Gyrodynamics and its Engineering Applications

485 Pages·1961·22.086 MB·English

by  Ronald N. Arnold and Leonard Maunder (Auth.)

GYRODYNAMICS
AND ITS ENGINEERING APPLICATIONS
BY
RONALD N. ARNOLD
Regius Professor of Engineering
University of Edinburgh
AND
LEONARD MAUNDER
Professor of Applied Mechanics
University of Durham
AN
ACADEMIC PRESS
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1961
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PREFACE
At first sight the behaviour of a gyroscope may appear incongruous.
It seems to take a mischievous delight in flouting the accepted laws of
nature. But bewilderment, in time, gives place to reason; what appears
an affront to intuition is theoretically predictable. With bodies of high
rotation one must learn to accept the apparent irrationality of rational
motion. But beyond its purely scientific interest, the gyroscope has
attributes of great utility. Never before have gyroscopic devices been
used so extensively, yet the number of works on the subject is surpris-
ingly small. This book may help to remedy the situation. It is intended
primarily for engineers, but its approach is mainly analytical, emphasis
being placed on scientific design rather than constructional detail.
Based on lectures given in the Post-Graduate School of Applied
Dynamics at Edinburgh University, it should prove of value to
University teachers, research workers and all who, from time to time,
wrestle with gyroscopic problems.
The book consists of sixteen chapters, of which the first seven are
concerned with classical dynamics. Vector notation is introduced as a
convenient method of representing generalities but, being used sparingly,
should cause little difficulty. The initial aim is to assemble certain
fundamental ideas governing the movement of bodies in three dimen-
sions. Motion with respect to moving axes is discussed in detail,
particular attention being given to the intangible Coriolis acceleration
which pervades gyroscopic theory. Thereafter, a study is made of
inertial characteristics of bodies and certain dynamical theorems,
followed by the historic problems of the motion of a free body and of a
symmetrical gyroscope under gravity.
The remaining nine chapters deal with gyroscopic applications,
mainly of a mechanical nature. They begin with a group of miscel-
laneous problems before embarking on a study of gyroscopic mechan-
isms. Complete chapters are devoted to stabilizers, the gyro-compass,
rate and integrating gyroscopes, gyro-verticals and gyroscopic suspen-
sions. Emphasis then moves to an examination of inertial navigation
while the closing chapters discuss the whirling of shafts and aircraft
gyrodynamics. Appendices are added, together with a group of
problems and answers intended for organized study.
v
VI PREFACE
An attempt has been made throughout to present the material in a
manner which stresses the physical concepts. Moreover, many sections
have been developed specifically for inclusion in this book and will not
be found in other literature.
Many publications, too numerous to mention, have been consulted
during the preparation and we wish to thank the following for their
courtesy in allowing us to reproduce information and diagrams :
The Ministry of Aviation for information included in Sections
11.2 and 12.7 to 12.9 (see footnote p. 323) and for Figs. 12.7,
12.9 (a) and 12.10 to 12.17.
The Institution of Mechanical Engineers for Figs. 9.2 to 9.4, 9.6 to
9.11 and 11.15to 11.20.
The American Society of Mechanical Engineers for Figs. 15.5 and
15.7.
The Royal Aeronautical Society and Dr. W. Cawood for Fig. 13.5.
Ferranti Ltd. for Fig. 13.8.
Dr. A. L. Rawlings for Fig. 10.12.
Many others have been of assistance in the publication of the book.
Our colleagues, particularly Prof. G. B. Warburton, Dr. J. D. Robson
and Dr. A. D. S. Barr, read and discussed much of the text, Miss Nora
Liddell dealt most efficiently with the burden of typing while Graham
Arnold, son of one of the authors, designed the book-jacket. The
publishers, moreover, were most helpful in consulting us during the
publication and in meeting our many requests. To all these we extend
our thanks.
Being conscious of human frailty, we realize that some errors must
inevitably have escaped detection in a work of this character. It will
be a kindness if readers will notify us of any mistakes which may come
to their notice.
Edinburgh, August 1961.
