, in the richness of their content and the combination of pro-
found theoretical development of the questions with an engineering approach to the
problems under consideration still remain unexcelled. It would not be an exaggera-
tion to say that the modern Soviet school of gyroscopists has grown up and is ?
developing precisely on the basis of the ideas that have been worked out and
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developed in these works.
It must, however, be noted that a number of textbooks and manuals on aviation
gyroscopy possess the shortcoming of being based on the description of individual
gyroscopic instruments without an appropriate methodological generalization. This
is all the more to be regretted because the above mentioned works by Krylov and
Bulgakov and a large number of works by other Soviet authors, which will be cited in
the present book, do provide adequate material for the generalizations and develop-
ments of the theory of aviation gyroscopic instruments that are necessary from the
pedagogical and methodological point of view.
An attempt has been made in this book to fill in this lacuna.
In the compilation of this book the works of the late A.S.Kozlov on the theory
of aviation gyroscopic instruments, published by VVIA imeni Zhukovskiy, have been
utilized.
In connection with the development of the theory and technology of gyroscopic
aviation instruments, during the last few years, it has become necessary to expand
the theoretical point of the book and to include in it a number of questions devoted
to the principles of operation of the most recent instruments. This work was done
by 11.S.Kozlov.
Professors V.A.Bodner, A.A.Krasovskiy, and G.O.Fridlender, together with in-
structor B.V.Komotskiy, took part in preparing this book for the press.
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INTRODUCTION
Of the instruments based on the utilization of the properties of the gyroscope,
the following are used in aviation:
1. Indicators of the aircraft position with respect to the earth, i.e., indi-
cators of the angles formed by the aircraft axes with the corresponding axes bound
to the earth. Such instruments include gyroscopic course indicators and gyrohori-
zons.
2. Indicators and devices for measuring angular velocities and rotary accelera-
tions of an aircraft about its axes. These instruments include the widely known
"turn indicator", that is, an indicator of angular velocity of rotation of an air-
craft about its normal axis. This group also includes devices for measuring angular
velocities and angular accelerations of the rotation of an aircraft about its axes,
which are used in autopilots.
Common to all gyroscopic instruments is the presence in each of them of a
rotor, rapidly rotating or vibrating, and possessing a sufficiently high moment of
inertia with respect to its axis of rotation or vibration. The properties inherent
in an instrument with such a rotor, and the phenomena thereby caused, will be termed
gyroscopic, and the rotor itself a gyroscope.
The rotor is caused to rotate either by means of an air jet, or by electrical
energy. Accordingly, gyroscopes are divided into pneumatic and electrical.
The rotors used in gyro instruments usually have three or two degrees of free-
dom, which are provided by the corresponding gimbal suspension with two frames or
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with a single frame.
In instruments of the former group, that is, in gyro angle-measuring instru-
ments, the frame of the gimbals, which serve as bearings for the rotor spindle, is
called the inner frame, while the frame serving as the bearing for the spindle of
the inner frame is termed the outer frame of the suspension. By adding to the num-
ber of degrees of freedom of the gimbals, still another degree of freedom of rota-
tion about the axis of the rotor, we obtain the total number of degrees of freedom
for the material parts of the rotor equal to three.
A rotor in gimbals is therefore usually called a gyroscone with_three degrees_
of freedom.
If in such a gyroscope the center of gravity of the system coincides with the
center of the hearing, then in this case it is called astatic, since the force of
gravity will have no effect on the position of the axis of its rotor. If in addi-
tion we assume the absence of friction in the bearings of the gimbals, and the
absence of systems imposing moments on the gyroscope as its position varies, then we
get a gyroscope that is usually called a "free gyroscope, since such an instrument
will remain free from the influence of any forces and moments, not only in any posi-
tion of the rotor axis, but also under any variation of the position of the base of
the suspension.
In instruments of the second group, which are designed to determine angular
velocities or to measure than, the suspension of rotor has one gimbal, the displace-
ment of which is resisted by a spring. lie shall term such gyroscopes, gyroscopes
with two degree of freedom'.
In an instrument designed for the simultaneous measurement of angular velocity
and angular acceleration, which are used in autopilots, the rotor gimbals have two
*They are sometimes called uprecessional", but we shall not use this term,
qince every gyroscope is, in fact, "precessional".
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frames, the displacement of which is resisted by a spring.
In instruments designed for the measurements of angles, moments are imposed on
the gyroscope with the object of giving the rotor axis selectivity with respect to
the direction to be determined by means of the given instrument.
This object is accomplished either by using the force of gravity of the gyro--
scope, i.e., by a corresponding displacement of the center of gravity with respect
to the point of support, or else by means of a system specially added to the gyro-
scope, which is termed a correction system.
According to the type of energy used in the correction system with the object
of producing the positional moments, these systems are divided into mechanical,
aerial, and electrical.
The problem of aviation gyroscopy has been and still remains, primarily connec-
ted with the need for determining the position of an aircraft with respect to the
earth axes, the geographic meridian and the vertical of the place, whether for
visual purposes or to make piloting automatic. In other words, this is the problem
of the artificial gyro horizon, i.e., of an instrument indicating the direction of
the geographic meridian.
Long before the need of such instruments for aviation was first felt, instru-
ments used for this purpose existed and were employed in marine navigation. But the
principles on which the marine gyrocompass (pendulum gyrocompass) and the marine
gyrohorizon (gyropendulum vertical) were based proved inapplicable for aviation.
This was due to three circumstances: the great accelerations in aircraft, the
unsuitability for the conditions of installation on an aircraft, of the dimensions
and weights of the pendulum gyrocompass and the gyropendulum vertical, and the dura-
tion of the transient states inherent in these instruments, a duration that made
them inapplicable to aviation conditions.
Other methods of determining the direction of the vertical and meridian also
exist. They are the use of a physical pendulum of one form or another to determine
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the vertical, and the use of the magnetic needle to determine the course. It must
be noted that the application of the magnetic needle, that is, of the magnetic com-
pass, has found its place in aviation, and has not lost its importance down to the
present day. But by itself it does not exhaustibly solve the problems of the avia-
tion compass. The inadequate stability of the compass card, the strong influence on
magnetic compass of turns, and of acceleration in general, the influence of units
with steel parts, varying their position with respect to the compass card with vary-
ing course of the aircraft, as well as the influence of stray magnetic fields due to
electrical and radio equipment, all these lead to the necessity either of replacing
or of supplementing the magnetic compass by a gyroscopic instrument free from the
above enumerated shortcomings of the magnetic compass. As for the use of the physi-
cal pendulum on the aircraft to determine the direction of the vertical, this did
not give a good account of itself at all, although attempts of this nature were,
indeed, made; in other words, the solution of the problem of building a gyroscopic
artificial horizon suitable for aviation practice was the only method of solving the
1)roblem of determining the direction of the vertical on aircraft.
The necessary solution of this problem and of the problem of the aviation gyro-
compass was found by using systems of correction for directional gyroscopes.
This path assured, first of all, the procurement of accuracy that was satisfac-
tory enough with the incomparably smaller weights and dimensions than are demanded
by the pendulum gyrocompass and the gyropendulum vertical. The question of the
duration of the transient states was also solved satisfactorily by this method. For
this reason gyro horizons (aviation horizons) with a correction system, and gyro-
compasses (course compasses) with a correction system, became parts of the indispen-
sible set of aircraft equipment and constitute the basis of the sensing system in
automatic pilots.
It is therefore natural that the theory of such gyroscopic instruments should
in ortant for aviation technology. It may be said in general that
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the theory of aviation gyro instruments is primarily the theory of directional gyro-
scopes with correction systems.
It is the exposition of questions of this theory that constitutes the task of
this textbook. While during the first stage rigorous demands were mit made on the
accuracy of the readings of gyroscopic directional instruments, as the range and
speed of flight increased, and as certain new problems appeared, these requirements
did increase.
The use of electric power to maintain the rotation of the rotor and for the
purposes of correction only partially satisfies the increased demands on the
accuracy of the readings of gyroscopic directional instruments. The point is that
the loads that are taken by directional gyroscopes have an unfavorable effect on
them. It is necessary to watch out for possible reduction of the reactions in such
gyroscopes, owing to the unavoidable connection with the indicating system or with
the automatic control system. And yet we have not been able to eliminate the harm-
ful action of these connections entirely.
As has been shown by the works of Soviet scientists, and, first of all, by B.V.
Bulgakov and his pupils (Bib1.13), a radical
ed by the power-stabilized multigyro systems
position in gyroscopic aviation engineering.
solution of this problem may be provid-
now beginning to occupy a prominent
We
have therefore
found it necessary
to dwell on a number of questions connected with such systems.
Gyroscopic measuring systems for the angular velocities of rotation of an air-
craft, termed by us gyroscopes, play a very important role in aviation.
A representative of this type of instruments is the widely known "turn indica-
tor". In reacting to the angular velocity of rotation of an aircraft, this instru-
ment, as it were, might be said to warn the pilot that the aircraft is entering a
state Of violation. Such instruments are of no less importance for automatic pilot-
ing devices. Their use as auxiliary sensing elements in automatic pilots makes it
possible to obtain artificial damping of the motions of the aircraft, which is of
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very essential importance from the point of view of assuring the necessary quality
of automatic piloting.
The acceleration-velocity gyroscope, which measures the angular velocity and
the angular acceleration of rotation of an aircraft, is also very important for the
automatic pilot.
This book also gives the principles of the theory of velocity and acceleration-
velocity gyroscopic instruments.
We shall now discuss certain historic moments in the development of aviation
gyroscopy of the USSR.
At the time when the problem of aviation gyroscopy had sufficiently matured,
the USSR already 2o,Pssed the necessary cadres for the solution of that problem.
The reason for this was that although there was no gyroscope industry, in the strict
sense of the word, in pre-revolutionary Russia, still the problems of E;yroscopy (and
in particular of aviation 7yroscopy) even then had attracted the attention of a
number of Russian scientists and inventors.
We might mention that as early as 1911 N.Ye.Zhukovskiy occupied himself with
these .rolilems, and precisely in the interests of aviation (13ib1.6).
The first aviation gyroscopes, in which the rotation of the rotor and the
correction were accomplished by pneumatic action; were developed and built in 1928-
1929 in the USSR for use in the automatic pilot.
The creation in 1936-1937 of completely electrified directional gyroscopes with
a correction system was a substantial achievement of Soviet designing thought and of
its aviation instrument industry. This work was performed in connection with the
creation of a combletely electric automatic pilot in these years, and it was the
princi-)al element in the solution of the problem of the electric automatic pilot.
In the problem of the all-electric positional gyroscopes itself, the principal ques-
tion was that of the electrical correction system. This question was solvedpn the
principles of the inductive correction mechanism proposed by Soviet inventors.
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The recent work of Soviet g-yroscopists in the field of electric aviation gyro?
scopes has been particularly successful, and as a result we now possess a. full set
of USSR electrouroscopic board instrwnents.
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CHAPTER I
GYROSCOPIC PHENOMNA AND PROPERTIES
Section 1.1. Visible Gyrosconic Phenomena
Consider the astatic pyroscope (Fig.1.1). The rotor, in the gimbals consisting
of the inner frame 3 and the outer frame 4, is called a gyroscope; in the astatic
gyroscope the center of gravity of the gyroscope coincides with the center 0, the
Point of intersection of the axes of the gimbal frames.
So lonr, as the rotor of the gyroscope is not rotating, we observe no phenomena
in this instrument that distinguish it from an ordinary nongyroscopic body, by which
we shall here and hereafter understand a body that does not possess a moment of
momentum before it is subjected to an exPeriment.
- Astatic Gyroscope
Thus the rotation of the base of
the gyroscope suspension 1 leads to the
variation in the position of the polar
axis of its rotor Oz, owing to the in-
fluence of friction in the suspension.
A moment applied to any of the frames
of the Cardan suspension leads to the
rotation of that frame about its axis
of rotation. A tap on the frame also
causes it to rotate in the direction of the tap.
Let us out the rotor of the astatic gyroscope in sufficiently ranid rotation
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about its polar axis Oz. We shall call this rotation the spin of the gyroscope
rotor. It is the presence of this spin that marks the transformation of this device
from a nongyroscopic body into a gyroscopic one. Let us then repeat all the opera-
tions that we performed with the rotor at rest. We shall now find the following
phenomena.
The Phenomenon Gyroscopic Rigidity
The gyroscope rotor axis acquires "rigidity": when the base of the suspension
rotates, the variation in the position of the polar axis Oz is unnoticeable,
although the friction in the axes of the gimbals, which have already varied its
position, still continues to act. Similarly, a tap on the gyroscope frame, with
force that would have been sufficient formerly to make that frame rotate several
times, now produces no visible effect.
The Phenomenon of Precession
The character of the motion of the gyroscope under the action of an applied
moment now changes: a moment applied to the outer frame of the gyroscope causes it
to rotate about the axis of the inner frame, and-, on the other hand, a moment
applied to the inner frame, causes the gyroscope to rotate about the axis of the
outer frame. When the direction of the applied moment is reversed, the sense of
rotation of the frames is likewise reversed.
This rotation of the gyroscope about the axes of its frames is called the pre-
cession of the gyroscope.
If we study the sense of the gyroscope rotor spin and the sense ofithe applied
moment, we obtain the following law of the sense of precession of the gyroscope:
the precession of the gyroscope w tends to make the vector of angular velocity of
pr
the rotor spin rt coincide with the vector of external moment L causing this pre-
cession (Fig.1.2).
It is easy to establish by further observation that, in addition to the
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Itextraoidinary" relation between the applied moment and the sense of rotation, pre-
cession also possesses other pronerties distinguishing it from the motions inherent
in nonzyroscopic bodies.
These peculiarities are as follows:
1. To a definite magnitude of the applied moment, at a definite angular velo-
city of spin, corresponds a definite value of the angular velocity of precession.
In nonfzyroscopic bodies, to a definite value of the applied moment there corresponds
a definite value of the angular acceleration.
As a consequence of this, with constant applied moment and constant angular
velocity of the spin, thc anFular velocity of precession will also be constant. In
nongyroscopic bodies, with a constant applied moment, the anular velocity of rota-
tion will increase.
ut from this it follows in turn that, in spite of the presence of a constant
applied moment, the energy of the gyroscopic system in the latter case still remains
unchanged, and this is natural enough: the plane of precession produced by the
anlied moment is perpendicular to the 'lane in which this moment acts. In other
words, the applied moment causing the precession does no work, that is, precession
is a motion nerformed without the expenditure of energy.
2. .lith cdnstant value of the applied moment, the anrtular velocity of preces-
5-Ion dIninishes with increasin7 angular velocity of the rotor snin, and increases
wl_th decrease in that anguJar velocity.
At constant anvlar velocity of rotor spin, the angular velocity of precession
increases with increase in the a-;;3ied moment, and decreases with decrease in the
an-,lied moment.
J. The value of the angular velocity of precession corresponding to given
va)ues of the applie,' moment and angular velocity of the rotor spin appears, instan-
taneously, wAh a jumo, on application of the moment; and in the same way, instan-
taneously and with a jumn, it disappears when the moment is removed.
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In this way, on variation of the value of the moment, the corresPonding varia-
tion in the angular velocity of precession takes place without a lag.
In other words, precession is ninertialess".
This peculiarity of precession is, however, in contradiction to the argument
that the energy of the system cannot vary instantaneously on application of a
moment, although the observed appearance of precession in this case means precisely
the instantaneous variation in the energy of the system.
It follows from this that the ninertialessnessu of precession is only an appar-
ent phenomenon, and that, in addition to precession, the applied moment must cause
some other motion as well, which is not perceptible by observation, during which the
moment does perform work assuring the increase of the energy of the system by the
value of the energy of precession. This motion, which is termed nutation, will be
considered in Sections 2.5, 2.6, and 2.7.
Consider the behavior of a wheel with a handle, so constructed that its shaft
is attached on bearings to the handle.
Let us impart a spin to the wheel, thus converting it into a gyroscopic body.
On the basis of the phenomenon of gyroscopic precession, established by observa-
tions, these motions would be expected to take place in a plane perpendicular to the
plane of action of the moment L of the force of gravity 'd (Fig.1.3). And this
actually does take place: when suspended on a thread attached to the 'handle, the
wheel does not fall, but rotates in a horizontal plane, and exactly in the same way,
when the handle is placed on a table, the wheel does not fall, but rotates, with
its axis describing a cone in space.
The Phenomenon of Gyroscopic Reaction
As we have seen above, an external moment causes a precessional moment in a
plane perpendicular to the plane in which the moment acts. According to the
DiAlambert principle, a system of external forces or moments acting on a body causes
with accelerations such that the system of inertial forces or
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moments is equal and opposite to the system of external forces and moments.
It follows from this that in precessional motion, the points of a body also
4
- Precession of Gyroscope Fig.1.3 - Precession under the Action of
the Moment of the Force of Gravity
move with accelerations. The moment of the forces of inertia due to these accelera-
tions is called the gyroscopic moment or moment of the gyroscopic reaction L.
(Fig .1.4)
Section 1.2. Relation between the Visible Gyroscopic Phenomena
The moments of friction in the suspension produce in the gyroscope at suffi-
ciently high sninning speeds so small a rate of precession that the deflection of
the rotor axis of the 'gyroscope rotor during a short time of observation is prac-
tically unnoticeable.
a? a
Pr
The angular velocity of precession
arising as a result of a tap on the
gyroscope, or, what is the same thing,
of the application to the gyroscope of
a finite external moment during a
negligibly short interval of time, may
have a considerable value, but the
angle of deviation of the rotor axis
due to this precessional motion will be practically imperceptible, since the time
Pi'-.1.4 - Relation between Precession
r
and Gyroscopic Reaction
ier the shock is negligibly small and equal to the time of applica-
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tion of the external moment.
Section 1.3. Physical Origin of the gyroscopic Reaction
As already remarked, the gyroscopic moment is the moment of the inertial forces
due to the accelerations that appear on the simultaneous existence of two rotations
of the gyroscope, rotation about the polar axis of the rotor, and rotation of the
axis itself.
Let us find these accelerations.
Let us take a rotor rotating about its polar axis at anoilar velocity ry.
Let us reproduce on the drawing
(Fig.1.5) the equatorial plane of this
rotor, by which we mean the plane per-
pendicular to the polar axis of the
rotor and passing through the center of
P, the support, that is, through the fixed
F- noint of the gyroscope. Let Ox and Oy
WI
Fig.1.5 - Velocity of Particles
of Rotor
be the mutually perpendicular axes ly-
ing.in the equatorial plane', but not
taking part in the spin 74. Let there
be, simultaneously with the' rota-
tion ri, a rotation about the axis Ox at angular velocity p. Let us consider
whether the particles of the rotor, in this case, will move with any acceleration
due to the gyroscopic moment. Let us assume, for simplicity of the reasoning,
that r? = const and p = const, that is, let us take a case when angular accelera-
tions are known to be absent.
The velocity Vi of any of the material points of the rotor is made up of the
component ui, perpendicular to the equatorial plane of the rotor, and the component
v. which lies in this plane (Fig.1.5).
Consider the behavior of the component u1, under the combined rotation of the
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tluns j,, w-111 he directed to the ri;.:ht in the first and fourth quadrants and to the
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--rroscw:le about the axes Oz and Ox.
At a ziven value of p, the magnitude of this component is determined, for each
nater'al particle of the rotor, by the distance of that particle from the axis Ox.
- ,7ehavfor of Components of
7elocity caused by Rotation alo:It
t"e Dquatorial
This distance, under the influence
? .
of the spin rt, Increases in the
first and third quadrants
(Fig.1.6), and decreases in the
second and fourth quadrants.
Therefore, during the interval of
time At = tu - t?2 the velocities
of the material particles travel-
in rz over certain areas in the
first and third quadrants, will
increase in modulus from the
values ' and'i.?to the values
lit 13
u."andl." rcsnectively.
11 113
This means that the particles
no7in- in the first and third quadrants will move at accelerations (jii)u and
dIrected, as indicated in that is, in the same sense in which the
velocities are directed in these quadrants.
Durinr: the same interval of time, the velocities ui of the material particles
passinc over certain areas in the second and fourth quadrants will decrease in modu-
'ds from the values
and '1,4' to the values u12" and ui4" respectively. This
-leans that the particles traveliruf in the second and fourth quadrants will move at
accelerat:ormO)and0.4)directedoppositetonevelocdtYu-in these quad-
i2 u 1
Start]nr: out from this, we ;et the result that in this case the accelera-
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left in the second and third.
Considerthebehariorofthecomponentw.when the gyroscope rotates simul-
taneously about the axes Oz and Ox.
Fig.1.7 - Behavior of the Components
of Velocjty Due to Rotation about
the Polar Axis
Under the influence of p, the
component, w. reverses its direction.
In order to represent more d.istinctly
the result of this change, let us re-
solve with respect to the axes Ox
and Oy (cf.Fig.1.7). It is easy to see
that the component w. of any particle
lx
does not change its direction under the
influence of p, which direction remains
parallel to itself, but that the com-
ponentw.receives an increment under
iy
the influence of p, causing a change in
the direction of the components TL '
iy
these increments being directed toward
the right in the first and fourth quadrants, and to the left in the second and third
quadrants. The accelerations j. with which the particles move will be directed in
the same way, that is, the accelerations jiv are in the same sense as the accelera-
tions jiu for all particles of the rotor (cf.Fig.1.7 and 1.8). Whence the total
acceleration ji with which the particles move, will be determined by the expression
Both these components appear, as we see, as a result of the simultaneous rota-
tion of the rotor about two axes, the polar axis and one of the equatorial axes;
one of these rotations gives rise to the corresponding component of the velocity of
the particles, while the other one produces the variation of this component in
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magnitude and direction.
Consequently the total acceleration is also the result of the simultaneous ro-
tation of the rotor about the two axes, the polar axis and one of the equatorial
IIIaxis.
?
It is commonly known that accelerations of this kind are termed rotational
(Coriolis force). These accelerations produce the forces of inertia fi directed
opposite to these accelerations
(cf.Fig.1.8). These forces of inertia
rib-
produce a moment of inertia directed,
as indicated on Fig.1.8. It is easy to
.0 I
st
- Direction of the Rotational
A'
see that this direction coincides
exactly with the direction of the gyro-
scopic moment. It may be asserted, on
this basis, that the gyroscopic moment
is the moment of the forces of inertia
due to the rotational accelerations
with which the particles of the gyro-
Accelerations and Forces of Inertia
scope rotor move when it rotates simul-
taneously about two axes, the polar axis and one of the equatorial axes.
Section 1.4. Mac,nitude of the Gyroscopic Moment
The component of the velocity of the particles ui (Fig.1.9) may be represented
by the acpression
sin i.
where Ri = radius-vector of the i-th particle;
yi = angle made by radius vector of i-th particle with the Ox axis at the
given instant of time.
Dut,
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o
6..
where yio = angle made by radius vector of i?th particle with Ox axis at initial
instant of time.
Whence we have:
Ip
? INK - RI sin Ti ph', cos Tirs
al di
Or, taking p = const and bearing in mind that Ri cos yi =x11 we get
=-- pr'.r.
Since the acceleration j. isis obtained on account of rotation of the vector w.
iy
about the axis Qx at angular velocity T, then, consequently it may be represented
by the expression
dirty)
I,. IP X w 01.
dr
_
whence, for the quantity jiw, bearing in mind that w. = R.I.' cos y. = rtx, pl.
we get (Fig.1.10);
? bar Prix.
i.e., an expression identical with the expression for jiu.
Thus we have
whence,forf?we get
JI
I. -- I.. ? pexs.
1. 2 rniper,
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The resultant moment L. of the inertfal forces due to these accelerations
al.c-2.t the axis Ox will he zero inder the condition of the symmetry of the rotor with
res?ect to either of the ea:latorial axes. L4.,) the resultant moment of these forces
aboat the axis 0y, will be directed as indicated on Fi7.1.11, and will equal:
?
;
?
IV" set tri
11, . ?
Fi-.1.9 - 1:annitude of the Component
of the Velocities Due to Rotation
abort. the allatorial Axis
- Resolution of the
Components of the Velocity Due to
Rotation about the Polar Axis
Since, under the condition of the symmetry of the rotor with respect to the
-olar axis:-
?
?
v rn V: ? rti R:
.11= ? AO ?
amIsinceneithernhorrtdonordarlx./we ret
where j -
I - 8
IrtII
1.
- polar moment of inertia of rotor.
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Thus, we see, the inertial gyroscopic moment due to the rotational accelera-
tions is the greater the higher the velocity of the rotor spin rt about its polar
axis, and the higher the velocity of rotation of the rotor axis p, which is in
agreement with our observations.
According to this, the rate of precession at a given external moment, equal in
magnitude to the gyroscopic moment, will be the lower, the higher the velocity of
the rotor spin.
From eq.(1.1) it follows, in addition
to this, that the gyroscopic moment like-
wise increases with the polar moment of
/7
inertia j. According to this, the preces-
Fig.1.11 - foments Produced by
Rotational Forces of Inertia
sion, other conditions being equal, will
be the smaller, the larger the moment of
inertia j.
In this deduction we assumed that the
rotor was plane. It is easy to see that
this conclusion is also true for rotors of
finite thickness. A rotor of finite
thickness may be divided into a number of
elementary disks, for each of which the gyroscopic moment will be calculated by the
same formula:
The resultant gyroscopic moment is determined by the formula
epEl, Jrp
since rt and p will be the same for all the elementary disks.
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CHAPTER. II
DERIVATION OF THE FUNDAMENTAL EQUATIONS OF MOTION OF THE
GYROSCOPE AND THEIR ANALYSIS
Section 2.1. The Kinetic Moment of the Symmetrical Gyroscope
Let a gyroscope possessing symmetry about both the polar axis and any of the
equatorial axes, rotate at angular velocity U1 about a certain arbitrary axis. In
connection with this rotation, the particles of the gyroscope will have the corre-
sponding velocities and momenta. The
sum of the moments of these momenta
1-z
with respect to the fixed point of the
\ .
\
gyroscope is tenned its kinetic moment.
-11111111
Let us bind to the gyroscope the
system of coordinates Oxyz, by placing
L
its origin at the fixed point of the
gyroscope, and matching the axis Oz
7
with the polar axis of the rotor.
Let us resolve the angular velo-
Fi2.2.1 - Kinetic Moment of a
Symmetrical Gyroscope
city U1into the polar component
and the equatorial component(7. Let
the velocity of the i-th particle of
Ri
therz-roscopewithradiusvector and mass mi .be equal to V. (Fir.2.7). The
kinetic moment of the Eyroscope G will then be represented by the aKpression
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O
Or, in view of the fact that
we get
where
(I:
(i,
? 11, ift,l' ,
s
R,
if rf (I: ?
[mi.: R,1.
Pd.
(2.1)
Let us resolve Ri into X izo, the component along the Oz axis, and the component
perpendicular to this axis, i.e., let us represent
where z is the orthogonal axis Oz and X. the projection of the vector R. on the
0
axis Oz.
or
Then the expression for Gz is rewritten in the following form:
I f
V arZ? ? PO
;
I ?
08 Ai iff ? $6111S2 NI ? Pu Imi,u ? Phi.
1-1
ginre Fm o x x.7 1 = 0 is in essence a vector product.
21
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Eakin- use, further; of tlIc formula for a double vector product, we ret
Op- mdt: (A, z..(1,1- Pit (A?zu. '?!)1 ? ms IV (pi.. ?1,1 -? c.. (Pis
? I
The first sus: of the expression so obtained is equal to zoro since ()\z p) = C
frarl the propert,7 of a scalar -rod,Ict;
v- ? I)
... ..t.t010`?
the condition of s7mmetr:- with respect to the nolar axis, since by virtue of the
symmetry for each narticle mi at the distance Pli from the axis Oz the onposite
point mt is found, which is the distance
from the axis Oz.
I:oreover, in view of the fact that the second term in the second sum is also
oval to zero from the property of a scalar product, we .set
v
where J = mt,P' - polar moment of inertia.
solimpiintotheequatorialcomponentL,directed alonc; w, and the compo-
nent Tz;, perpendicular to 7, i.e., represent:
P. s_ ? PI,
Then the exnression for Geq rewritten in the followinr; form:
? or
6, II a.' P2s) Im.." ? 11 ,.. P:0!
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1 I ?
v 1Pi I c.2, ? I m 0.? p:. I
! I I
111 Using the formula for a double vector product, we get
?
?
I
v (1 pj,) ?Ft.
I- I
I -.1
v (aV. p3 ?? us)i
? ? ? 1 .
I-)
The first sum in this expression is equal to zero, since (T -P2i) = 0 by the
-11
property of a scalar product;
by the condition of the same
symmetry with respect to the nolar axis, since by virtue of the symmetry for each
particle mi, the radius vector of which has the component Tiw, the opposite point
is found the radius vector of which has the component 1iw
? / ? In view
3.u.)
also of the fact that one of the tenns entering into the second sum is likewise
equal to zero from the property of a scalar product, we get
Ge
.
where = equatorial moment of inertia.
a I
Thus we have
Section 2.2. The Resal Theorem
(I- G:+ (It LI ?
(2.2)
Let us find the derivative of the expression for the kinetic moment, taken in
the general form eq.(2.1).
Differentiating eq.(2.1), we get
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I
II V
110 JP. i X M I ?- V Mj ?
Amol
St Ana n If
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Let us write a number of obvious equalities:
JR.
= t ,
Jt
by the property of a vector product;
m ) . ( , )eyiemo ( )interrial
' ,tt
where() external is tho resultant vector of external forces;
where (F.) internal
is the resultant vector of the internal forces acting on the
where )
external
internal
i-th particle
I1 te fttAl ? ) ;own.) )C*OrriAl ) irtittitAt
is the resultant moment of the external forces acting on the
i-th particle;
resultant moment of internal forces acting on the i-th particle
.1
( 1 ) )ertermit
1
where L - resultant moment of external forces acting on the body.
Eakins use of these equations, we get
do
(2.3)
?.ut dG IG is the velocity of the end of the vector of the kinetic moment G.
dt
Thus,
?
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i.e., the velocity of the end of the vector of kinetic moment G is equal in magni-
tude and direction to the moment of the external forces.
The relation so obtained is called the Resal theorem.
Let us apply the Resal theorem to the gyroscope.
At a sufficiently rapid proper rotation of the rotor, the equatorial component
_
of the kinetic moment of the gyroscope G, Is usually negligibly small by comparison
to its polar component G. In other words, it may be considered that the kinetic
moment of the urosco,-)e -d coincides in direction with the polar axis of rotation of
the gyroscone rotor.
Let us assume that the external moment L, directed as indicated in Fig.2.21 is
apnlied to a gyroscope having the kinetic moment -CI which we shall consider as co-
inciding with the axis of rotation of the rotor.
According to the Resal theorem, under the influence of this moment, the point
on the axis of the rotor corresponding to the end of the vector G, receives the
velocity U7, equal in magnitude and
direction to L. This velocity can be
obtained only under the condition that
the gyroscope is rotating with resect
(A) to a fixed noint at the angular velo-
city, directed as indicated in
Fig.2.2 - The Theorem of Koments
Fig.2.2.
It will be easily seen that this direction coincides with the direction of the
angular velocity of :recession described above and explained starting out from the
fact that the simultaneous existence of two rotations of the gyroscope about the
nolar and equatorial axes produces the corresronding external moment.
?
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Sect:on 2.3. Derivation of Annroximate Equations of Notion
of the G7roscooe in the Vector Form
As already stated, the eq,:atorial component of the kinetic momentTxeq of the
;yrosco7e is ne-.1i-iLly small by comnarison 'filth its :)olar comonent G,. For those
technical a.,..,lications with which we shall deal, the modlails Geq will be tens and
hundreds of the sands of times smaller than the modulus
On this basis, let us
ne-lect the eq!atorial com-onent in the calression for the kinetic moment (2.2).
Then the ex:'ression (2.2) is rewritten in the followi.n7: form:
6 (2.4)
where 7 is the ortho-pnal coincidin': with the polar axis of the rotor.
0
Differentiltin- eq.(2.4), we ret
i ,d1Jz 0
al di dl
This, on the basis of eq.(2.3)
j z 0 ? dr?
lit dl
Let -s consider 2 = const. Then we r:et
ji2ds?,
tit
d
allere ?2 - the velocity of the end of a unit vector bound to the ,olar axis of
d I
rutation of the rotor.
d
It is clear that the velocity can be obtained only as a res ;it of the ro-
di
tatun of the orth, i.e., of the rotation of the nolar axis of the rotor about an
axls not coincidinr with the orth itself. Such a rotation is the equatorial compo-
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Oft.
?
?
(11
nent of the rotations of the gyroscope Z. It follows from this that may be
Qt
represented by the expression
(1zo
=w,..
dt
YakinE use of this expression, we get
I .1r.1 I u, ? :?) I ( ? 9 ).
(2.5)
The relation so obtained is the vector form of the approximate equation of
motion of the gyroscope for the special case when the rotor spin may be taken as
constant. This special case is the principal case in the technical applications.
On the other hand, even the
approximation, from the point of view
to 10
of the technical applications, is of
very high accuracy, since it is based
fi;), Jos
on neglecting a quantity whose share is
measured in thousandths of a percent.
FiE.2.3 - The Law of precession
On this basis we may conclude that the
equation of motion of the gyroscope, eq.(2.5), describes these motions with suffi-
cient completeness.
It follows from eq.(2.5) that the motion of the gyroscope under the application
of the external moment L to it will be a rotation of the polar axis of the rotor of
the gyroscope about its equatorial axis at angular velocity w.
Let us solve eq.(2.5) with respect to 7i. For this purpose let us vectorially
mulbiply its right and left sides by that is, let us represent it in the
following form:
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Applying the formula for a double vector product to the right side of the
formula, we get
or
co XL.w (.7 P.!) ? (z w) ,
,?? L
????=
(2.6)
since (7o,
; .112) .17 --C2 by virtue of the fact that z0 and S2 are parallel, and
, 0 $
S! (z0,0) = 0 1,y virtue of the fact that z and To are perpendicular.
0
As follows from Fig.2.3, the direction of -to coincides e,cactly with the sense of
.)recession of the gyroscope, the sense which we observed on application of an
external moment and which was explained above from bhe rotational accelerations.
In this connection, we shall call the relation (2.6) the law of precession.
