A Treatise on Gyrostatics and Rotational Motion: Theory and Applications

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Macmillan and Company, limited, 1918 - Gyroscopes - 530 pages
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Contents

Elementary quantitative analysis of the motion of a top
68
Instantaneous axis axis of figure and axis of resultant a m
75
SECTION PACK 1 Gyrostats Equations of motion
80
A gyrostat under a constant couple Gyrostatic action of a body of any form e g an aeroplane propeller
82
Symmetrical equations of motion
85
Motion of a top reduced to motion of a particle
86
Effect on the precession of increasing or diminishing the applied couple Hurry ing and retarding precession The experiment of Sire
87
Greenhills graphical construction for tracing the effect of change of couple
89
Reaction of a ringguide or spacecone on a top
91
Explanation of the clinging of the axle of a top to a curved guide
92
Stability of the motion of a top or gyrostat
94
Rise and fall of a top
96
Occurrence of loops on the path
98
Motion of the axis between two close limiting circles
99
Tops rising and falling through a small range Strong and weak tops
100
Rouths graphical construction for rise and fall of a top
102
Rise and fall of a top started with initial precession
105
Motion under various starting conditions
106
CHAPTER VI
108
Determination of period of rise and fall
109
Error involved in approximations to the motion of a top
110
Theory of the upright or sleeping top
112
Stability of an upright top Graphical representation
113
Analytical discussion of the stability of an upright top
114
Time of passage of axis of weak top from limiting circle to any possible position
116
Periodic motion of a weak top
117
Path of a point on the axis of a weak top
118
Equations of motion of an upright top derived from first principles
119
Estimate of error in approximation to the motion
120
Explanation of an apparent anomaly
121
Azimuthal turning of a nearly upright top
122
Stable motion of a nearly upright top between two close limiting circles
123
Limiting case of stable motion of a nearly upright top
124
Interpretation of the discontinuity in p at the pole
125
CHAPTER VII
127
Reaction of a top or gyrostat on its support
129
Gyrostatic observation of the rotation of the earth Foucaults methods
132
Gyrostatic balance and gyrostatic dipping needle of Lord Kelvin
133
Gilberts barogyroscope
135
Gyrostat with axis vertical stable or unstable according to direction of azi muthal turning
137
Top supported by a string
139
Gyrostatic action of the wheels of vehicles etc Monorail cars
141
Gyrostatic action of the paddles and screws of steamers
144
Gyrostatic action of turbines
145
Steering of a bicycle or of a childs hoop
146
One spinning top supported by another Steady motion
147
Drift of a projectile
148
Turning action on a body moving in a fluid
149
Turning action of the water on a ship Why a ship carries a weather helm
151
Centre of effort of resistance of a fluid
152
Stability of a projectile in a fluid
153
1 flywheel clamped
156
2 flywheel undamped
157
Gyrostatic pendulum hung by an untwistable wire or a universal joint
158
CHAPTER VIII
161
A gyrostat hung by a steel wire
164
A gyrostat with two freedoms doubly unstable without spin
167
A gyrostatic system with two freedoms doubly stable without spin
170
Illustrations of the effect of spin on stability
171
A gyrostatic system illustrating doubly stable and doubly unstable arrangements
173
Gyrostatic control of the rolling of a ship Gyrostat frame clamped
175
Gyrostatic controller of the rolling of a ship Gyrostat frame undamped
178
Theory of two interlinked systems which are separately unstable Stability in presence of dissipative forces
180
CHAPTER IX
182
Steady vibrational motion of a gyrostatic chain under tension or under thrust
186
SECTION PAGE 3 Helical gyrostatio chain
187
more general discussion
189
A vertical gyrostatic chain under gravity
191
Gyrostat hung by a thread
192
dynamical explanation
196
gyrostatic analogy
197
THE EARTH AS A TOP PRECESSION AND NUTATION GYROSTATIC THEORY OF THE MOTION OF THE NODES OF THE MOONS ORBIT ...