CHAPTER 1
GENERAL INTRODUCTION
1.1 HISTORY
In a memoir read before the Academy of Sciences in Paris, 1852,f
which describes experiments relating to the movement of the Earth,
Foucault concludes:
"Comme tous ces phénomènes dépendent du mouvement de
la Terre et en sont des manifestations variées je propose de
nommer 'gyroscope' l'instrument unique qui m'a servi à les
constater."
Fig. 1.1. Diagrammatic illustration of a free gyroscope.
In this manner the word "gyroscope" was first introduced. To-day
it denotes a variety of mechanisms each having in common a rotating
mass. The conventional gyroscope, however, consists of a symmetrical
rotor spinning rapidly about its axis and free to rotate about one or
more perpendicular axes. Freedom of movement about an axis is
normally achieved by supporting the rotor in a gimbal and complete
freedom can be approached by using two gimbals, as illustrated in
Fig. 1.1. Friction at the pivots is always present in any practical case.
We shall see later that even the most delicate bearings introduce errors
in directional gyroscopes which can only be compensated by elaborate
t J. B. L. Foucault, Compt. Rend. Acad. Sei., Paris, 35, 1852, 421.
1
2 GENERAL INTRODUCTION 1.1
equipment. But the field of application of the gyroscope is wide and
an increasing variety of navigational instruments and control systems
are making use of its unusual characteristics.
At first sight the behaviour of a gyroscope may appear strange if
not indeed contrary to natural laws. But our increduUty arises mainly
from lack of experience. Had we been born to a world in which high-
speed rotation was a normal attribute of material bodies, gyroscopic
behaviour would have become intuitive. The gyroscope holds no
mysteries; it obeys meticulously the same dynamical laws which
govern the universe.
To present an adequate history of the gyroscope by way of intro-
duction is beyond the scope of this book. We shall attempt, however,
to outline the main theoretical contributions which explain its behaviour
and describe a few inventions which make use of its characteristics.
Much of the theoretical background is interwoven with the general
dynamics of rigid bodies which, though not directly concerned with
gyroscopes, forms the basis of gyroscopic theory.
The earliest appreciation of gyroscopic phenomena appears to date
to the time of Newton (1642-1727). It arose from a study of the motion
of our planet, which is itself a massive gyroscope. A description of
its motion will, in fact, help in understanding some of the essential
characteristics. The Earth approximates closely to a free gyroscope
for its axis remains almost fixed in the direction of the North Star,
Polaris, irrespective of its transit around the Sun. The direction of the
axis, however, had been changing slowly throughout the centuries, a
phenomenon known as ''precession of the Equinoxes". Its polar axis
is in the process of sweeping out a cone of 46° 54' 16" apex angle, one
circuit of which occupies 25,800 years (Fig. 1.2). Though extremely
slow, this motion is similar to the precession of a spinning top. It
arises from the gravitational moment to which the Earth is subjected
by the Sun as a combined result of its lack of sphericity and the
inclination of its axis to the ecliptic plane. A further periodic move-
ment is also present in which the Earth's axis describes a much smaller
cone whose diameter at the North pole is approximately 26 ft. This,
known as Eulerian motion, has an observed period of 428 days and
corresponds to the free oscillation or nutation of a gyroscope. The
above phenomena, illustrated in Pig. 1.2, are superposed on the orbital
motion of the Earth around the Sun. They would appear as described
only to an observer situated at the centre of the Earth who continued
to look in a fixed direction in space.|
f For an analytical treatment of the motion of the Earth see A. Gray, "Gyrostatics
and Rotational Motion", Macmillan, London, 1918, Chapters X and XI.
1.1 HISTORY 3
From the above we may identify three gyroscopic attributes, namely
directional stability, precession and nutation. It would, in fact, be
difficult to find any application of the gyroscope which does not
depend on one or more of these properties. For example, directional
stability, which may be regarded as the reluctance of a body to change
its orientation, provides a datum which is the basis of modern inertial
, \_23}27'8· ! /^ 26 ft die.