For the modulus 7.), we obtain, on the hasis of eq.(2.6),
L cul I 1 I %on . I )
(2.7)
In other words, the magnitude of the angular velocity of precession is propor-
tional to the equatorial component of the &Tiled moment and inversely proportional
to the kinetic moment of the uroscope.
Let us illustrate, starting out from the relations (2.7), for the law of pre-
cession, the f'insensitivity" of the gyroscope with respect to friction in the
Timbal s
Let as take Jci = 4000 ,g-cm/sec, a value corres,)onding to the kinetic moment of
the fryro scot le of the ryrohoriz,on. Let the moment of friction in the gimbals be
equal to 1 g-cm, a value that is in general exaggerated: according to the stan-
dards, the moment of friction in the gimbals must not exceed 0.5 g-cra for aviation
-1,rosco'es.
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Since it may be considered that the moment of friction in the gimbals lies in
the equatorial plane, we obtain, for the angular velocity of precession under the
influence of this moment, with the data taken by us:
(,) = 1/4000 rad/sec = 0.9 degree/min.
It is clear that such a rate of
rotation cannot be detected by simple
observation. The rotation of the
gyroscope as a result of such a rate
of rotation when the base of the
suspension is rotated is likewise not
detected, since in this rotation the
application of the moments of fric-
?
?
Fig.2.4 - Law of Gyroscopic Reaction
tion in any direction continues for a few seconds, in any case, not longer than a
few tens of seconds. For this reason the axis of the gyroscope appears to maintain
its position invariant in space.
TheinertialmomentL.,arising as a result of the rotation of the ,Tfroscone
about the equatorial axis of the rotor at angular velocity w with simultaneous ro-
tation of the rotor about the polar axis, and as a result of what we have called
gyroscosidc moment, is equal in magnitude to the external moment and is directed in
the opposite sense.
On the basis of what has been said, and of eq.(2.5), we obtain
I - I.JI2XwI.
?
As follows from Fiz.2.4, the sense of the gyroscopic moment of inertia L.,
according to eq.(2.8), exactly coincides with the sense for this moment fixed by
us, on the basis of the rotational accelerations.
We shall call the relation (2.C) the law of gyroscopic reaction.
Forthemodulus.we get, on the basis of eq.(2.8):
LJ,
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since
s-In (010)
since 7 = the eqlatorial component of
Thus, starting out from the aPproximate equation, the motions of the -yroscope
nay be treated as follows: on the application of an axternal noment to the rTro-
scope, there arises a Precession of the vrosco7,e, whose law is defined by eq.(2.6);
in this case there develo:s the ,:yroscopic moment of inertia defined by eq.(2.8).
Section 2.4. Anal7tic Form of Approximate Equations of Motions of the Gyroscope
-
Physically, the complete rotation of the r.7,yroscope Qi is made up of the rotor
spin -i;t, :)y which we mean the rotation of the rotor about its polar axis with re-
sect to the Inner frame of the r.imbals, of the rotation of the :Troscope in the
axes of the r:imbals, and of the rota-
tion of the base of the nimbals.
Let the resultant anular velocity
of the two latter rotations he equal
- Components of the Rotations
of the Gyroscope
30
to -6. Then Si]! which was represented
by us earlier in the form
may now be represented as:
r ? ? 5
By resolving 0 into an equatorial
component 7) and a on it with respect
to the polar axis of the rotor 7, we
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Then, for the total polar component f21, which we denote by C/, we get
r-f-r,
Let us connect with the gyroscope (Fig.2.5) the system of coordinates Chyz, by
matching the axis Oz with the polar axis of the rotor and placing the axes Ox, Cy
in the equatorial plane in such a way that they shall not take part in the proper
rotation of the rotor Tv.
Such a system of axes is called Resal axes. It is easy to convince oneself
that the axis of the inner frame of the gimbals may serve as one of the equatorial
axes of Resal, since this axis, like the axis of the gimbals, passes through the
fixed point of the gyroscope and at the same time lies in the equatorial plane.
Let us set up, starting out from eq.(2.5), approximate equations of motion of
the gyroscope in the analytic form, applicable to the system of Resal coordinate
axes selected by us.
On resolving the equatorial component of the end of the velocity of the rota-
tion of the gyroscope on Resal axes, we get
for
P VG ? 0.
Substituting this e)cpression in eq.(2.5), we get
\JC1z.: ? tiY.\:/9-z.:
Whence we have
1, ,JQq.
?
31
Y..
(2.9)
(2.10)
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The equations for the ,,yroscopic moment in the analytic form will be respec-
Lively of the form:
-JQq;
Section 2.5. Complete Form of the Equations of Motion of the Gyroscope
Let us take the complete exoression for the kinetic moment of a symmetrical
(-vroscop e, representini- the equatorial component in this expression as resolved
according to the Resal axes:
- ? .1,p.v. ? .1
Differentiatinf; this expression and bearing eq. (2.3) in mind, we get
dq d-
L J -t- .ro yo -1- Ju -0 -:-
dl dt di dt
JcP dx.el d-v?
dl
t .
dt
(2 .13 )
The derivatives of the orths are found from the considerations that all these
derivatives will be the velocities of the ends of these orths, which may be obtain-
ed only on account of the rotations of the orths about axes not coinciding with
then. Since the system of coordinates selected, and therefore also the system of
orths of the coordinate axes rotate at angular velocity 2, equal to
__ px. ? qy. ? rz.,
we obtain, on the basis of the above, the following expressions for the derivatives
of the orths
dZo
di (p.c.+ qy,)X ? py0 ? q.v..
dx? .(rze-r qyo)>: .v?= rye? qz?.
dl
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?
deva
di
(rzo+ pxo) x ye.= ? rxe+ pz.
On substituting these relations in eq.(2.19), then resolving L into components
along the Resal axes, we get the following system of equations by equating the co-
efficients of the same orths:
L'= j2q? j` qr.
Ly= Jed fq ? l2p+Jcpr.
(2.14)
(2.15)
,du
- ?
dt (2.16)
The equation so obtained are called modified Euler equations for the symmetri-
cal gyroscope to distinguish them from the classical Euler equations, introduced
with reference to the system of coordinates rigidly bound to the 94 'scope.
This form of complete equations of motion of the gyroscope is the one which is
usually employed in gyroscope theory, and has the advantage that it aflows a
graphically clear representation of the consequences of the presence in the gyro-
scope of the proper rotation T4, that is, of the factor that transforms a nongyro-
scopic body into a gyroscope.
Bearing in mind that the complete polar component of the angular velocity of
rotation of the gyroscope ci = r + r1, let us rewrite eq. (2.11) - eq.(2.16) in the
following form:
+(J? J.)qr Hy,
dl
Ly.. yd; ? (J ? J.)pr Hp,
tr_).
k di di !
where H = Jr? = proper kinetic moment of c-rroscope.
The first and second terms of eq.(2.17) and (2.18) would remain the same, even
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?
?
?
in the absence of the rotor spin, i.e., in the absence of the conditions transform-
inz: a solid body into a gyroscope in the real sense of this word. From their first
terms 're determine, as is clear, the moments of Inertia from the angular accelera-
tions about the axis of action of the moment, from their second terns, the moments
of inertia from the centripetal accelerations ta1:in5 place as a result of the rota-
.
tion of the -vroscone and angular velocity ).
The last terms of these equations are obtained only when the gyroscope rotor
is sninning. They give, as will be clear, the velocity of rotation about an axis
perpendicular to the direction of the component of external moment standing on the
left side cf the equations.
The role of the centripetal accelerations in all cases of interest to us will
be negligiIcly small, since the co-factors in the corresnonding terms will be pro-
ducts of the small anvlar velocities q and r. By rejecting them we find that in
the absence of rotor sPin, the external moment is expended only in imparting angular
acceleratons aott the axis of its action, as it should be for a nonmrroscopic
If, however, a rotor spin rT does occur, then, in addition to the angular
accelerations abo-t its axis of action, the external moment a.lplied to the gyroscope
-121 also produae an angular velocity abdut the actions .per:.endicalar in its direc-
tion to the mtion termed by U5 the :,recession of the ;:yroscope. Thus the complete
motion of the ::yroscope consists of two motions, one an ,Iordinary,r, inherentiin-all
bodies in general, including nongyroscooic bodies, and the second one, precession,
which Is peculiar to gyroscoPes alone.
Turninr: to the approximate eq,iations of motion (2.9) and (2.10), we note that,
according to these eqpatjons, the external moment produces only precession, and
with the same practically vantitative characteristics as the precession entering
into as a component of motion in the comp] etc equations, since, owing to the small-
ness of r in comParson to 0, we may put:
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./9 I (r ?r')-Jr-!f.
i.e., the approximate equations cover only the motions specifically inherent in
gyroscopes and not inherent in other bodies.
But we have already noted above that the degree of accuracy of the approximate
equations of motions of the gyroscope is very 'nigh for a sufficiently rapid rotor
spin.
If this is true, then it follows that the uordinaryu component of the motions
of the Eyroscopes, inherent in a gyroscope as in any nonsTvroscopic body, play a
very small role in comparison to the ugyroscopicu component of this motion, which is
inherent only to the gyroscope in the proper sense of this word.
The equations for the moment of inertia are written on the basis of eq.(2.17)-
(2.19) in the following form:
1,..-
? (2.20)
?
1 =I
'1:
--J.dg
di
_pH dr\
kilt di I
(2.21)
Rejecting the second terms of the right sides of eq.(2.20) and (2.21) as
negligibly small, we find that, in the absence of rotor spin, the external moment is
equal to the "ordinary" moment of inertia, that is, to the moment which any rotat-
ing body, including a nongyroscopic body, is able to develop. In the presence of
proper rotation of the gyroscope rotor, the external moment is equal to the sum of
the uordinaryu moment of inertia and the gyroscopic moment of inertia, which can be
developed only by a rotor possessing a considerable moment of momentum.
According to the approximate equations for the Moment of inertia (2.11) and
(2.12), the external moment is equal only to the gyroscopic moment. Consequently,
it follows from the high accuracy of the approximate equations that it is precisely
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to the gyroscopic moment of inertia that we must attribute the total moment of iner-
tia developed by the gyroscope.
410 Section 2.6. notion of Gyroscone under the Influence
of an Equatorial External Moment
Consider the character of the motion of the gyroscope when an external moment
L, actin7, let us assume, about the axis of the inner frame, with which we match the
axis Ox, is applied to it. We shall consider that in this case Ly = 0 Lz = 0
r = O. The restrictions adopted simplify the, character of the subsequent calcula-
tons, without substantially imairing the generality of the basic conclusions that
can be obtained.
Por the case we have taken, the equations of motions of the gyroscope, (2.14)
and (2.15), will take the following form:
dp
fly
? ?O.
git
(2.22)
(2.23)
- h'
where ---=
Jr ' r/ -
s3nce J > Jeq is always true. On differentiating once
Jeq Jeq
dq
eq. (2.22) and: substituting in it from eq.(2.23), we get
tit,
41- '
The solution of this equation, as is commonly known, is of the form:
I' I sin ? B
Eakin use further of eq. (2.22), we get
? A coc.tt ? B cin ut -4- .
Pseudoreular Precession
Let 'Is take initial conditions corresponding to the gyroscone at urestm under
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?
?
the application of an external moment, i.e., let us assume that, at t = 0, p = 0,
q = O. Under these initial conditions we obtain for the arbitrary constants
B A=.
As a result, the solution takes the followin7 form:
,
p sin , .
L
q= ( cOSitt).
(2.24)
(2.25)
which is represented by the graphs of p(t) and q(t) that are presented in Fig.2.6.
On considering the same case on the basis of the approximate equations of motion
(2.9) and (2.10), we put in them al = 11, or, what is the same thing, from the com-
plete equations (2.17) and (2.18), rejecting the terms representing the "ordinary"
motion in them, and leaving only the terms representing the precession of the 7yro-
scope, we obtain
Papprox 1.1 0
I.
qtp,OPDX
(2.26)
(2.27)
On comparing the approximate formulas of motion according to eq. (2.26) and
(2.27)t with the more exact formulas for (2.24) and (2.25), we convince ourselves
that the first two yield the constant component of the second two.
Denoting the angle of rotation of the polar axis of the gyroscope about the
equatorial axis Oy by a, and about the equatorial axis OK by
(2.24) and (2.25), the followinr::
L
2 I ? 'sintit + CI.
tp
we pet, by eqs.
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?
?
?
Let the initial conditions in the relation of an-les a and r he likewise zero,
4.e., let, at t = 0, a = L, r. = U. Then Le shall have
C1=0 Cg= .
ani as a result
3
I 0,c .,1).
wh,.re
(2.20
(2.29)
Thl,s the motjon of the 7yroscoe ]n the system -Inder study will consist of two
:iifferent, connonents:
-1) the first a = t?, of notion with a constant anular velocity about the
er:lato.:ial axis )er)endicular to the uirection of the lrpled moment; this comnonent
c?.11ed 1.,y us the nrecession of the -yroscone;
i) the second: a sin lit, = Tv (1 - cos 40, 1-ein- the periodic cscilla-
tions .oth alo,:t the axLs, coincidirp: with the direction of the anolied moment, and
a.,out the axis nerendici,lar to this direction, with a phase shift between these
oscillations eqaal to a/. shall cull the second comnonent of total motion the
-nitation of the -7roscone. It is otained as a result of the influence of terms of
the complete equations of motion of the :5rroscolle, which consist of angular accel-
erations caused n the yroscune :y the annlication of a moment to it. 4e have
termed their tens re)resent'np the uordinary" motion of the !Troscone in the sense
that the:,' would remain, even in the absence. of rotor snin, i.e., in the sense that
they are inherent in any non-yrosconic body. In the gyroscope, however, the effect
caused %)y these telms, as we have seen, is not at all uordinary": instead of a
definite rotation in the direction of action of the moment, as would occar in a non-
vroncodc Ludy, they cause nerjodic oscillations, both about the axis coincidinr;
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?
?
?
with the direction of the applied moment, and about the axis perpendicular to this
direction.
A
FiF.2.6 - Pseudoregular Precession - Path of End of Gyroscope Axis
of Gyroscope
in PseudoreFular Precession
As a result of the motions defined by eqs. (2.28) and (2.29), the axis of the
gyroscope will describe a certain path on a plane perpendicular to the original
position of the axis of the gyroscope, and located at unit distance from the center
of the suspension of the gyroscope. For the beginning of motion, when sin a a
and sin A = 0, this path may be described with sufficient accuracy by eq.(2.28) and
(2.29) in the rectangular system of the cooldinates OaR, the axis Ou of which is
parallel to the axis Ox, while the axis 09 is parallel to the axis Oy.
It is easy to see that this curve will be a cycloid with a radius equal to 4,m
i.e., that it will be equal to the amplitude of nutation (Fig.2.7).
Let us take for L a magnitude corresponding to the maximum value of the
correcting moment of the autopilot, amounting to 8 jr-cm, and the other data corre-
sponding to the characteristics of the gyrohorizon, i.e., let Jeq 1.0 g-cm.sec2,
H = 4000 g-cm-sec2. As a result we get the following expression for the amplitude
of the nutation 1pm
8 1
rad = 10 6 rad
16? 106
i.e., a quantity of no practical importance.
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?
In other words, the cycloid traced by the rotor axis will under normal condi-
tions have a radius so small that with a very high degree of accuracy it may be
ta1,en as a straight line.
The frequency of the periodic oscillations of the axis of the gyroscone rotor,
is measured in hundreds and thousands per sec. As a result of the negligible
value of the amplitude and of the high frequency, these periodic oscillations are
IL:)ercentible.
The motions of the gyroscope that have just been studied bears the name pseudo-
regular ',recession. In this term, the word nrev.plaru emphasizes the fact that film
thc practical point of view, the motion reduces in essence to a regular motion, that
is of uniform precession, while the word unseudolf emphasizes the fact that from the
theoretical point of view this motion contains not only nrecession hut also nuta-
t'on.
It should he akided to all that has been said that nutation dies away relatively
fart on account of frictdon in the suspension and of other resistances, after which
only -,recession remains, the existence of which is assured by the external moment
alplied to the eTroscone.
:,e-ular Precession
Let us take for the same case of the anplication of an external moment, the
followin: initial conditions: at t = 0, let p = 0, q = LA, i.e., let us assume
that about the axis the action or the applied moment there is a state of rest, and
al:o.lt the axis nernendicular to the applied moment, rotation already takes place at
an angular velocity equal to the constant component of the angular velocity of
nseudorei7ular ,recession due to that external moment.
For these initial conditions we have
A = 0; 7 = 0
411 and, consequently,
P = 0/
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?
?
that is, the motion will consist only of precession without nutation. Such a motion
is called regular precession.
As we see, by varying the initial conditions, that is, by taking them to corre7
spond to that motion of the gyroscope which is obtained after the damping of the
nutation, we obtain a motion which is without nutation from the very beginning.
It follows from this that nutation is the component of the motions of the gyro-
scope which, according to the initial conditions, may either occur or not occur. If
it does occur, then pseudoregular precession takes place. But in connection with
the damping of the nutation, this pseudoregular precession may be treated as a
motion passing over into regular precession and it is only this type of precession
that can exist for an indefinitely long time.
On applying the approximate equations of motion to this case, we obtain the
same result that was given in this case by the complete equations. This is natural
enough, since the nutation, which distinguishes the complete motion from the
approximate motion, has disappeared in this case in connection with a certain choice
of the initial conditions.
Influence of -a Shock
- Nutation after a Blow
-G10
Let us assume that as a result of
a blow in the neighborhood of the axis
Oy, the initial angular velocity 0_0
arises. In this case let p = O. Put-
ting L = 0 in eq.(2.25)4 we obtain for
these initial conditions:
A = -q0; B = 0
And, consequently,
sin F. t,
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q = cos U t?
Thus in this case the motion will consist of nutation alone (Fi,-,.2.0 without pre-
cession, tut this is understandable, since in this case the external moment, on
acco.mt of which precession may exist, is absent.
Section 2.7. Ener-etic Dalance of Lotions of the Gyroscone
Ath re7ular nrecession, the motion durinr thc course of all the time after the
initial instant will remain exactly the sane as it was at that nitial instant.
Altho-h precessfon still exists on account of the action of an applied
external moment, this action nevertheless does not chanie the ener:y of the system
'y comparison w.th the ener-y nossessed by the systam on the arinlication of the ax-
tel.nal
already rel-ari-ed, this is ax-laineu :y the fact that in this case the exter-
nal moment ca.:se5 notion in a -lane :.er)endic?lar to the dame in which acts.
Conseglently, m ass'Irfm- the existence of nrecession, the external moment does not
?:erforr. worl 1 and in this connection it cannot 1-0 a sot.rce of variation of the
eno of the system. -nt sfnce there arc likewise no other sources of enert.:y of
in thi-; case, it is inderstandahle that the enerc- of the s;stem should re-
am n unchann-ed the ez:,5tence, for as ion,- a :ler...od as 1.1a:,- 1:0 desired, of
-recession a.-td of the action of an external monent, that is, that it should
:t at tlio nit al inst:Lnt of Li2,e. t the irCtirll l_nstant of
;;:o ? 4as LInarted to ClQ r!::,-roscope,'which
r.eant the _Introduct2on of energy into the system that was exactly equal to the
tils ivo,1 u:ftern?:.1 nonnt is a'.3e to ,rovdc.
7n -..or.t:-.Lot to thr. :n 'tial conci;tions ].c.din- to the a)pearance of ro-,ular nre-
colon, 1)1(:. 'ulta-J conditions load3ur: to the a..earance of nseltdorci7ular preces-
.;:r,n QG not :A7017G t'AC intr[AlCt.on of add;.tional initial cner-y into the system.
un.;eritentl:, the enor7 of nsuldoreiar nrecession can he formed only on account of
I moment,. STAT
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Such work is actually performed by the external moment in nseudore5:ular preces-
sion in connection with the existence of a rotation about the axis of iis action.
The magnitude of this work nay be characterized by the angle of this shift, since
the external moment is constant.
AS follows from eq.(2.27), this angle, which always retains a positive sign,
LJ
varies from zero to a value equal to 2Tm = 2--LE, and has a constant component equal
H2
to ) i r i = LJeq . Consequently, the work of the external moment U1 = LS varies from
m H2
L2J L2Jen
zero to a value equal to 2?_2. for a constant component equal to ULc ? -1
112 112 . This
work, at each riven instant, according to the law of conservation of energy, must be
equal to the energy of pseudoregular precession U = (n2 +
ea
nr
Usin-: the expression eq.(2.24) and (2.25), we c:et for U :
Pr P
1.1.:
- (1 -- cos ,g).
/
2
q).
i.e., an expression identical with the expression for UL = LB, if ;.3 is taken accord-
ing to eq.(2.29).
The constant component of the energy of pseudoregular precession may in turn be
divided into the energy of precession nroper and the energy of nutation These
energies in our case will be equal to each other, and it is easy to convince oneself
of this. In fact, the energy of precession equals
Jo 1.2
2 2112
this corresponds to half the constant component of the work of the external moment.
On the damping of the nutation, the energy of nutation is dissipated, and the gyro-
J L2
scope now retains only the energy of proper precession equal to eq .
2112
The initial conditions corresponding to a blow mean the introduction of the
J q2
initial energy U0 = eq ? into the system. Since the subsequent motion is perform,-
9
ed without participation of external forces,
43
it follows that the energy of the
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system must remain invariant during the nutation following the blow and must remain
.1
equal to ' q 1 which, in reality, it does.
9
Section 2.(% Transition to the Equations of Motion with Respect to
Terrestrial Coordinate Axes
The basic content of the theory of aviation gyro instruments reduces down to
the determination of the laws of the establishment of the gyroscope in a position of
stable equ'librium in one system or the other of terrestrial coordinate axes. A
hnowled!-e of these laws allows us to determine the conditions under which a gyro in-
strtuaent is able to indicate the direction of these terrestrial coordinate axes or
to determine the rotary velocity of the aircraft with respect to the earth.
In both cases, to solve the problems so formulated, we must pass from the equa-
tions of motion in the Resal axes to the eqpations of motion in terrestrial axes.
Let us take the system of terrestrial coordinate axes On" (Fig.2.9). Let us
1-:atch one of these axes, let us say or, with the direction to be determined by the
aid of this gyro instrument: for r-yro horizons, it will be the direction of the
-brae vert4 cal, for coarse-indicatin: ,:yro instruments, the direction of the meri-
C.ian. Let us call this axis the principal terrestrial axis. Let us match, to other
let us say the axis On, the axis of rotation of the outer frame of the
ira-
hals, which is always located Parallel to some one of the aircraft axes in horizon-
tal flight. The direction of the third axis will then be determined according to a
right-hand system of coordnates.
Let us match with the oric;in of coordinates 0.Pn- the or-it-in of the system of
coordinates Oxyz. :le recall that the axis Oz of this system of coordinates has been
]:atched w:_th the -olar axis of the rotor, and that the axes Ox and ny lie on the
ecatorial plane and do not articipate in the rotor snin. As has been stated
the axis of rotation of the inner frame of the gimbals lies in the equator-
ial plane uf the .7yroscope, as the axis -.1assirr; through the center of the suspension
the axis Oz. On the other hand, this axis wal not partrte
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fo-
in the proper rotation of the rotor, but it will partici9ate in the other rotations
of the gyroscope, for example, about the axis of rotation of the frame of the sus-
r.ension, or in the rotations together with the base of the sus-ension. Consequently
the axis of the inner frame will satisfy the condition imposed on the equatorial
liesal axes and may be selected as one of such axes. Let it le the equatorial axis
Ox. The second equatorial axis Oy is then determined accord4.ng to a right-hand sys-
tem of coordinates.
Fir.2.9 - The Euler Angles
Draw the Plane zOn. Let this
plane, in a cjven position of the axis
Oz intersect the coordinate plane
in the line OA. Let its denote Ly a the
angle between the Princinal axis Or, and
the line OA, and by e the an-re between
the line OA and the axis Oz, selectiwT
the Positive sense of these angles by
the arrows shown in Fig.2.9. It is not
hard to see that these two an-les will
completely determine the rosition of
the axis Oz and the system of coordi-
nates OFnr, since the anrle Oz may be brought into an assigned position by a rota-
tion first in the :lane gOr, about the axis On by the an;le a, followed by a rotaton
in the plane z01 about the axis Ox by the angle A. By associating to the angles a
and fp, the anrle of rotation of the rotor in the equatorial plane on account of its
sAn, we obtain three degrees of freedom of the ryrosco-e.
The anrles a and 8 are, as is commonly known, called Euler angles. They are
selected by various methods, according to the form of the :roLlem to be solved. The
selection made by us is convenient in that a rotation in the --lanes E0r, and zOn is
ation about the axes of rotation of the outer and inner frames
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of the 'ynDSCO slls)ension, reslectively. The former results directly from the
fact that we have Latched the axis of :notat'on of the inner frame with the axis On,
(1-5 .er-endic .lar to the -lane ::Ur,while the latter results from the fact that we have
matc'ied the ax...s cf rotation of the inner frame with the axis Ctx, which is nerpen-
ctr ar to the lane zG.
It follcas from this that the angle a fs the an:-,le of rotation of the r,yrosco-,e
t the axis of rotation of the cuter fralne, while the am-le ].5 the an-le of
shift of the ?-;-rosco: e alo_:t the an(;le of rotation of the inner frame.
The -:)s!t2-ze.-ales of ri and 1: are directed, as shown in Fir-.2.9, that is, a,
an the --os!.t:-.rc: alon- the ner-at'%-e se) daxis C.
The syste--: of coordinates 0,5,1', always n.artic: ates in the rotation of the
earth, an:, LI nay also receive additional rotations on account of the
is lace:.ient of the aIrcraft ,Lith resect to the earth surface, since the direction
Of the vert ca. or :Ler'dian is afferent for aifferent ,eints of the ealth. Let, in
((...7' the a- re? ate, thc an- lar veloc .ties of a.:1 these rotations of the system of co-
crri- nates co:-.-:onents alon- the axis GF:, equal res-)ectively to (,),
-.7hcm the an-ular -.-eloc.ties of rotation of the -,yrescune, and q, with
't ?
res ect tc the axes Ox 0:?, are determined by the ex?Thressions
? 0 - C01(c. JC) - wit CoS (1, x)-1- cos kE,
(2.3u)
(2.31)
The cosinat of the Inr,les enter;n- into these a.k.nressions are riven in the
TO: o
Taule I
X Cosa 0 ? sin a
? sin a sin p cos p ? COS 2 SM p
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For small values of the anrles a and @, with which we shall for the most Part
be dealinr, this table is transfonaed into the followinr; one:
Tale II
x : 10 --a
y 01
On the basis of the last table, for small angles a and B, the expressions for
and q, taking account of the mobility of the system of coordinates 0, will take
the following form:
p=-? uJL-
q ei
(2.32)
(2.33)
The expression for Lx. and Ly in the general case may be re-,resented in the
followin!-- form:
L,i,
Ly Lity 114,
where I:1_, ,Fir L = moments produced IT the correction system cr thc systens of springs
-c
and damper devices;
Lpf;, Lp'0, = moments of forces of friction in the axis of rotation of the inner
and outer frames res ectively.
Section 2.9. Influence of Earth Rotation and of the Disnlacement
of the Aircraft with Resrect to the Earth
The essence of these phenomena, taken into account accordin5 to eqs.(2.30),
(2.31), or eqs.(2.32), (2.33), reduces to the following: the principal axis (X of
the system of coordinates 071'., which can be matched, as has been shown, either with
4-1 -
viLth the meridian of the Place, varies its direction in s.,STAT
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(0
This variaton occurs, on the one hand, as a result of the fact that the true ver-
tical and the local meridian also .artici,atc in the rotation of the earth, and on
the other hand, a result of the fact that with the displacement of the aircraft with
respect to the earth, the true vertical and the local meridian, whose direction must
bc detennined IT the aid of the gyro instruments, also vary their position.
It i s commonly known that theex-
,ression true local vertical is the tem e,-- /IC
/4.17
IWj / 4?- frf
a-n1:1ed to the direction of the action
of the force of gravity which practi-
cally coincides Wth the direction of the
earth radius ON (Fir.2.10). The deflec-
Y??
Fig.2.10 - Rotation of Plane of the Fi-.2.11 - Comnoncnts of the Rotation
Horizon
of the Plane of the Horizon
ton of the vertical from the direction of the rad is does not exceed units of
rliniates of arc. The prolongation of the vertical above Hz is called the zenith
line. The plane perpendicilar to the true vertical is called the nlane of the hori-
zon. The plane passinF through the point of the place 1. and the axis of rotation of
the earth, is called the nlane of the local meridian. The intersection between the
4ane of the horizon and the plane of the meridian is called the meridian, or men-
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The planes perpendicular to the axis of rotation of the earth are called planes
of the oarallels. Of them, the plane of the parallel passing through the center of
the earth 0 is called the plane of the equator. The angle between the true vertical
and the plane of the equator is called the local latitude.
As follows from Fig.2.10, the rotation of the plane of the horizon, and, with
it, of the vertical and meridian of the local vertical and local meridian in connec-
tion with the rotation of the earth, may be represented as the result of two rota-
tions, first, the rotation about the true vertical, during which the north point of
the meridian is shifted at first to the left, if viewed from the north when looking
northward (in the northern hemisphere), and second, the rotation about the meridian,
during which the eastern half of the plane of the horizon sinks while the western
half rises.
The angular velocity of the former rotation w2 = we sin T will be called the
vertical component of the earth rotation, while the angular velocity of the second
rotation w1 = w cos T will be called the horizontal component of the earth rota-
tion.
On resolving the speed of the aircraft V into its components along the meri-
dian Vc = V cos K and along the parallel VB = V sin K, where K is the course angle,
it becomes possible to treat the result of the flight of the aircraft as an addi-
tional rotation of the plane of the horizon, first about the axis of rotation of the
earth at angular velocity (013 = n
VB
Co S
(R = distance from aircraft to center of
Vc
earth), and second of angular velocity (Ac =?rt about the equatorial diameter of the
earth, perpendicular to the plane of the meridian of the given place (Fig.2.11).
Estimation of the Influence of Rotation of the Earth and the Speed of Flight
with Respect to the Vertical Position of the Principal Axis of the Gyroscope
In this case the principal akis Or of the system of coordinates OW, is matched
with the true vertical Mz. The axis On, with which the axis rotation of the outer
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frame is matched, will either coincide wf,th the d!rection of the velocity of
f1 'ht 7, or will be perpendicular to it, de,endn!_; on whether the axis of rotation
of the otter frame is parallel to the loncitudinal axis of the aircraft or to its
transverse axis. Here we shall consi-
C
der the an,-,le of drift of the aircraft
X_
r rur, /
I t ' - as equal to zero, that is, we shall.
- Jillowin; for the notation
of the Plane of the Horizon with
,,espect to the Gyro Horizon
consider the course eq zal to the route
an:-;3 e. It is not hard to see
(F-1,7.2.12) that if we take, let us say,
the former case as or 12asis, then we
obtain the latter case from the former
by increasiir, the course an-le of the
rcraft by the anc,le -1r .
On projecti.ng the correspondin
connonents (Fi:.2.12) onto the axis of
the syote7. we obtain the followini.; equations for the case when the axis of the
o :ter fra .0 f the 7,yrosco.se is parallel to the lon:it'idinal axis of the aircraft:
w.
01
sin A? wiicos? sin K? cos K ?1 sin K-
1sin- A
. V - cos? K,
0.1 cos K t- wi,cos ? cos K? w1 sin K = to, cos K
sin K c os K? cos Ksin K
? R
w1, s 11 -f? t?-:1 - sin K tg ?
V
wt. ? cal sin K-
11,= 01 cos K,
sin Ktgp.
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On substituting the relations so obtajned in eqs.(2.32) and (2.33); we get
p ? ? (ws sin K tg ir) ? sin K- ,
(2.34)
(2.35)
Allowinr, for the Influence of the Earth Rotation and of the Flight Sneed
jith Horizontal Position of the Principal Axis of the Rotor, Parallel to
the 1:eridian
In this case the axis 0 is matched with the meridian; the axis On with which
we shall consider the axis of rotation of the outer frame to be natched, will be
parallel to the no/nal axis of the aircraft. Consequently, for horizontal flight
(and it is for this that we are setting up the corresponding exnressions) it will
coincide with the zenith line Kz (Fig.2.13).
On projecting the saae components of the rotation as in the nreceding case,
onto the axes of the systen Or for a given arranr;ement of those axes, we get:
#0
Fi.2.13 - Allowing for the Rotation of
the Plane of the Horizon with Respect
to the Gyroscopic Course Indicator
51
or
V
w.=w
COs K;
.8 sin 7, (us ?
On
4- -V A - sin
R cos
WB COS or-- 0)1 +
V sin K
COS
P cost
V
w.= - cos K,
Vsin )
- sin 7.
Ras,
V
tu: - -S01 AC
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Sul?stituLing the relations so found in ogs.(2.32) and (2.33), we get
P
- . (mi + sin K) 2 + cosK.
R R
V' son
Y sin K) 3+(.i?sin ?.
R CO%
(2.36)
(2.37)
Allowin7 for the Effect of a Turn on the Gyro Horizon
In this case we may consider that the system of coordinates OW possesses only
V
a rotation a;)out the true vertical at angular velocity wu =--, where P = radius of
turn.
In other words, we need not reckon with the variation of the nosition of the
true vertical chring the time of the turn, owing to the earth rotation.
Assuinin-: this, we have
r-Jrm AM .
wo,
where the upper sign corresponds to a right turn and the lower sign to a left.
Sul,secilently,
p ? w..a.
with the same rule for the selection of the signs.
(2.38)
(2.39)
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CHAPTER III
BRIEF SURVEY OF PRINCIPLFS OF ACTION OF POSITIONAL
GYROSCOPES
Section 3.1. Astatic Gyroscopes
The principal advantage of gyroscopic positional instruments is their great
rigidity produced by the rapidly spinning rotor, usually mounted in the gimbals.
The motions of this rotor about the axis of the suspension may be sufficiently slow
owing to the increased proper kinetic moment of the rotor.
As a result of this the axis of the gyroscope rotor acquires stability with
respect to the action of disturbing forces. In this respect it is completely incom-
parable to the sensitive systems of positional instruments based on the use of other
nongyroscopic principles, for instance, with a magnetic needle or pendulum.
