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' ' l
APPLIED GYEODYNAMICS
WOHKS OF
ERVIN S. FERRY
PUBLISHED BY
JOHN WILEY & SONS, Inc.
Applied Gyrodynamics.
A text-book for students and 'engineers. Develops the
laws upon which gyroscopic devices are based, with many
applications and much illustrative material, as well as a
number of solved problems, riv -f 277 pages. 6 by 9.
218 figures. Cloth.
General Physics and its Application to Industry and
Everyday Life,
A text-book for college students in general and tech-
nical courses in which especial emphasis is laid on the
diverse relations of Physics to nature, agriculture,
engineering and the home. Much of the motivation
and illustrative material has not appeared heretofore in
any text-book.
Third Edition. Revised. 839 pa$es. 5J by 8. 603 fig-
ures. 1634 problems, many solved in detail. Cloth.
A Handbook of Physics Measurements.
By Erwin S. Ferry in Collaboration with O. W. Silvey,
G. W. Sherman, Jr., D. C. Duncan, and R. B. Abbott. A
self-contained manual of the theory and manipulation of
those measurements in Physics which experience has shown
to be most available for college and industrial laboratories.
Vol. I. Fundamental Measurements, Properties of
Matter and Optics. Second Edition, Revised,
xii-f 288 pages. 5} by 8. 162 figures. Cloth.
Vol. II. Vibratory Motion, Sound, Heat, Electricity
and Magnetism. Third Edition, Corrected.
x + 277 pages. 5Jby8, 143 figures. Cloth.
Sensitive Element of Sperry Gyrocompass, Mark X
(removed from binnacle).
APPLIED GYRODYNAMICS
FOR STUDENTS, ENGINEERS
AND USERS OF GYROSCOPIC APPARATUS
BY
ERVIN S. FERRY
Professor of Physics in Purdue University
FIRST REVISED EDITION
NEW YORK
JOHN WILEY & SONS, INC.
LONDON: CHAPMAN & HALL, LIMITED
1933
COPYBIQHT, 1932, 1933
BY ERVIN S. FERRY
All Rights Reserved
This book or any part thereof must not
be reproduced in any form without
the written permission of the publisher.
Printed in U. S. A.
Printing Composition and Plates Binding
F. H. GILSON CO. TECHNICAL COMPOSITION CO. STANHOPE BINDERY
BOSTON CAMBRIDGE: BOSTON
PREFACE
Gyrodynamics is the brain child of ihe pure mathematician.
From the beginning it was hedged about by a wall of differential
equations. Some engineers and physicists have broken through
this wall and have elicited the aid of gyrodynamics in producing
marvels greater than those of fabled Daedalus. Already, through
its aid, we have a device used on nearly all war ships and on many
of the larger merchant ships by means of which the ship can be
guided on any desired straight course without the aid of a helms-
man; another that will keep an airplane level and on a straight
course even though the aviator may be out of the cockpit making
some necessary adjustment to the engine; another that will keep
a Submarine torpedo on either a straight or a curved course and,
after the torpedo has run a predetermined distance, change the
course by a predetermined angle; another that will prevent the
rolling of a ship of any size even on a rough sea; another which,
carried on a train running at ordinary speed, will record all ir-
regularities of the track traversed. There are many important
instruments used in aviation, navigation and in the industries
that are based upon the principles of gyrodynamics. Stresses
are developed in the structure of ships, airplanes and other ve-
hicles carrying rotating machinery that can be computed and
guarded against only by a designer familiar with gyrodynamics.
The purpose of the present book is to bring gyrodynamics out
from behind the integral signs and to present it to the acquaintance
of engineers and students having the mathematical equipment of
the ordinary graduate of engineering or physics. This book is the
outgrowth of lecture notes of a course that has % been given for
several years. The first stage in the development of the course
was the collection of information of all gyroscopic devices of in-
dustrial importance. All of the United States patents and many
foreign patents granted on gyroscopic apparatus since 1900 have
been examined. Correspondence has been carried on with many
firms and individuals. The test of a ship stabilizer in Philadelphia
has been watched. Two factories making gyro-compasses in
Brooklyn and the gyro-compass school in the Brooklyn Navy
iii
IV PREFACE
Yard have been visited. Foreign-built gyro-compasses have been
inspected on ships in New York harbor. One summer vacation
of three months has been spent in attendance on the Sperry
Gyro-compass School. A trip was made to Annapolis and Wash-
ington for the purpose of obtaining information on the gyro
control of automobile torpidocs and a trip was made to Kiel,
Germany, for the purpose of examining the construction of the
Anschutz gyro-compass.
The second stage in the development of the course was the de-
duction of the laws and principles upon which depend the action
of the various devices considered, using therein only methods that
are understandable by students who are not specialists in mathe-
matics. Equations are derived that can be used for the solution
of numerical problems. Careful attention has been given to the
construction of clear definitions and diagrams.
The first chapter of the present book is preliminary to the sub-
ject of gyrodynamics and includes the definitions and laws of
physics assumed in the subsequent chapters. In the second chap-
ter are developed the laws of gyrodynamics upon which are based
the various gyroscopic devices used in industry. Many applica-
tions and much illustrative material are included as well as a num-
ber of solved problems designed to show methods of calculation.
The gyroscopic pendulum with such applications as the gyro-
horizon and the gyro-sextant are considered in the third chapter.
The fourth chapter is devoted to the consideration of anti-roll
devices for ships. In the fifth chapter the principles upon which
the gyro-compass depends are developed, and each of the five
makes of gyro-compass being installed on ships in 1931 is described
in some detail. The last chapter considers the dynamic stabiliza-
tion of a statically unstable body by means of a gyroscope. The
methods which have been used for stabilizing monorail cars are
described, and the reasons for the lack of success of the models
built are indicated.
It is my agreeable duty to acknowledge the aid received from
all to whom I have applied. For information concerning the gyro-
compass I am indebted to Mr. E. C. Sparling, compass engineer,
and to Mr. J. J. Brierly, head of the Gyro-compass School, of the
Sperry Gyroscope Company, Brooklyn, N. Y. ; to Mr. A. P. Davis,
chief engineer of the Anna Engineering Company, Brooklyn, N. Y. ;
to M. P. Vondorwahl, chief engineer of Anschutz & Co., Kiel; to
Mrs. S. G. Brown of the S. G. Brown Company, Limited, London,
PREFACE V
England, and to the Director of the Officine Galileo, Florence,
Italy. I am under deep obligation to Mr. F. P. Hodgkinson,
stabilizer engineer, and to Mr. L. F. Carter, recorder engineer,
of the^Sperry Gyroscope Company, and to Mr. William Dieter,
chief engineer of the E. W. Bliss Company, of Brooklyn, N. Y.
Professor Arthur Taber Jones of Smith College has been so good
as to give the manuscript of the present book the same careful
reading and scholarly criticism which he has bestowed on my
previous books. These criticisms have resulted in a great improve-
ment in the clearness and accuracy of the text.
EKVIN S. FEBKY.
LAFAYETTE, INDIANA, U. S. A.
September 1, 1931.
CONTENTS
Preface iii
CHAPTER I
DEFINITIONS AND PRINCIPLES OF ELEMENTARY DYNAMICS
1. Translation and Rotation
ART. PAGE
1. Linear Motion: Definitions and Units 1
2. Angular Motion: Definitions and Units 3
3. Composition and Resolution of Forces 5
4. Composition and Resolution of Uniform Linear Velocities and
Accelerations. Solved Problem 7
5. Composition and Resolution of Torques 10
6. Composition and Resolution of Angular Velocities, Accelerations
and Moments 10
7. The Relation between the Angular Velocity of a Body and the
Linear Velocity of any Point of the Body 11
8. The Relation between the Angular Acceleration of a Body and the
Linear Acceleration of any Point of the Body 11
9. Centripetal and Centrifugal Forces 12
10. The Dynamic Vertical and the Dynamic Horizontal 12
11. Moment of Inertia 13
12. Values of the Moments of Inertia of Certain Bodies 15
13. Axes of the Principal Moments of Inertia of a Body 15
14. Centripetal Forces Act ing on an Unsymmctrical Pendulum Bob.. . 16
15. The Relation between Torque and Angular Momentum 17
16. Conservation of Angular Momentum 20
17. Centroid, Center of Gravity and Center of Mass 21
2. Simple Harmonic Motion
18. Simple Harmonic Motion of Translation 22
19. The Period of a Simple Harmonic Motion of Translation 24
20. Simple Harmonic Motion of Rotation 25
21. The Physical Pendulum 28
22. The Conical Pendulum 29
23. Wave Motions and Wave Forms 30
24. Phase and Phase Angle 32
25. The Mean Value of the Product of Two Simple Harmonic Func-
tions of Equal Period 33
26. Oscillation of a Coupled System. Resonance 36
27. Damping of Vibrations 39
28. The Frahm Anti-Roll Tanks 42
vii
Viii CONTENTS
CHAPTER II
THE MOTION OF A SPINNING BODY UNDER THE ACTION OF A TORQUE
ART. PAGE
29. Degrees of Freedom 44
30. The Effect on the Motion of a Spinning Body Produced by an
External Force 45
31. Precession 48
32. Change in the Motion of a Ship Produced by Precession of the Shaft 50
33. Deviation of the Course of an Airplane Produced by Precessions of
the Propeller Shaft 51
34-35. Magnitude of the Torque Required to Maintain a Given Pre-
(sessional Velocity when there is Zero Motion about the Torque-
Axis, and when the Axes of Spin, of Torque and of Precession
are Perpendicular to One Another. Solved Problems 52
36. The Direction and Magnitude of the Torque Developed by a Spin-
ning Body when Rotated about an Axis Perpendicular to the
Spin-Axis. Solved Problems 60
37. The Period of Precession 66
38. The Kinetic Energy of a Processing Body 66
39. Nutation 67
40. The Effect of Hurrying and of Retarding the Precession of a
Spinning Body 69
41. Motion of the Spin- Axis Relative to the Earth 71
42. The Angular Velocity of the Spin-Axle of an Unconstrained Gyro-
scope with Respect to the Earth. Solved Problem 72
43. An Instrument for Measuring the Crookedness of a Well Casing. . 74
44. The Weston Centrifugal Drier 75
45. The Effect on the Direction of the Spin- Axis of a Top Produced by
Friction at the Peg 76
46. The Bonneau Airplane Inclinometer 76
47. The Sperry Airplane Horizon 78
48. The Drift of the Projectile from a Rifled Gun 80
49. The Effect of Revolving a Non-Pendulous Gyroscope with Two
Degrees of Rotational Freedom, about the Axis of the Sup-
pressed Rotational Freedom 82
50. The Pioneer Turn Indicator 85
51. The Clinging of a Spinning Body to a Guide in Contact with It ... 85
52. Components of the Torque Acting upon a Spinning Body Having
one Fixed Point, Relative to the Three Coordinate Axes of a
Rotating Frame of Reference 87
53. Components of the Torque Required to Maintain Constant Pre-
cession of a Body when the Precession Axis is Inclined to the
Spin-Axis 90
54. The Griffin Pulverizing Mill. Solved Problem 92
55. The Automobile Torpedo 94
56. The Pendulum-Controlled Depth Steering Gear 95
57. The Conditions that Must Be Fulfilled by the Horizontal Steering
Mechanism 96
CONTENTS ix
ART. PAGE
58. The Bliss-Leavitt Torpedo Steering Gear 97
59. Method of Compensating the Effect of the Rotation of the Earth 99
60. Devices for Changing the Course of a Torpedo 99
61. Airplane Cartography 100
62. Direct Control of the Axis of a Camera 101
63. Indirect Control of the Axis of a Camera 101
64. Control of the Line of Sight of a Camera 103
CHAPTER III
THE GYROSCOPIC PENDULUM OR PENDULOUS GYROSCOPE
1. General Properties
65. The Gyro-Pendulum 105
66. The Period and the Equivalent Length of a Gyroscopic Conical
Pendulum 105
67. The Inclination of the Precession Axis of a Gyroscopic Conical
Pendulum to the Vertical 106
68. The Period of the Undamped Vibration, back and forth through the
Meridian, of the Gyro-Axle of a Pendulous Gyroscope 108
69. The Torque with Which the Second Frame of a Gyroscope Resists
Angular Deflection Ill
70. The Length of the Simple Pendulum Which Has the Same Period
as an Oscillating Body to Which is Attached a Spinning Gyroscope 1 13
2. Gyro-Horizontals and Gyro-Verticals
71. Determination of the Latitude of a Place 115
72. Gyro-Horizons 116
73. The Schuler Gyro-Horizon 117
74. The Anschutz Gyro-Horizon 117
75. The Bonneau-LePrieur-Derrien Gyro-Sextant 119
76. Fleuriais Gyroscopic Octant 120
77. The Sperry Roll and Pitch Recorder 121
78. The Sperry Automatic Airplane Pilot 123
79. The Sperry-Carter Track Recorder 125
80. Directed Gun-Fire Control 128
81. Gun-Fire Directorscopes 129
CHAPTER IV
GYROSCOPIC ANTI-ROLL DEVICES FOR SHIPS
1. The Oscillation of a Ship in a Seaway
82. The Rolling of a Ship due to Waves 131
83. The Pitching of a Ship due to Waves 132
84. The Metacentric Height 132
85. The Experimental Determination of the Metacentric Height 133
86. The Period of the Rolling Motion of a Ship 134
87. Methods of Diminishing the Amplitude of Roll 135
X CONTENTS
ART. PAGE
2. The Inactive Type of Gyro Ship Stabilizer
88. The Effect on the Motion of a Swinging Pendulum Produced by an
Attached Gyroscope, (a) When the Precession of the Gyro-Axle
is Opposed by a Fractional Torque 136
89. The Effect of a Spinning Gyroscope on the Rolling of a Ship 140
90. The Schlick Ship Stabilizer 142
91. The Fieux Ship Stabilizer 144
3. The Active Type of Gyro Ship Stabilizer
92. The Effect on the Motion of a Swinging Pendulum Produced by an
Attached Gyroscope, (b) When the Gyro-Wheel is acted upon by
a Torque about an Axis Perpendicular to the Spin-Axis and the
Axis of Vibration of the Pendulum 146
93. The Sperry Ship Stabilizer 147
94. Operation of the Sperry Ship Stabilizer 149
95. The Braking System 151
96. Rolling of a Ship Produced by a Gyro 152
97. Admiral Taylor's Formula. Solved Problem 153
CHAPTER V
NAVIGATIONAL COMPASSES
1. The Various Types
98. The Altitude and Azimuth Method of Locating the Geographic
Meridian 162
99. The Directive Tendency of a Magnetic Compass 165
100. The Deviations of a Magnetic Compass on an Iron Ship 166
101. The Deviation of a Magnetic Compass Produced by a Rapid Turn 168
102. The Earth Inductor Compass 169
103. The Magneto Compass 169
104. The Sun Compass 170
105. The Apparent Motion of the Spin-Axle of an Unconstrained
Gyroscope Due to the Rotation of the Earth 172
106. The Meridian-Seeking Tendency of a Pendulous Gyroscope 173
107. The Meridian-Seeking Tendency of a Liquid-Controlled Non-
Pendulous Gyroscope 176
108. Making a Gyroscope into a Gyro-Compass 177
109. The Meridian-Seeking Torque Acting on a Gyro-Compass 178
2. The Natural Errors to Which the Gyro-Compass is Subject
110. The Latitude Error 182
111. The Error Due to the Velocity of the Ship. The Meridian-Steam-
ing Error 182
112. The Deflection of the Axle of a Gyro-Compass Produced by
Acceleration of the Ship's Velocity 184
113. The Period of a Gyro-Compass that will have Zero Ballistic Deflec-
tion Error when at a Particular Latitude 186
114. The Ballistic Damping Error 190
CONTENTS XI
ART. PAGE
115. The Compass Error Due to Rolling of the Ship when on an Inter-
cardinal Course. The Quadrantal or Rolling Error 191
116. Quadrantal Deflection Due to Lack of Symmetry of the Sensitive
Element 195
117. The Suppression of the Quadrantal Error 195
118. The Degree of Precision of Gyro-Co mpasses 196
3. The Sperry Gyro-Compass
119. The Principal Parts of the Master Compass 198
120. The Follow-Up System 202
121. The Method of Damping 203
122. The Magnitude of the Latitude Error for which Correction Must
Be made in the Sperry Gyro-Compass 205
123. Correction Mechanism for Velocity and Latitude Errors 206
124. Avoidance of the Ballistic Deflection Error 210
125. The Automatic Ballistic Damping Error Eliminator 212
126. Avoidance of the Quadrantal or Rolling Error 212
127. The Repeater System . 213
4. The Brown Gyro-Compass
128. Production of the Meridian-Seeking Torque 215
129. The Method of Damping 218
130. Absence of Latitude Error 220
131. The Meridian-Steaming Error 220
132. The Repeater System 221
133. The Ballistic Deflection Error 222
134. Prevention of the Quadrantal or Rolling Error 222
5. The Anschutz Gyro-Compass
135. The Sensitive Element of the Model of 1926 224
136. The Supporting System 227
137. Damping 228
138. The Meridian-Steaming Error 231
139. Prevention of the Ballistic Deflection Error 231
140. Avoidance of the Quadrantal or Rolling Error 232
141. The Follow-Up Repeater System 234
6. The Arma Gyro-Compass
142. The Sensitive Element 236
143. The Follow-Up System 238
144. The Course and Speed Error Corrector 241
145. Prevention of the Ballistic Deflection Error 244
146. Avoidance of the Ballistic Damping Error 246
147. Avoidance of the Quadrantal or Rolling Error 247
7. The Florentia Gyro-Compass
148. Arrangement of the Principal Parts of the Master Compass 247
149. The Follow-Up System 248
xil CONTENTS
ART. PAGE
150. Damping 250
151. The Latitude and Meridian-Steaming Error Corrector 251
152. Avoidance of the Ballistic Error 252
153. Avoidance of the Error Due to Rolling and Pitching of the Ship
When on Intercardinal Courses 252
CHAPTER VI
GYROSCOPIC STABILIZATION
1. General Principles
154. Static and Kinetic Stability 255
155. The Stability of a System Consisting of a Body Capable of Oscil-
lation and an Attached Processing Gyro-Wheel 255
156. Some Laws of Dynamic Stability 257
2. Gyroscopically Stabilized Monorail Cars
157. The Economy of Monorail Cars 259
158. The Principles upon Which Depend the Dynamic Stabilization
of Monorail Cars 259
159. The Effect of a Change in Linear Velocity on the Stability of a
Monorail Car that Carries a Single Statically Unstable Gyro-
scope with Vertical Spin-Axle 261
160. The Effect of a Change in Linear Velocity on the Stability of a
Monorail Car that Carries a Single Gyroscope with Horizontal
Spin- Axle Transverse to the Car 264
161. Methods for Increasing the Kinetic Stability of a Monorail Car
While the Car is Going Around Curves 265
162. The Schilovsky Monogyro Monorail Car 266
163. The Brennan Duogyro Monorail Car 268
164. The Scherl Duogyro Monorail Car of 1912 271
NOTATION USED
a Linear acceleration
a' Radial linear acceleration
a Angular acceleration
b Constant
c Constant
d t Internal diameter
do Outside diameter
id'z Mean value of the product of d\. and dz
F Force
g Acceleration due to gravity
H Metacentric height
h c Angular momentum with respect to an axis through c
h s Angular momentum with respect to the spin-axis
K Moment of inertia
K c Moment of inertia with respect to an axis through c
K s Moment of inertia with respect to the spin-axis
k Radius of gyration
L Torque
Z Length of the equivalent simple pendulum
X Latitude
m Mass
P Power
PH Horse-power
r Radius
R Resultant
S Torsional stiffness
i Time
T Period
v Linear velocity
v' Meridian component of velocity of ship
VQ Linear velocity at time zero
vt Linear velocity at time t
n v s Linear velocity of a body s with respect to a body n
w Angular velocity
w c Angular velocity about an axis through c
WQ Angular velocity at time zero
wt Angular velocity at time t
w f Mean roll velocity of a ship
W Work or energy
Wt Kinetic energy of translation
Wro Kinetic energy of rotation
x Linear distance or linear displacement
xiii
xiv NOTATION USED
Angle especially ballistic deflection
di Meridian-steaming error
<t>, 6 Angle or angular displacement
$ Mean angular displacement
\ Latitude
= Approximately equal to
S#2/ The sum of a series of terms of the form xy.
Different directed quantities are represented in diagrams by arrows having
heads of different shapes. In the case of directed angular quantities, the arrow-
head indicates that the torque, angular velocity, angular acceleration or angu-
lar momentum is in the clockwise direction about the length of the arrow as
axis as seen by an observer looking along the shaft of the arrow toward the
head. The arrow-heads used to represent the various directed quantities
are as follows:
force (F) angular momentum (h)
linear velocity (v) torque (L) E*
angular velocity (w) linear acceleration (a) +fr
APPLIED GYRODYNAMICS
CHAPTER I
DEFINITIONS AND PRINCIPLES OF ELEMENTARY
DYNAMICS
1. Translation and Rotation
1. Linear Motion. Definitions and Units. Linear displace-
ments are commonly measured in centimeters, meters, kilometers,
inches, feet, or miles of various lengths.
1 centimeter = 3937 inch
1 meter = 3.2809 feet
1 kilometer = 0.6214 statute mile
1 statute mile = 5280 feet
1 nautical mile = 6080 feet
1 inch = 2.5399 centimeters
1 foot = 0.3048 meter
1 statute mile = 1.6093 kilometers
Linear velocity is time-rate of linear displacement in any given
direction. The magnitude of velocity is called speed. Thus,
one might say that an automobile is moving with a velocity of 30
miles per hour north but that the automobile is capable of devel-
oping a speed of 50 miles per hour. A speed of one nautical mile
(6080 ft.) per hour is called a knot. If the linear displacement in
every equal time interval t has the constant value x, then the mag-
nitude of the constant linear velocity is
Linear acceleration is the time-rate of change of linear velocity.
There is a linear acceleration whenever either the magnitude or
the direction of a linear velocity changes. A body that in every
time interval t changes in linear speed from VQ to v t while maintain-
1
2 PRINCIPLES OF ELEMENTARY DYNAMICS
ing a constant direction of motion has a constant linear accelera-
tion in the direction of motion of the value
a _ y * "" Vo (<>\
a - t (4)
A body moving in a circular path of radius r with constant speed
v has an acceleration directed toward the center of the circle.
The magnitude of this so-called " radial acceleration " or " cen-
tripetal acceleration " is
(3)
r ^ '
Commonly employed units of linear acceleration are the centi-
meter per second per second, the foot per second per second, and
the mile per hour per second.
Any cause which either produces or tends to produce a linear
acceleration of the motion of a body is called force. Commonly
employed units of force are the dyne, gram weight, and pound
weight.
1 pound weight = 454 grams weight
At a place where the acceleration due to gravity is 980 centimeters
per second per second
1 gram weight = 980 dynes
Work is the accomplishment of a change in the position of a body
against an opposing force. The magnitude of the work is taken
to be equal to the product of the force overcome and the projection
of the displacement in the line of action of the force.
W = Fx (4)
The commonly employed units of work are the erg (dyne-centi-
meter), joule (10 7 ergs), and the foot-pound.
Energy is stored work. The amount of energy possessed by a
system of bodies is the amount of work it can do in passing from
its present condition to some standard condition.
The kinetic energy of translation of a body of mass m moving
with a linear speed v is
W t = %mv* (5)
Energy, being work, is measured in the same units as work.
Power is the time-rate of doing work. If a system does an
amount of work W in time t by opposing a constant force F
TRANSLATION AND ROTATION 3
through a distance x along the line of action of the force, the
power P has the value
!-_*-* (6)
The units of power are the erg per second, the watt (joule per sec-
ond), the foot-pound per second , and the horse-power ( = 550 foot-
pounds per second).
That property of a body which makes it necessary to use force
to produqe a linear acceleration in the motion of the body is called
inertia. Inertia is measured by the tendency of a body to keep
its linear velocity of constant magnitude and in an invariable
direction. Anything which possesses inertia is called matter.
The amount of matter in a body is called the mass of the body.
The ratio of the masses of two bodies is taken to be equal to the
ratio of the inertias of the bodies. The inertia of a body or
system of bodies equals the sum of the masses of its parts. Com-
monly employed units of mass are the gram, the pound, and the
slug or British engineering unit of mass. At a place where the
acceleration due to gravity is 32.1 ft. per sec. per sec.,
1 slug = 32.1 Ib.
The product of the mass of a body and its linear velocity is
called the linear momentum of the body. The units used arc the
gram-centimeter per second and the slug-foot per second. These
units have no name but are referred to as the centimeter-gram-
second absolute unit and the British en-
gineering unit of linear momentum, re-
spectively.
2. Angular Motion: Definitions and
Units. Angular displacements are com-
monly measured in degrees, radians or
revolutions. The degree is one-ninetieth part of a right angle.
The radian is the plane angle subtended at the center of a circle
by an arc equal to the radius of the circle. Thus, if the arc A B,
Fig. 1, is half as long as the radius CA, the angle < is one-half of
a radian. Whatever the length x of the arc, and the length r of
the radius
1*
= - radians (7)
r
1 revolution = 2 TT radians = 360.
1 radian = 57.3 = 3438'.
4 PRINCIPLES OF ELEMENTARY DYNAMICS
Angular velocity is the time-rate of change of angular displace-
ment in a given sense about a given axis. I The magnitude of angu-
lar velocity is called angular speed. Thus, one might say that a
top has an angular velocity of 200 revolutions per minute in the
clockwise direction about an axis inclined 30 degrees to the vertical
and 45 degrees west of the meridian plane. If the angular displace-
ment in every equal time interval t be constant and represented
by 0, then the magnitude of the constant angular velocity is
f (8)
Angular acceleration is the time-rate of change of angular
velocity. There is an angular acceleration whenever there is a
change either in the magnitude of the angular velocity or in the
direction of the axis of rotation. A body that in every equal time
interval t changes in angular speed from WQ to w t while the direction
of the axis of rotation remains unchanged, has a constant angular
acceleration about the fixed axis of the value
a - -*LZ? (9)
t
Commonly employed units of angular acceleration are the radian
per second per second and the revolution per minute per second.
Any cause which either produces or tends to produce an angular
acceleration of the motion of a body is called a torque or force
couple. A torque is equivalent to two equal, oppositely directed
forces, acting in parallel lines. The magnitude of a torque is
measured by the product of one of the forces and the perpendicular
distance between the lines of action of the two forces. The line
around which a torque either produces or tends to produce angular
acceleration is called the torque-axis. A. The magnitude of a torque
is expressed in centimeter-dynes, pound-feet, etc.
That property of a body because of which a torque is needed to
give to the body an angular acceleration is called moment of in-
ertia. Moment of inertia is measured by the tendency of a body
to keep its angular velocity of constant magnitude and about an
invariable axis of rotation. The moment of inertia of a particle
of mass m at a distance r from the axis of rotation is wr*. The
moment of inertia of a body about a given axis is numerically
equal to the sum of the products of the masses of the particles
composing the body and the squares of their respective distances
from the axis of rotation, y*
TRANSLATION AND ROTATION 5
Units of moment of inertia are the gram-centimeter 2 , the pound-
foot 2 , and the slug-foot 2 . This last unit is the British engineering
unit of moment of inertia.
When a body rotates against a constant opposing torque L
through an angular displacement $ radians, it does an amount of
work
W = L0 (10)
The kinetic energy of a body rotating with angular velocity w is
W ro = | Kw* (11)
where K represents the moment of inertia of the body with respect
to the axis of rotation. If K is measured in feet and slugs, and w
in radians per second, W ro will be expressed in foot-pounds.
If an angular displacement <t> be effected in time t with a constant
angular velocity w, then the power developed by the torque is
L W (12)
The product of the moment of inertia of a body with respect to a
given axis and the angular velocity of the body about the same axis
is called the angular momentum of the body with respect to the
given axis. Thus, in symbols, the angular momentum with respect
to an axis through c is
Jk = K c w c (13)
It can be shown that the sum of the moments of the linear momenta
of the elementary parts of a body equals the angular momentum
of the body. For this reason, angular momentum is also called
moment of momentum. It is sometimes called kinetic moment.
The units employed are the gram centimeter 2 -radian per second
and the slug foot 2 -radian per second. They have no names but
are referred to as the centimeter-gram-second absolute unit and
the British engineering unit of angular momentum, respectively.
3. Composition and Resolution of Forces. A force has both
direction and magnitude. It can be completely represented by a
straight line drawn in the direction in which the force acts, of a
length proportional to the magnitude of the force, and having an
arrow-head marked on the line pointing in the direction of the
action of the force. Two quantities represented by lines that in-
tersect are said to be concurrent. The motion of a body may be
due to the effect of two or more simultaneous forces. Two sys-
6 PRINCIPLES OF ELEMENTARY DYNAMICS
terns of forces having identical effects on the motion of a body are
said to be equivalent. A single force equivalent to two or more
simultaneous forces is called the resultant of the set. Two forces
which, acting together, are equivalent to a single force are called
components of the given force. The operation of finding the re-
sultant of a system of forces is called composition of forces.
The method of finding the resultant of two concurrent forces,
called the parallelogram law, may be stated as follows: // two
adjacent sides of a parallelogram represent in direction and in mag-
nitude two concurrent forces, both directed from the point of inter-
section, then the diagonal of the parallelogram drawn from this
intersection represents completely, in
direction and in magnitude, the re-
sultant of the two forces.
If the magnitudes of the two
A **" * - * P +' concurrent forces be represented
by Fi and F 2 , Fig. 2, and if the
FlG - 2 angle between them be 0, then
from the law of cosines the magnitude of the resultant R is given
by the equation
#2 = FJ + F 2 2 + 2 FJ 2 cos (14)
The angle between the line representing either one of the com-
ponent forces and the line representing the resultant can be ob-
tained from Fig. 2 and the law of sines,
2 singi sin ffi
__ _
sirT(180 -
Similarly,
. F 2 sin
sm ft =
. ^ i sin
sm ft =
(15)
By compounding the resultant of two concurrent forces with a
third force that is concurrent with this resultant, we can find the
resultant of all three forces.
The operation of finding the components in two given directions
of a force is called resolution of the force. The components of the
force F, Fig. 3, in the directions A and B, Fig. 4, are FA and FB }
Fig. 5; the components in the directions H and V, Fig. 4, are
TRANSLATION AND ROTATION
FS and Fy, Fig. 6. If H and V are at right angles to one another,
FH = F cos <f> and F v = F sin 0.
A force may be resolved into three components in assigned
directions. For instance, the components may be found in the
vertical, in the north-south, and in the east-west directions.
FIG. 3
B
Linear velocities, linear accelerations, and linear momenta can
be completely represented by directed straight lines as can forces.
The parallelogram law, used for forces, can also be used to com-
B
FIG. 5
FIG. 6
pound and resolve these quantities. However, since all velocity
is relative to some reference body, care must be exercised to keep
clearly in mind what object is moving and what object is taken as
the body of reference. One method by which this can be done
easily is illustrated in the following Article.
4. Composition and Resolution of Uniform Linear Velocities
and Accelerations. A passenger moving across a moving rail-
way carriage furnishes an example of what are called simultaneous
velocities. The passenger has a velocity relative to a point of the
carriage, and at the same time the carriage has a velocity with
respect to a point on the earth. These are called components of
the passenger's motion. The velocity of the passenger with refer-
ence to the earth is called his resultant velocity.
In order that we may keep clearly in mind what object is re-
8 PRINCIPLES OF ELEMENTARY DYNAMICS
garded as moving and what object is taken as the reference body,
it is convenient to use two subscripts. Let a subscript written at
the right indicate the moving body, and a subscript written at the
left indicate the frame of reference. Thus, the velocity of a tor-
FIG. 7
pedo with reference to the ship from which it is projected may be
written s v t , and the velocity of the ship with reference to the enemy
may be written n v s . In Fig. 7, OA represents s v t and OB represents
n v s . We notice that in these symbols for the velocities which we
have compounded, the right subscript of one velocity is the same
as the left subscript of the other, and that the resultant has the
two subscripts that are different. In our example, s vt and n v s
combined give the velocity of the torpedo with respect to the
enemy, n v t .
If we have given n vt and n v s and wish s vt we need only notice that
n v s sVn- Thus, we can compound n vt and s v n thereby ob-
taining 5 Vt.
The " torpedo director " consists of two rods variable in length
and direction which are linked together and to a third rod carrying
a telescope. The telescope points in the direction n vt when the
other two rods are adjusted to represent in direction and length
s vt and n v s . When the enemy ship arrives at C, Fig. 7, it is seen in
the telescope fastened to the rod OC. At this instant the torpedo
is launched in the direction and with the speed relative to the firing
ship represented by OA. The torpedo meets the enemy ship
at A.
If, however, we attempt to combine in this manner two velocities
which do not have a right subscript of one the same as a left
subscript of the other, we shall have a result which is without
meaning. For example, if we attempt to combine the velocity of
the torpedo with respect to the enemy and the velocity of the firing
ship with respect to the enemy, we would get the line OD', Fig. 8.
This line has no physical meaning.
Problem. Waves are moving across the sea with a velocity of one knot
westward, relative to the earth. The distance between the crests is 50 ft.
TRANSLATION AND ROTATION
9
Find the number of wave crests that in one hour pass under a ship which is
steaming at a velocity of 15 knots: (a) north-east; (6) south-west.
Ans. (a) 1909 crests; (6) 1739 crests.
Problem. An airplane has a supply of gasoline sufficient for six hours of
flying at a speed of 60 knots relative to the air. The wind has a velocity of
15 knots from the south. The pilot is ordered to proceed as far as possible
along a north-east course and then retrace his course to the starting point.
Find: (a) the course and ground speed to the turning point; (b) the course
and ground speed back; (c) the distance from the starting point to the turn-
ing point; (d) the time from the starting point to the turn.
FIG. 9
Outline of Solution. Draw OC, Fig. 9, representing the velocity of the wind
relative to the earth ( e v w = 15 mi. per hr.). From C describe two arcs of a
circle of a radius (CD == CE) representing the speed of the plane relative to
the wind ( w vp = 60 knots). Draw lines CD and CE to the points at which
these arcs intersect the ground course AB. Complete the parallelogram CG
and CH. The course out is along the line OH and that back is along OG.
The ground velocity ( e vp) is given by OE and the ground velocity back ( e vp f )
is represented by OD.
(a and 6) The data of the problem give the magnitude OH = OG together
with the angles COB and COD. Using the law of sines and the law of cosines
we find the values of <, <', OE which represents the ground velocity out
( e vp), and OD which represents the ground velocity back ( e vp f ).
(c) We have two expressions for the distance to the turning point:
10 PRINCIPLES OF ELEMENTARY DYNAMICS
where / represents the time occupied in going to the turning point and t'
represents the time occupied in returning. Then, the time of the round trip
gvp e vp cVp
Whence, the ground distance to the turn
e vp + e vp r
(d) The time from the starting point to the turn is given by the expression
6. Composition and Resolution of Torques. Three quantities
are required to specify completely a torque. They are the mag-
nitude of the torque, the direction of the torque-axis, and the sense
in which the torque acts around that axis, that is, whether it is
clockwise or counter-clockwise. A torque can be completely
represented by a straight line of a length proportional to the mag-
nitude of the torque, drawn in the direction of the torque-axis,
and carrying an arrow-head so placed that on looking along the
line in the direction of the arrow, the sense of the torque is clock-
wise.
The method of compounding concurrent torques is as follows:
Draw from a given point two lines that represent the two simultaneous
instantaneous torques. Complete a parallelogram on these two lines
as sides. Then, the diagonal of the parallelogram drawn from the
given point represents completely the resultant instantaneous torque.
6, Composition and Resolution of Angular Velocities, Accelera-
tions and Momenta. An angular velocity, an angular accelera-
tion or an angular momentum can be represented completely
by a straight line of a length proportional to the magnitude, drawn
in the direction of the axis and carrying an arrow-head which in-
dicates the sense of the angular quantity. In this book the
arrow-head is so placed that, on looking along the line in the di-
rection of the arrow, the sense of the angular velocity, acceleration
or momentum is clockwise,
Angular velocities, angular accelerations and angular momenta
are compounded and resolved by the parallelogram law exactly as
are linear velocities and accelerations.
Experiment. Set the Maxwell Top, shown in section in Fig. 10,
into spinning motion when the point of support is (a) below the
center of gravity of the top, (6) above the center of gravity. Ob-
TRANSLATION AND ROTATION
11
serve that the spin-axis of the body rotates in the direction of the
spin when the center of gravity is above the point of support, and
in the direction opposite the
spin when the center of gravity
is below the point of support.
For each case make an angular
momentum diagram showing
the angular momentum at the
beginning and end of a brief
time interval and the change
of angular momentum during
this interval.
7. The Relation between the FlG 10
Angular Velocity of a Body and the Linear Velocity of Any Point
of the Body. Consider a body, Fig. 11, rotating with uniform
angular velocity w about an axis through perpendicular to the
plane of the paper. Let XX' be a line fixed in space and PO a
line fixed in the body perpendicular to the axis of rotation. If
the body rotates steadily through the angle $ in t seconds, then
the angular speed is given, (1), by
X V ,.
w = r _ . = - - radians per sec.
t rt r
(16)
where x represents the arc of radius r subtended by the angle </>,
and v represents the linear speed of the point P. This equation
shows that the angular speed of a
body equals the linear velocity of any
point of the body divided by the dis-
tance of that point from the axis of
rotation.
8. The Relation between the Angu-
lar Acceleration of a Body and the
Linear Acceleration of Any Point of
FlQ. 11
the Body. If the body be acted upon by a torque, the angular
velocity about the torque axis will change in t seconds from a
value w to some value w t . The rate of change of angular velocity,
or the angular acceleration a, is
a
r
(17)
12 PRINCIPLES OF ELEMENTARY DYNAMICS
where a represents the tangential component of the linear accelera-
tion of a point distant r from the axis of rotation. Thus, in words,
the tangential component of the angular acceleration of a body
equals the linear acceleration of any point of the body divided by
the distance of that point from the axis of the acceleration.
9. Centripetal and Centrifugal Force. It is readily shown*
that, when a body is acted upon by a force of constant magnitude
having a line of action always in the same plane and always per-
pendicular to the direction of the motion, (1) the speed of the body
does not change, (2) the path of the body is a circle, (3) the body is
moving with a linear acceleration which is always directed toward
the center of the circle, and (4) the magnitude of this radial accelera-
tion is constant and equal to the square of the linear speed of the
body divided by the radius of its path.
According to Newton's First Law of Motion, a moving body will
continue to move in a straight line with constant speed until
acted upon by an external force. From the preceding paragraph,
when a body of mass m moves with constant speed v in the cir-
cumference of a circle of radius r, we see: (1) that there must be a
force acting upon the body, (2) this force must be directed toward
the center of the circle, and (3) the magnitude of the force is given
by
F c [= ma] = m V ~ (18)
The force required to overcome the inertia of a body in deflecting
it from a rectilinear path into a circular path is called centripetal
force. The agent which constrains the body to move in a circular
path is acted upon by a force directed away from the center of the
circle and has a magnitude equal to the force acting upon the body
that is moving in the circular path. This, frequently called centrif-
ugal force, is the reaction of the centripetal force and may be de-
fined as the resistance which the inertia of a body in motion opposes
to whatever deflects it from the rectilinear path. The centripetal
force acts upon the body that is moving in a circular path, and its
reaction, the centrifugal force, acts, not upon this body, but upon
the agent which constrains it to move in a circular path./
10. The Dynamic Vertical and the Dynamic Horizontal.
A plumb-bob that is at rest or in uniform linear motion relative
to the earth indicates the direction of the force of gravity at the
* Ferry's General Physics, Art. 66.
TRANSLATION AND ROTATION 13
place where it is situated. This direction is the true vertical.
When a plumb-bob is moving with a linear acceleration relative
to the earth, it assumes a direction called the dynamic vertical
corresponding to the given linear acceleration.
A plane normal to the true vertical is called a true horizontal.
A plane normal to a dynamic vertical is called a dynamic horizon-
tal corresponding to the given linear acceleration. The free sur-
face of an unaccelerated liquid is truly horizontal. The free
surface of a liquid in a vessel moving with linear acceleration is
not truly horizontal. When the speed of a locomotive is increas-
ing, the water gauge gives an indication that is too high; when the
speed is decreasing, the indication is too low. The free surface of
water in a rotating vessel is not truly horizontal.
11. Moment of Inertia. Experiment. With the arm out-
stretched, rotate back and forth an iron pipe on which are mounted
two iron balls, Fig. 12. Note that a considerable
torque, produced by twisting the wrist, must be given
the rod to change quickly the angular velocity,
whereas a smaller torque is required to produce a
smaller angular acceleration. Note that when the
spheres are close to the hand, a fairly small torque,
produced by twisting the wrist, will impart to the
apparatus a certain angular acceleration about the
outstretched arm as an axis. When the spheres are
at a distance of a foot or more from the hand, a much larger
torque is required to produce the same angular acceleration.
That property of a body because of which a torque is needed to
give the body an angular acceleration is called moment of inertia.
When a torque acts upon a body, it produces an angular accelera-
tion that is directly proportional to the torque and inversely pro-
portional to the moment of inertia of the body upon which it
acts. The moment of inertia is to be taken about the axis of the
torque. The angular acceleration is about the same axis. Or,
in symbols, L = K* (19)
The moment of inertia K of a body consisting of particles of
masses mi, m^, ra 3 , etc., at distances r i; r 2 , r 3 , etc., respectively, from
the axis of rotation is
K = Win 2 + Ws 2 + etc.
or, in briefer notation
K = r(wr 2 ). < (20)
14 PRINCIPLES OF ELEMENTARY DYNAMICS
If mass be expressed in slugs and distance in feet, the moment
of inertia will be expressed in British engineering units sometimes
called slug-feet 2 .
The moment of inertia of a body with respect to one axis has not
the same value as the moment of inertia of the same body with
respect to a different axis. The moment of inertia of a given body
with respect to a given axis is numerically equal to the mass of a
body which, if concentrated at unit distance from the axis of
rotation, would require the same torque as the original body to
produce the same angular acceleration. For any rigid body
revolving about any axis fixed in space, the moment of inertia with
respect to that axis is a constant quantity quite independent of
both the speed of rotation and the torque acting.
The distance from the axis of rotation at which the entire mass
of a body might be concentrated without altering the moment of
inertia of the body with respect to that axis, is called the radius
of gyration, or swing radius, of the body about the given axis.
If the entire mass M of the body were at the distance k from the
axis, the moment of inertia of the body with respect to this axis
would be
K = Mk* (21)
In this equation, k is the radius of gyration of the body.
Experiment. Clamp the frame of a bicycle in an upright po-
sition with the front wheel off the ground. With the wheel not
spinning, apply a torque to the handle-bars so as to impart to the
front wheel an angular acceleration about the handle-bar post as
an axis. Now set the front wheel spinning and apply a torque to
the handle-bars as before. Note that the torque to produce a
given angular acceleration is much greater when the wheel is
spinning than when it is not.
The torque required to impart to a spinning body a given angular
acceleration about an axis perpendicular to the spin-axis is greater
than the torque required when the body is not spinning. The ratio
of the torque applied to a spinning body about an axis perpendicu-
lar to the spin-axis, to the angular acceleration thereby produced
about the torque axis, is sometimes called the dynamic moment of
inertia of the spinning body with respect to the torque-axis.
The magnitude of the dynamic moment of inertia of a body with
respect to the torque-axis depends upon the angular momentum of
the part of the body that is spinning, with respect to the spin-axis.
TRANSLATION AND ROTATION 15
12. Values of the Moment of Inertia of Certain Bodies. The
moments of inertia of a body, having regular geometrical shape,
can be computed, but the moments of inertia of a body of irregular
shape are usually most easily determined by experiment. The
experimental determination is usually made by comparison with
a body of known moment of inertia. For such comparisons
cylinders and rings of known dimensions are convenient.
It is shown in books on Mechanics that the moment of inertia
of a uniform right solid cylinder of mass m and diameter d, about
its geometric axis, is
I ml* (22)
whereas about any axis parallel to the geometric axis and distant
p from it, the moment of inertia is
I md 2 + mp 2 (23)
If the cylinder has a length x, the moment of inertia about an axis
through the center and normal to the length is
m
K+S]
whereas about an axis coinciding with the diameter of one end, the
moment of inertia of the cylinder is
m TA + ^T ( 25 )
I JLO o I
The moment of inertia of a ring or right hollow circular cylinder
of outer diameter d and inner diameter rf,-, with respect to the
geometric axis, is
|- wK 2 + di 2 ] (26)
If the moment of inertia of a body of mass m about an axis
through the center of mass of the body is K c , then the moment
of inertia about any parallel axis distant p from the first axis is
K p = K c + mp* (27)
13. Axes of the Principal Moments of Inertia of a Body.
The moment of inertia of a sphere about an axis through the
center of mass equals the moment of inertia about any other axis
through the same point. The moment of inertia of a right cylinder
about an axis through the center of mass and perpendicular to the
length of the cylinder is greater than the moment of inertia about
16 PRINCIPLES OF ELEMENTARY DYNAMICS
any other axis through the same point if the length of the cylinder
is greater than the diameter. The moment of inertia of the cylin-
der about an axis through the center of mass and in the direction of
the length is less than that about any other axis. In the case of
most bodies, the values of the moments of inertia relative to their
mutually perpendicular axes are unequal.
It can be shown that the axis of greatest and that of least mo-
ment of inertia are perpendicular to one another. The perpen-
dicular axis through the center of mass of a body about which the
moment of inertia is maximum, that about which it is minimum,
and the other axis perpendicular to these two, are called the prin-
cipal momental axes of the body. The moments of inertia about
the principal momental axes are called the principal moments of
inertia of the body. If the principal moments of inertia of one
body or system are equal, respectively, to the principal moments
of inertia of another body or system, the two bodies or systems
are said to be equimomental.
For any rigid body there can be constructed an equimomental
system consisting of three slender and uniform rigid rods bisecting
each other at right angles, and coinciding in direction with the
principal momental axes of the given body.
14. Centripetal Forces Acting upon an Unsymmetrical Pen-
dulum Bob. Consider a pendulum having a bob that is capable
of rotation with negligible friction about the axis of the pendulum
rod and that is unsymmetrical with respect to the axis of the pen-
dulum rod. In Fig. 13, the pendulum bob is a rectangular bar
with the long axis perpendicular to the pendulum rod. The long
axis of the bob is inclined at an angle < to the plane of the knife-
edge and the pendulum rod. Consider A and A' at the centers
of the two ends of the bob at the instant when the pendulum is
passing through the equilibrium position. From A and A' draw
lines A B and A'B f perpendicular to the line CC' through the center
of the bob parallel to the knife-edge. From B and B' draw lines
BD and B'D' perpendicular to the knife-edge. The points D
and D f are the points about which oscillate the two particles at
A and A'.
In order that the particles at A and A f may rotate about DD f ,
the particles must be acted upon by centripetal forces F c and F c f
directed toward D and Z)', respectively. Each of the centripetal
forces F CJ etc., can be resolved into three components, one vertical,
one parallel to the axis of the bob, and one horizontal and per-
TRANSLATION AND ROTATION
17
pendicular to that axis. In the diagram, the horizontal com-
ponents acting on the particles A and A', perpendicular to the
axis of the rod, are marked h F c and h F c ', respectively. The hori-
zontal force acting on each particle necessary to cause the bob to
swing so that the long axis is inclined at an angle < to the plane
through the knife-edge and pendulum rod is non-existent when
there is zero friction between the bob and the pendulum rod. If
these horizontal forces are non-existent, the inertia of each particle
FIG. 13
of the bob will cause it to retreat as far as possible from the line CC".
Therefore the stable position of the loose bob is attained when the
long axis of the bob is perpendicular to the plane of the knife-
edge and pendulum rod.
An oscillating body tends to set itself in the position in which
its moment of inertia with respect to the vibration axis is maxi-
mum. A body having equal principal moments of inertia will
oscillate in any plane through the center of mass without any
tendency to turn.
Similarly, if a body, supported at the center of mass so as to be
capable of turning in any direction, be moved in a curved path, the
body will tend to turn so that the axis of minimum moment of
inertia is tangent to the path. A body that has equal principal
moments of inertia has no tendency to turn when moving in a
curved path.
/ 15. The Relation between Torque and Angular Momentum.
The product of the moment of inertia of a body K c with respect to
arT axis through a point c, and the angular velocity w c about the
same axis, is called the angular momentum of the body with respect
18 PRINCIPLES OF ELEMENTARY DYNAMICS
to the same axis. Thus, in symbols, the angular momentum with
respect to an axis through a point c
h c = K c w c
If a rigid body rotates with an angular acceleration a, about an
axis fixed in space, it must be acted upon by a torque (19):
L = K&
where K is the moment of inertia of the body with respect to the
axis of torque. Since angular acceleration is the rate of change of
angular velocity, we may write,
L[=*a] = ^? (28)
The numerator of the right-hand member of this equation is the
change of angular momentum. The right-hand member is the
time-rate of change of angular momentum. The above equation
shows that :
(a) The resultant torque acting on a rigid body about an axis
fixed in space, equals the time-rate of change of the angular mo-
mentum of the body relative to the given axis.
(6) There can be no increase or diminution of the angular
momentum of a body about an axis in space without the action
of a corresponding external torque about that axis. This theorem
is the analogue in rotation to Newton's First Law of Motion.
(c) When a torque acts upon a body, it produces an angular
acceleration, and a rate of change of the angular momentum of the
body, about the axis of torque, proportional to the torque.
(d) When a torque acts upon a body, there acts upon some other
body another torque of the same magnitude in the opposite di-
rection about the same axis. This theorem is the analogue in
rotation of Newton's Third Law of Motion.
(e) When the angular momentum of a body is constant about
any axis, then the resultant torque about that axis is zero. L?
Suppose that at a certain instant a body is spinning with con-
stant angular speed about an axis in the direction of the line h,
Fig. 14. The arrow-head on this line indicates the direction of
spin about the spin-axis, and the length of the line is proportional
to the product of the angular speed and the moment of inertia
of the body about the spin-axis. This line represents completely
the angular momentum of the body about the spin-axis at the
TRANSLATION AND ROTATION 19
chosen instant. Suppose that during a short time interval A
afterward the direction of the spin-axis changes by a small angle
A< and the magnitude of the angular momentum remains constant.
In Fig. 14, h' represents the angular momentum at the end of this
brief time interval, and Aft repre-
sents the change of angular mo-
mentum during this interval.
The rate of change of angular
momentum about the spin-axis is
. This is the value of the FIG. 14
torque about the axis Ah that is required to change the direction
of the angular momentum from htoh'. This torque produces a
rotation w p about an axis perpendicular to the plane of the torque-
axis and the spin-axis. Thus, a torque acting upon a spinning
body about an axis perpendicular to the spin-axis causes the spin-
axis to rotate about another axis perpendicular to the plane of the
axes of spin and of torque. This is analogous to the case, in
translation, in which a force of constant magnitude, acting per-
pendicular to the direction of motion of a body of constant linear
speed, causes the body to describe a circular path.^
Experiment. Stand on a rotatable table and hold above the
head a bicycle wheel with axle vertical. The wheel should be
provided with a rim consisting of 20 to 25 pounds of iron wire.
Set the wheel into rotation by pressing against the spokes with a
pointed stick. Note that while applying a torque to the wheel, the
rotatable table turns in the opposite direction.
Consider a rod suspended at the center of mass by a string.
Let the axis of the rod make an angle 6 to the vertical. Suppose
that a blow be imparted to one end of the rod in a direction per-
pendicular to the plane of the string and the axis of the rod. At
any instant there is an angular momentum h about an axis per-
pendicular to the rod and in the plane containing the string and
the axis of the rod. The angle 6 is changing and h is changing both
in magnitude and in direction. If a force couple of the proper
constant magnitude be applied about an axis which is always
perpendicular to the plane of h and the axis of the rod, then 6
will remain of constant magnitude. The rod now is rotating with
constant angular velocity about the string as axis, tracing the
surfaces of two cones having a common apex at the center of grav-
ity of the rod. This applied couple constitutes a centripetal
20 PRINCIPLES OF ELEMENTARY DYNAMICS
torque. It is producing angular momentum about its own axis
perpendicular to the plane containing h and the axis of the rod.
This angular momentum, combining with the h of the previous
position of the rod, gives the h in the new position. We have
here an example of the case in which the axis of the resultant angu-
lar momentum does not coincide with the axis of the resultant
angular velocity.
16. Conservation of Angular Momentum. In the special
case in which a rigid body is acted upon by a system of torques
the sum of which, relative to a given axis, is zero, we have (28) :
T K dw _ n
L ~ ~5T ~
That is, in this special case, the time-rate of change of angular
momentum is zero. In other words, the angular momentum is
constant. Therefore, if a rigid body is acted upon by a set of
external torques, the resultant of which about any assigned axis
fixed in space is zero, the angular momentum of the body relative
to the assigned axis is a constant quantity. From this equation it
follows that:
However the parts of any material system may act upon one an-
other, the total angular momentum of the system, about any axis fixed
in space, will remain constant, so long as the system is acted upon by
no outside torque. This theorem is called the Principle of the
Conservation of Angular Momentum.
If a circus performer wishes to turn a series of aerial somersaults,
he will jump from a springboard in such a way that he will give
himself a considerable angular velocity about a horizontal axis.
While turning the first somersault he will hold his body nearly
straight, but after that he will gradually draw his head and knees
together, thereby decreasing the moment of inertia of his body
about the axis of rotation. Since the angular momentum of his
body remains constant, when the moment of inertia is diminished
the angular velocity of his body is increased to such an extent that
he can make more somersaults before reaching the ground. If,
when he has nearly reached the ground, he sees that he will fall
face downward, he will straighten his body, thereby increasing his
moment of inertia, thus decreasing his angular velocity to such an
extent that he can alight on his feet, f /
Experiment. Set into rotation about a vertical axis the hori-
zontal rods carrying the two metal spheres, Fig. 15. Pull on the
TRANSLATION AND ROTATION
21
FIG. 15
handle H, thereby drawing the spheres close to the axis of rotation
and diminishing the moment of inertia of the rotating system.
Note that the angular velocity of the system is now much greater
than before. Release the
pull on the handle. The
spheres retreat from the
axis of rotation, thereby
increasing the moment of
inertia of the rotating sys-
tem. Note that the angu-
lar velocity diminishes.
During these operations no
torque was applied to the
rotating body about the
axis of rotation. Consequently, there was no change in the angu-
lar momentum of the system.
Although the work done in pulling the spheres toward the axis
of rotation does not change the angular momentum (Kw) of the
rotating system, it does increase the kinetic energy of rotation
(| Kw 2 ) of the system. While approaching the axis of rotation,
the spheres are moving not in a circular path but in a spiral path.
The force in the string is not perpendicular to the path. There
is a component of this force that acts upon the spheres in the di-
rection of the tangent. This tangential component of the force
produces an increase in the tangential velocity and also in the
angular velocity.
17. Centroid, Center of Gravity and Center of Mass. The
point of application of the resultant of a system of parallel forces
is called the centroid or center of the system of forces.
" If the action of terrestrial or other gravity on a rigid body is
reducible to a single force in a line passing always through one
point fixed relatively to the body, whatever be its position relative
to the earth or other attracting mass, that point is called the
center of gravity."* The only bodies which have true centers of
gravity are uniform spherical shells, uniform spheres, and spheres
whose density changes from the center to the circumference accord-
ing to some definite law. In the case of other bodies, the line of
action of the weight of the body does not pass through the same
point when the position of the body is changed. In engineering,
however, it is customary to assume that there is a definite point
* Thompson and Tait, Natural Philosophy, II, p. 78.
22 PRINCIPLES OF ELEMENTARY DYNAMICS
fixed in the body at which the weight of the body is applied. Thus,
it is assumed that every body has a center of gravity coincident
with the point at which the resultant of the gravitational forces
acting on the body would be applied if they were parallel.
If a body or system of bodies be conceived to be divided into
particles of equal mass, then that point, the distance of which from
any given plane is equal to the average distance from that plane of
all the constituent particles, is termed the center of mass, or center
of inertia, of the body or system of bodies.
If we represent by mi, m^, etc., the masses of the particles com-
posing a body of total mass M, and by x\, #2, etc., the respective
distances of these particles from any given plane, then it can be
shown that the distance of the center of mass of the body from the
given plane is , , ,
etc.
The center of mass has the following properties :
(a) The center of mass is coincident with the centroid.
(b) The weight of a body acts approximately at its center of mass.
(c) If the resultant of all the forces acting on a rigid body is a
single force the line of action of which passes through the center of
mass of the body, the motion of the body is without angular
acceleration.
(d) The center of mass of a body is so situated that the linear
motion of the body will not be changed if the total mass were
concentrated at this point and all the forces acting on the body
were transferred to this point without change of magnitude or
direction.
(e) The motion of the center of mass of any material system is
not affected by the internal forces between the parts of the system,
but only by external forces.
(/) No material system can of itself, without the action of ex-
ternal forces, change the motion of its center of mass.
(g) An unconstrained body, when acted upon by a system of
forces equivalent to a couple, will rotate about its center of mass
with constant angular acceleration.
2. Simple Harmonic Motion
18. Simple Harmonic Motion of Translation. When a body
moves with a reciprocating motion which has at every instant a
linear acceleration directed toward the center of the path and that
SIMPLE HARMONIC MOTION
23
varies directly with the distance of the moving body from that
point, the body is said to have a simple harmonic motion of trans-
lation.
The defining equation is
a = -cd (30)
where c is a positive constant. The negative sign indicates that
displacement d is measured from the equilibrium position whereas
acceleration a is directed toward the equilibrium position.
Any elastic body that is distorted within the limit for which
Hooke's Law is true, and then released, will thereafter vibrate
with simple harmonic motion of translation. The motions of
many other bodies are resultants of simple harmonic motions.
The time which elapses between two consecutive passages of the
oscillating body in the same direction through any given point of
its path is called the period of the motion. The maximum dis-
tance attained by the oscillating body from its position of equilib-
rium is called the amplitude of the motion.
It will now be shown that if a point moves with uniform speed
in the circumference of a circle, the projection of the point on any
straight line in the plane of the circle moves with simple harmonic
motion of translation. This fact is the basis of our most easy meth-
ods for finding values of the period of a simple harmonic motion of
translation as well as values for the displacement, acceleration and
velocity of a body vibrating with simple harmonic motion.
Let a particle P', Fig. 16, move with uniform speed in the
circumference of a circle P' A'B' y of radius r, and let P be the pro-
jection of this point on any right line
A B in the plane of the circle. As P f
moves with uniform speed in the cir-
cumference of the circle, its projection
P oscillates back and forth through a
middle position C between two extreme
positions A and B. The sort of mo-
tion described by P along the line A B
will now be investigated.
As the particle P' moves with uni-
form speed in the circumference of a jp IG
circle there is a constant acceleration
directed toward the center of the circle. In Fig. 16, this radial
acceleration a' is represented by the line P'Q. Let the compo-
24 PRINCIPLES OF ELEMENTARY DYNAMICS
nent, in the direction of the line AB, of the radial acceleration
a! be called a. This a is the acceleration of the point P. Let d
represent the displacement of P from the point C. We shall
count d positive when above C and negative when below. We
shall measure r outward from 0.
From the similar triangles P'QQi and P'OOi we have
a __ d
a! r
Representing the constant linear speed and angular speed of P f
by v and w, respectively, we have
a f ; and v = wr
r '
Eliminating a f and v from these three equations, we get
a = -w^d (31)
Since w is constant, it follows from this equation that the ac-
celeration of P is proportional to its distance from the center of
its path and is opposite in sign. That is, if a point moves with
uniform speed in the circumference of a circle, the projection of
the point on any straight line in the plane of the circle moves with
simple harmonic motion of translation.
19. The Period of a Simple Harmonic Motion of Translation.
The theorem we have just proved will now be used for the deter-
mination of the value of the constant c in the defining equation of
simple harmonic motion (30).
A comparison of (30) and (31) shows that c = w 2 . If the time
of one revolution of P', that is, the time of one complete vibration
of P, be denoted by T, we have
w = -j=j radians per unit of time.
Hence, if a body of mass m is moving with simple harmonic
motion of period T, then when the body is at a distance d from the
middle of its path, the acceleration is directed toward the middle
of the path and has the value
a[=- w *d]= -fe
SIMPLE HARMONIC MOTION 25
This body is urged toward the middle of the path by a force
(33)
It follows that the period of a simple harmonic motion of trans-
lation is
T = 2ir\/-~ (34)
the minus sign indicating that the displacement and the accelera-
tion are in opposite directions.
20. Simple Harmonic Motion of Rotation. When a body ro-
tates back and forth with a motion such that the angular accelera-
tion is always directed toward a position of equilibrium and is
always proportional to the angular displacement of the body from
that position, the body is said to have a simple harmonic motion
of rotation. The defining equation of simple harmonic motion of
rotation is , , .
a = b<t> (35)
where b is a positive constant.
When a body is vibrating about any axis with simple harmonic
motion of rotation, any point on a line fixed in the body, and not
on the axis of vibration, moves back and forth in the arc of a circle
with a linear acceleration of which the component tangent to the
arc has a magnitude which is proportional to the linear displace-
ment of the point, measured along the arc, from the position of
equilibrium. Hence the value of the period of a simple harmonic
motion of rotation as well as the value of the angular speed, ac-
celeration, and displacement at any time can be obtained from the
values of the period, linear speed, acceleration and displacement
of a point on a line fixed in the body.
Angular Acceleration and Period. Substituting in (32) values
of linear acceleration and linear displacement in terms of the cor-
responding angular quantities, (17) and (7), we find that
(36)
Solving for T and substituting for a its value from (19)
f (37)
where L is the torque acting upon a body of moment of inertia K
with respect to the axis of vibration, at the instant when the angu-
26
PRINCIPLES OF ELEMENTARY DYNAMICS
lar displacement of the body from the equilibrium position is 4>.
The symbol S represents the torsional stiffness, [L/<t>], of the body.
Angular Velocity. Figure 17 represents a body X that is vi-
brating with simple harmonic motion of rotation through an
amplitude <i> about an axis normal to the plane of the diagram
through the point 0. The point p of a line op, fixed in the body,
moves back and forth with simple harmonic motion of translation
in the arc A B. The distance A B is shown as a straight line A'B'
FIG. 17
FIG. 18
in Fig. 18. The simple harmonic motion of translation along
A'B' is also the motion of the projection P' of a point P that
moves with uniform speed in the circumference of a circle of diam-
eter equal to A'B f and in the same period as the simple harmonic
motion of rotation of the body X.