^-precession of equinoxes ^ ^ ^/ / ^ ^/
^<^(period 25,800yrs.) ^/ ^ ^^ polar rotation
^ ^ \^ ^ "1 ""^^^Ζ^><ίΓ (period 1day)
/ ^^^^^^yy ^ ^. ^^celestial pole
^>. /^ ^ Γ \ > / \^ y^^Eulerian motion
jf*. //^ \v \ (period 428 days)
I \ ^ \. / \ y^path of geometric
^v / >/ ^^ equator
Fig. 1.2. Gyroscopic motion of the Earth.
navigation. Also, it is found that rate of precession is proportional to
applied moment, and as the latter may be produced by the acceleration
of an offset mass, linear acceleration may be measured by angular
velocity and consequently linear velocity by angular displacement.
This integrating ability of the gyroscope is made use of in instruments
carried by rockets for recording their position in space.
A further characteristic depends on the reversible nature of action
and reaction. If forced to precess, a gyroscope exerts a reactive moment
proportional to the product of the velocities of spin and precession.
Moments of immense magnitude may thus be produced by the pre-
cession of high-speed rotors, a feature utilized in gyroscopic vibration
absorbers and in some ship stabilizers.
4 GENERAL INTRODUCTION 1.1
The mathematical foundation of gyroscopic behaviour must un-
doubtedly be ascribed to Euler (1707-1783). His initial work in this
field concerned the general motion of a rigid body for which he derived
a set of dynamical equations involving relations between applied
moment, inertia, angular acceleration and angular velocity. These,
known as Euler's equations, were stated with reference to axes fixed in
the body. Later he established the independence of rotation and
translation of a rigid body and devised what are termed the Eulerian
angles to define its orientation with respect to a fixed point. From this
background came his first direct contribution to gyrodynamics, the
general motion of a rigid body, fixed at a point and free from external
force.f This problem includes that of the free gyroscope and was to
occupy the attention of mathematicians for many years. The general
displacement of the body is, in fact, only expressible in terms of elliptic
functions which are mainly associated with the name of Jacobi ( 1804-
1851). Euler's later contributions included the inertial properties of
bodies, which led to the concepts of principal axes and momental
ellipsoid.
We may note here an important concept which first appears to have
been recognized by Clairaut in 1742, though credited much later to
Coriolis. This concerns the force to which a particle is subjected when
moving along an axis which is itself subjected to rotation. Although
this had not been neglected in earlier work, Clairaut indicated its
application to a moving frame of reference. Thus the force on the
particle may be considered as having three components, one which it
would have if attached to the rotating frame, another due to motion
relative to the frame and finally the so-called Coriolis component.
We shall see later that this conception has considerable importance in
gyrodynamics.
From the death of Euler in 1783 until the early part of the eighteenth
century, little was added to the theory of the gyroscope. A revival of
interest, however, is evident in the work of Poinsot (1777-1859) who
approached the subject by way of analogy. He demonstrated theoreti-
cally that if a free body, fixed at a point, were replaced by its momental
ellipsoid, the path of motion of the ellipsoid when rolled on a fixed
plane was identical to that of the body4 In this representation, the
distance of the plane from the fixed point was a function of the energy
and momentum of the body. At a later date, Sylvester§ showed that
f L. Euler, "Mémoires de Berlin", 1758, and "Theoria motus corporum solidorum seu
rigidorum", Rostock, 1765.
| L. Poinsot, "Théorie nouvelle de la rotation des corps", J. de Louisville, 16, 1851.
§ J. J. Sylvester, Phil. Trans. Roy. Soc. London, 1866.
1.1 HISTORY 5
if a solid homogeneous ellipsoid were used not only the path but the
time of its transit at eaoh position would be identical to that of the
actual body.
Many contributions of Poisson (1781-1840) are associated with
gyrodynamics. In particular, he appears to have been the first to
investigate the motion of the spinning top,f a much more complex
problem than that of the free gyroscope. Due to gravitational force a
top may perform a large variety of complicated motions and during
the latter half of the nineteenth century, much thought was devoted
to this subject. Poisson also made a comprehensive study related to
the work of Poinsot, which dealt with the rolling of bodies of various
shapes on a plane.