But, in contrast to the latter, gyroscopic positional instruments make it
necessary to take special measures to give them selectivity with respect to the
direction which they must indicate. The selectivity of one positional instrument or
another with respect to some position means the presence in that instrument of pro-
perties by virtue of which, when the sensitive system of the instrument deviates
from this position, a certain moment, called the positional moment, is applied to it
and returns it to the indicated position. A positional instrument thus possesses
selectivity with respect to that position which constitutes the position of stable
equilibrium for its sensitive system.
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The center of gravity of the astatic gyroscone is matched with the center of
the susnension, and, owing to this, the action of the force of gravity on the gyro-
scone is eliminated.
Thus any position of the astatic TIrroscone is an equilibrium position.
It is of substantial importance that the position of the gyroscope will vary,
during motion of the base of the suspension, owing to the action on the gyroscope of
the forces of friction in the axis of the susnension, and owing to a certain un-
avoidable residual imbalance of the gyroscope.
It is true that the rate of this change, at a nroner value of the kinetic
moment of the gyroscone, will be small. In this connection, in those cases where a
definite direction must be indicated for 2-3 min, it is entirely possible to use an
astatic gyroscone for these nur.loses, although in practice this is not done.
In this case, the fact that an astatic gyroscone is indifferent to the position
in which it is installed, is a substantial merit for this use, since, in this con-
nection, an astatic gyroscope may be utilized for the brief designation of any
direction with res-ect to the earth surface.
It must, however, be noted, that in this case the astatic gyroscope must
initially be set in the required direction, since the gyroscope itself is unable to
ufindu this direction.
Section 3.2. Position Gyroscopes with Displaced Center of Gravity
Two methods are used for giving cyroscooic position instruments the necessary
selectivity.
The first method, on which we shall dwell in this Section, is based on obtain-
inr, the necessary Positional moment by means of a definite displacement of the cen-
ter of gravity of the gyroscone with respect to the center of the suspension, that
is, by eliminating the astatic nature of the gyroscope.
In this case, when the gyroscope departs from the assigned direction, a posi-
ich will return the gyroscope to the assigned direction, will
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develop on account of the activity of gravity.
In technology two types of positional gyroscopes are in use, in which this same
method of obtaining selectivity, the gyropendulum and gyrocompass, are used.
Fir;.3.1 - Gyropendulum
I - Axis of inner frame; II - Axis of
Fig.3.2 - The Magnetic Gyrocompass
I - Axis of inner frame; II - Axis of
outer frame; 0 - Center of suspension; outer frame; 0 - Center of suspension;
0, _ Center of gravity
?
0 - Center of gravity
Gyronendulums serve to indicate the vertical. The axis of the gyrouendulum rotor is
held in the vertical direction by means of the downward displacement of the center
of gravity of the gyroscope 0 (Fig.3.1) along the spin axis from the center of sus-
pension 0. The gyrocompass serves to indicate the geograrhic meridian. The axis of
the gyroscope rotor is held in the direction of the meridian by the displacement of
the center of gravity of the gyroscope 0 below the center of the sus-)ension 0
(Fig.3.2) in the equatorial plane of the gyroscope.
A brief theory of the gyropendulum and gyrocompass is given in the following
Chapters, and here we have no intention of touching on this subject in any way. We
confine ourselves mainly to a few remarks of fundamental nature.
The equilibrium position for the spin axis of the gyropendulum will be the
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vertical of the locality (without taking account of the error owing to the earth ro-
tation and the displacement of the base of the gyroscope suspension).
The position of equilibrium for the rotor axis of the gyropendulum will be the
local vertical (not taking into account the errors due to the rotation of the earth
and the displacement of the base of the gyroscope suspension).
On deviation of the axis of the rotor from the vertical, the positional moment
of the force of gravity begins to act on the gyroscope. This force tends to restore
the rotor axis to the nlane nreceding the deviation, by moving it toward the verti-
cal. But under the action of this moment, the motion of the gyropendulum, according
to the law of Precession, begins not in the Plane of the deviation that has taken
place, but perpendicular to it.
Thus we get the result that the action of the positional moment in the gyropen-
dulum leads it to the liquidation of any violations that arise, and to the transfer
of such violation to a different plane. In the last analysis the spin axis of the
gyropendulum, instead of returning to the position of equilibrium, will undergo os-
cillations, describing a cone about the equilibrium position, having its vertex at
the fixed noint. Under the influence of friction in the suspensions, the radius of
this cone will diminish more and more, and ultimately the spin axis of the gyro
rotor will be matched with the angle of repose relative to the vertical. From the
technical aanect, however, this circumstance cannot be utilized, since by the condi-
tions of accuracy for the operation of the instrument, a more or less considerable
friction in the suspension must not be allowed, and at such friction as may be
allowed, too much time is required for damping of the precessional oscillations of
the gyrocomnass to take place on that account. In other words, in order to trans-
form the gyro,endulum into a -ositional instrument, it must be provided with some
means caieble of assuring the effective damping of its oscillations. Such methods
have indeed been worked out, but they considerably complicate the design of the in-
strument. And in a gyrocompass that does not use these special damping devices, the
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axis of the rotor, while it is returned to the meridian in case of its departure
from it, will also perform oscillations around that meridian of only a slightly more
complex form.
The equilibrium position of the rotor axis in the gyrocompass having a fixed
base with respect to the earth, is a direction in the plane of the meridian with a
certain entirely definite inclination to the plane of the horizon. On account of
this inclination, there develops a positional moment of the force of gravity, which
assures the precession of the gyroscope equal to the vertical component of the
earth rotation. In this case, the rotor axis of the gyroscope and the ilane of the
meridian will both be rotating at constant speed and will remain matched.
If the axis of the rotor of the gyrocompass, however, leaves the plane of the
meridian, the center of gravity of the gyrocompass will also leave this :lane. In
this case the rotation of the plane of the horizon due to the horizontal component
of the velocity of rotation of the earth will vary the angle between the gyroscope
axis and the horizon plane, which will lead to a change in the magnitude and direc-
tion of the positional moment due to gravity. When the rotor axis leaves the meri-
dian plane, the angle between the gyroscope axis and the horizon plane increases,
thus causing an increase in the rate of precession. This increase in the rate of
precession returns the rotor axis to the meridian plane. Owing to the fact that the
rate of precession of the gyroscope exceeds the rate of rotation of the meridian,
the rotor axis of the gyroscope will catch up with the meridian. As the meridian
advances, the angle between the gyroscope rotor axis and the horizon plane will
diminish, which will cause a corresponding reduction in the rate of Precession and
will again return the rotor axis to the meridian plane.
.Thus, in the gyrocompass as well, the gyroscope axis will describe a cone whose
center coincides with the center of the suspension, and whose axis coincides with
the equilibrium position of the gyroscope axis.
In exactly the same way as in the gyropendulum, by using damping devices of one
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kind or another, the damping of these oscillations of the gyrocompass can be assur-
ed, but exactly as in that case, this implies a substantial complication of the in-
strument design.
Another substantial disadvantage of both the gyropendulum and the gyrocompass
is that, owing to loss of astaticity, they become particularly subject to the action
of inertial disturbing forces. It must, finally, be recalled that these instruments
(esnecially the gyrocompass) possess considerable errors owing to the influence of
the earth rotation on them and owing to the displacements, with respect to the
earth, of the object on which they are installed.
:lhat had been said will ex-lain sufficiently why this method has not found
wides:-.read use in aviaticn practice.
Section 3.3. Positional gyroscopes with Correction System
Let us now dwell on the second method of giving gyrosco)ic Positional instru-
ments the necessary selectivity, based on the gyroscope itself remaining astatic,
111 but being sunplemented by what is called a correction system.
The operation of the correction system is based on the anolication of a posi-
tional moment to the gyroscone, when any deviation from its nosition takes place, in
such a way as to cause its sin axis to move (Fig.3.3).
The question whether deviation exists is answered, and its amount evaluated, by
comiaring the -osition actually occipied by the gyro with the position of some other
element having selectivity with resect to the direction which is to be indicated in
this case. For examnle, the position of the spin axis of a gyroscope indicating the
vertical is comnared for this nur:)ose with the nosition of a nendulum; the position
of the snin axis of a gyro indicating the meridian, with a magnetic needle, etc.
We shall henceforth term an element whose position is compared with the posi-
tion of the ryro the sensitive element of the correction system.
It is easy to see that in essence, a gyro correction system may be treated as
--?,,inf.nr of its ;,osition, and that in this sense it may be said that
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the method of making gyroscopic position instruments selective by means of a correc-
tion system is a method of automatic control of the position of an astatic gyro not
possessing selectivity, by the aid of a
special element that does possess such
selectivity.
In what does the technical meaning
of such a method of indicating an as-
signed direction reside? Would it not
be simpler to use directly, as the
positional instrument, the very same
element possessing selectivity by means
Fig.3.3 - Princinle of Operation of of which the correction is effected,
Radial Correction System or, in other words, by means of which
the automatic regulation of the gyro
(XA - Plane of deviation; e- Angle of
position is effected?
deviation; LK - Moment of correction;
The answer to this question is
toK - Angular velocity of correcting
that when we use a radial correction
precession'
system for a gyroscope we get a system
that combines not only the selectivity of the element by which the correction is
effected, but also the high rigidity or inertia inherent in the gyroscopes. This
means that the system as a whole will react much more weakly to disturbances than
the element effecting the correction itself.
Now let us consider, to be concrete, the system of a gyro-vertical indicator in
which a pendulum is used as the sensitive correcting member.
;le remark that during the initial period of the development of aviation there
were attempts to utilize pendulums of one kind or another to indicate the vertical,
but these attempts were unsuccessful, mainly owing to the fact that even the
,n oallpd linifnrm
fli ht of an aircraft still involves accelerations that vary
ibTAT
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mar,nitude and sign in a more or less random manner. Maneuvers involve such accel-
erations even more. As a result, the pendulum in flight is always under the influ-
ence of various disturbing inertial forces differing in direction and magnitude, and
a)
C)
b)
d) 3
e)
- Diagram of Formation of the Errors of Indicating System
without Gyro (a) and with Gyro (b)
a) Disturbing forces; b) Errors of indicating system; c) Disturbing forces;
d) Errors of corrector; e) Errors of indicating system
since the rigidity or inertness of the pendulum is relatively small, these forces
disturb its position rather substantially.
If the Pendulum is used directly as an indicator of the vertical, then the dis-
turbances in its position will mean the appearance of corresponding errors in its
readings. Dut if the pendulum serves instead as the sensitive member of a correc-
tion system, then the disturbances in this position will mean, from the very begin-
ning, only the application of the corresponding forces to the gyro.
In other words, while, in the former case, the chain of phenomena leading to
the appearance of errors due to the action of disturbing forces will consist of two
links, it will consist in the latter case of three links (Fig.3.4), of which the
third link, the gyroscope, involves a high degree of inertness.
Thus, although there will still be an ultimate disturbance in the nosition of
the gyro, that disturbance will be considerably less than the disturbances in the
than the disturbances of the pendulum.
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We shall illustrate this by an example.
On account of the disturbed oscillations of a pendulum, defined by the func-
tions y(t), let a disturbing moment proportional to these oscillations, i.e., equal
r
Fig.3.5 - Influence of Disturbances of Corrector
Y(t) - Disturbances of corrector; o(t) - Disturbances of gyroscope
to ky(t), where k is a factor of proportionality, act on the gyro in connection with
the disturbances of the correction system. According to the law of precession for
the disturbances of position of the gyro 6 under the influence of this moment, we
get
h(t)df,ti. '
where H = kinetic moment of gyro.
Assume that y(t) varies with time according to the diagram in 5 Fig 3?, . Then 6
will be determined by the mean area of the diagram of y(t), multiplied by the quan-
tity-k-, which may be made sufficiently small, on account of the sufficient value
of the kinetic moment of the gyro Ho.
61
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CHAPTER TV
ITEOPY OF THE GYROPE1MULUE
Section 4.1. 7ehavior of Undamped Gyronendulum on Fixed Base
without Allowinr' for the Earth Rotation
1.quations of Eotion
ie shall apply the term undamped 7.yro-)endulum to a gyroscope with three degrees
of freedom, whose olar axis is directed along the vertical, and whose center of
gravity is displaced upward or downward
alon7 the polar axis of the rotor Oz,
with respect to the center of the sup-
Fd-.4.1 - Gyro:ondulum on Fixed Base
port.
:le shall restrict our problem to
the study of the precessional motions
of a ryropendulum, disreflardjni; the ro-
tation as a component of the motion
which is of no practical importance in
this anplication, i.e., we shall take,
as
our basis, the ovations of motion
in the followinr,, form:
/It I,.
(4.1)
(4.2)
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Putting, in eqs.(2.34) and (2.35), the terms allowing for the influence of the
earth rotation and the flight speed as equal to zero, we get
q=3;
P? ?
(4.3)
(4.4)
For the moments andL when the center of gravity is displaced downward
4 yl
according to Fig.4.1, we have:
Li ? mg/cos(Cy )
Lv, mgl cos x) 4. cos p.
where the first terms are the positional moments due to the disnlacement of the cen-
ter of gravity; Lpq and Lo are moments of friction about the axis of rotation of
the inner and outer frames of the gimbals respectively. We shall take these moments
of friction as constant. To the upper signs correspond positive values of 8, & and
0 to the lower signs, negative values.
Making use of the values of cos (Y,07) and cos (,x) according to Table II of
Chapter 2, and taking cos fr1.= 1, we get
Lto.
Ly me- - nigh -f-
As a result, the equations of motion (4.1) and (4.2) for the gyropendulum, in
the case taken by us of a downward displacement of the center of gravity, take the
form
where
?
?4?
4-43 ? xpa.
H
63
(4.5)
(4.6)
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?
The signs of pi; and pa. are selected by the rule formulated for the selection of
signs for moments of friction.
Law of Motion of the Gyronendulum without Allowing for Friction in the Gimbals
Putting ivy = = 0 in eqs.(4.5) and (4.6), we rewrite these equations in the
following form:
(4.7)
(4.8)
Let us draw a sphere of radius equal to unit length about the fixed point of
the gyroscoPe. In its motion, the end of the spin axis will describe a certain path
on this sphere. It is the form of this path that will determine the law of motion
of the :-:yrc.
Let us take a part of this sphere near the intersection of its surface with the
-axis 0-, Let us take, further, the line of intersection between this sphere and the
coordinate plane as the coordinate axis Oa, and the line of its intersection
with the coordinate plane n0- as the coordinate axis 013. Such assumptions may be
considered correct only for small values of a and
As a result, the path described by the end of the spin axis on the sphere may
be a.7proximately treated as the curve F(a, ) = 0 in the coordinate plane 0 aCi, de-
fined by the solution of eqs.(4.7), (4.8).
By dividing eq.(4.7) by eq.(4.8) to eliminate dt, we get the following differ-
ential equation:
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IIIwhence, separating the variables and integrating, we have
?
2
where 0 is an arbitrary constant.
0
Assume that for t = 0, a =a0, H =00. Then for 90 we get
Thus the path sought is a circle with its center at the origin of coordinates,
and a radius equal to the initial disturbance. (Fig.4.2).
In other words, if, at the initial instant, the spin axis of the gyropendulum
was on true vertical, it will remain on
it. If, however, at the initial in-
stant, the spin axis of the gyropendu-
/7 N\\\
lum was deflected from the true verti-
t, cal by a certain angles 0, different
1
\\
J
);
true vertical in such a way that its
from 0, then it will rotate about the
end describes a circle in the coordi-
nate plane 00:0, while the axis itself
describes a cone with vertex at the
fixed point of the gyro.
It follows from eq.(4.7) that,
for the case we are investigating,
positive values of 0 correspond to positive values of112! while negative values
dt'
of p correspond to negative values of-A, that is, for all 9 > 0, a increases, while
dt
Fig.4.2 - Path of the End of the Spin
axis of Gyropendulum, Neglecting
Friction in the Gimbals
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?
This means that the end of the spin axis of the gyropendulum is displaced
clockwise along, the oath we have found.
The law of motion of the end of the spin axis so found may be treated as pre-
cession about the true vertical. The rate of this precession, which we shall denote
by will obviously be equal to the angular velocity of rotation of the radius
vector of the end of the spin axis in the coordinate nlane OaR.
On elininatinr: e from eqs.(4.7) and (4.8), we get
d'23
- 1.72 = O.
and on eliminatinr; a from them, we rr,et
dri ?
The solutions of these equations will be:
g fin sin (f.,?1 I 1.).
O.1).
where yG equals the initial angle of the radius vector of the end of the spin axis
ds
di
I,
sit
cos ? 1,.):-=
,
,leep sin (...,( 10) -
It follows from a comnarison of the expressions so obtained with eqs.(4.7)
(Ii..C) that
Thus the rate of regular precession is numerically equal to X, and it is
dLrected downward from the fixed noint of the Lyro.
On imvesticatfnr: this case on the basis of the complete equations of motion,
we would obtain, for the :nitial conditions adonted, one of the partial cases of
(cf. :-.V.Piulcakov 1 , paces 18-22), with a nuta-
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II
tion frequency 11 = j eq an with an amplitude smaller than the initial deviation 0
0
by a factor of .L. For (:),.e of the gyro horizons of the gyropendulum type, the quan-
tity1-1.? is of the order of 5 x 105. If we add the fact that nutation is very rapid-
ly dam:.ed, even by the resistance of the air without considering other factors, then
we reach the conclusions that it is entirely correct to neglect it.
On repeating the same study for an upward displacement of the center of gra-
vity, we get the same general result, with a single difference that in this case the
rotation of the spin axis will be counterclockwise.
This results from the fact that when the center of gravity is displaced upward,
the signs are reversed in the left sides of eqs.(4.7) and (4.8), and therefore a
will increase for all i3 < 0, while S will decrease for all a > O.
Law of Lotion of Gyronendulum Allowing, for Friction in the Gimbals
Let us rewrite the equations of motion (4.5) and (4.6) in the following form:
Let us introduce the new variables:
PIA 4:1) ?
a ma a p4,
(4.9)
(4.10)
where the index "ltt corresponds to the upper sign on the right side, and the index
(2) to the lower sign. It is easy to see that for all points except those at
which p& and pr; change their signs, we may write
5,
(4.11)
daL1==d7. (4.12)
On dividing eqs.(4.9) and (4.10) by each other and using (4.11) and (4.12), we
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?
?
get
???? Poo
Isid hi&
?
???? ????
11111.1 111.11
Or, on separatinr the variables and integrating:
2 + tP112 190.
1.
(4.13)
The equation of path so obtained is real for all values of a and 13 except for
those values at which pr. and p& change their signs, i.e., at which a1, Ili, pass
into a2, ;-;2 resPectively, or vice versa. At each such transition we will leave the
equation of path corresponding to the course of time before the transition, and find
a new one, selecting as the initial conditions for each successive piece of the
,,ath, the terminal values of the preceding one, i.e., in essence, adjusting each
successive Piece of the path to the preceding piece.
This change of sign and transition
takes place when et or 1.3 pass through
the 0 value, which takes place, accord-
ing to (4.9) and (4.10), at 8 -+ = 0
and a - to( = 0 respectively. Conse-
quently the boundaries of transition
(ft,.
al./Z.Z/Zhfd&
from al to a2, or vice versa, are the
straight lines fi = +- pf.i, and the boun-
daries (Fig.4.3) of the transition
from 131 to 52 or the reverse are the
straight lines a =
Je remark that the motion of the
end of the spin axis through all the
pieces of the path is clockwise in the
case under study.
- Path of End of Axis of a Gyro-
Pendulum with Downward Displacement of
Center of Gravity, Allowing for Friction
in the Gimbals
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This results from the fact that, by (4.9), a increases at positive p exceed-
ing 6, while for negative fi exceeding 6 in modulus, a decreases.
If for IBI ' p, lal > pa, then, turning to (4.10), we have a decrease in N at
positive a exceeding p(1, and an increase of H at negative a exceeding pa modulo. It
is not hard to see that all this means that the motion of the end of the spin axis
is clockwise on the path, and this is the motion of the spin axis along the path
even in the absence of all friction in the gimbals.
If, for INI pn, lal Pal then this means that the moment of friction exceeds
the moments due to the displacement of the center of gravity, and, consequently,
motion of the gyro is impossible.
Assume that at t = 0, a = ao > Ip&I > 0 N = No> IPI > 0 (cf. Fig.4.3). By
(4.9) and (4.10) we have for these cases & > 0; N < 0. Consequently the upper sign
must be taken for pa, and the lower sign for P, while the equation of the path
eq. (4.13) is written in the following form:
(2 + ($ P0)1'" +NY + (N?Pir?
which is the equation of the circle with center at the point (- p& PO and the
radius Ri = V(ao pO2 (rio - 6)2.
For f, =p, & passes through 0, and (3.13) ceases to be real. For the follow-
o'N.?
ing piece of the path a1 = a + Pa, must be replaced by a2 = a -Pain connection with
the change of sign of P&,. Taking as initial conditions for the new piece of path;
?
the terminal values of the latter, i.e., taking for t = tl, a = a1, and (3 = Pi, we
get
(2 - Pa )1 + ?dr (21 POt
(4.14)
which. is the equation of a circle with center at point (P&I PO and with radius
R2 = - pa (cf. Fig.4.3).
At a = pa, e passes through zero, and eq.(4.14) ceases to be real. For the
- ?
132 = 13 - c1(.3 must be replaced by 131 = 13 +P, in view of the
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?
?
chan;e of sign of 6. Then, selecting the initial conditions at finite values of
the solution (.4.14), i.e., taking for t = t2, a = p&l 1 = 02, we r?et
11 P.12 ? 4-%01: (,1 -1 P.1)2.
(4.15)
which is the equation of a circle with center at the noint (pa, -p0, R3 2+
(cf. Fi7.4.3).
At j = A "asses a-ain throu7h zero. On replacing in connection with
this a2 = a -pi 1T al = a + p and taking as the new initial conditions, finite
values of eq.(4.15), that is, taking, for t = t31 a = a3 and f-1 = -p, we ;et
11 ? ?.)? ? 0 f-f?31'?(23-1-Par.
which is the ecuation of a circle with center at the Point (a-pi, -ph) etc.
Thus, for each transition to the fo]loWng niece of the -ath, the center of the
circle, whose arc re2resents this :)iece, jumped clockwise to the adjacent angle of
the rectangle a'ccd (cf. Fig.4.3).
-.1e shall call this rectangle the rectangle of repose, and the angles p& and AN
the anrles of repose res,ectively of the outer and inner frames of the suspension.
As a result of this, the transition of the radius of the next circle will he
less than the radius of the nreceding circle 'r,y twice the angle of renose of the
outer or inner frames, denending on whether the center of the circle is disnlaced
narallel to axis of Oa or
In conserfzence, the resultant curve is represented by an involutional sPiral
(cf. Fig.4.3), consisting of arcs of circles of steadily diminishing radius, fitted
to each other.
The strairht lines on which the fitting is performed are marked by dashed lines
on Fig.4.3.
This process will be continued until the finite state of some piece of the tra-
jectory is in accordance with the conditions a i p60 W 6) which will mean the
imnossibility of frther motion.
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Thus the spiral will end at some point on the sides of the rectangle of repose.
For a gyropendulum with an upward displacement of the center of gravity, the
equations of motion will take the following form:
i? (2 Idl
Po).
On performing an analogous study, we may convince ourselves that the motion of
the end of the axis of the gyroscope over all the pieces of the path, will be coun-
terclockwise in this case. The
centers of the circles whose arcs represent pieces
of the path will also be displaced
counterclockwise at the corners of the
rectangle of repose.
As a result, at each transition,
Fig.4.4 - Fate of End of Spin Axis of
Gyropendulum with Upward Displacement
of the Center of Gravity, Allowing for
the Friction in the Gimbals
the radius of the next circle will not
A
decrease, but will increase by compari-
son with the radius of the preceding
circle at two angles of repose.
In this connection, the path, COM-
posed of the pieces, will notbe an in-
volutional but an evolutional spiral
(Fig.4.4), that is, with an upward dis-
placement of the center of gravity, the
deflection of the gyropendulum axis
under the influence of dry friction
will increase.
With the passage of time, the spin axis may occupy a horizontal position, and
then the upper gyropendulum may pass over into a lower gyropendulum.
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Section 4.2. Behavior of Gyronendulum in the Presence of a Hydraulic Arrester
The material such as given in this paragraph is taken with insubstantial modi-
fications from the book by B.V.Pulgakov.
Fi7,ure 4.5 (ives a diagram of the gyropendulum with hydraulic arrester in one
plane of oscillation.
As follows from its name, the hydraulic arrester is used to extinguish the pre-
cessional oscillations that can arise in the system under the action of disturbing
moments.
411 Fi7.4.5 - Diagram of Gyropendulum
?
with Hydraulic Arrester
As will be seen from the diagram,
the hydraulic arrester consists of two
vessels filled with liquid and connect-
ed by a capillary tube. The limited
rate of flow of the liquid through this
tube leacls to its nonuniform distribu-
tion between the vessels, which pro-
duces the moment N about the axis of
rotation of that frame of the gimbals with which these vessels are connected.
On the diagram of Fig.4.5, this frame happens to be the outer one. Its axis is
perpendicular to the plane of the drawing, and Ox is the axis of the inner frame.
As follows from this diagram, the moment may be renresented by the expression:
V
"a:S`:,
where a = distance between center of gravity of vessels and the axis of rotation;
S = cross-sectional area of vessels;
y = specific weight of liquid.
The pressure assuring the flow of the liouid through the capillary tube will be
n difference between the levels of the liquid over the true hori-
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zon in the tubes. If we take Poisseuille law that the rate of flow of a liquid is
proportional to this pressure, we get
(4.16)
Thus, in the presence of a hydraulic arrester, the expression for the moments
will be of the following form, if the moments of friction in the gimbals are
neglected:
mg15.
? mg13
On substituting these expressions in (4.1) and (4.2), likewise bearing in mind
(4.3) and (4.4), and associating them with the equations of motion (4.16), we ret
or
where
11: trWil
IIa mg1
- cita ? ? A.
?).p.o,
$+).3-1-(I?k)).-{.?. 0,
cs (1-1- ?)
mgl
?
k 1 ? .
mg1
On eliminating the variables 0 and T, we now obtain
C22 2o1 - 0.
(4.17)
(4.18) .
The. functions which together make up the solution of eq.(4.18), will die away
in the course of time if the roots of the characteristic equation (4.18) have a
nermtive real part. The satisfaction of this condition may be checked by means of
? .L.
iterion, without actually having to solve these equations.
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?
'According to this criterion, in an equation of the third order, all the roots of the
characteristic equation will have a negative real part if all the coefficients of
the equaticn are positive and if the followin7 inequality is satisfied
(4.19)
where A, 13, and C are respectively the coefficients of the second, first, and
zero-th derivative of the equation. In our case, the condition that the coeffi-
cients shall be positive is always satisfied for k > 0. As for inequality (4.19),
it will take the form:
1 - > 0
From the latter ineqwl'ty and from the inecrality k > 0 it follows that the dampin7
of the solution is assl_red by the condition
0 '111:S Ingl
(4.20)
nut if, in connection with the damning of the motion, assured by the condition
(4.20), q tends with the passace of time to anproach zero, then 5 and T will also
a:Troach zero, since the djfferential equations for these functions, which were con-
structed starting fro:. systan (4.17), will e entirely the same in structure and co-
efficients as eq.(4,1C) for q.
As a result, the nath of the. end of the spin is is transformed from the
circle that it wAs with an hdamped pendulum into an involutional spiral, with its
end at the orin of coordnates, that Ln the true vertical.
Section 4.3. Deviations of the Gyropendulqn
The cuilibrilm position of the snin axis of the Tiropendulum, allovrinr; for the
earth rotation and the disnlacaaent of the noint of susnension along the earth sur-
face, will not coincide, as will he shown below, with the true vertical. We shall
ten the denections of this equilihrin position from the true vertical the devia-
tions of the -:yronendulum.
Jo shall determine them for an undamped gyropendulum, ner;lecting the friction
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in the giMbals, since it would considerably complicate the analysis to allow for
these factors, without adding anything new in principle.
The Equations of Motion of the QyroDendulum on a MovinrY, Base
If, however, we take into consideration the moments of friction in the Eimbals
and the moments due to the hydraulic arrester, then for the moments Lk and
allowinc for the forces of inertia
caused by the longitudinal acceleration
of the aircraft, we fT,et (Fig.4.6):
ms
F1,7.4.6 - Gyropendulum on 1:ovini; Fase
11,,e sin K
by comparison with
1.1, mg la?m1V.
1? ? Mg la
On substitutinE these eKpressions
in (4.1) and (4.2), in which we also
substitute 7 and q according to (2.34)
and (2.35), and ner;lectinl the ey:pres-
sion
mg1
11
as a small quantity (which is true, even for high speeds and high latitudes), we get
the equation of motion of the 7yropendulum on a moving base in the following form:
or
cos K) mei ?
H sin K = ? mgla.
75
(4.21)
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?
V
% gm sln K
R *
(4.22)
Assume that V = const and k = const, then the equations of motion of the gyro-
pendulum will take the form:
y obi cos A.
,
. sin A ?
(4.23)
(4.24)
which differ from the form of the equations of motion with a fixed base only in
their constant rirht sides. It is commonly known that the equilibrium position of
any system whose motion is described by linear differential equations with constant
coefficients and with a constant right side is determined by the partial solution of
Lhese ovations. In this connection the deviations are determined by the partial
solutions of eq.(4.23) and (4.24), which partial solutions are of the form
p "" cos K.
A (4.25)
V
3. ust -' sin K ? . (4.26)
AR
Thus the deviations of the r7yropendulum consist of the deviations due to the
horizontal comnonents of the earth rotation, which we shall denote by (5c and the de-
viation due to the speed of flight, which we shall denote by 6v.
It follows from (3.25) and (3.26) that numerically
Z.4 = 1
Deviation Due to Earth Rotation
tot oil
( ? cos A ) ? ( sin r?l3
A
V
t
p
(4.27)
(4.28)
The deviation 6e represents the deflection of the spin axis from the true
-
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vertical in the plane of the true meridian (Fig.4.7), since - tan K.
Under the influence of this deviation a moment due to the displacement of the
Fig.4.7 - Deviation Due to Earth Fig.4.8 - Velocity Deviation
Rotation
center of gravity L , is Produced, equal in magnitude to
be
m g 1 sln 7, mgA.,
and directed perpendicular to the plane of the meridian, as shown in Fig.4.7.
This moment produces the precession 106e, which coincides in direction with the
horizontal component of the earth's rotation ut and equals it in magnitude, since
from (4.27) it follows that:
Velocity Deviation
The deviation 8. representa the deviation of the spin axis in the plane perpen-
(iirlilar to the direction of flight (Fig.4.8).
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Under the infl-lence of this deflection the moment Loy is formed, which is
directed as indicated in Fi-.4.E and is eval fn na.;nitide to
me sin 4%?-x me,.
Th:,s moment prochces the nrecossLon way, coincidin[: in direction with the ani-
lar velocit-: of rotation of the systen of coordinates oelnY.auin- to the velocity of
r_i:ht of the aircraft v'th res.:ect to the earth, and eq:al to it in mar7litIlde, for,
(4.2r)
.t. f,t? -
II
Condition of rond4stlr'2ance
Let 's find the cond'tion .1nder which (4.25) and (4.2A) will he partial solu-
t'ons of er_.(4.21) an.-1 (4.22) for / const.
Considerin- that 7 / const in the er_rressions for a. and P. we 7et
? ?
? -
4 ,=.0
rf
On st..t tri thesea-:.'lressions and also q. and in eq. (4.21) and (4.22),
-.ro -et
1't?
? ? K a) COSA ?s. - .
?R
I. V
? ,
:hence the rer;red conditcn is wren in the followinr form:
or, for the of the precoss;unal osc;llat;ons of the 7-ropendnlitm (the T,reces-
sion ar's:n- under the actjon of the nonent of the force of c-ravity alonr, the cone):
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T =. 2r 1
- i=84,4 mmm.
This condition is called the condition of nondisturbance (Schuler). When it is
satisfied, the behavior of the ryropendulum at V / const will not in any way differ
from its behavior for V = const.
It follows from this that by an appropriate choice of the design characteris-
tics of the Eyropendul.m, the influence of the lohrzitudinal accelerations on its
deviation may he eliminated.
- Influence of Accelerations
on the Gyropendulum
The physical meaning of this con-
dition is that when it is satisfied,
the precession of the gyropendulum
under the action of the moment of the
force of inertia, due to the lonEitu-
dinal acceleration, brings the spin axis
of the gyropendulam precisely into the
position corresnonding to the new value
of the velocity deviation.
In fact, the moment due to the
force of the inertia, produced by the
translational acceleration L'' is dir-
ected, as indicated in FiF.4.9, and is equal to
L t/W
The precession' , if the condition of nondisturbance is satisfied, is deter-
mined from the expression
61)
nct indicated in Fig.4.9.
mini ? V V
g iR
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It produces the increment of angle Ake equal to
!
w di dt I .
PR
AR ,I
As a resllt, the velocity deviation for each instant of time is determined by
the expression
at. =%,t.t, aat. ' (v. 1.) =
pp JR '
whore ovo and V0 are the values of the velocity deviation and the flight speed for
the begnninr, of the acceleration V.
Section 4.4. Effect of a Turn on the Gyronenduluin
Consider the behavior of the r:Iropendulum on a turn with respect to a system of
coordinates rotating about the vertical together with the aircraft, i.e., let us
determine the 1-ehavior of the gyropendultun with respect to an aircraft making a
turn.
If we do not consider the moments of friction in the gimbals nor the damping
moments, and if we neglect the earth rotation, then, considering the velocity vector
of the aircraft to be parallel to the axis of rotation of the inner frame, we find
the following ax.)ressions (Fig.4.10) for the moments 14, and Ly with respect to a
left turn:
mg!
L, , -mg la- ml V.,
Usinr, further, the aKpressions for p and q with respect to the left turn,
(2.3S) and (2.39) (cf. Chanter 2, Section 2.9), and substituting 14?, Ly, p and q in
ea.(4.1) and (4.2), we ret the eqtations of motion of the gyropendulum for the case
we are studying, in the following form:
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or
where
H (a w?P)
I1(+ ..a)-'
-
dl
(4.29)
(4.30
P7,e? !mu .8 4- (4.31)
A V..
Z:re= ? ?
so. 4 )k
(4.32)
For a right turn the sign of in (4.31) and .(4.32) must be changpd from posi-
tive to negative.
We shall consider the case when X - f21 < 4).3.
