Seely F.B. Analytical Mechanics for Engineers
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THE
GYROSCOPIC
COUPLE
481
disc. As soon as
precession
starts,
however,
that
is,
as soon
as
the
z-axis
has turned
through
the
angle d<f>,
the
change
AC
m
the
angular
momentum is
produced.
This
change
requires
a
couple
in
the vertical
plane (clockwise
as viewed
by
the
reader)
to
pre-
vent the
disc and
axle
from
rotating
counter-clockwise in
the
vertical
plane.
This
couple
is
supplied by
the two forces
W
having
a
moment
Wl.
Hence,
if
the
body
is
allowed
to
precess
about
the
FIG.
454.
2/-axis
with
angular
velocity
ft,
the external
couple
Wl
is resisted
by
the
gyroscopic
couple,
that
is,
W
from
which
the
velocity
of
precession
is
found to
be,
ft=
To,
uk
2
'
in which
k is
the
radius
of
gyration
of
the disc
with
respect
to the
z-axis.
If
precession
is
prevented,
a
couple (C
=
7coft)
must be set
up
in the
horizontal
plane,
ft here
being
the
angular
velocity,
at
any
instant, produced
by
Wl,
which
will
be
the same
whether
the
disc
rotates about
the
z-axis
or
not,
since
no
resistance is offered
to
the
482
IMPULSE AND
MOMENTUM
external
couple
unless
precession
is
allowed.
This
explains
why
a
heavy rotating flywheel
or
armature
on board a
ship,
with
its axis
horizontal
and
athwartship,
will
offer no
more
resistance to
the
rolling
of
the
ship
than
when it is
not
rotating.
The
bearing
of
the
axles, however,
must exert a
large
couple
C
=
7cofi in
a hori-
zontal
plane
which tends to
"
nose
"
the
ship
around. 2
here
represents
the
angular
velocity
of roll.
If
an external
couple
is
applied
in
a
horizontal
plane
to
increase
or
hurry
the
precession,
the disc and
axle
(Fig.
454a)
will
rise
since
7coO>TFZ.
This
principle
is
employed
in
the Brennan
mono-rail
car,
the
precession being
hurried
by
the
rolling
of the
axle of the
revolving flywheels,
extended,
on a
shelf attached to
the side
of the car.
This
principle
is also
used
in
the
"
active
type
"
of
gyroscope
for
stabilizing ships.
In
this case the
pre-
cession is hurried
by
means of a
precession engine
which acts
after
the
ship
has
rolled a
very
small amount thus
producing
a
gyro-
scopic
righting-couple
sufficient to
extinguish
the roll.
Since
the
roll is
checked
in
its
incipiency, only
a
small amount of work
is
done. The stresses
produced
in
the
hull
of
the
ship
are also
small
for
the same
reason,
and
hence,
the
weight
and
displacement
of
the
active
type
of
gyroscope
likewise
may
be small. For those
interested in
the various
ways
in
which
gyroscopic
motion
may
arise and
in
the
progress
which
has been made in the
application
of
the
gyroscope
to the
stabilizing
of
ships,
mono-rail
cars,
and
aeroplanes;
to the
guiding
of
torpedoes;
to
compasses;
and
to
the
Griffin
grinding mill,
the
following
references are
appended:
1.
Journal
of
the
Franklin
Institute,
May,
1913.
2.
American
Machinist, August
7
and
14,
1913.
3. Iron
Age,
December
1, 8, 15,
and
22,
1910.
4.
Scientific
American
Supplement,
January
26,
1907.
5.
Popular
Science
Monthly,
July,
1909.
6.
Scientific American,
December
18,
1915.
7. Journal
American
Society
of
Mechanical
Engineers,
Novem-
ber,
1920.
NOTE.
For
problems
on
the
gyroscope
see Art. 167.