The years which followed provided new approaches to gyroscopic
problems. Haywardf rewrote the Euler equations in terms of momen-
tum about a set of moving axes, Tait§ investigated the motion of the
free gyroscope by vector methods, while Routh studied the stability
of gyroscopic motion and gave a geometrical construction for determin-
ing the rise and fall of a spinning top.|| The contributions of Kelvin
were both practical and theoretical. He made a suggestion for using
a gyro-compass as early as 1883 and later developed analogies between
gyroscopic motion and the motion of electrons in magnetic fields.
The work of Klein may also be mentioned. He approached the motion
of a top by using parameters which later became known as the Cayley-
Klein parameters.
By the turn of the century gyroscopic theory was virtually com-
plete and since then emphasis has shifted to gyroscopic applications.
This has involved much theoretical work involving the solution of
specific problems rather than the discovery of new phenomena. The
gyroscope has become the province of the inventor rather than of the
mathematician.
The first practical application of the gyroscope appears to be due to
Serson who in 1744^f constructed a spinning rotor for indicating the
position of the horizon at sea. It was supported so as to be free from
disturbance by the motion of the ship and was the forerunner of the
gyroscopic horizon used in modern aircraft.
f S. Poisson, "Traité de Mécanique", 2nd ed., Paris, 1833.
J R. B. Hayward, "On a Direct Method of Estimating Velocities with Respect to Axes
Movable in Space", Camb. Phil. Trans., 10, Feb. 25, 1856.
§ P. G. Tait, "On Rotation of a Body about a Fixed Point", Proc. Roy. Soc. Edinburgh,
25, 1869, 279.
|| E. J. Routh, "Dynamics of a System of Rigid Bodies", Part II, 6th ed., 1905,
Articles 201-204. Macmillan, London. (Reprinted Dover Publications Inc., 1955.)
i[ "Gentleman's Magazine", 24, 1754. Also James Short: "An account of an Horizontal
Top Invented by Mr. Serson", Phil. Trans. Roy. Soc. London, 47, 1752.
6 GENERAL INTRODUCTION 1.2
The early part of the nineteenth century saw gyroscopes being used
in the teaching of dynamics and in 1819 Troughton produced an im-
proved gyroscopic horizon in which the centre of gravity of the rotor
could be adjusted accurately by means of screwed plugs. Many years
later such an instrument came into the possession of Sang. He per-
ceived that if the centre of gravity were made to coincide precisely
with the point of suspension, the rotor, free from gravitation torque,
would retain its orientation in space. By this means he hoped to show
the rotation of the Earth and read a paper on the subject in 1836
although not published until 1856.f Unfortunately, Sang had not the
resources to construct a sufficiently accurate rotor and his experiments
were inconclusive. Some years later, however, in 1852, FoucaultJ
constructed a gyroscope with which he successfully demonstrated
the Earth's rotation.
Since that date the gyroscope has been used in a variety of ways. It
has been used to navigate ships and rockets, to stabilize the rolling
of ships, to counter vibration and to operate innumerable control
mechanisms. Moreover, it is not limited by size or function. The
small directional unit of the gyro-compass operates by the same
principles as the massive rotor of the ship stabilizer.
It is clear that an appreciation of gyroscopic behaviour requires some
acquaintance with basic dynamical laws. Some patience will, therefore,
be required by the reader during the preliminary approach. Our
immediate objective will be to discover the torque which must be
exerted on a spinning rigid body to give its axis of spin some prescribed
motion. For this purpose, the body will be considered as made up of a
large number of small particles, each of which, due to the combined
motion, traces out some complicated path in space. At the outset,
therefore, it will be necessary to study the dynamics of a single particle
of matter and for this purpose we first consider the elements of vector
analysis.
1.2 INTRODUCTION TO VECTORS
The physical quantities used in gyroscopic theory normally fall into
two groups.§ In the first are the scalar quantities such as mass, time,
energy which can be completely defined by a given number of units,
e.g. 10-2 Kg.; 5-6 sec; 22 ft. lb. The second group contains the
vector quantities. These require in addition a statement of direction
t E. Sang, Trans. Roy. Scot. Soc. Arts, 4, 1856, 416.
X Compt. Rend. Acad. Sei., Paris, 35, 1852.
§ A third group including stress, strain and inertia may be represented by tensors.