Then, for a right turn, we have
Y?Sp"" D*), (4.33)
V.'.
??? A
(4.34)
Jt.
that is, the signs of u- and u- are different, while those of L* leftand
left right
X;ight are the sane.
On dividing (4.29) by (4.30), we obtain, omitting the indices Tor the X":
=
dp 24.10
whence, separating the variables and integrating, we have the followinF, equation of
the nath of the vertex of the s:An axis:
where A is the constant of integration
or
(2 + z*)' rp=b.
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Thus, dOrLn: a turn, the end of the axis describes the C.rcle about a
-0.1nt dislaccd frail the ori:In of coordinates alon- the axIs Ca by the
nivic a = -cart, and durinr a ri-ht turn by the an-,1c a =r'rht (Fir.4.11).
- Gyro und in a Tim
71-.4.11 - Path of Lnd of Sun Axis
of Gyroendul.l.m in a Turn ,
1c note that the a :-)arent 7ert.cal, that the direct:on of the resultant of
the force of 7rav:ty a:1( the centrfur.al force of inertia, durin3 a left turn, is
'n the coord*natcs of the -iane Oaf'. alonr the axis Oa by the an!--,le a =
vw
Xlef:t = - tan -1 --1!, that is, in the sane sense in which the center of the circle
re-)lesent.n- thc -.ath of the end of the spin axis 's disnlaced. :Iith a right turn,
thev, ")Crif7ht
a !7arent -rerttcal .5 ds-)lacc alonr, the same axis Oa by the angle a = .
tan-I --'21% that is in the sense o-,nosite the disnlacement of the center of the con-
ter of the circle re-)resentinr the :-)ath of the end of the snin axis.
The motion of the end of the sun axis a]onr its nath is clockwise durinr a
left t-rn and cointerclockwise durin-, a rir:ht turn.
This res'ilts directly from ec.(4.29) and (4.30), as well as from eq.(4.31) and
0, 22). In a left turn when ploft 0 for nositivc m the anrle a increases,
.
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while for negative R the angle a decreases; with a right turn, when u* 0 for
positive values of 6, the angle a decreases, and for negative values, it increases.
From eo.(4.29), (4.30), (4.31), and (4.33), it results that the angular velocity of
the motion of the gyrocompass axis along its path with respect to the aircraft takes
place at various angular velocities: on a left turn at anular velocity (0B + X, on
a right turn at angular velocity (An - X.
If X is selected with a magnitude such that it will satisfy the condition of
nondisturbance, then for all practically conceivable values of wtu rn' the value
of X may be neglected by comparison with wturn Doing this, we get
:I ? ----- tam
lejt
/
UI turn
?V
/ 101 Aid , -,R ?
Moreover, it should not be forgotten that the motion of the gyropendulum with
respect to the earth will be performed in a manner completely different, since we
must set up the system of coordinates OW' and must set up the conditions of motion
of the gyropendulum with respect to this fixed system of coordinates.
Examples. The gyropendulum gyro horizons desimled with a weight of the order
of 6 kg, have the following characteristics:
//.-- 1,7. 106 q cm sec,
mg1- 12:.0 g cm,
mg! 1
# ?7'4 10-3 --
?
H Sec.
If we take for the moments of friction in the gimbals Lp, a quantity of the
order of 5 g-cm, then for the angles of repose pa 6 = p we obtain
p ? ?0,23. ?
Intl
A,-,rinfion due to the earth rotation, 53 taking for itsits maximum
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value w1 =e' corresponding to a position on the equator, we get
113-0,b6?.
Let the aircraft be in flight at a speed of 540 km/hr = 150 m/sec. Then, for
the velocity deviation, we shall have
V
tv
PR
When the conditions of nondisturbance are satisfied, the deviations will take
the following values for the same values of (o3. in V:
P 3 3
Pt. ? LI
Let the aircraft perform a right turn with a 30? bank at a speed V = 540 km/hr
which corresnonds to (0 = 37.8 x 10-3 1/sec. The center of the path is displaced
turn
in this case from the true vertical, with the characteristics given, by the angle
leftI g
A V..?f? s.
/ (*I
If the conditions of nondisturbance are satisfied, for the same characteristics
of the turn, we have
is -.1.07 .
Oft
?
1.14
ruillti
The maximum deflection of the gyrohorizon from the true vertical during the
turn may be taken as a quantity equal to double the value of Y .
Let us assume that during a period of 30 sec, the aircraft varies its speed by
75 m/sec,' that is, that it moves with a translational acceleration V = 0.25 g.
:ftien the condition of nondisturbance is satisfied, the snin axis of the gyro-
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horizon, under the influence of this acceleration, will be deflected from the true
vertical in the plane perpendicular to the direction of flight by the angle A6v =
0.530, with an accuracy equal to the increment of velocity deviation Abv = 0.530 for
the velocity increment taken.
In the gyrohorizon under consideration, under the same conditions, the spin
axis will be deflected by an angle A05v, equal to
t' I
At the same time the increment of velocity deviation AOIr will be
Atv ? - ?0.(19'.
/R
i.e., the deflection of the spin axis under the influence of the acceleration con-
siderably exceeds the quantity corresponding to the increment of velocity deviation.
It follows from these data that a gyropendulum gyro horizon in which the condi-
tion of nondisturbance is completely satisfied, will have no error at all from the
longitudinal accelerations, and will have a small .error due to turns; as for the
deviations due to the rotation of the earth and the velocity deviation, these can
easily be taken into account.
The zyropendulum above considered has a considerable turning error. The longi-
tudinal acceleration also has a substantial influence on it-.
In other words, the deviation from the satisfaction of the condition of non-
disturbance in a gyropendulum gyro horizon increases the most unpleasant feature,
in the sense of the difficulties of estimating the error.
But, in order to satisfy the condition of nondisturbance, to decrease the value
of X = ME'
I the kinetic moment of the gyroscope together with the kinetic moment of
1-1
the entire installation, would have to be increased to several times the kinetic
moment of the gyro horizon under consideration. The weight would also haAre to be
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increased several times. nut, the weiftht of this instrument is already rather .7. reat,
it is '4-5 times as 7reat as the weight of a (:yrohorizon with a pendulum correction
systa-a.
In this connection, Tyrohorizons with a pend,11,1n correction system have proved
ncre corWctuent with res:)ect to satisfy-in7 the denands made on aviation instrunents.
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CHAPTER V ?
BASIC INFORMATION ABOUT THE THEORY OF THE GYROCOMPASS
Section 5.1. Behavior of the Undamped Gyrocompass on a Base
Fixed with Respect to the Earth
A gyroscope with three degrees of freedom whose center of gravity is displaced
downward with respect to the center of the support in the equatorial plane, along
the negative semiaxis 0y, is called an
undamped gyrocompass (Fig.5.1).
The influence of dry friction in
the gimbals will not be taken into
account, since it is very small in
actual gyrocompasses.
4
Fig.5.1 - Gyrocompass on Base Fixed
Then, for the moments L , L the
x y
expressions
Li= mg 1 Cos (.z) fug 10.
will hold (according to Fig.5.1).
with Respect to the Earth
On the basis of (2.36) and (2.37),
in which we must put V = 0, or directly on the basis of Fig.5.11 we have the
following formulas for p and q:
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-013.
q = a 4. ail? 1111 , 5.
On substituting these expressions in the general equations of motion of the
gyroscope, (3.1) and (3.2), we get
or
where
/ha' es - 1:0 ..-=171,c/ 15.
/1( ? wict) -
dt---= Q. A" ?OP
-
dl
one
I.
Iiz
(5.1)
(5.2)
We shall seek the path of the end of the spin axis of the gyrocompass, which is
described by it on the coordinate nlane 0 (x-; perpendicular to the meridian, i.e.,
the axis 0 with origin at the point of intersection between this coordinate plane
and the axis 0,-;
On dividing, for this purpose, eq.(4.1) by eq.(4.2), we obtain
- 11 ? 1?11 1111
dIA "13
Separatin,c,, the variables in the differential equation so obtained, and inte-
grating, we have the solution
3t ( 62
v,
? eel
? lot
or, dividing the right and left sides by 0(;, we have
rrss
. A ? so 1
+-. -
Fi7
0 .01
? mil
(5.?\
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go,Jght is an ellipse with the center displaced from the origin of
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w2
coordinates along the positive semiaxis 00 by the angle 0. = with a major
+1
?
semiaxis equal to 00 and a minor semiaxis that isV( wl
times smaller thane 1
0
X wl
wheree depends on the initial conditions.
0
At t = 0, let a= 0, 3= 134.= , (2)2 . Then, on the basis of (5.3), we have
A +(1)1
04=41%
i.e., if, at the initial instant, the rotor of the gyrocompass was in the plane of
the meridian, being raised above the plane of the horizon by the angle 0 = p*, then
it will remain in this position, thereby determining the position of the geographic
meridian.
If, at t = 0, CI= ao P =Po, then, on the basis of eq.(4.3), we have
1/7- ahl
?
.01
k -4 WO 1
13
that is, if at the initial instant, the axis of the gyrocompass rotor occupies an
arbitrary position, then it will rotate about the direction corresponding to the
stable state p = 0.. In this case (Fig.5.2), the end of the axis describes, on the
coordinate plane Oaf% an ellipse with
the center displaced from the origin
of coordinates by the angle p = 0.,
with the major semiaxis a =00, by
eq.(5.4), and a minor semiaxis
? Path of End of Axis of
Gyrocompass
b = 80 1 that is, equal to
X 4. (x)1.
b 11/ .1 all; +
oil +A
(A1 OD )2 ?
.
Thus, if an undamped gyropendulum performs a precessional motion about the
true vertical with a deflection from it depending on the initial conditions, then
the undamped gyrocompass will perform a precessional motion about the axis lying in
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is
the plane of the meridian, with a deviation from it which likewise denends on the
cond tions.
:y usin- 0.am-in- of the oscillations in a -yroPendulum, the character of its
motion can 1)e modified in such a way as to force the axis of its rotor to aPproach
the trio vert'cal, in an involit'onal snira2 .
-7 analo--, ,s*nr? .in- of t';o oscillations of the ryrocr-, aqs (by the same
method as in the -yro )enullIc or other.:ise), the character of the motion in the
-yrocompass may likew'sc Le modified in the direction of forcing its rotor axis to
alproach, in an involutional s)iral, to the direction above described in the plane
of the meridian.
Section 5.2. Influence of Yeloctv and Acceleration on the Behavior
of the G-rrocomnass
If we allow for the influence of acceleration on the displaced center of
-ravity, then we :et, for the moments LL and L, on the basis of Fir.5.3, the
-
5
follow n- equat Ions
1., mg1 cus( ? Inn, K..
-o? 1
(5.4)
On substitutin- the values of L, n, n, and q (from eqs.2.36 and 2.37) in the
x y
7eneral equations of motion of the ryroscone, eqs.(4.1) and (4.2), and neglecting
in
-- sin K in comnarison to and---- by comnarison with (,),, we get:
y
or
H (+ ci? (?01- (111)= m0 ? m11/ Cos K;
t.
I + wo ? cosh' ). 0.
+ wi)o w: ). cos K;
(ft
= ?w12,L cos K.
It
90
(5-5)
(5.6)
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Assume V = const, i.e., V = 0.
Then the equations of motion of the gyrocompass, eqs.(5.5) and (5.6), in which
the influence of the constant velocity of displacement with respect to the earth is
taken into account, will differ from eqs.(5.1) and (5.2), which were investigated in
the preceding section, only in the presence of an additional constant term in the
right side of the second equation. It follows from this that the solution of eqs.
(5.5) and (5.6) will differ from the preceding solution only in the displacement of
the center of the ellipse representing the path. This displacement is defined by
the partial solutions of eq.(4.5), (4.6), which are of the form:
? En
(5.7)
11.
A + le 1
It is clear that this displacement of the center of the ellipse from the meri-
dian, which means the displacement of the stable state of the gyrocompass from the
meridian, did not vary by angle but
did vary by angleoc: when the gyro-
compass was fixed with respect to the
earth, this displacement was equal to
zero, and, consequently, to the stable
state there corresponded the position
of the rotor axis of the gyrocompass in
Fig.5.3 --. Gyrocompass on Moving Base
91
the plane of the meridian. On allowing
for the rate of displacement with re-
spect to the earth, it is not equal to
zero, and consequently a deviation
from the plane of the meridian corre-
sponds to the stable state.
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This deviation is called the deviation of the gyrocompass.
As follows from eq.(5.7), it depends on the rate of displacement, the course,
and the latitude oC the place.
Let the speed of displacement V = 50 1n/hour, the latitude of the place (1) =
= 600, the course K = 0 or
7or Lllese data the deviation a. will be equal to
a* = 37.0?.
the sane time, for a course K = 900 or 270?, it will be equal to zero, i.e.,
'hen appliel the speeds of modern aircraft, this variation has so wide a range
tat it wonll 1-rri1y be possible to estimate it in practice with sufficient ac-
Let us find the condition under which the functions a*, p, would be partial
solutions of ens.(5.5) and (5.6), even for 7 const.
On substit: a* and 1,-;* in ens.(5.5) and (5.6), we get
V
0 - ? cos K?1 -R Cos K,
Rol
whence it folla.:s that the required condition will be written in the following form:
The condition so found also bears the name of the condition of nondisturbance
(Schuler). If it is satisfied, then the gyrocompass will have no other deviations
besides the velocity deviation.
It is eas- to show that the period of free oscillations of the gyrocompass can
be expressed sufficient accuracy by the expression
T 2i
,
y Awl
from which it follo-;-s that, on satisfaction of the conditions of nondisturbance, the
CY)
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period of free oscillations of a gyrocompass will equal:
T? 2w R 84,4
min
The period of damping of the oscillations will be correspondingly great. In
connection with this it takes a considerable time until the gyrocompass after start-
ing occupies the position of the stable state.
This circumstance is an additional and fundamental obstacle to the use of the
gyrocompass in aviation.
It is not hard to see that for ocean vessels, whose speed is relatively small,
and whose voyages are prolonged, these circumstances do not play so substantial a
role, and therefore the gyrocompass, as a compass whose readings do not depend on
the surrounding ferromagnetic masses, finds wide application there.
11
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CHAPTER VI
SCHEIB AND CHARACTERISTICS OF CORRECTION SYSTEPS ?
Section 6.1. Structural Scheme of Correction Systems
The correction of a positional gyroscope as a whole is most often accomplished
by means of two mutually inde-iendent systems, each of which is intended for one
specific degree of freedom of the gyroscope suspension, i.e., for one definite frame
of this sus;ension. There are very feu exceptions to this rule, and even these are
not of fundamental importance.
A structural diagram of the correction system can be drawn according to
Fig.6.1, where Dcl and Dc2 are signal transmitters, D and D are moment trans?
Ll L2
mitters (moment devices) of the correction systems respectively of the first (inner)
and the second (outer) frames of the gimbals, wi-ille the linkage between the signal
transmitter and the moment transnitter in each correction system is accomnlished by
the crossover method, in which each frame is corrected by the application of a
moment to the other frame. We note that, in certain cases, the signal transmitter
and the moment transmitter of the correction system are merged in a single design
unit.
As the signal transmitter, a certain distributing device is ordinarily used,
consisting of two elements, one of which is connected to the sensitive element of
the correction system, the other with the gyroscope frame to be corrected. Wheft
these elements are mismatched, energy of one form or another is transmitted to the
nnected to the given signal transmitter.
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By a moment transmitter we mean a certain actuating device which, under the
influence of the energy received from the signal transmitter, develops a Moment
about the axis of rotation of that gyroscope frame which is diagonal with respect
411 to the frame being corrected. The magnitude and sign of this moment depend on the
magnitude and sign of the mismatch or discrepancy between the elements of the signal
transmitter.
?
?
?
/"" - ????
-
? _ - _
We shall term the relation between
the moment developed by the transmitter
and the discrepancy in the signal
transmitter the correction character-
?
o.
example the radio gyrocompass, air
istic. Other devices can be used as
the sensing members of the correction
system, besides such instruments as the
pendulum and the magnetic needle, for
speed indicator, etc. The case of the
Fig.6.1 - System of Radial Correction so-called correction for perpendicu-
larity of the gyroscope axis to the
I, II - Axis of suspension; Del - Trans-
axis of the outer frame must be men-
mitter of signal; D - Transmitter of
12
correction moment to inner frame; Dc2 -
signal appears on deflection of the
tioned. In this case, the correcting
Signal transmitter; DL, - Transmitter
spin axis from the position perpendi -
of correction moment to outer frame
cular to the axis of the outer frame.
Depending on the kind of connection between the signal transmitter and the
gyroscope, we shall distinguish internal and external correction systems. By an in-
ternal correction system we shall mean a correction system in which there is a
direct connection between the signal transmitter and the gyroscope, and by an exter-
nal system, a system in which this connection is indirect. This type of connection
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is often used in the gyroscopes of autopilots. It reduces down essentially to hav-
ing the signal transmitter connected with the aircraft instead of being connected
with the gyroscope. In this case it must be borne in mind that the aircraft is un-
der the control of the gyroscope, which. in this case is the sensing member of the
autopilot. As a result, any departure of the gyroscope from the required direction
results in the departure of the aircraft as well, and, consequently, in the dis-
placement of the member of the signal transmitter that is connected with the air-
craft, with respect to the member of this transmitter that is connected with the
sensing member of the correcting system. This will mean that the corresponding
moment transmitter is put into operation.
The use of the external correction system is explained, first, by the fact that
in this case it is sometimes possible to obtain a more satisfactory design scheme of
correction, and secondly, and this is of more substantial importance, that an exter-
nal system of correction assures the liquidation of irregularities in the regula-
tion, or, in other words, provides automatic compensation of the action of a con-
stant disturbing moment on the aircraft. This latter is obtained as a result of the
fact that the correctioh of the gyroscope, and with it the variation in the position
of the aircraft, will continue until the aircraft occupies the only position in
which the discrepancy in the transmitter of the correction signal disappears. But
this means that any irregularity in the regulation will always be eliminated. The
position of the gyroscope, when an external system of correction is used, is so
selected that the signal fed from the gyroscope to the automatic pilot shall assure
the compensation of the constant disturbing moment.
Section 6.2. Schemes and Characteristics of Pneumatic Type Correction Systems
According to the form of energy used to excite the positioning moment, radial
correction systems are divided into, pneumatic, electric, and mechanical.
Figure 6.2 gives one of the schemes of a pneumatic type of external correction
system produced in the USSR, developed at the end of the 1930 for the lateral
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stabilization gyroscope of an autopilot. The signal transmitter here is the air
collector (1), connected with the aircraft. Two jets of the collector are covered
by slide valves, connected with the pendulum (2), whose axis coincides with the axis
of the frame being corrected.
Fig.6.2 - Diagram of Air Correction:
P - Air Pressure;
1 - Air collector; 2 - Pendulum with
shutters; 3 - Hemisphere with notches
7Fg.rI
?
".
Fig.6.3 - Mixed Characteristic of
Correction System
- Segment of proportional part;
- Steepness of proportional part of
characteristic; k - Moment of correc-
tion of constant part of characteristic
The slides are so oriented with respect to the jets of the air collector that,
on the deflection of the pendulum with respect to the air collector, the opening of
one air jet will increase and that of the other will decrease.
The moment transmitter is a hemisphere connected with the gyroscope and having
a grooved surface, against which the above described air jets impinge. If the air-
craft occupies a position in which there is no disagreement between the elements of
the signal transmitter (the air collector and the slide), that is, a position in
which the slides equally cover both of the collector jets, then no moment will act
on the gyroscope, since the pressures produced on the hemisphere by the air jets
will be the same, by virtue of the identical covering of these jets. If the air-
craft departs from the required position, then a discrepancy will arise between the
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elements of the signal transmitter, and as a result the slides will cover the two
collector jets differently. The pressures produced by the jets on the hemisphere
will be different, and a resultant moment will appear in connection with that
difference.
Under the action of the moment, the gyro will precess. In this case a signal
will he fed from the transmitter connected with the gyro to the autopilot, which
will then turn the aircraft into the necessary position.
It may be considered with a sufficient degree of accuracy that the correction
moment is proportional to the discrepancy in the signal transmitter, until one of
the air jet in the jets in the collector is completely covered and the other is
comnletely open. After this happens, the moment will remain constant. Thus, if we
denote the moment by L. and the discrepancy in the signal transmitter by o, we get
the characteristic of the correction for the given case, as shown in Fig.6.3.
This external correction system using a pendulum is used in this gyro only with
resect to the outer frame. The inner franc, however, is corrected by perpendicu-
larity of the spin axis to the axis of the outer frame. For this Purpose (Fig.6.4)
two shutters are rigidly attached to the inner frame, and on the outer frame, two
nozzles from the air collector. The reactive forces of the jets leaving the nozzles
act on the outer frame. .1hen the -;,yroscope axis is perpendicular to the axis of the
outer frame, the reactive forces of the jets will be the same, since the jets them-
selves will the same, and the moment Produced by these forces will be equal to
zero. lihen the perpendicular position is disturbed, the jet impinging on one shut-
ter will increase, and that on the other shutter will decrease, the equality of the
reactive fdrces of the jets will be impaired, and a resultant moment will appear
about the axis of the outer frame. .On account of this resultant moment, the inner
frame will begin to 2recess in the sense necessary.to restore the perpendicular
. .
position between the s jin axis and the axis of the outer frame.
In this way, in the case we have described, the signal transmitter is used
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directly as the moment transmitter, the elements of this transmitter being connect-
ed with the gyroscope, and, consequently, this system of correction is an internal
one.
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Fig.6.4 - Pneumatic Correction System Fig.6.5 - System of Pneumatic Correc-
for Inner Frame, Holding Spin Axis tion for Inner Frame in Directional
Perpendicular to Axis of Outer Frame Gyro
0I _ Outer frame; II - Inner frame;
1 - Air nozzles; 2 - Shutters
1 - Rotor; 2 - Flanges; 3 - Ail. noz-
zles; Fk - Corrective force; LI, -
Correction moment; wk - Angular velo-
city of corrective precession
The characteristic of this correction system will be the same in the working
range of angles of mismatch, as in the preceding case (Fig.6.3).
Figure 6.5 gives still another version of a correction system for the inner
frame with respect to the position of the rotor axis relative to the axis of the
outer frame, which is used in pneumatic directional gyrocompasses.
The moment of correction here is produced by means of fins on the rotor, and
of the air jet used to maintain the rotation of the rotor. If the spin axis of the
gyroscope occupies a position perpendicular to the axis of the outer frame, then
this jet will impinge only on slits in the rim of the rotor and will only produce a
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moment relative to the spin axis. If, however, this perpendicular position is dis-
turbed, then the jet will also impinge in part on one flange or the other, as a
result of which a certain moment about the axis of the outer frame will occur, thus
producing a correction of the inner frame. Instead of the flanges today,
scythe-shaped grooves are used, which produce an analogous effect.
The characteristic of this correc-
tion over the working area may, with a
sufficient degree of accuracy, be taken
art t9
as oroportional, that is, having the
form indicated in Fig.6.6.
///f Figure 6.7a shows still another
type of the system of pneumatic correc-
tion by maintaining a perpendicular
of Correction
position of the spin axis and axis of
outer frame. This system of correction is used in directional gyroscopic instru-
ments having electric gyromotors.
On the periphery of he rotor casing of the gyromotor, a series of slits (1)
has been cut.
When the rotor rotates, an increased air pressure is produced near its rim, and
the air passes through the slits (1) in the jets tangent to the surface of the
shell. The moment of the reactive forces of these air jets is balanced by the mo-
ment of the forces of reaction in the supports of the outer frame, when the rotor
axis is perpendicular to the axis of the outer frame.
When the rotor deviates from the position perpendicular to the axis of the out-
er frame, then the moment of the reactive forces produces the pair of forces FK and
the moment of correction L. The moment of correction LK causes the gyroscope to
precess to the neutral position indidated.
Figure 6.7b shows a modification of this correction system. Instead of slits,
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Fig.6.6 - Proportional Characteristics
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in this case the two nozzles (1), attached to the inner frame, are used. Jets of
nitrogen emerge from the nozzles on opposite sides. When the rotor axis is perpen-
dicular to the axis of the inner frame, the reactive force of the jets lies in the
Fig.6.7 - System of Pneumatic Correction of Inner Frame of Directional
Gyroscope with Electric Gyromotor
I - Slits in gyromotor casing; FK - Force of correction; Lk - Moment
410 of correction
plane of the axis of the outer frame. In this case there will be no moment relative
to the outer frame. When the axis of the rotor deviates from this position, the
force of the jets leaves the axis of the outer frame, and deviates by the quantity
rK, proportional to the sine to the angle of deflection of the spin axis. This pro-
duces the correction moment LK, which returns the spin axis to its previous posi-
tion.
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The system of correction shown in Fig.6.7b is used in the gyro of the
remote-reading electric gyromagnetic compass DGMK-3. In order to prevent the cor-
recting moment from varying with the altitude of flight, the gyroscope of the
DGMK-3 Is placed in an air-tight casing, so that the reactive force of the jets re-
mains constant.
Figure 6.8 gives a version of the pneumatic correction system where the signal
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transmitter likewise directly plays the role of the moment transmitter.
The principle of its design is as follows. To the inner frame of the gyro,
whose role in this particular case is played by the rotor casing (1), the air cham-
ber (2) is rigidly attached. For each frame, this chamber has two ports (3) to dis-
charge air from the chamber, which ports are arranged symmetrically with respect to
the axes of rotation of the frames.
This chamber is an element of the signal transmitter connected with the gyro.
But it is at the same time also the moment transmitter, since the reactive forces of
the release of the air jets flowing from the ports produce a moment about the axis
of rotation of the frame located diagonally to the frame being corrected. The
second element of the signal transmitter is provided by the pendulum shutters, sus-
pended on an axis parallel to the axis of rotation of the corrected frame. The
shutter covers the ports of the air chamber in such a way that if the axis of
symmetry of the ports coincides with the line of the edge of the shutters, then the
air jets issuing from the ports will be the same. If this coincidence is disturbed,
however, i.e., if a mismatch arises between the position of the corrected frame and
the nosition of the shutters, then the equality between these air jets is destroyed.
When the equality of the reactive forces is destroyed, a resultant moment of one
sign or the other appears.
As already stated, this moment will be diagonal with respect to the corrected
frame, that is, with respect to the plane in which the position of the gyro was dis-
turbed.
This system of pneumatic correction is very widely used. It is used both to
correct L7yro verticals and for horizontal correction of directional gyros.
Its characteristic is of the form shown in Fig.6.3, the width of the area of
-proportional correction 4) being determined by the width of the port and the length
of the pendulum shutters.
Figure 6.9 gives the arrangement of the pneumatic corrector used in the
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USSR-produced (1K-2 directional gyro for correcting the outer frame. The signal
transmitter here is the air collector (5) with its two nozzles placed on the rotor
casing and covered by the eccentric slide (7): The slide is connected with the mag-
i
netic system (6), whose axis of rotation is likewise attached to the rotor casing.
To the matched state of the trans-
mitter corresponds the equal covering
by this slide of the collector nozzles,
while to the mismatched state of the
transmitter corresponds unequal covering
of the two nozzles. In the matched
Fig.6.8 - Arrangement of Pneumatic
Corrector with Pendulum Slides
1 - Rotor casing; 2 - Air chamber;
3 - Air ports; 4 - Pendulum shutters;
4)- Mismatch between rotor axis and
shutter axis
Fig.6.9 - Arrangement of GMK-2
Pneumatic Correctois
1 - Rotor; 2 - Rotor casing (inner
frame); 3 - Outer, frame; 4 -Scale;
5 - Air collector; 6 - Magnet;
7 - Eccentric slide; 8 - Air chamber;
9 - Pendulum shutters; 10 - Shutter of
outer-frame corrector; 11 - Membrane of
pneumatic relay with plunger;
12 - Pneumatic relay
state, the same pressure reaches both the receiving nozzles, while in the mismatch-
ed state, there is a different pressure in each.
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An air chamber with two pairs of ports, attached to the inner frame (casing) of
the gyro, serves as the moment transmitter. One pair of the ports, the horizontal
pair, is covered by pendulum slides and serves to correct the inner frame of the
gyroscope, since the moment produced by the reactive forces of the discharge of the
air jets issuing from these ports acts about the axis of the inner frame. The
second pair of ports, the vertical ones, serves to correct the outer frame, since
the moment produced by the reactive
L. forces of the discharge of the air jets
Pi
L.
Fig.6.10 - Characteristics of Correc-
tion of Outer Frame of GMK-2
a - Characteristics of correction using
elastic membrane; b - Characteristic of
correction with inelastic membrane;
h - Path of membrane; hm - Maximum path
of membrane; 4)A- Hysteresis zone
from these ports acts about the axis of
the inner frame. The second pair of
ports is covered by shutters connected
by a plunger with the membrane of an
air relay, the chambers of which com-
municate with the receiving nozzles of
the above described signal transmitter.
In the matched state of this transmit-
ter, the pressures in the chambers are
the same, the membrane is in a neutral
position, and the shutters connected
with the membrane equally cover their
ports. With a mismatch in the signal
transmitter, the equality in the relay chambers is impaired and the membrane is
flexed toward one side or-the other. This displaces the slide, and the covering of
the ports becomes unequal, thus producing a moment of the corresponding sign.
Thus, for the correction of the inner frame, the air chamber in this case is at
the same time a signal transmitter and a moment transmitter. The characteristic of
the correction for this frame will be of the form shown in Fig.6.3. But for the
correction of the outer frame, it will only be a moment transmitter, connected with
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the signal transmitter by mans of a special pneumatic relay. The characteristic of
the correction will therefore depend, here as well, on the characteristics of the
relay. If the relay has an elastic membrane, then the relation between the path of
111 this membrane h and the mismatch 0 will be of the form given in Fig.6.10a, where OA
is the value of the error necessary to produce a pressure difference capable of
overcoming the force of dry friction of the relay plunger; here the right part of
the characteristic corresponds to the condition 0 > 0, while the left part corre-
sponds to the condition 0 < 0; hm is the maximum possible movement of the membrane.
Assuming further that the relations between the moment LK and the displacement
of the shutter is of a proportional nature up to full opening of the ports, and con-
sidering that the motion of the membrane does provide this full opening of the
ports, we get the result that the diagram of Fig.6.10a on a different scale repre-
sents the characteristic of correction in the case here being considered.
If the relay has an inelastic membrane, then the relation between the motion of
the membrane and the error will be different.
?
Until an error or mismatch equal to OA has accumulated, the membrane will be
motionless, and will then be displaced as far as the stop; the backward motion,
likewise as far as the stop, will be performed after an error of the opposite sign
equal to OA has accumulated. If we retain the same assumption with respect to the
dependence of the moment LK on the motion of the membrane, then we get a character-
istic of correction that in this case, as well, is of the hysteresis form given in
Fig.6.10b.
The methods and characteristics of a pneumatic corrector that have been here
considered are the most typical. Their principle merit resides in the simplicity of
their design, which is a point particularly applicable to arrangements using a pen-
dulum as the sensing member of the correction mechanism. It must be noted that in
those cases where the rotation Of the gyro rotor is likewise maintained by the
energy of an air jet, it is most advisable to use a pneumatic type correcto;. as
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well, and, in fact, this is usually done. However, the design simplicity of the
pneumatic corrector impelled the search for means of applying it to gyroscopes with
electrically driven rotors. Such designs for gyro verticals were particularly
favored in Germany. But such a solution is not correct for nonhermetic instruments.
' The reason is that a pneumatic correction system for nonhermetic instruments has the
disadvantage of hay:rig its efficiency de,end substantially on the flight altitude.
With a pneumatically driven rotor, this shortcoming will have less of an effect,
since in this case the rotor speed also decreases with increasing altitude, although
to a lesser extent, and the decreasing efficiency of the corrector is accompanied by
some reduction of the kinetic moment as well. But, with an electric drive, the
rotor speed will be :ractically independent of the altitude, and thus the reduced
efficiency of the corrector will make it im:.ermissibly sluggish at high altitudes,
or, if it is selected for :Ise under altitude conditions, it will become impermissi-
bly shar,_ near the ground.
The transition to electrically driven gyro rotors has been due to such advan-
tages of electric gyromotor as higher speed, and, consequently, smaller dimensions
for equal kinetic moment, elevated starting torque, facilitating the starting of
Qrromotors at a low temperature, convenience of installation (no air line), weak
altitude-dependence of the rotor speed, absence of the increased corrosion of parts
that is characteristic of gyro instruments with pneumatic drive, absence of the
heating owing to the heat given off by the gyromotors, etc.
Thus the transition to the electric drive in gyro instruments, with its sub-
stantial design and operating advantages, has made it necessary to develop electric
forms of radial correction as well.
Section 6.3. Circuits and Characteristics of Electric Systems of Radial Correction
Figure 6.11 gives the circuit of one of the versions developed in the USSR of
the electrical corrector of the induction type, as applied to one of the frames of
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The signal transmitter (cf. Fig.6.11a), consists of a contact roller connected
with the gyro frame being corrected, and a contact strip, connected with the sensor
of the corrector and divided into two parts by an insulating gap.
In the matched state of this transmitter, the contact roller is at the center
of the insulating strip; when it is mismatched, the roller is displaced from that
center. The moment transmitter consists of a system of two inductance coils, in the
slots of which there moves a current-conducting disc, connected with the axis of ro-
tation of the corresponding frame (i.e., located diagonally to the corrected frame).
As will be seen from the electrical circuit (Fig.6.11b), one of the coils is always
energized by one of the phrases of a triphase alternating current line. The second
coil, however, which we shall term the control coil, is connected to one of the re-
maining two phases of the line through the signal transmitter, depending on which
half of the contact blade of this transmitter is contacted by its contact roller.
Thus, so long as the contact roller remains on the insulated strip, that is, so long
as the signal transmitter is in the matched state, only one coil will be energized,
and, consequently, no moment will be developed on the driving disc. When the signal
transmitter is mismatched sufficiently to bring the contact roller onto either of
the conducting halves of the blade, both coils will be energized, and the alternat-
ing magnetic fluxes produced by these coils will be phase-shifted by 120?, in one
sense or the other, with respect to each other. Consequently a moment of one sign
or the other, and of constant magnitude, that is, not depending on the degree of
mismatching, will arise on the current-carrying disc on account of the interaction
of the eddy currents and fluxes, which are phase shifted with respect to each other.