INDEX
The numbers refer to
pages
Acceleration
angular,
242
components,
236,
245,
269
-
linear,
235
of
gravity,
5
-
relative,
266
tangential
and
normal,
236
uniform
circular,
251
rectilinear,
250
Angle
of
repose,
133
Angular momentum, 454, 456,
457
-
velocity,
229
Anti-resultant,
8
Balancing,
369
of
rotating
masses,
370
Bearings,
friction
in, 145, 148,
443
Bow's
notation, 7,
95
Cables,
102,
106
Catenary,
106
Center
of
gravity, defined,
173
by experiment,
178
parallel
force
system,
173
oscillation,
364
percussion, 333,
364
suspension,
364
Centrifugal couple,
371
-
force,
329,
370
Centroid, denned,
160
by integration,
163
graphical
method,
177
of
composite
figures
and
bodies,
168
Coefficient of
friction,
128
restitution,
472
rolling
resistance,
153
speed
fluctuation,
435
Composition, defined,
8
-
of
couples, 60,
62
-
forces, 28, 29, 34, 48, 53,
56,
65
Compound
pendulum,
363,
367
Conservation
of
energy,
439
momentum,
467
Conservative
system, 421,
440
Coriolis'
law,
270
Couple,
characteristics
of,
20
definition
of,
19
gyroscopic, 382,
478
moment
of,
19
resolution
of,
64
transformations
of,
22,
44
vector
representation
of,
20
D'Alembert's
principle,
305,
307, 326,
347
Dimensional
equations,
25
Displacement
angular,
224
linear,
222
relative,
263
Dynamics,
2
Dynamometer,
446
Dyne,
297
Effective
force,
303
Efficiency,
442
Energy,
420
conservation
of,
439
dissipation
of,
443
kinetic,
422
loss
of,
in
impact,
476
non-mechanical,
428
-
of
a
body,
424
particle,
422
-
potential,
421
483
484
INDEX
Equations
of
motion, defined,
288
for
a
particle,
298
plane
motion, 339,
340
rotation,
319
translation,
308
Equilibrant,
8
Equilibrium,
8
algebraic
conditions
of,
70
diagrams,
69
equations of,
69, 72, 73,
78,
82,
110
113,
114
graphical
conditions
of,
70
of
collinear
forces,
72
concurrent forces
in
a
plane,
73
space,
110
parallel
forces
in
a
plane,
78
space,
112
non
-
concurrent,
non
-
parallel
forces
in
a
plane,
82
non
-
concurrent,
non
-
parallel
forces
in
space,
114
three
forces,
75
Erg, 409,
417
First
moment,
160
Flexible
cables,
102,
106
Flywheel,
stresses
in,
356
Force,
1
characteristics
of,
4
component of,
8
conception
of,
1,
3
-
effective,
303
external
effect
of,
4
-
inertia,
307,
326,
345
measure
of,
4
moment
of,
14
polygon,
31, 36,
49
resolution
of,
11,
22
units
of,
4
vector
representation of,
6
Forces,
classification
of,
7
composition
of,
8
impulsive,
449
inertia, 326,
345
resultant
of a
system
of,
8,
28,
system
of,
7
triangle
of,
9
Friction, defined,
127
-
belt,
155
circle,
145
coefficient
of,
128
cone
of,
129
journal,
145
-
laws
of,
134
limiting,
128
of
inclined
planes,
135
screws,
142
wedges,
138
pivot,
148
rolling,
153
work
lost due
to,
443
Funicular
polygon,
37
Geepound,
296
Governors,
386
centrifugal
shaft,
391
inertia
shaft,
392
-
loaded,
388
-
Porter,
389
Rite's
inertia,
393
Guldinus,
theorem
of,
171
Gyroscope,
378
analysis
of forces
in,
379
precession,
axis
of,
383
velocity of, 383,
480
Gyroscopic
couple,
382,
478
Harmonic
motion,
253
Hoop tension,
356
Horsepower,
417
-
-hour,
418
Impact, 450,
471
loss of
energy in,
476
Impulse,
450
-
angular, 451,
452
components
of,
451
linear,
451
moment
of, 451,
452
Inertia,
denned,
289
-
forces, 307, 326,
345
moment
of,
180,
205
product
of,
194
INDEX
485
Instantaneous
center,