Suppose that the point P moves with uniform speed v e around
the circle and that in time t after passing the position Z), the radius
O'P has moved through an angle 6. At this instant, the component
velocity in the direction A 'B f , that is, the velocity of the projected
point P', has the magnitude
v t = v e cos
In Fig. 18, the linear velocity of the point P' when at the equi-
librium position is v e , and the value at t seconds later when the body
X has rotated through an angle <, is v t . Dividing each member of
this equation by the distance R of the point p from the axis of
vibration 0,
or, (16):
s-i
Wt W e COS
(38)
SIMPLE HARMONIC MOTION 27
where w e and Wt represent, respectively, the angular velocities of
the body X when moving through the equilibrium position, and
t seconds later.
In Fig. 18 the angle 6 equals the product of the angular velocity
of tha line O'P and the time t occupied in moving through this
angle. If time T is occupied in making one entire revolution about
0', then the angular velocity of O'P has the magnitude 2ir/T.
Hence, the angle
0= ^radians (39)
and (38) becomes
, A A ,
W t = W e COS - (40)
In this equation time is reckoned from the equilibrium position.
If it be reckoned from one end of the oscillation, that is ir/2 radi-
ans from the equilibrium position, we would have
. 2irt
= w e cos
(2irt TT\
I ~Y - 2 ) =
The maximum angular displacement from the equilibrium po-
sition is called the amplitude of the vibration. Another useful
formula for the angular velocity is one involving the period T,
angular amplitude <f> and angular displacement </>. From Fig. 18,
the linear velocity of a body moving with simple harmonic motion
of translation of period T and amplitude r at the instant when the
displacement from the equilibrium position is d has the value
r .. 2irr(O f O"\ 2ir A /~ 7 2 />IO v
t;/[= v e cos 6] = -y- \ f r)= -yry r - d ( 42 )
From Fig. 17, (7), and (16)
r = 3>R, d = <t>R and v t = w t R (43)
Substituting these values in (42) we find
'~* f (44)
At the equilibrium position, < = 0. When at this angular dis-
placement, the velocity w e has the magnitude, (44):
We ~ ~~m * radians per sec. (45)
28 PRINCIPLES OF ELEMENTARY DYNAMICS
If the angular amplitude, expressed in degrees, be represented by
$, (45) gives
2 7T $> $>
We = T 67~3 rad ' per SeC ' = 912 7 rad ' per sec ' ^ 46 ^
Angular Displacement. If a body is vibrating with simple
harmonic motion of translation of period T and amplitude r, the
linear displacement from the equilibrium position t seconds after
traversing the equilibrium position has the value, Fig. 18:
7 /> . ir TT ir /At7 .
d = r sm = r sin -yr = r cos! ^ -- y- I (47)
From (43), we find that when a body has a simple harmonic motion
of rotation with period T and amplitude <$>, the angular displace-
ment from its equilibrium position at time t is given by the ex-
pression
^ * * ^ 2?r ^ ^ (* 2 *A /MQN
= $ sm 6 = $ sm-y- = $ cosl 2 -- jr ) (48)
21. The Physical Pendulum. A compound or physical pen-
dulum consists of a suspended rigid body free to oscillate about a
horizontal axis. Consider the physical pendulum
AC (Fig. 19), consisting of a body of mass m sup-
ported on an axis normal to the plane of the dia-
gram and passing through the point A. Let the
center of gravity of the pendulum be at C. De-
note the distance AC by I. If the pendulum be
deflected from its equilibrium position through
an angle <, it will be acted upon by a torque which
tends to restore it to the equilibrium position and
which has a value
L = mg(BC) = mgl sin <j>
the negative sign indicating that the direction of the torque is
opposite that of the displacement.
If the displacement from the equilibrium position is small, sin <
is approximately equal to radians. In this case the above equa-
tion becomes
L = -mgl<t> (49)
Whence, at any instant, a pendulum displaced but a small dis-
tance from its equilibrium position is urged toward its equilibrium
SIMPLE HARMONIC MOTION 29
position by a torque nearly proportional to its angular displace-
ment from that position. Consequently, the angular motion of
such a pendulum is approximately simple harmonic motion of
rotation.
The period of vibration of a physical pendulum oscillating
through a small amplitude now will be determined. Substitu-
ting in the equation for the period of vibration of a simple har-
monic motion of rotation, (37), the value of the torque acting on a
compound pendulum displaced through a small angle from its
position of equilibrium, (49), we obtain for the value of the time
occupied by one complete vibration of a compound pendulum:
!F = 2ir\/ , (50)
V mgl
From this equation it is seen that when the amplitude of vibra-
tioix of a compound pendulum is so small that sin <t> may be replaced
by 0, the period of vibration of the pendulum is practically inde-
pendent of the amplitude of swing.
A heavy particle suspended by a string which is both inexten-
sible and massless, and which is capable of swinging in a vertical
plane, is called a simple or mathematical pendulum. Since the
moment of inertia with respect to the axis of oscillation of a simple
pendulum of length I and mass m is (21) :
K = mi 2
The period of vibration of a simple pendulum is (50) :
(51)
A pendulum consisting of a small spherical bob supported by a
very thin light inextensible string approximates closely to a
simple pendulum.
A rigid body free to oscillate about a horizontal axis that is
below the center of mass of the body is called an inverted pendulum.
One type of seismograph for recording earth tremors consists of
an inverted pendulum having the upper end normally held in
position by a set of springs.
22. The Conical Pendulum. If a body, suspended at a point
not coincident with the center of mass of the body, be given an
impulse directed to one side of the vertical line through the point
of support, each line of the body passing through the point of
30
PRINCIPLES OF ELEMENTARY DYNAMICS
support will sweep out a conical surface. The body will move
around and around with a definite period. A suspended body
which, owing to its weight, is capable of rotating in such a manner
that the line through the point of suspension and the center of
mass has constant angular velocity about the vertical line through
the point of suspension is called a conical pendulum.
An expression for the period of a conical pendulum now is to be
determined. In Fig. 20, a body of mass m with center of mass at
B is supported without friction at
A. When set into motion, the
body rotates so that the center of
mass traces a circle of radius r
about a center C vertically below
the point of suspension A.
The resultant force acting on
the body is the centripetal force
directed toward the center of the
circle. This is due to the weight
mg of the body, and the reaction
F of the support. Representing
FlG 20 *he distance AB by Z, the angle
BAG by 0, the linear speed of the
center of mass in the horizontal circle by v, and the period of a
revolution by T, the value of the centripetal force is
F c = [F sin <t>] =
Also,
F cos <[> = mg
Dividing each member of the former equation by the correspond-
ing member of the latter, we have:
sin </> __ 4 ?T 2 r __ 4 ir 2 1 sin
coM> T 2 g ~"
Whence, the period
ri cos <t>
~
(52)
23. Wave Motions and Wave Forms. A motion that goes
through the same series of changes at regularly recurring intervals
is called periodic. A periodic disturbance which is handed on
SIMPLE HARMONIC MOTION
31
successively from one portion of a medium to another is called a
wave motion.
It is convenient to represent graphically a periodic function by
a curve coordinating the function and time. Thus, if we plot
instantaneous values of the electromotive force in an alternating
current circuit against time, we obtain what is called the wave
form of the periodic alternating current. If we plot a ship's
instantaneous angle of roll against time we obtain the wave form
of the roll. If the quantity varies as the sine or the cosine of an
angle, then the wave form is called a sine curve, or a cosine curve,
respectively. Either a sine curve or a cosine curve is a harmonic
curve or harmonic wave form.
The value of the angular displacement from the equilibrium
position of a body vibrating with simple harmonic motion of ro-
tation is given by (48), where T is the period of the vibration, $
d
Y d' d"
^\P 'f
c
b
b'
S
c'
N
\f
by
s
c"
^-^1 9
a y-
/
\
/
X'
A'
B'
C r
tf
E'
F'
H'
/'
J'
K'
L'J
A"
B"
/*"
0"
i-f h
/
\
?
S
^J Js/ i
k
X
^^
^
S
J
Y' J
FIG. 21
is the amplitude, and </> is the angular displacement from the equi-
librium position at time t. The angular displacements made in
succeeding equal time intervals are proportional to the distances
ab, be, cdj de, ef, fg, etc., Fig. 21. To obtain these unequal dis-
tances, project onto the line YY' the positions at the end of equal
time intervals of a point P moving with uniform speed in the cir-
cumference of a circle (Art. 18). In the case here represented, the
circle, Fig. 21, is divided into twelve equal arcs, AB, BC, CD, etc.
On the line XX' mark thirteen points A', B', C', etc., at equal
distances apart. Through each of these points draw a line per-
pendicular to XX'. Draw a line parallel to XX' from each of the
points marked on the line YY'. Place a dot a' at the intersection
of the horizontal line through a and the vertical line through A'.
Similarly, place a dot b' at the intersection of the horizontal line
through 6 and the vertical line through 5', and so on for the re-
mainder of the intersections c', d', etc. The smooth curve drawn
through these dots is the wave form of the simple harmonic mo-
tion represented by (48). The period T is the time required to
32 PRINCIPLES OF ELEMENTARY DYNAMICS
T
make one complete vibration. At the time t = ^= 9 the angular
l^J
displacement < is proportional to the distance J3'6'. The angular
amplitude <f> is proportional to the distance D'd'.
The wave form due to a simple harmonic vibration is a sine curve
(or cosine curve).
24. Phase and Phase Angle. In the case of vibrations along a
horizontal line, forces, velocities, currents, etc., directed to the
right are usually termed positive; and displacements to the right
of the equilibrium point are termed positive. In the case of
vibrations along a vertical line, forces, currents, etc., directed
upward are usually termed positive; and displacements above the
equilibrium point are called positive. The phase of any periodi-
cally changing function is the fraction of a whole period of vi-
bration which has elapsed since the particular function last passed
the equilibrium position in the positive direction. Thus, if, in
Fig. 21, a body vibrates along the line YY' with simple harmonic
motion, and if, at intervals of one-twelfth of the period of vibra-
tion, the body is successively in the posit ions a, 6, c, rf, e, f, g, h, i,
j, k, I, a, etc., the phase of the displacement of the body when at
a is zero; when at c and moving in the positive direction, the phase
is J ; when at d, the phase is J ; when at c and moving in the nega-
tive direction the phase is 3 ; when at a and moving in the negative
direction it is J> etc.
If a body is rotating in a circle, the angular displacement of the
body from some reference position is called the phase angle of the
motion. Thus, in Fig. 21, when the point P is at B, the phase of
the displacement is ^ and its phase angle is radians, or 30; when
at D the phase is J and its phase angle is ~ radians, or 90. Since
2t
the motion on a straight line of the projection of a point that is
moving with uniform speed in the circumference of a circle is a
simple harmonic motion of translation, we may express phase in
terms of the phase angle of a point that is moving in the circum-
ference of a circle. Thus, in Fig. 21, when the body is at 6, and
moving in the positive direction, the phase of the motion is ^ and
7T
its phase angle is ~ radians or 30. In the equations of Art. 20,
the angle 9 = ~~- represents the phase angle of the vibration
at the time t.
SIMPLE HARMONIC MOTION
33
Two motions having phase angles that differ by zero or by any
integral multiple of 2 w radians are said to be in the same phase.
Two motions having phase angles that differ by TT radians are said
to be in opposite phases. In Fig. 21, the properties corresponding
to V and V are in the same phase, and those corresponding to d'
and d" are in the same phase. The properties corresponding to d'
and j are in opposite phase.
25. The Mean Value of the Product of Two Simple Harmonic
Functions of Equal Period. There are many cases in which it is
necessary to know the moan value of the product of two quantities
that are varying periodically. If, at any instant, a body rotating
with an angular speed w exerts a torque L, the power developed
at that instant equals wL. If the torque and the angular speed
are varying periodically, the mean power developed during one
cycle equals the average of a series of products of w and L during
the cycle.
Similarly, the mean power associated with an alternating electric
current during one cycle equals the average of a series of products
of instantaneous values of electromotive force and current during
the cycle.
The most important case is that in which the two periodic
quantities vary harmonically and are of the same period. Assume
two simple harmonic motions represented, respectively, by the
equations
d l = n sin and d 2 = r 2 sin (0 + 0)
where r\ and r 2 are the amplitudes of vibration, d\ and rf 2 are the
displacements from the equilibrium position, and 9 and (0 + 0)
" >^ y\
s
s
/
\
\
^^yA
lS\9\
/
\
s
1
/
/
\
\
"""^ "***^^ /
\
\
s
^
gj
I
1
\
\
s.
' ^S ^
S
^
-
/
\
i-<f!
Fia. 22
are the phase angles at the time t. Since the periods are the same,
is independent of time. In Fig. 22, the displacements of the two
vibrations at the end of a series of equal time intervals are in-
dicated by dots on the lines yy' and YY' 9 respectively. The wave
34 PRINCIPLES OF ELEMENTARY DYNAMICS
forms of the two vibrations are sine curves as shown. The product
of the two functions at any particular instant is*
did* = nr 2 sin sin (0 + 0)= ~f 2 [cos ft - cos (2 6 + 0)] (53)
^!
Since
it follows that during one-half cycle, that is, while t changes to
t + \ T, and becomes (0 + TT), the angle (2 + 0) becomes
2 ( ,- + TT) + ft. Whence, the change of argument of the last
cosine in (53) is 2 ir radians. The argument of the cosine is a
linear function of time. During this change of angle, the cosine
goes through all values, positive and negative, and the mean value
of cos (2 + ft) during any half period is zero. Consequently,
the mean value of the product of two simple harmonic functions of
equal period and of amplitudes r\ and r 2 is
d\d z = rir 2 cos ft (54)
where ft is the difference in phase of the two functions.
For example, if the two harmonically varying quantities rfi and
d 2 represent instantaneous values of electric current and electro-
motive force, respectively, then the power of an alternating cur-
rent during any half period is given by (54), where r and r z repre-
sent the maximum values of the current and electromotive force,
respectively, during the half period, and ft represents the difference
of phase between current and electromotive force. Or, if the two
quantities d\ and d 2 represent instantaneous values of angular
speed and torque, respectively, then the power transmitted to the
oscillating body during any integral number of half periods is
given by (54) where 7*1 and 7* 2 represent the maximum values of
the angular speed and torque, respectively, and ft represents the
difference of phase between angular speed and torque.
Equation (54) shows that:
(a) When two harmonic functions, di and d 2 , of equal period are
in the same phase, that is, when ft = 0, the mean value of the prod-
uct did<2 is maximum and equals one-half of the product of the
amplitudes of the two components.
* From the trigonometric relation :
sin x sin y = J [cos (x y) cos (x + y)]
SIMPLE HARMONIC MOTION 35
(6) When the two components are in quadrature, that is, when
the phase difference ft equals 90 or ~ radians, the mean value of
the product d\d 2 equals zero.
_ (c) When ft = w radians or 180, the mean value of the product
d\d 2 equals one-half of the product of the amplitudes of the two
components but with the negative sign.
These three cases are represented graphically in Figs. 23, 24 and
FIG. 24
25, respectively. In each figure, the harmonic curves shown in
full lines represent the equations
= TI sin -
and
, .
d 2 = r 2 sin
(+')
where ft is the difference in phase. In Fig. 23, the difference in
phase = 0; in Fig. 24, = ** radians or 90; in Fig. 25, ft = TT
radians or 180. In each figure, the or-
dinate of the dashed curve correspond-
ing to any particular instant equals
the product of the ordinates of the
two harmonic curves at that instant.
Thus, the dashed curves represent the
variation of the product, at every in-
stant, of di and d 2 . In case d\ and d 2
represent either electromotive force
and current or torque and angular
velocity, then the product is power. The dashed curve then shows
the variation of power with respect to time. The area between
FIG. 25
36 PRINCIPLES OF ELEMENTARY DYNAMICS
the power curve and the axis, for any time interval, is propor-
tional to the work done during the given interval. An area above
the axis represents positive work, that is work done by the elec-
tromotive force or by the torque; an area below the axis repre-
sents negative work, that is work done against the electromotive
force or against the applied torque. When the two harmonic
curves are of equal period and in quadrature, that is, when they
differ in phase by radians or 90, Fig. 24, the total work done
Zi
during one cycle is zero.
The quotient obtained by dividing the area under the power
curvo for one-half cycle, by the distance between the intersections
of this curve with the axis of the curve, equals the mean value of
the power during the half cycle. When the two curves are in
quadrature, Fig. 24, the mean power during a complete cycle is zero.
The phase difference between the current and the electromotive
force in an alternating current circuit can be altered by changing
either the capacitance or the inductance of the circuit. If the
current and the electromotive force in a circuit containing an
alternating current generator and motor are in the same phase,
then the generator is putting electric energy into the line and the
motor is transforming a maximum fraction of it into mechanical
work. If the current and electromotive force differ in phase by
one-quarter period, then we have a " wattless current " and the
work done during one complete cycle is zero. If the current and the
electromotive force are in opposite phase, then the motor is opera-
ting as a generator, opposing the electromotive force of the gen-
erator.
26. Oscillations of a Coupled System. Resonance. The
vibrations executed by a body which has been displaced from its
position of equilibrium and then released are called free vibrations.
The vibrations executed under the action of an external periodi-
cally varying force are called forced vibrations.
If the point of support of a vibrating pendulum is in periodic
motion, as when a pendulum is attached to another pendulum, or
when a pendulum is on a ship which is rolling with a constant
period, the pendulum will acquire a compound periodic motion
having two components of different periods. These periods will
be different from the periods of the freely vibrating pendulum or
of the freely vibrating support, except when the inertia of the
pendulum is negligible compared with the inertia of the support.
SIMPLE HARMONIC MOTION 37
The amplitude of the component oscillation of shorter period will
be smaller than the amplitude of the component oscillation of
longer period.*
Consider a group of pendulums hung from a horizontal rod
supported as shown in Fig. 26. Suppose that the pendulum A is
set swinging. When it is out on one side of its path it urges the
supporting rod out toward that same i ==- ==
side. When it is out on the other I T
side of its path it urges the support
out toward that side. The force act-
ing on the support varies harmoni-
cally. Soon the support and the
other pendulums are set into forced
vibration of nearly the same fre-
quency as the free vibration fre- C
qucncy of the exciting pendulum A . FlG - ^
If the length of the pendulum A is so much greater than the
lengths of the other pendulums that the frequency of the forced
vibration is much less than the natural vibration frequency of
these pendulums when vibrating freely, [hen they will vibrate with
the frequency of the forced vibration. If the frequency of the
forced vibration be increased by shortening the length of the
exciting pendulum A, for example, the amplitude of the vibration
of the other pendulums will increase. If the frequency of the
forced vibration equals the natural frequency of some pendulum
I), then this pendulum will be acted upon by the maximum force
in the direction of its motion when it is moving through its equi-
librium position. Consequently, the amplitude of vibration of D
will become large. The vibration produced when a periodically
varying force acts upon a body of nearly the same frequency of
vibration is called resonant forced vibration. The phenomenon
of the production of resonant forced vibrations is called resonance.
The body that excites the resonance is called the exciter, and the
one that is set into resonant forced vibration is called the resonator.
The frequency of the external force that produces maximum reso-
nance is called the critical frequency.
Resonance is a very important phenomenon in mechanics as
well as in electricity, sound and light. Some of the more important
aspects of the phenomenon that relate to mechanics are broadly
indicated in the present considerations.
* Edser, General Physics for Students, pp. 154-167.
38 PRINCIPLES OF ELEMENTARY DYNAMICS
The vibrations forced on the resonator lag behind the exciting
force. When the frequencies of the exciter and the resonator are
equal, the phase of the vibrations of the resonator are one-quarter
period behind the vibrations of the exciter, and the amplitudes of
the vibrations of the resonator are maximum. That is the reso-
nator is vibrating through the maximum amplitude when the
exciting force is zero. When the frequencies of the two vibrations
are nearly equal, a small change in the frequency of cither pro-
duces a considerable variation in the phase difference unless the
damping of the resonant vibrations is great.*
Energy is alternately introduced into and withdrawn from the
resonator by the exciter but the energy absorbed by the resonator
is greater than the energy omitted. The energy absorbed is
maximum when the frequency of the resonator is the same as that
of the exciter.
Consider the case in which the natural frequency of vibration of
D and the frequency of the forced vibration impressed upon it are
nearly equal. Suppose that one frequency is /h and the other is
/i 2 . Then, during this time interval, the two vibrations are in
phase (//i riv) times. That is, during this time interval, I ho
amplitude of vibration of the resultant motion will rise to a maxi-
mum and fall to a minimum (n\ n%) times. This is called the
phenomenon of beatx. If the pendulum A is set into vibration
when the periods of A and D are slightly different, then the ampli-
tude of vibration of D will increase to a maximum, fall to a mini-
mum, and repeat. As the energy increasing the amplitude of the
vibrations of D comes from the pendulum A, the amplitude of the
vibrations of the latter is minimum when the amplitude of D is
maximum. Energy of D is now imparted to A thereby increasing
the amplitude of its vibrations. Often the frequency of a ship's
roll is nearly equal to the frequency of the waves. When this
occurs, the amplitudes of successive rolls will become larger and
larger to a maximum, thereafter become smaller and smaller to a
minimum, and then repeat. As the frequency with which the
waves go under the ship depends upon the speed and course of the
ship relative to the direction of the waves, the amplitude of roll
can be altered by changing either the speed or the direction of the
ship.
When the frequency of the forced vibration is much greater
than the natural frequency of the body upon which it is impressed,
* Timoshenko, Vibration Problems in Engineering.
SIMPLE HARMONIC MOTION 39
the effect upon the latter is very small. A seismograph for record-
ing vibrations of the earth has a pendulum of such small frequency
compared to the frequency of the tremors to be measured that it
is not set into motion by rapid earthquake vibrations. The
instrument records displacements of the earth relative to a pendu-
lum of great moment of inertia and long period.
In this Article the case has been considered in which the periodic
force varies harmonically, that is, in which the force is always
directed toward the equilibrium position of the body and has a
magnitude directly proportional to the displacement of the body
from the equilibrium position. It should be noted, however, that
if, instead of varying harmonically, the external force starts and
stops suddenly but in a periodic manner, it will also set into reso-
nant vibration a body of a free period that is a submultiple of the
period of the external force. In this Article it has born assumed
that the frequency of the exciter is unaffected by the resonator.
The frequency of the exciter will be somewhat modified if either
the stiffness or mass of the resonator be not loss than that for the
exciter.
27. Damping of Vibrations. Whon linear vibrations arc
opposed by a force, or angular vibrations are opposed by a torque,
the amplitude of each swing becomes loss than the amplitude of
the preceding swing. The successive diminution of the ampli-
tudes of swing is called damping. If the body vibrating with
simple harmonic motion of rotation bo acted upon by a damping
torque which at each instant is proportional to the angular speed
of the body at that instant, then at all ordinary speeds the ratio
of the amplitudes of any two successive swings will be constant.
In this case the damping is constant.
Figures 27 to 31 represent a pendulous system consisting of a
weighted bicycle wheel to which is rigidly attached a pair of
connected tanks and partly filled with water or other liquid.
The wheel is capable of rotation with negligible friction about a
horizontal axle through the center. To produce a pendulum of
long period, the moment of inertia with respect to the axis of oscil-
lation is made large, and the torque about the axis of oscillation is
made small, (50). In the present piece of apparatus, the moment
of inertia is made large by replacing the ordinary tire of the bicycle
wheel by a coil of wire of some twenty-five pounds. A small
torque is produced by attaching a mass of a few ounces at a short
distance below the axis of oscillation.
40
PRINCIPLES OF ELEMENTARY DYNAMICS
If the pendulous system be tilted, and held in this tilted position,
liquid will flow from the higher tank to the lower, and thereafter
will flow back and forth several times with a definite period. If
the pendulous system be set into oscillation, and the frequency of
this oscillation be nearly the same as that of the liquid, then by the
time that the system is in its equilibrium position, there will be
an excess of liquid in the reservoir which was the lower when the
system was displaced from
the equilibrium position.
Thus, there will be a differ-
ence in phase between the
oscillation of the pendulous
system and the oscillation
of the liquid in the tanks.
The period of oscillation
of the liquid, as well as the
phase difference between
the oscillations of the liquid
and of the pendulous sys-
tem, can be regulated by
adjusting a valve V in either
of the tubes that connect
the two tanks. If the period
of flow of liquid back and
forth equals the period of
the pendulous system, and
if the velocity of the liquid is a quarter period ahead of that of
the pendulous system, then the following effects will occur.
Consider the action at an instant when the pendulous system has
a large deflection, and there are equal amounts of liquid in the two
tanks, Fig. 28. While the pendulous system is moving from this
position to the equilibrium position, an excess of liquid rises in the
tank that was the lower, the excess being maximum when the
pendulous system is in the equilibrium position, Fig. 29. During
this quarter period, the distance between the center of mass of the
pendulous system and the vertical line through the axis of vibra-
tion is being diminished. Hence, the torque urging the pendulous
system toward the equilibrium position is being diminished faster
than it would be if the liquid were not flowing.
While the pendulous system is moving from the equilibrium
position, Fig. 29, to the end of the swing, Fig. 30, the excess of
FIG. 27
SIMPLE HARMONIC MOTION
41
liquid in the tank on the advancing side of the pendulous system
produces a torque that opposes the motion of the pendulous system.
By the time that the pendulous system again has attained its
equilibrium position, a maximum excess of liquid has flowed into
the tank A, Fig. 31, which was the lower in Fig. 30. During mo-
tion, the torque urging the pendulous system towards the equi-
librium position is being diminished faster than it would be if the
FIG. 28
FIG. 29
FIG. :JO
FIG. :u
liquid were not flowing. Consequently, the amplitude of swing
is less than it would be if the liquid were not flowing.
Thus, the amplitude of vibration of the pendulous system is
being damped throughout the half cycle while the system is moving
from its equilibrium position to the end of its path and back.
During the next half cycle the amplitude of vibration is damped
in the same way. The damping of the amplitude of vibration
involves abstraction of energy from the vibrating pendulous sys-
tem. This energy is transformed into heat as the liquid and air
42 PRINCIPLES OF ELEMENTARY DYNAMICS
surge back and forth through the constricted passages connecting
the two tanks.
For the maximum rate of damping, the periods of the pendulous
system and of the oscillating liquid should be equal, and the
angular velocity of the pendulum and the torque acting on the
pendulum due to the oscillating liquid should differ in phase by
one-half period (Art. 25). These factors can be controlled within
certain limits by the valve V.
This method has been used for damping the roll of ships (Art.
28), and is used for damping the vibration of certain gyro-com-
passes (Art. 137).
28. The Frahm Anti-Roll Tanks. The rolling of a ship by
periodic waves can be diminished through resonance by an oscil-
lating mass of water surging back and forth from a tank on one
side of the ship to a tank on the other side. The frequencies of
sea waves vary from about five to eight per minute in deep water,
and from about eight to eleven in shallow water. The period at
any particular place does not remain constant for many minutes
in succession. The phase of the wave motion changes whenever
a wave crest topples over.
The Frahm anti-rolling device* consists of closed tanks placed
opposite to one another at the two sides of the ship. The tanks
are about half filled with water. In the earlier equipments de-
signed by Frahm, the tanks were placed inside the ship, the
lower parts of opposite tanks were joined by a water passage and
the upper parts joined by an air passage controlled by a valve.
In later equipments, the tanks have been placed on the exterior
of the hull like the " blisters " commonly used on warships
as a protection against torpedoes. The Hamburg- America line
steamers Albert Ballin, Deutschland, Hamburg and New York
are provided with such blisters extending above and below the
water line along about two-rthirds the length of the hull. The
blister on each side is divided into three parts by vertical parti-
tions. The upper parts of opposite tanks are joined by air pipes
provided with adjustable valves. The lower parts of opposite
tanks are not .joined by water pipes but open into the sea, Fig. 32.
The period of the water surging back and forth from one side to
the other can be adjusted within certain limits by adjusting the
air valves.
When the period of the waves is about the same as that of the
* U. S. Patent. Frahm, No. 970368, 1910; Frahm, No. 1007348, 1911.
SIMPLE HARMONIC MOTION
43
ship, the amplitude of roll of the ship will grow larger and larger
through resonance. It is the purpose of the tanks to prevent such
a building up of the roll amplitude. Suppose that the period of
oscillation of the water back and forth from one tank to the other
is about the same as the period of roll of the ship and that the ve-
locity of the water surging back and forth from one tank to the
other is one-half period behind the torque acting on the ship due
Outer
Opening
Outer
Opening
FIG. 32
to the waves. Under these conditions the surging water will
absorb energy from the rolling ship at the maximum rate (Art.
25). The required adjustment of period of surge of the water in
the tanks and of the phase difference between the angular velocity
of the ship and the torque acting on the ship, can be made by a
man operating the valve in the pipe connecting the tanks. The
maintenance of the proper adjustment is rendered very difficult
by the two facts that the phase of the torque acting on the ship is
changed when an oncoming wave topples over, and that the period
[>f the waves is constant for but short spaces of time.
It has been proposed to use a gyroscope to operate this valve
automatically.*
The weight of the Frahm anti-rolling tanks and contents is from
about three to five per cent of the weight of the ship, depending
upon the constants of the ship, f
* Hammond, No. 1700406, 1920.
f Robb, Studies in Naval Architecture, pp. 289-304.
CHAPTER II
THE MOTION OF A SPINNING BODY UNDER THE
ACTION OF A TORQUE
29. Degrees of Freedom. A locomotive, having a motion
limited to motion back and forth along a straight track, is said to
have one degree of freedom of translation. A block of ice on the
surface of a frozen lake has two degrees of freedom of translation.
A bird in the air can move in any direction and is said to have
three degrees of freedom of translation or to be unconstrained with
respect to linear motion.
A body that is rotating about an axis through the center of mass
of the body is said to be spinning. The flywheel of a stationary
engine is spinning about a fixed horizontal axis and is said to have
one degree of freedom of rotation. If the engine were on a turn-
table capable of rotating about an axis perpendicular to the spin-
axis, then the flywheel would be said to have two degrees of freedom
of rotation. If the flywheel were mounted so that the spin-axis
could rotate about two axes perpendicular to one another and also
to the spin-axis, then the wheel would be said to
have three degrees of freedom of rotation or to
be unconstrained with respect to rotation. A
wheel mounted so as to be capable of rotation
about five intersecting axes, Fig. 33, has three
degrees of freedom of rotation, whatever may be
the directions of the axes. A wheel mounted so
as to be capable of rotation about three inter-
secting axes has three degrees of freedom of rota-
tion except when all three axes are in the same
plane, Fig. 34. When all three axes of the mount-
ing are in the plane of the spin-axle, the wheel
cannot rotate about an axis perpendicular to that plane.
A ship's compass should stand upright however the ship may
either roll or pitch. The same is true of the oil lamps carried by
Sailing vessels. Each should be free to rotate relative to the ship
about an athwartship axis and also about a fore-and-aft axis, that is,
should have two degrees of rotational freedom about two perpendic-
44
MOTION OF A SPINNING BODY 45
ular horizontal axes. This result is attained by supporting the ap-
paratus pendulously by two knife-edges on a ring which in turn is
supported on two other knife-edges. The axes of the knife-edges
intersect at right angles to one another. Such supporting rings
are called gimbal rings or gimbals. The system of mounting is
often called Cardan's suspension and is in com-
mon use when a device must be supported so
as to have two degrees of rotational freedom.
The two inner rings of Fig. 34 constitute a
Cardan suspension for the gyro-wheel. The
gyro has a third degree of rotational freedom
about the spin-axis.
By means of a rapidly spinning wheel mounted
so as to have three degrees of rotational free-
dom, Foucault in 1852 demonstrated the rota-
tion of the earth. Since this apparatus exhibits the rotation of
any body on which it is placed, Foucault called it a " gyroscope."
If the direction of the spin-axle is changed, the moment of in-
ertia of the wheel opposes the angular displacement. When it is
desired to fix the attention on the property of the spinning wheel
to oppose any change in the direction of the spin-axle, the instru-
ment is called a " gyrostat." The spinning body is called the
gyro-wheel or gyro. That part of physics dealing with the laws
of motion of a spinning body under the action of a torque about
an axis inclined to the spin-axle is called gyrodynamics. Any
body or instrument that exhibits the laws of gyrodynamics may
be called a gyroscope. The instruments represented in Figs. 33
and 34 are called Bohnenberger's gyroscope of five frames, and
of three frames, respectively. By means of clamps, rotation of
one or more frames can be prevented, thereby reducing the num-
ber of degrees of rotational freedom of the gyro to two or one.
30. The Effect on the Motion of a Spinning Body Produced by
an External Torque. Experiment. Attach a small mass m to
the inner supporting frame of the gyroscope, Fig. 35, at a point
near one end of the gyro-axle. Set the gyro- wheel spinning in the
direction indicated in the figure by the arrow h s . At first, the
spin-axle will oscillate slightly up and down and also to the right
and left. Wait till the motion is steady and then observe that,
although the weight of m produces a torque about the axis L,
the dip of the gyro-axle about the torque-axis is not conspicuous.
Observe that so long as a constant external torque acts about an
46
MOTION OF A SPINNING BODY
axis perpendicular to the spin-axle, the spin-axle rotates with
constant angular velocity w p about an axis perpendicular to both
th(4 spin-axis and the torque-axis.
An outside torque acting upon an unconstrained spinning sym-
metrical body about the spin-axis changes the magnitude of the
spin-velocity but not the direction of
the spin-axis. An outside torque acting
about an axis perpendicular to the spin-
axis does not change the magnitude of
the spin-velocity but does change the
direction of the spin-axis in the manner
now to be explained. The cause of the
oscillations of the spin-axle before a
steady state is attained wilt be considered
in Art, 39.
We shall use two different methods of
explanation. In the first method, we
shall consider a symmetrical wheel, Fig.
36, spinning with angular velocity w s
about the axle of the wheel, and also
rotating with angular velocity w p about
an axis perpendicular to the spin-axis.
The resultant of these two angular veloci-
ties is an angular velocity represented in magnitude, direction of
axis and sense of rotation by the arrow labeled w r .
Each particle of the wheel is moving in a circular path about
some point on the axis of w r as
a center. In order that this
motion may continue, each par-
ticle must be acted upon by o
force directed toward the center
of its circular path (Art. 9). The
centripetal forces acting on two
particles A and A f situated at
points equally distant from the
axis of the resultant angular ve-
locity w r , are represented by the
arrows / and /'. These two forces constitute a couple acting in
the counter-clockwise direction about an axis perpendicular to the
axes of w s and w p . The sum of the centripetal couples that must
act upon all the particles composing the spinning body consti-
FIG. 35
FIG. 30
MOTION OF A SPINNING BODY
47
tutes a resultant centripetal torque L. A torque of this value
must be applied to the spinning body by an outside agent in order
that the body may rotate with angular velocity w p about an axis
perpendicular to the spin-axis.
If such a centripetal torque be not applied by some outside agent,
the inertia of each particle of the spinning body will cause it to
move tangentially to its circular path and the wheel will not turn
about any axis except the spin-axis. In case the face of the wheel
were not perpendicular to the spin-axis, a torque would act upon
the wheel tending to set the face of the wheel perpendicular to
the spin-axis. The magnitude of this torque equals the centrip-
etal couple required to hold the wheel inclined to the spin-axis.
Any rotating body tends to set
itself so that its axis of maxi-
mum moment of inertia is in
tho direction of the axis of ro-
tation.
In the second method of ex-
planation, we shall assume that
the wheel in Fig. 37 is spinning
at a constant rate in the direc-
tion indicated and, at the same
time, is turning at a constant
rate about the axis JD in the direction of an external torque L
of constant magnitude. Owing to the resultant of these two mo-
tions, particles in the quadrant GAD and in the quadrant OGJ
are approaching the axis JD while particles in the quadrants ODG
and OJA are receding from this axis.
When a particle approaches the axis JD } its moment of inertia
relative to that axis decreases. Since the rate of spin is constant,
the angular speed of each particle about the axis JD increases
when the particle is approaching this axis. The angular motion
about the axis JD of each particle in the quadrant OAD is accel-
erating toward the reader about the axis JD, while the motion of
each particle in the quadrant OGJ is accelerating away from the
reader. Thus, each particle in the quadrant OAD is acted upon
by an external force directed toward the reader, while each particle
in the quadrant OGJ is acted upon by a force directed away from
the reader.
When a particle recedes from the axis JD, its moment of inertia
relative to that axis increases. The rate of spin being constant, the
FIG. 37
48 MOTION OF A SPINNING BODY
angular speed of each particle about the axis JD decreases when the
particle is receding from this axis. The angular motion about the
axis JD of each particle in the quadrant ODG is decelerating away
from the reader (accelerating toward the reader), while the motion
of each particle in the quadrant OJA is accelerating away from the
reader. Thus, each particle in the quadrant ODG is acted upon
by an external force toward the reader, while each particle in the
quadrant OJA is acted upon by a force away from the reader.
The resultant of the forces acting on all the particles in the
quadrant GAD is represented by the line F\j Fig. 37. The result-
ants of the forces acting on the particles in each of the other three
quadrants are represented by the lines of equal length, F 2 , F* and
F^ respectively. These forces tend to turn the spin-axle about an
axis AG in the direction indicated by the arrow-head marked w p .
It has now been shown that whenever an external torque is applied
to a symmetrical unconstrained spinning body, the spin-axis tends to
become parallel to the torque-axis, and with the direction of spin in
the direction of the torque.
If the unconstrained spinning body be acted upon simultane-
ously by more than one torque, then the spin-axis tends to set it-
self parallel to the resultant torque.
In the case of a wheel mounted so that there is a fixed angle
between the spin-axis and the torque-axis, the torque-axis retreats
from the advancing spin-axis. The spin-axis continues to move
in the direction to make it become parallel to the torque-axis, but
it cannot become parallel on account of the rigidity of the con-
nection between the wheel and its supporting frame.
31. Precession. In the preceding Article it has been shown
that when a symmetrical unconstrained spinning body is acted upon
by an external torque about an axis perpendicular to the spin-axle,
the spin-axle will rotate about an axis perpendicular to the axes
of spin and of torque. For example, with the directions of spin
and of external torque as indicated by the arrows labeled h s and L
in Fig. 38, there would be a rotation of the spin-axle about the axis
AG in the direction indicated by the arrow labeled w p . It will
now be shown that this rotation of the spin-axle develops an in-
ternal torque that is in opposition to the applied external torque.
Because of the angular velocity %, a particle in the semicircle
ADGO is approaching the reader, and a particle in the semicircle
GJAO is receding from the reader. A particle at either A or G
has zero linear velocity and maximum linear acceleration perpen-
MOTION OF A SPINNING BODY
49
dicular to the face of the wheel. A particle at either D or J has
maximum linear velocity and zero linear acceleration perpendicular
to the face of the wheel.
While a particle of the wheel is moving from A to D its inertia
opposes the increase of its linear velocity toward the reader by an
internal force directed away from the reader. This force is maxi-
mum when the particle is at A and zero when it is at D. Thus,
all particles from A to D are urged away from the reader. As a
particle of the wheel goes from D to (/, its inertia opposes the
diminution of the linear velocity
toward the observer. Thus, all
particles from D to G are urged
toward the reader. As each
particle goes from G to J its
inertia opposes the increase of
linear velocity away from the
observer, and hence each par-
ticle from G to J is urged to-
ward the observer. As each
particle goes from J to A, its
inertia opposes the diminution
of linear velocity away from the observer by a force away from
the observer. Consequently, the angular velocity about the axis
AG develops an internal torque about the L-axis that is in opposi-
tion to the external torque about the same axis.
The resisting torque developed when a spinning symmetrical
unconstrained body is acted upon by an external torque about an
axis perpendicular to the spin-axis, is called gyroscopic resistance
or gyroscopic torque. Gyroscopic torques are internal couples which
change the direction of the spin-axle without performing work on
the body. The angular motion of the spin-axle accompanied by
a resisting torque is called precession. The axis about which the
spin-axle rotates is called the axis of precession. The angular
velocity of the spin-axle about the axis of precession is called the
precessional velocity. The precessional velocity is constant when
the external torque is of the same magnitude as the internal gyro-
scopic torque. The external torque that produces or maintains
precession is called the precessional couple or precessional torque.
Questions. 1. A contest sometimes seen in military carnivals is a race in
which each contestant rolls a heavy gun wheel around a course. Show that,
if the man is running alongside the wheel with the wheel on his right side,
50
MOTION OF A SPINNING BODY
(a) a downward force on the hub will steer the wheel to the left; (6) if the wheel
is making a turn to the right and tilting over toward the right, a push forward
on the hub will bring the wheel more nearly upright.
2. The face of a certain wheel is not perpendicular to the shaft to which it
is keyed. Show that when the shaft rotates it tends to process and the bear-
ings are acted upon by a torque about an axis which is not constant in di-
rection.
32. Change in the Motion of a Ship Produced by Precession of
the Shaft. Suppose that a wave exerts a torque on a side- wheel
steamer causing the steamer to roll to the starboard.*, In Fig.
39, the direction of the angular momentum of the paddle-wheels
and connecting shaft is indicated by h si and the direction of the
applied torque by L. The direction of the precessional velocity
thereby produced is indicated by w p . The precession causes the
bow to turn to the starboard. The development of the precession
w p causes a torque about the same
axis as the precession and in the
same sense. Since the spin-axis
tends to set itself parallel to this
torque-axis, and with the direction
of spin in the direction of the
torque, it follows that the roll to
starboard is opposed. A star-
board roll causes the starboard
paddle-wheel to dip deeper into
the water than the port paddle-wheel. This produces a turn-
ing of the bow toward the port, thereby opposing the starboard
turning due to precession.
A side-wheel steamer turning to the starboard under the action
of the rudder will heel to the port on account of precession.
In the case of a ship with screw propeller, the effects of rolling
and pitching are not the same as in the case of a side-wheel steamer.
So long as the shaft is horizontal, a side-wheel steamer is not
precessed by pitching and a screw propeller steamer is not pro-
cessed by rolling. Suppose that the angular momentum h s of
the shaft and propeller of a ship is in the direction indicated in
Fig. 40. If a wave exerts a torque in the direction represented by
L tending to raise the bow or lower the stern, the bow of the ship
will turn to starboard about a vertical axis with angular velocity
* The starboard side of a ship is the right-hand side and the port side is the
left-hand side when the observer has his back to the stern and his face toward
the bow.
FIG. 39
MOTION OF A SPINNING BODY
51
Wp. This precessional velocity implies a torque about the same
axis and in the same sense. Since the spin-axis tends to set itself
parallel to this torque-axis, and with the direction of spin in
the direction of the torque, it follows that the precession de-
creases the angle of the pitch
that produced it.
A side-wheel steamer will
roll less and pitch more than
a screw propeller steamer if
corresponding dimensions are
<>qual. 5 FlG 40
Show that when a screw
propeller steamer is at the same time rolling and pitching, the
bow is deflected back and forth out of the proper course of the
ship. This effect is called " yawing " or " nosing/ 7 *
33. Deviations of the Course of an Airplane Produced by Pre-
cession of the Propeller Shaft. Consider an airplane of which
the single propeller, shaft, and rotating parts of the engine have
considerable angular momentum h s with respect to the spin-axis,
Fig. 41. If no torque be applied tending to change the direction
of the spin-axis, the axle will maintain its direction in space, Art.
16. If a torque be applied in the direction represented by L,
by rudders or any other means, tending to tilt the head upward,
then the spin-axis with the attached airplane will precess with
an angular velocity repre-
sented by wfy. With the
directions of spin and of
torque as indicated in the
figure, this precession will
turn the head to the right.
If the precession be re-
tarded by steering to the
left, that is by applying a
torque opposite the direc-
tion of tfy, "then the spin-
axis will tend to set itself
parallel to the new torque-axis and the head will rise still more.
If, however, the precession be accelerated by steering to the right,
that is by applying a torque in the direction of w p , then the head
will be tilted downward. Thus it is seen that in order to turn an
* Suyehiro, " Yawing of Ships," Trans. Int. Nav. Art., 1920, pp. 93-101,
FIG. 41
52 MOTION OF A SPINNING BODY
airplane without causing the machine to tilt, it is necessary to use
not only the vertical rudder to produce the turn but also the hori-
zontal rudder to neutralize the tilt that otherwise would be pro-
duced by precession. If the airplane has two similar propellers
rotating in opposite directions with equal speed, then the preces-
sion will be zero. }
Questions. 1. Show that in the case of an airplane with a propeller rotating >
counter-clockwise as viewed by a person looking from aft forward, a turn to
the right tends to make the plane soar.
2. Show that an airplane, with a propeller rotating in the clockwise direction
as viewed from aft forward, can be turned to the right by elevating a horizontal
rudder at the stern.
3. Show that if an airplane makes a sharp turn about a vertical axis and
banks properly to furnish the required centripetal force, the airplane will
precess about a horizontal athwartship axis causing the nose to dip, even
though the engine is not running. Aviators sometimes ascribe this phenom-
enon to a " hole in the air."
4. In what direction would a horizontal rudder at the stern of an airplane
need to be tilted so as to prevent the airplane turning about an athwartship
axis while making a turn to the right, the rotation of the propeller being clock-
wise?
5. In what direction would a horizontal rudder at the stern need to be
tilted in order to make a turn to the right without using the vertical rudder,
the rotation of the propeller being clockwise?
6. Would the trailing edge of a vertical rudder need to be moved to port or
to starboard in order to make the bow of the airplane rise, the rotation of the
propeller being clockwise?
7. The Pitcairn-Cierva Autogiro is an airplane to which is added a set of
long vanes carried by a hub fastened rigidly to the airplane so that the vanes
can revolve above the cockpit about a vertical axis. Each vane is pivoted to
the hub in such a way that it can set itself at an angle of 90 or less to the axis
of rotation. In starting the airplane, both the propeller and the hub are ro-
tated by the engine in the clockwise direction as viewed by the pilot who is
behind the propeller and below the hub. As soon as the airplane is free of the
ground, the hub is disconnected from the engine. The motion of the airplane
through the air thereafter causes the vanes to rotate about the vertical axis
of the hub and also to rise like the ribs of an umbrella turned inside out.
Deduce the direction of any precession of the airplane while (a) the pro-
peller is being accelerated; (6) the right wing is being lifted by a gust of
wind.
34. Magnitude of the Torque Required to Maintain a Given
Precessional Velocity when there is Zero Motion about the Torque-
Axis, and when the Axes of Spin, of Torque and of Precession are
Perpendicular to One Another. Experiment. Stand on a rotat-
able stool while holding in the hands a bicycle wheel provided with
MOTION OF A SPINNING BODY
53
FIG. 42
a massive tire of lead or iron wire. While the wheel is spinning with
the axle horizontal, incline the right-hand end of the axle down-
ward, then upward. Observe the considerable torque required to
tilt the; axle. Also observe that the rotatable stool turns about a
vertical axis in one direction when the right-hand end of the axle
is being tilted downward and turns in the opposite
direction when the same end of the axle is being
tilted upward.
Experiment. When a gyro- wheel of the gyro-
scope in Fig. 42 is not spinning, push downward
against the point x with the rubber-tipped end of
a lead pencil. Observe the effect and the amount
of push exerted.
Fasten together the two outer rings of the gyro-
scope by means of a clamp C, Fig. 43. Set the
gyro-wheel spinning. Push downward against the
point x as before and observe the effect and the amount of push
exerted.
Detach the clamp so that the gyro-axle can precess. Set the
gyro-wheel spinning. Push downward against the point x as
before and again observe the effect and the amount
of push exerted.
Attach the small mass furnished to the frame
at x, Fig. 43. With the clamp C removed, set the
wheel spinning in the direction indicated. Observe
now that the torque due to the weight of the added
mass produces a precessional motion about a ver-
tical axis but that it produces inappreciable motion
of the spin-axle about the axis of the torque.
Observe that after the precession has become
uniform, the angular momentum produced by a
constant torque about an axis perpendicular to the spin-axle does
not increase the total angular momentum of the wheel but it
changes the direction of the axis of angular momentum. This
is analogous to the fact that centripetal force does not increase
the total linear momentum of a body traveling in a circular arc
but it changes the direction of the linear momentum, /
' We shall now deduce the relation between the precessional
velocity, the angular momentum about the spin-axis, and the
torque acting on a body that is symmetrical with respect to both
the spin-axis and the precession-axis. Represent the magnitude
FIG. 43
54
MOTION OF A SPINNING BODY
of the spin- velocity by w^ the moment of inertia of the wheel
with respect to the spin-axis by K 5J and the torque due to the
added weight by L.
In Fig. 44, the angular momentum of the gyro-wheel about the
spin-axis at one instant is represented by the line SiO, and at a
short interval of time At after-
ward by the lihe S 2 0. During
this time interval there has been
a small change of angular mo-
mentum represented by the line
SiSi perpendicular to the spin-
axis.
The time rate of change of
angular momentum, that is the
torque required to maintain the constant precessional velocity w p
of a body spinning with angular velocity w s about an axis per-
pendicular to the spin-axis and to the axis of precession, has the
value (28):
j 020 1
L = ~M
Now during the short time interval A, the vertical plane through
the spin-axis has rotated about a vertical axis through an angle
with a constant precessional velocity w p . Consequently, =
w p At. For a sufficiently short time interval, the chord $ 2 $i is
approximately equal to the arc 8281. Consequently,
FIG. 44
=
= w p At or S 2 S l = (S 2 0)w p At
Substituting this value of 8281 in the equation of torque,
Remembering that the angular momentum represented by S 2
has the magnitude K s w s [ = h s ], we find that the torque required
to maintain a constant velocity of precession about an axis per-
pendicular to the spin-axis has the value
L[= (SJ))w p ] = K s w s w p = h s w p (55)
where h s [= K s w s ] is the angular momentum of the body relative
to the spin-axis.
MOTION OF A SPINNING BODY 55
The value of the torque required to maintain constant preces-
sion about an axis that is not perpendicular to the spin-axis is
deduced in Art. 53.
Since for every torque there is a reacting torque of the same
magnitude about the same axis acting upon a different body, it
follows that if a body spinning with an angular speed w s rotates
about another axis with angular speed Wp there is developed a
reacting torque acting on the restraints having a value L given
by (55). The opposition which the axle of a rotating wheel offers
to being turned about the axis of an applied torque is called the
gyroscopic couple,, gyroscopic resistance, or the stabilizing property
of the gyroscope.
From (55) it is seen that
(a) If the torque L be zero, then the precession will be zero.
For example, if a wheel is mounted so as to be free to move about
any axis passing through its center of mass, then the spin-axle
remains fixed in space however the frame may be displaced. This
is the First Law of Gyrodynamics.
(6) When a constant torque L is acting on an unconstrained
symmetrical spinning body about an axis perpendicular to the spin-
axle, the spin-axis will prccess steadily about an axis perpendicular
to both the spin-axle and the torque-axis with an angular velocity
This is the Second Law of Gyrodynamics.
(c) When precession is prevented, that is, when w p = 0, then
the torque L = 0. Hence, a rotating body offers zero opposition to
a rotation of the axis of spin when precession is prevented.
36. The torque L produces an angular displacement about an
axis perpendicular to the torque-axis, but may produce zero dis-
placement about its own axis. Under these conditions, the torque
does zero work. This is similar to the fact that a centripetal force
acting perpendicular to the direction of the motion of a body mov-
ing in a circular path does no work on the body.//
Rotating dynamo or motor armatures, and the rotors of steam
turbines, mounted lengthwise of a ship, develop on the ship con-
siderable torques when the ship pitches, but not any when the ship
rolls. If the axes of rotation be athwartship, there will be torques
when the ship rolls, but not any when it pitches. With the mount-
56 MOTION OF A SPINNING BODY
ing in either direction, there will be a torque when the ship moves
in a curved course.
In the case of a locomotive rounding a curve, the change in di-
rection of the axles of the drive wheels develops a torque tending to
lift the drivers from the inside rail. When a side- wheel steamer
turns there is a similar torque. When a rapidly moving motor-
cycle makes a sharp turn, the axle of the flywheel develops such a
large gyroscopic couple that the rider is compelled to lean much
more toward the inside of the curve than he would if the machine
had no flywheel.
It should be kept in mind that (55) and all equations derived from
it apply only to bodies that are symmetrical with respect to the
spin-axis, that is, to bodies for which the moments of inertia about
all axes through, and perpendicular to, the spin-axis are equal.
It applies to rifled projectiles, electric generator and motor arma-
tures, pulleys, flywheels and turbine runners. / -
This equation requires modification to m&ke it applicable to
an unsymmetrical body, that is, one for which the moments of
inertia about all axes through, and perpendicular to, the spin-
axis are not equal. For such a body another term must be added
to (55) of the form (K\ K^w.Wp where KI and K 2 represent
the moments of inertia with respect to two perpendicular axes
that intersect the spin-axis at right angles. In the case of an air-
plane propeller of two blades, for example, the torque L is not
uniform throughout a revolution. A scheme to prevent vibration
of the machine being developed by this variable torque consists
in connecting the hub of the propeller to a fork on the end of the
engine shaft in such a manner that the hub is free to rock about a
pin perpendicular to the shaft and inclined to the long axis of the
propeller at an angle of about 45 degrees.*
The unique precessional velocity given by (57) is for the case in
which the precessional axis is perpendicular to the spin-axis. It
can be shown that in the case of a top spinning about an axis in-
clined to the vertical, there are two possible values of the preces-
sional velocity, depending upon the magnitude of the spin-velocity.
When the spin-velocity exceeds a certain value, the rapid preces-
sion is produced. Under this condition an increase of the external
torque applied to the spinning body results in a diminution of the
precessional velocity.!
* U. S. Patent. Messick, No. 1491997, 1924.
f Deimel, Mechanics of the Gyroscope, p. 89.
MOTION OF A SPINNING BODY 57
Problem. A certain motorcycle weighing 300 lb. has two wheels each of 15
lb., radius of gyration 12 in., and outside diameter 28 in. It has two disk-
shaped flywheels each of 20 lb. and diameter 10 in. rotating with the same
angular velocity as the wheels. The engine makes 3000 r.p.m. when the
machine is traveling 60 mi. per hr. The rider weighs my2
170 lb. The center of mass of the machine is 16 in.
above the ground and that of the rider is 36 in. above
the ground.
Find the angle through which the machine is tilted rag.
from the vertical when the machine has a speed of 60
mi. per hr. around a curve of 300-ft. radius due (a) to
centripetal effect alone; (6) to gyroscopic effect alone.
Solution. 60 mi. per hr. = 88 ft. per sec.
From Fig. 45,
m v 1
.~r "_ w _
Therefore = 38 40' inclination due to centripetal effect.
(6) The inclination 6 due to gyroscopic effect is given by the equation
gyroscopic force acting horizontally at the center of mass of the system
tan = : TT TT ; :
weight of the system
gyroscopic torque
_ height of center of mass above ground
weight of the system
The gyroscopic torque is given by the equation
L = KsW s wp
where the unprimed quantities refer to the motorcycle wheels and the primed
quantities refer to the flywheels. From (21)
K 5 [ = 2 mk*] = ^~~ = 0.94 slug-ft.'
From (22)
rr/ T 2 /wrH 2(20)5 2 nll , no
A' = - = xx U ino\2 =al1 slug-ft- 2
|_ 2 J 2 X 32.1(12) 2
Again,
speed of machine 88 nr ,
Ws = -* - = 77 = 75.b rad. per sec.
radius of wheel M
12
. 2 IT 3000 01 . ,
Ws ' = - = 314 rad. per sec.
t)U
speed of machine 88 ~ ork ,
wp = - L - r: . == 7^ = 0.29 rad. per sec.
p radius of curve 300
W p r = 0.29 rad. per sec.
58 MOTION OF A SPINNING BODY
Consequently, the total gyroscopic torque
L[= K s w s w p + Ks'ws'wp'} = 0.94(75.6)0.29 + 0.11(314)0.29 = 30.6 slug-ft. 2
The height A" of the center of mass of the man and machine above the ground
is, (29),
[170(36) + 300(16)]cos 00 n . . no n ..
A = L - - Wr ' - = 23.2 cos in. = 1.93 cos ft.
1 / U -f- oUU
Substituting these values in the equation for tan 6,
tan Q =
30.6 slug-ft. 2
1.93 cos ft. 0.03
470 Ib. wt. cos
Whence the inclination to the vertical due to gyroscopic effect is
e = sin" 1 0.3 = 1 40'
Problem, Attached to the motorcycle of the preceding problem is a side
car weighing 200 Ib. and having the center of mass 30 in. from the cycle and
16 in. above the ground. Considering both centripetal and gyroscopic effects,
find with what speed the machine must be moving around a curve of 100-ft.
radius in order that the sidecar may be on the verge of rising off the ground.
Solution. When the motor and sidecar go around a curve, the system is
acted upon by a torque tending to raise the sidecar. This torque is due to the
resultant effect of the centripetal force pulling the car out of a straight path
and the gyroscopic forces due to the change in direction of the spin-axes of the
two flywheels and the three wheels that roll along the ground. The resultant
torque tending to raise the sidecar is opposed by a torque due to the weight
of the car. Each of these torques is equivalent to the product of a horizontal
force along the radius of the circular path and the height of the center of mass
of the system above the ground. The horizontal force applied at the center of
mass of the entire moving system that is required to raise the wheel of the side-
car off the ground equals the sum of the centripetal force and the gyroscopic
force due to the three wheels on the ground and the two flywheels. We shall
now find the value of these four quantities and substitute them in the numera-
tor of the second member of the equation,
The horizontal counteracting _ the force required to raise the sidecar
force distance of the centroid above the ground
The torque, L f = 2^??1 = 500 Ib.-ft.
The distance X above the ground of the center of mass of the system consist-
ing of man, motorcycle and sidecar is, (29),
1TO X 36 + 800)06 + 200 Xl
Whence, the horizontal counteracting force
Tj _ 2U fa> _ , ?6 ft
MOTION OF A SPINNING BODY 59
The centripetal force
The gyroscopic force due to the three wheels rolling along the ground,
3(15)1 2 v v
\ a K0
L &
Fg a P = 32.1 lies IPO , aooflg v , lb
1.7o
The gyroscopic force due to the two flywheels
2,3000
= 0>0022 , lb
Substituting these values of F/*, F e , F g and F g ' in the equation stated in
words at the beginning of this solution,
284 = (0.208 + 0.0068 + 0.0022)0*
Whence, the speed with which the machine must move around the given
curve in order that the sidecar may be on the verge of rising off the ground is
v= 36.15 ft. per sec. = 24.6 mi. per hr.
Problem. A rotary airplane engine of 325 lb., and radius of gyration 14 in.,
makes 1300 r.p.m. when the plane has a speed of 150 mi. per hr. Find:
(a) the gyroscopic torque exerted on the plane when making a turn of 200-
ft. radius; (6) the force that must be applied at the tail when the distance
between the center of pressure on the tail and the center of mass of the airplane
is 15 ft.; (c) the mass that a similar engine of radius of gyration 14 in., would
need to have in order that under similar conditions the pilot could produce
a force on the tail amounting to 150 lb. wt.
Solution, (a) The torque is about a transverse axis and has a value
L = K s w s wp
where
Q * ^V= 13.7slug-ft. 2
27T-1300 i
Ws = = 136 rad. per sec.
ou
and
T rl 220 , t ,
Wfi\ = - = ^7: = 1.1 rad. per sec.
v L f J 2
Whence,
L \ = K s w s w p ] = 13.7(136)1.1 = 2050 Ib.-ft.
(b) The force on the tail
2050 Ib.-ft. = lb
15 ft. ~ . x w .
60
MOTION OF A SPINNING BODY
(c) The weight of the larger engine is
Lg
Lg
(150 X 15 lb.-ft.)32.l ft. per sec. per sec.
n ii ii i . i . i i =
( ft. j 136 rad. per sec. (1.1 rad. per sec.)
.,
ID. Wt.
Problem. The mass of a certain motorcycle is 350 lb., and the distance
between wheel axles is 5 ft. Each wheel has a mass of 15 lb., diameter of 2(>
in., and radius of gyration of 12 in. There is a flywheel of 45 lb., and radius
of gyration 5 in., with spin-axis in the direction of travel. The flywheel makes
3400 r.p.m. when the motorcycle is traveling (30 mi. per hr. The rider weighs
170 lb.
Neglecting the effect of any inclination of the machine from the vertical,
and assuming that when the machine is at rest the wheels exert equal forces
against the ground, find: (a) the gyroscopic torque tending to overturn the
machine when making a curve of 300-ft. radius; (b) the vertical components
of the reactions of the wheels against the ground at the same time.
Ans. (a) 22.3 lb.-ft.; (b) 255 and 205 lb. wt.
36. The Direction and Magnitude of the Torque Developed by
a Spinning Body when Rotating About an Axis Perpendicular to
the Spin-Axle. Consider a gyro-wheel with axle vertical and
angular momentum h s in the direction indicated in Fig. 46. Sup-
pose that the gyro-axle is rotating with ,angular velocity w in the
direction indicated. While the wheel is rotating, the particle at
A is raised, the particle at C is lowered, while the particles along
MOTION OF A SPINNING BODY 61
the axis of rotation of w are unaffected by the rotation about that
axis. All particles of the wheel are moving about the spin-axis h s .
While a particle of the wheel moves from the point in space A to
the point B, the component linear velocity of the particle changes
in the direction of the spin-axis from a finite value to zero. Sup-
pose that the points A, b, c, d, B, Fig. 47, are the positions of a,
particle after moving for equal intervals of time with constant
speed about the spin-axle perpendicular to the plane of the diagram.
If the plane of the wheel is at the same time rotating about an axis
DB, the change in the magnitude of the linear velocity in the
direction perpendicular to the plane of the wheel of a particle while
moving from A to 6 is represented by Aa', and the changes while
moving during succeeding equal intervals of time are represented
by 6fo', cc', <ld'. The rate of change of linear velocity increases
as the particle moves from A to B. The inertia of the particle
causes it to oppose this change of yelocity by a force proportional
to the rate of change of linear velocity of the particle. The forces
exerted by the particle when at various points between A and B
in opposition to the change of linear velocity normal to the plane
of the wheel, are represented by arrows in Fig. 46.