Since the insulating strip has certain finite dimensions, a certain value of
the mismatching is always required to cover this strip, and this value determines
the zone of insensitivity of the given correction system. Taking this circumstance
into account, and bearing in mind that after the roller passes onto the conducting
part of the blade, the moment transmitter will develop a moment of constant value, .
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we obtain the correction characteristic for the given case presented in Fig.6.12.
Figure 6.13 shows the arrangement for a considerably later version of the elec-
trical induction-type corrector, representing a different design solution of the
same ideas used in the version described above.
Fig.6.11 - Circuit of Induction.. Fig.6.12 - Constant Characteristic of
Electrical Corrector Electrical Corrector
I and II - Axes of suspension;
1 - Contact roller; 2 - Pendulum;
3 - Contact blade; 4 and 5 - Induction
coils; 6 - Current-carrying sector
e 6- Neutral zone of signal
transmitter of corrector
The signal transmitter here (Fig.6.13) is a bulb with a drop of mercury and
three contacts so arranged that the plane of displacement of the mercury drop shall
coincide with the plane of rocking of the corrected frame.
One of the contacts of the bulb is at its center, the other two at the edges.
In the matched state of this transmitter, the mercury drop is at the center of
the bulb and covers only the central contact; in the unmatched state, the drop moves
away from the center; and on operation, the mercury connects the central contact
with one of the edge contacts. We note that by mismatching of the signal trans-
'. mean its inclination with respect to the vertical.
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The moment transmitter here is a miniature induction motor with a
short-circuited rotor, seated on the axis of rotation of the corresponding frame
(that is, of the frame located diagonally across from the corrected frame). Exactly
as in the preceding case (cf.diagram of Fig.6.11b) one of the exciting windings of
the miniature induction motor is always energized by one of the phases of a triphase
supply line.
When the mercury connects the cen-
tral contact of the signal transmitter
with one of the extreme contacts, the
second phase of the miniature motor is
connected with one of the other two
phases of the supply line. As a re-
sult, a rotary field is produced in the
Fig.6.13 - Diagram of Electric Corrector miniature motor. The sense in which
with Mercury Switch
this field is directed will depend on
what phase has been connected to the
I, II - Axis of suspension
phase of the miniature motor. On this
1 - Bulb; 2 - Mercury; 3 and 4 - Wind-
account, a moment of one sign or the
ings of induction motor; 5 - Short
other is developed on the rotor. It
circuited rotor of induction motor
will be of constant magnitude (i.e., it
will not depend on the value of the error in the signal transmitter).
In order for the mercury drop to be able to roll from the central position and
to close the contacts, a certain inclination of the bulb is necessary. The mercury
drop obviously rolls in the opposite direction after the opposite inclination of the
bulb has' accumulated. When it rolls in this way, the mercury drop will naturally
not remain in the central position, but will pass to the opposite extreme position.
On this basis, the characteristic of the correction for the given case will
have the hysteresis form according to Fig.6.10,b.
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This system of correction can obviously be applied, and is, in fact, applied to
correct the outer and inner frame of a gyro vertical, and to correct the inner frame
of a directional gyro.
Figure 6.14 gives a diagram of the electrical connection of an induction-type
gyro vertibal. The signal transmitter here is a cup (Fig.6.14,a) with a
current-conducting liquid, an electrolyte, and four electrodes. The cup is rigidly
attached to the gyromotor casing. The electrodes are placed diagonally on the peri-
phery of the cup. In this case, each diagonal pair of peripheral electrodes is in-
tended for the correction of that frame of the gyro vertical whose axis of rotation
is perpendicular to the axis joining the given pair of peripheral electrodes.
The cup of the electrolytic transmitter is not completely filled with liquid so
that there is an air bubble. The electrolytic transmitter consists, as it were, of
a bubble level producing electric signals.
If the axis of the gyro rotor occupies the vertical position (Fig.6.14,a), then
this bubble will be in the center of the cup, and the resistance will be the same
between the central electrode, that is, the cup itself, and either of the peripheral
electrodes. When the gyro axis (Fig.6.141b) deflects from the vertical, the air
bubble will be displaced accordingly, and the resistances between the central elec-
trode and the two peripheral electrodes become unequal. The appearance of a differ-
ence between the resistances is expressed as the error signal in this transmitter.
Accordingly, on the circuit diagram of FiE.6.14,b, this transmitter is convention-
ally represented in the form of variable resistances between the central electrodes
and the corresponding peripheral electrodes.
A basic feature of the electrolytic transmitter is the practical absence of any
the smallest deflection from the vertical will produce a
change of resistances of the bulk conductor electrolyte. This feature of this sig-
nal transmitter of the corrector, and its design simplicity, are responsible for the
wide use of this type of correcting signal transmitter.
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Two-phase miniature induction motors are used in this correction system. One
of the phases of each of these miniature motors is always connected to a definite
Fig.6.14 - Corrector with Electrolytic Transmitter
and Induction Motor
phase of the supply line (cf. diagram in Fig.6.17,c), the other phase consists of
two windings connected in opposite directions and joined to the corresponding pair
of electrodes of the signal transmitter. The second phase of the supply line is
connected to the case. Thus, when the resistances between the upper electrode and
" I
-1-ctrodes are equal, the resultant ampere-turns of the second
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phase, and therefore also the moments at the shaft of the miniature motor, will be
zero; when these resistances, differ, however, resultant ampere turns in a definite
sense appear, thus leading to the
appearance of a moment in a definite
sense, proportional to these resultant
ampere-turns.
On the basis of the above, the
characteristic of this corrector may be
taken as analogous in form to the
characteristic of the pneumatic correc-
tor of the gyro vertical with pendulum
slides given in Fig.6.3..
The zone of proportionality in the
transmitter amounts to 0.50. For slopes over 0.50, one of the electrodes is com-
pletely covered with air, and the other with liquid. The resistance from the first
electrode will reach 5000 ohms, and that of the second 150 ohms. With greater
slopes, the value of these resistances will no longer vary.
The moment transmitters in the form of induction motors have the advantage of
allowing the correction system to operate when the axis of the suspension shifts
through all of 360?. The latter will be unavoidable, for instance, in correcting
directional gyroscopic instruments in azimuth.
The entire correction system is fed by a triphase alternating current source in
which the voltage phases are shifted by 120?. For the operation of the biphase
motors, the phase shift between the voltages of the excitation winding must be 900
.
To provide such a phase shift, part of the phase voltage of the phase A is fed to
the winding (3) (cf.Fig.6.14,c) while the line voltage between the phases B and C is
fed to the winding (4) and (5). These voltages, as follows from the voltage dia-
gram of the triphase system given in Fig.6.151 are shifted in phase by 90?. The
- Diagram of Voltages
of Triphase System
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winding (3) is connected in the circuit of one of the phases of the stator winding
of the motor for the gyro rotor. When the gyroscope is started up, starting
(a)
(a)
(b)
2
Itto cycles
(c)
a)
Fig.6.16 - Corrector for Electric Gyrohorizon
1 7 Electrolytic pickup; 2 - Solenoids; 3 - Movable cores of solenoids
(a) Solenoid; (b) Corrector switch; (c) Gyrohorizon gyromotor
currents of voltage higher than nominal flow in the stator winding, thus causing an
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increased rate of correction when the gyro starts. This shortens the time required
for the gyro to move to its normal position.
A certain complexity in the design of induction motors has led to the use of
moment transmitters of simpler design.
The correction system on one of the USSR electric gyro horizons consists of the
electrolytic pickup (1) (Fig.6.16) and the two solenoids (2) with shiftable cores,
connected according to the diagram of Fig.16,a. The solenoids are attached
(Fig.6.16,b) to the casing of the gyro rotor, and a displacement of their cores with
respect to the mean position (Fig.6.16,e) produces moments of unbalance, under the
action of which, the gyroscone precesses toward the vertical. The displacement of
the core of the solenoids takes place when the difference of current in the halves
of the winding connected with opposite electrodes of the pickup, reaches a certain
value. The characteristics of this correction system is close to that shown in
Fig.6.10,a, and ordinarily d2 = 0.5?.
Figure 6.17 gives a diagram of the electrical correction of the outer frame of
?
a directional gyro accomplished by a remote-reading magnetic compass. The signal
transmitter here is a disc connected with the axis of the outer frame and a brush
connected with the repeater of the remote-reading compass. On the diagram of
Fig.6.17, this is shown conventionally by the direct connection between the brush
and the magnetic needle. One side of the disc is made of conducting material, the
other side Of insulating material. The positive pole of a direct current source is
. I applied to the conducting part.
The moment transmitter used is a magnetoelectric device with its magnetic sys-
tem attached to the outer frame of the gyroscope while the coil is connected to the
axis of the inner frame. The coil is fed from the potentiometer of a two-coil
magnetoelectric relay. One of the coils of this relay is energized during the en-
tire time of operation of the correction system. The moment produced by the
ampere-turns of this coil holds the brushes against the potentiometer in a position
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?
?
?
such that the ampere-turns of the moment transmitter coil produce a moment causing
precession of the
nart of the disc. In this case the
gyro
and with it, also a shift of the signal transmitter disc in
the sense directed toward matching
between the brush and the conducting
part of the disc. The second coil
of the relay is fed from the brush
of the signal transmitter. In other
words, this coil will be energized
only when the above mentioned brush
is in contact with the conducting
or ampere turns of this second coil
Fig.6.17 - Diagram of Magnetoelectric
will create a moment ecceeding the
moment produced by the ampere-turns
Corrector of Outer Frame of Directional
of the first coil. Thus, as soon as
1 - Disc of gyro signal transmitter; the second coil is energized, the
2 - Brush of signal transmitter connect- brushes of the relay potentiometer
ed with repeater of remote magnetic pass over into the extreme opposite
compass; 3 - Corrector relay; 4 - Poten- position. The reverse direction of
tiometer of relay; 5 - Magnetic system of the correction moment and the sense
moment transmitter of corrector; 6 - Coil of precession of the gyroscope
of moment transmitter of corrector .change in accordance with this: now
the motion of the gyroscope will be
in the sense of matching the brush of the signal transmitter with the insulated part
of the disc.
Thus this system of correction in principle has no position of repose and has
no n utra] zones.
For this reason the gyroscope will not be damped in a definite position here,
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?
Fig.6.18 - Electrical Circuit of Magneto-
electric Corrector of Outer Frame of
Directional GTO
?
Fig.6.19 - Constant Characteristic of
Corrector of Hysteresis-Free Form with
neutral zone Equal to 0
Fig.6.20 - Diagram of Yagnetoelectric Corrector of Inner
Frame of Directional Gyro
I, II - Axis of inner and outer gyro frames
1 - Signal transmitter potentiometer connected with outer frame; 2 - Brushes of
signal transmitter connected with axis of inner, frame; 3 - Relay of corrector;
4 - Potentiometer of relay of corrector; 5 - Short-Circuited rotor of induction
motor; 6 - Winding of induction motor
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?
but instead it will oscillate about a certain position, in this case, about the
position of the brush, that is, about the magnetic meridian. Owing to the low rate
of correction, the amplitude of these self oscillations will be very small. Figure
6.18,a gives the electrical circuit of this corrector, and Fig.6.19 gives its
characteristic.
Figure 6.20 shows a diagram of the electrical corrector of the inner frame of
the same directional gyro. The signal transmitter here consists of a potentiometer
placed on the outer frame, and a pair of brushes connected with the axis of the
inner frame. Thus the correction of this frame is accomplished according to the
position of the rotor axis with respect to the axis of the outer frame. The
III
Fig.6.21 - Diagram of Electromechanical Corrector of Inner
Frame of Directional Gyro
I, II - Axes of inner and outer frames
1 - Contact strip of signal transmitter, connected with outer frame;
2 - Brush of signal transmitter connected with inner frame; 3 - Electro-
magnet; 4 - Discs of friction coupling; 5 - Electric motor; Rb - Ballast
resistors
nutuvlly To'-_tion of these ayes corrcspclias to thc metched state of
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the signal transmitter.
The moment transmitter is a three-phase miniature induction motor with
short-circuited rotor. One of the vertices of the triangle formed by the stator
winding is connected to one of the phases of the supply line during the entire time
of operation of the correction system.
The other two vertices are connected to two brushes mechanically connected to
the axis of a magnetoelectric relay which is fed by the notentiometer of the signal
transmitter. These brushes take off voltage from the potentiometer, connected on a
bridge circuit to the other two phases of the supply line.
When no current flows through the relay the voltage taken off by these brushes
is zero. When current appears in the relay winding, the brushes are displaced from
the electrical neutral, and will transmit the voltage to the stator windings of the
miniature induction motor. As a result, a moment in a definite sense appears, which
is proportional by mo(3ulus to the displacement of these brushes from the electrical
neutral, i.e., in the last analysis it is proportional to the deflection of the
rotor axis from a right angle with the axis of the outer frame.
In accordance with the above, the characteristic of the corrector described
will be of the form given in Fig.6.3.
Figure 6.21 gives a diagram of still another version of an inner frame
corrector.
The signal transmitter here is a contact blade with an insulating strip placed
on the outer frame, and a brush, connected with the axis of the inner frame. The
moment transmitter is an electric motor rotating two gearwheels, which may be engag-
ed by means of electromagnetic friction clutches, and systems of gears from the axis
of the outer frame.
These electromagnets are controlled from the signal transmitter by means of the
electric circuit of Fig.6.21. So-long as the brushes remain on the insulated strip
nr the sinal transmitter, both electromagnets are deenergized and the outer frame
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is not coupled to the electric motor. As soon as the brush is displaced onto either
conducting part of the blade, one electromagnet or the other is actuated. When this
happens, the electric motor is coupled
to the outer frame through two gears
rotating in opposite senses. This will
indicate that a moment of a definite
sense has been applied to the outer
JJ
frame. The magnitude of this moment
will be determined by the force of the
friction coupling, which in turn will
Fig.6.22 - Characteristic of
be determined by the force developed by
Electromechanical Corrector
the electromagnets, i.e., by the number
of its ampere-turns. As follows from the electrical circuit, the electromagnets are
fed at first through a ballast resistor, but then, with increasing mismatch, they
are fed directly, bypassing this ballast resistor.
Thus the corrector characteristic in this case will have a stepped form,
according to Fig.6.22.
Section 6.4. Mechanical Correction Systems
Of the mechanical systems we shall discuss only one friction correction system,
designed for correcting a gyrovertical. Figure 6.23 gives a diagram of this system
of correction. The correction sensor in it is a half-ring with an axis of rotation
coinciding with the axis of rotation of the frame to be corrected. The semicircle
has a slit with cork walls, through which passes a rotating roller connected by a
reducer with the rotor axis.
It is this roller that forms the second element of the correction system.
These elements of the correction system here act simultaneously as signal
transmitters and moment transmitters.
state of this correction system, the half ring is not pressed
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against the roller by any side of the slit.
To mismatch, there corresponds a position in which the half ring is pressed
against the roller by one side of the slit or the other, and therefore the force of
Fig.6.23 - Diagram of Friction Corrector of Gyrohorizon
I, II - Axes of gimbals
1 - Half-ring with friction slot; 2 - Rotating roller; fg - Component
410 of force of gravity pressing the wall of the slot of the half-ring
against the rotating roller; fK - Force of frictional adhesion,
frictional adhesion fK arises between them, which is proportional for the force of
this compression of the half ring against the roller, and which is directed along
the slot of the half ring.
Being applied to the axis of the rotor, this force exactly produces a preces-
sion of the gyro about the axis of rotation of the frame to be corrected and accor-
dingly, also about the axis of rotation of the correcting half ring.
The force that presses the half ring against the roller will be the projection
of the force of gravity of the half ring onto the normal to the rim of the roller.
Within the limits of small deflections of the half ring from the vertical, it may be
taken as proportional to these deflections. If we further consider the coefficient
the roller and the walls of the slot as constant, we get t?TAT
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result that the characteristic of this correction system will be proportional.
If the clearance between the roller and the half ring is taken into account,
then the characteristic of the corrector takes the form shown in Fig.6.23,b. At
angles of deflection of the gyrorotor axis OA and -OA, the half ring will be dis-
placed from one side of the roller to the other, and as a result the correction
moment will change its value with a jump.
As a whole, the gyrovertical has two correction systems of this type, one for
each of its frames.
To eliminate the influence of the moment due to the force of gravity, awing to
the force of the pressure of the half ,rings against the roller, counterweights are
provided, which impose on the gyro a moment equal and opposite to the moment due to
the force of gravity of the half rings.
We remark that if the roller does not rotate, then, in spite of the pressure of
the slot of the half ring against the roller, this will not lead to the appearance
of a force of frictional adhesion. In other words, when the rotation of the roller
stops, the correction system will be turned off.
Section 6.5. Analytic aoressions of the Correction Characteristics
As follows from the above exposition, the characteristics of correctors may be
divided into the following three groups, according to their form:
1. Proportional characteristics of hysteresis-free form (cf.Fig.6.6).
2. Constant characteristic of hysteresis-free form with zone of insensitivity
not equal to zero (Fig.6.12), and equal to zero (Fig.6.19), and constant character-
istics of hysteresis form (Fig.6.10,b).
3. ,Mixed characteristics of hysteresis-free (Fig.6.3) and hysteresis
(Fig.6.10,a) forms.
We might introduce the notion of the generalized characteristic of a corrector
of such a form that the above enumerated characteristics shall correspond to certain
special cases for it. The form of the characteristic according to Fig.6.241c sjAy
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?
?
satisfy such a condition.
The use of such a generalized characteristic in a theoretical analysis would
involve no difficulties of principle. By dividing such a characteristic into a
number of sections within which only some one law would operate, the behavior of the
gyro in these sections could easily be obtained.
It would then remain for us to adjust the results obtained for the individual
sections, and to obtain a general picture of the behavior. With the large number of
such adjustments, however, this method would be inconvenient in practice. Since not
c)
Fig.6.24 - Generalized Corrector Characteristics
a single one of the actual correction systems has a characteristic of such a
generalized form, it is advisable to conduct our investigation with respect to sim-
pler generalizations.
As such basic generalizations, we shall take the following:
a) proportional characteristic of the hysteresis form (Fig.6.240.);
b) constant characteristic of the hysteresis form (17ig.6.24,b).
Using these generalizations, it will be easy to obtain the law of gyroscope
behavior for other cases as well.
The analytic expressions of the corrector characteristics for the generaliza-
tions may be represented in the following form, distinguishing them by the literal
subscripts a and b respectively:
?
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?
km. k (6 ?0,signi),
Le b Ikas1gn8 at
(k sign at
where e - mismatch in signal transmitter of correction system;
k - slope of proportional characteristic of correction;
OA - zone of hysteresis;
k - value of constant correcting moment beyond zone of hysteresis;
kA - value of constant correcting moment in zone of hysteresis;
at 0.
at O 1, that is, with different efficiencies of the correction in the two
<
frames, will already denote the corresponding curvature of the paths (cf.Fig.7.3).
We note that the pictures of the paths so found may also be obtained directly
by eq.(7.16), if we use it as the equation of the tangents to the paths F(a, 13)= 0.
Let us find the locus of points which these tangents have the same angular co-
efficient, i.e., the locus of points where
de
?solt al?t const.
dS
Such loci of points are called isoclinic.
da
On substituting in eq.(7.16) according to eq.(7.26), we get
'
u
2 -
(7.26)
(7.27)
Thus, as already stated, the isoclines are straight lines passing through the
origin of coordinates and having an angular coefficient differing by a factor
1
of )-.) from the angular coefficient of the tangents.
Forv1 = 1, that is, at the same efficiency of correction on both frames, the
angular coefficients of the isoclines and those of the tangents coincide; this de-
notes coincidence with the isoclines of the paths themselves.
Assume v1 > 1, that is, that the correction of the outer frame is more effi-
cient than that of the inner frame.
In this case
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that is the angular coefficient of the tangent at each given point will be greater
than the angular coefficient of the isocline, which is determined, according to
(7.27), by joining a given point to the origin of coordinates. Thus the direction
of the tangent at each point is easily determined. But, by constructing the field
of tangents, it is easy to determine the approximate form of the paths (cf.Fig 7 3
It will be clear that in this case they will have a curvature toward the axis op.
Physically, this curvature is a consequence of the higher efficiency of the correc-
tion of the outer frame.
For v1 = 1, that is, in the case where the correction of the inner frame is
more efficient, the position will be the opposite, that is, in this case the angular
coefficient of the tangent at each given point will be less than the angular coeffi-
cient of the isocline. Accordingly, the paths will now be curved toward the axis
40 oa.
?
As for the duration of the build-up state, for vi 5 1, it may be determined by
the sane formula (7.25), by substituting el in it if el < e22 or e22 if E o. ?,o; :io; i 0 and 8 increases so long
as it does not vanish.
Thus in the steady state
8=0, 7. sin mt. STAT
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?
In this case the gyroscope does not perform its function of "averaging" the
readings of the sensitive element; the errors in the direction of the gyro are equal
to the errors of the sensitive element. But this is observed only at small ampli-
tudes and small frequencies of oscillation of the sensitive element. Thus, if the
angular velocity of the corrector amounts to 3?/min 0.05?/sec, then the frequency
of the disturbances equals 0.02 cps, and then the relation of eq.(8.10a) is satis-
fied, if
0.05 un
_
"4 2n.0.0. 0,4 degree
If the amplitude of the oscillations of the sensitive element is over 0.4? at
the same frequency, then the follow-up will be disturbed and the gyroscope will
oscillate at an amplitude smaller than the amplitude of the oscillations of the sen-
sitive elements.
Fig.8.2 - Influence of Periodic Disturbances in the System of Correction
with a Constant Correction Characteristic without Allowing for
the Friction in the Base
The second case is more difficult, since the amplitude of the oscillations ym
and the frequency win flight are usually such that ymw>wy.
Let us now consider the second case.
Obviously, if the condition:
lal 1..
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(8.11)
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is satisfied, then the oscillations of the correcting member will exert no influence
whatever on the behavior of the gyroscope, since in this case,
time, the following condition will be valid:
sign a=sign (a sin m1).
For
for any instant of
(8.12)
(8.13)
The condition of eq.(8.12) is now replaced by the following:
sign (2 - im ?t) 1, 1 t"
1? 1 jor
a> y,sincut
a < Tasimut
and the variation of a takes the form given in Fig.8.2.
Let us now take. some cycle of oscillations y (cf. Fig.8.2). Let a1 be the
value of a at the beginning of the cycle (time t1) a2 the value of a when i changes
its sign (time t2) and c,3, the value of a at the end of the cycle, or, what is the
sane thing, at the beginning of the following cycle (time t3) .
Let us write the obvious relations with respect to the reasoning for a1> 0:
or
Bearing in mind that
where
as= as w1(13-
33=al?mot [2 (11-1) ? (t3-11)).
t, ? 11. T+
11_ _r2.
T 2'`
I
Isl.. arc sin L ? arc sin -31) ?
Ta Ta
arc sin -421;
? Ta
174
(8.14)
(8.15)
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- 2- arc sin ?
Imo
we rewrite eq.(8.14) in the following form
.1)
21
(8.17)
(8.18)
It follows from eq.(8.18) that thej3ncrement ofa for a single cycle, which we
shall denote by bal, is determined by the expression
--;-) ? (8.19)
On the basis of eq.(8.19), as well as of eqs.(8.15), (8.16), and (8.17), the
following expression can be obtained fort()Kay for the mean value of the correction
rate over the cycle:
where
1 II I ?1 et
-
81/4 2- art 1411 ? -
- ? 2 - att NM ? 4- Oft MO
Tie III To
SI Cy as ? - ? ?
It 1 + C
? 1 as
(arc sin -- ? arc sin - --) .
214 la Ta'
(8.20)
So long as the numerator of eq.(8.20) still retains the positive sign, wK av
will be negative, and, consequently, a3 < (xi will hold. From this, and from the
condition of eq.(8.13), we get the chain of inequalities
and, consequently
(8.21)
I es..1ighti?
will ordinarily hold good.
For the value a = 1.7 calculated by us, the values of a31 and R Bl will be equal
ct.1=- 0,2587..
2.1=0,4381
2n
To a turn, at the angular velocityw= 1/sec taken in calculating a, there
60
will correspond for instance, an angle of bank of the order of 570 at a flight
speed of 540 km/hr.
P
It will be clear from this that for relatively deep turns,a Bl and B1 may
reach considerable values.
For turns at higher angular velocity and correspondingly steeper banks, the
values of au and t3 in will be smaller. Thus, for the case when a full turning
circle is made in 20 sec, a = 5. In this case we shall have
9.1=0.191
To this value of the angular velocity at the same flying speed of 540 km/hour,
there corresponds a bank of the order of 780. Thus we get aBi = 30 and 13B1 = 150,
as againsta = 14.7? and031 = 25? of the preceding case.
As follows from eqs.(9.18) and (9.20), the vertex of the gyroscope reaches this
equilibrium position after the lapse of the time
T-3 : - 4 I
that is, if we take = 0.06 1/sec, that is, after 50-70 sec. This means that with
a slower turn, for instance with a turn making a full circle in I min, the vertex
of the gyro practically reaches the equilibrium position already at the end of the
first turning circle. In slower turns, the vertex of the gyro may be set consider-
abl before the whole turning circle is completed. In a turn according to "-
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second example, three turning circles must be made for this. Let us find the path
of the gyro vertex to the equilibrium position.
For this purpose let us transfer in eqs.(9.18) and (9.20) respectivelya131
and,en, to the left sides, square the right and left sides of the equation so
obtained, and add them by parts. As a result we get
(2 2 )1 + ? '0.1)t
(9.24)
This eauation is the equation of a spiral in the coordinates Oap with its on-
at the point ( pi). The magnitude of the radius vector of the spiral de-
creases by an exponential law with the passage of time.
The angular velocity of rotation of the radius vector of the gyroscope vertex
will be equal, as is obvious from eqs.(9.18) and (9.20), to the angular velocity of
the turn raking Ilaking use of this fact, in the equations of the path, eq.(9.24), the
time p may be expressed in terms of the angle of rotation of the radius vector of
the vertex kp. Indeed, since this turn is accomplished at constant angular velocity
111 it follows that the time t required for rotation by the angle cp, will be deter-
mined by the formula
Eaking use of this relation and putting m = 0 for t = 0, we may rewrite the
equation of path, eq.(9.24), in the following form:
?where
2,
( 2 301)1 ? - %1)1 /Pe a ?
Or, in polar coordinates, with origin at the equilibrium point a B
13? Bl:
?
r?ile 61
r (2 2.1)? +
(9.25)
(9.26)
On determining the arbitrary constant A from the initial conditions for T = 0,
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r = no, we get
and, consequently, the equation of the path of the vertex will finally be written in
the following form:
r.R.e
Is
(9.27)
Thus the path of the vertex will be an involutional logarithmic spiral, ending
in the equilibrium point.
Let us assume that at the initial instant the gyro axis was at the true
vertical.
This means that
78
I rel 1+ az
Figure 9.5 gives the paths of the vertex for such initial conditions and the
value a = 2.0. It will be clear from them that during motion toward the equilibrium
position, the vertex will pass points with coordinates exceeding in modulus an
and 0131, but these latter deviations are in themselves so great that the use of a
gyrohorizon under such conditions during a turn becomes practically impossible. In
this connection, in those gyro horizons that use a system of correction with propor-
tional characteristics, the action of the system of correction along the angle a
during a turn is sometimes cut out. Since the duration of a turn usually does not
exceed 1-2 min, a gyro with the correction system cut off for this time will main-
tain its position sufficiently well. Although this measure involves certain compli-
cations in the design of the correction system, it is nevertheless not only entirely
rational; but also in this case entirely necessary. This behavior of the gyro may
be treated as a direct consequence of the composition of the precession due to the
resultant moment of the correction system and the rotation of the ystem of co-
ordinates 0a13, due to the rotation of the aircraft. 1
Let us now ne the value of the maximum turning errors. For this it STAT
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sufficient to find the coordinates of the first points of intersection of A and
with the path of the gyro vertex with the straight lines at which a = 0 and = 0,
respectively. The equations of these straight lines will be obtained if we put a = 0
and ? = 0 in eq.(9.16).
BO ? = 0 on the straight line
117=4.
and a = 0 on the straight line
Fig.9.5 - Path of Vertex of
Gyro Horizjn during a Turn
with Proportional Charac-
teristics of Correction,
without Allowing for Fric-
tion in the Gimbals
> 0 (Left Turn)
? 0 (Right Turn)
a a
Thes,straight lines are mutually perpendicular,
the straisht line a = 0 forming the angle (P with
the axis a (Fig.9.6), for which the following
equation
1
1 a . (9.28)
1 I 1 t a:
1 + -41;
is true, while the straight line 3 = 0 passes
through the origin of coordinates. Then the length
of the radius vector R, will be determined, accord-
ing to eq.(9.27), by the formula
R?c
and the length of the radius vector R2 by the formula
00.
r.
Rs Roe a ?
From evident geometrical considerations we get
mat' asi ? RI cos
t?, ? R2 Cos
expressions for a ni) I, 112 and cos p, we obtain, as for
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?
?
a simple transformation:
21) 2 ( 1 1- ae -- 381 ( 1 + /le
Mal )4a
?
ir at =I" Ts (1 e .,1(1 -re a
I ? al
Fis.9.6 - Determination of liaximum
Errors during a Turn for Propor-
tional Characteristic of Correction
re
el 4.
r:
(9.29)
(9.30)
Let us determine the value of a at
which a ,e = 13max? According to eqs.
(9.29) and (9.30), this value of a must be
determined from the equation
? el
...0.
a \
.1 (a) a 1 e 4 - - 1 ae-J "
This transcendental equation has one
root a = 1.26. For a < 1.26, a rnax > max,
while for a > 1.26 < , since
max ?ma..x
e ?0
2a
where a > 0 and f(a) is a continuously increasing function. It follows from the
expressions for a and ' that for the given value of yB, they are determined by
'max
the quantity
Put a = -then = 0, while a max Y
Put a = cc
Fora we shall have:
max
s\
11M mom lim 11(1+ at
is. ? ?-? ?
1 + al
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?
?
By analogy, for I-1 max,
iim p lim TOIL t
d? u? a I 4 al
. O.
Thus when a = -- varies from 0 to cc 1 there are limiting values of p max'
Let us determine the value of a at which, for an assigned el the limiting
value 0 will obtain. For this purpose, let us solve the equation
max
d?mag 0.
da
w
(1 i-e dr e ) (1 -I- al) -- (1 r
The equation so obtained is now transformed in the following way:
e (a ? 1)r. _
(a'?I.
1) a
This equation is satisfied by the root a 1.05.
Substituting a = 1605 in eqs.(9.29) and (9.30), we get
(3....)14?
In this case a takes the value defined by the expression
max
When a varies from 0 to cc, a . varies from B to 0, always remaining less in
modulus than yBy since
e?
damai
is
da
- _ al) _2a
2a 2
(1 + a2)1
- 0 and sign = 1; the straight line A2, to the condi-
tions 0 and sign d = 1; the
straight line n2, to & 0, > o will hold, lies below the
Fig.9.8 - Paths of Vertex of Gyro Horizon on Left Turn with
Proportional Characteristics of Correction, Allowing for
Friction in the Gimbals
1
Al and A2 - Loci of points where changes si.R.n; B, and B2 - Loci of
points where changes sign; C10 - Initial focus of spiral; Cli - Eocus
of spiral after intersection of vertex by straight line A at point a;
C12 - Focus of spiral after intersection of vertex by straight line 'B
at point a2, etc.
corresponding locus of points of A and B, and the region of points for which the
condition d 0 will be satisfied, and, consequently the straight
line A1 must be taken as the locus of points where cichanges its sign.
In connection with the signs of and f3 so established, the expressions for the
coordinates of the equilibrium position at the initial instant of time may be found
STAT
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?
?
by simultaneous solution of eqs.(9.37) and (9.38)
I
(2110Pe Sig Cie Ps? aPI).
-.- (Ts -- ) ?
? di?
(9.39)
(9.40
Thus, at the initial instant, the spiral will be drawn from the point C10, the
point of intersection of the straight line Al and 132, having coordinates in accord-
ance with eqs.(9.39) and (9.40). Let it be intersected in this case by the straight
line Al at the point al. This will mean a change of sign for i, and, in the first
place, a corresponding change of the expressions for the coordinates of the point
C11 of the equilibrium position, which now take the form:
(20 ), ? 1 *az Os? P, aP1).
0?1)111E` 7+1-1-e-; (*T. ?111- PI) (9.41)
(9.42)
?
and secondly, the change in the locus of points of change of sign of 0 from the
straight line D2 to the straight line Bl. It will be easily seen that as a result
of these changes, the length of the radius-vector of the spiral decreases
(cf.Fig.9.8).
During the further tracing of the spiral from the new center, let it intersect
at the point a2, the straight line B1, which is now the locus of points of the
change of sign of 3. As a result of this change, in the first place, the expres-
sions for the coordinates of the point C12 of the equilibrium position will again
change, and will now take the form
(a,1),-? + P2 + aPi).
4- all
(Pod-, (19 + PI )
? as a ,
(9.43)
(9.44)
and secondly, the locus of points of change of sign ofd will change from the
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?
straight line Al to the straight line A2. As will be clear, the coordinates of the
equilibrium position, that is, of the focus of the spiral, will already vary in this
case to such an extent that p will become positive (cf. the dashed segment in
?
Fig.9.8). But after the arrival at the straight line 131, the sign of p would have
to change from positive to negative.
A contradiction thus arises, from which it follows that a further motion of the
vertex (i.e., motion after it arrives at the point a2) will no longer be possible.
Consequently the point a2 in this case will be the position of rest of the vertex.
Thus, in this case of a left turn, the friction in the gimbals favors the more
rapid arrival of the gyro vertex at the equilibrium position, and also favors the
deformation of its path in the direction of reducing the maximum values of the
current deviations.
Let us now consider the case of a right turn. In connection with the change of
sign of a and iB, the locus of points where i and p change their signs are already
located in the positive quadrant of Oap, as indicated in Fig.9.9. In this case, the
region of points for which the condition d >0 is satisfied, as in the preceding
case, will lie above the corresponding straight lines A, while the region of points
for which the condition p < o will lie below those straight lines, the region of
points for which the condition p < 0 is satisfied will lie below the corresponding
straight line B, and the region of points for which the condition 0 < 0 is satisfied
will lie above those lines.