of
acceleration,
352
-
velocity,
284
Inclined
plane,
friction
of,
135
Joule,
409
Journal
friction,
145
Kilowatt,
417
Kilowatt-hour,
418
Kinematics, 2,
221
Kinetic
energy,
420
lost in
impact,
476
of a
body
having plane
motion,
426
rotation,
425
translation,
424
particle,
422
Kinetics, 2, 287,
303
Lami's
theorem,
74
Loss of
energy,
in
bearings,
434
in
impact,
476
Mass,
289
Mass-
center,
175
motion
of,
353
Moment-arm, 14,
19
Moment-center, 14,
87
Moment
equation,
limitations
of,
351
Moment of
inertia, 180,
205
about
inclined
axes,
198
, perpendicular
planes,
208
-
by
experiment, 220,
367
,
composite
areas,
190
, bodies,
218
,
graphical
methods,
202
,
integration,
184,
209
parallel
axis
theorem,
183,
207
polar,
181
principal axes,
199
Momentum,
453
-
angular,
454, 456,
457
a
vector
quantity, 453,
477
components
of,
454
conservation
of,
467
linear,
454,
455
moment
of,
454, 456,
457
Motion,
curvilinear,
222
gyroscopic, 378,
478
Newton's laws
of,
293
non-uniformly
accelerated,
256
of
a
projectile,
248
mass-center of
body, 353,
414
-
rigid bodies,
276
rectilinear,
222
-
relative,
262
simple harmonic,
253
uniformly
accelerated
curvilinear,
251
uniformly
accelerated
rectilinear,
250
Motion
graphs
acceleration-time,
261
distance-time,
228
-
speed-time,
241,
260
Newton's laws
of
motion,
293
Pappus,
theorem
of,
171
Parabolic
cable,
102
Parallelogram
law,
8
Particle,
221
equations
of motion
for,
297
Pendulum, compound, 363,
367
-
conical, 300,
387
simple,
360
torsion, 365,
368
Period of
oscillation, 362, 364,
366
Pivots,
friction
of,
148
loss of
energy in,
443
Plane
motion,
278,
336
Polar moment of
inertia,
181
Pole of
force
polygon,
36
Poundal,
297
Power,
417
special
equations
for,
418
Principal
axis,
199
Principle
of
impulse
and
momentum,
459,461
moments,
16,
38, 49, 57,
65
transmissibility,
8
work
and
energy,
430
Product
of
inertia,
194
parallel
axis
theorem
for,
196
486
INDEX
Projectile,
motion
of,
248
Prony brake,
446
Radius
of
gyration, 182,
207
Rays
of force
polygon,
36
Relative
motion,
262
Resolution,
8
of a
force,
11
couples,
64
directions
of,
87
Resultant,
8
Resultants
of force
systems,
28
--
collinear
forces,
28
concurrent
forces in
a
plane,
29
-----
space,
53
--
non
-
concurrent,
non
-
parallel
forces
in
a
plane,
48
------
space,
65
parallel
forces
in
a
plane,
34
space,
56
Reversed effective force for a
body,
307, 326,
345
Rigid body,
3
Rolling
resistance,
153
Rotation, 277,
315
Scalar
quantity,
6
Second
moment,
180
Sensitiveness
of
governors,
389
Slug,
296
Space diagram,
6
Speed,
226
Statically
indeterminate
force
sys-
tems,
69
Statics,
2
Stresses
in
trusses,
88
Strings
of funicular
polygon,
37
Tangential-effort diagram,
411
Theorems of
Pappus
and
Guldinus,
171
Translation,
276,
305
Triangle
law,
9
Trusses, analysis of,
88
graphical,
95
method of
joints,
89
sections,
90
Units,
294
absolute
system of,
297
gravitational
system of,
297
systematic system of,
297
Varignon's
Theorem,
16
Vector,
6
diagram,
6
quantity,
6
Velocity,
angular,
229
components
of,
231
linear,
226
-
relative,
264
Work and
energy,
principles of,
430
Work,
406
done
in
punching
hole
in
plate,
412
by
a force
system,
414
expressions
for,
406,
408
graphical
representation
of,
409
units
of,
409
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