While moving from B to C, the component of the velocity of the
particle normal to the plane of the wheel increases from zero at B
to a maximum at C. During the displacement from B to C, the
inertia of the particle causes it to oppose this change of velocity
at oach point of its path by a force which is in the direction opposite
to the velocity normal to the plane of the wheel. In Fig. 46, the
forces at different points from B to C are represented by arrows.
It is thus seen that each particle in the half of the wheel ABC
tends to rise, thereby producing a rotation of the wheel about the
torque-axis L. In a similar manner it can be shown that the half
of the wheel CD A tends to rotate downward about the same axis.
Therefore, when the axle of a spinning body is rotating about an
axis perpendicular to the spin-axis, a torque is developed on the body
about an axis perpendicular to the plane of the spin-axis and the axis
of rotation. The direction of this internal torque is opposite that of
the external torque which would need to be applied about the same
axis in order to produce a precession of the gyro-axle in the direction
in which the gyro-axis is rotating.
The reaction of this torque acts in the opposite direction about the
same axis on the agent that causes the rotation of the spin-axle.
When a torque is applied to an unspinning gyro about an axis
62
MOTION OF A SPINNING BODY
perpendicular to the spin-axle, the spin-axle tilts in the direction
of the torque. If the gyro is spinning and is unconstrained, the
spin-axle precesses. This precession develops an internal torque
that opposes the tilt which would occur if there were no precession.
About 1875 Sir Henry Bessemer organized a company to build
a steamer to carry passengers across the English Channel without
the discomfort of sea-sickness. Within the steamer was a large
cabin weighing 180 tons mounted so as to be capable of oscillation
about the fore-and-aft axis of the ship. A large wheel was mounted
so as to spin about a fixed axis perpendicular to the floor of the
cabin. The inventor assumed that the spin-axle would maintain
its vertical position and thereby maintain the floor of the cabin
nearly horizontal however the ship might roll. The device failed
because the wheel was mounted so that it could not precess.
Later, Schlick attained some success in preventing excessive roll
of a ship by means of a large wheel spin-
ning about an upright axis and capable
of precessing about an athwartship axis
(Art. 90).
Consider a pair of car- wheels, with
the connecting axle, moving around a
horizontal curve. Let the angular mo-
mentum h s of the system with respect
to the axle at any instant be repre-
sented by the line AB, Fig. 48, and the
equal angular momentum at a short
time A, afterward by AC. If the angular velocity about a vortical
axis due to the motion around the curve be w, then the angle
through which the axle turns in time Ai is
= w kt
The change in the angular momentum of the body, represented by
the line AD, has the magnitude,
h s 6 = h,w At
and the time-rate of change of angular momentum, or the gyro-
scopic torque acting on the spinning body about the axis AD, in
the clockwise direction, has the magnitude
L[= h,w] = K s w,w (58)
The reaction of this torque, that is the torque developed by a
spinning body of angular momentum A 3 , when turned with angular
MOTION OF A SPINNING BODY 63
velocity w about an axis perpendicular to the axes of h s and w,
acts in the opposite direction with the same magnitude on the
body that causes the change in the direction of the spin-axle:
L'[ = h s w] = K s w s w (59)
In the case of a car going around a curve, this reaction tends to
rotate the car away from the center of the curve.
If the spin-axle is not perpendicular to the axis about which the
spin-axle is turning, the torque is perpendicular to the axis of
turning and to the component of the spin-velocity perpendicular
to the axis of turning.
If w s and w are of the same sign, the torque will be positive; if
they are of opposite signs, the torque will be negative.
Problem. The shaft with the propeller of a certain torpedo boat has a
moment of inertia of 2000 lb.-ft. 2 with respect to the axis of the shaft. When
the shaft is rotating at an angular speed of 300 r.p.m., the torpedo boat makes
a half turn in 50 sec. Find the direction and magnitude of the torque devel-
oped on the bearings of the shaft.
Problem. A certain locomotive has four pairs of drive wheels, each pair
with the connecting axle having a mass of 4000 lb., a diameter of 64 in., and a
radius of gyration of 20 in. Find the value of the gyroscopic couple tending to
lift the wheels on one side of the locomotive when the locomotive is moving with
a speed of 60 mi. per hr. around a curve of 1000-ft. radius. Neglect any in-
clination of the axles to the horizontal.
Solution. From (58), the gyroscopic couple
T r v i A io v v 4 W& 2 *' 2
L[= A,M' 5 H = 4 Wfr 2 r |g ^ "~7#~~
where w is the mass of one pair of wheels with the connecting shaft, k is the
radius of gyration of the rotating system, v is the speed of the train, R is the
radius of the curve, and r is the radius of each drive wheel.
/4(4000) , W20 rA Y /60 X5280-4. V ( 12 * \
= (-32:1 " slugs AT2 ft V V-"3(3ocf~ ft - persec -J Urn. loooTtJ
= 4020 lb.-ft.
Problem. A steamship is propelled by three steam turbines rotating in the
same direction about a fore-and-aft axis at the rate of 200 r.p.m. The mo-
ments of inertia of the three rotors together with the connected shafts and
propellers are 1400, 700, and 1400 ton-foot 2 units, respectively. Find the
magnitude of the gyroscopic couple developed when the ship is pitching with
a maximum angular speed of 0.1 radian per second.
Solution. From (58), the torque L = K s w s w. Since one revolution equals
2 TT radians, the angular speed of the shafts is
200 (2 TT) ..
w s T~ radians per sec.
64
MOTION OF A SPINNING BODY
Whence, the torque developed by the pitching is
r r v i onnn / 1400 + 700 + 1400 , ., \ /400 * rad. \ / n , rad. \
L[= K s w s w] = 2000 I jr^-r slug-ft. 2 J I -^ I l 0.1 - 1
1 \ 32.1 & / \ 60 sec. /\ sec./
= 2000(228) Ib.-ft. = 228 ton-ft.
Problem. A certain passenger locomotive has a mass m = 293,000 lb.;
distance of the center of mass from the plane of the rails 6 = 7 ft.; each of the
four pilot wheels has a radius r\ = 20 in.; each of the six drivers has a radius
r 2 = 40 in.; each of the two trailer wheels has a radius r 3 = 26 in.; each pair
of pilot wheels with the connecting axle has a mass m\ 2100 lb.; each pair
of drivers with the connecting axle has a mass m^ 8000 lb. ; the two trailers
with the connecting axle have a mass m 3 2700 lb. ; the radius of gyration of
each pair of wheels with the connecting axle is 0.7 of the radius of the wheels.
The distance between the centers of the rails, i = 56.5 in. =4.71 ft.
FIG. 49
Assuming that the locomotive is moving around a curve of radius R 2000
ft., superelevated at an angle = 4, with a speed of 60 mi. per hr., find:
(a) the centripetal torque, with respect to an axis parallel to the rails and
midway between them, which tends to overturn the locomotive; (6) the gyro-
scopic torque, with respect to the same axis, due to the rotating wheels, which
tends to overturn the locomotive; (c) the gravitational torque, with respect
to the same axis, that opposes overturning, (d) With the locomotive running
at the same speed along a straight level roadway, find the direction and mag-
nitude of the gyroscopic torque developed when the wheels on one side pass
over a high section of rail 1000 ft. long with a maximum elevation 0.1 ft. high
at the middle, the other rail being horizontal.
Solution. Figure 49 represents a locomotive moving away from the ob-
server and making a right-hand curve.
(a) In order that the locomotive may move around the curve, there must
be a force acting at the center of mass directed horizontally toward the center
of the curve of the magnitude
Consequently, there must be a centripetal torque having a magnitude witl
respect to an axis perpendicular to the plane of the diagram through C, Fig. 49
mv* , . ,__
-^-bcose (60
MOTION OF A SPINNING BODY 65
If this torque be not applied, the locomotive tends to overturn, rotating away
from the center of the curve about a horizontal axis perpendicular to the
plane of the diagram. Substituting in (60) the data of the problem, we find
the value of the centripetal torque tending to overturn the locomotive to be
Ll = g^gdug.< 88f ^"- ) '7ft (0.098) = 246,600 Ib.-ft.
o*u. 1 ZUvMJ It.
(6) In order that a rotating axle may be turned about a vertical axis, it
must be acted upon by a gyroscopic torque toward the center of the curve,
about a horizontal axis perpendicular to the axle. The reaction of this torque
acts upon the locomotive in the opposite direction (Art. 36).
The value of this torque acting on the locomotive, due to all six pairs of
wheels, is
L 2 = (2 h s ' + 3 h s " + h s f ") w cos (61)
where h s ' represents the angular momentum of each of the pairs of pilot wheels
with connecting axle, and h s " and /?/" represent the angular momenta of the
drivers and trailers, respectively.
From the data of the problem:
2100 , /0.7(20)ft.V88ft. per sec. _ A1
= ^TT ! slugs I ^ ) 0f T = 4701
tj^i. 1 \ \2i / \J f .
Similarly, we find that
h s " = 35858 and h s '" - 7858
f v 1 88 ft. per sec. 0.998 A A/1/1 ,.
w cos = -75 cos = b^?r7i = 0.044 radian per sec.
[_ K J ZUUU it.
Hence the magnitude of the gyroscopic torque is
L = [2(4701) -f 3(35858) -f 7858]0.044 = 5493 Ib.-ft.
(c) The gravitational torque about a horizontal axis parallel to the rails is
clockwise and has the magnitude
LS = mgb sin 6
- 293000 Ib. wt. (7 ft.)0.07 = 143,570 Ib.-ft. (62)
(d) When one side of the locomotive
ascends a high spot, the axles rotate
about a horizontal axis parallel to the
straight rail with angular velocity w f ,
of the magnitude, Fig. 50:
w f = r = radians per sec. "^^^ *
* rf FIG. 50
The linear velocity of the train while ascending the elevation, Fig. 51,
v ~ ft. per sec.
cv
so that w 1 = -j radians per sec.
Consequently, when the wheel on one end of an axle ascends an elevated spot,
there is developed a gyroscopic torque about an axis perpendicular to the plane
66
MOTION OF A SPINNING BODY
of the axis of w f and of the spin-velocity w s . The direction of this torque about
its axis is opposite that of the torque which would need to be applied about
the same axis in order to produce a precession of the axle in the direction of
w'. This torque urges the ascending wheel in advance of the wheel on the
level rail. The magnitude of this gyroscopic torque due to a single pair of
wheels with connecting axle is
r r L ,i z. ev /OON
L 4 [ = h s iv ] h s -j (63)
FIG. 51
The torque developed by the six wheels on one side of the locomotive ascend-
ing the bump is
= [2(4701) + 3(35858) + 7858] - ~ = 462 Ih.-ft.
On descending the bump an equal torque is produced in the opposite direction
about the same axis. These torques tend to spread the rails.
37. The Period of Precession. The period of precession, that
is the time T required by the spin-axle to make one complete
revolution with constant angular velocity iv p about the axis of pre-
cession, is given by the expression
6[= Twp] = 2 TT radians
When the spin-axle and the precessional axis are perpendicular to
one another, the maintenance of the precessional velocity requires
a torque about an axis perpendicular to the spin-axle and the pre-
cession axis of the value, (55), L = K s w s w p . Consequently,
2j^f^s _ 2j7r/*,
L " L (b4 '
38. The Kinetic Energy of a Precessing Body. The kinetic
energy of a gyro-wheel of moment of inertia K s spinning with an
angular velocity w s is f jf^w 5 2 , (11). So long as w s remains con-
stant, the kinetic energy due to spinning is constant, whether there
is precession or not. If a mass m be applied to one end of the spin-
MOTION OF A SPINNING BODY
67
axle of the gyroscope, the spin-axle will dip through an angle <,,
and a torque L t will be developed about the torque-axis. The
spin-axle will precess about an axis perpendicular to the spin-axle-
and the torque-axis.
A torque imparts kinetic energy of rotation to a body only when
the torque produces angular velocity about the torque-axis. Con-
sequently, the torque L t imparts kinetic energy to the spinning
body only in producing the dip <. Hence, the kinetic energy of
precession equals the work done by the torque L t in producing the
dip and has a value equal to L t <t>. Therefore, the total kinetic
energy of the precessing body is
W,. =
=
(65)
The spin- velocity w s of a gyro-wheel is not altered by the applica-
tion of a torque that produces a precession of the gyro-axle.
In uniform precessional motion, torque produces angular mo-
mentum about the axis of precession but does no work about that
axis. This is analogous to the case of uniform circular motion in
which centripetal force produces linear momentum but does no
work in the direction of the linear momentum./'
39. Nutation. Experiment. Figure 52 represents a gyroscope
mounted as used by Fessel. By setting the counterpoise at differ-
ent places on the rod, torques about a horizontal axis can be pro-
duced of different magni-
tudes in either the clock-
wise or the counter-clock-
wise direction. By means
of the counterpoise, bal-
ance the gyroscope so that
the rod remains horizon-
tal. Set the gyro-wheel
spinning. Observe the
result. FlQ - 52
Stop the spinning. Move the counterpoise till it overbalances
the gyro. Set the gyro spinning. Hold the rod horizontal and
then release. Observe that the end dips and rises and at the same
time precesses with ununiform velocity. The nodding of the spin-
axle is called nutation.
Stop the spinning. Move the counterpoise till it is overbalanced
by the gyro. Set the gyro spinning. Hold the rod horizontal and
then release. Observe the result.
MOTION OF A SPINNING BODY
Any gyroscope exhibits nutation of the spin-axle on the sudden
application of a torque, but the long arm of FessePs mounting
renders the nodding especially conspicuous. The amplitude of
the nutational dip depends upon the amplitude and suddenness
of development of the gravity torque. Nutation would continue
indefinitely if there were no opposition to the dipping due to air
or bearing resistance, as for example in the case of the nutation
of the earth's axis. In the case of most gyroscopes the nutation is
quickly damped to zero, y
Suppose that the gyro is spinning at constant rate in the clock-
wise direction as seen when viewed from the counterpoise toward
the gyro. In addition, suppose that the position of the counter-
poise is such that there is a torque L in the clockwise direction
about a horizontal axis as indicated in Fig. 53. Note that there
is zero external torque about the vertical axis.
At the instant the gyro-frame is released, there is zero preces-
sion and the heavy end tilts downward below the horizontal. This
tilting gives a vertical com-
ponent A/i 5 of the angu-
lar momentum h s , directed
downward, Fig. 53. There
must be zero torque about
the vertical axis. Another
angular momentum must
be generated about the ver-
tical axis at the same rate
and in the direction op-
posite to the vertical component of h s . This is produced by the
development of angular velocity (and angular momentum) in
the direction indicated by w p . This precessional velocity is greater
than that which would have been produced if the spin-axle had
not been tilted so much.
i The gyroscope has greater kinetic energy when precessing than
when not precessing, though the total energy is constant. In
order that the total energy may remain constant, the accelerated
precessional velocity is accompanied by a rise of the heavy end of
the gyroscope.
The inertia of the system carries the heavy end above the equilib-
rium position. As the potential energy is increased by the eleva-
tion of the heavy end of the gyroscope, the kinetic energy of rota-
tion must decrease, that is, the precessional velocity must diminish.
FIG. 53
MOTION OF A SPINNING BODY 69
When the heavy end dips, the precessional velocity increases.
When the heavy end rises, the precessional velocity decreases.
After a few oscillations, the spin-axle will remain at a nearly con-
stant inclination ttf the vertical and the velocity of precession will
remain nearly uniform. The inclination of the spin-axle from the
vertical, at any instant, is called the angle of nutation.
Question. The counterpoise of Fig. 52 may be replaced by a little bucket of
sand provided with a hole in the bottom through which the sand may slowly
escape. Suppose that this bucket is placed near the end of the rod so as to
considerably overbalance the gyroscope, the gyro is set spinning, the rod is
held horizontal and then released. As the sand escapes, the counterpoise
will, after a time, lie overcome by the weight of the gyro. Describe the se-
quence of changes in precession and clip of the gyro-axle.
40. The Effect of Hurrying and of Retarding the Precession of a
Spinning Body. Experiment. Attach a small mass to the inner
frame of the gyroscope at the point x, Fig. 35, thereby producing
a torque L as indicated. Set the gyro-wheel spinning in the di-
rection indicated. Observe that the weighted side of the inner
frame dips slightly below the center of the gyro-wheel and that the
gyro-axle processes with an angular velocity in the direction repre-
sented by Wp. Push horizontally against the second gyro-frame
with the rubber tip of a lead pencil so that the spin-axle is moved
in the direction of its precession. Observe that the weighted side
of the inner gyro-frame rises. Now push on the second frame so
that the spin-axle is moved in the direction opposite its precession.
Observe that the weighted side of the inner frame sinks.
In each case the spin-axle moved so as to be more nearly parallel
to the axis of the new torque produced by the pushing on the second
frame, and with the direction of spin in the direction of the torque.
This is in accord with the Second Law of Gyrodynamics (Art. 34).
When an added torque causes an increase in the precessional
velocity, the angular momentum pro- n
duced by the added torque about the
precession axis is represented by the
line Op, Fig. 54. This angular momen-
tum, compounded with the angular
momentum OS of the gyro- wheel about FlG - 54
the spin-axis, gives a resultant angular momentum represented
by OR. Thus, it appears that the spin-axis tends to precess up-
ward through an angle about an axis through perpendicular
to the plane of OP and OS. By a similar analysis it can be
70 MOTION OF A SPINNING BODY
shown that an added torque that retards the original precession
will develop another precession that causes the spin-axis to dip
(Fig. 55).
The precession of the gyro brings into play a torque which neu-
tralizes the torque due to the overweight (Art. 36), and, so long
as the precessional velocity remains
constant, the overweight will not de-
scend. If the precessional velocity be
diminished, by pushing on the frame
for example, the torque due to pre-
' cession will be less than sufficient to
counteract the torque due to the overweight, and the overweight
will descend. If the precessional velocity be increased, the torque
due to precession will be more than sufficient to counteract the
torque due to the overweight, and the overweight will rise.
Air resistance and friction of bearings always produce a torque
about the precession axis in the direction opposite the precession.
It is for this reason that the spin-axle of a balanced gyro tends
to dip.
(a) When the precessional speed of the axle of a spinning gyro
is increased, the gyro is acted upon by an internal torque in oppo-
sition to the torque that produces the precession.
(6) When the precessional speed of the axle of a spinning gyro
is decreased, the gyro is acted upon by an internal torque in the
same direction as the torque that produces the precession.
(c) When an external torque is applied to the axle of a spinning
gyro in the direction of the precession, an internal torque is de-
veloped which acts iipon the gyro in opposition to the torque that
produces the precession.
(d) When an external torque is applied to the axle of a spinning
gyro in the direction opposite to that of the precession, an internal
torque is developed which acts upon the gyro in the same direction
as the torque that produces the precession.
While a spinning top is held on the palm of the hand, if the hand
be moved in a horizontal circle in the direction of the precession,
the top will tilt toward the vertical. " Hurry the precession, and
the top rises; retard the precession and the top falls." This
applies to a top spinning at an ordinary rate. If the spin-velocity
be so great that the fast precessional velocity occurs (Art. 35),
then if the precession be hurried, the spin-axle will tilt away from
the vertical.
MOTION OF A SPINNING BODY 71
As soon as a moving bicycle leans to the right, the rider turns
the machine about a vertical axis to the right. The spin-axes of
the wheels tend to set themselves parallel to this vertical torque-
axis, thereby tilting the machine upward. Usually the bicycle
will tilt beyond the vertical to the left ; the same series of operations
will be repeated in the opposite direction; the machine will oscil-
late back and forth about the position of equilibrium, and the
track on the ground will be serpentine. This gyroscopic righting
effect is less than the righting effect due to the tendency of the
center of mass to continue in the direction in which it was already
going.
The above law (a) is the basis of devices for preventing rolling
of a ship in a seaway (Arts. 92-97) and for preventing the over-
turning of a car when running on a single rail below the car (Chap.
VI). Law (6) is the basis of a device to cause a ship to roll when
it is desired to get off a sand bar, or to break a passage through a
frozen harbor /Art. 96).
41. Motion of the Spin-Axle Relative to the Earth. According
to the First Law of Gyrodynamics (Art. 34), the spin-axle of a
gyro mounted so that the intersection of the three axes of rotation
coincides with the center of mass will remain fixed in space, however
the frame may be displaced, until acted upon by a torque. If
the spin-axle is parallel to the axis of the earth, it will continue to
be parallel while the earth rotates. In general, however, it will
not remain in fixed relation to the earth. Consider a gyro with
spin-axle horizontal. If it be situated at either geographic pole,
it will rotate relative to the earth, making one revolution in a
horizontal plane in one day. If it be situated at the equator, in
the plane of the meridian, it will remain horizontal and in the
meridian line. If it be situated at the equator and in the plane
of the equator, it will remain in the plane of the equator but will
make one revolution in one day about an axis perpendicular to
the plane of the equator. At any latitude between and 90,
the spin-axle will rotate relative to the earth about a vertical axis
in the counter-clockwise direction as seen from above, and also
will dip about a horizontal axis.
Either of these two angular motions of the gyro-axle can be
prevented by the action of a suitable torque. If the torque causes
a precession just equal to the vertical component of the angular
velocity of the earth, then the spin-axle will turn with the earth
and will maintain its position with respect to the earth. Thus
72 MOTION OF A SPINNING BODY
suppose that at a certain latitude, the angular velocity of the gyro-
axle relative to the earth, about a vertical axis, is w radians per
second. Then motion of the gyro-axle in azimuth can be pre-
vented by the application of a torque about the east-west axis
that will produce a precessional velocity Wp w t given by the
Second Law of Gyrodynamics :
L wK s w s
where L is the torque expressed in pound-feet, K s is the moment
of inertia of the gyro relative to the spin-axle expressed in slug-
feet 2 , and w s is the spin- velocity expressed in radians per second.
If angular velocity of the spin-axle relative to the earth ex-
pressed in degrees per minute of time be represented by w', mo-
ment of inertia of the gyro relative to the spin-axle expressed in
pound-feet 2 be represented by AV, and spin- velocity expressed
in revolutions per minute be represented by n, then the above
equation may be written
L(= wK,w,} = 2 = 0.00000095 w' K,'n (66)
The vertical component, relative to the earth, of the angular
velocity of the spin-axle of a freely suspended gyro in latitude X is
w = w e sin X (68)
where iv e represents the angular velocity of the earth about the
geographic axis.
Since the angular velocity of the earth is [360 ^ 24 X 60] =
0.25 per minute of time,
w' 0.25 sin X degrees per minute of time. (69)
42. The Sperry Directional Gyro. The indications of a mag-
netic compass on an airplane can be relied upon only when the
airplane is steady and moving in a straight course with constant
speed. In taking a compass reading, the common practice is to
steady the airplane in a straight course when near the desired
compass heading and wait till the oscillations of the needle have
become sufficiently small. To maintain a straight course either
the ground must be visible or some gyroscopic turn-indicator must
be available.
The various uncertainties in the indications of the magnetic
compass have caused it to be largely displaced by the gyro-com-
pass for navigating marine vessels, but the considerable weight
MOTION OF A SPINNING BODY
73
and cost of the gyro-compass have prevented its adoption on
airplanes. Many of the advantages of the gyro-compass for
navigational purposes can be obtained by a simple gyroscopic
device which. is ckecked and reset at intervals against the indica-
tions of a magnetic compass.
The Sperry Directional Gyro consists of a freely suspended gyro
(7, Fig. 56, capable of spinning at a speed of some 12,000 revolutions
per minute about an axis which normally is horizontal. At this
speed of rotation, the spin-axle will maintain its direction in space
within three degrees for a quarter hour. The spin of the gyro is
maintained by air currents from two nozzles directed against
a
'l'MM (MM 1 ! 11 M r | T T T I T |
6 3 F 33 30
FIG. 56
buckets cut in the edge of the gyro. The air flow is produced by
a venturi as represented in Fig. 69, page 85. The path of air
through the instrument is indicated by the feathered arrows.
In using the directional gyro as a navigational instrument, one
method of procedure is first to put the airplane on the desired
course by reference to the compass, then push in the setting knob
A, thereby lifting the ring B and the lever C which, in turn,
brings the spin-axle horizontal and locks it in that position. Now,
the indicating card F of the instrument may be brought to the zero
position by twisting the setting knob which, in turn, rotates the
gear Z>, the azimuthal gear E and the attached indicating card.
Pulling out the setting knob frees the gyro from all constraint.
Then, the airplane can be kept in the desired course by steering
so that the card of the directional gyro continues to indicate zero.
74 MOTION OF A SPINNING BODY
Problem. A gyro of moment of inertia with respect to the spin-axis of 128
lb.-ft. 2 is spinning at latitude 45 N. with an angular speed of 2000 r.p.m.
The gyro is mounted so that the center of gravity of the wheel and that of
each of the supporting gimbal rings coincides with the intersection of the axes
of the knife-edges of the two rings. Find the mass and the position that an
added body must have so that when placed 4 in. from the center of mass of the
gyroscope it will cause the spin-axle to remain in the meridian plane.
43. An Instrument for Measuring the Crookedness of a Well
Casing. In drilling or in boring a deep well it is very difficult to
maintain the bore hole vertical or straight. Crooked holes may
prevent the rotation of the drill or may cause the drill to twist
off. The bottom of a well a couple of thousand feet deep may
be more than a hundred feet to one side of the vertical through
the top of the hole. It may be on the property of a proprietor
other than the one who owns the land on which the upper end of
the well is situated. Until recently no method has been known
to measure the direction and amount of the bends of a deep well
bore.
By means of the Surwcl gyroscopic clinograph, a permanent rec-
ord of the position, direction, length and amount of any bends in
a well casing can now be obtained. The apparatus is enclosed
within a cylindrical case several feet in length and five and one-
half inches outside diameter. This case contains a universal
spirit level, a universally mounted gyro with horizontal spin-axle,
a watch, a movie camera, miniature incandescent lamps, and bat-
teries of dry cells to spin the gyro and operate the lamps and movie
camera.
The upper spherical surface of the spirit level carries a series
of concentric circles spaced so as to indicate the inclination of the
axis of the case from the vertical in any direction. The bubble
cell is provided with an expansion coil which keeps the bubble
the same size whatever the temperature of the apparatus.
Any turning of the instrument about its geometrical axis while
being lowered into the well is indicated by the gyro which spins
about a horizontal axis at a speed of about 11,000 revolutions per
minute. The natural turning of the spin-axle relative to the earth,
due to the rotation of the earth, is neutralized by a precession of
equal speed in the opposite direction produced by the weight of a
small mass attached to the north bearing of the spin-axle (Art.
41). The vertical pivot of the gyro-casing carries a pointer which
indicates the direction, in azimuth, of the spin-axle. Besides the
MOTION OF A SPINNING BODY 75
two hands of the watch there is a pointer that marks the time the
apparatus started down the well.
The instrument is lowered into the well at a known constant
rate. The movie camera takes a series of pictures at a regular
interval which depends upon the speed with which the instrument
is lowered. The first picture is taken as the instrument starts
down the well. Each picture shows the time as well as the in-
clination of the instrument and the amount the instrument may
have twisted about its axis from its original position, at the in-
stant the picture was taken. From the views of the three scales
shown in any one picture the distance of the apparatus from the
top of the well at any instant can be determined, as well as the
direction and magnitude of the inclination from the vertical of
the bore hole at that instant.
44. The Weston Centrifugal Drier. In laundries water is
removed from wet clothes by a " centrifugal drier " consisting of
a metal cylinder with perforated sides capable of rotating at high
speed about a vertical axis. In most models, the perforated cylin-
der is rigidly fastened on the upper end of a vertical shaft. If
the wet clothes are not packed uniformly about the axis of rotation
the shaft will be subjected to considerable stresses when rotated
at high speed.
In the Weston centrifugal machine used to separate molasses
from sugar crystals, this difficulty is avoided by hanging the per-
forated cylinder from a universal
joint on the lower endW a vertical
shaft. In case the center of mass
of the suspended system is not on / ,/, .
the axis of rotation, it will move
away from the vertical[line through
the point of support, Fig. 57. The
suspended system now is acted
upon by a gravitational torque in
the direction represented by the
symbol at L in the figure. When FlG ' 57
the suspended system is rotating in the direction represented by
h s , it will precess in the direction represented by w p .
A pin, in line with the axis of the perforated cylinder, fits loosely
in a block of metal B that can slide around on the bottom of an
outer cylinder. When the axis of the perforated cylinder is not
vertical, this block is dragged around in a circular path, thereby
76
MOTION OF A SPINNING BODY
Fia. 58
developing a torque on the suspended system. At the instant
represented in the figure, this frictional torque is represented by
Lf. At all times the torque due to friction will be in the direction
opposite to the precession w p . It follows, from law (d), Art. 40, that
the suspended system is acted upon by a torque in the same di-
rection as the gravitational torque L. Consequently, the sus-
pended system precesses so as to hang more nearly vertical.
45. The Effect on the Direction of the Spin-Axis of a Top Pro-
duced by Friction at the Peg. Figure 58 represents the rounded
peg of a top that is spin-
ning about an axis which
is inclined to the vertical
and is in the vertical plane
XZ. The center of mass
of the top is at C.
Let h s represent the angu-
lar momentum of the top
relative to the spin-axis.
The force of friction /i at
the point of contact of the
peg with the ground is shown parallel to the OF axis and is di-
rected away from the reader. An equal parallel force / 2 acts in
the opposite direction at the center of mass C. These two forces
constitute a force couple having a lever arm x. The moment of
this couple is about an axis in the vertical plane and is in the di-
rection indicated by L.
Since the spin-axis tends to set itself parallel to and in the same
direction as the torque-axis, it follows that h s turns about an axis
parallel to YO in the counter-clockwise direction as viewed by the
reader. As h s becomes more nearly vertical, x and therefore L
decreases in value.
If the spin-axis were inclined to the left of the vertical, the
direction of the torque due to friction would be in the direction
opposite that represented in the figure, and the spin-axis would
tend to become vertical as before.
Thus, on account of the friction between the ground and the
rounded peg of a spinning top, the spin-axis of the top tends to
become vertical. Owing to this effect, the upper end of the pre-
cessing axis of the top traces a converging spiral.
46. The Bonneau Airplane Inclinometer. There is great need
of an instrument that will furnish a line which will remain truly
MOTION OF A SPINNING BODY
77
vertical when on a ship or airplane subject to large linear or
angular accelerations. The Bonneau inclinometer* is a top having
a spherical peg the center of which coincides with the center of
gravity of the top, Figs. 59 and 60. The peg stands in a spherical
cavity of less curvature. The top is free
of gravitational torque and precessional
motion. Centripetal and other lateral forces
do not deflect the spin-axle unless they dis-
place the pivot in its bearing. If such a
displacement occurs, the rounded support-
ing peg accelerates any precessional velocity
thereby causing the spin-axle to assume a practically vertical
position (Art. 45). The top is maintained in rotation by two
streams of air impinging on short blades cut in the periphery.
The driving air currents are produced by the motion of the air-
plane through the air.
In one model, there are two parallel circular stripes about the
periphery of the top and another similar stripe about the cylin-
Fio. 59
drical glass housing, Fig. 61. When the airplane is horizontal, the
stripe on the housing is parallel to the two stripes on the top.
In another model, the indication is made by a light pointer fast-
ened to the top coaxially with the spin-axis. On the spherical
cover glass over the top there are circles and radial lines by means
of which the observer can read the inclination of the housing to
the vertical gyro-axle.
* U. S. Patent. Bonneau, No. 1435580, 1922.
78
MOTION OF A SPINNING BODY
47* The Sperry Airplane Horizon. It is impossible for an
aviator surrounded by dense clouds to determine without instru-
ments whether the airplane is on even keel, is inclined to the hori-
zontal, or even whether he and the airplane are upside down.
The Sperry airplane horizon is an instrument designed to indicate
the angle between the keel of the airplane and the horizon. It
consists of a gyro spinning at about 12,000 r.p.m. about an axis
which normally is vertical. The gyro casing is mounted in a
gimbal ring having axes parallel to and perpendicular to the longi-
tudinal axis of the airplane. The intersection of the gimbal axes
is at a short distance above the center of mass of the gyro and
casing.
FIG. 62a
FIG. 626
The rotation of the gyro is maintained by the impact of two jets
of air directed tangentially against a row of buckets cut in the edge
of the gyro. The air is set into motion by a venturi which draws
air from the surroundings through ducts in one of the gimbal
trunnions, the gimbal ring and the two nozzles. A venturi used
in a similar manner is illustrated in Fig. 69.
The exhaust air escapes from four ports, spaced 90 apart,
through the lower part of the gyro casing. Hanging pendulum-
wise in front of each port is a vane or shutter which varies the
effective area of the port behind it as the casing tilts. When
the gyro is vertical, each port is covered to the same degree and the
reactions on the casing due to air escaping from the four ports are
balanced. When, however, the gyro-axle is not vertical, the ports
are covered by the pendulous vanes in unequal degree and the
MOTION OF A SPINNING BODY 79
reactions due to the escaping air are not balanced. With the gyro
tilted as in Fig. 62a, the port B is not covered by its pendulous
vane X whereas the other three ports are covered by their re-
spective vanes. The reaction of the escaping air urges the lower
end of the gyro casing away from the reader, thereby producing
a torque in the direction represented by the directed line L.
This torque, when the gyro is spinning in the direction represented
by the directed line h s , results in a precession w p which brings the
spin-axle into the vertical position without oscillation.
The face of the Sperry airplane horizon has a vertical dial F,
Fig. 626, fastened rigidly to the gimbal ring. The upper part of
the dial is colored blue to represent the sky and the lower part is
colored grey to represent the earth. The dial is encircled by a
mask which is fastened to the case of the instrument. The mask
carries a divided circular scale and also a silhouette of an airplane
as viewed from the tail.
When the airplane banks, that is, tilts about a fore-and-aft axis,
the case of the instrument rotates relative to the gimbal ring and to
the gyro spin-axis. The angle of bank is indicated by the position
of the pointer P marked on the air-earth dial, relative to the cir-
cular scale S attached to the instrument case.
When the nose of the airplane is directed downward, the actual
horizon appears to the pilot to rise, and when the airplane is
climbing, the actual horizon appears to sink. A white bar //,
in front of the dial of the instrument, fastened to a link which is
pivotted to the gimbal and to the case of the instrument, rises and
sinks relative to the center of the dial, exactly as the actual horizon
appears to rise and sink relative to the nose of the airplane. Fig.
626 indicates a 30 left bank and level fore-and-aft axis of the
airplane.
The horizon bar defines a horizontal plane. The angle between
this plane and the keel of the airplane depends upon the loading of
the airplane. For example, when the airplane is flying a level
course with tail high, the Sperry airplane horizon indicates a slight
dive, whereas when the airplane is loaded to such an extent that it
rides tail low, the horizon bar indicates a slight climb.
When the airplane is making a turn, the pendulous shutters on
the gyro casing would cause an unbalanced reaction on the casing
even though the spin-axle were vertical. This results in a slight
precession of the spin-axle and tilt of the horizon bar which, how-
ever, ceases as soon as the airplane resumes a straight course.
The position of the horizon bar used as normal is that when the
80 MOTION OF A SPINNING BODY
airplane is in straight level flight with its present load. By the aid
of standard turn indicator and rate-of-climb indicator, the pilot
knows when the airplane is in straight level flight. Knowing the
normal position of the horizon bar, the pilot can use the Sperry
horizon to inform him of the altitude of his airplane during night
or blind flying when the invisibility of the actual horizon would
render flying difficult and often dangerous.
48. The Drift of a Projectile from a Rifled Gun. If a pro-
jectile fired from a gun were uninfluenced by any force except
its weight, the center of
mass of the projectile
would describe a parab-
ola having a vertical axis
(Ferry's General Physics,
p. 68). If the bore of
FlG * 63 the gun were not rifled,
there would be no tendency for the projectile to rotate about any
axis. If the projectile were cylindrical, the axis would remain
parallel to the axis of the gun throughout the flight of the projec-
tile, as represented in Fig. 63.
In the actual case, however, the projectile moves through air
and is acted upon by air resistance, which profoundly modifies
the motion. Air resistance lifts the nose of the projectile. If the
bore of the gun were smooth, the nose of the projectile might be
lifted to such an extent that the projectile would strike broadside-
on or might tumble. For this reason, the projectiles used with
smooth-bore guns are spherical. By rifling the bore of the gun,
the projectile is given an angular velocity of spin about the long
axis of as much as 3000 revolutions per second for infantry arms.
The large projectiles of the coast artillery spin much more slowly,
in some cases as slowly as 100 revolutions per second. The pro-
jectiles from American guns rotate in the clockwise direction as
viewed from the gun. Owing to the " rigidity of the spin-axis/'
a spinning long projectile has a much greater range than if it were
not spinning, but there is a lateral deflection. With projectiles
of the same model, this lateral drift is to the right when the di-
rection of spin is clockwise and to the left when the spin is counter-
clockwise. The amount of drift depends upon the shape and upon
the angular momentum of the projectile about the spin-axis.
The drift amounts to as much as one yard in a 1000-yard range
and eleven yards in a 3000-yard range. Drifts as great as 2000
feet have been observed in a 20-mile range. Drift is caused by a
MOTION OF A SPINNING BODY 81
combination of the direct effects of air pressure and air friction,
and by gyroscopic effects produced by torques developed by air
resistance and air friction. There is no complete explanation of
the observed facts of drift of projectiles, but it is probable that the
following actions are important causes.
Due to the rotation of the earth, a body moving with linear
velocity over the surface of the earth will be deflected, relative to
an observer on the earth, out of its original path. The vertical
component of the linear velocity of a projectile is deflected west-
ward when the projectile is ascending and is deflected eastward
when descending. A body thrown vertically upward will fall
west of the place of projection. The horizontal component of
the linear velocity of a projectile is deflected to the right in north-
ern latitudes and to the
left in southern latitudes.
These deflections are not
due to forces acting on
the projectile but are due
solely to the rotation of
the body to which the
linear velocity of the pro-
jectile is referred.
Consider a projectile spinning in the clockwise direction about the
axis of figure. There is a force due to air friction opposing the spin.
When the axis of the projectile is inclined to the trajectory, this
air friction is greatest on the under side of the nose and the pro-
jectile " rolls off " toward the right. Since the friction is greater
on the nose than on the rear end, the front end of the projectile
rolls to the right more rapidly than the rear end. The wind now
striking the left side of the projectile causes a drift to the right.
Again, suppose that the axis of figure of the projectile is in-
clined to the trajectory TR, at an angle 0, Fig. 64. The resultant
air resistance is a force F parallel to the trajectory and applied at
a point called the center of effort. Usually the shape of a pro-
jectile is so designed that the center of effort E is on the axis of
figure and in advance of the center of mass M. Resolving F
into two components, one parallel and one normal to the axis of
the figure, we have F p = F cos 6 and F n = F sin 6. The former
component retards the motion of the projectile. The latter has a
moment L about the center of mass, L = xF n xF sin 0, where x
represents the distance ME. This torque xF sin 6 acts about an
82 MOTION OF A SPINNING BODY
axis through M perpendicular to the plane of the axis of figure and
the trajectory, tending to raise the nose of the projectile. The
angular momentum of the projectile about the axis of figure is so
great, however, that this torque produces negligible deflection
about the torque-axis but does produce a processional velocity
about an axis perpendicular to the torque-axis and the spin-axis.
In Fig. 65, the angular momentum h s , the torque L, and the pre-
cessional velocity w p are represented by directed lines. Since the
spin-axis tends to set itself parallel to the torque-axis, with the
direction of spin in the direction of the torque, it follows that,
owing l,o air pressure on the under side of the nose of the projectile,
2 the nose is deflected to the
right along a line perpen-
dicular to the plane contain-
ing the trajectory and the
spin-axis.
-* Again, the friction of air
against the lower side of the
nose of the spinning projec-
tile produces a torque oppos-
() ^ ing spinning, about an axis
CD, Fig. 64, nearly parallel to the plane containing the trajec-
tory and the spin-axis. On account of this torque, the spin-axis
AB precesses about an axis perpendicular to the plane of the
spin-axis and torque-axis, tending to become parallel with and in
the same direction as the torque-axis CD. Thus, the nose of the
projectile is lowered and the spin-axis is made more nearly tangent
to the trajectory.
The result of the drift and the drop is a rotation of the axis of the
projectile in the clockwise direction about the trajectory. The
rotation of the axis of the projectile would be counter-clockwise
if either the direction of spin were counter-clockwise or the center
of effort were behind the center of mass.*
49. The Effect of Revolving a Non-Pendulous Gyroscope with
Two Degrees of Rotational Freedom about the Axis of the Sup-
pressed Rotational Freedom. A gyroscope in which the center
of mass of the gyro and supporting frame is at the intersection of
the axes of rotation is said to be non-pendulous. If the center of
mass is below the intersection of the axes of rotation, the gyroscope
* For a fuller discussion see Crantz, Lehrbuch der Ballistik, 1925; Hermann,
Exterior Ballistics; Moulton, New Methods in Exterior Ballistics, 1926.
MOTION OF A SPINNING BODY
83
is said to be pendulous; if it is above, the gyroscope is said to be
antipendulous or top-heavy. '
Experiment. Clamp the middle ring of the gyroscope, Fig. 34,
so that rotation of this ring about the vertical axis is prevented.
As the gyro now can rotate about two axes perpendicular to one
another but not about a third perpendicular to the plane of the
other two, the gyro is said to have two degrees of rotational free-
dom. Set the gyro spinning and place the gyroscope on a rotatable
platform. Rotate the platform in the clockwise direction about
the vertical axis. Observe that the spin-axle of
the gyro sets itself parallel to the vertical axis
with the direction of spin in the clockwise direc-
tion. Now rotate the platform in the counter-
clockwise direction. Observe that the gyro
turns over so that the spin-axle is again parallel
to the vertical axis, and the direction of spin is
in the counter-clockwise direction.
When the gyro-axle is revolved, a torque acts
upon the gyro relative to the axis about which rota-
tion is suppressed. The spin-axis tends to set itself
parallel to this torque axis, and with the direction of
spin in the direction of the torque, according to the
Second Law of Gyrodynamics (Art. 34).
Experiment. The gyro of Fig. 66, has but two de-
grees of rotational freedom. The arrow is fixed rigidly
to the frame supporting the wheel so as always to be in
the same direction as the gyro-axle. If a person hold-
ing the handle of the instrument horizontal rotates
himself about one heel, the gyro-axle with the attached
arrow will set itself vertical. When he turns about the
heel in the opposite direction, the wheel with the at-
tached arrow will turn a somersault. In each case the
axle becomes parallel to the axis of rotation of the
entire instrument and with the spin of the wheel in the same di-
rection as the rotation of the instrument.
Experiment. Figure 67 represents a knife-edge pendulum to
one end of which is attached a gyroscope in a frame that cannot
rotate about an axis parallel to the knife-edge. Set the pendulum
swinging and observe that the gyro-axle sets itself parallel to the
knife-edge and with the direction of spin in the same sense as the
angular velocity of the pendulum about the knife-edge.
FIG. 67
84
MOTION OF A SPINNING BODY
Experiment. Figure 68 represents a gyroscope attached to a
system capable of angular oscillation about a vertical axis. The
ring carrying the gyro is incapable of rotation about a vertical axis
independently of the outer frame. Set the gyro spinning and set
the whole system into angular oscillation about the vertical axis.
Observe that the spin-axle sets itself parallel to the axis of oscil-
lation and with the direction of spin in the same sense as the angu-
lar velocity of the system.
// a gyroscope be rotated about an axis about which the turning
of the gyro-axle is prevented, the gyro-axle tends to set itself parallel
to the axis of rotation with the spin of
the gyro-wheel in the same sense as the
rotation of the gyroscope.
The same action will occur if one
degree of rotational freedom is par-
tially suppressed. Consider a gyro-
scope with the two gimbal axes hori-
zontal and the spin-axis inclined to the
vertical. On rotating the gyroscope
about a vertical axis, the spin-axis
continues in a fixed direction in space.
In order that it may maintain its di-
rection in space, it must rotate with
respect to the frame of the gyroscope,
about a vertical axis. If rotation
about either gimbal axis be opposed by
appreciable friction, motion of the spin-
axle with respect to the gyroscope frame, about a vertical axis, is
opposed by a torque about the vertical axis. When this occurs,
the spin-axle revolves about the vertical axis in a converging spiral
till it becomes vertical with the direction of spin in the direction of
the torque. When there is appreciable friction at one or both of
the gimbal axes and the gyroscope is rotated about a vertical axis,
the equilibrium position of the spin-axle is vertical whether the
center of mass of the gyro is above the intersection of the gimbal
axes, is at the intersection, or is below it. If the apparatus is on
an airplane making a turn about a vertical axis, the spin-axle will
set itself in the direction of the apparent vertical. The action
of this apparatus is the basis of devices* for producing a vertical
line of reference on moving bodies.
* U. S. Patent. J. and J. G. Gray, No. 1308783, 1919.
FIG. 68
MOTION OF A SPINNING BODY
85
FIG. 69
50. The Pioneer Turn Indicator. Several types of turn in-
dicators are based on the principle that if a gyroscope be rotated
about an axis about which turning of the gyro-axle is prevented, the
axle will tend to set itself parallel to the axis of rotation with the
spin of the gyro-wheel in .
the same sense as the ro- ^ -
tation of the gyroscope.
The Pioneer turn indi-
cator,* widely used by
aviators, consists of a
gyro- wheel spinning about
a horizontal axis trans-
verse to the body of the
airplane. The gyro-frame
is capable of rotation
ftbout the fore-and-aft axis
of the airplane A B, Fig.
69. It cannot rotate,
relative to the airplane,
about a vertical axis. If the airplane turns about a vertical
axis, the gyro-axle tends to set itself vertical with the direction
of spin in the direction the airplane is turning about the verti-
cal axis. This turning is opposed by a spring
S. The resulting turning of the gyro-frame
about the fore-and-aft axis of the airplane
is transmitted to a pointer P by means of
a pin N and fork F. A nozzle directed
toward the row of blades is open to the out-
side air. The case is exhausted of air by a
tube connected to a Venturi nozzle V at-
tached to the outside of the airplane body.
51. The Clinging of a Spinning Body to a
Guide in Contact with it. Experiment.
The axle of the Maxwell Top shown in Fig.
Fio. 70 70? is adjusted till the point of support coin-
cides with the center of mass of the top. This is done so that
gravitational forces need not be considered. Keeping the axle
vertical by means of a stick with a round hole in one end, set the
top spinning in the usual manner by means of a string. With-
draw the stick, leaving the axle vertical. The top spins steadily
* U. S. Patent. Colvin, No. 1660152, 1928.
86
MOTION OF A SPINNING BODY
about the vertical axis. Pull the upper end of the axle over into
contact with the guide (7, and then release the axle. Now the end
of the axle rolls or slides around against one side of the guide,
clinging tightly to it, dashes around the end of the guide, still
clinging to it, and rolls or slides along the other side of the guide.
Suppose that the top has an angular momentum about the spin-
axis of constant magnitude. When the spinning axle is brought in-
to contact with the guide there is developed on the axle a force of
friction represented by the arrow /, Fig. 71. This force, in com-
FIG. 71
FIG. 72
bination with an equal, parallel, and oppositely directed force
acting on the lower end of the axle, constitutes a torque LI. Under
the action of this torque, the axle precesses against the guide and
pushes against it with considerable force. The reaction of this force
against the rotating axle, in combination with an equal, parallel,
and oppositely directed force acting on the lower end of the axle,
constitutes a second torque L 2 . Precession developed by this
torque causes the upper end of the axle to slide along the guide
from the position A toward the position B.
When the axle is on the opposite side of the guide, the friction
against the guide develops a torque acting on the axle represented
by I//, Fig. 72. The axle presses against the guide. The reaction
MOTION OF A SPINNING BODY
87
of this force develops a torque acting on the axle represented by L^.
The spin-axle processes from the position OC toward the position
OD.
In this Article we have considered a spinning body with the
fixed point coinciding with the center of mass of the body. Pres-
sure is exerted also against the guide when the fixed point of the
spinning body is either below or above the center of mass. An
important application of this last case is made in the Griffin pul-
verizing mill, Art. 54.
52. Components of the Torque Acting upon a Spinning Body
Having one Fixed Point, Relative to the Three Coordinate Axes
of a Rotating Frame of Reference. In Fig. 73, the line h repre-
FIG. 73
sents the angular momentum of a spinning body. The axis of this
angular momentum passes through the origin of a system of co-
ordinate axes OABC. The components of h about the axes of
this reference frame are h A , h B and he, respectively. Let this ref-
erence frame OABC be capable of rotation with angular velocity
w about an axis through the origin of a set of rectangular axes
OXYZj fixed in space. Denote the projections of w on the axes
of the rotatable frame by w Aj W B and we, respectively. It is
required to find expressions for the torque, that is the rate of
change of the angular momentum of the given body, at any in-
stant, with respect to the moving axes OA, OB and OC.
The torque acting on the spinning body about the axis OA, at
any instant, is made up of the sum of four parts the rate of
change of the angular momentum about the axis OA of the angular
momentum h if the movable frame were at rest relative to the
fixed frame of reference, and the three torques about the axis OA
88 MOTION OF A SPINNING BODY
contributed by the component angular velocities W A , WB and we
of the rotatable frame of reference. Since the components of the
angular momentum of the spinning body about the three rotatable
axes do not depend upon the position of the fixed axes, the rate
of change of these quantities does not depend upon the position of
the rotatable frame relative to the fixed axes of coordinates. The
determination of these rates of change will be simplest when the
two frames coincide.
We shall now find each of the four parts which together make
up the component torque about the axis OA acting upon the spin-
ning body. First, note that if the rotatable frame were at rest,
the rate of change about the axis OA of the angular momentum of
the spinning body would be ~. Second, note that while the
component velocity WA about the axis AO rotates the angular
momenta h B and he about AO, the projections of h B and he are
always zero. Consequently, a constant angular velocity WA con-
tributes zero rate of change of angular momentum about AO.
Third, consider the contribution of rate of change of angular
momentum about the axis AO, due to the component angular
velocity we of the movable reference frame. Imagine the com-
ponent of h in the plane of AOB, Fig. 73, and represented b}^ the
line h' y Fig. 74, to be rotating with angular velocity we about an
axis through 0, perpendicular to the plane AY) Y. When h' makes
an angle </> with AO, the component of h f about AO is /?/ cos <.
The rate of change of this component of the angular momentum
relative to the axis AO is
h' sin 0-TT = h' sin </> we
(it
When = 0, this rate of change equals zero, that is, the rotation of
hA contributes zero effect. When </> = 90, sin <f> = 1, h f = h B ;
that is, the rate of change of the angular momentum about the
axis AO contributed by the rotation of h B equals -~h B wc. This
is the entire rate of change relative to OA due to we*
Fourth, consider the rate of change of angular momentum about
the axis AO, due to the component angular velocity W B of the
movable reference frame. Imagine the component of h in the
plane of AOC, Fig. 73, and represented by the line h", Fig. 75,
to be rotating with angular velocity W B about an axis through 0,
perpendicular to the plane XOZ. When h" makes an angle <
MOTION OF A SPINNING BODY
89
with AOj the component of A" about AO is /i" cos </>. The rate
of change of this component of the angular momentum relative
to the axis AO is
-h" sin -jj = h" sin W B
When 0-0, the rate of change equals zero, that is, the rotation
of h A contributes zero effect. When = 90, sin 0=1, h" = h c \
that is, the rate of change of the angular momentum about the
FIG. 74
FIG. 75
axis AO contributed by the rotation of he equals hcw B . This is
the entire rate of change relative to OA due to W B .
Collecting the values of the four torques composing the com-
ponent acting upon the given body about the axis OA, and repre-
senting this component by the symbol LA, we have,
dt
-
(70)
Proceeding in the same manner, we find the component torques
acting upon the given body about the axes OB and OC to be, re-
spectively,
L B = -^ - +
LC = 7 +
h A w c - h c w A
h A w B
(71)
(72)
The signs of the last two terms of (70), (71) and (72) may be
checked by the following considerations.
90 MOTION OF A SPINNING BODY
With the angular velocity of the movable reference frame ABCO,
relative to the fixed frame of coordinate axes XYZO, as indicated
in the diagrams, the rotation of the rigid frame ABCO produces
changes in the magnitudes of the component angular momenta
about the axes OA, OB, and OC. When the component angular
momentum hs, Fig. 74, is turned with angular velocity we, there is
a, change of angular momentum about the axis OA in the direction
opposite tiA. Thus, the rate of change of momentum hgwc> that is,
the torque about the axis OA contributed by the angular velocity
we is negative. Again, in Fig. 75, the turning of he with angular
velocity WB produces angular momentum about the axis OA in the
direction of h A , so that the torque hcw B is positive.
Proceeding in the same manner we find that the torque h A wc
about the axis OR is positive whereas the torque h c w A about the
same axis is negative. Similarly, the torque h B w A about the axis
OC can be shown to be positive and the torque h A w B to be negative.
When the spin- velocity is constant and about the A -axis, and the
precessional velocity is about the 5-axis, and the resultant external
torque is about the C-axis of a fixed system of coordinate axes
OA BC, then equations (70), (71), and (72) reduce to the expression
for the Second Law of Gyrodynamics (55). This is evident if we
substitute in (70), (71), and (72),
h A = hs, (constant), he = 0, and h B = K B w Bj (constant),
WB ~ wp and WA = we =
53. Components of the Torque Required to Maintain Constant
Precession of a Body when the Precession-Axis is Inclined to the
Spin-Axis. Consider a symmetrical body spinning about an
axis CO fixed with respect to a rigid frame of reference OABC,
Fig. 76. The spin-axle CO with the attached rigid reference frame
rotates with angular velocity w~ about an axis ZO of a fixed system
of rectangular coordinates OXYZ. The spin-axle is at a constant
inclination 6 to the axis ZO. At the instant considered, let AO
be in the plane ZOC. Then BO is perpendicular to ZO. The
required values of the components with respect to the axes AO,
BO, and CO, respectively, of the torque required to maintain a
constant velocity of precession of a symmetrical body spinning
at constant angular speed w s about an axis inclined at a constant
angle to the axis of precession, are obtained from (70), (71), and
(72) by substitution of the proper values now to be determined.
Since CO of the moving reference frame rotates with angular
MOTION OF A SPINNING BODY
91
velocity w, about ZO of the fixed system of coordinates, the com-
ponents of w z about AO, BO, and CO are, respectively:
W A = -iv, sin 0, W B = 0, and w c = w- cos (73)
Now we shall determine the components of the angular mo-
mentum of the spinning body relative to the axes AO, BO, and
CO. The component angular
velocity of the spinning body
about any line equals the sum
of the components of u\ and
w z about that line. The com-
ponent angular velocities of
the spinning body relative to
the rectangular axes AO, BO,
and CO, arc (0 w z sin 0),
(0 + 0), and (w s + w z cos 0),
respectively. Representing
the moments of inertia of the
spinning body relative to the axes AO, BO, and CO by K A , K B ,
and K c , respectively, the values of the angular momenta of the
spinning body about these axes arc seen to be
h A = K A w z sin 9, h B = 0, and he = KC(W S + w~ cos 6)
(74)
Assuming that the velocities of spin and precession are const ant
and the position of the spin-axle relative to the axes AO, BO, and
CO, is constant, it follows that
FIG. 7(>
dh
dh
dh c
and -=
(75)
Substituting in (70), (71), and (72) the values given in (73), (74),
and (75), wc find that the torques due to the external forces acting
upon the spinning body about the axes AO, BO, and CO are, re-
spectively,
L c =
Z 2 sin 6 cos 6 + KC(W S + w z cos 6)w z sin (76)
Consequently, for steady spin and steady precession about an axis
inclined to the axis of spin, there must be zero torque about any
line in the plane of the spin and precession axes, but there must
92
MOTION OF A SPINNING BODY
be a torque about a line normal to the plane of these axes given
by (76).
When the spin-axle is perpendicular to the precession axis, 6 =
90, sin = 1, cos = 0, and (76) becomes the Second Law of
Gyrodynarnics,
L B = K c w s w z = h s w z (77)
When = 0, sin = 0, and the torque LB 0.
The results of Arts. 52 and 53 apply to a top spinning about a
fixed point, the Griffin pulverizing mill, and many other cases.
64. The Griffin Pulverizing Mill. One form of mill for pul-
verizing cement clinker, ores and other hard materials consists of
one or more massive rollers hang-
ing vertically from the ends of a
rotating arm. Owing to rotation,
each roll presses against the inside
of the enclosing pulverizing ring
with a centrifugal force propor-
tional to the square of the velocity
of the roll relative to the pulveriz-
ing ring.
In the Griffin mill, a roll is hung
by a universal joint from the end
of a vertical shaft, Fig. 77. If the
shaft with the rigidly attached roll
be set rotating while its axis is
vertical, the axis will remain ver-
tical. If, however, the roll be
moved over to the inside of the
ring, it will roll around the inner
face and push against the pulverizing ring with a centrifugal force
and also a force due to gyroscopic torque. Because of the re-
sultant of these two actions the force against the pulverizing
ring is greater than it is in the case of the centrifugal roll mill.
Paddles on the lower side of the roll toss the material between
the roller and the pulverizing ring.
Problem. The roll of a Griffin pulverizing mill weighs 880 Ib. and is 8 in.
thick. The diameter of the upper face is somewhat greater than that of the
lower face and the mean diameter is 23 in. The roll is fastened rigidly on the
end of a shaft having a diameter of 5.75 in. and mass of 600 Ib. The length
of the shaft from the point of suspension to the upper face of the roll is 6 ft.
The roll moves around the inside wall of a pulverizing ring having a diameter
FIG. 77
ERRATUM
REPLACING THE SOLUTION OF THE PROBLEM, PP. 93-94
of 40 in. Find the force with which the roll presses against the pulverizing
ring when the pulley is making 105 r.p.m.
tiolittiott. The torque acting upon the roll and shaft due to the rotation
about the axis of the shaft, AC, Fig. 7S, and the rotation of the axis of the
shaft about a fixed vertical axis, 0#, is given by r <
(70). Let
in, mass of the roll = '~r^y-- ^ -7.1 slugs
f .1 ir ) - . , i
///$ = mass of the shaft - .7.) y ~ IS. 7 slugs
/ r - length of roll - ().(>(> ft.
/ s - length of shaft = ft.
<1 r = diameter of roll = 1.9 ft.
d s - diameter of shaft = 0.4S ft.
'^ = moment of inertia of the shaft and roll
about an axis, O.I, coinciding \\ith the
diameter of one end, (21) and (27),
TV r g
FKJ. 7S
= 1331 siug-ft, 1 '
KC - moment of inertia of the shaft and roll about the axis of the shaft,
<)(', (22),
= i MS.7 X 0.23 -f 27.1 X 3.0) - 12.9 slug-ft. 1 '
deflection of the roll shaft from the vertical - sin" 1
( ' M - " f) |
V <> /
y ?/' 3 = angular velocity, about a vertical axis, of the azimuth plane through
the roll shaft and the point of suspension, relative to the pulverizing
ring. In (7(>) this is represented by ir z .
z w s = angular velocity of the roll shaft relative to the rotating a/imuth
f)lane. In (70) this is represented by //v
r wp = angular velocity of the pulley about its axis, relative to the pul-
verizing ring = 10/> r.p.m.
The value of rW now will be determined. If the azimuthal plane through
the pulley shaft and roll shaft were not rotating, then whil^ the roll moves with-
out slipping once around the inside of the pulverizing ring, it would make R/r
turns relative to the azimuthal plane. But while the roll makes one circuit of
the pulverizing ring, the a^irnuthal plane makes one revolution in the opposite
direction. Thus, the number of revolutions made by a roll of radius / while
rolling once, without slipping, around the inside of a ring of radius R is
( ~ 1 J , relative to the rotating azinmthal plane. Hence, during one revolu-
tion of the driving pulley, the roll makes 73 -- revolutions relative to the
azirnuthal plane. The product of this fraction and the angular velocity of
the driving pulley, relative to the pulverizing ring, equals the angular velocity
of the azimuthal plane relative to the ring,
= 23.3 radians per second.
The negative sign is used to indicate that the motion of the azimuthal plane is
in the direction opposite that of the driving pulley.
The value of z ws now will be determined. While the center of the roll is
going once around the pulverizing ring, the roll makes r ^z revolutions rela-
tive to the azimuthal plane. TTence, the angular velocity of the roll shaft rela-
tive to the azimuthal plane
/j> /j> i 4 ir~ MV.S,
r rn,= R _ -. rtr* J - 4() _ ,, KM - -,SS r.p.m.
= 40.C) radians per second.
Substituting the data and values now found in (7(>), the magnitude of the
torque acting upon the roll and shaft due to the rotations,
L B - - 1331 r-23.3) 2 o.i2 x 0.99 + 12.9 ( to.(> - 23.3 x 0.99)
(-23.3 X0.12) = -80420 Ib.-ft.
The force due to this torque acting on the roll perpendicular to the roll shaf
n -S()42olb.-ft. ,o,..., i, 4
1 =-lT337t = - 1 ^ nh - wt -
The pulverizing ring is acted upon by the horizontal cornponerr
of the reaction of this force
F r - 13()53cos() 40' - 1351711). wt,
As the roll shaft is deflected from the vertical through an angle
6, there is a horizontal force pulling the roll away from the rinj
of the value, Fig. 79,
(SSO X (133 + 000 X 3.33) sin t <0
e _ - _ - r - = 142 lb. wt.
.:,,
FIG 79 Consequently, the total force with which the roll pushes agains
the pulverizing ring is
F f -f = 13517 - 142 - 1337511). wt.
MOTION OF A SPINNING BODY
93
of 40 in. Find the force with which the roll presses against the pulverizing
ring when the roll shaft is making 165 r.p.m.
Solution. The torque acting upon the roll and shaft due to the rotation
about the axis of the shaft, AC, Fig. 78, and the rotation of the axis of the
shaft about a fixed vertical axis, OZ y is given by
(76). Let
, . , 800 Ib. _ . .
m f mass of the roll = --r- = 27.4 slugs
*j2i. 1
t ^ u ^ 600 Ib. to ,_ ,
m s mass of the shaft = -,^-r~ =18.7 slugs
l r = length of roll = 0.66 ft.
Is = length of shaft = 6 ft.
d r ~ diameter of roll = 1.9 ft.
d s diameter of shaft = 0.48 ft.