For the same initial'conditions as in the preceding case, we shall in this case
have for the initial instant of time d > 0, 13 < 0, the straight lines A2 and Bl as
the loci of points of change of sign, and the following expressions for the co-
ordinates of the equilibrium position of the points C10
(ail) + (?i. ?
,
O *
ai 1aal I Ts +Ps -P1-) ?
* ? a
209
1
(9.45)
(9.46)
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It will be seen from Fiz.9.9 that the vertex of the gyro, after travelinz the
very short distance separating the straj.cht line Fp which is the locus of points of
- Paths of 7ertex of Gyro Horizon on Ri-ht Turn with
Pronortional Characteristics of Correction, Allowinu for
Friction in the Gimbals
Al, A2 - Loci.s of points of change of sjgn of d; -1, 2 - Locus
of points of change of sign of 9; C10 - Initial focus of spiral;
Cil - Ff)cus of spiral berinning at point al; C - Focus of spiral
egjnning at point a2; etc.
change of sign of ?from its initial position as taken by us, will intersect this
straight 1:,ne, and in connecton wjth this the sir;r1 of 3 will chane from positive
to ne-ative, which vill mean that the locus of points of change of sign of d will
change from the straight line A2 to the straight line Al, and the expressions for
the coordinates of the equilibrium of the point Cu will change to the following:
-,????
(2.1)' .1 as ( P.? aP,),
a 1
(.1). (in P2 ? ?P1
a ) ?
, 1 4 al
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(9.47)
(9.48)
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1.7
ft. in connection with which the locus of points of change of sign of 0 ^ will change from
the straight line B1 to the straight line 132, and the expressions for the coordi-
As a result of this change, the length of the radius vector of the vertex of
the gyroscope will be increased.
On intersection with the straight line Al, a change of sign of a takes place,
nates of the equilibrium position of the pOnts 012 will change to the following
+ 1+40? ( + P, aPI). (9.49)
P.
_-_-1;;;- (7. ? Pt + -1; Pi) ? (9.50)
With this change, the length of the radius vector of the vertex will likewise
increase. Let us assume that there is subsequently intersection with the straight
line B2' which will mean the change of the locus of points of change of sign from
the straight line Al to the straight line A2 and the change of the expressions for
the coordinates of the equilibrium position of the point C13 to the following:
(2.1 + ;-; - P, api),
:40 (T.
(9.51)
(9.52)
as a result of which the length of the radius vector of the vertex is again increas-
ed, and so on.
The motion of the vertex of the gyro in the plane Oap leads to the establish-
ment of a self-oscillatory closed cycle embracing the rectangle of repose. Let us
derive the conditions that determine the parameter of the cycle. As the parameter
of the cycle, let us take the distance from the point A of the intersection of the
cycle with the straight line Al to the vertex C of the rectangle of repose. Let us
denote this distance by rnp. Let us also denote the sides of the rectangle of re-
pose as follows:
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1 2101 oi
CI CI 11110 Oro ) ) '
I ,2 COS I 4- a? cos,
1 2tes_
11 1311" CliC" 1(.1) _
I COO - a') cos
2pi
11,
2pi
c.
)1 11
then, according to what has been set forth and eq. (9.27), we shall have the follow-
' 4
1
111:7 s:Fstem 9f equations describing the motion of the gyro vertex along the limit
cycle:
Acne' ? (ACII-f- b) e;
CC 1, . BC13e' (BC 11-1- C)
DC3.= CC11e2' (CC11-1- b) e;
AC11? DCliem ??(DC3,,i- C)
Since for a ri7ht turn yr aBl, and the maximum deviation of
the gyro vertex from the origin of coordinates, when moving along the limit cycle,
is determined according to eq.(9.60) from the following formula:
= +
a= op ma
r.
. 6)3 LP
On replacing a py ___ and p by , we Fet
Eli
(finp)mat
4 L. T. 4 L
I ?
z2 ? x at/
(9.62)
(9.63)
Let us determine the value of 6 for which Cr3np)max takes the minimum value for
0
. Yao
a given ratio ? For this purpose, let us set the derivative of(Pnp)max with
w,
Do
respect to e equal to 0:
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For e we shall have
0
d(Papbnat Tim IL,
de mw ,4i
11
I s:
4L
;
4?10.
4 1. 4
? - ? fa ;
'ego n cot/
8 S
p.
it cif
(9.64)
(9.65)
Let us now determine the value of the maximum deviation of the gyro vertex from
the origin of coordinates on motion along the limit cycle on of(9 ) for the
'nP max
Yi
selected value of e and the other values of -- . For this purpose, let us substi-
(9)3
tute in eq.(9.63) for Eo its value as determined adcording from eq. (9.64):
-.. 7.
+ 4L _,fr.1i leo-
r?
cif 40. sit/ r
After simple transformations we get
. To T.
if'IL. moss
nil V las
ooso
(9.66)
Y3
Figure 9.10 shows the variation of the value of on variation of w from 0
(dB
to - m. In fact, as wp - 0,
as
...V
afCtg - -
11111 X
??? ? II ? 0
els
Y3
IwEI m / decreases monotone, and approaches 0 as lw 1
,? B
(013
If foriJ
y:,
wo we take its maximum value A, obtained in the actual range of '1B and
Bo
cc
Y13
aircraft speeds V, then (13np) will vary with the variation of within the
STAT
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'1 4 mit s
nu- On,),?.:21 A
I .4
where B - minim= value of
Y.k.)
for , then
0
Y
to3
for 0-
_, max'
(9.67)
If, however, we take the value B
2 If BV ) st?
nif P m
A 4 B Iii
n ,11.
(9.60
Taking the inequaL_ty > B into account, we have the followinr:, inequalities
for
o
- Gra:ths of ?Yo = f CAB)
,
?io
for v and vrain
max
A 4 B
? 2 I
? B,
) A '
(9.('9)
(9.70)
since the arithmetic mean is Freater
than the Feometric mean.
Further,
B
2 I A' -A .
'I -4
(9.71)
From the inequalities (9.69), (9.70),
and (9.71), we obtain
A ?
A > A
? 2 I 2 B. (9.72)
)11' ) A
The conclusion may be drawn on the basis of eqs.(9.67), (9.68) and (9.72) that to
red-ace the turninr: errors, the value of E should be selected for the maxi= value
0
V
max
of the ratio , equal to about
,
Then from eqs.(9.(4) and (9.65) we cet
41.. A'
r:// Vms%
218
(9.73)
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ON,
jilLp
". VOISI
(Pridniz 4
g
(9.74)
To illustrate these conclusions we present an example. Let = 0.000314
1/sec, and let the speed vary from 100 m/sec to 300 m/sec.
/Y3 ,
In this case let us assume that 1.---) YB
= 10 see,(__
\ ?
Y3 w3 nun Y3 )(flax
we select E according 0 to__)we shall have, by eq.(9.66):
6)d 5
min
= 30 sec. Then, if
.0.0,0063 1/sec..,
for ????-? =--10 sec.
for1-"--=--30 sec.
If we select 60 according top) , we shall obtain, analogously: e
B max
0.0036 1/sec
for
for
.= 10 s ec . (11,dou 0,1390. 7,9?.
40.
.111- 30
sec.
Since for YD? , by eq. (9.65), PB10 = R?, it follows that the limit cycle will
31) Y3
practically pass through the origin of coordinates (Fig.9.11). If the ratio T)
0
decreases on account of the decrease in
Fig.9.11 - Limit Cycles:
YBO
Curve 1, for
(,)DO
Curve 2, for T.
- < ?
ims oes
for any duration of turn.
219
the velocity v or the increase of con,
then this will lead to the decrease
of Bl with the radius of the limit
cycle being maintained practically the
same. In this case the origin of co-
ordinates comes inside the limit cycle.
The vertex of the gyro will approach
the limit cycle from inside, while the
value of the maximum error will not
exceed the values given in eq.(9.74)
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- . -
Turning Deviations of G ro Horizon with Constant Characteristic of Correction
On substituting in eq.(9.1) the expressions for p and q according to eq.(9.10)
and the expressions for and 1:ir according to the conditions of the constant char-
acteristic, taking it as being hysteresis-free and without a zone of insensitivity,
and also bearinp the conditions of eqs.(9.11) and (9.12) in mind, we obtain the
equations of motion in the following form:
or
as") mni ki Sign (a + '1' Ili sign Ps
H at -4- %a) sign, --1-Nsign a.
- !sign + ? gigs 01.
0 ? "ea 0.2 P + 1149 slia el.
(9.75)
(9.75a)
where ni, j = 1 the second digit of the subscript indicating the number of the
Li
moment of friction Lw the first digit the number of the moment of cOrrection K.
Let us for the time being reject the terms characterizing the influence of
friction, and let us rewrite the equations so obtained in the following form:
o ? ? wit sign Os T.),)
m., ? 40,1 sign p. (9.76)
Assumin7,, as usual, that the operation of differentiation does not include
those moments of time when sign ( + N) and sign 0 undergo a break in. continuity,
that is, when a + y:18 and chanpe their signs, we find it possible to operate an
eq.(9.76) as on linear equations.
Taking this into account, let us eliminate the variable from the first equa-
tion by the aid of the second equation. As a result we obtain
where
2 2
a + 40.2 ?w.bi sign 0,
220
(9.77)
(9.78)
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(lb
.
The integral of this equation will be the expression
a=A sin (ti,/ S) a.1,
where
A and 6 are arbitrary constants.
where
(9.79)
-b1 sign (9.80)
On substituting the solution eq.(9.79) in the first of eqs.(9.76), we find
p =A cos (.,1+ %.)
sign (a
b2 .
0.. (9.83)
(9.81)
(9.82)
? Thus both a and (3 are periodic functions in this case.
In other words, in contrast to the preceding case of proportional correction
characteristics, in this case, with constant correction characteristics, there
exists no position of stable equilibrium for the gyro axis in the prcess of turning.
We shall specially dwell on this circumstance somewhat later.
Let us transfer a32 and 0132 to the left side of eq.(9.79) and eq.(9.81) respec-
tively, square the equations so obtained, by parts, 8,nd add these parts, as a result
of which we obtain
(a?aa)t ? L: '5.212 ??1'
(9.84)
The equation so obtained will be the equation of the path of the gyro vertic. in
Oari coordinates. As will be clear it is the equation of a circle with aB21 F'B2 as
the coordinates of its center.
However, before making use of this equation to construct the path of the
vertex, we must first determine the locus of points passage through which involves
a break in the continuity of sign (a + y B) and sign
Let the vertex of the gyro be located at the initial instant of time at the
origin of coordinates 0(10 (that is, the gyro axis was in the local vertical). In
?al instant of time, for a left turn y > 0, the condition
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sign(a+70=1
will obtain, and, consequently, on the basis of eq.(9.76), the condition
and the corresponding accumulation of the negative value of a.
The positive term 13a will therefore accumulate in eq.(9.76) for
Put motion
alon the axis op cannot at first take place, since on appearance of a small posi-
tive value !of 3, on account of the positive f3, the negative term w will appear
with a jump in the expression for and this term will at first exceed in modulus
the positive terrawBa. Therefore will vary with a jump from its positive value to
some negative value, and on account of this, the positive value of F, which appears
at first, and is as small as nay be desired, will be eliminated.
Things will proceed in this way until the condition
lost.
begins to be satisfied, that is, until a reaches in modulus a value satisfying the
condition
satisfyin:z this condition the existence of the already discontinuous func-
tion
1.10).
will be assured; which will mean the accumulation of the continuous function
as well.
In this way, in the range of values
(9.85)
the vertex of the gyro will move along the axis Oa toward the negative values of a .
7eginning with the value
-1 --1--
be determined by e4.(9.84) for
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222
(9.86)
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sign(z -10-1 * sign
Let us find the time that will elapse from the instant the turn begins until
110 the-condition
is satisfied.
The motion of the gyro vertex during .his time will obey the law
? 00 di,
or, bearing in mind the fact thatwia is constant,
a= ?omit.
Whence, after replacing a according to eq.(9.86) and introducing the notation t1 for
the required time interval, we obtain
1
i.e., the required time interval will be equal to the part of the period of the
turn.
The
The initial conditions for determining the arbitrary constant for the next part
of the path will be
g.b1.
3?=0.
whence, bearing in mind eqs.(9.80) and (9.82), we get
A' IP
If, in addition to the above, we take, as is usual, and occurs,
(9.87)
= (()K2 =
which means bl = b2 = b, and assume that during the motion of the vertex of the .gyro
around the circle, the coordinate of this vertex remains in modulus less than yB,
then we get the result that the remaining part of the path will be a circle inscrib-
ed in the quadrant 00 with the coordinates (-b, b) for the center, where b is con-
sidered a positive quantity (Fig.9.12a). For a right turn, the center of the circle
will have the coordinates (+b, b).
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- ? -
If the condition
o,-tains, considering that 1-,1 and b2 are quantities essentially positive, then the
path will likewise 1,e a circle with (Tb1, b2) as the coordinates of the center, tan-
gent to enly one axis Oct and either not reaching axis o(R, for bl > b2 or else inter-
sectinr- It (for bl < b2).
4,4J ?. 0
?
b, = b2
> 0
6, 61
() 2. 0
SI, <
uis< 0
a)
a
<
- Paths of 'flotion of Vertex of Gyrohorizon with Constant Characteristics
of correction on Ilbtion of the Vertex of the Gyroscope from the Origin
of Coordinates
Let 'is find the locus of possible positions of the centers of the circle so
foluld with varying (dm limited by the condition that
By eliminatingwt from eqs.(9.E0) and (9.E2), for this purpose, we net
where the upper si-n corresponds to a left turn and the lower sign to a right turn.
ThUs these loci will be straight lines with an angular coefficient equal in
modulus to unity, with increasing distance of the points of the straight lines from
the origin of coordinates corresponding to decrease bf wB (Fig.9.13).
2211.
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-
(-?
As w decreases, the radius of the circle increases, reaching, at4)B kp, its
YB
maximum value, equal to 2(Fig.9.14). Taking account of eqs.(9.78) and (9.83), we
shall have for bb kp:
"a 'Igo
T.
2 ?
wV
Since we may take 6.13/ --- (for banks that are not very great), we shall have
sus Kr ? To
V
2.0/
ForB> kp' the maximum turning errors are determined by the formula
6)13
ax max= 2.
ca ? re a 0
tra*
0.1
Fig.9.13 - Locus of Point of Centers of
Circles Described by the Vertex of Gyro
Horizon during a Turn, with Constant
Characteristics of Correction, Allowing
for Friction
(9.88a)
Fig.9.14 - Path of Motion of Vertex of
Gyro Horizon with Constant Characteris-
tics of Correction when the Gyro Vertex
Moves from an Arbitrary Point
For 6)B 0.
Up to the values
(9.102)
motions will take place along the Oa axis in the positive sense, and then along the
circle. In this case, the initial expressions for the coordinates of the center of
this circle will be of the form
(9.102a)
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0.21 - b (1 +
6.1(
We remark that for a right turn, w B > 0 and
(9.102b)
= b < 0. The first to be intersect-
(013
1) ed, as in the preceding case, will be the locus of points of change of sign of d,
defined by an equation of the following form (cf. straight line A1), (Fig.9.16).
:n). (9.103)
As a result of this change of sign of d, the expression for the coordinate of
the center of the circle (11B2)p2 will take the form
0180;1'6 ?b(1?n),
(9.102c)
which will mean the increase of the component of the radius of the circle along the
axis Oct by the quantity Act.
Fig.9.16 - Limit Cycle of Motion of Vertex of Gyro Horizon during Right
LP
Turn with Constant Characteristics of Correction and? > 0.37
Lk
When, during the following motion, the path of the vertex intersects the locus
of points of change of sign of p, whose equation will be of the form (cf. straight
line B2):
a-
- hi I to
(9.104)
the expression for the coordinate of the center of the circle (5B2)1p2 will now
change to the following;
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(p a);,. b (1 ?is),
(9.102d)
which will mean that the component of the radius of the circle along the axis op
will increase by the quantity Apo etc.
Fig.9.17 ? Path of Motion of Vertex of-Gyro Horizon during a
Right Turn with Constant Characteristics of Friction
,p
and >0.37
-k
On continuing this construction further, we may obtain the entire path of
motion of the gyro vertex as has been done in Figs.9.16 and 9.17. Here, depending
on the ratio?, we may obtain a closed stable cycle (Fig.(9.16), if < 0.37, or
LI. LK
Lp
a divergent path (Fig.9.17), > 0.37 (Bibl.5).
LK
I
The value of the critical ratio -A = 0.37, at which a stable cycle is still
Lk
possible, may be determined by equating the segment AC15 and the radius
5 ? which
wR
may be expressed in terms of b7 and n.
The values of the maximum errors in motion along the limit cycle during a right
turn may be determined on Fig.9.16:
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(9.105)
(9.106)
These formulas will hold until the limit cycle intersects the straight
2 use ?
26 (1+ n).
- -26( 1 + 2n).
line Y B = 0. If this intersection takes place,
right turn may be determined from Fig.9.18:
?
then the maximum errors during a
ye.
3,...? 7. ? 2I'n
(9.107)
(9.108)
I
If -,p-- >0.37, then, as has al-
1-1.
ready been stated, the path of the gyro
vertex will be divergent. Even this
path, however, will still tend to its
limit cycle after a few revolutions.
The values of the maximum errors here
will increase by more than in the
I'P
case < 0.37.
Fig.9.18 - Limit Cycle during a Right
The case of *1 has not been con-
Turn as it Intersects the Straight Lk
sidered in detail, since in practice
Line a +yB = 0
< 0.37 is usually the case.
11,
Mixed Characteristic of Correction
Fundamental studies of the behavior of a gyro horizon with mixed correction
characteristic during a turn have been made, and in particular, by S.S.Tikhmenev.
It is on these studies that the theory given below is based.
According to the ratios between the deviations of the gyro axes a and the
angle of bank in the turn yB, and the values of the zones of proportionality(
pi
and 4,2 of the mixed characteristic, the following four cases may hold:
*- is411; 1>CDI, 2nd case
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1st case
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a +1.>01; a?
3rd case
4th case
The 1st and 2nd cases correspond to the cases already investigated for charac-
teristics that are only proportional (1st case) and only constant (2nd case); the
3rd and 4th cases are now to be investigated.
We remarked that the 4th case must be considered the most real of all these
cases, and we shall therefore initiate our investigation with that case. This means
that during all the notions of the gyro vertex along the Oaaxis, the law of the
constant characteristic will hold, and along the 09 axis, the law of the propor-
tional characteristic.
The equations of motion for this case are written in the following form
or
where
?K, sign (a + + Li., sign O.}
+ was) = ?K1---!.1sIgn?
= w3 ? Isign (2 + -id ?nil sign 01,1
g(?(212 ? pi sign ei),
a1 =.
II
'(9.109)
(9.110)
Let us reject, for simplicity, the terms allowing for friction in the gimbals.
Equation (9.110) is then rewritten in the following form
? ? ai11 sign (2 +
1,3 + %a+ ?O.
Assuming as usual that the operations will not include the instants of time at
which sign (x + yB) changes its sign, and eliminating the variable 0 from the first
equation by the aid of the second equation, we get
????????????u.t
: h
2 + tia ? 641 ses ? a.1 sign (2
al
235
(9.112)
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?
The integral of this equation will be
est
Ae 2 sin (mut r
where A and 6 are arbitrary constants,
4a1
1284 --- sign (2 + i.)?
On substituting eq.(9.113) in the first of eqs.(9.11), we obtain
where
-
At' 2 [m cos (mu)," + ? --sin (mosat + a) .1+ p.4,
2al
th sign (a +
(9.113)
(9.116)
(9.117)
It follows from eqs.(9.113) and (9.115) that in the fourth case which we are
studying here there exists an equilibrium position for the gyro vertex, as for the
case of the proportional characteristics, but with a time constant of the transient
state twice as long as in the case of the proportional characteristics alone.
As will be seen from eqs.(9.115) and (9.117), the coordinate of this equili-
brium position, 0 B4 coincides identically with the expression for the coordinate p
B2
of the center of the circle described by the gyro vertex with constant characteris-
tics of correction while the coordinate aB4 is less than the coordinate aof this
center by a factor of al (here and hereafter we shall put al > 1).
As before, the coordinate pB4 will be the same for a left and right turn; while
the coordinate aB4 will be negative for a left turn and positive for a right turn.
Let us find the locus of points of this equilibrium position. For this pur-
pose, after eliminating the angular velocity of the term for this purpose, from
wh, 6)3
eqs.(9.115) and (9.117), and bearing in mind that b2 - , al = ---, we get:
(or,
? al sign (a + 1)
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or, bearing in mind that wu = e2(1)2,
a
111 that is, this locus will be a parabola with its vertex at the point 0 (Fig.9.19).
In this case, to the points more remote from the vertex of the parabola, the smaller
values in modulus of the angular velo-
cities of turn wB will correspond.
.From the equation of the coordi-
nates of the steady value of the turn-
ing deviations for .the case under con-
sideration, with coordinates of the
center of the circle described by the
vertex of the gyro for constant charac-
teristics of correction (second case),
and from the coordinates of the steady
position of the vertex for proportional
characteristics (first case) it follows
that in this case the turning devia-
tions will be minimum, since
a)
Fig.9.19 - Locus of Equilibrium
Position of Vertex of Gyro during Turn
in the Case of Various Correction
Characteristics: Locus IV Corresponds
to Constant Characteristic with Respect
to Angle a and to Proportional Charac-
teristic with Respect to Angle (3;
Locus III Corresponds to the Opposite
Case
a) Case IV; b) Case III
will be more complex than the path of the
< :2.11
..'=.s 01. $ 1. Let us determine the limit
which amax and will approach as a -.-cc. For-this we remark that according to
max
eq. (9.174) ?/- = - __-.t1 0 and t2 -.0 as a - -cc, since the period of
L1
4
motion along the limit cycle approaches zero when the angular velocity increases
without limit.
Then, by making use of the LtHospital rule, we get
iirticamis limPisaga"-LIP. (9.210)
The values ofa and(?) for lal > I differ little from their values
max max
as a -
CC
On comparing these values of the radii of the'limit cycle of a right turn with
lateral correction turned on (cf. eq.(9.61) and with that correction turned off
eq. (9.210) we see that turning off the lateral correction of a right turn leads to
the increase of the radius of the limit cycle. Figure 9.26 shows the limit cycles
of maximum radius for a given rate of correction which are obtained on a right turn
with correction turned off (curve 1), and with the lateral correction turned on
(curve 2).
This does not mean that turning off the correction for a right turn is imper-
missible. For small values of the angular velocity on a right turn the turning de-
viations with correction turned off during the time of transition to the establish-
ment of the limit cycle, the deviation will be less than with the corrector turned
on. Moreover, if the gyro axis at the beginning of a right turn coincidep with the
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?
?
?
vertical (ad =rid =a0 =00 = 0), then it will remain in this position (cf.eq.
(9.167)) since there will be no cause that will operate to bring the gyro away from
the vertical.
Fig.9.26 - Limit Cycles of Ilaximum
Radius for a Given Rate of Correction
with Proportional Characteristic with
Lateral Correction Turned On (Curve 1)
and with Lateral Correction Turned Off
(Curve 2)
Let us now consider the motion of
the vertex of the gyro during a turn with
the lateral corrector turned off, in the
case of a constant correction character-
istic. The equations of motion for this
case, with the same correction parameters
on both axes and the same moments of
friction in the axes of the gimbals, are
obtained from eq.(9.75), if we reject the
term k2 sign (a + YB) in the first equa-
tion:
min sign
(9.211)
vi,?,,(sign it sign s). (9.212)
Lk
where wk. = . rate of correction;
'P
n = - ratio between moment of friction and moment of correction.
The equation of the path of the gyro vertex during a turn with lateral correct-
or turned off is the equation of the arc of a circle, as when the lateral corrector
is on (eq. 9.91) but with its own values for the centers a2 andB2* 9 ?
B '
where
(a?c111)14-(f-0,1)tal'Rs. (9.213)
2157 - 'IN 'sign p n sign al.
262
(9.214)
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R - radius of circle determined by the initial position of the gyro vertex.
Fig.9.27 - Path of Motion of Vertex of Gyro during a Left Turn with
Constant Correction Characteristics and Lateral
Correction Turned Off
The coordinates of the centers of the circles lie on the corners of the squares
ABCD and EFGH (Fig,.9.27). The loci of the points of change of the centers of the
circles lie on the straight lines:
1) 3= 0, i.e., we + (o/c (sign 3 + n sign = 0 - the straight lines AB, CD,
EF, and GH;
2) 0 = 0, i.e., the axis Oa;
3) a = 0, i.e., (0343 w K n sign p = o - the straight lines AD, BC.
Let us first consider the behavior of the vertex of the gyro during a left
turn:> 0. Let it be situated at the point M on the axis Oa. For this point,
WB
6)1
> wK. For this reason it follows from eq.(9.212) that c;$ < 0, but in this case, 3a
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by eq.(9.211) we get the result that i < 0 as well the vertex of the gyro enters the
region (1) (Fig.9.27), where 13< 0, 13< 0, and a < 0, for which the center of the
circle by eqs.(9.214) and (9.215) will be the point A with coordinates
(314)A .?
(Pa)A N.
4?1
(9.216)
On the straight line AP the sign of p changes, since we becomes less than wx.
As a result of this, the coordinate OB2 of the center of the circle also changes.
Thus the center of the circle for the region (2) becomes the point B with the co-
ordinates
0..)a ? a.
sis
(9.217)
On the straight line BC the sign of changes, and the point C with the coordinates:
(9.218)
becomes the center of the circle for region (3). On the axis Oa the sign of 13
changes and the point E, with the coordinates
(2.1), 1=1
(P r
I?ft (l+n),
(9.219)
now becomes the center of the circle for region (4). On the straight line EF, the
sign of f3ichanges, and the point F, with the coordinates
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?
?
? 641 (1 4- Pohl
ak.
'11a.
now becomes the center of the circle for region
(5)?
(9.220)
Finally, on the straight line
AD the sign of a changes, and the point G, with the coordinates
..? (1?a),
(Po)o
(9.221)
now becomes the center of the circle.
By constructing the path of the gyro vertex, we satisfy ourselves that, as a
result of the change in the centers of the circles, the radii of the circles
diminish by jumps, both on account of the action of the moments of friction in the
axis, and on account of the action of the longitudinal correcting moment. As a
result, the vertex of the gyro reaches the square of repose ABCD or EFGH. We
observe that the points of equilibrium of the gyro vertex will be the segment KL,
the upper half of the square EFGH, the lower half of the square ABCD. In fact,
after the vertex of the gyro reaches the boundaries of these half square or the
segment KL, this type of substitution of centers of the circle does take, place,
which would be expected to lead to a reversal of the sense of rotation of the gyro
vertex with respect to the center of the circle, which is impossible. Physically
this means that, on the appearance of a deviation of the gyro vertex from these
boundaries, there appears from the halves ofithe square and from the segment KL in
any dii.ection a correcting moment and moments of friction of such sign ais to return
the vertex of the gyro to its previous position.
Thus on deviations from the segment KL we have
fore 3 and pwiil have different
> IwpFt
and there -
signs, by eq.(9.212), that is, any deviation along
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?
?
:1
the 0. axis that appears will be eliminated. On deviation from the side of the
square FG upward, we will have
7 Kai( I co, 1
??(). iaK KM( - 01.7
It follows from the above that the maximum value of the steady errors during a left
turn with the correction turned off is determined by the formulas
Oast = nt
YON
(9.222)
(9.223)
From a comparison of Fis.9.15 and 9.27 and of eqs.(9.98) - (9.101) and eqs.
(9.222) - (9.223), it follows that turning off the lateral correction on a left turn
considerably reduces the turning errors.
Fig.9.28 - Limit Cycle on Right Turn with Constant Correction
Characteristics and Lateral Correction Turned Off
Uith a right turn it may be established by an analogous method that the cen-
ters of the circles here will be the vertices A, DI Cp Ep Hp GI in succession.
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constructing the path of the gyro vertex (cf.Fig.9.28), we may make sure that the
passage of the center of the circle from one corner of the square to thp other cor-
ner leads to an increase to the radius of the circle, while the passage of the cent-
er of the circle from the corner of one square to the corner of another square leads
to a decrease in the radius of the circle'. The former transition is due to the
change of signs of the moments of friction in the gimbals, the second transition is
due to the change of sign of the longitudinal correcting moment. The action of
these two opposite factors leads to the establishment of the limit cycle. Let us
determine the radius of the circle r1 with vertex at the point A for the limit
cycle, at assigned values oft0B, wic, and n. It is obvious that the limit cycle will
become established at r4 = rl.
Let us write the obvious geometrical equalities:
2 ----
um
r: 1. MAP 1- NE' (MG? NL)? -r NE' (1/ r2 - 1,1 -- 2 -- -
2 ?
4 '
? 3 so2 lop' ea I
n ri 4- 4
_ _ _
up. sou ? 102
If(ri -r 4 n) ? n' ? 2 ) -
. .2 ?
?
On equating r4 and ri, after simple transformations we obtain the following
equation for r ?
1'
Whence we find:
r
I 4n?--1) ) (40? 1) 4- 16/0 ? 7nt+ 1=0.
?2n+
1//r1::ii.-E1 I -1
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Fig.9.29 ? Relation between Radius of
Lo
Limit Cycle r1 and n =
Fig.9.30 ? Path of Motion of Gyro
Vertex during Right Turn with Lateral
'P
Correction for = 0.48
11%
Fig.9.31 ? Paths of Motion of Vertex of Gyro during Right Turn
with Lateral Correction Off for < 0.48
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?
?
Lp
On Fig.9.29, r1 has been plotted against the ratio n =--- . For n = 0.48, the
LK
radius of the limit cycle equals half the side of the square e,. In this case the
limit cycle is the boundary cycle (Fig.9.30): all the paths tPat are inside it, are
closed in the rectangles of repose for the segment KL, all the paths located outside
WI?
the limit cycle, wind about that limit cycle. For 0 < n < 0.48 r1 < the limit
wB
cycle does not exist, since, for the closure of the cycle, it is necessary that
wh
In this case all the paths are closed in the rectangle of repose
w
(Fig.9.31). When n increases from 0.48 to 0.5, the radius of the limit cycle in-
creases from zero to infinity. For n = 0.5, the limit cycle does not exist, and
there is a small region of the plane in which the paths meet and are closed in the
zone of repose, while in the rest of the area, the paths are divergent. It follows
from what has been said that when n = 0.48 is selected, the value of the steady
turning errors with the lateral correction turned off on a right turn will be the
same as for a left turn, and will be determined by the formulas eqs.(9.222) and
(9.223). On comparing these quantities with the values of the maximum turning
errors on a right turn with operating lateral correction, the conclusion may be
drawn that turning off the correction when its characteristic is constant decreases
the turning errors. All that need be done is to assure a ratio between moment of
friction and moment of correction of not more than 0.48. In practice this ratio
usually does not exceed 0.1 - 0.2.
Section 9.3. Ballistic Deviations
General Remarks
By ballistic deviations we shall mean here the displacements of the gyro axis
due to the action of longitudinal accelerations on it. By the positional correction
of ballistic deviations we shall mean deviations due to the action of accelerations
on the pendulum serving as the sensitive member of the corrector of the frame whose
axis of rotation is perpendicular to the direction of flight.
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With oscillatory accelerations, oscillations of this pendulum will also take
place. Since the influence of the oscillations of the sensitive member of the cor-
rector has already been elucidated in Chapter 4, we shall confine ourselves here
merely to elucidating the influences of the constant accelerations to which the air-
craft is subjected during the course of some finite interval of time. We shall here
consider that the pendulum of the corrector for the other frame is not subjected to
disturbance.
In view of the foregoing, the expressions for the mismatches in the trans-
mitters of the correction system, in the case of the longitudinal arrangement of the
axis of the outer frame, may be written in the following form
where
51=9-11%.
arc tg - - - .
We shall not take into account the moments of friction in the gimbals, since
their influence in this case is no,t manifested in any new effect.
Let us also take into account, in the equations of motion of the gyroscope, the
influence of the displacement of the aircraft with respect to the earth. Then the
equations of motion of the gyroscope will take the form
/13 (2):
V
1$ + g).
Eallistic Deviations with a Proportional Characteristic
(9.224)
In this case, the equations of motion eqs.(9.224) may be rewritten in the
ttin for simplicity, Ic1 = k2 = k):
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ci+samO,
- LI
V1_ V)
kk's g
(9.225)
The partial solutions of eq.(9.225)jfor V = const, which represent the steady
deviation in the case under study, are written in the following form:
2. 1E10'
V
-( v
gl
,(9.226)
that is, with a variable a there will be no deviation, but with a variable 0 there
will be, and it will consist of the velocity deviation 5v = ---V-, with which we are
already familiar, and the new ballistic deviation:
V
MEI
1(9.227)
equal to the angle of deviations of deflection of a pendulum from the true vertical
under the influence of the acceleration V.
Let us confine ourselves in our further investigation to the equation with re-
spect to the variable Its total integral is written in the form
(9.228)
Let, for t = 0, 13= -6v. Then, for the arbitrary constant B we obtain
(9.229)
II--i
Let us now introduce the variable A5 = 0 +v1 which we shall call the current
'value of the ballistic deviation. Using this variable and the value found for the
arbitrary constant, let us rewrite eq.(9.228) in the following form:
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?
?
.1%,v?Lt (1 ? "
(9.230)
Thus the current value of the ballistic deviation 6* tends here to its steady
value according to a law of increasing exponent with a time constant inverse in mag-
nitude to the single velocity of precession E$ and consequently, the increase of de-
viation will be particularly intense at the initial instant of time, after which it
will slow down. A value practically equal to 45;1., that is, equal to the deflection
of the pendulum, is reached by A51:r after the lapse of the time
3
I
For E = 0.06 1/sec, Ty = 50 sec.
Under actual flying conditions, however, so prolonged a state of acceleration
is improbable, or in other words, under actual flight conditions, the ballistic de-
viation will not reach its full value. It is therefore interesting to elucidate the
ability of the gyroscope to accumulate ballistic deviations over the given time of
action of the acceleraetion-c; or in connection with the given increment of flight
speed V.
It is clear that, other conditions being equal, this ability will be less, the
smaller the value of the unit rate of precession C. On the other hand, at constant
acceleration, the time of action of the acceleration will be determined by the ex-
pression
V
(9.231)
that is, will be inversely proportional to the acceleration.