KA. = moment of inertia of the shaft and roll
about an axis, OA, coinciding with the
diameter of one end, (25) and (27),
- 1222 slug-ft. 2
Kc = moment of inertia of the shaft and roll about the axis of the shaft,
OC, (22),
= | (18.7 X 0.23 + 27.4 X 3.6) = 12.9 slug-ft. 2
(I (5(5 _ 0.95 \
; 1
b /
= 6 40'.
T w z angular velocity, about a vertical axis, of the azimuth plane through
the roll shaft and the point of suspension, relative to the pulverizing
ring. In (76) this is represented by w z .
zWs = angular velocity of the roll shaft relative to the rotating azimuth
plane. In (76) this is represented by iv s .
r w s = angular velocity of the roll shaft about its axis, relative to the pul-
verizing ring
165
= -~ 2 TT = 17.2 radians per second.
The values of f w z and z w s now will be determined. The number of revolu-
tions made about the axis of the roll shaft by a roll of radius r while rolling
once, without slipping, against the inside of a ring of radius R is
R t
n 1
r
Therefore,
[rUh
= -T
T
= -23 radians per sec.
(78)
The negative sign is used to indicate that r w z is in the sense opposite that of r
94 MOTION OF A SPINNING BODY
The resultant of the angular velocities r iv z and z w s is r w s . (See Arts. 4 and 6.)
From Fig. 78 arid the parallelogram law, (Art. 3),
rWs 2 = rWz 2 + 2 r Wz zW s COS </> + zU>s 2
Substituting in this equation the data of the problem as well as the value
of r wz from (78), and solving the resulting equation for Z w 5y
z w s = 15.5 radians per sec. (79)
Substituting the values now found in (76), the value of the torque acting
upon the roll and shaft due to the rotations
LB = -1222(-23) 2 0.12 X 0.99 + 12,9(15.5 - 23 X 0.99) f- 23 X 0.12)
- -76450 Ib.-ft.
The force due to this torque acting on the roll perpendicular to the roll shaft
-76450 lb.-ft. ton^u *
p = T^TTfi = 12077 Ib. wt.
b.33 ft.
The pulverizing ring is acted upon by the horizontal component of the re-
action of this force
F f = 12077 cos 6 40' - 119561b.wt.
A-s the roll shaft is deflected from the vertical through an angle
0, there is a horizontal force pulling the roll away from the ring
of the value, Fig. 79,
, (880 X 6.33 -f 600 X 3.33) sin 6 1 . _ n
J TT^JT) ^ = I"*" * l) - W ' '
6.33 cos
Consequently, the total force with which the roll pushes against
the pulverizing ring is
F r -/ = H956 - 142 = 11814 Ib. wt.
55. The Automobile Torpedo. A torpedo that is
operated by its own power and is steered automatically
by a self-contained apparatus is called an automobile torpedo.
We have torpedoes 22 feet long and 21 inches in diameter that
have a speed of 40 knots when submerged and have an effective
range of 8000 yards. Such a machine consists of some 3000 parts,
costs in the neighborhood of $16,000 and is capable of destroying
the largest battleship.
An automatically operating gyroscopic steering device can be set
so that after the torpedo leaves the vessel the torpedo will maintain
an assigned course which may be either straight or bent. The
torpedo can be maintained at an assigned distance below the sur-
face of the sea and caused to sink if it does not strike a target.
All modern automobile torpedoes are operated by compressed
gas. The large ones are equipped with two engines of some 700
MOTION OF A SPINNING BODY
95
horse-power, each operating a propeller. The two propellers
revolve with equal speed in Opposite directions on the after ends
of two concentric tubular shafts. The exhaust gases escape
through the inner tubular shaft. Some European countries use
reciprocating engines, usually of four cylinders arranged radially
in a plane perpendicular to the shaft.* The U. S. Navy uses
rotary engines.
A reservoir some 12 feet long contains air at a pressure of about
2800 pounds per square inch. After being reduced in pressure to
FIG. 80
about 350 pounds per square inch, the air is led into a combustion
chamber or " pot " where the temperature is raised by a flame of
alcohol or gasoline sprayed into the pot. The fuel spray is ignited
by a cordite fuse which in turn is ignited by the explosion of a
percussion cap on the launching of the torpedo. The temperature
of the pot and contents is prevented from rising too high by spray-
ing into this pot a stream of pure water in addition to the air and
fuel. The water flashes into superheated steam thereby adding
considerable energy to the working substance.
Gas from the pot operates not only the main motors but also
the two steering motors and the turbine that starts the gyro and the
one that maintains the spinning.
56. The Pendulum-Controlled Depth Steering Gear. A hori-
zontal rudder RH, Fig. 80, operated by piston in a cylinder M
maintains the torpedo horizontal at a predetermined depth in the
water. The displacement of the piston is controlled by a valve
which in turn is controlled by the joint action of a pendulum P
and two connected hydrostats, one on either side of the propeller
shaft.
* Dumas, " La Torpille Automobile," Le Genie Civil, p. 401 (1915).
96 MOTION OF A SPINNING BODY
Each hydrostat H has a diaphragm / at the lower end of a ver-
tical cylinder. The under side is exposed to the pressure of the
sea. Any desired pressure is applied to the upper side by adjusting
the tension of a spiral spring that fills the vertical cylinder. If
the torpedo sinks below the depth corresponding to the pressure
for which the spring is set, the sea pressure pushes the diaphragm
upward and the attached bell crank KLC rotates in the clockwise
direction about a fixed shaft L thereby moving the valve rod DE
in the direction indicated in the figure.
If the nose of the torpedo dips, the pendulum pulls the rod AB
in the direction indicated by the adjacent arrow, and the valve-
rod DE is pushed in the direction indicated in the figure.
57. The Conditions that Must Be Fulfilled by the Horizontal
Steering Mechanism. No device connected with the automobile
torpedo has required so much study and experimentation as that
employed to maintain a fixed course however the torpedo may
be buffeted by waves. All practical devices depend upon the
11 rigidity of the spin-axis " in space of a gyro of three degrees of
rotational freedom so long as the gyroscope is unacted upon by
any outside torque.
The gyro must be mounted in gimbal rings. There must be a
mechanism to keep the spin-axle pointing in a predetermined di-
rection so long as the torpedo is within the launching tube. There
must be a starting motor that will get the speed of the gyro up tc
a high value by the time the torpedo leaves the launching tube.
There must be a device that will disconnect the locking device
as soon as the torpedo enters the water and that will at that in-
stant substitute for the starting motor another motor which wil
maintain the spin-velocity without interfering with the movemen
of the gyro-axle relative to the torpedo. A torque must be ap
plied to the gyro that will produce a processional velocity of th
gyro-axle equal and opposite to the angular velocity of the gyro
axle relative to the earth (Art. 41).
While the torpedo is within the launching tube, the gyro of th
locked gyroscope is speeded up to about 10,000 revolutions pe
minute by means of a separate turbine motor geared to the gyre
axle. The torpedo is projected by a charge of cordite. Abou
the time the torpedo strikes the water, the gyroscope is unlockec
the starting motor is disconnected, and two streams of high prei
sure gas are directed tangentially into shallow buckets cut into it
edge of the gyro.
MOTION OF A SPINNING BODY
97
If the torpedo axis is deflected from the direction it had when the
gyroscope was unlocked, the angle between the torpedo axis and the
gyro spin-axis is changed. The displacement of the gyro spin-
axle relative to the torpedo axis is employed to operate a valve
which controls the passage of compressed gas to one side or the
other of the piston of a steering motor connected to the vertical
rudders. In the transmission of the considerable force required
to operate this valve, no appreciable torque must be applied to the
gyroscope. Otherwise, the gyro-axle would be caused to precess
and no longer remain in a fixed direction in space.
FIG. 81
58. The Bliss-Leavitt Torpedo Steering Gear. These con-
ditions have been met in the Bliss-Leavitt torpedo steering mecha-
nism based on the patents of F. M. Leavitt and Wm. Dieter.*
The Bliss-Leavitt torpedo is used in the
United States and other navies.
The operation of the device can be un-
derstood from an inspection of Fig. 81,
which is a simplified plan view of the
mechanism for controlling the vertical rud-
ders. The spin-axis of the gyro is hori-
zontal, Fig. 82, the axis of rotation of the
inner gimbal ring is horizontal and per- FIG. 82
pendicular to the spin-axis; the axis of rotation of the outer
gimbal ring is vertical. Attached rigidly to the upper side of the
* U. 8. Patents. Leavitt, No. 741083, 1903; No. 768291, 1904; No.
795045, 1905; No. 785424, 1905; No. 814969, 1906; No. 901355, 1908;
No. 925709, 1909; No. 925710, 1909; No. 1080116, 1913; No. 1145025, 1915;
No. 1197134, 1916; No. 1291031, 1919.
Dieter, No. 1148154, 1915; No. 1233761, 1917; No. 1318980, 1919; No.
1402745, 1922; No. 1440822, 1923.
98
MOTION OF A SPINNING BODY
outer gimbal ring is a horizontal disk having a cam on the edge.
This cam comprises a rectangular part C, Fig. 82, concentric
with the edge of the disk, and a spiral part c, like one turn of the
square thread of a jackscrew. A light tappet, t, capable of angu-
lar motion about a vertical pivot, oscillates rapidly toward and
away from the cam along the radius of the cam-disk parallel to
the longitudinal axis of the torpedo. The tappet has two fingers
at different levels.
Normally, that is when the torpedo is on a straight course, the
axis of the torpedo and the radius of the cam-disk through the
FIG. 83
center of the square part of the cam are parallel to the gyro spin-
axle. In this case, when the tappet moves up to the cam-disk,
the two fingers strike the square part C of the cam and the tappet
will be turned till its axis is parallel to the radius of the cam-disk
that goes through the center of the square part of the cam. This
position of the tappet is preserved during the reverse stroke. On
the reverse stroke, the arm of the tappet will move into the gap
between the two pallets p and p', Fig. 81. Just as soon, however,
as the torpedo axis departs slightly from parallelism to the gyro
spin-axis, one finger of the oscillating tappet strikes the cam and
the other strikes the edge of the disk. As the cam is farther from
the center of the disk than is the edge of the disk, the tappet will
be tilted to the left or right of the torpedo axis. On the reverse
stroke, the arm of the tappet will strike one of the pallets p or p',
MOTION OF A SPINNING BODY 99
tilt the two connected elbow levers and move the valve rod v
either forward or backward. The displacement of the valve
rod allows compressed gas from the combustion pot to enter
the steering motor E on one side of the piston or the other,
thereby turning the vertical rudder either hard to port or hard
to starboard. Figure 83 gives a view of the assembled control
mechanism.
59. Method of Compensating the Effect of the Rotation of the
Earth. During the time that a torpedo is making a run of ten
minutes, the earth rotates 2.5. If the gyro-axle remains fixed in
space, its direction relative to the earth changes by an amount that
depends upon the latitude and also upon the direction in which the
torpedo is moving. The deflection of the spin-axle in azimuth
can be compensated by the application of a torque that will main-
tain a precessiorial velocity equal and opposite to the proper com-
ponent of the angular velocity of the earth. The required torque
is produced by the weight of a small nut that can be moved along
a screw fastened to the inner gimbal ring and extending in the
direction of the gyro-axle. To obtain the correct torque, the gyro-
scope is mounted on a stand, the gyro set spinning at the correct
speed, and the position of the adjusting nut changed till the gyro-
axle remains stationary in azimuth. This adjustment also com-
pensates for any lack of balance of the gyroscope that would pro-
duce a precession.
The spin-velocity of a torpedo gyro is not constant throughout
a long run. The precessional velocity due to any given torque
varies inversely with the spin- velocity (57). Consequently, the
adjusting nut is placed so that the mean precession of the gyro
equals the angular velocity of the gyro-axle due to the rotation of
the earth at the given place. If the adjustment is correct for a
ten-minute run at a given latitude, then for a run of shorter dura-
tion, or at a place nearer the polo, the torpedo will be deflected
to the right.
60. Devices for Changing the Course of a Torpedo. Several
methods have been devised for causing an automobile torpedo to
make a turn of any predetermined angle and thereafter proceed
along a straight course. The Leavitt* method of " angle fire "
or " curved fire " is to turn the cam-disk from its zero position on
the second frame of the gyroscope through the required angle in the
proper direction. The rudder is thereby held hard over till it
* U. S. Patent. Leavitt, No. 925710, 1909.
100 MOTION OF A SPINNING BODY
has steered the torpedo through the predetermined angle, that is
till the line from the center of the cam-disk to the center of the
square part of the cam is parallel to the axis of the torpedo.
From that moment, the torpedo will be steered in a straight course.
The Dieter* method of angle fire is to keep the cam-disk fastened
to the gyroscope so that the line from the center of the disk to the
center of the square part of the cam remains parallel to the gyro
spin-axle, and turn the entire gyroscope through the required angle
about an axis perpendicular to the torpedo axes. The operation is
similar to that of the preceding method.
Other methods for producing angle fire have been devised by
Kaselowsky, Waldron, Patterson, Blountf and others.
A torpedo proceeding along a straight course has but one chance
to hit a given target. If, however, after reaching the neighbor-
hood of the target, the torpedo be caused to go around and around
in circular paths, then there are more chances of a hit. Devices
to cause a torpedo to proceed on a straight course for a predeter-
mined distance and then execute circular paths have been developed
by several inventors. J
61. Airplane Cartography. During the Great War methods
were developed by which military maps were made from a series
of photographs taken by a camera mounted on an airplane. As
the airplane moved back and forth in parallel paths, the camera
took a series of overlapping views on a long film which afterward
could be cut up and fastened together so as to form a mosaic of
the district covered. The points which were common to over-
lapping parts of successive pictures served to bring the separate
pictures into register and also to show any difference in scale of
successive pictures.
Since the war, the method has been greatly improved and its
use extended to civil operations. It is estimated that over rough
terrain one camera on an airplane can take in one hour the pictures
for a reconnaissance map that would be as accurate and detailed
as the data that would be taken in one month by a party of one
hundred surveyors.
In order that all the pictures in a strip may be on the same
* U. S. Patent. Dieter, No. 1153678, 1915.
f U. S. Patents. Kaselowsky, No. 661535, 1900; Waldron, No. 983467,
1911; Patterson, No. 1332302, 1920; Blount, No. 1527777, 1925.
J U. S. Patents. Dieter, Nos. 1303038 and 1303044, 1919; Meitner and
May, No. 1401628, 1921; Trenor, No. 1517873, 1924; Bevans, No. 1527775,
1925.
MOTION OF A SPINNING BODY 101
scale, the distance between the camera and the ground would need
to be constant. It is, however, impossible and unnecessary to
maintain this distance constant. Any difference in the scale of
adjacent pictures is shown in the overlapping portions. The
pictures that are out of scale are enlarged or diminished to the
required degree by photography and the new pictures used in the
mosaic.
In order that the photographs may be without distortion, the
external axis of the camera must be maintained in the direction of
the radius of the earth. If the camera be suspended pendulum-
wise, the axis will be deflected from the vertical when the velocity
of the airplane changes in either magnitude or direction. It is
practically impossible to keep the velocity constant under usual
atmospheric conditions.
62. Direct Control of the Direction of the Axis of a Camera.
The first aviator photographers had no means to correct the de-
flection of a camera produced by acceleration of the airplane, except
by manual adjustment. The camera was supported pendulously
and moved back as soon as a deflection was observed. Some avi-
ator photographers still use the same method. This method,
however, brings the camera back to only approximately the
correct position after a noticeable deflection has occurred. The
pictures taken while the original deflection is occurring and while
it is being corrected are distorted.
Probably the most obvious plan to prevent the deflection of the
camera is to apply the First Law of Gyrodynamics, by attaching to
the camera one or two gyroscopes.* This plan is faulty in that
when a gyroscope exerts a torque in erecting a camera, the gyro
itself is acted upon by a torque, and this torque causes the spin-
axle and the attached camera to turn about an axis perpendic-
ular to the torque-axis. Thus, the camera is given an undesired
displacement. Again, even though the spin-axle were directed
toward the center of the earth at one instant and the direction of
the spin-axle remained fixed in space, the spin-axle will not point
to the center of the earth at later instants. Consequently, the
pictures taken at later instants will be distorted.
63. Indirect Control of the Direction of the Axis of a Camera.
A camera or other device can be tilted by a torque produced by a
motor that is started in either direction, stopped or reversed, by a
* U. S. Patents. Fairchild, No. 1546372, 1925; Lucian, No. 1634950, 1927;
Titterington, No. 1645079, 1927.
102
MOriON OF A SPINNING BODY
gyroscope. Fairchild and Morton* have devised an apparatus
for this purpose that includes a gyroscope the axis of which con-
tinues to point toward the center of the earth. The gyroscope
controls two motors without itself being acted upon by sufficient
torque to disturb the direction of the spin-axle.
The camera C is fastened rigidly to one end of a frame A A',
Fig. 84, capable of rotation about two horizontal axes, perpendicu-
lar to one another, through the point 0. The casing G of a gyro
with vertical spin-axle is supported non-pendulously in gimbal rings
on the other end of the frame. The gyro-casing carries a hori-
zontal ring R capable of being turned about the spin-axle. Fast-
Fiu. $4
ened to this ring is a threaded rod /) in the direction of a radius of
the ring. A mass m can be moved along this rod.
Suppose that the instrument is in the northern hemisphere, the
spin-axle is vertical and the gyro is spinning in the clockwise di-
rection as viewed from above. After a time, although the spin-axle
will be pointing still in the same direction in space as before, it will
be pointing to the east of the center of the earth because the earth
has rotated meantime from the west toward the east. The di-
rection of the spin-axle can be maintained vertical by processing
the spin-axle about a horizontal axis in the meridian plane of the
earth, with an angular velocity equal to that of the earth and in the
opposite direction. The required precessional velocity can be
produced by the application of the proper torque about an east-
west axis. With the direction of spin as specified above, the de-
sired torque is developed when the mass m is at the proper distance
to the north of the spin-axle. By this device the spin-axle is
caused to maintain a practically vertical position.
* U. S. Patents. Fairchild and Morton, No. 1559688, 1925; No. 1679354,
1928.
MOTION OF A SPINNING BODY 103
One method by which the gyro may be used to keep the camera
axis pointing to the center of the earth involves the use of two
motors controlled by currents induced by any motion of the cam-
era relative to the gyro. One motor can turn the frame about an
axis A A' and the other about a horizontal axis perpendicular to
A A' . A fiat coil P, with axis vertical, is fastened to the upper
face of the gyro-casing. This is traversed by a
high-frequency alternating current of a few mil-
liamperes produced by the rotation of an arma-
ture forming part of the gyro. Four other flat
coils with axes vertical are supported above the coil
P by a bracket fastened to the frame. The five
coils are represented in perspective in Fig. 84a. FIG. 84a
The coils Si and 82 constitute part of a secondary circuit that
includes a three-electrode vacuum tube, transformer, condensers,
direct current motor and battery. The coils $/ and S% constitute
part of another secondary circuit. So long as the primary coil
P is equally distant from the four secondary coils, zero electro-
motive force is induced in each secondary circuit and neither motor
starts. This is the condition when the camera is vertical.
If the end A of the frame tilts upward, the system of four
secondary coils becomes inclined to the horizontal plane of the
coil P, Fig. 84a, coil P now is nearer 82 than to 8} and it is equally
distant from Si and 8%. An electromotive force is being induced
in 8182 whereas zero electromotive force is being induced in 81*82'.
The motor in the 8182 circuit starts and tilts the frame till the plane
of the secondary coils is parallel to the plane of the primary coil P,
that is, till the camera axis becomes vertical.
In the same manner, if the frame becomes tilted about an axis
A A' , the distance between the primary coil and the two secondary
coils Si and S 2 ' becomes unequal. The motor in the Si 82 circuit
starts and tilts the frame back till the camera axis is again vertical.
The currents in the primary and secondary coils are so minute that
these actions produce an inappreciable force on the gyro and there-
fore no precession.
64. Control of the Line of Sight of a Camera. The camera,
instead of being held vertical over the ground to be photographed,
may be mounted rigidly along the fore-and-aft axis of the airplane
and a vertical beam of light from the ground reflected into the
lens system. This can be done by either a plane mirror or a to-
tally reflecting prism placed in front of the lens system. In order
104 MOTION OF A SPINNING BODY
that the image formed by light from an object vertically below the
airplane may remain fixed in position on the sensitive film when
the airplane turns about either a longitudinal or transverse axis,
the reflector must be turned in the opposite direction to that in
which the airplane turns, and to half the extent. This result can
be produced by a gyroscope attached to the reflector.*
* IT. S. Patents. Sperry, No. 1688559, 1928; Henderson, No. 1709314,
1929.
CHAPTER III
THE GYROSCOPIC PENDULUM OR PENDULOUS
GYROSCOPE
1. General Properties
65. The Gyro-Pendulum, A gyroscope mounted so that the
center of gravity is either below or above the intersection of two
horizontal perpendicular axes about which the system can oscil-
late is called a gyroscopic pendulum, gyro-pendulum, or pendulous
gyroscope. Since a gyro-pendulum with the center of mass below
the point of support has greater stability than a compound pen-
dulum of equal mass, it is much used for stabilizing cameras,
telescopes and other instruments subject to accelerations on ships
and airplanes. Some forms of gyro-compasses and ship stabilizers
are pendulous gyroscopes. Inverted gyro-pendulums have been
used to stabilize vehicles that are statically unstable such as
vehicles designed to operate on a single rail. A gyro-pendulum
may be arranged to oscillate in one plane like an ordinary pen-
dulum, or may be arranged to oscillate as a conical pendulum,
A gyroscope fastened to an oscillating body so as to apply a
periodic torque to the body, may be arranged in such a manner that
the successive vibrations of the oscillating body may be either
increased or diminished. Such results depend upon the principle
proved in Art. 25. In case a periodic torque acts upon an oscil-
lating body of the same frequency, (a) energy will be imparted to
the oscillating body at the maximum rale, and the amplitude of vi-
bration will increase at the maximum rale, when the torque is in
phase with the angular velocity of the oscillating body] (b) energy
will be abstracted from the oscillating body, and the amplitude of
vibration will diminish at the maximum rate, when the torque is in
opposite phase to the angular velocity of the oscillating body.
66. The Period and the Equivalent Length of a Gyroscopic
Conical Pendulum. In Fig. 85, the center of gravity of the ro-
tating system of weight mg is at a distance I from the point of
support C. The gravitational torque L is counter-clockwise
about a horizontal axis through C perpendicular to the plane of the
diagram. The angular momentum about the spin-axle is repre-
105
106
THE GYROSCOPIC PENDULUM
sen ted by h s . The precessional velocity w p is about a vertical axis.
If the inclination of the spin-axle to the vertical be represented by
0, then the external torque L is given by
L = mgl sin d
From the Second Law of Gyrodynamics,
Art. (34), this external torque equals the
product of the precessional velocity and the
component of the angular momentum about
an axis perpendicular to the torque-axis and
the precession axis. Thus
mgl sin = w p h s sin 6
Hence the precessional velocity
mo mgl
W P = iT~
FIG. 85 h s
The period of the precessional motion, that is the period of the
gyroscopic conical pendulum
2 7T~| 2_7T/? 5
VKjl
T
w f .
(80)
If the radius of gyration of the processing system, with respect to
the spin-axis, be A* A , the above expression may be put into the
forms
<= 2 ^ (81)
f 2jr/?,l =
L m $ \
The length of a simple pendulum of the same period, from (51),
(80), and (81) is
l = T \ = A* . = *>X (82)
e 4 7T 2 mH 2 g gl' 2
67. The Inclination of the Precession Axis of a Gyroscopic
Conical Pendulum to the Vertical. A spinning gyro on the earth
has two component angular velocities, one w s , about the spin-axis
of the gyro, and another, w c , about the axis of the earth. The
centrifugal couple tending to bring the spin-axis into parallelism
with the axis of the earth is about an axis perpendicular to the
plane of the spin-axis and the axis of the earth. The axis of the
resulting precession is inclined to the vertical at a small angle now
to be determined.
In Fig. 86, the point G represents the center of mass of a gyro
on the earth at latitude X; the line AG is the spin-axis of the gyro
GENERAL PROPERTIES
107
and D is the point of support ; the line VC is the true vertical and
H H f is the true horizontal at (?; the line NS is parallel to the axis
of the earth. The spin-axis is inclined to the axis of the earth at
an angle 0.
The gyroscopic torque has the magnitude given in (58). It
also equals the product of some force GF parallel to the axis of the
earth and a lever arm DQ[= DG sin 0].
Or
h s w c sinO - GF(DGsin 0)
Whence, the force parallel to the earth's axis has the value
_ hsU>e
= T
(83)
where I represents the distance
DO.
The gyro is also acted upon by
its weight mg, represented in the
figure by the line GW. The re-
sultant GK of the two forces GF
and GW is the apparent force of
gravity acting on the gyro. Its
direction is the apparent vertical
about which the gyro-axle pre-
cesses. The angle i between the true vertical VC and the axis
V R about which the gyro-axle processes is given by the equation
GJ~ GFcos\
FIG. S(>
Since i is very small, we may write tan i = i. Since JF is very
small, compared to FR(=GW), it may be neglected and the above
equation may be written
GF cos X /0>IN
i = - (84)
mg
From (83), this becomes
i =
h s w e cos X
mgl
Now the external torque acting on the gyroscope is mgl sin 0,
where </> is the angle between the spin-axis and the vertical. The
component of the angular momentum about an axis perpendicular
108
THE GYROSCOPIC PENDULUM
to the vertical is h s sin </>. Hence, from the Second Law of Gyro-
dynamics, (Art. 36),
mgl sin </> = h s w p sin <f>
Substituting the value of mgl from this equation in (84), we obtain
for the angle between the precession axis and the vertical
w
(85)
The angular velocity of the earth
w e [ = 2 ir radians por day] = oi\^i\(\ radians per sec.
OvJ,AvJvJ
and the angular velocity of precession
27T ,.
Wp ~7p- radians per sec.,
where T represents the period of precession.
Substituting in (85) these values of w e and w p , and remembering
that
1 radian = 3438 minutes of angle,
TcosX .. f T 7 cos X (3438)
1 = 86,400 radums r 8MOO~
T cos X . . - , on .
= -^ j-7- minutes of arc (86)
-ZO. JLTC
Thus, when the period of precession is 25.14 seconds of time, the
angle i at the equator is one minute of arc.
68. The Period of the Undamped
Vibration, Back and Forth Through
the Meridian, of the Gyro-Axle of
a Pendulous Gyroscope. Figure
87 represents the gyro-axle AO of a
pendulous gyroscope referred to
rectangular coordinate axes OV y
ON, and OE that extend vertically
upward, northward and eastward.
The gyro-axis is in the vertical plane
VOII. It is inclined to the me-
ridian plane VO N at the angle </> and
to the horizontal plane EO N at the angle 6. The center of mass
of the gyro and frame is at G and the point of suspension of the
moving system is at D. The distance DG will be represented by
the symbol L The gyro-axis AO and the line DG are perpendicu-
FIG. 87
GENERAL PROPERTIES 109
lar to one another. The weight ing of the moving system has a
lever arm BG about an axis through D perpendicular to the plane
VOH and produces a torque about this axis of the value L =
mgl sin 6.
Representing the angular velocity of the moving system about
a vertical axis V fixed in space by w', and the angular momentum
of the gyro-wheel with respect to a horizontal axis OH by h/,
then when the precession is steady we have
L = h/w'
Since the velocity of precession of the moving system about a
vertical axis fixed to the earth is d<j)/dt, the velocity of precession
about a vertical axis fixed in space equals the
sum of d<b/dt and the component velocity of the
earth about the vertical at the given place.
If w e represents the angular velocity of the earth
about its axis, then the component angular velo-
city 0/>, Fig. 88, about a vertical axis through
a point D at latitude X is w e sin X. Hence the
velocity of precession about a vertical axis fixed
in space is
w f = -jr + w e sin X
Cit
The component of h s about OH perpendicular to 0V, Fig. 87, is
h s ' = h s cos 6
Consequently, for an axis fixed in space through D and perpendicu-
lar to the vertical plane VO PI,
[L = ] mgl sin 0[= h s 'w'] = h s cos o(-ji + w e sin X j
When 9 is so small that the difference between cos and unity may
be neglected and also the difference between sin 6 and 6 radians,
we may write
mgl 6 = h s I -~ + w e sin X J (87)
The left-hand side of this equation is the gravitational torque
which tends to cause the gyro-axle to become horizontal. The
right-hand side is the gyroscopic resistance offered by the gyro-
axle opposing the tendency to dip.
Again, the velocity of precession about a horizontal axis OH' 9
or N H, Fig. 87, fixed to the earth and perpendicular to the spin-
110 THE GYROSCOPIC PENDULUM
axis OAj is dd/dt. The component angular velocity of the earth
about a horizontal axis DV, Fig. 88, in the meridian plane, is
w e cos X. This is the component angular velocity of the earth
about ON, Fig. 87. The component of this about OH' or the
parallel line N H is w e cos X sin 0. Hence, the velocity of preces-
sion about a horizontal line O H r fixed in space is
dO . _ . ,
~j- + W e COS X Sin
The torque about the axis perpendicular to OH r and OA, that is
about the axis DG, is zero. Consequently,
= h s l-jj + w e cos X sin J
Differentiating (87) we have
o , r/ 2
Here are two equations, one obtained by considering precession
about a vertical axis and the other by considering precession about
a horizontal axis. Each equation involves both and </>. Now
we shall eliminate and from the resulting equation study the
motion of the spin-axis in the horizontal plane.
JC\
On combining the last two equations by eliminating -^ , we find
h s d 2 .
t -IT? = We cos X sin
mgl dt 2
which can be put into the form
(mql
~ -
d' 2 <j> (mql
~ jTj = I ~i -
at* \ n s
^ .
W e COS X 1 Sin
In order to abbreviate the labor of repeatedly copying the quan-
tity within the parenthesis, we will represent it by the symbol A.
When <{> is small, sin = radians and the above equation assumes
the form
dt* ~
Multiplying both sides by 2 -- and integrating, we obtain
GENERAL PROPERTIES 111
For convenience in subsequent integration, let C = AB 1 . Then
Extracting the square root,
^ = VAVB^--^ or -.-^--^ = rf* VI
tit V&- - <$>-
Mcasuring time from the instant, when <j> = 0, that is, when the
gyro-axle is in the meridian, and integrating the last equation, we
obtain
sin-' t = iVA + d
When / 0, </> = and hence the first member is zero and Ci = 0.
Whence
4> = B sin t V~A
which is of the same form as the equation of a simple harmonic
motion of rotation, (48),
= $ sin It -*\
Therefore, the undamped motion of the gyro-axle of a pendulous
gyroscope back and forth across the meridian is approximately
simple harmonic and of the period T . - .
V A
Replacing the value of A in this equation
/ -
4 Imgl
V
cos X V miw e cos
Among other things, this equation shows that the period of the
gyro-axle of a pendulous gyroscope back and forth across the me-
ridian is increased by diminishing the distance between the point
of support and the center of gravity. This fact is utilized in the
design of various gyro-apparatus.
69. The Torque with Which the Second Frame of a Gyroscope
Resists Angular Deflection. Consider a gyro-wheel G of spin-
velocity w s and moment of inertia K s with respect to the spin-axis,
mounted in two co planar frames A and 5, Fig. 89. Let the axis
pp f pass through the center of gravity of the wheel and inner frame.
112
THE GYROSCOPIC PENDULUM
If the outer frame be turned through a small angle 6 with angular
j/\
velocity -~r about an axis at C normal to the plane of the frames
while the gyro-wheel is spinning and the two frames are clamped
together, the inner frame will be acted upon by a torque about the
/ i x rr dO
axis pp equal to K s w s -7- .
FIG. 89
If the two frames are not clamped together, the inner frame will
precess through an angle < in time t, about the axis pp', with an
angular acceleration -^ . If <t> remains small, we have from (19)
and (58)
,., d' 2 d> rr . dO
where K p is the moment of inertia of the inner frame together with
the gyro-wheel, with respect to the axis pp' about which the inner
frame is turning.
Let us measure angles and time from the instant when the two
frames are in the same plane. At this instant t = and = 0.
Integrating the above expression between the limits and t, we
have, when 6 is small,
or
~dt
dt
K*
(89)
GENERAL PROPERTIES 113
If the inner frame were clamped so that the gyro could not
precess, a certain torque would be required to give the apparatus
a chosen angulai acceleration. When the gyro is precessing, the
centrifugal couple must be balanced by an additional torque.
In order to give the apparatus the chosen angular acceleration it is
necessary to apply an outside torque which is the sum of these two
components. That is,
Substituting in this equation the value of , found above,
(IL
This is the torque about an axis normal to the precession axis that
is required to turn the inner frame together with the gyro-wheel
with an angular acceleration.
When the inner frame with the gyro-wheel is turned with con-
stant velocity, the first term in the right-hand member of the above
equation is zero. Consequently, when the gyro-axle of a precess-
ing gyro-wheel is turned with constant angular velocity through
a small angle about an axis perpendicular to the precession axis,
the processing wheel exerts a torque on the restraints of the value
_ _
j c - ^ - -,
l\.p A p
where K p is the moment of inertia of the gyroscope with respect to
the precession axis.
70. The Length of the Simple Pendulum That Has the Same
Period as an Oscillating Body to Which is Attached a Spinning
Gyroscope. When the gyroscope is not spinning, the pendulous
body has a period given by (50). A simple pendulum has a period
given by (51). If the simple pendulum has the same period as the
pendulous body,
where H represents the distance from the center of mass of the
pendulous body to the knife-edge, m is the mass of the pendulous
body, K c is the moment of inertia of the pendulous body with
respect to the axis of oscillation, and I is the length of the simple
pendulum that has the same period as the given pendulous body.
114 THE GYROSCOPIC PENDULUM
When the pendulous body is inclined to the vertical at an angle 0,
the total torque K c ai acting upon it is due to gravitational forces
and has the value, (19 and 49):
K c a.i = mgH sin 6
where the negative sign indicates that angular displacements are
in the direction opposite the torque. When 6 is so small that sin 6
may be replaced by 6 radians, the total torque about the knife-
edge
K&! [= mfc 2 ai] = -mgHd (93)
or the magnitude of
Substituting in (92) this value of m //, we find the length of the
simple pendulum that has the same period as the oscillating body
to which is attached an unspinning gyroscope to be
(94)
^
When the gyro is spinning and the spin-axle is processing about
a horizontal axis perpendicular to the knife-edge, the total torque
acting on the pendulum about the knife-edge is made up of two
parts, one due to gravity and another due to precession. The
torque due to precession equals h s w p . From (89)
where represents the angle of precession when the pendulum is
deflected from the equilibrium position. Hence, the total torque
acting on the pendulum about the knife-edge, when the spinning
gyro is precessing about a horizontal axis perpendicular to the
knife-edge, has the value
Kc&z = mall 6 ~r qQlmH + ^- ) (95)
K c \ gl\ c /
where fy represents the angular acceleration of the system and K c
represents the moment of inertia, both with respect to the knife-
edge. The negative signs indicate that 6 is measured in the di-
rection opposite the angular acceleration.
Whence, the length k of the simple pendulum which has the
GYRO-HORIZONTALS AND GYRO-VERTICALS 115
same period as a statically stable body that is carrying a spinning
and precessing gyroscope has the value (94 and 95)
'[ =
(96)
A comparison of (94) with (96) shows that when the attached
gyroscope is spinning, the length of the simple pendulum equiva-
lent to the given gyro-pendulum becomes shorter than when the
gyro is not spinning, that is the period of vibration of the gyro-
pendulum becomes shorter.
2. Gyro- Horizontals and Gyro-Verticals
71. Determination of the Latitude of a Place. The latitude of
a place is the angular distance of the place from the equator,
measured on a meridian. It is expressed in
degrees, minutes, and seconds north or south
of the equator. In Fig. 90, EQ represents
the equator of the earth and NS the polar
axis. HO is the horizontal at the point X.
The angle XCE is the latitude of X.
The latitude can be found from an obser-
vation of the altitude above the horizon of
the sun, a planet, or a star. The sun crosses
the meridian at noon. Suppose that at this
time the light from the sun to the point A" follows the line AX,
and that the declination of the sun, that is the angular distance
of the sun above the celestial equator, is <5. Since the altitude of
the sun is AX II, Fig. 90, we see that the latitude of X is given by
X - 90 - AXH + d
Altitudes of heavenly bodies above the horizon are measured by
means of sextants or octants. The declinations of the principal
heavenly bodies at various times, as well as the times at which they
cross the meridian, are given in the Nautical Almanac.
Again, suppose that the altitude of the pole star be observed.
From the figure it is seen that if the pole star were on the polar
axis of the earth, then the altitude of the pole P'XO would equal
the latitude X. The angular distance of the pole star from the
polar axis of the earth, at various times, is given in the Nautical
Almanac.
116 THE GYROSCOPIC PENDULUM
From the data for any star as given in the Nautical Almanac,
together with the observed altitude at any known time not too
remote from the time of crossing the meridian, the latitude of the
observer can be computed.
72. Gyro-Horizons. The latitude of a place is determined
from the altitude of a celestial body. For the measurement of the
altitude of a celestial body we must have a horizontal surface of
reference. The horizon plane is the most convenient reference
plane when the horizon is visible. On land we could use also the free
surface of an unaccelerated liquid, or a plane normal to an unac-
celerated plumb-bob. On a ship at sea neither of the two latter
devices is available because each is subject to any force that
accelerates the motion of the ship, such as the forces that produce
rolling, pitching, or a change in either speed or course.
These forces produce less effect on the direction of the spin-
axle of a gyro-pendulum that is normally vertical than upon the
direction of either a liquid surface or a plumb-bob. A single im-
pulse deflects the spin-axle in the direction of the force to a slight
extent and it also produces a motion of precession having a definite
period. If the period of precession of the spin-axle is great com-
pared with the period of the accelcrative forces producing rolling
or pitching, the effect of the alternating impulses during a complete
period of precession will be nearly neutralized. If the period of
precession is not less than one hour, the deflection of the spin-axle
of a gyro-pendulum from the vertical, produced by rolling or pitch-
ing of the ship, will be very small. This small error due to pre-
cession is eliminated almost completely by using the mean position
of the spin-axle at instants separated by a half period of precession.
For this procedure it is desirable that the period of precession be
as small as possible. The required long period is secured by mount-
ing the gyro so that the center of gravity of the processing system
is at a short distance below the point of support (Art. 68). The
desired short interval between two instants separated by a half
period requires that the center of gravity of the processing system
be at a great distance below the point of support.
The first gyro-horizon was made in the middle of the eighteenth
century by an English instrument-maker named Serson. It con-
sisted of a thick disk balanced at a point above its center of mass
and capable of spinning on the upper end of a vertical pin. When
the disk was maintained in rapid rotation, the spin-axis quickly
assumed a practically vertical position and the upper polished sur-
GYRO-HORIZONTALS AND GYRO-VERTICALS
117
face of the disk became practically horizontal. It was proposed
that this surface be used as a base from which to measure the alti-
tude of heavenly bodies when the horizon is obscured by clouds or
haze. The apparatus was tested on a British warship but, un-
fortunately, the ship went down and both the instrument and the
inventor were lost. Since that time, gyroscopic horizons b ave
received little attention till quite recently.
73. The Schuler Gyro-Horizon. This instrument* consists
of a gyro mounted in a spherical bowl that floats in a vessel of
liquid contained in an outer bowl, Fig. 91. The inner bowl is
1
FIG. 91
kept centered by a pointed rod extending upward from the bot-
tom of the outer bowl. The outer bowl is carried on gimbals.
A plane mirror M is normal to the spin-axle. The center of gravity
of the gyro with the inner bowl is at such a distance below the cen-
ter of buoyancy that the period of precession of the spin-axle is
about 85 minutes.
74. The Anschutz Gyro-Horizon. This instrument! has a
mirror M, Fig. 92, mounted on a gyro-casing supported on gim-
bals, the axes of which are at different levels above the center of
gravity of the gyro and casing. Normally, the spin-axle is ver-
tical. The housing carrying the gimbal axes can be turned in
azimuth till the gimbal axis A A nearer the center of gravity of the
gyro and casing is in line with the heavenly body under observa-
tion. As forces acting at the center of gravity and producing
torques about the axis A A nearer the center of gravity have a
short lever arm, the precession which they produce about the other
gimbal axis BB is small. Consequently, the torque produces
* U. S. Patents. Schuler, No. 1480637, 1924; No. 1735058, 1929.
t U. S. Patent. Anschutz, No. 1141099, 1915.
118
THE GYROSCOPIC PENDULUM
only a small tilting of the spin-axis in the vertical plane of observa-
tion. A torque about the axis BB, further from the center of
gravity, produces a precession of short period, as required for a
quick succession of half-period observations.
The amplitude of oscillation of the spin-axle is damped by the
friction of a mass of liquid moving back and forth in an annular
trough D fastened to the gyro-casing concentric with the spin-axle.
FIG. 92
Tilting of the spin-axle, due to the rotation of the earth, can be
neutralized by a torque capable of producing an equal precession
toward the east (Art. 41). The required precession is produced by
the weight of a nut C that can be adjusted in position in azimuth
and also in distance from the spin-axis. This nut must be either
north or south of the spin-axle, depending on the direction of spin,
and at a distance from the spin-axle depending upon the latitude.
In setting the instrument for measuring the altitude of a celestial
body, the instrument is turned till the gimbal axis nearer the center
of gravity of the gyro and casing is in the vertical plane of the body
under observation, the horizontal ring carrying the adjusting nut
C is rotated till the nut is in the geographical meridian and on the
proper side of the spin-axle, and the nut is moved horizontally to
the scale position corresponding to the latitude of the place.
GYRO-HORIZONTALS AND GYRO-VERTICALS
119
75. The Bonneau-LePrieur-Derrien Gyro-Sextant. This is
a light instrument designed for the use of aviators. It consists
of a sextant to which is attached a gyro-horizon. Without refer-
ence to a visible horizon, altitudes can be measured with it to a
precision within about fifteen minutes of angle. Attached to the
sextant frame is a case in which is a top G, Fig. 93, capable of
rapid rotation. The top is provided with a row of blades about the
periphery. It is set into rotation and maintained at a high speed
by means of either a few strokes of a bicycle pump or by gas es-
caping from one of the little steel capsules called " sparklets "
filled with liquefied carbon dioxide and which are sometimes used
to aerate beverages. The upper surface of the top is a plane
mirror which maintains a
nearly horizontal position
while the top is spinning
rapidly about a vertical
axis NG.
Pivoted to the center R
of the divided arc of the
frame of the instrument is
an index arm / which car-
ries at one end a vernier
scale V and at the other
end a plane mirror B. r , ,..,
r IG. 93
The mirror is divided into
two parts, one part being silvered and the other part being trans-
parent. The surface of the mirror is in the plane of the axis of
the pivot and is perpendicular to the axis of the index arm. The
axis of the telescope is perpendicular to and intersects the axis
of the pivot. The divisions of the scale are reckoned from a zero
on a line through the pivot perpendicular to the axis of the
telescope.
When the telescope is pointed to a star, light from the star tra-
verses the transparent part of the index mirror and forms an image
in the focal plane of the telescope eye-piece. Light from the star
is also incident on the horizontal gyro-mirror G. The index arm
can be turned so that light after reflection from the gyro-mirror and
the index mirror will form another image superposed on the first.
Since A'G and A T are two parallel lines cut by GB, we have the
angle
A'GB = GET or \ (A'GB) = | (GET) = NGB
120 THE GYROSCOPIC PENDULUM
When the light after reflection from the two mirrors coincides
with the axis of the telescope, the angle
GBV = VBT = \GBT
Whence, the angle NGB = GBV
Consequently, NG and BV are parallel, and the two mirrors are
parallel. Since the spin-axis NG of the top is practically vertical,
the two mirrors are practically horizontal.
From the construction of the divided circle, the line BO from
the pivot to the zero line is perpendicular to AT and A'G. And
since BV is perpendicular to the horizontal HG, it follows that the
altitude of the star A 'GH = OBV.
76. The Fleuriais Gyroscopic Octant. This is an octant (or a
sextant) to which is attached a top with a flat surface normal
to the spin-axis and with the
center of mass below the peg.
The recent models carry on
_\\ (L^^-b the flat surface, Fig. 94, a con-
r verging lens L and a piece of
plane clear glass G perpendicu-
FlG - 4 lar to the plane of the diagram.
One face of this glass is ruled with a series of parallel horizontal
lines. The ruled face is in the focal plane of the lens. Light
from the ruled lines, after traversing the lens L, the fixed half-
silvered mirror C and the objective lens of the telescope T, forms
an image of the rulings in the focal plane of the eye-lens. In
the focal plane of the eye-lens there is a cross hair perpendicular
to the plane of the frame of the instrument, that is, the cross hair
is nearly horizontal. When the plane of the frame of the instru-
ment and the axis of the top are vertical, light from the middle
ruling traverses the axis of the telescope and the image of this ruling
coincides with the cross hair. The axis of the telescope is now
horizontal.
An eye of an observer at the eye-lens of the telescope perceives
the image of the rulings every time light from the rulings proceeds
along the axis of the telescope. The image is produced once during
each revolution of the top. The impression of an image on the
human retina persists for about one-tenth of a second after the
cessation of the exciting cause, the duration depending upon the
brightness of the light. Consequently, if the top be rotated at a
sufficiently high speed, the image will appear to be continuous.
GYRO-HORIZONTALS AND GYRO-VERTICALS 121
The top is enclosed in an air-tight case K, Fig. 95, provided with
a pair of windows in line with the axis of the telescope. When the
instrument is used during the daytime the ruled glass plate is il-
lumined by daylight entering the window farthest from the tele-
scope. When used at night, the ruled plate is illumined by a tiny
incandescent lamp operated by a pocket dry battery.
The required high spin-velocity is produced by a current of air
blowing against a row of blades around the periphery of the top.
A hand-operated exhaust pump is connected to a tube E and the
case is evacuated to a pressure of from six to eight centimeters of
mercury. The pressure is indicated by an aneroid barometer on
top of the case. On now opening the stop-cock F, air rushes into
the case and against the bladevS on the periphery of the top. The
top quickly attains a speed of more than 7000 revolutions per
minute. The stop-cock F is again closed and the pump operated
till the pressure is again reduced to about six centimeters of mer-
cury, the stop-cock E is closed and the pump detached. The top
will continue to rotate in the vacuous space for several minutes
without the spin-velocity diminishing as much as 25 per cent.
This time is long enough to take the required observation.
If the spinning wheel rested on a sharp point, there would be a
small angle between the spin-axis and the vertical that could be
computed from (86). In the actual instrument, however, this
angle is reduced to a negligible value by rounding off the peg that
supports the spinning wheel (Art. 45).
77. The Sperry Roll and Pitch Recorder. This instrument is
used to record the angles of roll and of pitch of a ship in a seaway,
122
THE GYROSCOPIC PENDULUM
together with the periods of the oscillations. It is a pendulous
gyroscope* mounted so as to be capable of rotation about two
perpendicular intersecting axes that normally are horizontal.
The degree of pendulousness is such that the instrument has high
dynamic stability together with a period of vibration that is many
times greater than the period of either the roll or the pitch of any
ship. Hence the direction of the spin-axle is practically unaffected
by the oscillations of the ship. The vibrations of the instrument
FIG. 96
are damped by the motion of a mass of liquid contained in an an-
nular passage concentric with the spin-axis. The passage is con-
stricted so that the flow of liquid back and forth will be sufficiently
out of phase with the oscillations of the gyro to produce the desired
damping effect.
Figure 96 shows the instrument as mounted on a ship with the
outer gimbal axis athwartship and the inner one in the fore-and-
aft direction. The pen nearest the divided scale records the angle
of roll on a strip of paper that is moved parallel to the keel of the
ship at a constant rate by means of a clock mounted on the bed
* U. S. Patent. Sperry, No. 1399032, 1921.
GYRO-HORIZONTALS AND GYRO- VERTICALS 123
plate. The pen to the left records angles of pitch and the pen to the
right indicates time intervals.
The roll-recording pen is held by a horizontal arm fastened to a
vertical counterbalanced circular loop capable of rotation about
pivots in line with the fore-and-aft gimbal axis of the instrument.
The concave side of the upper part of this loop is deeply grooved.
Into this groove projects a rod fastened to the top of the gyro-case
and in line with the spin-axis. When the ship rolls, the spin-axle
and the loop remain in the vertical plane while the paper carrier
moves athwartship under the pen. If at the same time the ship
pitches, the groove in the loop permits motion of the loop relative
to the paper carrier without affecting the movement of the paper
relative to the roll-recording pen.
The pitch-recording pen is supported by a parallel motion de-
vice that permits motion of the pen only in the athwartship
direction. It carries two horizontal guides that for part of their
length are inclined at about 45 degrees to the fore-and-aft axis
of the ship. In the space between the two guides is the upper end
of a vertical rod, the lower end of which is fastened to the outer
gimbal ring. When the ship pitches, the parallel motion device
moves relative to the gyro spin-axle along the fore-and-aft axis
of the ship. In so doing, the guides are pushed in the athwart-
ship direction by the rod extending upward from the outer gimbal
ring. Thus the attached pen traces a curve that indicates the
angle of pitch with respect to time.
78. The Sperry Automatic Airplane Pilot. This apparatus
maintains an airplane on any predetermined straight course as
long as desired. It is especially useful in " blind flying " through
opaque clouds. The aviator may leave the cockpit for any pur-
pose, may even walk out to the end of a wing, with confidence that
the machine will continue in its course on a horizontal keel. If
he were to faint for a brief period the automatic pilot would keep
the machine in a straight course on a horizontal keel. He can
disconnect the automatic steering device instantly whenever he
decides to take control.
The automatic pilot comprises two universally mounted gyros,
one with spin-axle vertical and the other with spin-axle horizon-
tal. The first keeps the airplane horizontal by controlling the posi-
tion of horizontal rudders. The second maintains the course of
the airplane in azimuth by controlling the position of vertical
rudders. The power required to spin the gyros is supplied by the
124
THE GYROSCOPIC PENDULUM
electric system of the airplane, while the power to operate the
rudders is supplied by air turbines which rotate when the airplane
is in motion.
The directional gyro, that is the gyro which by operating vertical
rudders keeps the machine on its straight course, is shown in Figs.
97 and 98. If the airplane turns in azimuth, the gyro-frame turns
relative to the spin-axle; two contactor plates attached to the frame
slide along a trolley not shown in the figures; an electrically oper-
ated clutch connects one of the air turbines to the vertical rudder
thereby automatically turning the airplane back into the course.
FIG. 97
The frictional forces acting at the bearings of the gyro are so
small that the spin-axle will remain fixed in space for several
minutes, but after a time the spin-axle will be appreciably out of
its original direction. So long as the airplane is on a straight course
the degree of constancy in direction of the spin-axle can be checked
by comparison with a spirit level and magnetic compass. When
the spin-axle does become tilted from the horizontal, an air-blast
escaping from an orifice in the gyro-casing 0, Fig. 97, produces a
torque about a vertical axis, which torque automatically brings
the spin-axle back into a horizontal plane. When the spin-axle
is horizontal, the orifice is covered by a shutter S and the above-
described torque is not produced.
When the aviator observes that the spin-axle has moved in
GYRO-HORIZONTALS AND GYRO- VERTICALS
125
azimuth from the original setting, he presses an electric button/
thereby energizing one of a pair of curved solenoids A, A', Fig.
98, attached to the gyro-frame. This causes a curved iron core
attached to the gyro-casing to be pulled toward the middle of
the solenoid, thereby developing a torque on the casing, about a
horizontal axis, in the proper direction to cause the spin-axle to
precess about a vertical axis back into the original direction.
This operation moves the contactor attached to the gyro-casing,
relative to the trolley, thereby causing one of the air turbines to
turn a vertical rudder in the proper direction to bring the keel of
FIG. 98
the airplane parallel to the new position of the spin-axle, that is, to
bring it into the original course.
There are two electric buttons, curved solenoids and curved
cores, one system to cause a turning of the airplane in the clock-
wise direction and another to cause a turning in the counter-clock-
wise direction.
The second gyro operates in a manner similar to the directional
gyro, above described, except that this gyro with vertical spin-
axle operates horizontal rudders so as to maintain the axis of the
airplane in a horizontal position.
79. The Sperry-Carter Track Recorder. This is a group of
instruments* that while carried on a test car moving at train speed
* Railway Engineering and Maintenance, Vol. 23, p. 316.
126
THE GYROSCOPIC PENDULUM
will make a chart of the magnitudes and positions of all irregu-
larities of the track. On the paper are drawn curves coordinating
distance and time, distance and differences in the elevation of the
two rails under the car as well as the positions and magnitudes of
rail spreads and rail depressions.
Differences in the elevation of the two rails are recorded by means
of a gyroscopic apparatus mounted on a table supported by one
of the car axles, Fig. 99. The function of the gyro is to furnish,
at any instant, a vertical plane parallel to the rails, with reference
to which can be measured any elevation of one rail above the other.
The gyro is mounted with the
spin-axle normally parallel to the
car axles. The gyro-casing is
supported in a ring jR, Fig. 100,
which normally is vertical. The
gyro-casing is capable of turning
about a vertical axis, and the
ring is capable of turning about
a horizontal axis parallel to the
rails. The center of gravity of
the gyro and of all parts attached
to it coincides with the intersec-
tion of these three axes. Hence,
a change in either the direction
or magnitude of the velocity of the car produces no torque on the
gyroscopic system and consequently no change in the direction
of the spin-axis in space.
When the car axle tilts from the horizontal, the frame of the
gyroscope tilts with respect to the spin-axle, thereby causing a
paper record roll to move under the pen p perpendicularly to the
plane of the diagram.
For accurate indications, the indicator post / must be main-
tained in a vertical plane perpendicular to the normal position of
the car axle. The indicator post is maintained in this vertical
plane by means of a pendulum P and a pair of current-carrying
solenoids SS. These solenoids are fastened to the ring R. Pro-
jecting into each solenoid is one end of a soft iron core having the
other end fastened to the gyro-casing. When either of the sole-
noids is traversed by a current, the far end of its core is drawn
inside the solenoid thereby producing a torque on the gyro-casing
about a vertical axis.
FIG, 99
GYRO-HORIZONTALS AND GYRO-VERTICALS
127
If, for any reason, the indicator post becomes deflected out of a
vertical plane perpendicular to the car axle, that is, out of the
plane of the diagram, Fig. 100, the contact arm C moves relative
to a trolley wheel T on the upper end of the pendulum P. An
electric circuit is thereby completed through one of the solenoids
S, a torque acts on the gyro-casing about the vertical axis, and the
ring R precesses about a horizontal axis till the indicator rod is
parallel to the pendulum.
Since the plane of vibration of the pendulum is perpendicular
to the direction of motion of the car, no change in velocity of the
FIG. 100
car while it is moving on a straight track will deflect the pendulum
from the vertical. When, however, the car is moving around a
curve, the pendulum bob will retreat from the center of the curve,
thereby deflecting the pendulum from the vertical. Consequently,
during the time that the car is going around a curve, the elec-
tric circuit controlled by the pendulum must be kept open.
An arrangement of contacts automatically keeps the pendulum
circuit open while the car is going around a curve. During
this time, the direction of the spin-axle is fixed in space because
the gyro is mounted so as to be free of any torque due to any
change in either the direction or magnitude of the velocity of
the car.
While the car is going around a curve, the spin-axle of the gyro-
scope tends to turn relative to the car about a vertical axis. This is
due to the fixity in space of the spin-axle. Again, the friction of
the pen on the paper produces a torque about the horizontal axis
128 THE GYROSCOPIC PENDULUM
of rotation of the ring. This torque also tends to produce pre-
cession of the spin-axle about the vertical axis.
The spin-axle should be kept as nearly parallel to the car axles
as possible because this is the position in which the gyroscope is
least affected by pen friction. For this reason, just as soon as the
spin-axle is deflected from parallelism to the car axle by as much as
two degrees, a torque is applied to the ring R about the horizontal
axis and of sufficient magnitude to prevent the turning of the spin-
axle about the vertical that otherwise would occur. The required
torque is produced by a pull on one or the other of two soft iron
cores, each having one end attached to the ring R and the other
end within one of two solenoids D fastened to the frame of the
apparatus. The circuit through one solenoid or the other is made
or broken as the trolley T\ moves from one contact plate across a
strip of insulation to another contact plate on the contact arm C.
By adding another control pendulum capable of turning about a
horizontal axis perpendicular to the one used in this apparatus, an
apparatus could be produced that would furnish a horizontal plane
which could be used wherever an artificial horizon is required.
80. Directed Gun-Fire Control. It is a complex problem to
make the observations and computations from which a gun on a
pitching and rolling ship may be directed so that the projectile
may strike another ship that is moving with respect to the first
ship and that is invisible to the gunners. The target must be
visible to an observer who is in communication with computers,
and the latter must be able to transmit orders to the gunners.
The observer may be on a mast above the smoke of battle, or he
may be in an airplane at such a height that he can see the target
which may be below the horizon of the gunners.
The observer notes the direction of the line from the target to
the mother ship, the range, that is the distance between the target
and the ship, and he estimates the course and speed of the target.
These data are communicated to the computers. They know the
course and speed of their own ship, the drift of the projectile from
each gun (Art. 48), as well as the approximate deflection produced
by the wind. There is an additional deflection when the axis of
the gun trunnions is not horizontal. If the right end of the trun-
nion-axis dips, the projectile will be deflected to the right of the
line of sight by an amount depending on the angle of dip. All
angles in azimuth must be measured from a base line which is
fixed relative to the earth. This base line is indicated by a gyro-
GYRO-HORIZONTALS AND GYRO- VERTICALS 129
compass of the highest degree of precision known to the art.
Such an instrument is called a " gun-fire control compass, " where-
as an instrument of a precision sufficient for navigational purposes,
but not for gun-fire control, is called a " navigational compass."
These data are set up on the dials of a computing machine called
a " range clock " or " range and bearing keeper." This instru-
ment then indicates automatically and continuously what the
range and the bearing of the target will be at any assigned sub*
sequent instant so long as the data remain unchanged.*
The appropriate angles of elevation and of train are transmitted:
to indicating instruments situated at the guns. The gunners
keep the gun pointed and trained in the directions given by the
indicators. When the ship has rolled to the proper angle the gun
is fired, either automatically or manually, as desired. An observer
called a " spotter/' stationed either on a mast or in an airplane,
notes where the projectile strikes and informs the computers.
The angles transmitted to the gunners are revised till the target is
hit.
As the various gun-fire control computing machines do not de-
pend on gyro-dynamics, they need not be described here. Various
sighting instruments and automatic firing mechanisms, however,
are stabilized by gyro-pendulums.
81. Gun-Fire Directorscopes. For a given angle of gun eleva-
tion with respect to the deck of a ship, there is but one angle of
roll of the ship at which a projectile fired from the gun will strike
the target. The optical system of a sighting telescope on a ship
may be stabilized by a gyro-pendulum so that the image of a dis-
tant target will remain stationary in the field of view however the
ship may roll. A co-acting firing mechanism may be set so that
when the ship has rolled to an assigned angle with respect to the
horizontal, one or more guns will be discharged. If the guns were
correctly loaded, correctly pointed with respect to the deck of the
ship, and correctly trained with respect to the meridian, the pro-
jectiles will hit the target. An apparatus consisting of one or
more sighting instruments combined with a co-acting firing mecha-
nism, designed to discharge, in the proper direction and at the
proper instant in the roll of the ship, the guns controlled by the
apparatus, is called a gun-fire direct orscope.
* U. S. Patents. Ford, No. 1370204, 1921; No. 1450585, 1923, No.
1472590, 1923; No. 1484823, 1924. Meitner, No. 1455799, 1923. E. Sperry,,
No. 1296439, 1919; No. 1356505, 1920; No. 1755340, 1930.
130 THE GYROSCOPIC PENDULUM
Angles in azimuth can be laid off relative to the N-S line of a
gyro-compass, and angles in elevation with respect to a gyro-
vertical or gyro-horizontal forming part of the apparatus. An
electric circuit through the gun, gyro-compass and firing mecha-
nism may be closed automatically at the instant the gun axis
makes the proper angle to the N-S line of the gyro-compass, and
the proper angle of elevation relative to a gyro-pendulum forming
part of the firing mechanism.*
* U. S. Patents. Schneider, No. 1507209, 1924; Radford, No. 1531132,
1925; Ford, No. 1597031, 1926; Grouse, No. 1689327, 1928.
CHAPTER IV
GYROSCOPIC ANTI-ROLL DEVICES FOR SHIPS
1. The Oscillation of a Ship in a Seaway
82. The Rolling of a Ship Due to Waves. From Archimedes'
Principle, a body either wholly or partially immersed in a fluid is
buoyed up by a force equal to the weight of the fluid displaced.
The weight of a body acts at a point called the center of gravity
of the body. The buoyant force acts at a point called the center
of buoyancy. The center of buoyancy is at the center of mass of
the fluid displaced.
If the center of gravity of a completely immersed body is below
the center of buoyancy, the body is statically stable, that is if
this body be tilted and then released it will recover its former
position. If the center of gravity of a completely immersed body
is above the center of buoyancy, the body is unstable; that is
if this body be tilted and then released, it will not recover its
former position but will turn over till the center of gravity is below
the center of buoyancy.
A floating body, however, may be statically stable when the
center of gravity is above the center of buoyancy. Figure 101
represents a ship standing upright in still water with the center of
gravity of the vessel at G and the center of buoyancy at B. Sup-
pose that a wave moves under the ship from the left to the right.
At the instant represented in Fig. 102, the center of buoyancy has
moved to the left of the line of action of the weight F g . The
weight F g and the buoyant force Fb now constitute a couple having
a lever arm x. This couple causes the ship to " heel " or roll,
in the clockwise direction. If the water surface is again level when
the ship has rolled into the position shown in Fig. 103, the center
of buoyancy is to the right of the line of action of the weight.
Now the couple tends to roll the ship back into the upright po-
sition. The advancing wave emerging from under the ship adds
a couple in the same direction.
The angular amplitude of vibration of a ship produced by a single
wave is always small. If, however, a series of waves pass under
131
132
ANTI-ROLL DEVICES FOR SHIPS
the ship at regular intervals, and the period of the oncoming waves
is nearly equal to the period of roll of the ship, then the angular
amplitude of roll quickly builds up to a considerable value. The
building up of a large amplitude of oscillation by the cumulative
effect of a periodic disturbance having nearly the same period as
that of the vibrating body is called resonance (Art. 26). The large
angle of roll, so common with ships at sea, occurs when the period
of the oncoming waves is nearly equal to a natural period of oscil-
lation of the ship. The period at which the oncoming waves meet
the boat depends on how fast they travel, the distance from one
FIG. 101
G
I
I
I
FIG. 102
FIG.
wave to the next, the course of the boat across the waves, and the
speed of the boat.
83. The Pitching of a Ship Due to Waves. If either the bow
or the stern of a ship with a single propeller be suddenly raised or
lowered, the precession thereby produced will deflect the ship's
course either to the right or to the left. This " yawing " or
" nosing " from side to side results in the pushing aside of a larger
mass of water by the moving ship and a consequent loss of power
and speed. Pitching increases the difficulty of maintaining the
ship's course and increases the fuel consumption. The yawing
produced by pitching can be neutralized by the use of two similar
propellers and shafts rotating in opposite directions. This device,
however, does not diminish the up-and-down pitching motion
with the accompanying discomfort to passengers, danger of injury
to freight, and strains within the frame of the ship, nor does it
reduce the natural yaw due to a wave not striking the bow and
stern at the same time.
84. The Metacentric Height. When a vessel floats upright,
the center of gravity G and the center of buoyancy B are on the
same vertical central line AA', Fig. 104. When the vessel is
heeled over, the center of buoyancy is displaced to one side of the
THE OSCILLATION OF A SHIP IN A SEAWAY
133
central line. So long as the heeling does not exceed about 10 degrees,
the line of action of the buoyant force intersects the central line A A'
near the same point M . That point in a floating body slightly dis-
placed from equilibrium, through which the
resultant upward force of the displaced liquid
intersects the vertical through the center of
buoyancy when the body is not displaced, is
called the metacenter of the body. If a ship be
slightly tilted from the equilibrium position,
the intersection M, of the line of action of the
force of buoyancy and the line perpendicular
to the decks of the ship through the center of
gravity, coincides with the metacenter of the
ship. The metacenter for rolling does not coin-
cide usually with the metacenter for pitching.
If the metacenter is above the center of grav-
ity, the vessel is stable; if it is below the center of gravity, the vessel
is unstable; if it coincides with the center of gravity, the vessel is
statically neutral. The distance MG from the metacenter to the
center of gravity is called the metacentric height of the vessel. The
degree of stability of a vessel depends upon the metacentric height.
A vessel with a metacentric height that is large in comparison
with the size of the vessel has a large righting moment, a short
period of roll, is very stable, but will
roll quickly through large angles and will
change its direction of roll with a jerk.
Such a vessel is difficult to steer, requires
an excessive amount of fuel for a given
speed, and is uncomfortable for crew and
passengers. Many cargo steamers carry-
ing coal or ore are uncomfortable and hard
to handle because of their great metacen-
tric height. If this cargo could be dis-
tributed so as to raise the center of gravity
of the vessel the severity of these troubles
would be diminished.
85. The Experimental Determination of
the Metacentric Height. The metacentric height of a vessel of
known tonnage displacement can be determined from an observa-
tion of the tilting produced by shifting a known weight of crew
or ballast for a known distance across the deck. In Fig. 105, a
FIG. 105
134 ANTI-ROLL DEVICES FOR SHIPS
vessel of known weight F g has been tilted from the upright posi-
tion through the small angle by moving a load F across the deck
through a distance x 1 '. The center of gravity of the vessel with
cargo is at G, the center of buoyancy at B, and the metacenter
at M . The small angle through which the ship has been tilted
is measured by means of a plumb-bob of length I hung at A.
Represent by the symbol b the distance along the deck that the
plumb-bob has moved when the vessel is tilted. Represent the
metacentric height MG by the symbol //.
Now the moment of the tilting force equals the moment of the
righting force. Taking moments about the metacenter and assum-
ing that the angle of tilting is so small that x' and 6 are approxi-
mately equal to their horizontal projections,
Fx' = F g x = F g H sin 4 = F g H ~
Whence, the metacentric height
86. The Period of the Rolling Motion of a Ship. By the
period is meant the time of a complete back-and-forth oscillation.
When a vessel has been rolled through an angle < there is acting
upon the vessel a righting torque, Fig. 105, having the value
L = F g x = FJI sin
where H is the metacentric height.
This equation shows that when the angle of roll </> is so small that
sin </> may be replaced by radians, then the restoring torque is
directly proportional to <p. This is the law of simple harmonic
motion of rotation (Art. 20). Consequently, if a vessel be rolled
through a small angle from its equilibrium position, and if it be
unacted upon by any torque except the righting torque, it will
roll back and forth with a periodic motion that is approximately
simple harmonic motion of rotation. On substituting in (37)
the value of the torque L = F g ll$ y we find that the period of
such a simple harmonic motion of rotation is
(98)
where K is the moment of inertia of the vessel relative to the axis
about which the rolling occurs. The negative sign indicates that
THE OSCILLATION OF A SHIP IN A SEAWAY 135
< is measured in the direction opposite to L. Wind and f Fictional
forces, and the forces due to the impacts of succeeding waves,
produce such additional torques that the rolling motion is not
simple harmonic and the value of the period cannot be computed
with precision.
If, however, the rolling of a ship be assumed to be simple har-
monic, then the H in (96) represents the metacentric height. In
this case, if a ship carries a gyroscope of great angular momentum
and with the spin-axle and the precession axis perpendicular to
the axis of roll, then the period of roll is altered as it would be by
(h 2 \
// -\ ^pj- Equation (98) shows that
an increase of the metacentric height of a ship produces a decrease
in the period of roll. Consequently, the rolling of a ship can be
caused to be quicker by mounting on the ship a large gyro capable
of spinning and processing as above indicated.
87. Methods of Diminishing the Amplitude of Roll. Roll
increases stresses in the ship's structure and engines; it increases
the effective area of cross-section of ship that must be pushed
through the water, and consequently the fuel consumption; it
decreases speed; it decreases the comfort of passengers and crew;
it decreases the accuracy, range and rapidity of fire from naval
vessels. These are ample reasons for the serious efforts made to
diminish or suppress roll. Anti-roll devices are commonly called
" ship stabilizers " because by their use the tilting of the ship from
the equilibrium position by an applied torque is diminished.
The most obvious device is to attach long planks lengthwise on
the outside of the hull below the water line. These so-called
" bilge keels " decrease the roll but they also decrease the speed
of the ship. To avoid the excessive friction through the water
when the stabilizing effect of the bilge keels is not needed, it has
been proposed to use fins that can be moved in and out of longi-
tudinal slits through the hull of the ship.* When the ship is rolling,
the fins would be protruded; when the ship is not rolling the fins
would be withdrawn.
Frahm's anti-roll tanks (Art. 28) have been used to a consider-
able extent, although their mass and the difficulty in adjusting the
size of the connecting passage to proper operation, have limited
the use of the Frahm system.
* U. S. Patents. Thompson and Schein, No. 1475460, 1923; Motora, No.
1533328, 1925; KSfeli, No. 1751278, 1930.
136
ANTI-ROLL DEVICES FOR SHIPS
Several schemes have been devised, and some of them developed
as far as the initial experimental stage, in which the property
of the axle of a spinning gyro-wheel of maintaining its direction in
space has been used to control an engine which in turn would move
large masses to the higher side of a rolling ship. The magnitudes
of the masses that would need to be moved has caused these
suggestions to be neglected.
At the present time gyroscopic anti-roll devices are in successful
use. Thev are much used on yachts and also to some extent on
large ships. The action of these devices is based
on the fact, proven in Art. 25, that if a body oscil-
lating with a periodic angular motion be acted upon
by a torque of the same period, and in the opposite
phase to the velocity of the oscillating body, then
the system producing the periodic torque will absorb
energy from the oscillating body.
2. The Inactive Type of Gyro Ship Stabilizer
88. The Effect on the Motion of a Swinging
Pendulum Produced by an Attached Gyroscope:
(a) When the Precession of the Gyro-Axle is Op-
posed by a Frictional Torque. Experiment. The
pendulum shown in Fig. 106 carries a gyro-wheel
mounted so as to have two degrees of rotational
freedom. The gyro- wheel spins about a nearly ver-
tical axis through the center of mass of the wheel
and attached frame, and it is capable of rotation
FIG. 106 about a perpendicular axis A A 1 in the plane of
vibration of the pendulum. For this experiment, the wheel and
its supporting frame are lowered till the latter axis is above the
center of gravity of the gyro- wheel and its supporting frame. The
gyro-frame is prevented from rotating too far out of the plane of
vibration of the pendulum by means of stops. A brake B is pro-
vided by means of which the precession of the gyro-axle can be
opposed by a torque of any desired magnitude.
Release the brake. Set the pendulum into oscillation while the
gyro-wheel is spinning. Observe that there is negligible damping
of the amplitude of oscillation of the pendulum when the preces-
sion is unopposed by the friction brake.
Release the thumbscrew that clamps the gyroscope to the pen-
dulum. Set the pendulum swinging while the wheel is spinning.
THE INACTIVE TYPE OF GYRO SHIP STABILIZER 137
Observe that the gyroscope twists back and forth about the axis
of the pendulum as the pendulum oscillates back and forth.
The angular acceleration of the motion of the pendulum is
accompanied by a torque on the gyro-frame about an axis parallel
to the knife-edge. This torque is maximum when the pendulum
is at the end of its swing and is zero when the pendulum is at the
v
FIG. 107
FIG. 108
middle of its swing. It produces a precession of the gyro-axle
about an axis A A', Fig. 107, perpendicular to the knife-edge.
This precession produces a torque L p on the pendulum about an
axis perpendicular to the spin-axle and to the axis of precession.
At any instant the magnitude of this torque is L p = h s w Pg , where
h s and w pg represent the instantaneous values of the angular mo-
mentum of the gyro-wheel about the spin-axis, and the angular
velocity of precession, respectively. Hence, when the gyro-axle
is passing through its equilibrium position, the gyroscopic torque
138
ANTI-ROLL DEVICES FOR SHIPS
acting on the pendulum is maximum and consequently the de-
flecting torque on the pendulum is maximum.
The swinging of the pendulum and the precession of the gyro-axle
have a common period. When the velocity of precession is maxi-
mum, the angle of precession is zero, Figs. 107 and 108. Con-
sequently, when the deflection 6 of the pendulum from the vertical
is maximum, is zero. The variation of 6 with respect to time is
represented by the curve marked 6 in Fig. 109. The variation of
with respect to time is represented to a fair degree of accuracy
by the curve marked 0.
FIG. 109
FIG. 110
The angular velocity of the pendulum due to gravity is zero
when the inclination of the pendulum to the vertical is maximum,
and the velocity is maximum when the inclination is zero. The
relation between the angular velocity of the pendulum and time is
represented approximately by the curve marked w p .
At the instant when the pendulum is at the end of a swing,
is maximum, the gyro-axle is passing through its mid-position
(0 = 0), the speed of precession Wp K is maximum, and the torque
LP acting on the pendulum due to the attached gyroscope is
maximum.
When the angle between the gyro-axle and the vertical is 0, Fig.
108, there is a vertical component of the torque acting on the pen-
dulum due to the precession of the attached gyroscope of the value,
Fig. 110.
L v LP sin = h s Wp g sin
This vertical component tends to twist the pendulum about the
axis of the pendulum, as was observed in the experiment. The
component about a horizontal axis parallel to the knife-edge is
= Lp cos = h s w pg cos
THE INACTIVE TYPE OF GYRO SHIP STABILIZER 139
The variation of Lhp with respect to time is represented with a
certain degree of precision by the curve marked L hPy Fig. 109.
The power imparted to the pendulum by the gyroscope at any
instant equals the product of the torque acting on the pendulum
at that instant due to the gyroscope L/,p and the angular velocity
of the pendulum, w p . The average value of the power during one
complete vibration is given by a curve of products of the instan-
taneous values of Lj t p and w p . It will be observed that the curves
of Lj lP and w p are in quadrature, that is, they differ in phase by
90 degrees or a quarter of a period. Now the average value of the
product of two sine curves of the same frequency is zero when the
curves are in quadrature (Art. 25). That is, if the relation be-
tween Lhp and time, and that between w p and time, were accu-
rately represented by sine curves in quadrature, then the average
power applied to the pendulum throughout one vibration by
the precessing gyroscope would be zero. Consequently, under
the conditions specified, the vibrations of the pendulum are
undamped.
If, however, the phase difference between the Lhp curve and the
w p curve were greater than 90 degrees, then the power imparted to
the pendulum by the precessing gyroscope would be negative, that
is, energy would be abstracted from the pendulum. The more
nearly the two curves are to being in opposite phase the greater
will be the damping. They may be brought more nearly into this
condition by retarding the LHP curve relative to the w p curve.
This can be done by opposing the motion of precession w pg by a
friction brake. Friction at the bearings A A', Fig. 107, constitutes
a torque about that axis in the direction opposite the precession.
This torque produces a precession of the inner gyro-frame about
an axis perpendicular to the plane of the diagram which opposes
the velocity of the pendulum and increases the phase difference
between L h p and w p . The power absorbed from the pendulum is
now no longer zero. The vibrations of the pendulum are damped.
The amplitude of vibration of the pendulum would not be damped
if there were no opposition to the precession of the gyro.
The method here indicated for bringing variations of the pre-
cessional velocities of the gyro-axle of a spinning gyroscope more
nearly into opposite phase with the variations of the torques
acting on an oscillating body that carries the gyroscope is the basis
of the action of the ship stabilizers of the inactive type designed by
Schlick, Fieux, and others.
140
ANTI-ROLL DEVICES FOR SHIPS
89. The Effect of a Spinning Gyroscope on the Rolling of a
Ship. Consider a ship on which is mounted a gyro-wheel spin-
ning about a vertical axis. Let the frame supporting the gyro-
wheel be capable of rotating about an athwartship axis. When the
ship rolls, the gyro-axle precesses about this axis A A', Fig. 111.
The effect of the precession of the gyro-axle on the rolling motion
of the ship is the same as that of the gyroscope on the motion of
the pendulum considered in the preceding Article. If the gyro-
axle precesses freely, there is zero damping effect; if the preces-
sion is opposed by a moderate frictional torque, damping of the
amplitude of roll is produced. The amount of damping would be
maximum if the torque opposing precession at each instant were
proportional to the speed of precession at that instant. Steering
is not affected by a gyro-wheel with vertical spin-axis. The period
of roll of the ship is decreased, (Art. 86).
A gyro- wheel spinning about an athwartship axis and capable oi
rotating about an axis A A' perpendicular to the elecks of the ship,
Fig. 112, will precess back and forth about this axis as the ship
rolls from side to side. As in the case of the gyro with vertical
spin-axle, the amplitude of roll of the ship will be damped if the
precessional motion be opposed by a frictional torque.
Suppose the direction of spin is that represented by the line h s ,
Fig. 113. Then while the ship is rolling to starboard with angular
velocity represented by the symbol w, the gyro-frame and ship
will be subjected to a torque about an axis perpendicular to h s
and w in the direction represented by the line L, (Art. 36). Owing
to this torque the ship steers to starboard. Similarly a roll to
THE INACTIVE TYPE OF GYRO SHIP STABILIZER 141
port causes the ship to steer to port.* For this reason a single
gyro-wheel with spin-axle horizontal cannot be used to stabilize
a rolling or pitching ship.
This yawing can be prevented by using two gyro-wheels G\
and (7 2 that are spinning in opposite directions about horizontal
\A
FIG. 113
FIG. 114
FIG. 115
athwartship axes, Fig. 114. These gyro- wheels are capable of ro-
tation about vertical axes xi and #2, respectively. When the ship
rolls, the two gyros precess in opposite directions about these
axes. As the spin velocities are also in opposite directions, the
torques developed by a roll of the ship are in the same direction.
* Suyehiro, " Yawing of Ships," Trans. Inst. Nav. Arch., 1920, pp. 93-101.
142
ANTI-ROLL DEVICES FOR SHIPS
If the processional motion of each is opposed by a moderate fric-
tional torque, the amplitude of ship's roll will be damped.
In Fig. 115, the velocity of roll at some instant is represented
by w. From Art. 36, the torque acting on the frame of Gi about a
vertical axis is in the clockwise direction as represented by the sym-
bol L], and that acting on the frame of (j 2 is in the counter-clockwise
direction as represented by the symbol L 2 . As these torques are
equal and oppositely directed, the pair of precessing gyro-wheels
produces zero effect on the steering of the ship.
FIG. 116
90. The Schlick Ship Stabilizer. This consists of a gyro-wheel
of great moment of inertia* spinning about a vertical axis and
capable of precessing about a horizontal athwartship axis AA f ,
Fig. 116. The precession axis is above the center of gravity of the
gyro-wheel and casing. The period of the apparatus when swing-
ing as a gyro-pendulum is made approximately the same as the
period of roll of the ship in a calm sea. The inclination of the
gyro-axle from the equilibrium position is maximum when the
rolling ship is in its equilibrium position, and it is zero when the
ship is at the end of a roll. When there is no opposition to pre-
cessing, there is a phase difference of 90 degrees between the pre-
cossional velocity of the undamped gyro-axle and the gyroscopic
torque acting on the ship (Art. 89). In this case there is zero
* U. S. Patent. Schlick, No. 769493, 1904.
THE INACTIVE TYPE OF GYRO SHIP STABIILIZER 143
absorption of energy from the ship's roll and consequently no
damping of the amplitude of oscillation.
To produce an absorption of energy of the ship's roll, that is,
a damping of the amplitude of roll, the phase difference between
the processional velocity and the torque acting on the ship due to
the precessing gyro-wheel must be more than 90 degrees. In the
Schlick ship stabilizer, the phase difference is increased by means
of a brake* applied to the precession axle. A considerable part
of the energy of the waves is used in producing precession of the
gyro-axle and then is transformed into heat at the brakes.
Professor A. Foppel has shownf that if precession of the gyro-
axlo be opposed by the proper frictional torque, and if the moment
of inertia of the gyro-wheel about t he spin-axis has the value
, (99 )
5 w s ^ '
then, during the time of one roll, a Schlick ship stabilizer will re-
duce the velocity of roll through the equilibrium position to 0.283
of the value it would have when the gyro-wheel was not operat-
ing. In this equation, <I> is the maximum amplitude of roll with-
out stabilizer, F g is the weight of the ship, // is the metacentric
height, K is the moment of inertia of the ship about the axis of
roll, and w s is the spin velocity of the gyro- wheel.
An essential part of this device for diminishing ship's roll is
the brake. The brake torque at each instant should be propor-
tional to the precessional velocity. Schlick has used band brakes
and hydraulic brakes. When the brakes are not applied, the zero
precessional velocity of the gyro, W PK , and the zero angle of roll of
the ship, 0, occur at the same instant, Fig. 109. Even with moder-
ate damping they occur at near the same instant. A ship must
roll through an appreciable angle before the precessional velocity
of a Schlick gyro-stabilizer, and the torque thereby produced,
become of sufficient magnitude to produce an effective opposition
to the roll.
* U. S. Patent. Schlick and Wurl, No. 944511, 1909.
f In the appendix to an article by Otto Schlick, " The Gyroscopic Effect
by Flywheels on Board Ship/ 7 Trans. Inst. Naval Architects, 1904, p. 117.
A determination of the constants of a Schlick ship stabilizer by Foppel's
method has been given in some detail by Professor John Perry in Nature,
77, 447 (1908).
Greenhill, Report on Gyroscopic Theory, pp. 34-36.
144
ANTI-ROLL DEVICES FOR SHIPS
The Schlick ship stabilizer shortens the period of the ship's
roll by increasing the metacentric height by the amount (Art. 86):
Gear
Segments
mK2g
The quantity within the parentheses represents the rotational
energy of the gyro, K s represents the moment of inertia of the gyro
with respect to the spin-axis, and m and K c refer to the entire ship.
A ship gyro-stabilizer of the inactive type is effective only when
the rolling of the ship is periodic. As a matter of fact, the rolling
of a ship in a seaway remains
periodic but for brief intervals of
time. The Schlick device has
not been built since the inven-
tion of ship stabilizers of the
active type (Art. 93).
91. The Fieux Ship Stabilizer.
This consists of two electri-
cally driven gyro-wheels of great
moment of inertia spinning in op-
posite directions about axes that
are horizontal and athwartship,
Figs. 117, 118. The two gyro-
frames are capable of precessing
about vertical axes. The two
gyro-casings are geared together
so that at any instant the velocities of precession of the two gyros
are equal and in the opposite direction. This arrangement avoids
any effect of the gyros on the steering of the ship. As shown in
Art. 89, the stabilizing torques developed by the two gyros are in
the same sense about the fore-and-aft axis of the ship.
The outstanding feature of the Fieux* stabilizer is the hydraulic
brake mechanism employed to absorb the energy of the ship's
roll. This device has the following important properties:
(a) It permits the maximum angle of precession.
(fe) It produces zero brake effect when the precession is changing
in direction.
(c) It produces maximum opposition to precession when the
velocity of precession is maximum and diminishes the brake effect
progressively as the precessional velocity decreases.
* Revue Maritime, 1924, p. 351; 1925, p. 180.
Fia. 117
THE INACTIVE TYPE OF GYRO SHIP STABILIZER 145
Figure 119 represents a vertical section through the stabilizer
in a fore-and-aft plane, and also a horizontal section through ab.
The brake mechanism is in a tank forming the base of the appa-
FIQ. 118
YIQ. 119
ratus. This base is filled with a mixture of glycerine and water.
Keyed to the lower end of the vertical precession axle x of each
gyro are two vertical vanes Fi, F 2 , which move back and forth
146 ANTI-ROLL DEVICES FOR SHIPS
within a cylindrical enclosure when the gyro precesses. The
vanes are pierced by openings controlled by gates connected
by an arm pivoted at B. The opposition to the flow of liquid
through the openings depends upon the pressure of the gates
against the vanes, and this depends upon the speed of precession
of the gyro. The braking power is also adjustable by varying the
leakage between the vanes and the walls of the enclosing cylinder.
Two sections of the enclosing cylinder are hinged at HI and // 2 .
By turning the screws S\ or 82, these sections can be moved in or
out, thereby changing the opposition offered to the motion of the
vanes through the liquid. This adjustment would be made by
hand and by an amount depending on the violence of the sea.
3. The Active Type of Gyro Ship Stabilizer
92. The Effect on the Motion of a Swinging Pendulum Pro-
duced by an Attached Gyroscope: (b) When the Gyro-Wheel
is Acted upon by an Outside Torque about an Axis Perpendicular
to the Spin-Axis and the Axis of Vibra-
tion of the Pendulum. Experiment.
The pendulum shown in Fig. 106 carries
a gyro-wheel mounted so as to have two
degrees of rotational freedom. The gyro-
wheel spins about an axis perpendicular
to the axis of vibration of the pendulum
FIG. 120 , , . u , j-i
and can rotate about a perpendicular
axis in the plane of vibration of the pendulum. In the present
experiment, the latter axis may be either on, above, or below
the center of gravity of the gyro-wheel and attached frame.
Hook one end of a stiff wire into an eyelet on the upper part
of the frame that carries the gyro-wheel. While the pendulum is
at rest, and the gyro-wheel spinning, pull the upper end of the gyro-
axle out of the plane of the fixed outer frame. If the gyro-wheel
is spinning in the clockwise direction as viewed from above, the
pendulum bob will be given a slight motion to the right. This is
in accord with the rule given in Art. 36. It is illustrated in Fig.
120, in which the line h s represents the angular momentum of the
gyro-wheel about the spin-axis, w represents the angular velocity
produced by the pull on the supporting frame, and L represents
the gyroscopic torque thereby produced.
Pull the upper side of the gyro-frame every time the pendulum
bob starts to move toward the right from the left end of its path,
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 147
and push it every time the pendulum starts to move toward the
left from the right end of its path. After a few oscillations, the
pendulum will be vibrating with a considerable amplitude of swing.
The building up of the amplitude of swing is due to the pendulum's
being acted upon by a series of separate torques occurring with
the same period as the natural period of the pendulum and always
in the same direction as the vibration of the pendulum. This is an
example of resonance (Art. 26).
Again, while the pendulum is swinging and the gyro-wheel is
spinning clockwise as before, pull the upper end of the gyro-shaft
every time the pendulum bob starts to move toward the left from
the end of its path and push it every time the pendulum bob starts
to move toward the right from the other end of its path. This
procedure causes the pendulum vibrations to become reduced to
zero after a few swings. In these two cases, observe that the
direction of the torque applied to the gyro-wheel that increases
the amplitude of vibration of the attached pendulum is opposite
the direction of the precession of the gyro-axle produced by the
motion of the pendulum, and that the direction of the torque
applied to the gyro-wheel that damps the amplitude of vibration
of the attached pendulum is the same as the direction of the pre-
cession of the gyro-axle produced by the motion of the pendulum.
Consequently, in the case of a gyro-wheel having two degrees of
rotational freedom mounted on a pendulum in such a manner that
the gyro-axle, the axis of precession and the axis about which the
pendulum vibrates are mutually perpendicular to one another, the
amplitude of vibration of the pendulum will be increased if the gyro-
axle be rotated by an outside torque in the direction opposite to the
precession, whereas the amplitude of vibration of the pendulum
will be decreased if the gyro-axle be rotated by an outside torque in
the same direction as the precession.
In the first case, the periodic gyroscopic torque L, Fig. 120,
acting on the pendulum is in phase with the angular velocity of the
pendulum. The pendulum is absorbing energy at each vibration
and the amplitude of vibration increases (Art. 65). In the second
case, the periodic gyroscopic torque applied to the pendulum
and the angular velocity of the pendulum are in opposite phase.
The pendulum is losing energy at each vibration with a conse-
quent diminution of the amplitude of vibration.
93. The Sperry Ship Stabilizer. The Sperry ship stabilizer
is designed to neutralize each roll increment soon after its incep-
148
ANTI-ROLL DEVICES FOR SHIPS
tion, thereby preventing the angle of roll becoming large. Its
effectiveness is not dependent on the period of roll being constant.
It applies the torque developed by the rotation of the spin-axle
of a gyro of great moment of inertia, to the damping of the
roll of a ship. The gyroscopic torque opposing roll is caused to
be in the opposite phase to the velocity of roll. Consequently
(Art. 65), the power absorbed from the rolling ship and the damping
of roll thereby produced are maxima.
The Sperry ship stabilizer comprises two electrically driven
gyroscopes, a powerful electromagnetic brake, an electric motor,
two motor generators and a switchboard. One of the gyroscopes
Mam
Uni
A C Gen
& C. Generator
FIG. 121
is small and of three degrees of rotational freedom. The spin-
axle processes when the ship rolls through even such a small angle
as a single degree. The other gyroscope is very large and of two
degrees of rotational freedom.
The arrangement of the parts of the device as seen on looking
from the stern toward the bow of the vessel is as sketched in Fig.
121. The main or stabilizing gyro (7, of great moment of inertia
with respect to the vortical spin-axis is non-pendulous and is
mounted so as to be capable of procession about an athwartship
axis. This gyro is spun by alternating current supplied by a
turbogenerator as indicated in the diagram. The spin-axle can
be rotated forward or back about an athwartship axis by means
of the " procession motor " M geared to the casing of the stabilizing
gyro. The procession motor is operated by current from a direct
current generator.
The direction of rotation of the precession motor and of the
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 149
connected main gyro-casing is controlled by the small uncon-
strained gyro. The spin-axis of the control gyro is horizontal and
athwartship, in a casing that can rotate about a vertical axis A,
Fig. 122, perpendicular to the plane of the diagram. Centralizing
springs &i6 2 are attached to a pin (h at one side of the casing.
Another pin r/ 2 attached to the casing can play back and forth
between two electric contacts e\ and e%. When the pin rf 2 is in
contact with ci, a current from the direct current generator rotates
the precession motor M in one direction. When contact is made
with e 2 , the armature of the precession motor rotates in the opposite
direction. (In the elevation, Fig. 121, the contacts e\ and e* are
shown one above the other. Really, one is behind the other
as indicated in the plan view, Fig. 122.)
FIG. 122
Fia. 123
The amount and speed of rotation of the spin-axle of the stabi-
lizing gyro G are regulated by a magnetic brake on the armature
shaft of the precession motor. This brake has strong springs that
seize the armature shaft, thereby preventing rotation of the sta-
bilizing gyro-axle except when the pressure of the springs is
opposed by the pull of an electromagnet. The brake coils are in
series with the precession motor. Consequently, when current is
cut off the precession motor, the magnetic brake seizes the arma-
ture shaft and prevents rotation of the stabilizing gyro about the
athwartship gudgeons FF, Fig. 121.
94. Operation of the Sperry Ship Stabilizer. Suppose that
the ship rolls to port even so little as one degree of angle, thereby
producing on the horizontal spin-axle of the control gyro a torque
L, about a horizontal fore-and-aft axis, Fig. 123. Then this axle
will precess in the direction Wp, thereby causing the pin d 2 to
move over and make contact with e\. This completes an electric
circuit through the precession motor and magnetic brake. As
the back electromotive force of the motor when at rest is very
150 ANTI-ROLL DEVICES FOR SHIPS
small, the initial current through the motor and magnetic brake
is large. The brake is completely released, the motor starts very
quickly and the stabilizing gyro-axle moves about the athwartship
axis with high angular acceleration in the same direction as the
precession due to the torque produced by the ship's roll. The
magnitude of the angular acceleration depends upon the design
of the motor and attached apparatus. About one-half second is
required for the control gyro to operate and about one second for
the precession motor to acquire full speed.
While the precession motor is gaining in speed, the back elec-
tromotive force increases in value, thereby reducing the current
in the motor and in the magnetic brake. When the precessional
speed of the spin-axle of the stabilizing gyro has the required value,
it is maintained practically constant for the required length of
time by allowing the proper amount of current to traverse the
brake coils.
After the stabilizing gyro-axle has processed to near the end of its
swing, the brake effect is suddenly increased, thereby bringing the
precessional speed to zero when the gyro-axle has made the speci-
fied maximum displacement from the central position. At this
instant nearly one-half period should have elapsed since the spin-
axle of the stabilizing gyro was in the central position
When the vessel starts to roll in the opposite direction, the spin-
axle of the control gyro precesses in the opposite direction, thereby
making an electric contact which causes the precession motor to
rotate in the direction opposite to its previous rotation. The
same succession of actions above described is now repeated.
The spin-axle of the stabilizing gyro moves back and forth in
opposite phase to the velocity of the ship's roll even though the
period of roll is not constant. Nearly all the energy that would
exhibit itself in the roll of the ship is converted into heat, and the
roll is quenched soon after its inception. If any single wave
would produce a roll less than that which the stabilizer can pre-
vent, then the control gyro automatically reduces the arc of pre-
cession and stops the precession as soon as this single roll has been
neutralized. On the other hand, if any single wave would produce
a roll greater than that which the stabilizer can neutralize, then
the ship will roll through a diminished angle. If the succeeding
waves are small, the roll will be entirely quenched
The complete stabilizer equipment of the Sperry active type
installed in a large yacht is represented in Fig. 124. It weighs
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 151
about 3 per cent as much as the ship. The Japanese 10,000-ton
airplane carrier, Hosho, the 2000-ton Italian flotilla leader, Piga-
fetta, and many yachts of various sizes up to 5500 tons are stabi-
lized by similar equipments. The stabilizer for a yacht costs in
the neighborhood of one-fortieth as much as the yacht. The
largest ship stabilizer ever built is on the Lloyd Sebaudo liner,
Conte di Savoia. The ship has a displacement of 45,000 tons,
metacentric height of 2.2 feet, and period of roll of about 24 sec-
FIG. 124
onds. The stabilizer consists of three complete units each having
a main gyro of 215,000 pounds, diameter 13 feet, moment of in-
ertia 4,700,000 pound-feet 2 , and speed of 910 revolutions per min-
ute. The entire equipment has a weight of about 624 tons and
cost about $700,000. It will prevent the roll of the ship exceed-
ing three degrees from its upright position.
95. The Braking System. This is a highly important part of
any ship stabilizer. The system used in the Sperry active type
now will be briefly described. Rolling of the ship produces on the
stabilizing gyro a torque that is proportional to the spin velocity
of the gyro and the angle of roll of the ship at the given instant.
152 ANTI-ROLL DEVICES FOR SHIPS
The torques acting on the stabilizing gyro due to rolling of the ship
are called pawive moments. These torques are greatest when the
gyro is locked so that it cannot precess.
The gyroscopic torque opposing roll is maximum when the spin-
axle of the stabilizing gyro is in the central position. When the
spin-axle tilts from that position, the torque diminishes in a man-
ner which depends upon the angle of tilt. Since only small torques
are developed when the tilt is large, it is common practice to limit
the precession to 60 degrees on each side of the central position.
When the stabilizing gyro approaches the end of the permitted
angle of precession it causes a " limit switch " to open the direct
current generator shunt field. At about the same instant the
control gyro also opens the same circuit. Farther precession of
the stabilizing gyro drives the precession motor as a generator.
Thus, kinetic energy of the stabilizing gyro is first transformed into
electric energy and then into heat. This action is called dynamic
braking. The combination of dynamic braking and the friction
braking produced by the magnetic brake can bring the precessional
speed to zero in less than one second of time.
The following additional braking effect is possible but is seldom
used. When the control gyro releases the magnetic brake, the
precession motor armature starts with high acceleration. With
increase of precessional speed of the stabilizing gyro there is a
decrease in the load on the precession motor. The passive moment
due to the rolling of the ship increases the angular speed of the
stabilizing gyro about the athwartship gudgeons. The angular
speed of the stabilizing gyro about the precession axis may become
so great that the precession motor acts as a generator. In case
this action occurs, the direction of current in the direct current
circuit reverses, thereby causing the direct current generator to
operate as a motor. These actions would result in a braking
effect that would diminish the velocity of precession. The op-
position to the motion of precession produced by the precession
motor acting as a generator is called regenerative braking.*
96. Rolling of a Ship Produced by a Gyro. In certain cases
it is desirable to be able to cause a ship to roll. For example, a
* Further information regarding the Sperry active type ship stabilizer
may be found in the following U. S. Patents:
Sperry, No. 1150311, 1915; No. 1232619, 1917; No. 1452482, 1923; No.
1558514, 1925. Schein, No. 1605289, 1926; No. 1617309, 1927; No. 1655800,
1928.
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 153
ship aground may be able to free itself if it can roll toward deeper
water; a ship stuck in mud may be able to free itself by rolling
back and forth till the keel is sufficiently loosened; a ship may be
able to break a passage through ice by rolling against the ice sheet.
A vessel equipped with a stabilizer of the active type will roll
with gradually increasing amplitude if there be applied to the main
gyro a periodic torque that is in opposite phase to the precession
of the spin-axle that would be produced by the natural rolling of
the ship in the direction in which rolling is desired (Art. 92).
By changing electric connections at the switchboard, the apparatus
may be caused to operate either as a reducer of rolling or as a
producer of rolling.
97. Admiral Taylor's Formula. The effect of waves on the
rolling of a ship is cumulative, each wave contributing a small
effect till the total angle of roll may become Inrge. By neutra-
lizing the small increments of roll, the angle of roll will not become
large. Admiral D. W. Taylor, U. S. N., has shown that a roll
increment <f> can be neutralized by a gyro-wheel having a moment
of inertia relative to its spin-axis of the value
AV = 1225 </>/> IIT (1QO)
iL
where K/ is expressed in pound-foot 2 units, is the difference in
degrees between two successive amplitudes of roll in the same
direction, D is the displacement of the ship in tons, // is the
motacentric height in foot, T is the poriod of roll in seconds, and
n is the spin- velocity of the gyro in revolutions per minute. The
required moment of inertia may be due either to a single gyro or
to two or more gyros.
The derivation of this equation never has been published, but it
may be seen in the Archives of the U. S. Navy Department in
Washington.
Problem. It is required to compute the principal elements of a ship sta-
bilizer of the non-pendulous active type that will quench a roll increment
= 5 ? where < is the difference between two successive angles of roll in the
same direction. The ship has a displacement D = 2200 tons, rnetacentric
height // = 2.5 ft., period of roll T - 13 sec.
The diameter of the gyro-wheel is to be 8 ft. On account of the limit of
fiber strength of steel, the peripheral speed must not exceed 33,000 ft. per min.
Stops are used that limit the amplitude of precession to 00 on each side of
the equilibrium position. A magnetic brake and a motor rotating at 500
154 ANTI-ROLL DEVICES FOR SHIPS
r.p.m. are used to control the precessional speed of the gyro-axle. The full
power of the motor together with the torque due to the ship's roll are used to
accelerate the precessional velocity for 1.25 sec. after the start of precession:
During the first quarter second, the acceleration is 0.3 rad. per sec. per sec.
Assume that there is a lag of 0.5 sec. between the inception of a roll and the
inception of precession. The precessional velocity attained in about 1.25
sec. after the inception of precession is maintained constant by the use of a
motor and brakes for such a length of time that a constant deceleration, then
developed, will cause the gyro-axle to come to rest 120 from one end of a
swing in the 6.5 sec. of one-half period.
The precession motor is connected to the gyro-casing by gears having diame-
ters in the ratio of 1 to 100. Assume that the efficiency of the gears is 80 per
cent, and that the moment of inertia of the gyro-wheel, casing, and gear with
respect to the precession axis is nine times that of the gyro-wheel with respect
to the spin-axle.
Compute: (a) The moment of inertia and the mass which the gyro-wheel
must have if it consists of a uniform disk 8 ft. in diameter; (6) values of the
roll velocity of the ship at quarter-second intervals throughout one-half cycle
of roll where the maximum amplitude of roll is 2 from the vertical; (c) the
horse-power of the motor which, making 500 r.p.m., will produce a mean ac-
celeration of the gyro-axle of 0.3 rad. per sec. per sec. during the first quarter-
second of precession; (d) values of the precessional velocity and angular dis-
placement of the gyro-axle at the end of each quarter-second interval through-
out one half-cycle; (V) the time at which the final deceleration of the preces-
sional velocity of the gyro-axle should begin; (/) the gyroscopic torque opposing
roll at the end of each half -second during one half-cycle of roll.
(g) Construct a table with the following data arranged in consecutive col-
umns: 1^ time of roll reckoned from the end of an oscillation; tp, time of
precession reckoned from the end of an oscillation; w r , instantaneous roll
velocity; w r , mean roll velocity during the preceding time interval; wp, in-
stantaneous velocity; <', angular displacement of the gyro-axle from end of
oscillation; </>/, displacement of gyro-axle from equilibrium position; <J>,
mean angular displacement of gyro-axle during preceding time-interval;
L#, gyroscopic torque opposing roll.
(h) Plot on the same time-axis, curves coordinating
w r and t r , </>/ and / f , L K and t r
Solution, (a) The Required Moment of Inertia of the Gyro-Wheel. -
From the equation of IT. S. Admiral D. W. Taylor, the required moment of
inertia of gyro-wheel and axle, with respect to the spin-axis, expressed in
pound-feet 2 , is
, = 122507)//T
Substituting in this equation the data of the problem:
IT ' *225(5 C H2200 tons) (2.5 ft.) (13 sec.)
K s = = 333350 Ib.-ft.'
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 155
The mass of a uniform disk 8 ft. in diameter that will have the above mo-
ment of inertia, from (22) :
(b) Velocities of Ship Roll at Instants One Quarter-Second Apart through-
out a Roll of Amplitude Two Degrees and Period Thirteen Seconds. In
the case of a body vibrating with simple harmonic motion of rotation of
period T, the angular velocity at time / after leaving the end of an oscillation
(41), is
. 2wt
w t = w e sin -yr
where w e represents the velocity when traversing the equilibrium position.
If the amplitude of vibration measured from the equilibrium position be <f>
radians or <,
[<f> 2 7Td> "I <
= 2 TT Y T = 57~3~f I = cj~i2T T ratlians P er sec ' ( 102 )
Consequently,
.0 / j \
wt = qTo~T s * n (360 if ) radians per sec. (103)
We shall assume that during one roll of a Nhip, the motion is simple har-
monic. Then, if the stabilizer keeps the ship within 2 degrees of the vertical,
the velocity of roll at time tr of the ship considered in this problem is
c\ ^f\0 /
w * ~ 7f\ iowi^ 8m ~~TQ~ ~ 0.017 sin (27.7 t r ) radians per sec. (104)
(y. \2i} (loj io
The mean roll velocity during an interval while the instantaneous roll
velocity changes at a uniform rate from w r ' to w f " Ls
Wr = w r ' + | (w r " ~ W/) (105)
Values of the instantaneous roll velocities at the end of half-second intervals
throughout a half-cycle of roll are given in column 3 of the table (p. 161).
Values of the mean roll velocities during each half-second interval are given
in column 4.
(c) The Horse-Power of the Motor Which, Making 5(X) R.P.M., will Pro-
duce a Mean Acceleration of the Gyro-Axle of 0.3 Radian per Second per
Second During the First Quarter-Second of Precession. The mean angular
acceleration during any time interval
a- T >
3 - K?
where Lp represents the mean torque acting upon the precessing system with
respect to the axis of the gudgeons about which the system precesses, and Kp r
represents the moment of inertia of the same system with respect to the same
axis. The total torque is the sum of the torque, L\ t due to the precession mo-
tor, the torque, L 2 , due to the ship's roll and the torque, L 3 , due to the center
of mass of the system being not on the axis of precession.
156 ANTI-ROLL DEVICES FOR SHIPS
Since the stabilizer of the present problem is non-pendulous, the torque
L z 0, and we may write
i, - ^* (106)
Since the motor is connected to the gyro-casing by gears of 80 per cent
efficiency, and the ratio of the number of teeth on the motor shaft to the num-
ber on the gear attached to the gyro-casing is 1 : 100, the torque at the gyro
due to the precession motor is
L! = (0.8) (100) Li'
where LI is the torque at the motor.
The power of the precession motor expressed in horse-power is
P. = L
h 33000
whence,
r , 33000 Pj t ln - D ,
Ll = 2(3 J = 10 ' 5 Ph at
and
Li [= 80 Li'] = 840 PH at the gyro
Now we shall find the mean torque L 2 acting on the gyro due to the ship's
roll. When the mean roll velocity of the ship is w>, and the gyro-axle is in the
equilibrium position, the mean torque acting on the gyro has the value,
L 2 = K s w s wr
where Ks is the moment of inertia of the gyro-wheel with respect to the spin-
axis expressed in slug-feet 2 and velocities are expressed in radians per sec.
If moment of inertia expressed in pound-feet 2 is represented by K s ' t and spin-
velocity when expressed in revolutions per minute is represented by n y then
the preceding equation assumes the form:
T __ KS 2-n-n-- _ K s 'nw r
' "" 3271 "GO" Wr ~ ~307~
The mean tor(|iie acting on the gyro during the time that the mean velocity
of roll is 77> and the mean displacement of the gyro-axle from the equilibrium
position is <I>, has the value
Yf/wvcos* 333350(1 31 0) _ ~ 1yl0 o,mn- ^:
Tj7v7~~ = - "in- - Wf cos * = 1422400 -M> cos <f>
From column 4 of the table (p. 101), the mean roll velocity w r of the ship
while the gyro-axle has preceded for 0.25 sec. from the end of a swing is 0.00504
radian per sec.
The mean angular displacement * of the gyro-axle from the equilibrium
position during any time interval (V t) is
* = GO - -J- (</ + 02') (107)
where the quantities within the parenthesis represent the values of the in-
stantaneous angular displacement of the gyro-axle at the beginning and end
of the time interval. At zero time, 0i' =0. At time 0.25 sec., <&/ will not
be greater than one degree. Suppose that we assume it to be one degree.
Whether this assumed value be half the correct value or two times that value
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 157
will make an inappreciable difference in the computed value of the horse-power
required to produce the required acceleration of the gyro-axle. Also, as in
the subsequent calculations, we shall use a rounded-ofY number for the horse-
power instead of the computed value, it will be safe to assume ir the present
computation that at time 0.25 see. 0/ 1. In this case:
4>~ = 60 - 5 (<// + 0,') - GO - I (</> + 1) - 59.5 (108)
Hence,
L 2 = 1122400 i/yeos* = 1422400(0.00501)0.507 - 3635 Ib.-ft.
From the assumptions of the problem,
AV[= 9 AY] - 9(333350) lb.-ft. 2 = 3,000,000 lb.-ft. 2
and the acceleration of the gyro-axle during the first quarter-second is a = 0.3
radian per sec.
Substituting in (100), these values of a/,, L } , L and Kp', we have
3<>35)32J
Q = _-
" 3,000,000
Whence, the power required of the precession motor is
Ph = 29 II. P.
In the subsequent computations we shall use 30 1 1. P.
(d) Values of the Angular Velocity and Displacement of the Gyro-Axle at
the End of Each Quarter-Second Interval throughout One Half-Cycle. ~-
Representing the velocity at the beginning and at the end of a time interval t
by WQ and wp t respectively,
( L *+ L *\ , /^M />/, + ! 122 100 ?/;, cos TiA
( - K , }t = w n + ( - K >
For the first quarter-second interval / = 0.25 sec., /r = and
w r = 0.00504 rad. per sec.
Hence,
, .0504 cos /rt orV)0 . .
w p = + __-___ (0.25)32.1 rad. per sec.
We shall now find the mean angular displacement *t j of the gyro-axle from the
equilibrium position during this interval. Our method will be to make a guess
of the angle moved through during this interval and then check the accuracy
of the guess. First we shall test the guess that during this time interval the
gyro-axle precesses one degree. In this case the mean displacement from the
equilibrium, from (108), would be 59.5 and the angular velocity at the end of
the interval would be, from the preceding equation
wp = 0.077 rad. per sec.
An inappreciable error will be made if we assume that during this brief time
interval the angular acceleration is constant. In that case, the displacement
from the end of a swing would be
<' = 4>\ 4- I w pt radians
= + j [0.077(0.25)57.3] - 0.55
and the mean angular displacement from the equilibrium position would be
J = 60 - HO + 0.55) = 59.72
158 ANTI-ROLL DEVICES FOR SHIPS
This value is slightly different from that obtained from the assumption in
(108). Our guess of one degree displacement during the first quarter-second
was too high. The displacement was more nearly 0.55. If now we go
through a computation as above with 0.55 as the assumed value of <', we
shall find that this value will be checked very closely. This is the value to
be used.
In the same manner we find values for instantaneous precessional velocity
at the end of each quarter-second interval during the first 1.25 sec. of pre-
cession, the angular displacement of the gyro-axle from the end of an oscil-
lation and from the equilibrium position at the end of each quarter-second in-
terval, and the mean angular displacement during these quarter-second in-
tervals. The values found are given in columns 5, 6, 7, and 8 of the table.
After 1.25 sec. from the beginning of precession, the brake is applied so as
to maintain the precessional velocity constant till the instant at which the
velocity is to be given a constant deceleration sufficient to bring the gyro-axle
to the end of the 120 swing in 6.5 sec. from the time it was at the other end
of the swing. While the velocity of precession is constant, that is, from tp =
1.25 sec. till the instant when the final rapid deceleration is started, values of
0', </>/, and 4> can be obtained from the equations:
0' = 0/ + wpt 57.3 (1 10)
0/ = 60 -0' degree (111)
* = 00 - -J (0i' + 2 ') degree (112)
For example, when tp = 1.25 sec.:
0' = 14.78 + 0.4333(0.25)57.3 - 20.99
0/ = 60 - 20.99 = 39.01 = 39 V
* = 60 - i (20.99 + 14.78) = 42.12 = 42 20'
(e) The Time at Which the Final Deceleration of the Precessional Velocity
of the Gyro-Axle Should Begin. As stated in the problem, the angular ve-
locity of the gyro-axle 1.25 sec. after leaving one end of a swing is to be main-
tained constant until the gyro-axle has come to near the other end of the swing.
When the gyro-axle has rotated for the proper time, the brake effect is suddenly
increased, thereby bringing the velocity to zero at the end of the 6.5-sec. half-
period. We shall now determine the number of seconds after the inception
of precession, when a constant deceleration must begin in order 'that the
gyro-axle may attain a displacement of 120 from the other end of the swing
in 0.5 sec.
From column 5 of the table (p. 161), we see that 1.25 sec. after the inception
of precession, or 1.75 sec. after the inception of the roll, the angular velocity
of the gyro-axle is 0.4333 rad. per sec. This is represented by the point J5,
Fig. 125. Suppose that the constant deceleration must begin at some point
C, which is t\ sec. later than B. Represent by t 2 the time interval from C
to the end of the 6.5-sec. half-cycle at D, when the gyro-axle has traversed
120 to this end of a swing. Then, reckoning time from the inception of the
roll:
fc + fe = (6.5 - 1.75) sec. = 4.75 sec.
ti 4.75 sec. 2
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 159
From column G of the table (p. 161), the angle through which the gyro-axle
moves during the time ti + 1 2 is
120.0 - 14.78 = 105.22 = 1.836 rad.
Since the deceleration is uniform, the mean velocity of precession during the
time tz is \ (0.4333) rad. per sec. Hence,
1.836 rad. = (0.4333)*i + I (0.4333)fc
= (0.4333)(4.75 - fe) + (0.2166) 2 = 2.058 - 0.2166 h
Consequently,
(2.058 1.836) rad. , _
^ _. _. ' _ in
' f\ i -*-!//-> i i .v/
0.2 Ibb rad. per sec.
Therefore, the final deceleration of the precession of the gyro-axle must be
started (6.5 1.0) sec. = 5.5 sec. after the inception of the roll or 5.0 sec.
after the inception of precession.
rad
sec.
o.4
JO.Z
oO.I
1 234567
Time after inception of roll in sees
FIG. 125
If 2 comes out with a negative sign, we would know that with the present
constant angular velocity of the gyro-axle during the middle part of a half-
cycle, the gyro-axle would need to precess for longer than the half-period be-
fore it would reach the end of an oscillation. In this case we would need to
increase the velocity during the middle of the half-cycle by allowing the
initial acceleration to last for a longer time than that allowed in the problem.
Again, if k is positive, but so small that the final deceleration would need to be
completed in too short a time, then an impracticable brake torque may be
required. In this case, also, we would allow the initial acceleration to last for
a fraction of a second longer than that previously allowed.
The value of ^ should be from 0.5 sec. to 1.0 sec. For a gyro-wheel of the
size here used, the value of t 2 we have obtained is larger than strictly necessary.
We could have stopped the initial acceleration a little sooner.
(/) The Values of the Gyroscopic Torque Opposing Roll at the End of
Each Half -Second During One Half-Cycle of Roll. When the precessing
gyro-axle is in the equilibrium position, the torque exercised by the gyroscope
on the ship is
[ = K s w s wp] =
K s r 2 TTU
(333350)1310
- 307 - "
wp = 1422400 wp
160
ANTI-ROLL DEVICES FOR SHIPS
When the precessing gyro-axle is inclined at the angle </ to the equilibrium
position, the gyroscopic torque opposing the ship's roll is
L g = 1422400 wp cos </
(113)
FIG. 126
On substituting in this equation values of the precessional velocity and dis-
placement of the gyro-axle from the equilibrium position as given in col-
umns 5 and 7, we obtain the values of gyroscopic torque opposing the ship's
roll given in column 9 of the table (p. 161). These values are plotted against
time after the inception of roll, in the curve, Fig. 126.
THE ACTIVE TYPE OF GYRO SHIP STABILIZER 161
1
2
3
4
5
6
7
8
9
1 *3
*-?
o>
*>
A fl
bO
Time from end of ro!
(starting with ship a
maximum angle)
Time of precession
measured from max
displacement
(Precession lags 0.5 se<
Instantaneous roll
velocity
(Eq. 104)
Mean roll velocity
during preceding tin
interval
(Eq. 105)
Instantaneous pre-
cessional velocity
(Eq. 109)
Angular displacemer
of gyro-axle from en
of oscillation
(Eq. 110)
Displacement of gyr
axle from equilibriui
position
(Eq. Ill)
Mean angular dis-
placement of gyro-
axle during precedin
itime interval
(Eq. 112)
Gyroscopic torque
opposing roll
(Eq. 113)
^
tr
tp
y
uy
wp
*'
<t>t
*
Lg
seconds
seconds
rad. per
rad. per
rad. per
defireoH
degrees
degrees
Ib.-ft.
soc.
sec.
aoo.
00
00000
00
25
00205
00102
50
00406
00301
0000
60
75
0.25
0.00602
00504
0770
55
59 27'
59 43'
5.6X10.4
1 00
0.50
0.00788
00695
1581
2 24
57 46'
58 37'
12 00
1 25
75
00962
00875
2439
5 12
54 53'
56 19'
20 00
1 50
1.00
01129
01045
3353
9 27
50 44'
52 d 49'
30 20
1 75
1.25
01272
01200
4333
14 78
45 13'
47 59'
43 4
2 00
1.50
0140
01336
0.4333
20 99
39 1 '
42 20'
47 9
2 50
2 00
0159
0149
4333
33 41
26 35'
32 48'
55 1
3 00
2 50
0169
0164
4333
45 83
14 10'
20 23'
59.7
3 50
3.00
0169
0.0169
4333
58 25
145'
7 58'
61 6
4 00
3 50
0159
0164
4333
70 67
-10 40'
-4 28'
60 5
4 50
4 00
0140
0149
4333
83 09
-23 5'
16 53'
56 6
5 00
4 50
01129
01264
4333
95 51
-35 31'
29 18'
50 2
5 50
5 00
00804
4333
107 29
-47 17'
41 24'
41.8
uniform
aooel.
6.50
6 00
00000
120
-60 00'
00
CHAPTER V
NAVIGATIONAL COMPASSES
1. The Various Types
98. The Altitude Azimuth Method of Locating the Geographic
Meridian. From the known position of a celestial body, to-
gether with simple astronomical observations, the direction of the
true north at the place of observation can be determined. Accu-
rately predicted positions of various celestial bodies as well as the
times at which certain celestial phenomena will occur are given in
Nautical Almanacs published annually by the governments of our
maritime nations. Useful tables and all the standard methods of
determining navigational quantities are given in Bowditch's
American Practical Navigator published by the Hydrographic
Office of the U. S. Navy.
The geographic meridian of a place on the earth is the great
circle of the earth that passes through the given place and the
poles of the earth. This line is in the true north-south direction
at the given place. The line in the direction of the horizontal
component of the magnetic field of the earth at any given place is
called the magnetic meridian at that place. At very few places is
the magnetic meridian in the plane of the geographic meridian.
A magnetic needle unaffected by any force except that due to the
earth's magnetic field sets itself in the plane of the magnetic merid-
ian at the given place. The compass bearing of any object is the
angle at the center of the compass card between the magnetic
axis of the compass needles and the straight line from the center of
the card to the given object.
A celestial body may be located by two quantities. There are
three pairs of such quantities commonly employed to locate an
object in the sky. They are called altitude and azimuth, declina-
tion and right ascension, celestial latitude and celestial longitude.
The celestial sphere is an imaginary sphere of infinite radius
onto which, to an observer on the earth, the celestial bodies appear
to be projected. Figure 127 represents the celestial sphere, drawn
as though it were of finite radius. The small circle represents the
162
THE VARIOUS TYPES
163
earth with the poles marked N and S, respectively. The celestial
poles are the points (XX f ) at which the prolongation of the earth's
axis NS intersects the celestial sphere. The celestial equator or
equinoctial is the great circle (QEQ'W) formed by the intersection
of the celestial sphere and the plane of the earth's equator. The
great circle H n EH 5 W is the celestial horizon for an observer at A.
The celestial meridian, declination circle, or hour circle of any
celestial body F is the great circle of the celestial sphere (XYX f )
passing through the given body and the celestial poles. The
celestial meridian of the place A
on the earth is ZXQ'X'Q. The
point Z at which a vertical line
from a place A on the earth inter-
sects the celestial sphere is called
the zenith of the given place. The
half of a celestial meridian which
lies on the same side of the equi-
noctial as the zenith is called the
upper branch; the other half is
called the lower branch.
At a given place A, the polar
angle or hour angle of a celestial
body Y is the angle 4> at the celestial polos between the meridian
of the place and the meridian of the celestial body. The azimuth
or true bearing of a celestial body Y is the angle between the me-
ridian of the observer at A and the vertical great circle passing
through the body. In Fig. 127 the azimuth 6' of the body Y
measured from the north is X'ZY, while measured from the south
it is X'Z' Y. Azimuths are to be reckoned from the north in north
latitudes and from the south in south latitudes. The altitude of
any body is its angular distance YD (or angle YAD) from the
horizon of the observer, measured upon the vertical great circle
through the given body. The zenith distance of a body is its
angular distance YZ (or angle YAZ) from the zenith, measured
on the vertical great circle through the body. Zenith distance is
the complement of the altitude. The declination of a body Y
is its angular distance F Y (or angle FA Y) from the equinoctial,
measured on the celestial meridian of the given body. It is desig-
nated as north (+) or south ( ), according to the direction of the
body from the equinoctial. The polar distance of a celestial body
Y is its angular distance YX (or angle YAX) from the pole, meas-
164 NAVIGATIONAL COMPASSES
ured on the celestial meridian passing through the given body.
If the polar distance and the declination are measured from the
same pole, the polar distance equals 90 declination; if they are
measured from opposite poles, the polar distance equals 90 +
declination.
The hour angle of the sun relative to some chosen place on the
earth is called the local apparent time at the given place. Ap-
parent time is expressed either in hours or in degrees. Thus, we
may speak of the sun being a certain number of hours, minutes
and seconds east (or west) of Greenwich. One hour equals 15
degrees of arc. As the apparent motion of the sun relative to the
earth increases and decreases in the course of a year, it is con-
venient to think of a fictitious " mean sun " that is assumed to
have a uniform motion relative to the earth. The Civil Day
begins at the instant of transit of the mean sun across the lower
branch of the meridian of the observer, that is, at midnight.
Clocks indicate civil time, that is, the hour angle of the mean sun,
measured from the lower branch of the meridian of some selected
place on the earth Greenwich, for example. At any instant,
the difference between the apparent and the mean time, that is,
the difference between the hour angles of the apparent and the
mean sun, is the equation of time. Equations of time are tabu-
lated in the Nautical Almanac for every even hour of Greenwich
civil time throughout the year. Knowing his longitude from
Greenwich, an observer with a clock keeping Greenwich civil time
can find his local civil time. Then, from the equation of time
given in the Nautical Almanac, he can find his local apparent
time.
Whenever the sun is visible, the compass bearing can be ob-
served. If we know the three sides of the spherical triangle XYZ,
we can compute the azimuth or true bearing of the sun, measured
from the north, by means of one of the standard equations of
spherical trigonometry,
Cog2 i e = rinjfrJjL=.l)jgS(* + y+.g) (n4)
sin x sin y
where x, y, and z represent the sides of the triangle opposite the
corners X, Y, and Z, respectively. The difference between the
computed azimuth and the observed compass bearing of the sun
is the angle between the north-south lines of the compass card and
the geographical meridian where the compass is situated.
THE VARIOUS TYPES 165
From Fig. 127 we find the following values for the quantities
in this equation :
sin^[x + y - z] = sin [(90 - latitude) + (90 - altitude) -
polar distance]
- sin [90 - (lat. + alt. + p.d.)]
= cos \ (lat. + alt. + p.d.)
sin %[x + y + z] = sin [90 - \ (lat. + alt. + p.d.) + p.d.]
/lat. + alt. + p.d.
= cos
1
sin x sin (90 lat.) cos lat.
1 1 1
= sec. lat.
= sec. alt.
sin y sin (90 alt.) cos alt.
Substituting these values in (114):
, n , , ,, , , , /lat. + alt. + p.d. , \
= cos 2 (lat. + alt. + p.d.) cos ( L p.d.)
X (sec. lat.) (sec. alt.) (115)
cos 2 -J- a =
The computation of the left-hand member can be simplified by
expressing 6 in times of its supplement 0', that is, by reckoning
azimuths from the south instead of from the north. Thus,
cos 2 \ - cos 2 * (180 - YZX') = cos 2 (90 - 0') = sin 2 \ tf
= \ vcrsin 0' = hav (180 - 0).
Half versines of angles, or " haversincs " as they are called, are
tabulated in Bowditch's American Practical Navigator and in
other books on navigation.
99. The Directive Tendency of a Magnetic Compass. A
magnetic compass needle tends to set itself in the direction of the
horizontal component of the magnetic field where it is situated.
The magnetic poles of the earth do not coincide with the geographic
poles. The magnetic north pole is situated in Boothia Peninsula,
Canada, at latitude about 70 N., longitude about 96| W. more
than a thousand miles from the geographic north pole. The mag-
netic south pole is at latitude about 73} S., longitude about
147| E. The number of degrees of angle between the geographic
meridian at a particular place and the axis of a compass needle
free to turn in a horizontal plane is called the magnetic declina-
tion at the particular place.
A line connecting all adjacent points on the earth at which the
magnetic declination is zero is called an agonic line. One agonic
166 NAVIGATIONAL COMPASSES
line is an irregular curve which in the western hemisphere extends
from longitude about 96 W., at latitude 70 N., to longitude
about 28 W., at latitude 70 S.; and in the eastern hemisphere
extends from longitude about 28 E., at latitude 70 N., to longitude
about 138 E., at latitude 70 S. A compass needle on an agonic
line places itself in the geographic meridian, that is, points true
north and south. At various places on the earth, a compass
needle that is uninfluenced by any magnetic field except that of the
earth will show declinations as great as 180 degrees. In the state
of Maine the compass points about 25 degrees west of the geo-
graphic meridian, and in the state of Washington it points to the
east an equal amount. The compass declination at any point on
the earth changes with time. At New York harbor the declina-
tion now is about 11 degrees and is increasing at the rate of 6
minutes per year. The magnetic declinations are known for all
parts of civilized lands and navigated areas. They are not known,
however, for large areas within the arctic and antarctic zones.
The directive tendency of a magnetic compass depends upon the
magnitude of the horizontal component of the earth's magnetic
field. At no place within either the arctic or the antarctic zone
is the horizontal component greater than about one-half the value
in New York. Within considerable areas it is nearly zero.