At the same time, the limiting value to which the ballistic deviation tends
will increase with increasing acceleration.
In order to elucidate the ultimate result of this contradictory influence of
urrent value of the ballistic deviation in the case under
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?
?
?
study, let us substitute in the solution eq.(9.230) the time T..r according to eq.
(9.231), and let us then expand this right side of eq.(9.230) into a series and
transform the solution eq.(9.230) to the following form
telf. 1 OIVIJI. 1 eavl__
V 2' V' Y 3! VI ? ?
(9.232)
It will be seen from the expression so obtained that the ballistic deviation
increases in modulus with the increase in modulus of the increment of velocity AV,
and that for a given AV it will be the larger, the larger in modulus the accelera-
tion V is, since in this case the terms in parenthesis in the right side of eq.
(9.232) will be the closer to the quantity
For the compensation of the ballistic error on account of the increment of the
velocity deviation, the condition
V V
-----0.
R: g
must be satisfied.
Integration of this equation gives thP following law of variation of velocity
at which this compensation is observed:
= t'oe .
Taking = 0.06 1/sec, we find that the compensation of the ballistic error is
possible only an exponential law of increase of velocity with a time constant
equal to 10 hrs. Thus the practical accomplishment of this compensation is unreal,
owing to the different orders of magnitude of the ballistic and velocity deviations.
For a numerical evaluation of the ballistic deviation, let us take the case of
the increment of velocity PIT = 30 m/sec with accelerations of 3 and 6 m/s
6 = 0.06 1/sec. We now obtain, by eq.(9.230), taking t =T.-
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?
?
t - 3 trisic' "at 7.6;
t mine ..8.80.
It will be clear that the errors obtained in this case will be rather consider-
able.
The Constant of the Correction Characteristic
As in the preceding case, it will be sufficient to consider only the equation
of moment with respect to the variable p. In this case it will be rewritten in the
following form:
V?
sign ?10 - ?
?
(9.233)
According to this equation, at = 0, precession of the inner frame occurs in
the sense of matching the gyro axis with the position of the apparent vertical.
The influence of the flight speed V is manifested in this case in the slowing
or acceleration of this precession to an insubstantial extent, and we shall there-
fore not take it into account.
The current value of the ballistic deviation is determined in this case by the
expression
V?v; sign V,
where 1.71. is the time of action of the acceleration.
Or, taking V = const and expressing
we get
AV
P., ? - /
V ? ?
V
274
(9.234)
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?
?
As will be clear, in this case the ballistic deviation will be proportional to
the increment of velocity AV, and inversely proportional to the acceleration 1, that
is, in contrast to the preceding case, it will be the smaller, the greater the
acceleration with which the accumulation of the velocity increment AV takes place.
Let us take for V those same data of the numerical example that were u.sed
above, and let us take wK = 6?/min, AV = 30 misec. In this case we get
v . t rnpria..^ 1?;
; _1 0
= 5
?b rnktiv- ?
As will be clear, with a constant characteristic, the ballistic deviations will
be considerably smaller than with a proportional characteristic.
Let us determine the value of the constant acceleration VKp' for which the
ballistic deviation for an assigned YV reaches its maximum value , equal to the
V
angle of inclination of the apparent vertical k :
since
From this we find:
A V
ing
Vim
(9.235)
(9.236)
? ?
For V > VKp the gyro axis will not succeed in getting as far as the position
of the apparent vertical
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?
AV AV
Ina < P... - ?
? Illp
(9.237)
For V , VK, the axis of the gyro will succeed in reaching the position of the
apparent vertical
?
?
0 - mat ?
g
Thus, for an assimed value of AV, there is a critical value of the constant
acceleration VKp at which the ballistic deviation is maximum.
dith a variable acceleration, the maximum ballistic deviation for an assigned
\V is obtained in the case when the velocity of rotation of the apparent vertical is
equal to the velocity of precession of the gyro. In this case the axis of the gyro
will follow the apparent vertical as it is displaced. For this case, the equality
will hold, since
and
No.t
= I" dt ?g.1tdt.
gw,11,
then the time of increment of velocity is determined from the formula
V 2AV
==
Kula.
Then the maximum value of the ballistic deviation is determined from the formula
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1
%se( "at 2?IVEra
?
that is, it is ir2 times as great as at constant acceleration.
Mixed Correction Characteristic
(9.238)
If the acceleration V is considerable, then the deflection of the apparent
vertical from the true vertical on account of the action of the acceleration will be
considerably greater than the zone of proportionality of the mixed characteristic.
,Consequently the behavior of the gyro will be determined mainly by the constant part
1 of the mixed characteristic, that is, it will be about the same as for constant
correction characteristics.
?
At low accelerations, when
the behavior of the gyro will be the same as with .a proportional characteristic.
But, as already established, with a: propo/Lional characteristic, for a given accel-
eration, the ballistic deviation will be smaller the smaller the acceleration is.
Thus with a mixed characteristic the favorable aspects of both the proportional
and constant characteristics of the correction are to a certain extent both used
with respect to the ballistic deviation.
Section 9.4. Gyro Horizons with Electric Drive
At the present time gyro horizons with an electrically driven gyroscope rotor
are widely used. Gyro horizons with a pneumatic drive, in view of the worsening of
their characteristics in flight at high altitudes, are employed very rarely.
The indication system of the instrument is made in two versions; 1) in the
411 form of a motionless aircraft silhouette, connected with the body of the instrument,
and a movable line imitating the horizon, which is connected with the gyro; and
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2) in the form of a movable aircraft silhouette, connected with the gyroscope, and
a timed horizon line.
The AGK-47B Gyro Horizon
The gyro horizons with corrector and electrically driven gyro rotor include the
type AGK-47B gyro horizon, whose setup and operation are described below. The
AGK-47B gyro horizon is a combination instrument, the single case of which contains
not only a gyro horizon but also a turn indicator and a side-slip indicator. The
instrument needs a triphase current, 36v, 400 cycles. The source of this current is
usually a special PAG-1F transformer.
Fig.9.32 - Parts of Gyro Ebtor of Gyro Horizon
The .yro horizon consists directly of a gyroscopic unit, a correcting unit, an
arrester, and an indicating system.
The axis of rotation of tha outer frame of the gimbals is located parallel to
the transverse axis of the aircraft. The gyro motor casing, whose axis is parallel
to the longitudinal axis of the aircraft, serves as the inner frame of the gimbals.
The gyro motor of the gyre horizon (Fig.9.32) is a triphase asynchronous elec-
tric motor. The stator of the gyro motor (Fig.9.32,c) is attached to the cover of
the casing.
The rotor of the gyro motor (cf.Fig.9.32,b) consists of a massive steel fly-
wheel into which a block of electrotechnical iron with a short circuited winding is
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pressed. The rotor of the gyro motor is made in the form of a bell within which the
stator is placed. This form of rotor makes it possible to obtain the maximum moment
of inertia for the given dimensions and weight of the gyro motor (about 0.9 g.-cm
sec2). The rate of rotation of the rotor of the gyro motor is about 20,000 rpm.
The instrument uses a corrector cons sting of a liquid level electrolytic
Switch of the corrector and two solenoids with cores. Its operation is described in
Chapter 6.
In order to diminish the turning errors, the gyroscope axis is located not
vertically but'inclined forward in the direction of flight by 2?. For this purpose
an inclined boss is provided at the bottom of the gyro motor casing, to which the
sensitive correction element is attached by its base. Obviously bubbles in the
level will be located symmetricelly with respect to their contacts with the indicat-
ed inclination of the gyro axis.
The static balancing of the gyro units is effected with the cores in the middle
position of the solenoids. The current is fed to the gyro motor and the correcting
device through brushes and contact rings placed on the axes of the frames of the
gimbals. The indicating system of the gyro horizon consists of a "horizon linen
connected with the instrument casing, and of an aircraft silhouette, connected with
the gyro unit.
For the bank readings on the instrument to correspond to the actual bank, the
aircraft silhouette is connected with the inner frame of the gimbals through a gear
drive with a gear ratio of 1 : 1. When the aircraft banks, the "horizon linen,
connected with the instrument case, rotates relatively to the initial position by
the angular bank of the aircraft. The outer frame of the gyroscopic gimbals is also
rotated by the angle of bank. If the aircraft silhouette were attached directly to
the axis of the inner frame, then in this case it would be rotated with respect to
the line of the "horizon" by the angle of bank in the direction opposite the actual
bank of the aircraft, and consequently the instrument would yield the reverse read-
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?
ings, but since the aircraft outline is connected with the axis of the inner frame
of the gimbals through the above mentioned transmission, it would therefore turn
- Indicating System of
Gyro Horizon
??0:0..1 t
FiE.9.34 - Outer View of Gyro Horizon
1 - Handle for moving index of horizon;
1 - Horizon line; 2 - Aircraft 2 - Arrester
silhouette; 3 - Inner frame of gimbals
(Casing of gyro motor); 4 - Outer
frame of gimbals; 5 - Gear drive;
6 - Solenoids
when the aircraft banks with respect to the "horizon lineWby the angle of bank, in
the direction corresponding to the actual bank of the aircraft. This is illustrated
by Fig.9.33.
In addition, a scale of longitudinal inclinations and a scale of banks are
attached to the gyro unit. For convenient use of the instrument, the line of the
horizon may be shifted with respect to the instrument case by means of a crank
mechanism, whose handle is brought out on the face side of the instrument
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A special mechanical device is used to arrest the gyro-unit. The maximum time
necessary for arresting an operating instrument does not exceed 15 sec.
AGB-1 Bomber Gyro Horizon
The AGB-1 type electric gyro horizon is another example of the gyro horizon
with radial correction (Fig.9.35), which is used on heavy aircraft. On the face
part of the gyro horizon, a side-slip indicator is placed.
?
?
Fig.9.35 - a) External View of Type AGB-1 Gyro Horizon; b) Face View
1 - Front flange; 2 - Aircraft outline; 3 - Vertical pressure; 4 - Horizon line;
5 - Spherical screen; 6 - Bank scale; 7 - Ball of side-slip indicator; 8 - Scale
division of side-slip indicator; 9 - Handle for blocking corrector switch;
10 - Handle of angle-of-attack mechanism
The gyro horizon consists of a gyroscopic unit, a correcting mechanism, an in-
dicating.system, a starting system, and a system for starting the corrector, and
cutting out the corrector during turns.
Figure 9.36 gives a schematic diagram showing the arrangement of the gimbals on
the aircraft. The axis of rotation of the outer frame of the gyroscope gimbals is
located parallel to the longitudinal axis of the aircraft, while the axis of the
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?
inner frame (the gyro motor casing)
This arrangement of the gimbal axes
c)
Fif;.9.36 - Schematic Diagram of
Arrangement of Gyro Horizon on
Aircraft
is parallel to the lateral axis of the aircraft.
allows
a) Axis of measurement of angle of
bank; b) Axis of measurement of anEle
of pitch; c) Direction of longitudinal
axis of aircraft; d) Trace of gyro:-
scope axis
frame of the gimbals. The stator
the cover of the casing.
of the
measurement of the true angle of pitch
(dive or climb) during a bank, that is,
it allows measurement of the angle be-
tween the longitudinal axis of the air-
craft and the plane of the horizon, and
meas-irement of the true angle of bank,
the angle of rotation of the aircraft
about its longitudinal axis when the
angle of pitch is not zero. The range
of measurement of the angles is 1800 in
bank and 160? in pitch. The principal
feature of the gyro horizon is the high
accuracy of its readings during a turn,
which is obtained by cutting out the
lateral correction of the gyroscope.
The gyro unit of the instrument (Fig.
9.37) contains the gyro motor (Figs.
9.38 and 9.39) which is placed inside a
casing. The casing is also the inner
gyro motor (2) (Fig.9.38) is attached to
The rotor (1) is a massive steel flywheel with the block
(6) of electrotechnical steel and a short circuited winding, embracing the stator on
the outside, pressed into it. The moment of inertia of the rotor is 1.9 g - cm -
sec2. The speed of the rotor is 21,000 - 23,000 rpm. The kinetic moment of the
gyroscope amounts to about 4000 g - cm - sec. The motor of the rotor is triphase
and synchronous, with one pair of poles and star-connected windings. It is fed by
a triphase 36 volt 400 cycle alternating current. The line current of the motor
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18 19 20
6
9
11
14 0
\ s
"m
Jab ft--16j
AR
MRZ.-a7
m=
/2 /3 /4
?? ?
? all ? .
=LoyII 4 BE
4
b)
c)
a - View from above
1 - Junction box; 2 - Lock nut; 3 - Disc;
4 - Pin; 5 - Nut; 6 - Shaft of gyro unit;
7 - Screw; 8 - Body of gyro motor; 9 - Gyro
motor; 10 - Cover of gyro motor; 11 - Bush-
ing: 12 - Lead balancing weight; 13 - Brack-
et; ih - Balancing screw; 15 - Rotor of
electric motor of lateral corrector;
16 - Screw; 17 - Pressure reading;
18 - Stand; 19 - Contact group: 20 - Nut
b - Side view
1 - Cover of gyro motor; 2 - Axis of gyro
unit; 3 - Body of gyro motor; L1 - Liquid
switch; 5 - Screw; 6 - Rotor of electric
motor; 7 - Balancing screw; 8 - Bracket;
9 - Bushing; 10 - Nut; 11 - Pin; 12 - Nut;
13 - Disc; 14 - Gyro motor
Fig.9.37 - Gyro Unit of Gyrg Horizon
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Fig.9.38 - Gyro Motor
1 - Gyroscope rotor; 2 - Stator of electric motor; 3 - Bushing;
4 - Cover of gyro motor; 5 - Screw; 6 - Electrotechnical steel block
with short circuited winding; 7 - Cover of gyro motor; 8 - Bushing;
9 - Shaft of gyro motor; 10 - Bearing; 11 - Nut
Fig.9.39 - Outside View of Gyro Motor
284
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phase is about 0.35 amp. The time taken by the gyroscope to reach full speed is
2 min. During operation the gyro motor may be heated to not over 450 C. In the
I.
Fig.9.40 - Kinematic Scheme of Gyro Horizon
1 - Gyro unit; 2 - Gimbal frame; 3 - Rotor of lateral corrector electric
uotor; 4 - Stator of electric motor; 5 - Spherical screen; 6 - Handle of
angle-of-attack mechanism; 7 - Vertical scale division; 8 - Side-slip
indicator; 9 - Aircraft outline; 10 - Bank scale; 11 - Signal flag;
12 - Handle for blocking corrector switch; 13 - Front flange; 14 - Strip-
horizon line; 15 - Current leads; 16 - Casing; 17 - Carrier; 18 - Lever;
19 - Bracket; 20 - Stator of electric motor of longitudinal corrector;
21 - Rotor of electric motor; 22 - Brushes of switch of lateral corrector;
23 - Leads; 24 - Cutout disc for lateral corrector; 25 - Bimetallic relay
a) Direction of flight
lower part of the gyro unit a pendulum switch is attached, forming the sensitive
element of the corrector. In design and characteristics it does not differ from the
sensitive element of the corrector of the AGK-47B gyro horizon (cf.Fig.6.14). The
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1
gyro-unit (1) (Fig.9.40) is suspended on bearings in the gimbal frame (2). The
rotor of the electric motor of the lateral corrector (3) is rigidly connected to it.
The st6tor of the electric motor is attached to the gimbal frame (2), which is
suspended on bearings in the body (16) of the instrument. In addition, the rotor of
the longitudinal correction electric motor (21), the disc (24) for cutting out the
lateral correction during a turn, the spherical screen (5) with vertical scale (7)
for reading' off the lateral bank, and three horizontal scales for reading off the
angle of pitch (0 and I 100) are all rigidly attached to the frame (2).
The electric current is supplied to the gyro motor and the correction system by
means of the moment-free power inputs (15) and (23). The moments of friction of
these inputs are negligibly small, since the pointed contacts are located on the
axis of rotation on the gimbals. The strip (14) with lever (18) reproduces the line
of the horizon. The lever (18) is suspended in the brackets (19) attached to the
frame (2). The tang (17), attached on the casing of the gyro unit enters the slot
of the lever (18). In the horizontal position of the aircraft, the lever (18) with
the strip (14) are perpendicular to the axis of the rotor to the rotor shaft of the
gyro motor, and are consequently also in the horizontal plane. The aircraft outline
(8) is connected with the instrument case (16) through a gear drive which is con-
trolled by the lever (6) for setting the angle of attack. When the aircraft is in-
clined, the body and outer frame of the gimbals are also inclined with respect to
the vertical shaft of the rotor of the gyro motor. These inclinations are reproduc-
ed by the strip-horizon (14) with respect to the aircraft outline (9). Figure 9.41
shows the readings of the gyro horizon during various maneuvers of the aircraft.
The electric motors of the longitudinal and lateral correctors consist of
asynchronous biphase electric motors (Fig.9.42). The operation of the correction
system is described in Chapter 6 on pages 74-76. Three windings are placed in the
rotor grooves: one excitation,winding and two control windings with the windings in
411 opposite senses. The reversal of the correcting moment is effected by switching on
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9-8000Z0000ZOnl?1701-0-1-8dCW-V10 91-/90/?1,0Z eseeiej -104 penaiddv /Woo pazwes - ;Jed LII108!4!sseloaCI
N.)
?
a)
N ? V
A
???????!
o ? ) ii4C;
i'?>7
.4 I.
?
g)
d)
?
Fig.9.41 - Readings of Gyro Horizon during Various Maneuvers of Aircraft -
a - Horizontal flight; b - Gaining altitude; c - Descent; d - Left turn;
e - Right turn; f , Left turn-and gaining altitude; g - left turn and descent;
h - Right turn and gaining altitude; i - Right turn and descent
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one control winding or the other. The stator consists of a block of electrotechni-
cal steel with grooves cast in aluminum alloy and forming a short-circuited stator
winding.
Fig.9.42 - Outer View of Electric Correction Motor
1 - Rotor; 2 - Stator
The maximum torque developed by the electric motor is 7 - 9 g - cm, and the
diameter of its rotor is 61 mm. The resistance of the excitation winding is
5.2 I 0.5 ohms and that of each' of the control windings is 420 ! 50 ohms.
Figure 9.43 gives the electrical circuit of the gyro horizon. The current is
supplied to the gyro motor (1) from the series converter PAG-1F through the term!-
!
nals ABC of the plug connector of the gyro horizon. The central contact of the
liquid switch (2) is connected to phase A of the gyro motor. The excitation wind-
ings of the electric motors (3) and (4) of the correction are connected in series
with the phase windings B and C of the electric motor of the gyroscope. This method
of connecting the windings increases the torques of the correcting motors on start-
ing the instrument by 2 - 2.5 times as a result of the starting currents of the gyro
motor flowing through them. This considerably shortens the time required for the
. t?
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gyroscope to reach the vertical operating position on starting.
The lateral correcting motor (4) is turned off, during a bank, by the contact
disc (5), located on the shaft of the outer frame of the gimbals. When the aircraft
banks by more than 7 - 90, the lateral correction is cut out. Two current-carrying
sectors on this disc, with angles up to 180, are electrically connected with the
common lead-out (8) of the electric correcting motor (4). In horizontal
flight, the brushes (9), connected with the body of the instrument, are located in
the middle of the sectors of the disc (5). In this case, the control winding and
the motor (4) are connected with phase B and develop the normal correcting moments.
If the bank of the aircraft exceeds 90, then the brushes (9) break contact with the
current carrying sectors of the disc (5), and the motor (4) is switched off.
The corrector cutoff is blocked during the period of starting the instrument by
means of the bimetallic relay (7) and the mechanism (6), which are connected with
the object of duplication.
At the instant of starting the instrument, the bimetallic relay is in the
closed state and the control windings of the electric motor (4) are connected to the
contact of this relay to phase B. In this case the lateral corrector of the gyro-
scope operates at any lateral inclinations of the gyroscope. At the same time the
starting mechanism (6), whose closed contacts duplicate the bimetallic relay (7), is
turned on. This duplication allows the instrument to be started even if the bi-
metallic relay should fail for some reason, or if the pilot should forget to turn on
the starter mechanism (6).
The winding on the bimetallic relay, with a resistance of 1500 ? 50 ohms, be-
comes heated, and in 45 - 180 sec after the current is turned on, it breaks the cir-
cuit connecting the phase B with the terminal G of the snap connector. During this
time the gyro spindle is able to establish itself in the vertical position, owing to
the forced operation of the correction mechanism.
To assure stability of the operating time of the bimetallic relay in an ambient
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temperatUre range from +500 to -60? C, two bimetallic plates are installed in it.
:Jhen the temperature of the air varies, both plates are flexed parallel to each
other, maintaining the contact force constant at 25 to 30 g.. When the heating wind-
ing is switched onto one of the bimetallic plates, this plate becomes heated and de-
formed. ,The contact pressure continuously weakens, and after 40 - 150 sec, the con-
tacts are separated.
Rl
ii 14-*
? 9
R1
Fig.9.43 - Electrical Circuit of Gyro Horizon
a - Gyro motor winding; 2 - Liquid pendulum switch; 3 - Electric motor for
longitudinal corrector; 4 - Electric motor for lateral corrector; 5 - Disc
for cutting out corrector; 6 - Mechanism for blocking corrector cutout;
7 - 3imetallic relay; 8 - Common leadout of control winding; 9 - Brushes;
Ri, R2, R3 - Resistors
The signal flag (11), connected with the handle (12), appears in the zone of
visibility of the front part of the instrument (Fig.9.441a) when the cutoff of the
lateral correction is blocked. When the block is removed, that is, when the switch
is in a neutral position, the signal flag disappears (Fig.9.44,b).
The resistors R1 = 500 ohms and R2 = 240 ohms serve to produce the necessary
ics for the correction motors. The resistor R3 - 5.1 ohms is a
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spark arrester in the circuit of the brushes (9) and the disc sectors (5). The ter-
minal G of the snap connector serves to test the operating time of the relay (7) and
the angle of cutoff of the lateral corrector. With this object a voltmeter is con-
nected between the terminals B and G. If the brushes (9) are on the current carry-
ing sectors of the disc (5), then the voltmeter will show a voltage drop of about
18 volts across resistor R2. When the brushes are displaced away from sectors, the
voltmeter shows the full phase voltage of about 36 - 40 volts.
Figure 9.45 shows the design of the instrument.
Fig.9.44 - Front View of Gyro Horizon before Starting (a)
and After Establishment of the Gyro (b)
The technical characteristics of the gyro horizon are as follows:
1. Errors in determining bank and pitch in rectilinear flight, not over 10.
2. Error in readings due to turns lasting up to 6 min and with banks from 10 to
8000 amount to 1 - 20.
3. Errors in readings arising on additional turns of aircraft by 30 - 60? with
banks of 5 - 70, reach 40.
4. Direct current used by instruments when supplied by the PAG-1F converter,
2.5 amp.
5. Alternating current used by each phase not more than 0.4 amp.
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bi
Fig.9.45 - General View of AGB-1 Gyro Horizon
a - Top view; b - Side view
I1
16
1 - Bearing; 2 - Special disc; 3 - Nut; 4 - Body plug; 5 - Glass; 6 - Ring;
7 - Stator of longitudinal corrector electric motor; 8 - Stand; 9 - Screw;
10 - Pin; 11 - Disc; 12 - Plug; 13 - Plug yoke; 14 - Washer; 15 - Rear cover;
16 - Casing; 17 - Gimbal connection; 18 - Body; 19 - Gasket; 20 - Gasket
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6. Time to prepare instrument for use: at ambient temperatures of +20 - +50?C,
not more than .3 min, and at -60?C not more than 6 min. (An aircraft with a gyro
horizon is allowed to take off only after the lapse of this period of preparation of
the instrument.)
7. Maximum altitude of instrument, up to 20,000 in.
8. Weight of instrument, not over 2 kg.
9. Instrument is vibration proof for horizontal vibration with accelerations up
to 0.6 g, and for vertical vibration with accelerations up to 1.5 g in the frequency
range of 10 to 80 cycles.
The mechanism of the gyro horizon is covered by the casing (16) (Fig.9.45) with
the rubber washer (19), keeping the instrument dustproof. With this same object,
gaskets are placed between the casing (16) and the socket yoke (13). The gyro unit
is balanced by means of lead weights and balancing screws. The axial play of the
gimbal connection (17) is accomplished by gaskets inserted between the plug (4) and
the body (18) of the instrument. The face side of the instrument is covered by the
glass (5) attached to the front cover by the split ring (6).
The installed diameter of the gyro horizon is 110 mm, the length of the body is
156 mm. The attachment to the instrument board is accomplished by four 113 x 15
screws screwed into locking nuts of the front flange.
? '
The AGI-1 Fighter Gyro Horizon
In stunt flying, the shaft of the gyro rotor may coincide with the shalt of the
outer frame. In this case the gyro will lose one degree of freedom and will there-
fore lose its stability and will be blocked thereby interfering with the normal
operation of the gyro horizon. Various devices are used to prevent the coincidence
of these axes.
Figure 9.46 shows another version of this device in a gyro vertical with stop.
On the casing (5) of the gyro rotor is installed the rod (2) with the disc (3)
at the end. Through the outer axis of the gimbals is passed the stop (1) which, STAT
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when the axis of the gyroscope rotor dangerously approaches the shaft of the inner
frame, will make contact' with the disc (3). The force acting on the disc will pro-
duce a moment whose vector is directed along the axis of the inner frame of the
Fig.9.46 - Gyro Vertical with Stop
b)
1 - Stop; 2 - Rod; 3 - Disc; 4 - Hollow gimbal shaft; 5 - Casing;
6 - Outer frame
a) Axis of gyroscope rotor; b) Longitudinal axis of aircraft
gimbals. The precession due to this moment is directed in such a way that the disc
(3) slides along the surface of the rest of the stop (1) and runs around it, thus
eliminating the coincidence between the gyroscope spindle and the outer frame of the
gimbals. Obviously the stop (1) will force the gyroscope spindle to deflect from
the vertical by a certain angle, which is a shortcoming of this device.
Another example of a device that eliminates the superimposition of the axes, is
the gyro horizon in which the axis of the gyroscope rotor does not lie in the plane
perpendicular to the axis of the inner frame of the gimbals and of the outer frame
passing through its axis, but is inclined by angle of 30 with respect to this plane.
The type ACI-1 gyro horizon for a fighter aircraft is an improved example of
such designs (Fig.9.47).
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In this gyro horizon, the angle between the axis of the gimbals and the axis of
the gyroscope rotor is automatically held at 900 during any maneuvers of the air-
craft, as a result of which the gyroscope maintains its maximum stability constant.
For this purpose, the gyro unit (1) (Fig.9.48) and the gimbals (2) are suspended on
bearings in the servo frame (3), which is automatically maintained by the electric
motor (6) in a position such that the axis II-II shall be perpendicular to the prin-
cipal axis of rotation of the gyro. The sensitive element registering any disturb-
ance in the mutually perpendicular position of the axes, and which switches on the
actuating electric motor (6) to restore the mutually perpendicular position of the
axes, consists of the switch (4) attached to the outer gimbals (2), and the sliding
contact (5), connected with the gyro unit. The rate of actuation of the servo frame
is taken considerably greater than the maximum angular velocity of the banking
maneuvers of the aircraft, thanks to which the mutually perpendicular position of
the axes and the maximum stability of the gyro are maintained during any maneuvers
0 of the aircraft whatsoever.
The readings of bank 'and pitch of the aircraft are measured from the position
of the aircraft outline (7), which is connected with the body of the instrument,
with respect to the spherical scale (8) connected with the gyro unit. Consequently,
the scale (8) for reading off the longitudinal and lateral banks does not vary its
position with respect to the ground. The angle of Ditch is measured about the hori-
zontal axis II-II, the angle of bank is measured about the axis I-I located parallel
to the longitudinal axis of the aircraft. This position of the axes of the gyro
system allows measurements of the true angles of pitch and bank. The spherical
scale (8) is painted in colors: the upper hemisphere is colored brown, the lower
blue. When the aircraft dives, the aircraft outline enters the upper hemisphere;
this hemisphere has numbered scale divisions for reading off the angles of pitch
from 0 to 360?. The accuracy of the reading of the angles of bank decreases as the
angle of pitch increases.
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Fig.9.47 - a) Outer View of Gyro Horizon,
Type AGI-1, for Fighter Aircraft; b) Face
View
1 - Body; 2 - Aircraft outline;
3 - Spherical scale; 4 - Index-horizon;
5 - Pitch scale; 6 - Bank scale;
7 - Sideslip indicator; 8 - Index of
sideslip indicator; 9 - Handle for shift-
ing aircraft outline; 10 - Starting
?
button
297
Thus, for instance, with an
angle of pitch up to 70?, the error
in bank does not exceed 30; with an
angle of pitch of 75? the error
amounts to 3 - 4?; with an angle of
pitch of 80? the error reaches 6?.
In a vertical dive or climb (angle of
pitch I90?), the longitudinal axis of
the aircraft coincides with the
direction of the principal axis of
the gyro, and t:he gyro horizon loses
its sensitivity to the angle of bank.
Consequently, in diving or climbing
at angles of around 80 - 90?, the
bank cannot be followed with the gyro
horizon.
To eliminate the turning errors,
the AGI-1 gyro horizon, like the
AGB-1 gyro horizon, is provided with
a cutout for the corrector, on turns
with an angle of bank of more than
13?. Owing to this cutout, the error
after a turn does not exceed 30
.
The electrical circuit as given
by Fig.9.49 for the AGI-1 gyro hori-
zon has much in common with the cir-
cuit of Fig.9.43. The electric motor
(1) of the gyroscope, thel liquid
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pendulum switch (2), and the electric correction motors (3) and (4), are the same as
in the AGP-1 ..yro horizon. The power supply for the electric circuit of the gyro
horizon is through the terminals ABC
from a type PAG-1F converter. The
excitation windings of the correction
motors (3) and (4) are connected in
series with the phases of the elec-
tric motor of the gyroscope, which
provides for the forced state of
operation of these motors on starting
the instrument. The operation of the
- Schematic Kinematic Diagram
correction system of the gyroscopes
of AGI-1
is the same in AGI-1 and AGB-1. The
1 - Gyro unit; 2 - Gimbals; 3 - Servo excitation winding of the actuating
frame; 4 - Switch for electric motor; motor of the servo frame (frame (3)
5 - Contact , of switch; 6 - Electric in Fig.9.48) is connected between
actuating motor; 7 - Aircraft outline; phases A and B. The middle point of
- Spherical scale the control windings is connected to
a) Direction of flight phase A. The ends of the control
windings are brought out to the
switch (5). The contact of the switch, connected with the case of the gyro motor,
is connected to the phase B and feeds this phase to the first or second control
winding of the motor (6), depending on the side to which the outer gimbal axis of
the gyroscope deflects from the perpendicular position to the shaft of the rotor of
the gyro horizon. The electric motor (6) actuates the servo frame and restores the
perpendicular position between the axes. The switch (7), located on the outer shaft
of the gimbal, switches the ends of the control windings of the electric motor (6)
and provides the correct sense of rotation of the electric motor in the case of the
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actuation of the servn frame when making a Nesterov loop.
The ballast resistor 10 limits the rate of precession of the gyroscope in the
longitudinal direction to 3 ? 5.5 ?Ablin. At ,temperatures below -35?C this resistor
8
Fig.9.49 ? Schematic Electrical Diagram of AGI-1
1 _ Electric motor of gyroscopes; 2 - Liquid pendulum switch; 3 ? Longitudinal
correction electric motor; 4 ? Electric motor of lateral correction; 5 ? Switch
of electric motor of actuation of servo frame; 6 ? Electric motor of actuation
of servo frame; 7 ? Switch; 8 and 9 ? Spark arresting resistors of 3300 ohms
(0.25 w); 10 - Ballast resistor, 300-430 ohms (0.5 w); 11 - Bimetallic switch;
12 - Cutout of corrector on turns; 13 - Spark arrestor resistor, 5100 ohms
(0.25 w); 14 - Bimetallic relay; 15 - Heating element; 16 - Ballast resistor,
100 ohms (1 w)
is closed by the bimetallic switch (11), thus accelerating the starting of the in-
strument under low temperature conditions.
The cutoff switch for the lateral corrector (12), operates when the aircraft
banks more than 13?. The ballast resistor (16) limits the current in the control
windings of the electric motor and of the lateral corrector. The rate of preTA
c---4on
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of the gyroscope in the lateral direction is as high as 3.5 - 7 Vmin.
In starting the instrument, the corrector cutoff switch (12) is blocked by the
bimetallic time relay (14), and the control windings of the electric motor (4) of
the lateral corrector are connected to the terminal 13 through the closed contacts of
relay (14), regardless of the initial position of the gyro unit or the magnitude of
the angle of bank. In 40 - 150 sec after the heating element (15) is turned on, it
causes deformation of one of the bimetallic plates of relay (14), thus separating
the contacts, unblocking the corrector cutoff switch, and bringing the gyro horizon
from the starting into the operating state. The heating element is always energized
after that, and a clearance of the order of 0.4 - 0.8 mm is established between the
contacts of relay (14).
The terminal G in the snap connector is used, as in the gyro horizon AGB-1, for
testing the operating time of the relay (14) and the angle of bank at which the cor-
rection is switched cff.
Figure 9.50 gives a structural diagram of the AGI-1 gyro horizon. The gyro
motor (1), with liquid pendulum switch, is suspended on bearings in the gimbal (4).
On the shaft of the gyro unit is attached the spherical scale (6), the cut-in switch
(5) for the mutually perpendicular position of the axes, and the rotor of the elec-
tric motor (2) for the longitudinal correction. The stator of this electric motor
is located on the frame (4).
The frame (4) is suspended on bearings attached in the servo frame (7).
The stator of the electric motor (3) of the lateral corrector is mounted in the
- servo frame (7), while the rotor is attached to the frame (4). The servo frame sus-
pended in the body rests on its front part on the three supporting bearings (8) and
in the rear part on one radial ball bearing. The electric actuating motor is
attached to the instrument case and is connected with the servo frame through a
reducer with a gear ratio of 1 : 16.3. On the rear part of the instrument, on the
IIIshaft of the servo frame, is mounted the switch (15), cutting out the corrector
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during turns.
To shorten the starting time of the instrument the push-button mechanism (10)
is employed to switch on the power supply of the gyro horizon.