100. The Deviations of a Magnetic Compass on an Iron Ship.
While a ship is being built, the magnetic field of the earth causes
the iron structure to become a big magnet with the north-seeking
pole toward the north. The various hammering operations facili-
tate the magnetic induction. If the ship is heading north while
being built, there will be developed a north-seeking pole at the
lower part of the bow and a south-seeking pole at the upper part
of the stern. The steel parts of the ship will retain a part of this
magnetism after the ship is launched and is pointing in any di-
rection. A ship that was built with the keel north and south will
give no deviations due to this subpermanent magnetism when
pointing either north or south. As the ship is rotated 360 degrees,
the compass will deviate to the east in one semicircle and westerly
in the other (Fig. 128). Deflections of the compass from the mag-
netic meridian due to this cause are called semicircular deviations.
In many cases, semicircular deviations amount to as much as 20
degrees. The semicircular deviation of a compass changes with
change of geographical position. If the bow of the ship has been
toward the east when building, a north-seeking pole would be
THE VARIOUS TYPES
167
developed on the port side. The permanent magnetism of the
steel of the ship would give deviations represented by Fig. 129.
The soft iron of a ship becomes magnetized by the earth's
magnetic field in which it is situated. As a ship is headed in
different directions, the magnetic poles induced in horizontal
masses of iron change their position relative to the ship. The
compass deviations due to this temporary magnetism are easterly
while the head of the ship is pointing in the quadrant from magnetic
-
N
FIG. 128
N
FIG. 130
north to east (Fig. 130), and also while the ship is pointing from
magnetic south to west. They arc westerly while the ship is
pointing from east to south and also when it is pointing from west
to north. These deviations are called quadrantal deviations.
In many cases, quadrantal deviations are as much as 10 degrees.
The quadrantal deviation of a compass depends upon the direction
of the fore-and-aft axis of the ship relative to the magnetic meridian
at the place where the ship is situated.
When an iron ship either rolls or pitches, the compass is affected
by changes in the vertical induction of elongated soft iron masses
as well as by changes at the compass of the subpermanent magnetic
field of the steel parts of the ship. This so-called " heeling error "
may amount to more than one degree of compass deviation per
degree rolling or pitching of the ship.
A large part of the total deviation of the compass due to fixed
masses of iron and steel can be eliminated by properly placed bar
magnets, soft iron rods and spheres in the neighborhood of the
needle. The residual deviations can be determined experimentally
and plotted on a curve for the use of the navigator. However,
as the direction and intensity of the earth's field are subject to
168 NAVIGATIONAL COMPASSES
variations and as the magnetic condition of an iron ship is not
constant, the compass indications must be checked frequently
against the direction of the geographic meridian as given by the
pole star or other celestial body. The directive force acting on
the needle of an adjusted magnetic compass is very feeble on the
deck of a war ship and is nearly zero within a submarine. It is
profoundly altered when large guns and turrets are changed in
position.
101. The Deviation of a Magnetic Compass Produced by a
Rapid Turn. If a hardened steel needle be balanced on a pivot,
and afterwards magnetized, the north-seeking end will dip below
the pivot when the needle is in northern latitudes. To use the
magnetized needle as a compass it is kept horizontal by the addi-
tion of a counterpoise to the south-seeking end. The center of
mass of the needle is no longer vertically below the pivot but is
toward the south-seeking end of the needle. If the compass needle
is on an airplane making a rapid turn, the center of mass of the
noodle will lag behind the north-seeking end, thereby causing the
north-seeking end to move in the direction the airplane is turning.
Thus, when an airplane is executing a rapid turn, the angle of turn
indicated by the compass readings is less than the angle actually
turned.
If an airplane while moving northward in northern latitudes
executes a rapid turn eastward, the compass indicates that the air-
plane is pointing west of the true course. The angular speed of
the noodle may be greater than the angular speed of the airplane.
In this case, the airplane will appear to be turning westward when
really it is turning eastward.
An airplane cannot make a turn unless a force be applied toward
the center of the curve. To produce this centripetal force the air-
plane is tilted about a fore-and-aft axis, the underside of the wings
being directed away from the center of the curve. This operation
is called " banking. " In case the angle of bank, that is, the tilt
of the wings from the horizontal, is insufficient to develop the
required horizontal thrust against the wings toward the center
of the curve, the airplane will not follow a circular path but will
slide off along a diverging spiral. When this occurs, the north-
seeking end of the compass needle will be deflected in the direction
opposite that in which the airplane is turning.
While an airplane is making a turn, the magnetic compass may
deflect through a considerable angle either in the direction of the
THE VARIOUS TYPES 169
turn or in the opposite direction. The needle may even spin in one
direction if the turn is sudden and the angle of banking is correct, or
in the opposite direction if the angle of banking is much too small.
102. The Earth Inductor Compass. A magnetic compass on
the instrument board of an airplane is subject to deviations due to
proximity to the large mass of magnetized steel and unmagnetizcd
iron of the motor. It is also subject to errors when the airplane
makes a turn (Art. 101). Close attention is required to observe
the compass card indication to distinguish between an indica-
tion of 43 degrees from one of 48 degrees, for example. It would be
better, especially in long-distance flying, if an easily seen index
would show simply whether the airplane were on the desired course,
or were to the right or to the left.
The earth inductor compass was devised to avoid deviation due
to the iron of the motor and also to diminish the difficulty and un-
certainty in observing indications of the course. It is essentially
a direct-current generator consisting of a coreless armature ro-
tating in the magnetic field of the earth and connected to a milli-
voltmeter on the instrument board. The armature is placed so
far astern that it is outside of the magnetic field due to the iron of
the motor. It hangs pendulum-wise, with shaft vertical, from a
universal joint connected to the vertical shaft of either an air-driven
or an electrically driven motor. The brushes can be turned
around the commutator by rotating a dial on the instrument board.
The controller dial is graduated in degrees.
When the axis of commutation, that is, the line joining the points
of contact of the brushes and the commutator, is perpendicular
to the magnetic meridian, no electromotive force is induced in the
rotating armature and the indicating millivoltmeter needle is in
the zero position. When the line joining the brushes is turned
about the armature shaft from this position, the indicating in-
strument deflects to one side or to the other depending upon the
direction in which the brush holder was turned. If it be desired,
for example, to fly on a course 30 degrees east of north, the pilot
sets the controller disk at 30 degrees east of north and thereafter
maintains the plane in the direction that will cause the indicator
to remain on the zero mark.
103. The Magneto Compass. Greater sensitivity than is
possible in the earth inductor compass is obtained in Tear's mag-
neto compass,* in which elongated pole pieces, PI and P 2 , of high
* General Electric Review, xxxii, 1929, p. 190.
170
NAVIGATIONAL COMPASSES
FIG. 131
permeability and low coercive force, are placed on opposite sides
of the armature A, Fig. 131. The armature is rotated by a motor
about an axis fixed with respect to the airplane. The longitudinal
axis of the pole pieces passes through
the center of the armature and is
maintained horizontal however the
airplane may tilt. The axis of the
pole pieces can be turned about a
vertical axis by means of a flexible
cable connected to a graduated dial
on the instrument board.
When the axis of the pole pieces is perpendicular to the mag-
netic meridian, no electromotive force is induced in the armature
and the indicating millivoltmeter needle is at the zero position.
When the axis of the pole pieces is not in this position, the needle
of the indicating instrument will be deflected. The deflection is
independent of the position of the brushes on the commutator C.
With the magneto compass, a course is sot and maintained
exactly as with the earth inductor compass.
104. The Sun Compass. In regions where deviations of the
earth's magnetic field from the geographic meridian are unknown,
and in regions where the intensity of the horizontal component of
the earth's magnetic field is too weak to control the direction of
a magnetic compass needle, the sun compass is available for in-
dicating directions so long as the sun is visible. The Bumstead
sun compass consists of a clock with a single hand that makes the
circuit of the face once in twenty-four hours, Fig. 132. The clock
is mounted so that it can be turned about horizontal pivots set in
two brackets attached rigidly to a horizontal disk. The edge of
the disk is marked off into degrees and cardinal points like a com-
pass card. The disk with the attached clock can be turned in
azimuth with respect to a lubber line on the base plate. The clock
hand carries a shadow-pin and screen by means of which the clock
can be turned till the hand points to the sun. At points in the
arctic or antarctic zones, the sun remains continuously above the
horizon for many days at a time. Suppose that a sun compass in
the arctic zone is on the meridian of Greenwich, with the clock
face parallel to the sun's rays, the clock set for Greenwich sun time
and the hand pointing toward the sun, A, Fig. 133. Then the
noon-midnight line of the clock face will be in the meridian plane
of Greenwich. While the earth is rotating about its axis in the
THE VARIOUS TYPES
171
counter-clockwise direction as viewed by an observer above the
north pole, it carries the instrument into the positions A, B, D, F.
Throughout the twenty-four hours of a complete rotation of the
FIG. 132
earth, the clock hand continues to point toward the sun so long
as the instrument is on the meridian of Greenwich and the clock
is set for Greenwich time.
When the instrument is on any meridian and the clock is set
for the local time of that meridian, then if the instrument be
turned so that the clock hand
points toward the sun, the noon-
midnight line will lie in the merid-
ian plane of the place where the
instrument is situated, C and
E, Fig. 133. At any particular
place, the clock face must be in-
clined to the horizontal at an
angle of (90 - latitude). To
facilitate the making of this set-
ting, there is a divided arc at-
tached to the clock marked off
to give this difference for all IG *
latitudes at which the instrument would be employed.
In using the instrument, the clock is set for local sun time, the
172
NAVIGATIONAL COMPASSES
FIG. 134
face inclined to the horizontal at the proper angle and the clock
with the attached disk turned with respect to the lubber line
through an angle equal to that of the desired course from the
meridian. Then, the proper course is maintained by steering
the airplane or ship so that
the shadow of the shadow-
pin is maintained along the
axis of the clock hand.
When the course is not ex-
actly north or south, the ap-
parent time changes. In this
case either the clock may be
reset after each change of a
few degrees longitude, or it
may be set permanently to
the apparent time of the
middle meridian.
105. The Apparent Motion
of the Spin-Axle of an Un-
constrained Gyroscope Due to the Rotation of the Earth. Ac-
cording to the First Law of Gyrodynamics (Art. 36), the spin-axle
of an unconstrained spinning gyro, unacted upon by any torque,
remains fixed in space. This law is also called the law of rigidity
of plane of the gyro, or tho law
of the fixity in space of the
spin-axle. If the gyro-axle of
a spinning gyro at the equator
is parallel to the geographic axis
of the earth, then the axle will
remain horizontal and in the
meridian plane. If the gyro-
axle of a spinning gyro at the
equator is horizontal in the
east-west position, as at 0, Fig.
134, then, although the gyro-
axle preserves its direction in
space as the earth rotates, it appears to an observer on the earth
to make one complete turn each twenty-four hours about the hori-
zontal axis in the meridian plane.
If an unconstrained spinning gyro be situated between the
equator and either pole, then the spin-axle will appear to an ob-
Fia. 135
THE VARIOUS TYPES 173
server on the earth to move about both a horizontal and a vertical
axis, Fig. 135. At two times during twenty-four hours, the gyro-
axle is horizontal and at two times it is in the meridian plane.
The motion of the spin-axle relative to the earth is about a cone
having the center of the gyro as apex.
The spin-axle of an unconstrained gyroscope continues to be
directed toward the same fixed star but it does not continue to be
directed toward the same fixed point on the earth. The fixity of
the spin-axle in space of an unconstrained gyroscope is inadequate
for the production of an instrument that will indicate directions
on the earth.
106. The Meridian-Seeking Tendency of a Pendulous Gyro-
scope. It has been shown that if a gyroscope be rotated about
an axis about which the turning of the gyro-axlo is prevented
(Art. 49), the axle will set itself parallel to the axis of rotation with
the spin of the gyro-wheel in the same direction as the rotation of
the gyroscope. It might bo imagined that this effect would be
adequate for causing the axle of a spinning gyro-wheel, with one
degree of angular freedom suppressed, to sot itself parallel to the
earth's axis. For example, if a spinning gyro-wheel on the earth
is mounted so that angular motion about every horizontal axis is
suppressed, the spin-axle will tend to set itself parallel to the earth's
axis and will turn toward the meridian plane. If, however, the
instrument is on a moving ship, the spin-axle will tend to sot itself
parallel to the axis of the resultant of the angular velocity of the
earth and that of the ship with respect to the earth. Since the
angular velocity of the ship is often much greater than that of the
earth, a gyro-wheel mounted in this manner would not give cor-
rect indications. To be of value for use as a compass, the gyro-
axle (a) should be urged toward the meridian by a torque sufficient
to bring it into the meridian within a reasonable length of time
after being set into spinning motion, (6) should quickly return to
the meridian when displaced therefrom, (c) should be nearly hori-
zontal when in the meridian.
If, at some instant, the spin-axle of a gyroscope north of the
equator is nearly horizontal and in the meridian plane, then at
succeeding instants the north-seeking end will have an apparent
motion away from the meridian toward the east. In order that
the gyro-axle may maintain its position in the meridian, the north-
seeking end must be given a westerly precessional velocity equal
to the vertical component of the earth's angular velocity. This
174
NAVIGATIONAL COMPASSES
required precession can be produced by a torque about a hori-
zontal axis which tilts the gyro-axle from the horizontal plane.
Two different methods are now in use to produce this tilting torque
on the spin-axles of gyro-compasses. One is by making the sen-
sitive element pendulous. In the following Article the other
method will be considered in which the tilting torque is produced
by a moving mass of liquid.
Now it will be shown how the weight of a mass attached to the
supporting frame of a spinning gyro-wheel below the center of the
gyro- wheel, will cause the spin-axle to precess toward the me-
ridian in which the gyroscope
is situated. Consider a gyro-
scope with horizontal axle
pointing east and west, X, Fig.
136. In this diagram, it is
imagined that we are looking
down on the northern hemi-
sphere of the earth as from
an airship. Suppose that the
freedom of the gyroscope about
the horizontal axis is restricted
by hanging a mass m on the
lower side of the supporting
frame. We shall call a mass
suspended from the support-
ing framo of a gyro-wheel "a pendulous mass." The pendulous
mass will be pulled toward the center of the earth. This pull will
produce zero torque when the gyro-axle is horizontal as at X.
While the earth rotates, the spin-axle of the gyro-wheel tends to
maintain its position in space with the result that, when the gyro-
scope has reached a position F, the axle is no longer horizontal.
The weight F of the pendulous mass exerts a torque in the counter-
clockwise direction about an axis perpendicular to the plane of the
diagram. The line L, representing the torque, is directed upward
from the plane of the diagram. Suppose that the direction of spin
is as represented by the line h s . Then, from the law of precession,
the spin-axle tends to become parallel to the torque-axis with the
direction of spin in the direction of the torque. In the present
case, the direction of the torque is the same as the direction of
rotation of the earth. Therefore, the gyro-axle of the pendulous
gyroscope tends to become parallel to the earth's axis, with the
FIG.
THE VARIOUS TYPES
175
direction of spin in the same direction as the rotation of the earth.
In the same manner it can be shown that if the center of mass of
the gyroscope is above the point of support, the gyro-axle tends to
become parallel to the earth's axis, with the spin in the opposite
direction to the rotation of the earth.
When the spin-axle reaches the meridian plane (position Z,
Fig. 136), the north-seeking end of the gyro-axle is at its maxi-
mum elevation above the horizon and the weight of the pendulous
mass is exerting its maximum torque thereby producing a maxi-
mum velocity of precession. The gyro-axle will cross the meridian
FIG. 137
plane and, continuing its angular motion beyond the meridian
plane, the axle will dip and pass through the horizontal position.
During the dipping, the torque due to the weight of the pendulous
mass decreases till it is zero when the spin-axle is horizontal. At
this instant, the precessional velocity is zero and the spin-axle
is at its greatest angular displacement from its original direction
at X. Continuing its dipping, the north-seeking end of the spin-
axle passes through the horizontal plane, raising the pendulous
mass and thereby developing a torque in the opposite direction.
The north-seeking end of the spin-axle rises, again crosses the
meridian plane and repeats its motion back and forth. The
oscillation of the spin-axle of a gyro-compass back and forth
through the meridian plane involves an exchange of energy be-
tween the compass and the earth.
It has now been shown that the deflection of the spin-axle of a
176
NAVIGATIONAL COMPASSES
pendulous gyroscope from the horizontal develops a torque that
produces a precession of the spin-axle toward the meridian plane.
If the spin-axle be displaced out of the meridian plane, the north-
seeking end will oscillate back and forth around an elliptical orbit
abcda, Fig. 137. This figure represents the path of the prolonga-
tion of the gyro-axle on a vertical plane as seen by an observer
looking from the south toward the north. If the motion be
undamped, the angular amplitude of each oscillation will equal
the original angular displacement from the meridian. The period
of vibration depends upon the torque acting on the oscillating
system, the angular speed of the spinning gyro- wheel and upon the
moment of inertia of the os-
cillating system. It is always
made to be about 84 minutes
(Art. 113).
107. The Meridian-Seek-
ing Tendency of a Liquid-
Controlled Non-Pendulous
Gyroscope. Attached to
the sides of the frame sup-
porting the gyro-wheel, G,
Fig. 138, are reservoirs joined
by a small tube.* This figure
represents the view as seen by
an observer above the north
pole of the earth. With the
gyro-axle horizontal, the two
reservoirs are filled with mercury, up to the level of the middle
of the gyro-axle. The center of gravity of the gyroscope is raised
to the center of gravity of the gyro- wheel by means of an adjust-
able counterpoise above each reservoir. This mercury-filled sys-
tem is called a " mercury ballistic."
Suppose that at a particular instant the spin-axle is horizontal
and east and west as shown at X, Fig. 138. In Fig. 137, a line
along the spin-axle intersects the vertical plane XZ at the point a,
east of north. While the gyroscope is carried by the rotation of
the earth to the position F, Fig. 138, the spin-axle tends to pre-
serve its direction in space, thereby causing the reservoir B to be
raised above B f relative to the horizontal plane at B. Conse-
quently, some mercury flows from the reservoir B to the reservoir
* U. S. Patent. Harrison and Rawlings, No. 1362940, 1920.
FIG. 138
THE VARIOUS TYPES 177
B', thereby producing a torque about a horizontal axis perpendicu-
lar to the plane of the figure and directed away from the reader.
This torque produces a precession that causes the spin-axle to
tend to set itself parallel to the torque-axis and with the direction
of spin in the direction of the torque. As this torque continues,
the spin-axle becomes parallel to the meridian plane, the tilt of the
spin-axle becomes maximum and the direction of spin is opposite
the direction of rotation of the earth. A lino along the spin-axle
now intersects the vertical plane at 6, Fig. 137. The spin-axle
crosses the meridian plane and becomes less tilted from the hori-
zontal. The spin-axle becomes horizontal when the inclination
to the meridian plane is maximum. A line along the spin-axle
now intersects the vertical plane at c. Owing to the rotation of
the earth, the dip will continue past the horizontal and mercury
will flow toward the reservoir at B, thereby causing a precession
in the reverse direction. During this precession, a line along the
spin-axle traces a curve cda on the vertical plane.
The cycle of motions causes the end B r to trace an elliptical path
with major axis horizontal, east and west, and minor axis vertical.
The end B is the north-seeking end of the gyro-axle. If there be
no damping, this motion is repeated with undiminishing amplitude.
The period is made to be about 84 minutes (Art. 113). From the
preceding consideration it is seen that the direction of spin of the
liquid-controlled gyroscope is opposite to the direction of rotation
of the earth, whereas the direction of spin of the pendulous gyro-
scope is the same as the direction of rotation of the earth.
108. Making a Gyroscope into a Gyro-Compass. In Arts.
106 and 107 it has been shown that, owing to the rotation of the
earth, the spin-axle of a gyroscope, with rotational freedom about
a horizontal axis partially suppressed, will trace the surface of an
elliptical cone. The apex of the cone is at the center of mass of
the gyro. The axis of the cone is horizontal and in the meridian
plane. The maximum amplitude of deviation of the spin-axle
from the meridian may be many degrees. If the gyroscope were
on the earth or on a stationary ship, the meridian could be located
by taking the mean of the extreme positions of the spin-axle.
The length of time required for this determination is so great that
the gyroscope, as described, is quite useless for determining the
course of a ship at sea.
To. make a gyroscope into a gyro-compass, means must be pro-
vided to damp the oscillations to such an extent that very quickly
178 NAVIGATIONAL COMPASSES
the spin-axle will move into and remain in the meridian plane.
The required damping can be effected by a torque that will either
(a) diminish the tilt of the spin-axle from the horizontal, or (b) op-
pose the horizontal motion of the spin-axle. The oscillation of
the spin-axle of a gyro spinning in the same direction as the ro-
tation of the earth can be damped by means of a viscous liquid.
The magnitude of the damping torque should be proportional to
the instantaneous angular speed of the gyro-axle about the axis
of the torque.
109. The Meridian-Seeking Torque Acting on a Gyro-Compass.
The spin-axle of a gyro-compass is kept approximately hori-
zontal by either a pendulous mass or a mercury ballistic. Freedom
of rotation of the gyro about a horizontal axis perpendicular to the
spin-axle is partially suppressed by a torque that is developed when
the spin-axle is turned. If a gyroscope with one degree of rota-
tional freedom either wholly or partially suppressed be rotated, the
spin-axle will tend to set itself in the direction of the axis of ro-
tation (Art. 49). Consequently, the spin-axle of a stationary gyro-
compass on the earth tends to set itself parallel to the horizontal
component of the earth's angular velocity at the place where the
compass is situated.
When the spin-axle and the axis about which the instrument is
being rotated are perpendicular to one another, the torque tending
to turn the gyro-axle has the value (Art. 36) :
L =- K s w s w e = h s w e (116)
where w e represents the angular velocity of the earth about the
polar-axis.
The magnitude of the torque urging the spin-axle toward the
meridian plane when the gyro-compass is at latitude X, and the
spin-axle is inclined to the meridian plane at the angle (/>, is ob-
tained by substituting for w e the value of the component of the
angular velocity of the earth with respect to a horizontal line per-
pendicular to the vertical plane that contains the spin-axis. In
Fig. 139, the spin-axle of the gyro is in the direction AO, with
respect tq a system of rectangular coordinates consisting of a
vertical VO, a horizontal line MO in the meridian, and a hori-
zontal east-and-west line EO. A vertical plane containing the
spin-axle intersects the horizontal plane MOE in the line HO.
The vertical plane VOII is inclined to the meridian plane VOM
at the angle <.
THE VARIOUS TYPES
179
At a point on the earth at latitude X, Fig. 140, the component
angular velocity of the earth with respect to a horizontal axis OB
in the meridian plane at O is w e cos X. In Fig. 139, this component
is represented by the line OB. The component of OB about a
horizontal axis PB perpendicular to the vertical plane VOII
containing the spin-axle of the gyro-wheel is
OB sin = w e cos X sin <
v
FIG. 139
Substituting this value for w e in (116) we have
L = h s w c cos X sin $
(117)
This is the magnitude of the torque urging the spin-axle of a sta-
tionary gyro-compass toward the meridian. It depends upon the
angular momentum of the gyro, upon the latitude and upon the
angle between the spin-axle and the meridian. The meridian-seek-
ing torque is so small, especially when the gyro-compass is at a
place of high latitude and the spin-axle makes a small angle with
the meridian, that unusual methods must be adopted to reduce
the opposing torque due to the method of suspending the sensitive
system.
For the passage? under the ice to the North Pole, Sir Hubert
Wilkins' Nautilus was equipped with a gyro-compass. Now the
directive tendency of the sensitive element of a gyro-compass
produced by a constant precession toward the meridian is too
small to be effective at latitudes greater than about 85 degrees.
Consequently, when near the Pole, the compass gyro was discon-
nected from the precessing device and the gyro left as free of all
restraint as possible. In this condition, the spin-axle holds its
direction in space instead of maintaining its position in a meridian
of the earth. The spin-axle moves away from the meridian, to
which the spin-axle was parallel at the instant the gyro was dis-
180 NAVIGATIONAL COMPASSES
connected from the processing device, with an angular velocity
given by (68), Art. 42.
The spin-axle of a gyro-compass is preferably horizontal, be-
cause (a) the directive force is proportional to the horizontal com-
ponent of the angular momentum of the gyro, (6) the compass is
used to measure bearings, that is, angles on a horizontal plane.
If the spin-axle were inclined to the horizontal, it might be possible
to project the direction of the axle on a horizontal plane if we
could assume a given plane to be horizontal. The uncertainty in
the horizontality of a given plane would introduce an appreciable
error in the determination of either bearings or the meridian plane.
A gyro-compass having a gyro of large angular momentum is
less affected by small disturbing torques than is one having a
gyro of smaller angular momentum.
If the compass be on a vehicle that could move with such a
linear velocity that its angular velocity about the earth's axis is
equal and opposite the angular velocity of the earth, then the
meridian-seeking torque would be zero. The relation between
the linear velocity of the vehicle in knots and the latitude at which
the directive tendency of the gyro-axle will be zero can be easily
obtained. Thus, remembering that one knot is a speed of one
nautical mile per hour and that a nautical mile is 6 1 of 3 -J- of
the earth's equatorial circumference, it follows that a point on the
.i r i r (360 X 60) ftnn
equator moves with a linear speed ot ^ = 900 knots.
-^4
The motion is eastward. At latitude X, a point on the earth has a
velocity of 900 cos X knots eastward. So that a gyro-compass will
have zero directive tendency when on a vehicle moving westward
at v knots at latitude X if
v = 900 cos X
For example, if an airship is flying westward at 90 knots, the
gyro-compass will have zero directive tendency at latitude X
given by the equation 90 - 900 cos X. In this case X = 80 10'.
It will now be shown that, on account of the tilt of the spin-
axle when in the resting position, any great change in the spin-
velocity of the gyro will result in a deflection in azimuth of the spin-
axle. The gyro-wheel constitutes the rotor and the gyro-casing
constitutes the field magnet of an electric motor. When the gyro
is spinning, there is a reacting torque applied to the casing. So
long as the spin-velocity is constant this torque is balanced by
THE VARIOUS TYPES 181
windage and f Fictional torque; but if the spin- velocity should
change, the torque acting on the casing is unbalanced.
If the spin is in the same direction as the rotation of the earth
(pendulous gyro-compass), then when the spin- velocity is acceler-
ating there will be a torque acting on the gyro-casing tending to
turn the casing in the direction opposite that of the rotation of
the earth. If the north-seeking end of the gyro-axle is tilting
upward, the vertical component of this torque will produce a
slight easterly deviation of the north-seeking end of the spin-axle.
When the spin- velocity of the same compass is decelerating, there
will be a westerly deviation.
In the case of a gyro-compass having the spin in the direction
opposite to the rotation of the earth (mercury ballistic compass),
changes in the spin-velocity result in deviations in the directions
opposite those for the pendulous gyro-compass.
PROBLEMS
1. The Brown gyro-compass has a single gyro-wheel of 4.5 Ib. wt., diameter
4 in., spinning at 15,000 r.p.m. Assuming that the radius of gyration is 0.74
of the radius, find the meridian-seeking torque, in grain-inches, at latitude 40
when the spin-axle is inclined 1 to the meridian.
2. The Sperry Mark VI gyro-compass has a single gyro-wheel of 54 Ib. wt.,
diameter 10 in., spinning at 6000 r.p.m. Assuming that the radius of gyration
is 0.75 of the radius, find the meridian-seeking torque in grain-inches at lati-
tude 40 when the spin-axle is inclined 1 to the meridian.
3. The Sperry Mark X compass has a single wheel of moment of inertia
22.2 lb.-ft. 2 , spinning at 10,000 r.p.m. Find the meridian-seeking torque in
grain-inches at latitude 40 when the spin-axle is inclined to the meridian at 1.
4. One model of Arma compass has two gyro-wheels rotating in the same
direction with axles inclined to one another at 60, the apex of the angle being
toward the south. Suppose that each gyro-wheel weighs 5 Ib. 2 oz., has diam-
eter of 5.12 in. and spins at 20,000 r.p.m., and that the radius of gyration is
0.74 of the radius of the wheel. Find the meridian-seeking torque in grain-
inches when the compass is at latitude 40 and bisector of the angle between
the two spin-axles at an angle of 1 to the meridian.
6. An early model of Anschiitz gyro-compass consisted of three gyro-wheels
each of mass 5 Ib. 2 oz., diameter 5.12 in., spinning at 20,000 r.p.m. These
gyro-wheels were arranged at the apexes of an equilateral triangle. The axle
of the south gyro-wheel made equal angles to the sides of the triangle while the
axles of the other two gyros were along the sides of the triangle. A determina-
tion gives 0.13 lb.-ft. 2 for the moment of inertia of each gyro about its spin-axle.
Find the meridian-seeking torque of the system in grain-inches when at lati-
tude 40 and the axis of the south gyro is inclined 1 to the meridian.
182 NAVIGATIONAL COMPASSES
2. The Natural Errors to which the Gyro-Compass is Subject
110. The Latitude Error. Owing to the rotation of the earth,
combined with the tendency of the spin-axle of a gyroscope to
maintain its direction in space, the end of the axle of a gyro-com-
pass moves back and forth across the meridian plane and also up
and down across the horizontal plane through the center of the
gyro- wheel (Arts. 106, 107). If the gyro-compass is north of the
equator and if the gyro-axle is horizontal and pointing east of
the meridian plane, the north-seeking end of the spin-axle will
rise and move westward. The tilting of the spin-axle is necessary
for the production of the torque required to cause the gyro-axle
to seek the meridian. At any latitude the tilt of the spin-axle
of an undamped compass, when the axle is in the meridian plane, is
just sufficient to produce the rate of precession about a vertical
axis required to maintain the axle in the meridian.
Any damping action that opposes the natural tilting of the spin-
axle will result in a lower rate of precession than that required to
maintain the axle in the meridian. The resting position of the
spin-axle will then be deviated from the meridian plane by an angle
called the latitude error or the damping error. If the method used
for damping does not involve an opposition to tilting of the gyro-
axle, then there will be zero latitude error. When there is latitude
error, its magnitude depends upon the latitude and upon the con-
stants of the compass. The Florentia and the Sperry gyro-com-
passes are subject to a latitude error. This latitude error is com-
pensated automatically by a device that forms part of the compass.
111. The Error Due to the Velocity of the Ship. The Meridian-
Steaming Error. A gyro-compass on board a ship at sea is sub-
ject to deflecting forces due to the motion of translation of the ship
and also to angular motions of steering, rolling and pitching. The
deflection of the spin-axle from the meridian due to the linear
velocity of the ship will now be considered.
Suppose that the ship is moving with a linear velocity v s on a
course inclined at an angle 9 to the meridian. Then the component
in the direction of the meridian is v s cos 0. The angular velocity
of the ship with respect to the earth about an east-and-west axis-
has the value
Vs cos
ew s - R
where R represents the radius of the earth.
NATURAL ERRORS
183
If we represent the angular velocity of the earth about its axis,
with respect to some fixed line in space, by pw e , then the component
angular velocity of the earth relative to a horizontal axis in the
meridian plane at a place of latitude X (Fig. 141) is
m W e = p W e COS X
FIG. 141
In Fig. 142, the two angular velocities e w s and m w e are repre-
sented by the lines OX and OF, respectively. The resultant
angular velocity of the ship with respect to a horizontal axis in
the meridian plane, m w sj is represented by OR. The spin-axle
of the gyro-compass sets itself parallel to the axis of the resultant
angular velocity of the ship. In this figure
tan 8l = = T ^L (118)
m W e RpW e COS X
This deflection, 81, called the meridian or north-steaming error,
is westerly for a northern course, and easterly for a southern course.
It is independent of the construction of the compass. In Art.
113 it will be shown how the appropriate meridian-steaming de-
flection can be produced, without oscillation, just as soon as any
change occurs in either the speed or the course of the ship.
If the linear speed of the ship be expressed as v knots, then
v s = 6080 v ft. per hour. R = 20,925,000 ft.
P w e 1 rev. per day = -^j- radian per hour = 0.26 rad. per hour.
Hence,
6080 feet per hour vcos0
tan 5i - ^20^,Wo7^^26rad. per hr.) cos X
Neglecting the small difference between Si and tan Si, and remem-
bering that 1 radian = 57.3 degrees, we have the value of the
jneridian-steaming error:
Si = 0.0011
V COS
cos X
radians = 0.063
v cos
cos X
degrees (119)
184
NAVIGATIONAL COMPASSES
Problem. A ship at latitude 60 N. is steaming north-east with a speed of
20 knots. Find the deflection of the spin-axle of the gyro-compass from the
geographical meridian due to the meridian-steaming error.
Solution. From (119), since = 45 and A = 60,
0.063
20 (0.707)
0.5
= 1.8 west of the meridian
112. The Deflection of the Axle of a Gyro-Compass Produced
by Acceleration of the Ship's Velocity. The Ballistic Deflection
Error. Experiment.
The pivots supporting
the inner frame of the
gyroscope, Fig, 143, are
horizontal. A mass m
attached rigidly to this
frame and hanging below
the gyro-wheel tends to
keep the gyro-axle hori-
zontal when the frame is
at rest. Place the gyro-
scope on the intersection
of two lines drawn at
right angles to one an-
other on the table, and
marked S-N and W-E,
respectively. Place the
spin-axle parallel to the
S-N line, and set the gyro-
wheel spinning in the di-
rection indicated by the
arrow h s .
(a) Slide the instrument quickly along the table in the direction
S-N. During the time the motion of the frame is being accelerated
in the direction S-N, the pendulous mass m hangs back, thereby
developing a torque about an axis parallel to the E-W line in the
direction indicated by the arrow L. The spin-axis turns toward
the torque-axis about a vertical axis in the direction represented
by the arrow Wp.
(b) Accelerate the motion of the gyro-frame in the direction
N-S or decelerate the motion in the direction S-N. Note that
during the change in velocity the spin-axle precesses in the direction
opposite that developed in the preceding case (a).
NATURAL ERRORS 185
(c) Push the gyroscope in the direction S-N with constant
velocity and then suddenly move it toward either the right or left.
Note that, during the change in the direction of the velocity, the
spin-axle precesses in the same direction as in case (6) when the
motion in the direction S-N was decelerated.
(d) Now accelerate the motion of the gyro-frame in either the
direction E-W or W-E, that is, perpendicular to the spin-axis.
No change in the direction of the spin-axis is produced.
Question. Suppose that we have a non-pendulous mercury ballistic gyro-
compass with the spin of the gyro in the direction opposite that considered
above. Find the direction of the deflection of the spin-axle that would be
produced if the instrument were to be accelerated in each of the ways considered
above.
So long as the velocity of a ship is constant, the spin-axle of the
gyro-compass will have a resting position at a small angle from the
meridian of a magnitude depending upon the velocity and latitude
of the ship (Arts. 110 and 111). If the velocity of the ship is
changing while the ship is on any heading except perpendicular
to the spin-axle of the gyro-compass, then during the time the
velocity is changing, the spin-axle will be deflected from the rest-
ing position which the spin-axle had when the ship was moving
with its first velocity. This deflection will be constant while the
acceleration remains constant. The deflection of the spin-axle
from its resting position, produced by an acceleration of the me-
ridian component of the ship's velocity, is called the ballistic de-
flection.
A diminution of velocity in the meridian is produced not only
by diminishing the speed in the meridian but also by changing the
heading either eastward or westward. In every case, the ballistic
deflection produced by any acceleration of the ship's velocity is
in the same direction as the change in the direction of the resting
position of the gyro-axle, on account of the changing meridian-
steaming error.
Consider a ship steaming north at 10 knots. Because of the
meridian-steaming error, the resting position of the spin-axle will
be at an angle NOX, west of the meridian, Fig. 144. Suppose
that now the speed on the same heading be increased to 20 knots.
The resting position of the spin-axle corresponding to the new
velocity is shifted to YO. Suppose that during the time the speed
was changing from 10 knots to 20 knots, the ballistic torque acting
on the gyro-frame caused the gyro-axle to precess into the position
186 NAVIGATIONAL COMPASSES
ZO. The ballistic deflection is XOZ. The angle YOZ by which
the gyro-axle either overshoots or undershoots the correct settling
position for the particular latitude, course and speed, is called the
ballistic deflection error. With any change in the meridian com-
ponent of the ship's velocity, the spin-axle will be deflected in the
same direction as the movement of the gyro-axle produced by the
change in the meridian-steaming error. When the acceleration of
the ship's velocity ceases, the gyro-axle oscil-
lates about the resting position YO. There
will be zero ballistic deflection error if XOZ
= XOYj that is, if the ballistic deflection
equals the change in the meridian-steaming
error.
In order that the compass card may be in
its true resting position throughout the time
the velocity of the ship is changing, the com-
pass must be forcibly precessed, during this time, to the resting
position proper to the speed, course and latitude at the end of the
acceleration.
If the gyro were not spinning, the spin-axle would tilt through-
out the time the velocity of the ship is accelerating. If, however,
the gyro were spinning, the " rigidity of plane " would oppose
tilting. In this case, the pendulous mass would be acted upon by
a force which would develop precession about a vertical axis.
If the precessional velocity were of the proper magnitude, it would
cause the compass to process, during the time the velocity of the
ship is accelerating, from the resting position proper to the speed,
course and latitude at the beginning of acceleration, to the resting
position proper to the speed, course and latitude at the end of
the acceleration. Under this condition, there would be zero
ballistic deflection error.
113. The Period of a Gyro-Compass That Will Have Zero
Ballistic Deflection Error When at a Definite Latitude. The
magnitude of the ballistic deflection depends upon the torque that
opposes tilting of the sensitive element. In the case of a gyro-
compass of the pendulous type at a given latitude, the torque that
opposes tilting depends upon the degree of pendulousness of the
sensitive element. In the case of a liquid-controlled non-pendulous
gyro-compass at a given latitude, it depends upon the weight of
liquid that is moved from one side of the gyro-case to the other and
upon the distance between the opposite mercury reservoirs. Each
NATURAL ERRORS
187
of these factors affects the period of vibration of the spin-axle
back and forth through the meridian. It will now be proved that
if the period of the sensitive element has a certain value, which
for all latitudes is not far from 84 minutes, then the spin-axis
of the gyro-compass will move, without oscillation, to the resting
position appropriate to the given speed and course, just as soon as
any change occurs in either the speed or course.
Imagine a symmetrical body supported at the center of mass so
as to be capable of turning about this point in any direction. An
acceleration of the motion of the body in any
direction will produce no turning of the body.
Imagine, farther, that a mass m be attached
to the body at a distance x from the center of
mass (7, Fig. 145. When the motion of the
system is constant, the mass m is acted upon
by a force mg vertically downward and by an
equal force vertically upward. If the linear
motion of the system is subject to a constant
acceleration a in a horizontal direction to the
right, the pendulous part m of the system must
be acted upon by a force ma in the direction of
the acceleration. This force is the resultant of the weight mg
vertically downward and a force F directed toward the point C.
This force F is inclined to the vertical at an angle </>, given by the
relation
tan <{> =
ma
a
mg g
(120)
When the horizontal distance of m from the vertical line through
C is 6, the system is acted upon by a torque about an axis perpen-
dicular to the plane of the diagram of the magnitude
L = mgb = mgx sin <
When $ is so small that sin may be replaced by tan <, we have,
on substituting the value of sin < from the above equation, in
(120):
L = mxa
Suppose that the suspended system includes a gyro spinning
about a nearly horizontal axis. We are now dealing with a pen-
dulous gyro-compass. If the ship carrying the gyro-compass be
accelerated while on a meridian course, the pendulous part of the
188 NAVIGATIONAL COMPASSES
sensitive element will be acted upon by a torque as above. If,
however, the course be perpendicular to the spin-axle, the torque
will be zero. Representing the meridian component of the mean
acceleration of the ship during the short time t by a, there will
act upon the gyro a torque L which produces a precessional ve-
locity about the vertical axis of the magnitude,
mxa
[_ 771 __ mx
~h s \ h^
In this short time t the meridian component of the velocity of
the ship and of the compass changes by an amount at. During
this time, the gyro-axle turns through the angle
of . mxat mx , , /im\
|8[= w p t\ = h - = (t;, - %) (121)
Where /3 is the " ballistic deflection " and (v t # ) is the change
in the meridian component of the velocity of the ship in time t.
When the meridian component of the velocity of the ship
changes from Vo to v t , the resting position of the gyro-axle is de-
flected through an angle (6/ Si), from
the position Si, Fig. 144. Now /3 depends
only upon the constants of the compass
and the linear acceleration of the ship,
whereas the meridian-steaming errors de-
FlG> 14t) pend upon latitude and velocity. The
ballistic deflection /3 and the deflections Si and S/ due to changes
in the meridian-steaming error are in the same direction. Con-
sequently, a gyro-compass can be designed with such a period T Q
that, when the instrument is at some particular latitude, the bal-
listic deflection shall equal the change in the meridian-steaming
error (Si' 61) produced by the change in the velocity of the ship.
If the ship, starting from rest, moves along a meridian through
a distance dz in time dt y with constant linear acceleration a, the
velocity at the end of time t will be
dz
-j- = at
dt
During the motion, the latitude of the ship has changed by the
amount, Fig. 146:
NATURAL ERRORS 189
where R is the radius of the earth. The rate of change of latitude
is
d\ __ (h __ v' _ at
~dl ~ Rdt~~ R~li
where [= at] is the meridian velocity v f of the ship at the end of
time t. Substituting in (121) the value of at from (122), we obtain
for the ballistic deflection
_mxli d\
P - ~h7 dt (123)
Before equating this value of the ballistic deflection and the
value of the meridian-steaming error, we shall express the latter
in terms of the time rate of change of latitude. Since the meridian-
steaming error di is so small that we may neglect the difference
between 61 and tan Si, (118) may be written
6, = _^_
Rw e cos X
where the meridian component v s cos 6 of the velocity of the ship
is represented by v' and the symbol p w e is abbreviated to w e .
On substituting in this equation the value of v' from (122), we
have for the meridian-steaming error
S, = V (124)
w e cos A at
Now equating the ballistic deflection (123) and the meridian-
steaming error (124),
MxR = 1_
h s w e cos X
or h s = mxRw c cos X (125)
When the angular momentum has the value given by this equa-
tion, the spin-axle will move without oscillation to the resting
position proper to the particular latitude and final velocity. On
substituting this value in (88) we find that the period of the azi-
muthal vibration back and forth through the meridian of the
undamped compass that produces zero ballistic deflection error
has the value
To = 2^17^ (126)
190 NAVIGATIONAL COMPASSES
At the equator R = 20,925,000 ft., and g = 32.086 ft. per sec.
per sec. Hence, at the equator, 7 7 o[= 5068 sec.] = 84.4 min.
Since at other latitudes the radius of curvature of the earth is
slightly less and g is slightly greater than at the equator, the period
of the compass at places either north or south of the equator should
be somewhat less than at the equator.
Since (125) must hold when the period of azimuthal vibration
of the undamped compass is such that there is zero ballistic de-
flection error, we see that, at any latitude X, the period can be
adjusted to the desired value by varying either h s or mx. Hence,
a gyro-compass at latitude X will have zero ballistic error when
either h s or mx is adjusted so that the ratio
^ = Rw e cos X (127)
mx
The mx in this equation is called the " pendulous factor/'
The period of every gyro-compass is made of such a value that
when the instrument is at a selected latitude, the ballistic error
is zero. The desired period is obtained by having proper values
of the angular momentum of the gyro and of the pendulousness of
the sensitive element or the mass of active mercury in the ballistic.
114. The Ballistic Damping Error. While the meridian com-
ponent of a ship's velocity is being accelerated, either by a change
of speed or of course, there will be a tilting of the gyro-compass
except when the acceleration is perpendicular to the spin-axle.
This is due to forces impressed by the damping mechanism and to
the precession in azimuth produced by the pendulousness. When
the degree of pendulousness is correct for a particular latitude,
then the gyro-compass will give zero ballistic deflection error at that
latitude during the time the ship's velocity is accelerating. At
any other latitude, the gyro-compass will show a ballistic deflection
error during the time the velocity of the ship is accelerating.
The tilt of the spin-axle associated with the acceleration of the
ship's velocity is superposed on the tilt due to the rotation of the
earth. The torque producing this additional tilt ceases when the
acceleration ceases but the spin-axle starts oscillating back and
forth through the equilibrium position of the spin-axle for the
particular velocity and latitude of the ship.
The oscillation of the gyro-compass may continue for an hour
or more after the acceleration of the ship's velocity has ceased.
The device employed to damp the oscillation exerts a torque on the
NATURAL ERRORS 191
spin-axle which disp/aces the settling point from the normal equi-
librium position. The maximum deflection of the spin-axle from
the equilibrium position due to this cause is called the maximum
ballistic damping error. In the case of any gyro-compass, the
damping of the vibration of the gyro-axle produces an error after
the ship has completed a change in course or speed. This bal-
listic damping error is most marked after the ship, steaming at full
speed, has completed a turn of 90 degrees or more. It attains a
maximum value in about 20 minutes after the velocity of the ship
has become constant. The magnitude of the damping error in-
creases with increase in the acceleration of the ship. Much
greater accelerations are produced by sudden turns than by any
possible change in the speed of the ship. Since this error is small
while the ship is on a straight course but may be large when the
ship is making a turn, it is also called the ballistic turning error.
It is also called the damping acceleration error.
The ballistic damping error and the accompanying oscillation of
the spin-axle can be prevented by stopping the operation of the
damping device during the turning of the ship.
115. The Compass Error Due to Rolling of a Ship When on an
Intercardinal Course. The Quadrantal or Rolling Error. A
pendulous gyro-compass on a ship that is rolling or pitching is
acted upon by a force that has a maximum value at the end of a
roll or pitch, and another force that has a maximum value when the
pendulous system is passing through its equilibrium position.
The first of these forces will be considered in the present Article.
The second will be considered in the following Article.
A ship's compass is supported in a Cardan gimbal mounting
consisting of two horizontal rings, one capable of rotation about an
axis parallel to the keel of the ship and the other about a trans-
verse axis. In so far as freedom to turn in any direction is con-
cerned, a gyro-compass on board ship is equivalent to a gyro-wheel
mounted in five rings. For simplicity of representation, in the
present Article, the pendulous gyro-compass will be represented
by a gyro-wheel in a casing free to turn about any axis through its
center and carrying an additional mass attached to the lower side
of the casing. This mass that causes the gyro and casing to be
pendulous we shall call " the pendulous mass."
Consider the effect on the direction of a gyro-compass produced
by rolling or pitching of a ship about an axis that is perpendicular
to the spin-axle. In Fig. 147, the keel of the ship is east and
192
NAVIGATIONAL COMPASSES
west, perpendicular to the plane of the diagram the bow of the
ship pointing away from the reader. Suppose that the ship os-
cillates back and forth about a horizontal axis perpendicular to
the plane of the diagram through O. At the end of a swing, the
inertia of the pendulous mass m carries the mass beyond the
equilibrium position, thereby developing a torque on the gyro-
wheel about an axis perpendicular to the plane of the diagram
indicated by the sign (*) or () marked on the diagram. If the
direction of the spin velocity is that represented by the arrow
marked h sj then, when the compass has reached the end of a swing
at A, the north-seeking end of the spin-axle will be deflected away
from the reader toward the west. When the gyro-compass comes
to the other end of an oscillation at C, the north-seeking end of the
spin-axle will be deflected toward the east. A succession of back
and forth swings of the ship about a horizontal axis produces a
succession of vibrations of the gyro-axle but no permanent de-
flection. Thus, when the ship is pointing east or west, rolling
produces a vibration of the gyro-axle but no fixed deflection from
the meridian. Similarly, when the ship is pointing north or south,
pitching produces a vibration but no fixed deflection of the gyro-
compass. The period of vibration of the spin-axle is that of the
oscillation of the ship. As this is never greater than about 15
seconds whereas the period of the entire compass is several min-
utes, there is no resonant vibration of the spin-axle. Consequently,
the amplitude of vibration of the spin-axle due to rolling and pitch-
ing is small.
Suppose that the pendulous gyro-compass is on board a ship that
NATURAL ERRORS
193
is either rolling or pitching about an axis through parallel to the
spin-axle and perpendicular to the plane of Fig. 148. While the
ship oscillates back and forth through the equilibrium position, the
pendulous mass m moves
back and forth with the
same period. When the
pendulous mass comes to
the end of its swing, its in-
ertia carries it beyond its
equilibrium position there-
by rotating the gyro-casing
in the direction indicated
by the curved arrow drawn
from m. Since this rotation
is about an axis parallel
to the spin-axle it produces
zero effect on the direction of the spin-axle. Thus, when a ship is
on either a north or a south course, rolling produces no deflection
of the gyro-compass. When on an east or west course, pitching
produces no deflection.
In the preceding paragraphs, it has been shown that when a
ship is steaming on a cardinal course, north, east, south, or west,
neither rolling nor pitching
produces a deviation of the
gyro-compass from the me-
ridian. Now, we shall con-
sider the effect on the direc-
tion of the spin-axle of a
pendulous gyro-compass pro-
duced by rolling of the ship
when steaming on an inter-
cardinal course. To fix the
ideas, suppose that the course
is NW-SE as indicated in
Fig. 149, The rolling of the
ship causes the compass to
move back and forth along
a path A B, perpendicular to the course, and causes the pendulous
mass to move back and forth relative to the true vertical through
the center of the gyro. After a roll to the north-east, the center
of mass of the pendulous mass will be at a point c east of the true
194 NAVIGATIONAL COMPASSES
vertical through the center of the gyro and back of the plane of
the diagram. A force F will act at this point in the direction of
the roll. After a roll to the south-west, the center of mass of
the pendulous mass will be at a point c' west of the true vertical
through the center of the gyro and back of the plane of the dia-
gram. A force F' will act as this point in the direction of the
roll.
Each of the two forces F and F' may be replaced by three com-
ponents, one in the north-south line, one in the east-west line and
another in the vertical line. The components F e and F w , in the
oast-west line, are equal, oppositely directed, and perpendicular
to the gyro spin-axle. These forces produce no net torque on the
spin-axle. The vertical components produce torques about the
spin-axle but these torques pro-
duce no deflection of a horizon-
tal spin-axle.
After a roll to the north-east,
the meridian component, F n of
the F acting at c has a lever
arm equal to the distance be-
tween its line of action and the
spin-axle of the gyro. It pro-
duces a torque L on the gyro,
about a true vertical axis through the center of the gyro, in the
counter-clockwise direction as seen when viewed from above.
After a roll to the south-west, the component F s of the force
F' applied at c' acts on the gyro with a torque L', in the coun-
ter-clockwise direction, about a true vertical axis through the
center of the gyro. Thus, when the ship rolls in either direction,
the gyro is acted upon by a counter-clockwise torque about a true
vortical axis.
Since the direction of spin of a pendulous gyro-compass is clock-
wise when viewed from the south toward the north, and since the
spin-axle tends to set itself parallel to the torque-axis and with the
direction of spin in the direction of the torque, Art. 106, it follows
that the north-seeking end of the spin-axle of the compass here
considered will tilt upward while the ship rolls. The upward
tilt raises the center of mass of the pendulous mass and moves it
to the north of a vertical line through the center of the gyro.
Figure 150 represents a view as seen when looking from the east
toward the west. The weight mg of the pendulous mass develops
NATURAL ERRORS 195
a torque L\ which, in turn, produces a precessional motion wp.
As a result, the north-seeking end of the spin-axle precesses west-
ward.
It has now been shown that a pendulous gyro-compass on a
rolling ship headed either north-west or south-east is deflected
toward the west. In the same manner it can be shown that a
pendulous compass on a ship headed either north-east or south-
west will be deflected toward the east when the ship rolls. The
spin-axle of a pendulous gyro-compass on a rolling ship steaming
on any intercardinal course is deflected in the direction to make
it more nearly parallel to the keel of the ship. The deflection
due to pitching is in the direction opposite that due to rolling for a
heading in any direction.
116. Quadrantal Deflection Due to Lack of Symmetry of the
Sensitive Element. It has been shown that, if the mass of a
swinging pendulum bob is not distributed symmetrically with
respect to the pendulum axis, the bob will tend to set itself in the
position in which its moment of inertia, with respect to the vibration
axis is maximum (Art. 14).
In the case of all gyro-corn passes having a sensitive element con-
sisting of a single gyro, inert masses must, be added to the sensitive
element in order that the* moment of inertia of the suspended
system with respect to every horizontal axis through the center
of mass may be equal. The mass, however, of a sensitive element
comprising two or more gyros can be distributed symmetrically
about the line through the 1 center of mass and the center of oscil-
lation. Neither a compass consist ing of a single gyro with properly
adjusted balancing masses nor a compass consisting of two or more
properly placed gyros is subject to deflections due to centripetal
forces developed during rolling and pitching of the ship.
117. The Suppression of the Quadrantal Error. No feature
of the gyro-compass has so taxed the ingenuity of physicists and
designers as the avoidance or elimination of the quadrantal error.
In Art. 115 it has been shown that the quadrantal error is due to
the cumulative effect of vertical components of the precession of
the spin-axle produced by the swinging of the compass system from
side to side as the ship rolls or pitches. The quadrantal error
would be avoided : (a) if the suspended compass system remained
vertical however the ship rolls or pitches; (6) if the impulses im-
parted to the swinging gyro-frame by the rolling and pitching of
the ship were applied only when the line through the center of the
196 NAVIGATIONAL COMPASSES
gyro-wheel and the center of gravity of the pendulous mass is
vertical ; (c) if the point of application of the force on the gyro due
to the pendulous mass were shifted back and forth from one side
of the vertical to the other in such a manner that a vertical pre-
cession is thereby produced which is equal and opposite to that due
to the rolling or pitching of the ship; (rf) if the natural period of
oscillation of the pendulous system were much greater than the
period of oscillation of the ship; (e) if the sensitive element were
non-pendulous.
118. The Degree of Precision of Gyro-Compasses. In order
that a compass may furnish a satisfactory base line for gun-fire
control, it must have a greater degree of precision than is required
of a compass used only for navigating the ship. Large naval
vessels are equipped with gyrocompasses of the most careful
design and workmanship known to the art. Such compasses are
called " gun-fire control " compasses. The less severe require-
ments of merchant vessels and naval vessels of the smaller sizes
are met adequately by instruments of a- somewhat simpler type
called " navigational compasses/' Of the gyro-compasses men-
tioned in the following Articles, the Anna Mark IV and the Sperry
Mark X instruments are " gun-fire control " compasses. The
others are " navigational " compasses.
The tests of gyro-compasses submitted for use in the United
States Navy include measurements of the undamped period and the
damping percentage of the sensitive element, the change of set-
tling point when the voltage impressed on the gyro-motors changes,
and the change of settling point when the binnacle is rotated
about a vertical axis or when rocked about various horizontal
axes. The tests of the changes of the settling point produced
by rotation and rocking of the binnacle are made by means of a
machine called a " scoresby/' from the name of the inventor.
This machine consists of a platform that can be rotated in either
direction about a vertical axis and that can be rocked with a given
period and with a given amplitude about a horizontal axis in any
azimuth. Figure 151 represents a scoresby rocking a gyro-com-
pass G as in an actual test.
If .r and y represent two successive amplitudes of swing in the
/> <jj
same direction, then * - may be called the fractional damping
(. y\
1 would be called the damping percentage. The
x )
NATURAL ERRORS
197
damping percentage is calculated from the amplitudes of the
first three peaks as follows divide one hundred times the ampli-
tude of the second peak by the amplitude of the first peak and
subtract the quotient from one hundred; divide one hundred times
the amplitude of the third peak by the amplitude of the second
peak and subtract the quotient from one hundred. The mean
of these two results is the mean percentage of damping. If the
compass be precessed either toward
the east or toward the west 30 de-
grees from the settling point and
then released, the damping percent-
age must not be less than 60 per
cent nor more than 80 per cent.
When rotated about a vertical
axis at a uniform rate of 3 degrees
per second, a gyro-compass of the
" navigational " grade must not
show an error greater than 0.5 de-
gree during a run of twenty hours,
and a gyro-compass of the " gun-
fire control " grade must not show
an error greater than 0.2 degree.
When rocked back and forth
about a horizontal axis through an
angle of 15 degrees from the vertical
with a period of 9 seconds a gyro-compass of the " navigational "
grade must not show an error greater than 0.6 degree during a
run of forty-eight hours, whatever may be the heading, and a
gyro-compass of the " gun-fire control' ' grade must not show an
error greater than 0.3 degree.
When rocked back and forth about a horizontal axis through an
angle of 40 degrees from the vertical with a period of 9 seconds a
gyro-compass of the " navigational " grade must not show an
error greater than 1.0 degree during a run of forty-eight hours,
whatever may be the heading, and a gyro-compass of the " gun-
fire control " grade must not show an error greater than 0.5
degree.
When the voltage impressed on the gyro-motor (or motors) is
changed from normal to 10 per cent more or less than normal, the
maximum settling point error of a gyro-compass of the " naviga-
tional " grade must not exceed 0.5 degree, and for a gyro-compass
FIG. 151
198 NAVIGATIONAL COMPASSES
of the " gun-fire control " grade the error must not exceed 0.2
degree.
A gyro-compass equipment includes the master compass, an
electric motor-generator, storage battery and switchboard, to-
gether with repeater compasses. Many installations include a
course recorder, a radio direction finder, and an automatic pilot
operated by the master compass. By means of a gyro-compass
controlled automatic pilot, the ship may be kept in a straight
course as long as may be desired without the aid of a helmsman.
The master compass and electric plant are installed between decks.
Then the ship is steered from a regular repeater compass, and
bearings are taken from two other repeater compasses at the ends
of the officers' bridge.
The* magnetic compass is a simple and relatively cheap instru-
ment, but the errors to which it is subject can be but partially
eliminated or allowed for. On the other hand, the gyro-compass
is a complicated and expensive outfit, but the errors to which it
is subject can be completely eliminated automatically. A stand-
ard magnetic compass for a large ship costs about $500. The
cost of a master gyro-compass such as is found on merchant ships
and the smaller naval vessel costs about $3000. The cost of a
gyro-compass equipment as used on merchant ships, consisting of
the master compass, electric plant, three repeaters and course
recorder, costs about $5500. An automatic pilot adds about
$2200 to the cost of the compass equipment. A single gyro-com-
pass equipment, of the highest grade, such as is used for gun-fire
control on large battle ships, costs from three to four times as
much as one of the " navigational " grade used on merchant ships.
It is usual to have two complete " gun-fire control " gyro-compass
equipments on a large battle ship.
3. The Spcrry Gyro-Compass
119. The Principal Parts of the Master Compass. Since 1920
all gyro-compasses made by the Sperry Gyroscope Company*
have been single-wheel instruments of the liquid-controlled non-
pendulous type (Art. 107). The various models differ in directive
torque, degree of precision, and in details of design, but they are
similar in all essential respects. The models commonly employed
on mercantile vessels are designated Mark VI and Mark VIII.
The gyro-wheel of each is 10 inches in diameter, weighs 54 pounds,
* Made in Brooklyn, N. Y., in London, England, and in Tokio, Japan.
THE SPERRY GYRO-COMPASS
199
and rotates in the open air at a speed of 6000 revolutions per min-
ute. The two models differ in that the gyro-wheel of Mark VI
constitutes the armature of a direct current motor and the field
coils are imbedded in the gyro-casing, whereas the gyro- wheel of
Mark VIII constitutes the rotor of a three-phase alternating cur-
rent motor and the stator coils are imbedded in the gyro-casing.
Mark V and Mark X are designed for use on naval vessels. The
gyro-wheel of the Mark V is 12 inches in diameter, weighs 45 pounds
and rotates at 8600 revolutions per minute. The gyro-wheel of
the Mark X Gun-Fire Control Gyro-Compass* is 13 g inches in
diameter, weighs 122 pounds and rotates at 10,000 revolutions per
minute. The gyro-wheels of Mark V and Mark X rotate in a
partial vacuum of about 28 inches of mercury below atmospheric
pressure.
A simplified diagram showing the fundamental parts of the
Sperry gyro-compass, as seen when looking toward the compass
from the south, is represented in Fig. 152. The gyro-wheel,
spinning about a nearly horizontal axis perpendicular to the
plane of the diagram, is enclosed by the gyro-casing G. The gyro-
casing is supported on horizontal bearings by a vertical ring R
which, in turn, is suspended by a system of vertical wires from the
top of a tube extending upward from an outer ring P called a
" phantom/ 7 The phantom is supported by ball bearings on a
" spider " S supported on gimbal rings carried by the binnacle
XX. The phantom carries the compass card C, and the " azi-
* See Frontispiece.
200
NAVIGATIONAL COMPASSES
FIG. 153
FIG. 154
THE SPERRY GYRO-COMPASS
201
FIG. 155
FIG, 166
202
NAVIGATIONAL COMPASSES
muth gear " Z, by means of which the phantom may be rotated so
as to bring it into the plane of the vertical ring. The " mercury
ballistic " B\Bz is fastened to a frame pivotted to this phantom.
It is loosely connected to the gyro-casing by a pin e placed a little
to the east of the vertical line through the center of mass of the
suspended system.
The gyro- wheel, casing and vertical ring constitute the " sen-
sitive element," Fig. 153, which is acted upon by a meridian-
seeking torque due to the rotation of the earth. The phantom
element is shown in Fig. 154. The supporting spider is shown in
Fig. 155. The complete gyro-compass, Mark VIII, removed from
the binnacle is shown in Fig. 156. In this compass, the mercury
ballistic consists of two pairs of mercury reser-
voir B\B\ and B^B^ each pair connected by a
small tube.
120. The Follow-Up System. When the
sensitive element becomes turned out of the
meridian plane in either direction through an
angle even so small as one-quarter of a degree,
an electric contact is made which causes a
small reversing motor, M, Fig. 155, to rotate
the azimuth gear, Z, Fig. 154, through an
equal angle in the same direction. Thus the
phantom element follows the sensitive element,
thereby preventing any appreciable twisting
of the suspending wires. In this manner, the
suspending wires never develop an appreciable
torque in opposition to the meridian-seeking
torque acting on the compass. The follow-up system neutralizes
the effect of frictional forces that oppose motion about the vertical
axis.
A pair of contactor plates, insulated from each other, is carried
on each of the opposite sides of the phantom element, Q, Fig. 154.
Lightly pressing against each pair of contractor plates is a small
trolley wheel, T', Fig. 153, borne by the supporting ring. The
electric connections of the follow-up system are shown in Fig. 157.
In this figure, A represents the armature and FF' represent the
field coils of a reversing motor fastened to the spider. The two
pairs of contactor plates X Y and X f Y' are attached to the opposite
sides of the phantom element, and the two trolleys TT f are at-
tached to opposite sides of the sensitive element. Since the
Fia. 157
THE SPERRY GYRO-COMPASS 203
motor is geared to the azimuth gear Z, Fig. 154, it is called the
" azimuth motor." When the trolleys touch the contactor plates
XX', as indicated in the diagram, current from B follows the
course BTXFAB, and the motor armature together with the
connected phantom element and system of contactor plates rotate
in one direction. This rotation continues till the contactor plates
YY' touch the trolleys TT f . Then the current follows the path
BT'Y'F' AB. The direction of rotation of the azimuth motor and
connected phantom element now is opposite that in the former
case. The phantom element " hunts " back and forth through
an angle of less than a degree about the sensitive element as mid-
position. This oscillation results in elimination of lag in the in-
dications of the sensitive element.
121. The Method of Damping. If there were no damping, the
north-seeking end of the gyro-axle would move in an elliptical
orbit in a plane perpendicular to the meridian plane, Fig. 137.
The amplitude of oscillation will be damped if either the vertical or
the horizontal component be reduced. If both components be re-
duced, the damping will be greater. In tin* Sperry gyro-compass
both components are reduced by an arrangement that causes the
major axis of the elliptical path traced by the end of the precessing
spin-axle to be inclined to the horizontal. The arrangement con-
sists in connecting the mercury ballistic loosely to the gyro-casing
by means of a pin that is slightly to one side of the vertical line
through the center of mass of the gyro-casing and wheel.
As shown in Art. 107, if the mercury ballistic were connected
to the gyro-case by a pin vertically below the center of the gyro-
wheel, then the prolongation of the north-seeking end of the gyro-
axle would trace on a vertical plane XZ, Fig. 158, a path which is
an ellipse with major axis horizontal. Now we shall consider the
effect of connecting the mercury ballistic to the gyro-case by a
pin, e, to the east of the vertical through the center of the gyro.
Suppose at some instant the spin-axle is horizontal and the north-
seeking end is directed toward a point a, Fig. 158, east of north.
Owing to the rotation of the earth, mercury flows from the more
easterly reservoirs to the more westerly (Art. 107). This causes
the phantom P to press against the pin e and so produces a torque
about an axis oL, perpendicular to the line oe. The torque pro-
duces a precession about the line oe, inclined to the vertical at an
angle 6. Since the axis of precession is inclined to the vertical, the
prolongation of the spin-axle will cross the meridian at a point b'
204
NAVIGATIONAL COMPASSES
nearer the horizontal than it would if the precession axis were
vertical, Fig. 158. It will cross the horizontal at a point c' nearer
the meridian than it would if the precession axis were vertical.
It will cross the meridian again at a point d' nearer the horizontal
than it would if the precession axis were vertical. In fact, the
prolongation of the north-seeking end of the spin-axle traces on
a vertical plane XZ, perpendicular to the meridian plane, a
spiral that converges to a point on this plane very close to the
intersection of the meridian and the horizontal plane through
the center of mass of the
gyro. When the compass
is in northern latitudes,
this point is slightly above
the horizontal; when in
southern latitudes, it is
slightly below. The prin-
cipal axis of the spiral is
inclined to the horizontal
plane and is fixed with
reference to the precessing
gyro- wheel. The degree
of damping is controlled
by the distance between
the pin e and the vertical
through the center of the
gyro. The amplitude of
each swing of the gyro-axle
is about one-third that of
the preceding swing.
In Fig. 158, the torque
acting on the gyro-wheel is represented by L. This torque pro-
duces a precessional velocity represented by w pj toward b' . If
the mercury ballistic is maintained always in the same position
relative to the gyro-axle, then the projection of the gyro-axle will
continue to leave the elliptical path and trace a converging spiral
having a major axis inclined to that of the ellipse. The con-
stancy of the position of the mercury ballistic relative to the gyro-
wheel is maintained by means of the follow-up system. This
causes the ballistic to follow every movement in azimuth of
the spin-axle. By means of the ballistic, supported by the phan-
tom, and the eccentric connection with the gyro-case and mer-
FIG. 158
THE SPERRY GYRO-COMPASS
205
cury ballistic, the spin-axle will quickly settle to a position which
is nearly horizontal and in the geographic meridian.
122. The Magnitude of the Latitude Error for Which Cor-
rection Must Be Made in the Sperry Gyro-Compass. When the
gyro-compass is at any latitude there is a vertical component of the
earth's angular velocity w e of the value, Fig. 159,
MM = w e sin X
I^IG. 159
It follows that, in order that the spin-axle may remain in the
meridian, the spin-axle must be caused to process about a vertical
axis with an angular velocity equal in magnitude to w cv but in the
opposite direction. This precession requires a torque about a
horizontal axis. The requirement of a torque about a horizontal
axis, and the additional requirement that for damping there must
at the same time be a torque about the vertical axis, are fulfilled
in the Sperry Mark V compass and later models by connecting the
mercury ballistic to a point of the gyro-casing slightly to the east
of the point vertically below the center of the gyro-wheel, as we
have seen in Arts. 107 and 121. By this scheme, the applied
torque is about an axis making a small angle to the horizontal, and
the processional velocity Wp thereby produced is about an axis in-
clined at the angle 6 to the vertical.
The horizontal component of the torque produces a precessional
velocity w pv about a vertical axis, Fig. 160. The vertical com-
ponent of the torque produces a precession w P / t about a horizontal
axis perpendicular to the gyro-axle of the value
u>ph = Wfr tan
If the spin-axle of the compass is to remain in the meridian,
the vertical component of the velocity of precession must equal
the vertical component of the earth's rotation. That is,
206 NAVIGATIONAL COMPASSES
Substituting in this equation the values above,
= w c sin X
tan
= w e sin X tan (128)
In order that the gyro-compass may remain in equilibrium,
the resting position of the gyro-axle, aa f , Fig. 161, must make an
angle <5 2 with the meridian plane NS such that
the component w P h of the processional velocity
of the spin-axle about a horizontal axis hh' per-
pendicular to the spin-axle, will equal the com-
; ponent of the earth's rotation with respect to the
same axis. In Figs. 159 and 161, the horizontal
component in the meridian plane of the angular
velocity of the earth is represented by w e h. Thus,
we wish to have
Wp/ t = w f h sin 5 2
From Fig. 159
Weh = W e COS X
From those last, two equations and (128) we can eliminate w eh ,
Wph and w e , and obtain
sin X tan = cos X sin <5 2
For all compasses designed for the same angle 6, tan 6 is a con-
stant. If <5 2 be small, we may replace sin <5 2 by <5 2 . Under these
conditions, the correction of a Sperry compass due to latitude has
the value
<5 2 = tan tan X = b tan X (129)
where b represents the constant tan 6. This correction must be
added to or subtracted from the compass reading, depending upon
whether is positive or negative and whether the compass is
north or south of the equator. In northern latitudes the deflection
is east of the meridian; in southern latitudes it is west of the
meridian. At latitude 50, the latitude error is about 2.
123. Correction Mechanism for Velocity and Latitude Errors.
The compass card is surrounded by a coplanar ring called the
" lubber ring " on which is engraved a line called the " lubber
line " parallel to the keel of the ship. If the spin-axle is in the
meridian plane, the angle between the N-S line of the compass
THE SPERRY GYRO-COMPASS 207
card and the lubber line gives the direction of the ship. The cards
of gyro-compasses are marked in degrees, from to 360.
A person could tell the correct time from the reading of a clock
known to be five minutes fast by subtracting from the observed
reading the five-minute error. He could tell the correct time
without the necessity of keeping the error in mind if the clock
face were rotated through 30 degrees in the clockwise direction
about the spindle which carries the two hands. This procedure
might not be regarded as correcting the clock error but it would
at any rate make the clock indicate the correct time. In a similar
manner the meridian-steaming error and the latitude error of the
Sperry gyro-compass are allowed for automatically by a shift of
the lubber line.
The total deviation of the gyro-axle from the meridian due to
the meridian-steaming error together with the latitude error is
(119 and 129):
5 1 + fc = c08 J6 ta nX (130)
COS A
where a and b are easily determined constants, v is the speed of
the ship in knots, is the course measured in degrees from the
meridian and X is the latitude. The first term in the right-hand
member of (130) results in an eastward deflection of the north-
seeking end of the spin-axle when the ship is steaming southward
and a westward deflection when steaming northward. The second
term results in an eastward deflection when the ship is in northern
latitudes and a westward deflection when in southern latitudes.
The device used on the Sperry gyro-compass consists of two
dials, a cam and a system of rods that maintain the proper shift
of the lubber line after the dials have been set for the latitude and
speed of the ship. Figure 162 is a diagram of the device that
is used on compasses of the models designated Mark II and Mark
V. The lubber ring X is turned relative to the compass card Y
by the system connected to the pin Z. The course-corrector ring R
is fastened rigidly to the phantom and consequently remains in
constant relation to the gyro-axle of the compass. Its plane is
inclined to the horizontal plane. The east-and-west diameter of
the ring is horizontal. The higher side of the ring is directly
above the north-seeking end of the gyro-axle of the compass. A
roller C on one end of the rod AC engages in a groove in the edge
of the ring. The end A of the rod AC and the two dials Di and
208
NAVIGATIONAL COMPASSES
-D 2 are fixed to the spider. When the course of the ship changes, the
ring and the gyro-axle preserve their direction in space whereas
the rod AC and all the remainder of the correction device turns
with the ship. When the ship with the rod AC turns relative to
the ring R, the roller C moves either up or down as it follows the
groove in the ring. When the course of the ship is inclined to the
Speed
Corrector
FIG. 1(52
meridian, the roller is displaced vertically from the horizontal
plane through the center of the ring by an amount x which now
will be determined. Figure 163 represents the speed-corrector
ring R, in elevation and in plan, when the course of the ship
makes an angle with the meridian. The inclination of the
plane of the corrector ring to the horizontal
is 0. From the figure :
X = d tan </> = r cos tan </>
Since r tan </> is constant, the above equation
shows that the vertical displacement x varies
directly with the cosine of the course of
the ship. This displacement produces a
corresponding displacement of the pin E
fastened to the dial DI.
A slot in the arc of a circle extends from
the center to near the edge of the speed-corrector dial DI. The
radius of the arc equals the distance between the two pins at the
ends of the rod GIL When the speed corrector dial is rotated
from the position shown in the diagram, the rod GH and the pin
Z are moved along. The amount of displacement of the lubber
ring thereby produced depends upon the distance v of the pin G
FIG. 163
THE SPERRY GYRO-COMPASS 209
from the center of the dial D\. After the pin G has been clamped
at the position corresponding to the speed v of the ship, the lubber
ring will be displaced automatically by the amount required to
allow for the quantity v in (130). Figure 162 is for a due westerly
course. In this case there is zero correction for speed.
The pin E can move freely along the radial slit in the speed cor-
rector dial. When the course of the ship is in the east-west line,
the slot is horizontal as shown in the diagram. The position of
the pin E in this slot is determined by the position of the pin F
in the latitude-corrector dial D 2 . The distance of the pin F from
the center of the latitude corrector dial is made proportional
to the constant a in (130). The length of the rod EF equals the
distance between the centers of the two dials. When the latitude
corrector dial is set for zero latitude, the pins E and F are on the
straight line through the centers of the two dials.
When the index is placed opposite the number
representing the latitude of the ship, this arrange-
ment makes allowance for the quantity ----- in
1 J cos X
(130).
One end of the rod JL can slide in a fixed guide
P. This rod is fastened to the latitude-correc-
tion dial Z>2 by means of a pin and clamp K that
permit rotation of the dial. The distance of the
pin from the center of the dial is made propor- ' IG *
tional to the constant b in (130). If while the end // of the rod
HJ remains fixed, the index of the latitude correction dial be set
at the number corresponding to the latitude X of the ship, that is
the clamp K moved to a new position K', then the edge of the
lubber ring will be moved from the zero position through a dis-
tance proportional to KK'[= b tan X], Fig. 164. Thus proper
allowance would be made for the second term in the right mem-
ber of (130).
The speed and latitude corrector used on the Sperry gyro-
compass Mark VI and VIII is much simpler in design than that
just described. Two blocks, xx', Fig. 165, are attached to the
lubber ring. The other part of the device, NK, is attached to
the spider. The two parts are connected by a stud Z projecting
from a nut that can be moved back and forth by means of the
knurled headed screw Y. The face of the nut is marked off into
divisions representing latitudes. By setting the division corre-
210
NAVIGATIONAL COMPASSES
spending to the latitude opposite the lubber line, the lubber line
is shifted relative to the spider by an amount proportional to the
quantity 6 tan X (130).
FIG. Ki5
FIG. 1(56
The course corrector cam consists of a circular groove in the
under face of the azimuth gear, Z, Fig. 154. The groove is eccen-
tric relative to the azimuth gear. In this groove fits a roller C
on the end of a bent lever PQ, Figs. 165 and 166. The other end
of the bent lever fits into the lower end of a vertical lever R T,
Fig. 166, capable of rotation about a pin 8. This pin can be
moved up or down by means of a screw, K, Figs. 155 and 165,
thereby changing the lever arm 8T, Fig. 166. A displacement of
the end T rotates the lever UV about a fixed pin (/, thereby shift-
ing the lubber ring attached to XX' by an amount which depends
upon the lengths of the various lever arms. The lever PQ allows
for the quantity a cos 9. The allowance for - r is made by a
proper setting of the pin S. This setting is effected by adjusting
the screw K, Fig. 165, until the correct latitude reading on the
horizontal scale M is on the correct speed curve engraved on the
face plate N, Figs. 156 and 165. The speed curves are obtained
empirically.
By thus shifting the lubber line to the proper amount, the
ship will point correctly but the compass will not. In taking
bearings of heavenly bodies, or of points on the shore, it is neces-
sary to know true directions. True directions can be gbtained by
means of a repeater compass electrically connected to a trans-
mitting device mounted on the lubber ring of the master compass
(Art. 127).
124. Avoidance of the Ballistic Deflection Error. In the
Sperry gyro-compass models designated Mark V, Mark VI, Mark
VII and Mark VIII, the period of vibration of the sensitive ele-
THE SPERRY GYRO-COMPASS 211
ment required to avoid the ballistic deflection error is secured by
regulating the amount of mercury that flows from one side of the
sensitive element to the other.
The mercury ballistics of the Sperry Mark VI and Mark VIII
gyro-compasses have two separate reservoirs K {) /? 2 , Figs. 152,
156 and 158, on the north side of the phantom element, connected
by two tubes to correspondingly situated reservoirs on the south
side. The mass of mercury that can pass from one side to the
other is such that the ballistic error is zero when the ship is at
latitude 40 and is not great at any latitude.
The Mark V gyro-compass has a single mercury reservoir on
the north side of the phantom element and another on the south
side. These reservoirs are divided into compartments and pro-
vided with valves which can be adjusted to give four different
masses of active mercury. With one weight of active mercury,
the torque that opposes tilting is such that the ballistic error is
very small at all latitudes between and 35; with another weight-,
the error is very small at all latitudes from 30 to 50; with an-
other weight, the error is very small at all latitudes from 45 to
62; with another weight, the error is small at all latitudes from
60 to 70; in each case the ballistic error is zero at the mean lati-
tude.
The period of the Sperry gyro-compass models designated Mark
IX, Mark X and Mark XI is adjusted to the value proper to any
given latitude by changing the lever arms of the 1 weights of the
active liquid in the mercury ballistic. This result is accomplished
by moving four mercury reservoirs to the proper distance from t he
vertical axis through the center of mass of the suspended system.
Each pair of reservoirs is fastened on opposite ends of a jointed
tube similar to the jointed gas-light brackets sometimes used over
work benches. By means of a worm and gear provided with a
divided scale the reservoirs can be moved to the position proper
to the known latitude. This device can be recognized in the
Frontispiece. Observe the horizontal shaft carrying a worm at
each end, on the right-hand side of the sensitive element. Each
worm engages in a horizontal gear to which is rigidly attached a
cylindrical reservoir. The reservoir attached to the gear that
engages with the worm on the left-hand end of this horizontal
shaft is only partially obscured by the gear. Observe the small
tube joining this reservoir with another on the left-hand side of
the sensitive element.
212 NAVIGATIONAL COMPASSES
125. The Automatic Ballistic Damping Error Eliminator.
The Mark X gyro-compass is provided with an automatic damping
eliminator which moves the pin e, Figs. 152 and 158, to a position
vertically below the center of mass of the gyro-wheel when the
ship has made a turn of as much as 15 degrees. Thereafter there
is zero damping. The operation of the device is as follows:
While the ship is making a turn, the compass binnacle with the
attached spider element turns with respect to the phantom ele-
ment and the gyro-axle. The large azimuth gear attached to the
phantom element is in mesh with a small pinion attached to the
spider element. The gear ratio is 3500 : 1. The pinion is con-
nected to a shaft on which is mounted a group of three balls at-
tached to three springs and a collar capable of sliding along the
shaft. When the shaft and balls rotate, the balls fly outward
thereby moving the collar along the shaft. When the ship is
turned rapidly through an angle as great as 15 degrees, the sliding
collar closes the electric circuit of a magnet which then moves the
eccentric pin to a position vertically below the center of mass of
the gyro-wheel. The damping now ceases and further ballistic
damping error is prevented. When the turning of the ship ceases,
the ball governor ceases to rotate, the electric circuit to the elimi-
nator magnet is broken, the eccentric pin returns to its normal
position and damping is renewed.
126. Avoidance of the Quadrantal or Rolling Error. The
sensitive element of the Sperry mercury ballistic gyro-compass is
non-pendulous both when the mercury ballistic is in position and
when it is detached. So long as the spin-axle is horizontal, all
solid masses arc balanced with respect to the horizontal axis.
Forces due to rolling or pitching tend to accelerate these solid
masses but they also will be balanced, and, consequently will pro-
duce zero deflection of the spin-axle. However, the meridian
component of forces due to rolling or pitching will cause mercury
to flow from one reservoir to the other, thereby causing the center
of mass of the sensitive element to move to one side of the vertical
line through the point of support. The oscillation of the mercury
causes the sensitive element to act like a pendulum and to have
a quadrantal error. By adding a small mass of metal to the top
of each mercury reservoir, the sensitive element is made slightly
anti-pendulous or top-heavy, thereby diminishing the quadrantal
error produced at the end of each roll or pitch.
The dimensions of the apparatus arc so selected that there is a
THE SPERRY GYRO-COMPASS
213
phase difference of nearly one-quarter period between the oscil-
lation of the sensitive element and the mercury from one reservoir
to the other. As the residual slight deflections are not cumulative,
they produce zero resultant deflection of the spin-axis.
127. The Repeater System. The latitude error and the me-
ridian-steaming error produce a deflection of the gyro-axle out of
the meridian plane (Arts. 110 and 111). It follows that the card
of the master compass does not give true directions relative to the
meridian. When, however, the lubber line is shifted relative to
the compass card through an angle equal to the latitude and
meridian-steaming errors, the course of the ship relative to the
Jronsrni
Operated bu Master
Compass , ,
' Supply Line
FIG. 167
meridian is properly indicated. The master compass gives true
headings but not true bearings. The angle which any horizontal
line through the compass makes with the meridian plane equals the
angle between the given line and the gyro-axle plus the angle
through which the lubber line is shifted from its zero position.
The sum of these two angles is the bearing of any object on the
given line. The addition of these two angles is made automatically
by a repeater system consisting of a transmitter T, Figs. 155 and
156. The transmitter is electrically connected to repeater com-
passes, course recorder, automatic pilot, radio direction finder
or other devices situated at convenient places on the ship.
The transmitter of the Mark VI and Mark VIII models is
represented diagrammatically in the left side of Fig. 167. It
consists of a collector ring r, twelve terminal blocks x, y, z, ar-
ranged in a circle concentric with the collector ring, and a contact-
making transmitter arm capable of rotating about the central
shaft. The transmitter arm is bent. The shaft s about which the
arm rotates is at the elbow. On each end of the transmitter arm
214 NAVIGATIONAL COMPASSKS
is a trolley which presses against the circle of contact blocks.
At the elbow are brushes which press against the collector ring.
The shaft s fastened to the transmitter arm is connected by a
gear train to the azimuth gear of the compass. When the ship
and the lubber ring turn with respect to the sensitive element, the
transmitter arm rotates inside of the circle of terminal blocks.
Each repeater compass, course recorder or other auxiliary oper-
ated by the master compass has a repeater motor which is repre-
sented diagrammatically on the right side of Fig. 167. The motor
has three pairs of poles and a soft iron armature A. When one
pair of diametrically opposite poles is energized, the soft iron
armature sets itself in line with that diameter, thereby rotating
any compass card, course recorder or other auxiliary attached to
the armature. When the transmitter arm rotates in the counter-
clockwise direction, contacts are made in the sequence x, y, z,
x, ?/, etc., and the repeater armature rotates in the same direction.
When the transmitter arm rotates in the clockwise direction, con-
tacts are made in the sequence x, z, y, x, z, etc., and the repeater
armature rotates in the same direction.
With the transmitter arm in the position shown in the diagram,
current goes from the direct current supply line to the collector
ring r, to two x contact blocks, to the motor field coils XX' and back
to the generator. The armature A sets itself in the direction XX'.
If the transmitter arm rotates clockwise till one trolley is in con-
tact with an x block while the other trolley is still in contact with
a y block, then both the YY' field coil and the XX' field coil
will be magnetized, and the armature A will set itself midway
between them. Thus while the* transmitter arm is making one
revolution, the repeater armature makes one revolution in twelve
steps.
The gear ratio between the repeater motor armature and the
connected repeater compass card of the Mark VI and the Mark
VIII Sperry gyro-compasses is 1 to 180. Consequently one
revolution of the repeater motor armature rotates the connected
repeater compass card two degrees. Therefore each electric im-
pulse from the transmitter produces a rotation of the repeater com-
pass card of one-twelfth of two degrees or ten minutes of angle.
That is, a Mark V, Mark VI or Mark VIII gyro-compass repeater
will indicate a change in the ship's course of five minutes of angle.
The Mark X indicates a change in course of one-hundredth of a
degree.
THE BROWN GYRO-COMPASS
215
Since the transmitter is attached to the lubber ring, the cor-
rections made for latitude and velocity by shifting the lubber
ring are included in the indications transmitted to the repeaters.
1. The Brown Gyro-Compass
128. Production of the Meridian-Seeking Torque. The
Brown gyro-compass is an instrument developed in 1916, of the
pendulous type, with a single gyro-wheel of 4.25 Ib. spinning at
FIG. 168
about 15,000 revolutions per minute.* The master compass,
removed from the binnacle, is shown in Figs. 168 and 174. The
gyro-casing is carried by horizontal trunnions mounted in bear-
ings on a ring R, Figs. 169 and 174, which always is nearly vertical.
The vertical ring is capable of turning about a vertical shaft ex-
tending above and below the ring. The entire sensitive element
is carried by an outer frame suspended from the gimbal rings of
the binnacle not shown in the figures. The sensitive element
does not rest on a solid bearing but upon a column of oil at high
pressure under the lower shaft. Friction about a vertical axis is
reduced to almost zero by rapid pulsations of oil pressure which
move the sensitive element up and down through an amplitude of
* Made by S. G. Brown, Ltd., London, England.
216
NAVIGATIONAL COMPASSES
about one-eighth inch some 180 times per minute. The compass
card is attached to the vertical ring and the lubber ring is attached
to the outer frame.
The unspinning gyro-wheel and casing constitute a non-pen-
dulous system with respect to the horizontal trunnions connecting
it to the vertical ring. The vertical ring and supporting frame
constitute a pendulous system capable of oscillating about a hori-
zontal axis. The required meridian-seeking torque is produced
by displacements of oil between
two connected " control " or
" working bottles " Ci, C 2 , Fig.
169, fastened to the gyro-casing
G. The displacement of oil is
due to an air-blast issuing from
an upward directed nozzle Z fixed
in the vertical supporting ring R.
The rotation of the gyro-wheel
produces a strong blower effect.
The trunnions carrying the gyro-
casing are tubular, and one of
them conducts air to the nozzle
Z. When the ring is vertical and
the spin-axle is horizontal, this
air-blast enters the two sides of
the " air-box " A equally and produces equal pressures on the sur-
faces of the oil in the two control bottles. When the gyro-axle
is tilted as in Fig. 169, oil is forced from the lower to the higher
control bottle, thereby developing a torque on the gyro-system
about an axis perpendicular to the plane of the diagram and in the
direction represented by the symbol at L. If the direction of
spin is in the direction represented by the arrow h s , then the gyro-
axle will process about a vertical axis in the direction represented
by the arrow w p .
Now consider the effect of the rotation of the earth on the
direction of the spin-axle of a Brown gyro-compass. Imagine
that we are above the northern hemisphere of the earth looking at
a Brown gyro-compass at X (Fig. 170). Suppose that the spin-
axle is horizontal, not in the meridian plane, and that the direction
of spin is that represented by the arrow marked h s . The pendulous
frame and ring are vertical. Because of the rotation of the earth,
the instrument will move into the position Y. The pendulous
FIG.
THE BROWN GYRO-COMPASS
217
frame and ring continue to be vertical and the direction of the
spin-axle continues in the same direction in space that it was when
the instrument was at X. The resulting displacement of the gyro-
casing with respect to the vertical frame causes an excess of air to
enter the western control bottle and consequently an excess of oil
to enter the eastern bottle. A torque is thereby produced about
FIG. 170
an axis perpendicular to the spin-axis and in the direction repre-
sented by the symbol at L, Fig. 170. As the spin-axle moves
toward parallelism with this torque-axis, it moves toward the
meridian plane with the direction of spin in the direction of the
rotation of the earth. The displacement of oil from one control
bottle to the other causes this non-pendulous sensitive element to
act as a pendulously mounted gyro. For this reason it is common
to speak of the " degree of pendulousness " of the Brown sensitive
218
NAVIGATIONAL COMPASSES
element. As shown in Art. 106, the spin-axle crosses the meridian
plane with the north-seeking end above the horizontal plane
through the center of the gyro-wheel. It does not remain in that
position, but slowly describes the surface of an elliptical cone having
as axis a horizontal line in the meridian plane through the center of
the gyro-wheel. The control bottles and air system constitute
an air pressure liquid relay that causes the statically non-pendulous
sensitive element to precess into the meridian plane with the di-
rection of spin of the gyro in the direction of rotation of the earth.
129. The Method of Damping. The
oscillation of the spin-axle back and forth
across the meridian plane is due to the
tilting of the spin-axle from the horizon-
tal. The oscillations can be damped by
a torque about a horizontal axis oppos-
ing the torque produced by the pendu-
lous vertical ring and supporting frame.
This counter torque will be most effective
in producing damping if, at every in-
stant, it is proportional to the angular
velocity of the sensitive element about
the vertical axis.
In the Brown gyro-compass this result
is accomplished by the displacement of
a mass of oil back and forth between two
metal bottles fastened to the opposite faces of the gyro-casing,
beside the control bottles. The upper ends of these " damping
bottles " are connected by small pipes to an air-box A above the
upturned air-nozzle Z, Fig. 171. Note that the connections of the
air-pipes from the control bottles to the air-box, Fig. 169, are not
the same as the connections of the air-pipes from the damping
bottles to the air-box, Fig. 171. A needle valve V controls the
speed of flow of oil from one damping bottle to the other.
The operation of the damping device is as follows : Suppose that
the spin-axle is horizontal and at its maximum angular displace-
ment to the east of the meridian. The north-seeking end is at a,
Fig. 137. The oil is at the same level in the control bottles, and
also in the damping bottles. As the earth carries the gyro-compass
from the position X to the position F, Fig. 170, the spin-axle tilts,
oil is forced from the lower control bottle into the upper, and the
spin-axle turns toward the meridian. When the meridian plane is
FIG. 171
THE BROWN GYRO-COMPASS
219
Gyro Wheel Case
being crossed, the tilt of the spin-axle is maximum, the amount of
oil in the upper control bottle is maximum, and the angular ve-
locity of the spin-axle about a vertical axis is maximum.
Meantime, oil has been moving slowly from the upper clamping
bottle into the lower. So long as the spin-axle is nearly horizontal
there is nearly the same amount of oil in the two damping bottles
and consequently there is little opposition to the torque due to the
control bottles. By the time the spin-axle is crossing the meridian
plane with maximum speed, the difference in the amount of oil in
the two damping bottles is nearly maximum and the opposition
to the motion of the spin-axle is considerable. The transfer of
oil from the upper to the lower damping bottle produces a dimi-
nution of the pendulousness
of the sensitive element and,
consequently, a diminution
of the deflection of the gyro-
axle.
The power acting on the
sensitive element at any in-
stant by the damping system
equals the product of the
angular velocity of the sensi-
tive element and the torque
opposing vibration at that
instant. To produce damp-
ing, the phase difference be-
tween the torque and the angular velocity must be more than 90
degrees (Art. 88). The phase difference can be changed by regu-
lating the flow of oil through the needle valve.
When the needle valve has been properly adjusted, oil flows
so slowly from the upper damping bottle to the lower that the oil
in these bottles has not become at the same level at the time when
the spin-axle is horizontal. This difference in head causes the
flow to continue in the same direction, notwithstanding the op-
posing air-blast, till the sensitive element is near the end of the
swing. Throughout this time, the oil in the damping bottles
produces a torque in opposition to the torque due to the pendu-
lousness of the sensitive element. Hence, the deflections in azi-
muth of the spin-axle are diminished. After two or three vibra-
tions, the spin-axle will come to a resting position. If no torque
acts upon the sensitive element other than the ones already con-
ax P/pes ^ "~^ North Control &ott/e
FIG. 172
220 NAVIGATIONAL COMPASSES
sidered, the resting position of the spin-axle will be in the meridian
plane.
We Jiave described the Brown damping device as though there
were two air-boxes and air-blast nozzles, one of each for the con-
trol bottles and one for the damping bottles. In fact, however,
the control bottles and the damping bottles are piped to a single
To C, To D
FIG. 173
air-box, Fig. 172. The air-box is shown in detail in Fig. 173.
The pipes are joined to the north working and to the north damp-
ing bottles, and to the south working and south damping bottles
as indicated.
130. Absence of Latitude Error. In northern latitudes, the
north-seeking end of the gyro-axle will have a resting position
above the horizontal plane through the center of the gyro-wheel.
In southern latitudes, it will have a resting position below the
horizontal. The natural tilt of the spin-axle is the amount re-
quired "to cause precession at the rate at which the meridian is
turning, that is the amount required to cause the axle to precess
into the resting position after it has been displaced from that
position. If the torque due to the pendulousness of the gyro-com-
pass be small, the spin-axle will have the same resting position
that it would have if the pendulousness were greater, but the speed
with which the spin-axle processes will be slower.
The method employed to damp the vibrations of the Brown gyro-
compass is equivalent to a diminution of the degree of pendulous-
ness of the gyro-system while it is oscillating through the meridian
plane. It does not oppose the tilt of the gyro-axle required to
precess the gyro-axle to the normal resting position. It follows
that the Brown gyro-compass is without latitude error (Art. 110).
131. The Meridian-Steaming Error. Every gyro-compass is
subject to an error depending upon the meridian component of the
velocity of the ship -(Art. 111). No means are provided for cor-
recting the indication of the Brown master gyro-compass, but the
meridian-steaming error is allowed for in the repeater compasses
THE BROWN GYRO-COMPASS 221
by an eccentric mounting of the repeater cards. The proper
degree of eccentricity in the mounting of the repeater cards is made
by setting dials for the given latitude, speed, and course.
132. The Repeater System. The pendulous system of the
Brown gyro-compass, Type A, is shown in Figs. 168 and 174. The
gyro-wheel, gyro-casing with the attached two control bottles and
the two damping bottles, the vertical supporting ring R and the
attached compass card C, constitute the sensitive element. The
outer frame F carrying the lubber ring L and the motor-driven oil
pump P hangs on the Cardan
rings of the binnacle and turns
with the ship.
The angle between the lubber
line and the north-south line
of the compass card is to be
transmitted to and indicated by
the repeater compasses, course
recorder and other repeater de-
vices. A horizontal gear ring,
Ay Fig. 168, of about the same
diameter as the compass card,
is mounted on the outer frame
concentric with the lubber ring.
The angle indicated by the repeaters is to be the angle between
the lubber line and an index line on the gear ring when this index
line is parallel to the north-south line of the compass card. The
gear ring A can be rotated about a vertical axis by a reversible
motor M fastened to the outer frame. Frictionless contact be-
tween the sensitive element and the gear ring is effected by an
air-blast from the gyro-casing against two plungers of a pneumatic
electric contact maker.
The gyro-casing is supported by two hollow trunnions carried
by knife-edges on the vertical ring. An air-blast produced by
the rotating gyro-wheel passes through one of these trunnions,
emerges from a nozzle fixed to the vertical ring and enters the air
box connected to the control and to the damping bottles. An-
other air-blast traverses the other trunnion, emerges from a second
nozzle B attached to the vertical ring and blows against two ad-
jacent plungers of the pneumatic electric contact maker D fastened
to the lower face of the follow-up gear-ring. The two plungers are
connected to a rocking arm that makes an electric contact causing
222 NAVIGATIONAL COMPASSES
the reversible motor to rotate in one direction when one plunger
is pushed in, and in the reverse direction when the other plunger
is pushed in. If the ship is headed in the direction of the spin-
axle, the air-blast strikes the two plungers equally and no electric
contact is made. As soon as the ship begins to turn, the supporting
frame F turns relative to the sensitive element, thereby moving
the two plungers relative to the air-blast nozzle B. One plunger
is pushed more strongly than the other, the reversible motor
starts and rotates the follow-up gear till the two plungers are
pushed equally. The angle through which the follow-up gear-
ring has been rotated is the angle the ship's head is turned from
the north-south line. The follow-up gear is electrically connected
to step-by-step motors which rotate the cards of the repeaters
through the same angle that the follow-up gear has been rotated.
133. The Ballistic Deflection Error. The resting position of
the spin-axle of a compass depends upon the latitude and the
meridian component of the velocity of the ship (Art. 111). When
the velocity of the ship changes either in direction or in magnitude,
the spin-axle is deflected from its resting position and oscillates
about a new resting position. The magnitude of the so-called
ballistic deflection depends upon the latitude, the meridian com-
ponent of the linear acceleration of the ship's velocity, and upon the
period of vibration of the gyro-axle back and forth across the
meridian plane.
In Art. 113, it is shown that if the period of vibration of the
sensitive element is of the proper value, the ballistic deflection
error will be zero and the spin-axle will move without oscillation
into the resting position proper for the new velocity of the ship.
As so much sailing is done at latitudes about 40, it is common
practice to adjust the period of the Brown compass for zero bal-
listic deflection error at this latitude.
134. Prevention of the Quadrantal or Rolling Error. While
a ship is rolling or pitching, a jerk is imparted to the pendulous
system at the end of each roll or pitch. It is shown in Art. 115
that these jerks produce zero deflection of the gyro-axle when the
ship is headed on a cardinal course, north, east, south, or west, but
that an error is produced when the ship is on an intercardinal or
quadrantal course, unless means be taken to prevent it. Any
error in the indication of a gyro-compass that might be produced
by these jerks can be avoided by delaying the action of the jerks
on the sensitive element till the axis of suspension is vertical.
THE BROWN GYRO-COMPASS 223
In the Brown gyro-compass this result is produced by motion of
the oil in the control bottles.
Owing to the rotation of the earth, the spin-axle of the compass
tilts out of the horizontal plane. Oil is forced into the control
bottle on the upper side of the gyro-case. As the earth rotates
on its axis so very slowly, the flow of oil due to this cause ceases
as soon as the tilting of the gyro-axle ceases.
When the ship rolls, the gyro-compass is forced into oscillation
about the fore-and-aft axis of the ship with a period equal to that
of the roll. The mean angular velocity of the compass relative to
the spin-axle is some two thousand times the mean angular velocity
of the earth about its axis. Now, the velocity of oil from one
control bottle to the other is so great that the kinetic energy of the
oil causes the flow to continue after the tilting of the spin-axle has
ceased. Hence, the oscillation of the oil from one control bottle
to the other lags behind the oscillation of the compass and behind
the oscillation of the ship. The lag is made such that, when the
ship is at its maximum inclination to the vertical, tin* oil is at the
same level in the two control bottles; when the ship is upright and
the air-blast is equally divided at the two compartments of the*
air-box, the oil is at its maximum height in the control bottle on the
side of the gyro-wheel from which the ship lias just righted itself.
Thus, the jerks imparted to the compass at the end of a roll do
not affect the gyro-wheel till the ship is on even keel, a quarter
period later.
If the ship is heading on a quadrant al course, Fig. 149, the jerks
imparted to the compass when the ship is at the end of a roll are not
received by the sensitive element of the Brown gyro-compass till
the axle of the vertical supporting ring is vertical. Hence, the
linear motion of the north-seeking end of the spin-axle produced
by these jerks can have no vertical component. There will be
zero deflection from the meridian due to the rolling (Art. 117, a).
If the moment of inertia of an oscillating pendulous system is
not the same with respect to all horizontal axes through the point
of support, the system will tend to rotate about a vertical axis
till the moment of inertia of the system is maximum with respect
to the axis about which it is oscillating (Art. 14). A gyro-compass
having unequal moments of inertia with respect to different hori-
zontal axes through the point of support would exhibit quadrantal
error. The moment of inertia of the Brown gyro-compass is
made of the same value with respect to all horizontal axes through
224
NAVIGATIONAL COMPASSES
the point of support by attaching to each side of the vertical sup-
porting ring a compensating mass W, Fig. 174.
5. The Anschutz Gyro-Compass
136. The Sensitive Element of the Model of 1926. The first
seaworthy gyro-compass was produced in 1908 by Dr. Hermann
Anschutz at Kiel, Germany. The model of 1926 has two gyro-
wheels each with the spin-axle at an angle of about 30 degrees to
the meridian. Each gyro-casing is capable of a slight rotation
about a vertical axis. The two are connected by geared arcs so
that, when the sensitive element is in equilibrium, the angles are
equal between the meridian and the two gyro-axles.
Until 1926 the sensitive element was supported by a float in a
vessel of mercury. When this method was used, the turning of
FIG. 175
FIG. 176
the sensitive element about a vertical axis was opposed by a small
but unconstant torque due to the surface tension at the mercury-
air-metal line of contact. This slight drag is avoided in the model
of 1926 by the very ingenious scheme of enclosing the gyro-wheels
and damping device in a hermetically sealed spherical globe that is
kept poised within an outer spherical shell filled with an electro-
lyte consisting of a mixture of acidulated water and glycerine.
This globe with the enclosed apparatus constitutes the sensitive
element.* The sensitive element is slightly pendulous.
The gyro-globe is about ten inches in diameter, and the outer
spherical shell has an inside diameter of about ten and one-half
inches. Both the globe and the outer spherical shell are made of
thin metal. The outside of the gyro-globe and the inside of the
spherical shell are covered by a thin layer of hard rubber vulcanized
on the metal shells. The gyro-globe has round electrodes, pi, p 2 ,
Fig. 175, at the upper and lower poles, and also an equatorial
* U. S. Patent. Anschutz, No. 1589039, 1926.
THE ANSCHttTZ GYRO-COMPASS 225
electrode A, A'. This equatorial electrode is divided by two
vertical gaps. One of these halves is further divided into two by
a horizontal gap. All three parts are electrically connected.
The spherical shell is provided with two polar electrodes P if P 2 ,
and two ring-shaped electrodes K y K'. The electrodes consist
of thin layers of hard rubber charged with carbon and vulcanized
on the metal globes. All electrodes are inlaid in the insulating
hard rubber coatings of the outside of the gyro-globe, or the inside
of the spherical shell, so as to be flush with the spherical surfaces.
The volume of the gyro-globe is such that the weight of the
liquid displaced by it is nearly equal to the weight of the globe.
Any tendency of the gyro-globe to sink or to move laterally from
the central position within the surrounding sphere is prevented by
the magnetic force of repulsion developed by the interaction of the
magnetic field about an alternate current carrying coil (7, Figs.
176 and 177, within the globe, and the magnetic field of eddy
currents induced in a conducting saucer-shaped electrode P%
forming part of the lower side of the outer spherical shell. The
centralizing coil produces a conical repelling field directed toward
the center of the gyro-globe.
The centralizing coil C and the three-phase motor of the two
gyros Gi, G^ Fig. 177, are joined to the equatorial electrode A
and to the two polar electrodes, as indicated in Fig. 176. A three-
phase current passes from the electrodes P], P% and K, K f on the
inside of the outer spherical shell, through the thin layer of electro-
lyte, to the corresponding electrodes pi, p%, and A, A' on the out-
side of the gyro-globe, Fig. 175.
Lubrication of the moving parts within the gyro-globe is effected
by wicks dipping into a pool of oil in the bottom of the globe,
x, x', Fig. 177a.
The gyro-globe is exhausted of air and then filled with dry hydro-
gen at atmospheric pressure. The use of hydrogen instead of air
results in several advantages: (a) the oil required for the operation
of the apparatus within the globe suffers no chemical change even
during months of continuous service; (6) since windage loss is
proportional to the density of the surrounding gas, the windage
loss with hydrogen is about one-fourteenth that which would occur
if the globe were filled with air at the same pressure; (c) since the
thermal conductivity of hydrogen is about seven times that of air
and the diffusivity is about four times that of air, it follows that
hydrogen displaces air in any porous insulating medium and there-
THE ANSCHUTZ GYRO-COMPASS
227
by increases the thermal conductivity of the porous material;
(d) the high thermal conductivity of hydrogen improves the rate
of transfer of heat across the film of contact; (e) since hydrogen
displaces the oxygen that otherwise would be in the pores of the
insulating material, damage of the insulation due to corona dis-
charge is prevented.
136. The Supporting System. The spherical shell B contain-
ing the sensitive element A is hung by six arms, DD, Figs. 178 and
179, to a vertical spindle S supported by ball bearings on the top
of a bowl E which in turn is carried by the gimbal rings of the
FIG. 178
FIG. 179
binnacle F. Attached to the vertical spindle are a horizontal gear
H connected by a pinion to a reversing motor M, the compass
card C, and five collector rings / which serve to connect electrically
the master compass to the outside part of the equipment.
Wires from the collector rings extend down the supporting arms
DD to the outer spherical shell B. One collector ring is connected
to the two equatorial electrodes K K f , another is connected to the
polar electrode PI, and another to the polar electrode P 2 . The
other two collector rings are connected to two electrodes L\, L 2
attached to the spherical shell midway between the two equatorial
electrodes KK'. The electrodes LI and L 2 are called the " follow-
up " electrodes.
A top view of the compass removed from the binnacle is shown
228 NAVIGATIONAL COMPASSES
in Fig. 180. A group of springs connecting the compass to one of
the gimbal rings, and a rod of felt between the springs and the com-
pass, serve to reduce vibration. The thermometer at the left,
and the two tubes that extend from the top plate of the compass
to the trunnion in the foreground, are parts of the system that
controls the temperature of the liquid in the outer containing vessel.
FIG. 180
137. Damping. Two different methods are used on different
models of the Anschiitz gyro-compass for producing damping of the
vibrations of the sensitive element. In the first method to be
described, an annular tube is fastened, with the plane of the ring
horizontal, to the frame that carries the gyros. The tube is partly
filled with oil. If the sensitive element tilts, oil flows from one
part of the tube to the part diametrically opposite. The flow is
retarded by a set of partitions across the bore of the tube, each
partition being pierced by an aperture of predetermined size.
The operation of this damping system is as follows. When the
meridian-seeking axis of the sensitive element moves east of the
meridian, the north point of the compass card will rise till the
gravitational torque due to the tilted pendulous system precesses
the axis back to the meridian (Art. 106). The north point of the
compass card continues to move after traversing the meridian,
decreasing its upward tilt as it moves west of the meridian. After
reaching a maximum displacement west it moves eastward and
dips below the horizontal. After traversing the meridian in the
eastward direction, the dip decreases and becomes zero when the
maximum deflection eastward is produced.
THE ANSCHtlTZ GYRO-COMPASS 229
While the sensitive element is tilting up and down the oil in the
damping device moves back and forth. While the north-seeking
end of the sensitive element is above the horizontal, that is while
the north point of the compass card is moving westward, there is
an excess of oil in the south side of the damping device. Similarly,
there is an excess of oil in the north side of the damping device
while the north point of the compass card is moving eastward.
The torque produced by the ununiform distribution of oil in the
damping device is, at all times, in the direction opposite the gravi-
tational torque due to the tilted pendulous sensitive element.
Thus the tendency to precess, due to the tilting of the oscillating
sensitive element is diminished at every instant by a torque due
to the displaced oil in the damping device. This torque does not
oppose precession but reduces the effective pendulousness of the
sensitive element.
In the second method, the oscillations of the sensitive ele-
ment are damped by a pair of tiny Frahm anti-roll tanks, 7\,
T<2, Fig. 177, fastened to the inner surface of the gyro-globe.
Consider a gyro-wheel and casing to which are attached two con-
nected oil tanks forming a pendulous system capable of oscilla-
tion about an axle through x, Fig. 181. The reservoirs are half
filled with oil. If the connecting tubes were stopped up or if
they were sufficiently large in diameter, the device would produce
no damping of the vibrations of the pendulous system. If, how-
ever, there be a considerable opposition to the flow of oil, then
when one reservoir is higher than the other, oil will flow slowly
from the higher to the lower reservoir. By the time that the
pendulous system is vertical there will be an excess of oil in the
reservoir that was the lower, Fig. 182. Later, there will be a flow
in the opposite direction, followed by a flow back and forth at a
definite period. The period, as well as the phase difference be-
tween the oscillation of the oil and of the pendulous system, can
be regulated by an adjustment of a valve in either of the tubes that
connect the two reservoirs.
If the period of flow back and forth equals the period of vibra-
tion of the pendulous system, and if there is a phase difference of
a quarter period between the motion of the oil and the motion of
the pendulous system, the following effects will occur. When the
axle of the spinning gyro-wheel is in the meridian plane, the north-
seeking end of the axle will be higher than the south-seeking end
and will be moving westward (Art. 106). At some such instant,
230
NAVIGATIONAL COMPASSES
there will be equal amounts of oil in the two reservoirs, Fig. 181.
Since oil is flowing from the upper reservoir to the lower, the dis-
tance between the center of gravity of the pendulous system and
the vertical line through its axis of vibration is being diminished.
Hence the torque urging the pendulous system toward the equi-
librium position is being diminished faster than it would be if the
oil were not flowing. By the time the north-seeking end of the
spin-axle has reached its maximum displacement toward the west
and the spin-axle is horizontal, the oil has reached its maximum
height in the southern reservoir, Fig. 182.
FIG. 181
FIG. 182
FIG. 183
FIG. 184
At the end of another quarter vibration of the pendulous sys-
tem, Fig. 183, the spin-axle is in the meridian plane, the north-
seeking end is dipping below the horizontal and the quantities of
oil in the two reservoirs are equal. During this motion, the oil in
the reservoirs diminishes the pendulous effect of the pendulous
system and the spin-axle dips more slowly than it would have done
if the oil were not flowing. By the time that the spin-axle again
has become horizontal, Fig. 184, and its north-seeking end has
reached its maximum displacement toward the east, the oil has
risen in the northern reservoir to its maximum height. During
this motion, the torque urging the pendulous system toward the
equilibrium position is being diminished faster than it would be if
the oil were not flowing. Consequently the amplitude of swing
toward the east across the meridian plane is less than it would be
if the oil were not flowing.
It is left as an exercise for the student to show that when the
flow of liquid is adjusted as above specified, then the periodic
torque acting on the gyro due to the surging liquid is in nearly
opposite phase to the angular velocity of the tilting spin-axle.
THE ANSCHtTZ GYRO-COMPASS 231
This is the condition of nearly maximum damping (Arts. 25 and
27).
138. The Meridian-Steaming Error. The meridian-steaming
or north-steaming error occurs whenever a gyro-compass is being
carried with a velocity which has a meridian component, that is
whenever the ship is steaming on any course except directly east
or west.
The magnitude of this error depends upon the course, the speed,
and the latitude. As it does not depend upon the design of the
instrument, it cannot be avoided. Its magnitude under various
conditions can be computed and allowed for. A table of values of
the meridian-steaming error is attached to the Anschtitz compass.
The errors may be taken account of by moving the lubber ring of
the master compass and of the repeaters through the angle corre-
sponding to the known course, speed, and latitude.
In northern latitudes the north-seeking end of the axle of a
gyro-compass should tilt upward, and in southern latitudes it
should tilt downward, through just the proper angle 1 , to cause the
axle to process toward the meridian plane at the rate required to
keep the axle in the meridian plane. Any opposition to the proper
tilting of the gyro-axle produced by damping will cause the rate
of precession to be less than that required to keep the gyro-axle
in the meridian plane. The settling position will be at an angle
to the meridian plane called the latitude error.
The methods of damping the vibrations of the sensitive ele-
ment of the Anschtitz gyro-compass of 1926 do not. reduce the
tilt required, at the given latitude, to cause the axis of the sen-
sitive element to precess into the meridian plane. In fact, the
tilt must be greater than it would need to be if the damping device
were not used, before the reduced effective pendulousness of the
sensitive element will generate the necessary rate of precession at
the particular latitude. There is no torque opposing the precession.
Consequently, any latitude error is avoided (Art. 110).
139. Prevention of the Ballistic Deflection Error. When a
ship is steaming in any direction except on an east-west heading,
the gyro-compass is deflected from the meridian plane by an angle
called the meridian-steaming or north-steaming error (Art. 111).
The resting position of the north-south line of the sensitive system
depends upon the latitude and the meridian component of the
velocity of the ship. If the velocity of the ship be changed either
in direction or in magnitude, the north-south line of the sensitive
232 NAVIGATIONAL COMPASSES
system may not move at once to the resting position proper to the
new velocity but may oscillate for some time. The angle between
the new resting position and the resting position corresponding to
the final velocity is called the ballistic deflection. In Art. 113
it is shown that a gyro-compass will be without ballistic de-
flection error at the equator if the undamped period has a value
given by (126). In the same Article it is shown that if the
compass be moved to any other latitude, the north-south line of
the sensitive element will move without oscillation to the resting
position proper to the final velocity of the ship so long as the
quantity
K S W S
___
mx cos X
r
=
is maintained constant.
In the gyro-compasses already described, the period is adjusted
to the proper value by regulating the pendulousness mx of the
sensitive element. In the Anschiitz and in the Arma gyro-com-
passes, the desired constancy of this quantity is maintained by
varying the spin- velocity w s of the gyro as the cosine of the latitude
varies. Each gyro of the Anschiitz compass is the rotor of a three-
phase motor operated by a current of 330 cycles per second. By
varying the inductance of two of the motor windings, the speed can
be changed as gradually as may be desired from about 17,000 to
about 30,000 revolutions per minute. By setting a dial for the
given latitude, the gyros will have the proper angular momentum
to produce the required period of vibration of the sensitive element.
Details of a similar device used on the Arma gyro-compass as
well as the means provided to prevent this change in spin-velocity
affecting appreciably the direction of the spin-axle are given in
Art. 145.
140. Avoidance of the Quadrantal Rolling Error. Suppose
that a ship on an intercardinal course carries a pendulous gyro-
compass. When the ship is on even keel, the center of mass of
the sensitive element of the compass is at C, Fig. 185. When the
ship rolls, the center of mass of the sensitive element rotates readily
about a north-south axis but not about an east-west axis. When
the ship is at the end of a roll to port, the center of mass of the
sensitive element has rotated to C' and is acted upon by force F f .
When the ship is at the end of a roll to starboard, the center of mass
of the sensitive element has rotated to C" and is acted upon by a
THE ANSCHtTZ GYRO-COMPASS
233
force F" '. The center of mass moves back and forth along a line
perpendicular to the spin-axle.
The meridian components of these athwartship forces, produced
by the shifting of the center of mass along the east-west line, pro-
duce a tilt of the spin-axle and also an azimuthal deflection when
the ship is on an intercardinal heading (Arts. 115 and 117). This
rolling, quadrantal or intercardinal error will be reduced to zero
if there be no shifting of the center of mass
of the pendulous sensitive element when the
ship rolls. There will be only a very small
back-and-forth displacement of this center of
mass when the period of vibration of the
sensitive element east and west is so much
greater than the period of the ship's roll that
the amplitude of the vibration forced upon
the sensitive element is small.
In the Anschiitz gyro-compass, the re-
quired long period of vibration of the sensi-
tive element about a horizontal meridian axis
is effected by having the sensitive clement comprise two gyros
with the spin-axles always nearly horizontal. The principle
involved will now be described. Each gyro is capable of precessing
through a limited angle about a
vertical axis. The plane of the axis
of precession is in the normal north-
south plane of the sensitive ele-
ment. The two gyro-casings are
connected by cams and springs so
that, at all times, the two spin-axles
make equal angles with the vertical
plane through the precession axis,
Fig. 186. The angle between the
spin-axles is always about 60 de-
grees.
If a change be made in the direction of the spin-axle of a gyro
free to precess, a gyroscopic torque opposes the change. This
opposing torque is proportional to the component of the angular
momentum of the gyro with respect to an axis perpendicular to
the axis about which the spin-axle is turned. With two gyros
each of angular momentum h s and the spin-axes horizontal and
inclined 30 degrees to the meridian, there is a resultant angular
234 NAVIGATIONAL COMPASSES
momentum of the system with respect to the north-south line, of
the value 2 h s cos 30 = 1.73 A,, Fig. 187. Along an east-west
line, the angular momentum of the western gyro is h s sin 30,
while that of the eastern gyro is h s sin 30, the plus and minus
signs referring to the directions of spin. The opposition to roll
would be the same if the direction of spin of both gyros were re-
versed. Thus the opposition to roll about a horizontal north-
south axis is proportional to 2 h s sin 30 = h s , and the opposition
about an east- west line is proportional to 1.73 h s . These torques
are so great that when the system is given jerks by the rolling or
pitching of the ship, the period of the vibrations of the sensitive
element thereby produced is from five to ten
times as great as the period of the succession
of jerks. This great difference of period pre-
"*^ 4"* eludes forced resonant vibration of the gyro-
\ / element in tune with the vibration of the ship.
Nj/ - The jerks imparted to (he pendulous system
by the rolling or pitching of the ship produce
a slight horizontal oscillation, but a negligible
tilting of the line from the point of support
to the center of mass of the pendulous system. Hence there is
negligible quadrantal error. The centralizing device also opposes
any rocking motion of the sensitive element.
141. The Follow-Up Repeater System. As the sensitive ele-
ment does not touch the supporting system which carries the com-
pass card, a follow-up device must be employed to maintain the
compass card in constant relation to the sensitive element. The
scheme employed comprises a reversing motor, M, Fig. 178,
pinioned to a horizontal gear // fastened to the supporting system.
The method of controlling the motor will be described by the aid
of Figs. 188 and 189, which represent the gyro-globe and spherical
shell in elevation, and in plan, respectively. In Fig. 188, the two
electrodes KK' are shown as though they were separated by a
greater distance than the two electrodes A f ', A f . In the actual
apparatus, this is not the case.
When the sensitive element with spinning gyro-wheels is in
equilibrium and the north-south line of the compass card is in
the meridian, the two " follow-up " electrodes, LI, L 2 , on the in-
side of the spherical shell, face the centers of the corresponding
vertical gaps between the equatorial bands on the gyro-globe. The
three lines X, Y y Z, of a three-phase current circuit, are joined,
THE ANSCHtlTZ GYRO-COMPASS
235
respectively, to the pair of equatorial electrodes on the inside of
the outer spherical shell, to the upper polar electrode PI and to
the lower polar electrode P 2 . The broad equatorial electrode
A on the outside of the gyro-sphere and the two polar electrodes
Pi 9 p% are joined to the three terminals of the two three-phase
gyro-motors. The broad and the two narrow electrodes on the
gyro-sphere are connected together.
r
Fia. 188
FIG. 189
At the instant when X is at a higher potential than Y and Z,
current from X goes to KK r . There it divides, a part crosses
the electrolyte to A, traverses the gyro-motors to p\, the electro-
lyte to PI and thence goes to the line Y. Meanwhile the other
part crosses the electrolyte to A', passes to A, traverses the gyro-
motor to pi, the electrolyte to Pi and thence to the line Y. These
two paths are of different resistance.
With the spherical shell in some such position relative to the
gyro-globe as that indicated in Figs. 188 and 189, the electro-
motive forces at LI and L 2 will be in the same phase. When the
spherical shell becomes slightly turned from this position, by
the turning of the ship for example, the electromotive forces at the
follow-up electrodes LI and L 2 will be in different phase. The differ-
ence in phase will be reversed when the deflection of the spherical
shell from the equilibrium position is reversed in direction. Thus,
with any given phase difference between the lines X and F, the
wire QiQ-2 will be traversed by a current which in magnitude and in
direction depends upon the angular displacement, from the equi-
librium position, of the spherical shell relative to the gyro-globe.
The variations of this current are caused to produce greater varia-
tions in the current of a three-electrode tube amplifying circuit.
236 NAVIGATIONAL COMPASSES
The terminals of the amplifying circuit and the line Z, Fig. 189, are
joined to the terminals of a three-phase reversing motor Af, Fig.
178.* By these means, the hollow sphere is turned till the follow-
up electrodes are in the equilibrium position, that is, till the north-
south line of the compass card is in the geographic meridian. The
same lines operate repeater compasses, a course recorder, an auto-
matic pilot, or other repeater devices.
6. The Arma Gyro-Compass
142. The Sensitive Element. The Mark IV Model 1 gyro-
compass made by the Arma Engineering Company of Brooklyn,
New York, Fig. 190, has two gyros capable of precession about
FIG. 190
vertical axes. The two gyro-casings are held by a bell crank and
springs so that the angle between the spin-axles is kept constant,
Fig. 191. When the sensitive element is in the resting position,
the spin-axles are inclined about 40 degrees from the meridian.
The gyros spin at about 12,000 revolutions per minute in an
atmosphere of helium under low pressure. The advantages in the
use of helium are the same as in the use of hydrogen (Art. 135).
* U. S. Patent. Anschutz, No. 1586233, 1926. " The New Anschutz,"
published by the Nederlandsche Technische Handel Maatschappy " Giro,"
The Hague, Holland.
THE ARMA GYRO-COMPASS 237
The frame that supports the gyro-casings also carries two con-
nected oil tanks, D l} D 2 , Fig. 191. One of these tanks is on the
north and the other is on the south side of the frame when the
element is in the resting position. The motion of oil back and
forth from one tank to the other quickly damps vibration of the
sensitive element (Art. 137).
Figure 192 represents the two principal units of the master
compass pulled apart vertically. The gyros and attachments are
FIG. 191
supported by a hollow metal globe, S, that floats in a bowl of mer-
cury, H. The globe together with the two gyros Gi and (? 2 , with
the entire supported system, constitute the sensitive element.
Any change in friction that might be produced by variations in the
surface tension at the line of contact of mercury, air and metal is
prevented by a continuous vibration of the mercury maintained by
a small motor fastened to the bowl and to the binnacle.
The compass card A, a horizontal gear and horizontal arm PP f
are fastened rigidly to a vertical tubular shaft R. This system
constitutes the phantom element. The phantom is carried by the
compass frame. The latter is hung by springs from the inner
gimbal rings of the binnacle. The phantom element is capable of
being rotated about a vertical axis by a follow-up motor M. The
238
NAVIGATIONAL COMPASSES
tubular shaft R carries a group of slip rings against which press
brushes for transmitting currents to the various electric devices
attached to the phantom and to the sensitive element.
Three of the electric conductors for the operation of the sen-
sitive element are within the tubular shaft projecting downward
from the phantom and terminating in three pins near the center of
the supporting globe. One pin is central and rests in a conical
FIG. 192
cavity filled with mercury in a block of insulating material fastened
to the supporting globe at its center. The other two pins dip into
annular grooves filled with mercury, in the same block.
143. The Follow-Up System. There is no mechanical con-
nection between the compass card and the sensitive element.
When the north-south line of the card becomes out of the meridian
plane, it is brought into line with the meridian axis of the sensitive
element by a reversible motor M controlled by an alternating cur-
rent induced in two pairs of coils B, /?', C, C', Fig. 192, attached to
the phantom. This induced current is due to rapidly changing
magnetic fields set up by four alternating current magnets, E, E f ,
F, F', attached to the sensitive element.
The turns of the follow-up coils B and B f are in a horizontal
plane above two alternating current magnets E and E f that pro-
ject upward from the top of the sensitive element. The two coils
THE ARMA GYRO-COMPASS 239
are wound in opposite directions. The turns of another pair of
follow-up coils 0, C" are wound in opposite directions in a vertical
plane and normally are in front of horizontal alternating current
magnets F and F f .
As shown in Fig. 193, a 120- volt, 60-cyclo current energizes the
primaries of two transformers, the secondaries of which are con-
nected to the four alternating current magnets E y E' , F, F' fastened
to the sensitive element. Current from the same line is rectified
and led to the mid-point of the field coils of the follow-up motor.
The amplifying and rectifying circuit consists of three triple-
electrode vacuum tubes with the necessary transformer, con-
denser and resistances. In the diagram is shown an extra set of
three tubes to guard against interruption of the operation of (he
system in case a tube should fail. In the subsequent description,
the tube shown at the left end of each set will be called the " input
tube " of that set, and the other tubes of each sol will be called
the first and the second " output M tubes, respectively. A 110-
volt direct current energizes the armature of the reversible, follow-
up motor and also the 4 plate circuit of the input amplifying lubes.
The field coils of the follow-up motor are wound in opposite
directions so that, if no current traverses them except the recti-
fied current from the alternating current supply lino, the two field
pole strengths will be equal and of the same sign. Under this
condition there will be zero torque tending to rotate the armature*.
The current induced in the follow-up control coils, after being
rectified and amplified, is superposed on the rectified current in
the follow-up motor field coils from the alternating supply line.