Fig.9.50 - Structural Diagram of AGI -1
1 - Gyro motor with liquid pendulum switch; 2 - Electric motor for longitudinal
corrector; 3 - Electric motor for lateral corrector; 4 - Gimbal frame; 5 Switch
for setting axes in mutually perpendicular position; 6 - Spherical scale;
7 - Servo frame; 8 - Support bearings of servo frame; 9 - Cam surface of servo
frame; 10 - Push-button mechanism; 11 - Electric motor operating servo frame;
12 - Aircraft outline; 13 - Mechanism for shifting aircraft outline; 14 - Lateral
banking scale; 15 - Cutout switch for corrector on turns; 16 - Side slip indicator
When the button (10) is pressed, the force is transmitted through a bearing on
the face side of the servo frame, made in the form of the cam (9). The profile of
the face cam is so selected as to assure the setting of the servo frame frpm any
arbitrary position to its normal horizontal position. The displacement of the air-
vertical plane is effected by means of the mechanism (13)STAT
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n-
.AI
0 8 It! 11 12 1.1 F43 0
Fig.9.51 - General View of AGI-1 (View from Top)
1 - Gyro unit; 2 - Glass; 3 - Spherical scale; 4 - Aircraft outline; 5 - Bimetallic switch; 6 - Ball
bearing of push-button starting mechanism; 7 - Starting push-button; 8 - Instrument casing; 9 - Gimbal
frame; 10 - Rubber gasket; 11 - Ball bearing; 12 - Brush unit and contact rings; 13 - Unit of servo
frame; 14,15 - Stator and rotor of electric motor for longitudinal correction; 16 - Unit of plate;
17 - Spark-suppressor and ballast resistors; 18 - Bearing of shaft of servo frame; 19 - Unit of current-
carrying brushes; 20 - Current-collector rings; 21 - Spindle of servo frame; 22 - Disc of corrector cut-
off; 23 - Casing; 2/4. - rechanism of bimetallic time relay; 25,26 - Rotor and stator of electric motor for
lateral corrector; 27 - Ball bearing; 28 - Ring with supporting bearings for servo frame; 29 - Unit of
front flange; 30 - Handle of mechanism for shifting aircraft outline
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Fig.9.52 - General View of AGI -1 (Side View)
1 - Stator of electric motor of gyroscope; 2 - Rotor of electric motor and gyroscope; 3 - Side-slip
indicator; I. - Attachment of front flange; 5 - Liquid pendulum switch; 6 - Starting pendulum;
7 - Yoke for snap connector; 8 - Rubber gasket; 9 - Disc of correction cutout; 10 - Collector; ?
11 - Shaft of servo frame; 12 - Bearing; 13 - Electric motor actuating servo frame; 14 - Casing;
15 - Attachment of plate; 16 - Reducer; 17 - Attachment of servo frame; 18 - Spherical scale;
19 - Rubber gasket; 20 - Case of instrument; 21 - Ring with bearings
9-8000Z0000ZOnl?1701,0-1-8dCll-V10 91-/90/?1,0z 8S8I8I -104 penaiddv Ado Pez!PeS LII108!4!sseloaCI
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re.
whose handle is located in the left part of the front flange of the body.
The angles of bank are read off on scale (14), the angles of pitch on the
spherical scale (6). The side-slip indicator (16) is placed at the bottom of the
411
front flange of the case.
Firures 9.51, 9.52 and 9.53 show the structure of gyro horizon AGI-1.
The technical data of the gyro motor and pendulum liquid switch are the same as
those of the analogous units of the AGB-1. The longitudinal corrector motor
develops a torque of not less than 7 r; - cm. The DC resistance of its excitation
winding is 5 ohms 1 0.5 ohm; the DC resistance of each control winding is 180 ohms 1
le ohms. The electric motor for the lateral corrector develops a torque of about
5.5 g - Cu; the direct current resistance of these windings is: excitation winding,
12.5 ohms 1 1 ohm; and for each control winding, 300 ohms 30 ohms.
The heating element of the bimetallic time relay has a DC resistance of
1500 ohms 1 50 ohms. The electric motor operating the servo frame is standard, of
type DID-0.5. Its maximum static torque is about 5.5 s - cm. Its idling speed is
not less than 13,000 rpm. The direct current resistance of the excitation winding
is 70 ohms 1 7 ohms. The resistance of each control winding is 260 ohms ? 26 ohms.
?
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Frincipal Characteristics of AGI-1 Gyro Horizon
1. Error of readings of angles of pitch and-bank in state of horizontal
irht, 10.
2. Errors of instrment:
a) after turn with bank of over 150, not over 30;
U) after stunt flying, not over 5?.
3. Range of measurement of angles of pitch and bank, unlimited.
4. P0Wer supply of instrument, triphase, 36 v, 400 cycle alternating current.
Alternating current (line) drawn, not over 0.6 arr.
5. Altitude erfo rmanc e of instrument to 20,000 m.
nge of operation from
304
+50?C to -60?C.
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7. Weight of instrument not over 2.6 kg.
8. Time when instrument ready for operation after switching on:
at temperatures from +500 to -30?C 2 min
at temperatures from -300 to -60?C
3 min
Fig.9.53 - Plate Unit (Rear View)
1 - Bearing; 2 - Electric actuating motor with reducer; 3 - Bimetallic relay;
4 - Yoke of plug connector; 5, 6, 7 - Spark-suppressor resistors; 8 -'Ballast
resistor, 100 ohms (1 w); 9 - Assembly boxes
- The instrument allows supervision of the state of horizontal flight, bringing
the aircraft into horizontal flight on loss of spatial orientation in bank and
_pitch, and allows supervision of the accuracy of execution of all maneuvers by an
"I
'unt maneuvers.
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au'ins night flirhts, the colored luminescent coating of the indiCating
elements cf the gyro horizon provides easily perceived orientation in bank and pitch.
?????
Fig.9.54 - Readings of AGI-1 Gyro Horizon in Performing a Nesterov Loop
Figures 9.54, 9.55, and 9.56 give examples of the readings of the gyro horizon
during axection of various maneuvers.
The AGI-1 gyro horizon is installed on a shock absorbing mounting with an
allowable load factor of 1.3 g. The instrument board should be located perpendicu-
lar to the longitudinal axis of the aircraft with an accuracy of 1 10. The gyro
horizon is attached to the instrument board by four M5 x 15 screws with half-round
STAT
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heads.
?
goa,,
? Ar.
0
? V?:.?
?41 ij
Fig.9.55 - Readings of AGI-1 Gyro Horizon during a Right Barrel Turn
1,4 r iL
?
?vfris;r44;.-rarric-r-?--.7-71,-raw-iwymocipwwgipaRikikpotyr.icir,
lings of AGI-1 Gyro Horizon during a Nesterav Half Loop
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The PAG-1F converter supplying the gyro horizon with power should have a 15 amp
fuse for the starting current and a 3 amp fuse for the operating current.
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CHAPTER 10
DEVIATIONS OF COURSE GYROSCOPES
Section 10.1. The Directional Gyro (GPK)
General Remarks
A course gyroscope whose outer frame has a constant precession designed to com-
pensate the effect of the earth rotation, while the inner frame has a radial correc-
tion with one characteristic or another, accomplished either with a pendulum or by
the position with respect to the outer frame, is called a directional gyro. Usually
the outer frame, that is, the frame which is not corrected, is used as the working
frame in the directional gyro, that is, till frame used to attach the compass card,
or for connection with the corresponding transmitter of the autopilot. Its position
is oriented before it is turned on by rotating the arrested gyroscope by means of
the corresponding handle until the readings of the directional gyro card agree with
the readings of the magnetic compass card. After this has been accomplished, the
arrester is released. By virtue of the properties of gyroscopic rigidity, for a
limited period of time, of the order of 10 - 15 min, the directional gyrol will in-
dicate, with a certain degree of accuracy, the course of the aircraft, that is, it
will serve as a compass. When this period of time has elapsed, the readings of the
directional gyro are again verified from the readings of the magnetic compass and
the appropriate corrections made in its position on the basis of these readings.
This principle of utilization of the directional gyro is connected with the
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fact that the exact holding of the aircraft course still does not mean the exact
holding of the assigned route line, owing to the influence of drift, which is always
known only approximately. This circumstance forces periodic checks, every 10 - 15
min, of the actual maintenance of the assigned route, followed by the appropriate
corrections to the assigned course. When this is done the position of the direc-
tional gyro is incidentally corrected.
We shall consider that, at the initial instant, the corresponding correction
has been made, and that in connection with this correction the spin axis is located,
at the initial instant, sufficiently close to the plane of the meridian. Its non-
coincidence with this plane and its subsequent deflection from it by the angle of
shift of the outer frame will be what will characterize the deviation of the direc-
tional gyro.
Deviation of Directional Gyro in Rectilinear Flight
? sion
The starting equations of motion will, as before, be the equations of preces-
Hq L ?
? lip
For p and q we have, respectively,(cf. eq.(2.36) and eq.(2.37):
VV
P ? 0 ?(?1 sin A.)2 + cos K.
qmoB2 ?+ sInk),+ (w,+ t? sin A"
- -)sin f,
k ? , (Os?
(loa)
wherefol = horizontal component of angular velocity of earth rotation, wi = we
cos 99;
we = angular velocity of earth rotation;
V = flight speed;
rc raft;
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= latitude of place;
R = radius of earth.
The expressions for the moments Lk and I. will be taken in the following form:
k2+ sign
- (p) - 2. ,sign
(16.3)
where k2 - the moment, constant in magnitude and direction, designed to compensate
qin
the effect of the earth rotation;
component of angular velocity of rotation of aircraft about the axis of
the outer frame, i.e., along the normal axis of the aircraft; since the
axis of the outer frame of the directional gyro is located parallel to
the normal axis of the aircraft, then, for horizontal flight,n will
represent the angular velocity of yawing of the aircraft;
f(p) - moment of correction of inner frame;
- component of angular velocity of rotation of aircraft about the axis of
the inner frame Ox.
The form of the expression selected for Itc and Ly thus take into account the
influence of friction in the suspension, not only in connection with the rotation of
the gyro about the axes of the gimbals, but also in connection with the rotation of
the aircraft about these axes. The actual picture of the origin of the moments of
friction in the axes is precisely what is connected with the sign of the resultant
velocity of rotation of the corresponding bearing and the spindle lying in it, with
respect to each other.
On substituting eqs.(10.2) and (10.3) in (10.1), we get
V s i n K '1
ti + V Sin K ) + (w3+ R co, , I sin ,1"`
.113+ Loign (0 ? 4+):
II [0 -f--(.0,1 +V sin k) a. ?? - V " R K.- 1 -= - i()?
R
? L.,sign (ci ? .%..).
311
(10.4)
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We shall consider in the future that the characteristic of correction f(0) is
proportional. Then eq.(10.4) may be rewritten in the following form:
where
d I all + V sln K)p -f-(.21.
k
V . ,
-( w -
Iii A i a ? ap -4- 23.
i l
1 f k
(10.5)
Id' ale ?(03+ RV- csios" Ai.) sin 4-1.17 sign (0 ?40.
(10.6)
0 V cm L
sign(a-- (10.7)4
?1
?
Let us confine ourselves to a qualitative analysis of the paths of the vertex
of the gyro, by constructing the field of tangents to these paths.
On dividing one of eqs.(10.5) by the other, we get
4 V On K) WI
eh
?
d? V
(-1 i? sin ) IP-- VI
?
(10.8)
The equation so obtained is the equation of the slopes with respect to the axis
O of the tangents to the paths of the vertex of the gyro.
Let us find the locus of points at which the tangents to the paths have one and
the same inclination v. As already mentioned, such loci are called isoclinic. For
this purpose it is sufficient to equate the right side of eq.(10.8) to the quan?
tity v. On doing this, and solving the equation so obtained with respect to a, we
get the required equation of the isoclines in the following form:
...1-v1 a y
2= P+ (10.9)
????
111 where
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It follows from this equation that the isoclines will be straight lines.
Let us find further the coordinates of the intersection of the isoclines for
tangents of two different slopes, vl and v2. For this purpose it is sufficient to
solve simultaneously the two equations obtained after substituting in the equation
of isocline eq.(10.9) av equal respectively to vi and v2, i.e., the equations
2+
401
12 I
? r
so Viral
p .
.2
?I PI I
(10.10
Whence the coordinates of the intersections of the required intersection of the
isoclines ao and 00 are determined by the expressions
, a S21
cte -t-
"! "'I ?
p
It follows from these expressions that the coordinates of the points of the
intersection of isoclines do not depend on the slope of the tangents to which these
isoclines refer, and, in particular, that with unvaried values of f;i1 f)2, col and e,
all the isoclines will intersect in a single point.
This circumstance very greatly simplifies the problem of constructing the field
of tangents to the paths of the vertex of the gyro; in order to determine the slope
of the tangents to a path at a given point, it is sufficient to draw a ptraight
line connecting this point with the point of intersection of the isocline, prolong
the straight line to the intersection with the 0 3axis, and determine the tangent
of the angle between this line and the Op axis.
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Let us assume that this tangent is equal to Then, in order to determine
the angle between the Op axis and the tangents to the paths vi, to which the
isocline so drawn relates, we have, by eq.(10.9), the following equation:
via
!I I =
whence, by solving this equation with respect to vi, we obtain
(10.13)
(10.14)
Let us now return to the expressions for the coordinates of the point of inter-
section of the isoclines.
Equation (10.11) for a0 consists of two terms, of which the second is of an
ni
6 n
order which is times higher than the order of the second, since and are
(,)1 4)1
e .
of about the same order. The quantity . is of the order of one thousand, if we
(01
have in mind for E the usual order of 0.05 to 0.10 1/sec. It follows from this that
in the expression for ao, the first term can, with a high degree of accuracy, be
neglected by comparison with the second term. Moreover, the coordinate 0 will be
so much greater in modulus than the coordinate ao, that, in the neighborhood of
small values a and R, to which the theory here developed relates exclusively, the
slopes of the isoclines will be relatively greater in modulus.
For isoclines I (Fig.10.1) = the inclination to the OP axis equals 90?,
and, consequently, according to eq.(10.13), the angle between the Opaxis and the
tangent to, the path will be equal to 00 (cf.Fig.10.1).
With increasing angle between the positive semiaxis Op and the isocline,
takes a negative value, which gradually diminishes in modulus as this angle in-
creases. In connection with this, the inclinations of the tangent to the Op will
increase.,
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At points with isoclines having an inclination equal to tan-10, where
a
?
(10.15)
the inclinations of the tangents to the path will be equal to 90?. It is easy to
see that there will only be one such isocline, namely the one that is obtained by
joining the point of intersection of the isoclines to the origin of coordinates (if
in eq.10.11 we neglect the first summand).
We shall call this isocline the boundary isocline (isocline 3 in Fig.10.1).
We remark that it follows, from what has been said of the order of magnitude of
E
, that the boundary isocline will in practice coincide with the axis Oa.
Wi
In this way the motion of the vertex of the gyroscope from the points lying
above the boundary isocline with which we have been dealing up to now, may be char-
acterized as motion toward the boundary isocline, or, practically, motion toward the
Oa axis.
At the points lying below the boundary isocline, the inequality
(10.16)
will obviously hold, and therefore the slopes of the tangents to the path of these
points will be less than 90?. In other words, the motion of the gyro from these
points will also be directed toward the boundary isocline, that is, practically to-
ward the Oa axis. It follows that, under any initial conditions, the motion of the
gyro vertex along the boundary isocline will ultimately be established.
We remark that it follows from what has been said on the relation between
and wi, and from the form of eq.(10.14), that at all points, except for the imme-
diate neighborhood of the boundary isoclines, the tangents to the paths, and con-
sequently the paths themselves will be close to lines parallel to the Op axis.
For the velocity of the motion of the gyroscope vertex, U, in this case, we
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get, by eq. (10.5):
?
?
Lig-A 0.11141 202+ (w2-- 4)-11 ' (10.17)
s.
Fig.10.1 ? Paths of Motion of Top of Directional Gyro without Allowing
for Friction in the Gimbals, with Proportional Characteristic
By the condition of eq.(10.15), on motion along the boundary isocline,
On substituting thie value of f3 in eq.(10.17), and bearing eq.(10.15) in mind,
we obtain
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'2 )2
2 :_g3
(10.19)
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or
U
(10.20)
In this case, the components of the velocity along the Oa and 013 axes will be
respectively:
us It 1219
U. at UV
(10.21)
(10.22)
In the subsequent argument we shall assume friction in the gimbals to he
absent.
Then the projections of the velocity U on the Oa axis will be determined,
according to eqs.(10.6) and (10.21), by the expressions
LI a
= ? w., + V sink ) sin T.
// R cos ,
(10.23)
that is, it will be determined by the difference between the precession of the gyro
owing to the action of 'the moment of compensation, and the precession due to the
vertical component of the earth rotation.
If, in particular,
then we get
.sn
(
613
II k cos I sin ?.
=i)
(10.24)
Consequently, if in this case, at the initial instant of time, the gyro axis
was brought into the plane of the meridian, then it will remain in that plane, that
will perform its function for an indefinitely long time.
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Obviously such a condition, for a given value of k21 will be satisfied only for
some definite latitude at which eq. (10.24) will hold. At other latitudes however,
Ua 0, will already hold, and with the passage of time the gyro vertex will depart
from the position originally given it with respect to the earth.
In particular, if k2 = 0, then the gyro vertex will depart from the originally
assigned position at the rate
%in A
?
R en.
Let us now elucidate the role of friction in the gimbals.
Let us do this at first, assuming the absence of compensation of the earth ro-
tation, and not taking into account the role of the aircraft rotation.
In this case the expressions for ni and ci2 will take the following form:
Since, usually
L.
sign ? ? v ?1,, A .1
I II R cog; 14in
-4';
Ws= ? sign a +
In A\ . ""
sit' ?
// I R co% // R
it follows that the signs of and f2 will be determined by the signs of the angu-
lar velocities 0 and a respectively.
In connection with this, the point of the intersection of the isoclines will
be, according to eqs.(10.11) and (10.12), for (.3 >0, in quadrant IV of the plane
Oaf>, (Fig.10.2), and for fi >
1
AI. 11?11.?
Fig.10.7 - Electrical Circuit of Remote Gyro Magnetic Compass
the points of the output (4) of the potentiometer of the magnetic transmitter is
obviously determined by the position of the brushes on the given potentiometer with
respect to the output points, that is, by the aircraft course, and by the poten-
tials taken off from the potentiometer of the gyro unit, that is, by the position
of the brushes and the potentiometer of the gyro unit.
1-":"'
4) b)
Fig.10.8 - Distribution of Potential on Potentiometers of
Distant Transmission
a) 27 v; b) Feed, 27 v
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The voltage from the points of lead-out of the potentiometer of the magnetic
transmitter is fed to the amplifier (5), and from the amplifier output to the
actuating motor (6) located in the upper part of the gyro system. If this voltage
is not equal to zero, then the actuating motor of the gyro unit is brought into ro-
tation, and its rotation is transmitted to the brushholder through a reducer with a
high gear ratio. The sense of the rotation depends on the sign of the voltage fed
to the amplifier.
Thus, in rotating, the actuating motor displaces the brushes on the potentio-
meter of the gyro unit, which leads to a change in their potentials, and, conse-
quently, to variations in the readings of the indicators. The displacement of the
brushes of the gyro unit will continue'until the voltage between the lead-out points
of the potentiometer of the magnetic transmitter becomes equal to zero. From a
consideration of the distribution of the potentials on the potentiometers of the
gyro units and the magnetic transmitter (Fig.10.8), it will be clear that for a
given position of the brushes on the potentiometer of the magnetic transmitter,
that is, on a given course, the voltage between the lead-out points will be equal to
zero at a completely determinate position of the brushes on the potentiometer of the
gyro unit. To put it more precisely, the position of the brushes on the potentio-
meter of the gyro unit with respect to the lead-out points differs in this case from
the position of the brushes on the potentiometer of the magnetic transmitter with
respect to the lead-out points by a constant angle equal to 900.
Since the position of the brushes on the potentiometer of the magnetic trans-
mitter is determined by the course of the aircraft, it follows from what has been
said above that the position of the brushes on the potentiometer of the gyro unit
will also be determined by the course of the aircraft. Consequently, the voltages
taken from the potentiometer of the gyro unit and fed to the indicators will like-
wise be determined by the caurse of the aircraft.
In this way, it may be stated, in short, that the brushes of the potentiometer
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of the gyro unit follow the position of the brushes on the potentiometer of the
magnetic transmitter.
When the aircraft makes a turn, (we assume that before the turn the brushes
were in the matched state) the brushes in the gyro unit will rotate with respect to
the potentiometer by an angle equal to the angle of turn of the aircraft, and the
indicators will show, without a lag, the new course and the angle by which the air-
craft has turned.
If the magnetic system of the transmitter during a,turn were not entrained,
and there were no turning error, then the brushes in the transmitter would rotate
with respect to the potentiometer by the same angle of turn, and the system of the
brushes would still remain in the matched state. But the entrainment and turning
error of the magnetic system of the transmitter lead to mismatching in the position
of the brushes, and the brushes of the gyro unit will begin to be displaced along
the potentiometer, tending once more to reach the matched state. This leads to the
appearance of error in the instrument readings. It will be easily seen that this
error will be determined by the rate of displacement of the brushes along the poten-
tiometer of the gyro unit and by the duration of the turn. This rate is low and
amounts on the average to 2 - 30 a minute. Consequently, during a minute of turn,
the instrument can accumulate an error not exceeding 30, while the entrainment of
the magnetic system may have the value of several tens of degrees.
It is well known that in rectilinear flight the magnetic system of a compass
undergoes oscillation. In view of the low rate of matching of the brushes of the
gyro unit, these oscillations will not be able to be transmitted to the indicators,
and the indicators will therefore show the mean compass course.
The drift of the gyroscope in azimuth does not lead to the appearance of sub-
stantial errors in the instrument readings. The rate of drift of the gyroscope in
azimuth does not exceed 10 a minute, and, consequently, the mismatching in the
position of the brushes due to the drift of the gyroscope (to the rotation of the
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potentiometer of the gyro unit) will be liquidated by the rotation of the brushes
of the gyro unit, whose rate of displacement is greater than the rate of drift of
the gyro. The brushes of the gyro unit "overtake" the potentiometer, which drifts
together with the gyro.
For the rapid matching of the brushes, for example in starting up the instru-
ment, when the angle of mismatch may reach 180?, the gear ratio in the reducer is
changed, thus providing a rate of matching of the brushes equal to 20? a second.
The gear ratio of the reducer is varied by means of the electromagnet (7) (cf.Fig.
10.7), mounted in the upper part of the gyro unit. The electromagnet is turned on
by pressing the matching button (8) (cf.Fig.10.7).
The gyroscope used in the remote gyro magnetic compass has a correction system
based on the mutually perpendicular position of the rotor axis and the axis of the
outer frame. The operation of this correction system is described in Chapter 6 on
page 101.
The gyro motor is a triphase asynchronous motor with short circuited rotor.
The rotor speed is about 22,000 rpm. The case of the gyro unit is air-tight. Its
inner cavity is filled with nitrogen. This protects the parts of the instrument
from corrosion and assures normal operation of the gyroscope correction system at
any altitude.
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CHAPTER XI
VELOCITY AND ACCFTERATION VELOCITY GYROSCOPES
Section 11.1. Principle of Operation
By velocity gyros we mean gyros intended to detect and measure the angular
velocity of rotation.
Fig.11.1 - Diagram of Turn Indicator
I - Axis of inner frame
1 - Counteracting (centering) spring; 2 - Damper; 3 - Indicating system
forc7)c, vector of angular velocity being measured; L - Applied gyroscopic
moment; Irip - Counteracting moment
These instruments are also often called "precession gyroscopes", and certain
types of these instruments are termed turn indicators or damping gyroscopes.
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One of the most widely used types of velocity gyros is the turn indicator,
which indicates the angular velocity of an aircraft turn, and is a part of the set of
piloting-navigational instruments. It consists of a gyro with two degrees of free-
dom; one of rotation of the rotor and the other rotation of the suspension, with the
\ degree of freedom of the suspension limited by the centering spring and damper
(Fig.11.1).
illFig.11.2 - Diagram of Acceleration-Velocity
Gyroscope
I and II - Axis of suspension
1 - Counteracting spring of inner frame;
2 - Damper; 3 - Counteracting spring of
outer frame; 4 - Indicating system;
H - Kinetic moment;(7c - Vector of angular
velocity being measured; -1,1 - Acting
(gyroscopic) moment about axis of inner
MMI
frame, L1 - Counteracting moment about
axis of inner frame; 112 and ;pi - Effec-
tive moments about axis of outer frame;
- Counteracting moment about axis of
Lnp2
338
It may also be treated as a
gyro with three degrees of freedom,
whose outer frame is the case of
the instrument, which is connected
with the aircraft. In the litera-
ture, the following explanation of
the principle of action of the turn
indicator is usually given.
Its action is based on the
utilization of the gyroscopic
moment. The direction of the axis
of the lacking degree of freedom is
matched with the axis about which
the angular velocity is to be
measured. Thus rotation about this
axis will be imparted by constraint
to the rotor. Under the action of
the moment of gyroscopic reaction
so arising, the frame of the sus-
pension will rotate until the act-
ing gyroscopic moment is balanced
by the counteracting moment of the
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spring. This takes place at a completely determinate angle of shift of the frame of
the suspension, which is a measure of the gyroscopic moment as well, and conse-
quently, also of the angular velocity of rotation of the rotor due to the appearance
of this gyroscopic moment.
The damper, as usual, is intended to extinguish the oscillations of the frame.
This method of explaining the principle of operation of the turn indicator has
the shortcoming that here the moment of gyroscopic reaction, constituting the moment
of the forces of inertia, appears as the cause of the rotational motion of the frame.
Another method of explaining the operation of the turn indicator is as follows.
When the case rotates at the angular velocity wc (cf.Fig.11.1), a moment whose
vector coincides with the vector we acts on the gyro through the bearings of the
frame. This moment causes precession of the gyro, tending to match the vector of
kinetic moment with the vector of external moment. As a result the frame rotates
about the axis. The tension of the spring due to the rotation of the frame produces
a moment causing precession of the gyroscope in the sense of rotation of the air-
craft. In the steady state, the moment of the spring produces a rate of precession
equal to the rate of rotation of the aircraft. In this case the moment of pressure
from the bearings on the gyroscopes disappears. In the quantitative aspect, both
these treatments of the principle of operation of the velocity gyroscope are
naturally analogous.
The acceleration-velocity gyroscope (cf.diagram on Fig.11.2) is intended to
measure both angular velocity and angular acceleration. This is accomplished, in
contrast to the velocity gyroscope, by not completely depriving the
acceleration-velocity gyroscope of its freedom of rotation about the axis of
measurement, since the connection with the casing is not rigid but elastic.
At constant rate of rotation of the aircraft about the measurement axis, this
connection assures the establishment of precisely the same constant rateof rotation
of the gyro about the same measurement axis.
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The inner frame of the gyro will react
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to this velocity in exactly the same way as in the ordinary velocity gyro, by rota-
tion by an angle corresponding to the given value of the angular velocity of rota-
tion of the aircraft.
In this case, in the steady state, the spring of the outer frame will not be
deformed, but the moment of the spring of the inner frame produces a precession of
the gyroscope, in the sense of rotation of the aircraft, at an angular velocity
equal to the angular rate of turn. A peculiarity of this device is that the moment
of the spring of the inner frame and the gyroscopic moment balancing it are the in-
ternal moments for the unit outer frame-inner frame-rotor. They therefore do not
produce deformations of the spring of the outer frame, which is connected with the
instrument casing, and consequently also with the aircraft.
When the angular rate of turn varies, the angle of deflection of the inner frame
must obviously also vary, since it is precisely this angle that is the measure of
the angular rate of turn. The more rapidly the angular rate of turn varies, the more
411 rapidly will the inner frame rotate about its axis, i.e., the rate of rotation of
the inner frame will be proportional to the angular acceleration of the aircraft
when the angular rate of turn varies. In this rotation the gyroscopic moment mani-
fests itself. Its vectsor is directed along the axis of the outer frame. The modu-
lus of this gyroscopic moment, however, will be proportional to the rate of rotation
of the inner frame, that is, proportional to the angular acceleration of the air-
craft. Obviously this gyroscopic moment will modify the motion of the gyroscope
about the axis of the outer frame with respect to the aircraft until the moment from
the spring of the outer frame balances it. In this case the outer frame will de-
flect from its normal position with respect to the casing (the aircraft) by an angle
proportional to the gyroscopic moment, and consequently, to the angular acceleration
of the aircraft.
If we use, instead of two springs, as in the acceleration-velocity gyro (cf.
410 Fig.11.2), only one spring (Fig.11.3), which is able to produce moments both about
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the axis of the inner frame and about the axis of the outer frame, then the deflec-
tion of the outer frame will now be proportional not only to the angular accelera-
tion, but will also be a function of the angular rate of turn.
For this purpose the spring must
be attached at one end to the inner
frame, and at the other end to the in-
strument casing. Let us have a steady
turn; then the inner frame will rotate
about its axis. In this case the
spring will be stretched. The tension
of the spring will produce a moment not
only about the axis of the inner frame,
but also about the axis of the outer
frame. This will produce a deflection
of the outer frame by an angle which is
a function of the angle of deflection of the inner frame, that is, a function of the
angular rate of turn.
Naturally, when the aircraft rotates at an angular acceleration, the angle of
deflection of the inner frame will vary. In this case a gyroscopic moment will
arise, thereby producing an additional deflection of the outer frame with respect to
the aircraft by an angle such that this gyroscopic moment will be balanced by the
spring.
Thus the angle of deflection of the outer frame will be proportional both to
the angular velocity, and to the angular acceleration, in a definite ratio between
them. This ratio depends on the ratio of the arms ri and r2, that is, on the design
parameters of the device.
Fig.11.3 - Diagram of
Acceleration-Velocity Gyroscope with One
Counteracting Spring
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j3ection 11.2. Motion of Frame of Velocity Gzroscope
Equations of Motion
Ip this case we shall start out from the complete equations of motion of the
gyroscope and shall reject in them only the terms characterizing the influence of
the centripetal accelerations, rewriting them in the following form:
Is Adr:4 lig.. LA.
1"ddit
where 'eq and 'eq are the equatorial moments of inertia of the gyro with respect
x y
to the Qx and Oy axes respectively.
As already pointed out, a velocity gyro may be treated as a gyro with three
degrees of freedom, the outer frame of which is the casing of the instrument. For
turn indicators and velocity gyroscopes of autopilots, the casings are rigidly con-
nected with the aircraft. In essence, it is the aircraft that performs the role of
the outer frame for these instruments.
It follows from this that, in eq.(11.1), q is determined by the projection of
the angular velocity of the aircraft about the axis of measurement wc on the axis
Oy; the equatorial moment of inertia 'eq y is determined by the moment of inertia of
the aircraft itself with respect to the axis of measurement; and Ly is determined by
the moments acting on the aircraft about that same axis. In other words, the second
of the eqs.(11.1) will relate practically to the rotations of the aircraft, that is,
to what is, essentially, the assigned object of measurement and does not need any
investigation in this case. In this case the second term of the left side of this
,equation, representing the gyroscopic moment which arises in connection with the
rotation of the gyro about the Ox axis, shows what the influence of the motions of
the gyro will be on the motion of the aircraft. It is perfectly clear that this
"Influence, in view of the smallness of the value of this gyroscopic moment by corn-
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parison with the moment of inertia of the aircraft itself, will be negligibly sma11.
Consequently, the second of eqs.(11.1) may be simply rejected.
Let us take the original system of earth-bound coordinate axes 0,1:71e, which we
shall orient in the following manner (Fig.11.4): we matcb the axis Ot: with the axis
of measurement, the axis On with the axis of rotation of the gyro frame, and there-
by, with one of the axes of the aircraft. JThis latter procedure means that the
system of axes OW will rotate about the axis of measurement, that is, about the
axis (X, at the angular rate of turn of the aircraft wc.
Let the other two axes of the aircraft, which we shall denote by (:)'e and OEe,
deviate from the 0t and OF, axes respectively by the angle of bank y. According to
the arrangement of the instrument (cf.Fig.11.1), to the neutral position of the
gyroscope frame, and, accordingly, to the zero reading, will correspond the position
in which the system of gyroscope axes Oxyz will coincide with the system of aircraft
axes, C&nee, respectively. We shall denote by the letter 13 the angle of deviation
of the gyroscope from this position owing to rotation about the axis of the frame
and shall consider that sense to be positive that brings the rotor axis of the gyro-
scope Oz closer to the axis (:). It is this angle 0 that, in this case, will be the
measure of the angular velocity we.
The moments acting on the gyroscope will consist of the moments produced by the
deformation of the spring and the resistance of the damper, as well as the moment
produced by the friction in the bearings of the shaft of the frame.
We shall consider for simplicity that rotation at the angular velocity we about
the vertical is the only rotation of the aircraft, that is, that the aircraft is
performing a turn in the horizontal plane.
Then, on the basis of the above, we may write the following expressions
4/....(cos(7--11).
h?, ;NA +Li, sign 0.
343
STAT
f''
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STAT
Declassified in Part - Sanitized Copy Approved for Release 2013/05/16: CIA-RDP81-01043R002000020008-5
?
?
where
k - coefficient of rigidity of the spring, which we shall take as constant;
sp
' md - coefficient of damping;
L1 - moment of friction acting about the axis of the frame.
Fig.11.4 - Position of Axes of Velocity Gyro
Consequently, omitting hereafter the second subscript in the symbol for the
equatorial moment of inertia, the equation of motion of the gyroscope frame is now
written in the following form:
J.; ? ? ksp We cos IT?) -Li sign
(11.5)
or, replacing sin p by the value of p in radians, and replacing cos 0 by unity,
which is permissible, since p usually does not exceed 10 - 200:
/.1 ? ni II tu, sin y) =II w,co: 1-1.11 sign (11.6)
Influence of Damping and of Rigidity of Centering Spring
We shall consider
0,=--const
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?
and shall omit, in the right side, the term characterizing the influence of friction,
having in mind a subsequent return to the consideration of the influence of the de?
pendence of we on the time and the influence of friction.
Let us rewrite eq. (11.6), under these assumptions, in the following form
where
p-1-2Deio+4,?.411?, (11.7)
m,
=
I' 4 (nip?Hure sin 7)
, /hp - /hoc sin 7
Ili UM
I.
p ? == AOC COS
ik5p.. Heft sin 7
If we assume that the condition
D