The follow-up control coils are connected in the grid circuit of the
input tube. The plate circuit includes the primary windings
of the interstage transformers. When the ship changes course, an
alternating electromotive force is induced in the follow-up con-
trol coil connected to the grid circuit. This results in an alter-
nating characteristic being impressed on the direct current flowing
in the plate* circuit and an alternating electromotive force being
induced in the interstage transformer secondaries. These sec-
ondary windings are connected with the circuit in such a manner
that the grid of one output tube becomes more positive with respect
to its filament at a given instant, and the grid of the other tube
becomes more negative. Thus the current in one follow-up
motor field coil will be strengthened and the current in the other
field weakened. Consequently the armatures will rotate and
240
NAVIGATIONAL COMPASSES
DAMPING
CUT-OI/T
SWITCH
n
DAMPING
CUT-OUT CONTACT
SECOND
AC MAGNET
TRANSFORMER
FIRST
I A.C MAGNET
I/TRANSFORMER,
FIG. 193
THE ARM A GYRO-COMPASS 241
turn the phantom element till the intersections of the three pairs
of follow-up control coils are opposite the corresponding alter-
nating current control magnets. When the ship changes course
in the opposite direction, the follow-up motor rotates in the op-
posite direction thereby again bringing the phantom with the con-
nected compass card into coincidence with the sensitive element.
The pair of horizontal follow-up control coils lilt' are of larger
diameter and are separated by a wider gap than the two pairs of
vertical follow-up control coils C and (?'. When the north-south
line of the compass card and the connected phantom dement are
displaced through a considerable angle from the meridian-seeking
axis of the sensitive element, the small vertical follow-up con-
trol coils will be beyond the range of induction of the follow-up
alternating current magnets, but an alternating current will be
induced in the large follow-up control coils. This current will
cause the follow-up motor to turn the phantom dement till the
small vertical follow-up control coils are so close to the correspond-
ing alternating current magnets that currents will be induced in
their coils also. This action results in the phantom dement
being turned till the north-south line of the compass card is parallel
to the meridian-seeking axis of the sensitive element. The large
horizontal follow-up control coils produce a coarse adjustment, and
the small vertical follow-up coils furnish the fine adjust mcnt.
Geared to the follow-up system is the transmit tor system for
the operation of repeater compasses, course recorder and automatic
pilot. Also geared to the follow-up system is a device that makes
the proper correction for the course and speed error.
144. The Course and Speed Error Corrector. The gyro-
compass on a moving ship is subject to an error which depends upon
the latitude and upon the meridian component of the velocity of
the ship (Art. 111). The compass is deflected toward the west
when the meridian component of the velocity of the ship is di-
rected toward the north, and toward the east when the meridian
component is directed toward the south. This error is called the
meridian-steaming error, the north-steaming error, and also the
course and speed error. The method used in the Anna compass to
correct this error consists in setting the phantom element so that
the line joining two diametrically opposite follow-up coils makes
with the line joining corresponding alternate current control mag-
nets an angle equal to and in the same direction as the deflection
of the sensitive element due to the course and the speed of the
242
NAVIGATIONAL COMPASSES
ship. Then, when the follow-up system turns the phantom, with
the connected compass card, till the follow-up coils are in line with
the corresponding control magnets, the compass card will show no
rror due to the course and speed of the ship although the meridian-
seeking axis of the sensitive element is deflected from the me-
ridian.
In Figs. 194 and 195 the circle a represents a gear fastened to
1he axle (hat carries the compass card; 6, a gear fastened to the
upper end of the vertical axle of the phantom element; c and r/,
two gears capable of turning about axles fastened to a plate at-
FIG. 1<H
(ached to the compass card; e, a pinion connecting the follow-up
motor and the compass card. Fastened to the gear d is a fork //'.
If the prongs of the fork be moved nearer to or farther from e,
the attached gear d is turned, and the connected phantom gear
b is turned in the same direction. Such a displacement of the
prongs of the fork can be produced by moving forward or back a
nearly fridionless wheel (j on the end of a screw h which is ro-
tatable in a nut fastened to the compass frame.
Suppose that the meridian component of the ship's velocity is
directed north. Then, the meridian-seeking axis of the sensitive
demon t will be deflected toward the west through an angle 61,
depending upon the latitude, speed and course of the ship (124).
Values of this angle for various conditions are given in tables.
If no correction be made, the follow-up system will rotate the
north-south line of the compass card through the same angle in the
same direction.
THE ARMA GYRO-COMPASS
243
Suppose, now, that the proper angle corresponding to 61 be set
up on the dial i attached to the screw that moves the wheel g
forward or back. For a north meridian component of ship's
velocity, the dial-setting will turn the fork and the attached gear d
counter-clockwise. The connected phantom gear 6 will be turned
in the same direction, thereby rotating the line connecting the
attached follow-up coils CC", Fig. 192, through the angle 61 in the
FIG. 195
counter-clockwise direction away from the line connecting the
control magnets FF'. Since the line of action of the force with
which the screw pushes against the fork passes through the axis
of rotation of the compass card, this force is without effect on the
indication of the compass card.
Before the dial was set, the line connecting the follow-up coils
CC' on the phantom element coincided with the line connecting the
244 NAVIGATIONAL COMPASSES
control magnets FF' on the sensitive element. Also, both the
sensitive element and the compass card were deflected from the
meridian through an angle d\ in the counter-clockwise direction.
The setting of the dial rotated the line connecting the follow-up
coils CC' through an angle 61 in the counter-clockwise direction
away from the line connecting the control magnets FF' without
changing the direction of the north-south line of the compass card.
The follow-up system immediately started to rotate the pinion e y
thereby turning both the compass card and the phantom geared to
it through the angle <$i in the clockwise direction, that is, till the
line connecting the follow-up coils CC/ was brought again into
coincidence with the line connecting the control magnets FF'.
By this rotation of the compass card, the north-south line was
brought into the meridian. The meridian-seeking axis of the sen-
sitive element is still at an angle d\ to the meridian.
145. Prevention of the Ballistic Deflection Error. A gyro-
compass will be without ballistic deflection error at the equator if
the undamped period has a value given by (12(3). If the compass
be moved to any other latitude X, the compass will continue to be
without, ballistic deflection error if
_ A >< r_ Kw}
^ i /tciM
7HX COS X
be maintained constant (Art. 113).
The desired constancy can be maintained by varying the spin
velocity u\ as the cosine of the latitude varies. The spin-velocity
of the gyros of the Anna compass is changed by adjusting the speed
of the motor generator that energizes the stator coils of the two
gyro-motors. In Fig. 190, /), K are the terminals of the direct
current motor that rotates the three-phase generator XYZ which
operates the two gyro-motors (n and GV The armature A of the
direct current motor is connected in series with a variable rheostat
R. The rot at able arm of the rheostat is keyed to the shaft of a
small reversible two-phase " torque motor " T. The rotor of the
torque motor consists of a squirrel cage armature without com-
mutator or slip rings. When the two stationary coils of the motor
are traversed by two alternating currents in different phase, a
rotating magnetic field is developed which produces a torque on the
rotor. Rotation of the rotor is opposed by the torque developed
by a spring, one end of which is fastened to the shaft and the
other end attached to a fixed support. The rheostat arm turns
THE ARMA GYRO-COMPASS
245
till these two torques balance one another. The angle of turn is
less than one revolution.
The two field coils of the torque motor are connected to a
tuned resonant circuit J KLX consisting of a variable inductance
Af, resistance hi and fixed capacitances d and C 2 . A choke coil
/? 2 prevents a too rapid increase of current. The resonant fre-
quency of the circuit is inversely proportional to the square root
of the inductance. It is possible to vary the frequency of this
circuit within wide limits, and consequently vary the resistance
of the rheostat If, the current in the series field coil /S of the direct
FIG. 11)0
current motor, the speed of this motor, the electromotive force of
the alternator XYZ and therefore the spin-velocity of the two
gyros G\ and (7 2 .
In setting the dial fastened to the variable inductance M
for a particular latitude, the resonant frequency of the tuned cir-
cuit is made to correspond to the speed at which the motor gen-
erator should run when the compass is at that latitude. If the
speed is too low, the reversible torque motor T will turn in the
direction to reduce the resistance of the rheostat R. If the speed
is too high, the reversible motor will turn in the opposite direction,
thereby increasing the rheostat resistance. At the correct motor-
generator speed, the potential difference at the terminals of the
two field windings of the torque motor are in phase and the torque
motor stops.
A tilt of the spin-axle of a gyro-compass is necessary in order
that the spin-axle may remain in the meridian when at any latitude
either north or south of the equator (Art. 106). In Art. 109 it has
been shown that if the spin- velocity of a gyro-compass be con-
240 NAVIGATIONAL COMPASSES
sidcrably altered while the spin-axle is tilted from the horizontal,
the spin-axle will deflect in azimuth. The method used in both
the Anschiitz arid in the Arma gyro-compasses for avoiding the
ballistic deflection error by changing the spin-velocity of the gyros
as the latitude of the ship changes must result in a certain azi-
muthal deflection. This deflection is reduced to a negligible
amount, however, if the proper change in spin-velocity is made
corresponding to each small change in latitude. At latitudes in
the neighborhood of 40 the azimuthal deflection is about 0.2
degree for changes of spin-velocity corresponding to 5-degree
changes in latitude.
146. Avoidance of the Ballistic Damping Error. When the
meridian component of the velocity of the ship changes either in
direction or in magnitude, the inertia of the oil in the damping
tanks causes oil to move in the direction opposite that of the
acceleration of the ship. For example, if a ship while steaming
northward either suddenly stops or makes a quick turn, there is an
acceleration of the ship toward the south and oil will move from
the south damping tank to the northern one. If the oil could
flow freely from one tank to the other, the oil would become at
the same level in the two tanks when the acceleration ceased.
The constricted passage connecting the two tanks retards equaliza-
tion however, and develops a precession of the sensitive element
away from the normal resting position. This results in a ballistic
damping or ballistic turning error (Art. 114).
This error is avoided by preventing damping of vibration of the
sensitive element during the time that the velocity of the ship is
changing. The greatest acceleration produced by any practical
change of speed of a ship on a straight course is so small that the
ballistic damping error is negligible so long as the course of the
ship is unchanged. The acceleration may be so great when a ship
steaming at considerable speed makes a sudden change of course
that the turning of a ship may produce a large ballistic damping
error. All devices for avoiding the ballistic damping error are
designed to prevent damping throughout the time that the ship
is making a turn.
The Arma damping cut-out device consists of an electric sole-
noid-operated valve in the oil line between the two damping tanks.
The solenoid is energized by a unidirectional intermittent current
that is controlled by a contact switch mounted on the shaft of the
phantom element. This cut-out contact switch consists of a con-
THE FLORENTIA GYRO-COMPASS 247
tact stud attached to the compass frame and a fork attached to a
collar that can turn with a certain amount of friction about the
phantom shaft. When the ship starts to turn in either direction,
the friction between the cut-out switch and the phantom shaft
carries the switch around and brings one prong or the other
into contact with the fixed contact stud. This completes the
electric circuit through the solenoid and causes the cut-out valve
to close. If the ship had turned in the opposite direction, the
other prong would have made contact and the valve would be
closed as before. The distance between the prongs is such that
contact is always made before the ship has turned more than about
10 degrees. After the ship has ceased turning, the yawing of the
ship back and forth quickly breaks the switch contact, and normal
damping is resumed.
147. Avoidance of the Quadrantal or Rolling Error. A com-
pass with a single gyro is subject to an error when on a rolling
ship on an intercardinal course (Arts. 115, 116). This error is
avoided in the Arma, as in the Anschiitz gyro-compass, by the use
of two gyros with the spin-axles inclined to one another (Art.
140).
7. The Florentia Gyro-Compass
148. Arrangement of the Principal Parts of the Florentia
Master Compass. The sensitive element of this instrument,*
made by the Officine Galileo, Florence, Italy, consists of a single
meridian-seeking gyro-wheel in a case, (7, Fig. 197, hanging from a
hollow ring F that floats in an annular trough of mercury T.
The gyro has a mass of about 5 kg. and a moment of inertia of
about 70,000 gm. cm' 2 . It rotates at a speed of 20,000 revolutions
per minute.
The mercury trough T hangs from a vertical spindle carried by
a ball-bearing on the horizontal frame or spider D. The spider is
suspended by girnbal rings, R\Ri, from the top of the binnacle H.
The lubber line is engraved on an arm L fastened to the spider.
The compass card C is fastened on top of the spindle. The rigidly
connected system from the mercury trough to the compass card
constitutes the phantom element, P. The phantom element is
stabilized by a second gyro SO within a casing directly below the
compass card. By means of a reversible motor M, pinioned to a
* U. S. Patent. Martienssen, No. 1493213, 1924.
248 NAVIGATIONAL COMPASSES
horizontal gear on the phantom clement P, the latter is kept in
fixed position relative to the meridian-seeking axle of G.
The sensitive element >S is kept central with respect to the an-
nular trough by two pins p', p. The upper pin is attached to the
phantom and projects into a small ring fastened to the sensitive
element. The lower pin is attached to the gyro-casing G and pro-
jects into a slot q in the end of an arm fastened to the phantom
element. The center of mass of the sensitive element is about
0.8 cm. below the center of buoyancy. The Florentia gyro-com-
pass is slightly pendulous.
The gyro of the sensitive element is the rotor of a three-phase
motor. One of the three phases is led to the motor through the
supporting spindle. The other two are led through two flexible
platinum strips that extend from the phantom to the gyro-casing.
149. The Folio w-Up System. The upper end of a vertical
helical spring is suspended from the upper plate of the sensitive
element. On the lower end of the spring is a pair of silver balls,
Fig. 198. These balls hang between a pair of vertical metal plates
X, X', Fig. 199, mounted on a frame carried by the phantom ele-
ment.
When the north-south line of the compass card is in the vertical
THE FLORENTIA GYRO-COMPASS
249
plane of the spin-axle of the meridian-seeking gyro, the balls are
midway between the two vertical plates. When the phantom
turns relative to the spin-axle of the meridian-seeking gyro, one
of the balls comes into contact with one of the vertical plates,
thereby completing an electric circuit through the follow-up motor
FIG. 198
FIG. 199
M mounted on the spider. The motor rotates the phantom till
neither ball touches a contact plate, that is, till the north-south
line of the compass card is in the vertical plane through the spin-
axle of the meridian-seeking gyro. Really, the motor rotates the
phantom slightly beyond the equilibrium position, so that the
other vertical plate is brought into contact with the hanging balls
and the direction of rotation of the motor is reversed. Thus, the
compass card is caused to " hunt " back and forth through an
250
NAVIGATIONAL COMPASSES
amplitude of a degree or two, thereby avoiding any chance for the
phantom to stick.
The position of the north-south line of the compass card relative
to the lubber line is transmitted to the course recorder, and to
repeater compasses, by means of a transmitter commutator as in
the case of other gyro-compasses.
150. Damping. Damping of the oscillations of the spin-axle
is effected by a method that causes the follow-up motor to absorb
energy from the oscillating element.* The pin p, Fig. 197, which
extends downward from the lower side of the gyro-casing projects
into a slot q in an arm attached to the phantom element. The
east side of the sensitive element is over-
weighted so that the pin presses against the
west jaw of the slot. In Fig. 197, this over-
weighting is indicated by a mass A on the east
side of the sensitive element.
The spin-axle of a pendulous gyroscope
tends to become parallel to the earth's axis,
with the direction of spin in the same sense
as the direction of rotation of the earth (Art.
106). It follows that when the spin-axle is
in equilibrium, the sense of rotation of the
gyro is clockwise as seen by an observer look-
ing along the axle from south toward the north.
When the spin-axle of the Florentia gyro-compass oscillates,
the phantom follows, in azimuth, the motion of the sensitive ele-
ment. The north-south line of the compass card and the parallel
slot are in the vertical plane of the spin-axle only when the sensitive
element is in the resting position. At other times the vertical
plane through the slot is slightly behind the vortical plane through
the spin-axle. Consider the case when the north-seeking end of
the spin-axle is east of and moving toward the meridian plane.
In Fig. 200, the spin-velocity is represented by w^ the cross-section
of the vertical pin by p, and the slot in the arm attached to the
phantom element by q. Since the east side of the phantom is over-
balanced, the east jaw of the slot pushes on the pin with a force
represented by F. The component / of this force, parallel to the
spin-axle, develops a torque on the sensitive element represented
by L. As the spin-axle tends to set itself parallel to the torque-
* Oscar Martienssen, " Eine neue Methode zur Dampfung der Schwingungen
eines Kreiselkompasses," Physik. Zeit. 29, 295-300 (1928).
THE FLORENTIA GYRO-COMPASS 251
axis and with the direction of spin in the direction of the torque, it
follows that the oscillation of the spin-axle is opposed. The pin
p projecting downward from the gyro-casing is acted upon by
forces that oppose both the motion of the spin-axle in azimuth and
the motion in elevation. These oppositions to the motion of the
spin-axle produce rapid damping of the vibration of the sensitive
element.
If the sensitive element were overbalanced on the west side, the
amplitude of successive swings of the spin-axle of the meridian-
seeking gyro would become greater and greater.
151. The Latitude and Meridian-Steaming Error Corrector.
Since the method employed to damp the vibration of the sensitive
element of the Florentia gyro-compass opposes tilting of the axle
of the meridian-seeking gyro, this com-
pass is subject to the latitude error (Art.
110). In common with all gyro-com-
passes it* is subject to the meridian-
steaming error (Art. 111). These two
errors are corrected by a device that
turns the phantom, relative to the spin-
axle of the sensitive element, through
such an angle that the north-south line
of the compass card is free of both errors
although the spin-axle of the sensitive
element is not in the meridian plane. * IG - 201
The pair of silver balls, Fig. 198, suspended from the top of the
sensitive element, hang between two vertical plates, X, X', on one
end of a crank, KK' ', Figs. 199 and 201, mounted on a frame at-
tached to the phantom. A helical spring causes the upper end of
the crank to press against the edge of a ring, F, Figs. 197 and 199,
fastened flat against the under side of the spider. The position of
the ring can be adjusted so as to be eccentric to the vertical axis
of the sensitive element and phantom. This eccentric ring con-
stitutes a cosine cam similar to the one used on the Sperry Mark
VI and Mark VIII gyro-compasses (Art. 123). The degree of
eccentricity is controlled by the position of an arm on the upper
face of the spider. This correction arm is graduated for latitude
and is capable of being moved over a flat scale carrying speed
curves.
When the correction arm is set for a given latitude, speed, and
course, the cam determines the proper position of the crank, and
252 NAVIGATIONAL COMPASSES
consequently the position of the contact plates, relative to the
north-south axis of the compass card. The follow-up motor, in
circuit with the contact plates and the pair of balls between them,
now maintains such an angle between the north-south axis of the
compass card and the spin-axle of the sensitive element that the
latitude and the meridian-steaming errors of the sensitive element
do riot appear in the compass card indication. The probability
of any failure to operate is reduced by the addition of a duplicate
corrector system consisting of a second cosine cam Y r together with
an additional pair of contact plates and pair of silver balls placed
diametrically opposite the ones just described.
162. Avoidance of the Ballistic Deflection Error. In the
Florentia, as well as in all other gyro-compasses, the deflection of
the spin-axle of the sensitive element from the meridian, that
would be produced when there is a change in either the speed or
course of the ship, is avoided by designing the instrument so that
the period of the undamped azimuthal vibration back and forth
is about 84 minutes when at the equator (Art. 113). At other
latitudes the period should be less. The period of the Florentia
gyro-compass is fixed by the makers so as to reduce to a negligible
value the ballistic deflection error in the region in which most
sailings are to be made. There is no device on the gyro-compass
of 1924 by which the navigator can alter the period, nor any de-
vice for avoiding the ballistic damping error (Art. 114).
153. Avoidance of the Error Due to Rolling and Pitching of
the Ship When on Intercardinal Courses. Deflection of the spin-
axle of the sensitive element from the meridian produced by rolling
and pitching of the ship is much reduced by keeping the entire
suspended system always nearly vertical, however the ship may
roll or pitch (Arts. 115 and 117). This result is accomplished by
making the phantom into a gyro-pendulum of long period.* The
period of the conical oscillation of the phantom element about a
vertical axis is increased to about 40 seconds by means of a gyro,
SG, Fig. 197, having a mass of about 10 kg. and a moment of in-
ertia of about 405,000 gm. cm. 2 This stabilizing gyro not only
increases the period of vibration of the phantom but also greatly
reduces the amplitude of vibration. A top view of the entire
instrument is given in Fig. 202.
The stabilizing gyro opposes any torque that tends to rotate
the phantom about any axis inclined to its spin-axle. Consider
* U. S. Patent. Martienssen, No. 1493214, 1924.
THE FLORENTIA GYRO-COMPASS 253
the effect of a torque that tends to rotate the phantom element
about an axis perpendicular to the plane of the diagram, Fig. 197.
Such a torque causes the spin-axle of the stabilizing gyro to precess
slowly about a horizontal axis in the plane of the diagram. The
annular mercury trough is thereby slightly tilted but the mercury
FIG.
surface remains practically horizontal. Consequently, the di-
rection of the meridian-seeking gyro axle is unaffected by the pre-
cession or by the torque that produced it.
QUESTIONS
1. Would a gyroscope which is entirely free from friction about its separate
axes align its spin-axis with the rotational axis of the earth when the gyro-
axle is given an initial easterly or westerly displacement? Explain.
2. What are the controlling forces which convert a gyroscope into a me-
ridian-seeking device?
3. Show why the spin-axle of a pendulous gyro-compass tends to set itself
in the meridian plane and with the direction of spin in the same direction that
the earth rotates, that is, clockwise as seen from the south.
4. Show why the spin-axle of a liquid-controlled non-pendulous gyro-com-
pass tends to set itself in the meridian plane and with the direction of spin in
the direction opposite the rotation of the earth, that is, counter-clock wise as
seen from the south.
6. Show that each end of the spin-axle of an undamped gyro-compass moves
in an elliptical path.
6. Describe three methods employed to damp the vibration of gyro-com-
passes back and forth across the meridian plane.
7. Show that the damping device used on the Anschtitz gyro-compass is
254 NAVIGATIONAL COMPASSES
applicable only to a compass that spins in the same direction as the rotation
of the earth.
8. Why are the .spinning wheels of gyro-compasses always of great angular
momentum?
8. Deduce an expression for the magnitude of the directive force acting upon
a gyro-coin pass. Where, upon the earth's surface, does the gyro-compass
possess its maximum and where its minimum directive force?
10. Suppose that, in assembling a gyro-compass, the gyro-wheel were
mounted slightly too far toward the south. What would be the effect upon
the direction of the settling position of the spin-axle of the gyro?
11. Show that if the spin-velocity of a Sperry gyro-compass diminishes
considerably from the correct value, the compass will deflect westward of the
meridian plane when the compass is in northern latitudes.
12. In going from the equator to latitude 60, the spin-velocity of both the
Anschiitx and the Anna gyro-compass is reduced 50 per cent. Why? State
the reason why this change is not accompanied by a deflection of the spin-
axle out of the meridian plane.
13. Describe the cause of the " latitude error." Give the names of the
gyro-compasses in which it exists and show how it is compensated.
14. Describe the cause of the " meridian-steaming " or " north-steaming "
error and show how it is compensated.
16. Describe the cause of the " ballistic deflection " error and show that
this error will not occur if the period of vibration of the spin-axle has the
proper value.
16. Find the period of a gyro-compass that will have zero ballistic deflection
error when the compass is at a given latitude.
17. Show what is meant by the " ballistic damping error " and state how
it can be prevented.
18. Describe the cause of the " quadrantal " or " rolling " error and give
two methods by which it is suppressed.
19. The mercury ballistic of the Sperry gyro-compass is supported by a
" phantom element." Why could it not be attached directly to the gyro-
casing?
20. The mercury ballistic of the Sperry gyro-compass is loosely connected
to the gyro-casing by a, pin that is a little to the east of the vertical line through
the center of the gyro. Show what would be the effect upon the compass if
this eccentric connection were offset to the west of the vertical axis rather
than to the east.
21. Describe the met hods used in three different makes of gyro-compass for
causing the position in azimuth of the compass card of the master compass to
remain in fixed relation with respect to the direction of the spin-axle of the
gyro.
22. Name a natural error to which gyro-compasses are subject, that is
avoided in each of the following types: (a) pendulous sensitive element,
(6) non-pendulous element provided with a mercury ballistic, (c) two-gyro
sensitive element. Explain how each type avoids the error named.
CHAPTER VI
GYROSCOPIC STABILIZATION
1. General Principle*
154. Static and Kinetic Stability. If a body, after suffering
a slight angular displacement, recovers its former position, the body
is said to be stable: If, after suffering a slight angular displace-
ment, the body departs further and further from its former position,
the body is said to be unstable. If, however, after suffering a
slight angular displacement, there is no tendency of the body either
to recover its former condition or to depart further from it, the
body is said to be in neutral or indifferent stability. A body that
is stable is also said to have positive stability and one that- is un-
stable is said to have negative stability. The degree of stability
of a body is measured by the amount of work necessary to effect
a permanent change in the position of the body.
When a whipping-top is standing on its flat end it is statically
stable; when standing on its peg and not spinning, it is unstable;
when lying on its side, it is neutral. If, however, the top be given
an angular displacement when spinning with constant speed, the
top will not tumble over as it would if not spinning but will oscil-
late and eventually regain very nearly the condition it had before
it was disturbed. A top, when spinning, is said to be kinetically
or dynamically stable. Since, in this case, the top is acted upon
by an unbalanced torque, the top is not in equilibrium.
A steady motion is said to be kinetically or dynamically stable
if, when slightly disturbed, it oscillates in such a manner that the
vibratory deviation from steady motion approaches steady motion
as a limit when the disturbance approaches zero.
155. The Stability of a System Consisting of a Body Capable of
Oscillation and an Attached Precessing Gyro-Wheel. Repre-
sent the mass of the system by ra, the radius of gyration with re-
spect to the axis of oscillation by fc, and the distance from the cen-
ter of mass to the point of support by H. Represent by 6 the
angle at any instant between any line fixed in the system and the
position of this line when the system is in its equilibrium position.
255
256
GYROSCOPIC STABILIZATION
(a) A Statically Stable Gyro on a Statically Stable Body Capable
of Oscillation under the Influence of Gravity, When the ampli-
tude of oscillation is small and the gyro-wheel is not spinning,
the equation of motion of the system of mass m is
mk 2 -T^ = mgflO (131)
where t is the time occupied in moving through the angle 9.
If the gyro-wheel is spinning, there is an additional torque
acting on the system (Art. 69) in the same direction as that due
to gravity. When the spin-axis is
perpendicular to the axis of oscilla-
tion, the equation of motion of the
system is (90) arid (131),
L c '= -
(132)
FIG. 203
where K c represents the moment of
inertia of the gyroscope with respect
to the axis about which the system
oscillates.
Since no change of the magnitude
of 6 produces a reversal of the direc-
tion of the resultant torque acting
on the system, it follows that the
system Us not only statically stable
but also always dynamically stable. \
(b) A .Statically [Bistable Gyro on a Statically Unstable Oscil-
lating Body. Experiment. Figure 203 represents a system in-
cluding a gyro-wheel G that can either hang pendulously and oscil-
late about a rod cc f or can be turned into the position shown in the
figure with the center of gravity above the axis of support. When
in the position shown in the figure, if the system be slightly dis-
placed about the axis cc f while the wheel is not spinning, the system
will continue to rotate for ISO degrees till the center of gravity is
below the line cc' . The system is sometimes called a stilt top or
inverted gyroscopic pendulum. It is statically unstable when in
the position shown.
With the gyroscope anti-pendulous and the entire system un-
stable, set the gyro-wheel spinning. Now push on the gyro-frame,
thereby producing a small displacement about the axis cc'. The
system does not flop over. It is dynamically stable.
GENERAL PRINCIPLES 257
In the present case, the torque due to gravity is in the same
direction as the gyroscopic torque which was developed by the
push and which produced an angular acceleration about the axis
cc' . The torque acting on the system is, (Art. 69):
L c " = mgfld v-
The direction of the resultant torque depends upon the value of 0.
This system is dynamically stable when
Vs "*-- > mgllO
l\ c
T\
Multiplying each side of this inequality by - -~ , we see that the
condition of dynamical stability assumes the form:
f (wgtf\
:\ 2 )
2 > K
The left-hand side of this inequality represents the kinetic energy
of rotation of the gyro-wheel. The quantity within the paren-
thesis represents one-half of the change in the potential energy of
the system in falling through the distance //. Thus, an oscil-
lating system that is statically unstable can be rendered dynami-
cally stable by means of a statically unstable gyroscope capable
of precessing about an axis perpendicular to the axis of oscillation
of the system, if the kinetic energy of rotation of the gyro-wheel
is greater than K c / K s times one-half of the potential energy lost
by the system in rotating about the axis of oscillation from a ver-
tical position to a horizontal position.
This principle is the basis of all methods proposed for stabilizing
monorail cars.
166. Some Laws of Dynamic Stability. There are several
laws of dynamic stability to which we shall need to refer in sub-
sequent considerations. The following can be proved by analytic
methods and can be illustrated readily by the apparatus shown in
Fig. 203.
(a) If the inner fraxne and spinning wheel of a gyroscope be
statically stable w r ith respect to a horizontal axis perpendicular
to the spin-axle, and if at the same time the second frame be also
statically stable with respect to a horizontal axis perpendicular
to the first, then the system will be dynamically stable and the
two axes will precess in the direction opposite the spin (Art. 155, a).
258 GYROSCOPIC STABILIZATION
(6) If the inner frame and spinning wheel of a gyroscope be
statically unstable with respect to a horizontal axis perpendicular
to the spin-axle, and if at the same time the second frame be also
statically unstable with respect to a horizontal axis perpendicular
to the first, then the system may be dynamically stable, and when
it is dynamically stable the two axes will precess in the direction of
the spin (Art. 155, b).
(c) Precession of a statically stable gyroscopic system is sup-
pressed by retarding 1he precession till the system is in its equi-
librium position.
(d) Precession of a statically unstable gyroscopic system is
suppressed by accelerating the precession till the system is in its
equilibrium position.
(e) Hurrying the precession of a top causes the top to rise against
the torque due to gravity. Hurrying the precession of a statically
unstable gyroscope causes the spin-axle to rotate about the axis
of torque in the direction opposite that of the torque (Art. 40).
(/) Retarding the precession of a top causes the top to fall.
Retarding the precession of a statically unstable gyroscope causes
the spin-axle to rotate about the axis of torque in the direction of
the torque (Art. 40).
(g) A gyro-wheel incapable of precessing produces no effect
on the stability of a body to which it may be attached (Art. 34).
(h) A body that is statically stable and subject to periodic
oscillations can be rendered dynamically stable by retarding the
precessional velocity of the vertical gyro-axle of an attached gyro-
scope that is either statically stable or neutral (Art. 88).*
(/) A body that is statically unstable and subject to oscillations
can be rendered dynamically stable by accelerating the precessional
velocity of the vertical gyro-axle of an attached unstable gyro-
scope.*
(j) A body consisting of two gyroscopically coupled systems
capable of oscillating about horizontal axes cannot be dynamically
stable unless both are either statically stable or statically unstable. f
(k) The overloading of one side of a system that is statically
unstable, but that has been rendered dynamically stable by means
of a spinning gyroscope, causes the overloaded side to rise and the
center of gravity of the system to oscillate across a vertical line
through the point of support of the system.
* Bogaert, L'Kffet gyrostatique et ses applications (1912), Art. 68.
f Deimel, Mechanics of the Gyroscope (1929), Art. 101.
GYROSCOPICALLY STABILIZED MONORAIL CARS 259
(I) When a statically unstable body rendered dynamically
stable by a statically unstable gyroscope with vertical spin-axle is
moved around a curve in the direction of the spin of the gyro-wheel,
the system becomes more stable; when revolved in the direction
opposite the spin, the system becomes less stable (Art. 159).
(m) When a statically unstable body to which is attached a
gyroscope of neutral stability and with vertical spin-axle goes
around a curve in the direction of the spin of the gyro-wheel, the
entire system becomes both statically and dynamically unstable.
When the system goes around the curve in the direction opposite
the spin, the system oscillates about an equilibrium position.
(n) When a statically stable body to which is attached a gyro-
scope with vertical spin-axle and positive static stability is rotated
in the direction of the gyro- wheel, the stability of the 1 system is
increased; whereas if the rotation be in the direction opposite the
.spin, then the stability of the system is decreased.
2. Gyroscopically Stabilized Monorail Cars
'157. The Economy of Monorail Cars. Cars that would safely
run at high speed on a single rail on the ground would effect con-
siderable economies over birail cars, not only in original cost of
installation but also in maintenance of way and in operation.
The ability of a birail locomotive to go up a grade is limited by the
coefficient of static friction between the edge of the wheels and the
top of the rails. Grades are seldom over two per cent. Five
per cent is considered very high. Wheels with double flanges,
slightly tapered, would give a greater frictional force but could not
be used on a curve with birail cars. They could, however, be
used on even a sharp curve with monorail cars, thereby enabling
steeper grades to be traversed by monorail cars than by birail
cars. A monorailway could be built in a terrain requiring curves
and grades too sharp for an ordinary birail railway. There would
be many places where a monorailway would be used if we were
certain as to the dynamic stability of the statically unstable cars,*
A considerable amount of thought has been devoted to the dy-
namic stabilization of monorail cars but at the present time there
is no commercial line in operation.
158. The Principles upon Which Depend the Dynamic Stabi-
lization of Monorail Cars. The dynamic stabilization of a mono-
rail car requires that a torque acting upon the car from out-
260 GYROSCOPIC STABILIZATION
side shall be neutralized by a torque produced within. In all
methods thus far used for causing a monorail car to erect itself
after becoming tilted from the vertical, the tilting causes the spin-
axle of a gyroscope to precess and the velocity of precession is
caused to be accelerated by some torque. The differences in the
various schemes have been in the number of gyroscopes employed,
the direction of the spin-axles relative to the rail and in the method
used to produce the acceleration of the spin-axles.
Consider a monorail car in which is mounted a single gyroscope
with vertical spin-axle capable of precessing about an axis trans-
verse to the car, Fig. 204. Sup-
pose that the gyro and casing
are mounted so as to constitute
a statically unstable system.
Just as soon as the car tilts
from the vertical, the spin-axle
FIG. 204 FIG. 205
begins to precess. If the spin velocity is in the direction indicated
by h s , the direction of the precession is as indicated by w p . Just
as soon as the gyro-axle is deflected from the vertical, the gyro and
casing begin to fall in the direction of the gyroscopic precession,
thereby increasing the velocity about the precession axis in a given
time by an amount w', Fig. 205. A righting torque L f now acts on
the car (Art. 36). If this righting torque is large enough, it brings
the car back toward the upright position in spite of the gravita-
tional torque that tends to tip it over further. But if the righting
torque is large enough to bring the car to the upright position it is
likely to carry it beyond that position.
The action is similar if the velocity of the spin-axle about the
GYROSCOPICALLY STABILIZED MONORAIL CARS 261
precession axis is made greater than the precession velocity due to
the gravitational torque. The angular velocity of the spin-axle
may be increased by a motor or other outside agent. There is
now an unneutralized righting torque, L". For maximum effect,
the acceleration of the precessional velocity produced by the motor
should stop when the free precession stops. The direction of the
acceleration should reverse when the direction of procession re-
verses. The righting torque L" should comprise two components,
one that is proportional to the instantaneous precessional velocity,
and one that is proportional to the angle of precession at that
instant.
159. The Effect of a Change in Linear Velocity on the Stability
of a Monorail Car that Carries a Single Statically Unstable Gyro-
scope with Vertical Spin-Axle. First, consider the effect of a
change in the magnitude of the ve-
locity of the car. If the speed of
the car be increased when moving
from left to right, on a straight mil,
Fig. 206, the center of gravity of
the gyro-system will hang backward
causing the upper end of the gyro-
axle to tilt backward. The tilting
is in the direction indicated by w.
Suppose that the gyro spins in the
direction indicated by h s . Accom-
panying the tilting backward of the spin-axle is a torque acting
on the gyro in the direction indicated by L (Art. 36). This
torque tilts the gyro and the attached car to one side of the verti-
cal through the rail. If the linear velocity of the car be do-
creased, the spin-axle will tilt forward arid the car will tilt in the
direction opposite that when the velocity of the car is increased.
When the linear velocity of the car becomes constant, the car will
oscillate from side to side about its position of equilibrium.
Now, consider the effect upon the stability of the car produced
by motion around a curve. Figure 207 represents a monorail
car, going away from the reader and carrying a statically unstable
gyroscope. If the gyro were not spinning and tho car started to
go around a curve, the car would tilt away from the center of the
curve, thereby exerting a gravitational torque L on the gyro. If
the gyro is spinning in the direction indicated by the arrow h s ,
this torque causes the spin-axle to precess in the direction indicated
202 GYROSCOPIC STABILIZATION
by the arrow Wp. Suppose that the angular velocity of the spin-
axle about the precession axis is accelerated on account of the
static unstability of the gyroscope and that the increase in angular
velocity of the gyro about the precession axis due to this cause is
w', then there is a torque L', Fig. 208, tending to tilt the car toward
the center of the curve (Art. 36). This torque is due to the dis-
placement of the center of gravity of the car and that of the gyro-
scope from the vertical through the rail. If the gyro has a suffi-
FIG. 207 FIG. 208
ciently great angular momentum h sy the car will lean beyond the
upright position toward the center of the curve.
The degree of kinetic stability of the car effected by the gyro-
scope depends upon the direction and magnitude of the velocity
of the car around the curve. Figures 209 and 210 represent the
car going away from the reader and making a right turn around a
curve with center at the right-hand side of the diagram. Figure
209 represents the car tilting away from the center of the curve and
Fig. 210 represents the car tilting toward the center of the curve.
The angular velocity w of the car about the curve is in the same
sense as the vertical component of the spin-velocity of the gyro.
The component of the angular momentum of the gyro in the di-
rection perpendicular to the axis of turning is h s '. When this
component is turned with angular velocity w, a torque L acts on
the gyro-wheel in the direction tending to rotate the spin-axle to-
ward a vertical position (Art. 36). As the gyro cannot rotate
about the axis of L independently of the car, this torque acts on
the car and opposes the tilting of the car from the vertical.
Figures 211 and 212 represent the car going away from the reader
GYROSCOPICALLY STABILIZED MONORAIL CARS 263
and making a left turn around a curve with center at the left-
hand side of the diagram. The angular velocity w of the car
about the curve is in the opposite sense to the vertical compo-
nent of the spin-velocity of the gyro. Following the method of
Fm. 210
FIG. 211
FIG. 212
the preceding paragraph, we arrive at the conclusion that in this
case a torque acts on the car tending to increase the tilt from
the vertical.
Thus it appears that a monorail car stabilized by a gyro-wheel
264
GYROSCOPIC STABILIZATION
with vertical axle tilts whenever the velocity of the car is changed
either in magnitude or direction. The tilt is greater when the car
moves around a curve in the direction opposite that of the spin of
the gyro-wheel than when it moves around a similar curve in the
same direction as the spin of the gyro-wheel.
160. The Effect of a Change in Linear Velocity on the Stability
of a Monorail Car that Carries a Single Gyroscope with Hori-
zontal Spin-Axle Transverse to the Car. When the car goes
FIG. 213
FIG. 214
around a curve and the gyro is not spinning, the car tilts away from
the center of the curve. When the car is making a right turn, if
the direction of spin of the gyro is as indicated by h s , Fig. 213, a
torque acts on the gyro in the direction indicated by L (Art. 41).
Since the gyro cannot turn, relative to the car, about the axis of
this torque, the car is acted upon by a torque which tends to di-
minish the tilt away from the center of the curve. If the direction
of spin of the gyro were reversed, the car would be acted upon by a
torque tending to increase the tilt away from the center of the
curve.
When the car is making a left turn, if the direction of spin of
the gyro is as indicated by h sy Fig. 214, a torque acts on the car
which tends to diminish the tilt away from the center of the curve.
If the direction of spin were reversed the car would be acted upon
by a torque tending to increase the tilt away from the center of the
curve.
GYROSCOPICALLY STABILIZED MONORAIL CARS 265
161. Methods for Increasing the Kinetic Stability of a Mono-
rail Car while the Car is Going around a Curve. A monorail
car can be maintained kinetically stable if at every instant the pre-
cessional velocity of the statically unstable gyro is of the proper
value and in the proper direction. In order that a curve may be
traversed safely by a monorail car, the processional velocity of the
spin-axle must have a value that is different from that required
when the car is moving along a straight level track. The pre-
cessional velocity can be changed by applying a torque about the
axis of precession. A variable torque may be applied by an out-
side motor as in the case of the active type of ship anti-roll device
(Art. 93). A variable torque may be produced also by varying
the degree of statical instability of the gyro system.
Statical instability of the gyro system can be produced either
by having the point of support below the center of gravity or by
applying springs that tend to pull over the system. The free
precession of the gyro-axle and the tilting of the car are slower
when the angular momentum of the gyro is increased, and quicker
when the degree of static instability of the gyro system is increased.
Mr. Peter P. Schilovsky, who has devoted much attention to
monorail cars, produces the required degree of kinetic stability of
the car while it is making a turn by altering the degree of static
instability of the gyroscope according to the angular velocity of
tilting of the car.*
Another method for avoiding any change in the degree of kinetic
instability of a monorail car when changing the direction of motion
is to employ two gyros spinning in opposite directions about axes
which normally are perpendicular to the track. The normal di-
rection of the spin-axles may be either vertical or horizontal. A
view of the first case as seen when looking down on a horizontal
plane is represented in Fig. 215. The two gyro-casings, with the
gyro spin-axles vertical, are represented by G and G f . Suppose
that, owing to motion around a curve, wind pressure on the car,
ununiform loading, or any other cause, the side A of the car is
moved downward. Torques L and U act on the gyros G and G',
respectively. When spinning in the directions indicated by the
symbols h s and h/, the gyros will precess in opposite directions as
indicated by Wp and WP'. The two gyro-casings are coupled to-
gether by two gear segments g and g f so that the precessional
* Schilovsky, The Gyroscope: Its Practical Construction and Applications,
p. 224, Spon and Chamberlain, London and New York.
266
GYROSCOPIC STABILIZATION
speeds at any instant are equal in magnitude. If, now, these
preccssional speeds be accelerated to the same degree, equal torques
will be developed on the two gyros and on the attached car in
directions opposite those indicated by L and L f (Art. 36). If the
angular momenta of the gyros and the accelerations of the pre-
cessional velocities be sufficiently great, the car will be righted.
The case in which the spin-axles are normally horizontal is
FIG. 215
FIG. 210
represented in Fig. 216. As in the preceding case, the two gyro-
axles are coupled together by two gear segments so that the pro-
cessional velocities are equal in magnitude and opposite in di-
rection. Suppose that owing to motion around a curve, or any
othor cause, the side A of the car is moved downward. Then the
two gyro-axles will precess in the directions indicated by Wp and
Wp'j respectively. If, now, these precessional velocities be acceler-
ated to the same degree, equal torques will be developed on the
two gyros, and on the attached car, in the directions opposite those
indicated by L and L', Fig. 216. If the constants of the gyros are
of the proper value, the car will be righted.*
162. The Schilovsky Monogyro Monorail Car of 1915.
Consider a monorail car on which is mounted a single statically
unstable gyro spinning about an axis that normally is vertical and
capable of precession about a horizontal axis transverse to the rail.
When the car tilts to one side, the spin-axle of the gyro precesses
either forward or backward. Suppose that a motor accelerates the
precession. There is now a torque which opposes the present
tilting of the car and tends to make the car tilt in the opposite
direction (Art. 158). This overbalancing torque sets up a counter-
* Cousins, " The .Stability of Gyroscopic Single Track Vehicles/' Engineer-
ing (1913), pp. 678, 711 and 781, is a noteworthy analytical treatment.
GYROSCOPICALLY STABILIZED MONORAIL CARS 267
active force which tends to rotate the spin-axle in the direction
opposite to the precession and beyond the vertical position. When
the car leans in the direction opposite the original tilt, the direction
of precession is reversed. Suppose that now the speed of preces-
sion in this direction is accelerated by torques which the motor
applies to the gyro. The above series of operations is repeated
and the car oscillates from one side of the vertical to the other.
If the impulses on the gyro due to the motor are applied at the
proper times, are in the proper directions and are of the proper
strengths, the motor will absorb energy from the oscillating car,
FIG. 217
damp the amplitude of the oscillations and cause the car to become
kinetically stable.
The device for stabilizing monorail cars proposed in 1915 by
Schilovsky* is represented diagrammatically in Fig. 217. In this
much simplified diagram, the various parts of the device arc not
drawn to scale, the gyro being relatively much larger than here
indicated. The gyro G spins about an axis that normally is verti-
cal. It is mounted so as to be statically unstable and capable of
precessing about a horizontal axis transverse to the rail. The
shaft of the precession motor M rotates continuously with con-
stant speed.
Suppose that the precession motor shaft rotates in the clock-
wise direction and that the direction of spin of the gyro is as in-
dicated by the symbol h s . When the car tilts toward the reader,
* U. S. Patent. Schilovsky, No. 1137234, 1915.
268 GYROSCOPIC STABILIZATION
the spin-axle precesses in the counter-clockwise direction as in-
dicated by the symbol wp. At the same time a heavy plumb-bob P
pushes toward the reader the link A and the vertical arm of the
bell crank B, thereby pushing downward the yoke C and the at-
tached slotted segment D which is fastened to C by a pivot E.
This action causes gear teeth on the upper edge of the segmental
slot to engage with a gear on the shaft of the precession motor,
thereby causing a thrust on the link F and an acceleration of the
precession of the spin-axle. The accelerated precession develops
a torque that tilts the car to the other side of the vertical. The
precession of the spin-axle now reverses in direction, the plumb-
bob pulls on the bell crank in the opposite direction and the
slotted segment disengages from the motor shaft. Immediately
afterward the teeth on the lower edge of the segmental slot engage
the gear on the motor shaft, thereby accelerating the present pre-
cession of the gyro spin-axle and developing a torque that again
causes the car to tilt upward and beyond the vertical. Thus the
car oscillates back and forth through the vertical, losing energy of
vibration to the precession motor.
With this device, if the constants of the apparatus are of the
proper values, the tilt produced when going around a curve is over-
come to the same degree and in the same manner as a tilt due to
wind pressure or any other cause when the car is moving along a
straight level track.
163. The Brennan Duogyro Monorail Car. The first mono-
rail car stabilized by two gyros, spinning in opposite directions
about axes that normally are perpendicular to the track, was
designed and built by Louis Brennan. The stabilizing system of
Brennan's monorail car of 1905 consists of two gyros G and G',
Fig. 218, spinning about axes that normally are horizontal and
transverse to the car.* Each gyro is mounted in a casing capable
of precessing about a vertical axis. The two casings are coupled
together by two segments of gears g and g f so that at any instant the
precessional velocities of the two gyros will be equal and in opposite
directions. The coupled casings are mounted in a frame capable
of rotation through a small angle about an axis through the center
of gravity C of the gyro system and parallel to the track. The spin-
axle of each gyro carries a rigidly attached roller a, a', and a loose
roller 6, &', respectively. At a short distance below each of the
* U. 8. Patent. Brennan, No. 796893, 1905. Reveille, Dynamique des
Solides (1923), pp. 398-406.
GYROSCOPICALLY STABILIZED MONORAIL CARS 269
keyed rollers a and a', is a short shelf d and d f , respectively. Under
each of the loose rollers 6, &', is a similar shelf e and e', respectively.
A plan view of these shelves is shown in the lower part of the dia-
gram. When the car floor is horizontal, the spin-axles are perpen-
dicular to the rail, and each of the keyed rollers a and o! is almost
in contact with the corresponding shelf d and d', respectively, at
a point near one end of the shelf. The other shelves do not extend
9'
FIG. 218
far enough to be under the loose rollers when the spin-axles are
perpendicular to the rail.
Consider a monorail car with the gyros spinning in the directions
indicated by h s and h/, respectively, Fig. 218. Suppose that for
any reason the car tilts to the left. This tilt produces a torque
on the spin-axles of the two gyros in the directions indicated by
L and I/, respectively. The spin-axles precess in the directions
indicated by w p and w p ', respectively. The tilting of the car to
the left brings the shelf d' into contact with the keyed roller a 1 .
Because of the friction between this roller and shelf, the roller
advances along the shelf toward the reader, thereby accelerating
the precession w p r of the gyro G' and also accelerating the preces-
sion w p of the coupled gyro G. This acceleration is accompanied
by a torque on the two gyros in the directions opposite the torques
L and L' that produced the original precession (Art. 40). This
270
GYROSCOPIC STABILIZATION
torque is transferred to the attached frame and tends to right the
car. The magnitude of this righting torque depends upon the
angular momentum of the gyros and the acceleration of their
processional velocity.*
While the keyed roller a' is moving in contact with the shelf
d f toward the reader, the other keyed roller a is moving off the
shelf d toward the reader. The loose roller now is above the shelf
e but not touching it. The righting torque tilts the car to the
right beyond the vertical position, thereby causing the shelf e
to press upward against the loose roller b. The torque thereby pro-
duced causes the roller a to process away
from the reader and to bring it onto the
shelf d. At the same time, the gearing moves
a' and V also away from the reader. During
this motion neither a' nor b f touches a shelf.
This back-and-forth motion of the car is re-
peated through smaller and smaller ampli-
tudes of oscillation till the car stands upright
or till the car is acted upon by another dis-
turbing torque.
Brennan's monorail car of 191(5 is provided
with a more effective device to control the
processional velocity of the gyros. Since the
velocity of precession is proportional to the
FIG. 219
torque tending to tilt the car, the righting torque should be pro-
portional to the processional velocity. Also, the righting torque
should be proportional to the length of time that the disturbing
torque acts. In Brennan's monorail car of 1916 there is applied
about the precession axes of the gyros a torque which, at any in-
stant, is nearly proportional to the velocity of procession, to the
angular displacement of the spin-axles from the central position,
and to the frictional torque opposing precession.
Figure 219 shows the two gyros, as seen by an observer looking
down upon them, spinning in opposite directions about axes
normally horizontal and perpendicular to the track, and capable
of processing about vertical axes. Rigidly attached to the gyro-
casings G and G' are gear segments g and g f . These gear segments
mesh with a double rack r connecting two pistons capable of being
pushed back and forth by air pressure in two cylinders C and C'.
* Eddy, " The Mechanical Principles of Brennan's Monorail Car," J.
Franklin Inst. (1910), p. 467.
GYROSCOPICALLY STABILIZED MONORAIL CARS 271
The admission of compressed air into these cylinders and the ex-
haust from them are controlled by a compound air valve, not
shown in the diagram, operated by a device attached to the upper
end of the precession axle D of one of the gyros.*
As soon as the car tilts, the spin-axles of the two gyros begin to
precess in opposite directions. The rotation of the precession
axle of G' causes the attached mechanical device to exert a force
on the end of a rod, to the other end of which is a compound air
valve connected into the service lines of the servomotor cylinders.
This force is the resultant of a force that is proportional to the
precessional velocity, to a force that is proportional to the angular
displacement of the gyros from their central position, and to the
friction around the precessional axes. This resultant force opens
the valves of the servomotor to a degree proportional to the force,
thereby accelerating the precessional velocity by an amount pro-
portional to the torque that tilted the car from the vertical. This
acceleration of the precessional velocity is accompanied by a
torque adequate to erect the car if the dimensions of the apparatus
are of the proper values. The car is furnished with a device to
hold the car upright when the gyros are not spinning, f
164. The Scherl Duogyro Monorail Car of 1912. The sta-
bilizing device consists of two gyros, spinning in opposite directions
about axes that normally are vertical, and capable of processing
about axes that are transverse to the longitudinal axis of the car.
The two gyro-casings are coupled together by a pair of gear seg-
ments so that the velocities of precession of the two gyro-axles, at
any instant, are equal in magnitude and opposite in direction.
There is a motor that accelerates the precession at the proper times.
This precession motor is of the hydraulic type, consisting of a
cylinder and piston together with a set of valves that are con-
trolled by the rocking movements of the gyro system. Hydraulic
pressure is produced by an electrically driven pump. A device} is
provided so that, in case there should be an interruption of the
main current that operates the gyro and precession motors, the
continued rotation of the gyros will cause the gyro-motors to act
as electric generators, the current thereby produced operating
the stabilizing system for a considerable time. As soon as the
speed of rotation of the gyros falls to such a value that the stability
* U. S. Patent. Brennan, No. 1183530, 1916.
t U. S. Patent. Brennan, No. 1019942, 1912.
t U. S. Patent. Falcke, No. 1048817, 1912.
272 GYROSCOPIC STABILIZATION
of the car is endangered, a magnetic device releases a set of supports
that will hold the car upright.*
When the car tilts to one side, the gyro spin-axles precess toward
one another; when the car tilts to the other side the spin-axles
precess away from one another. In either case, the tilting of the
spin-axles operates valves which control the operation of the hy-
draulic precession motor piston. The movement of this piston
accelerates the precessional velocity of the two connected gyros.
The result is a torque which tilts the car upward and, possibly,
past the vertical position.
* U. S. Patent. Scherl, No. 959077, 1910.
INDEX
(The numbers refer to pages)
> Acceleration, angular, 4, 10
linear, 1, 7
radial, 2
relation between angular and linear,
11
Agonic line, 165
Airplane, gyroscopic actions on, 51,
59
Airplane cartography, 100
Airplane gyroscopic pilot, 123
Angular momenta, 5, 10, 17, 20
Anschiitz gyro-compass, 224-235
Anschiitz gyro-horizon, 117
Anti-roll devices for ships, 42, 131-
161
Arma gyro-compass, 236-247
Autogiro, 52
B
Ballistic damping or turning error,
190
Ballistic damping or turning elimina-
tor, 212, 246
Ballistic deflection error, 184, 186
avoided, 210, 222, 231, 244, 252
Beats, 38
Bessemer's steamer with gyroscope,
62
Bliss-Leavitt torpedo stearing gear,
97
Bonneau airplane inclinometer, 76
Bonneau-LePrieur-Derrien gyro-sex-
tant, 119
Brennan monorail cars of 1905 and
1916, 268
Brown gyro-compass, 170
Bumstead sun compass, 170
Camera control on an airplane, 101-
104
Cardan suspension, 45
Carter's track recorder, 125
Cartography, airplane, 100
Center of gravity, 21
Center of mass, 21
Centrifugal drier, Westori's, 75
Centripetal and centrifugal forces,
12, 16
Centroid, 21
Clinging of spinning body to guide, 85
Clinograph for measuring crooked-
ness of well casings, 74
Compass, gyroscopic, 162
inductor, 169
magnetic, 165
magneto, 169
sun, 170
Couple, force, 4
Coupled systems, 36
Course and speed errors of gyro-
compasses, 182
avoided, in Anschiitz compass, 231
in Arma compass, 241
in Brown compass, 220
in Florentia compass, 251
in Sperry compass, 206
D
Damping of vibrations, 39
in Anschiitz gyro-compass, 228
in Arma gyro-compass, 237
in Brown gyro-compass, 218
in Florentia gyro-compass, 250
in Sperry gyro-compass, 203
Degrees of freedom, 44
274
INDEX
Deviations of magnetic compass, 166,
168
Directed gun-fire control, 128
Directorscopes, gun-fire, 129
Drift of projectiles, 80
Dynamic stability, defined, 255
some laws of, 257
Dyne, 2
E
Earth inductor compass, 169
Earth's axis and spin-axis of gyro-
scope, 71-74
Energy, 2
of precessing body, 66
Erg, 2
Errors to which a gyro-compass is
subject, 182-196
F
Fieux ship stabilizer, 144
Fleuriais gyroscopic octant, 120
Florentia gyro-compass, 247-253
Follow-up repeater system of An-
schiitz gyro-compass, 234
of Arma gyro-compass, 238
of Brown gyro-compass, 221
of Florentia gyro-compass, 248
of Sperry gyro-compass, 202
Force, 2, 5
centripetal and centrifugal, 12, 16
Frahm anti-roll tanks, 42
Freedom, degrees of, 44
Friction at peg of top causes rise of
spin-axis, 76
G
Gimbals, 45
Griffin pulverizing mill, 92
Gun-fire control, 128
Gun-fire control compasses, 196
Gun-fire control directorscope, 129
Gyration, radius of, 14
Gyro, gyroscope, gyrostat, defined, 45
Gyro-compass, Anschlitz, 224-235
Arma, 236-247
Brown, 215-223
Gyro-compass, Florentia, 247-253
meridian-seeking tendency of, 173-
181
Sperry, 198-214
Gyro-compass is subject to errors,
182-196
Gyrodynamics, first and second laws
of, 55
Gyro-horizontals, 115-128
Gyro-pendulum, 105
of the same period as a certain
simple pendulum, 113
Gyroscope modifies rolling of a ship,
140
Gyroscopic conical pendulum, 105-
108
Gyroscopic torque or resistance, 49
Gyro-verticals, 115
H
Horizon, Sperry airplane, 78
Horizontals, dynamic and true, 12, 13
Horse-power, 3
Hurrying and retarding precession, 69
Inclinometer, Bonneau's, 76
Inductor compass, 169
Inertia, 3
moment of, 4, 13-15
Intercardinal error, of gyro-com-
passes, 191
suppression of, 195, 212, 222, 232,
247, 252
K
Kinetic energy of a precessing body,
66
Kinetic stability, defined, 255
of monorail cars, 259-272
some laws of, 257
Latitude, determination of, 115
Latitude error of gyro-compasses,
182, 205
INDEX
275
Latitude error of gyro-compasses,
avoided, 220, 231, 241
corrected, 206, 251
Leavitt torpedo steering gear, 07
Locomotive, gyroscope couple acting
on, 63-66
M
Magnetic compass, deviations, pro-
duced by a rapid turn, 168
when on an iron ship, 166
directive tendency of, 165
Magneto compass, 169
Mass, 3
Maxwell top, 10
Meridian, magnetic, 162
Meridian method of locating the
geographic, 162
Meridian-seeking tendency, of a liq-
uid-controlled gyroscope, 17(5
of a magnetic compass, 165
of a pendulous gyroscope, 173
Meridian-seeking torque acting on a
gyro-compass, 1 78
Meridian-steaming error, of gyro-
compasses, 182
Anschiitz, 231
Arma, 241
Brown, 220
Florentia, 251
Sperry, 206
Metacentric height, 132
Moments of inertia, 4, 13, 15
dynamic, 14
principal, 15
Momentum, angular, 5, 10, 17, 20
Monorail cars, gyroscopic stabiliza-
tion of, 257-272
Brennan, 268
Scherl, 271
Schilovsky, 266
Motorcycle, gyroscopic forces acting
on a, 57
N
North-steaming error, of gyro-com-
passes, 182
Anschiitz, 231
North-steaming error, Arma, 241
Brown, 220
Florentia, 251
Sperry, 206
Nutation, 67
O
Octant, Fleuriais gyroscopic, 120
P
Pendulous gyroscope, 105
meridian-seeking tendency of, 173
period of, 108-111
Pendulous mass, 191
Pendulum, conical, 29
forces acting on an unsymmotrical,
16
mathematical or simple, 29
physical, 28
with attached gyroscope, 136, 140
Period, of a conical pendulum, 30
of a gyro-compass, 180-190
of a mathematical and of a physical
pendulum, 29
of simple harmonic motion of rota-
tion, 25
of the roll of a ship, 134
Phase and phase angle, 32
Pioneer turn indicator, 85
Pitching of a ship due to waves, 132
Port side of a ship, denned, 50
Power, 2
Precessing body, kinetic; energy of, 66
Precession, 48
period of, 66
Precessional torque, 49
Precessional velocity maintained by a
torque, 55
Projectile, drift of a, 80
Q
Quadrantal deflection, 191, 195
Quadrantal error, avoided, in trie
Anschiitz gyro-compass, 232
in the Arma gyro-compass, 247
in the Brown gyro-compass, 222
276
INDEX
Quadrantal error, avoided, in the
Florentia gyro-compass, 252
in the Sperry gyro-compass, 212
R
Radian, 3
Repeater system, of the Anschiitz
gyro-compass, 234
of the Arma gyro-compass, 238
of the Brown gyro-compass, 221
of the Florentia gyro-compass,
248
of the Sperry gyro-compass, 213
Resonance, 30
Retarding and hurrying of precession,
69
Revolving a gyroscope having two
degrees of freedom, 82
Roll and pitch recorder, 121
Rolling error, 191
avoided, in the Anschiitz gyro-corn-
pass, 232
in the Arma, gyro-compass, 247
in the Brown gyro-compass, 222
in the Florentia gyro-compass,
252
in the Sperry gyro-compass, 212
Rolling of a ship, by waves, 131
methods of diminishing, 135
period of, 134
produced by a gyroscope, 152
Rotation, simple harmonic motion of,
25
ScherFs monorail car of 1912, 271
Schlick's ship stabilizer, 142
Schilovsky's monorail car of 1915, 266
Schuler's gyro-horizon, 117
Sextants, gyroscopic, 119, 121
Ship stabilizer, of the active type, 146
Sperry's, 146-160
of the inactive type, 136
Fieux, 144
Schlick, 142
Side-wheel steamer, gyroscopic torque
acting upon, 50
Simple harmonic functions, mean
value of products of, 33
Simple harmonic motion, of rotation,
25
of translation, 22
Slug, 3
Sperry-Carter track recorder, 125
Sperry's airplane horizon, 78
Sperry's airplane pilot, 123
Sperry's gyro-compass, 198-215
Sperry's roll and pitch recorder, 121
Sperry's ship stabilizer, 147-160
Spin-axis of a gyroscope and the
earth's axis, 71, 74, 172
Spinning, defined, 44
Stability, static and kinetic, 255
Stabilization by gyroscopes, 255-272
Starboard of a ship, defined, 50
Steamship, gyroscopic torque acting
on, 63
Sun compass, 170
Swing radius, 14
Tanks, Frahm anti-roll, 42
Taylor's formula for ship stabiliza-
tion, 1.53
Torpedo control, 94-100
Torque, 4, 10
Torque and angular momentum, 17
Torque developed by rotation of the
spin-axis, 60
Torque effect on a spinning body, 45,
52, 87, 90
Torque opposing turning of the sec-
ond frame of a gyroscope,
111
Torque required to maintain con-
stant precession, 52
Top, rising of the spin-axis of, 76
Track recorder, Sperry-Carter, 125
Turn indicator, Pioneer, 85
Vertical, the true and the dynamic,
12, 13
Velocity, angular, 4, 10
INDEX
277
Velocity, linear, 1, 7
relation between angular and linear,
11
Vibrations, damping of, 39
free and forced, 36
W
Wave motion, 30, 131-135
Weston centrifugal drier, 75
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