199
Calculation of attractive forces
201
Couples applied to the earth by the attraction of the sun and moon
202
Solar couple on the earth
204
Mean angular speed of precession
206
Precession due to lunar attraction
207
bodycone and spacecone
209
results of theory and observation
212
Precession and nutation from instant to instant
214
Graphical representation of the variable parts of 6 and p
216
Mean couple tending to bring the equator and the moons orbit into coincidence
217
Gyrostatic theory of the regression of the nodes of the moons orbit
219
Estimation of periodic term
221
Effect of the equatorial belt of the earth on the motion of the moons nodes
222
Remarks on the gyrostatic calculation of the motion of the moons nodes
223
CHAPTER XI
226
Positions of the axis of figure instantaneous axis and axis of resultant a m
227
Influence of the internal constitution of the earth
228
Period of free precession for an earth ab rigid as steel
229
Rise or fall of the earths surface in Eulerian precession
230
Effect of annual transfers on accentuating free precession
231
Comparison of terms
233
Numerical illustrations
234
Systematic observations of variation of latitude at different observatories
235
Diagrams and tables of results
237
CHAPTER XII
239
Quaternion property of Eulers parameters
240
Kleins parameters
242
Relations of elliptic integrals
244
Formulae for the numerical calculation of elliptic integrals of the first and second kinds
245
Landens transformation
246
Convergent series for elliptic integrals _
248
Calculation of complete elliptic integrals by gseries Numerical example
249
Numerical calculation for an actual top
251
different speeds of rotation
252
Numerical determination of the axes of an actual top to the vertical
253
Calculation of complete and incomplete elliptic integrals of the second kind
254
Numerical examples
255
Formulae for the azimuthal angle ip
257
Elliptic integral expressions for the angles and f
258
Numerical calculation of time in terms of the angle 6 for an actual top
259
Numerical calculation of elliptic integrals of the third kind
261
Numerical calculation of p for any step in time
262
CHAPTER XIII
264
theoretical discussion
266
Limits of instability of a prolate liquid ellipsoid of revolution
267
1 case of no forces
269
2 case of ex traneous forces
271
case of a golf ball
272
Drift of an elongated fast spinning projectile
274
Action of friction
275
Graphical representation of the motion
276
Position of centre of effort for an air ship
278
Motion of a ring in a perfect fluid Equation of energy
279
Impulse of the motion of a solid in a fluid
280
Vibrations of a ring moving in a fluid Quadrantal pendulum
282
Elliptic integral discussion of finite oscillations
283
Graphical representation of the motion of a ring in a fluid
284
Stability of a body rotating in a fluid
285
CHAPTER XIV
287
Discussion of results of theory
288
1 t small 2 t great
289
Calculation of the small terms in the equations Discussion of cases
290
Motion of a flat disk The boomerang
291
General explanation of the motion of a boomerang Specification of forces
292
Trajectory of a light disk in its own plane
293
Difficulties of a complete theory Approximate constants of the motion
294
Motion with change of direction of the plane of motion
295
CHAPTER XV
297
Calculation of the azimuthal motion
299
Azimuthal angle described by pendulum from one limiting circle to the other exceeds x and is less than jr
300
the limiting circles are nearly coincident with the equatorial circle
302
Pendulum nearly vertical Theorem of Lagrange
303
The azimuthal angle from one limiting circle to the other proved less than rr
304
Possibility of motion of rise and fall on a surface of revolution with axis vertical
305
Motion of a particle between two close circles on a surface of revolution
307
initial motion in the plane of the equatorial circle
308
A simple pendulum oscillating through a finite range
309
Discussion of the force along the supporting cord or rod of a simple pendulum
311
Graphic representation of finite pendulum motion
312
Motion of a particle on a concave surface of revolution
325
Motion of a particle on a paraboloid with axis vertical
326
Cases integrable by elliptic functions
328
Reaction of the surface on the rolling ball
330
Comparison of photographs of the paths of an actual pendulum with the theoretical paths
331
CHAPTER XVI
332
Expanding or contracting bodies Mean axes
334
Expanding or contracting body unacted on by force
338
Rigid body containing a flywheel and turning about an axle
339
Gilberts barogyroscope
342
oscillations about steady motion
343
elliptic function discussion
345
case in which the arms reach the upward vertical
346
a liquid filament in a revolving vertical circular tube
347
a ball containing a gyrostat and rolling without slipping on a horizontal table
349
A rolling ball containing a gyrostat Track on the table
351
A rolling ball containing a gyrostat
353
A rolling ball containing a gyrostat
354
A rolling ball containing a gyrostat Method of solution by direct reference to first principles
356
A cylinder containing a gyrostat and rolling on a horizontal piano
357
CHAPTER XVII
358
Stability of a body spinning about a nearly vertical axis
359
lines of curvature of the surface at contact not parallel to the principal axes through the centroid ai
361
Experiments with a top supported on an adjustable curved surface _ 63
363
Effect of oscillations in producing azimuthal turning of the body
365
Summary of results
366
Other particular solutions
368
Tshapliguines integral
369
The HessSchiff equations of motion of a top
370
The HessSchiff equations are inapplicable to a symmetrical top under gravity
372
Case of resultant a m of constant amount
373
Motions when S0
374
Motion when S0 Pendulum motions
375
Motion when S 0 Distinction between cases
376
A homogeneous ellipsoid spinning on a horizontal plane
377
Stability of any solid with a principal axis normal to the horizontal surface
379
CHAPTER XVIII
382
Varying motion of a top on a rounded peg
384
A top supported on a circular edge round the axis of figure
385
A top on a rounded peg and containing a flywheel
386
Problem of a disk or hoop on a horizontal plane
388
A coin spinning on a table
389
calculation from first principles
391
Centre of gravity of a top raised by friction
394
Condition of minimum kinetic energy
395
Minimum kinetic energy is a necessary but not a sufficient condition for the erection of a top
396
Summary of conditions for the rising of a top
398
Numerical examples
399
A top in form of a sphere loaded symmetrically about a diameter
400
A uniform sphere loaded symmetrically
401
Further examples
402
CHAPTER XIX
404
Generalised coordinates
405
Lagranges equations of motion
406
Conditions fulfilled by generalised coordinates
407
Proof of Eulers equations by vectors
408
β ΟεηβΓβΙίδΟίΙ πιοιηεηία Ηαιηίΐΐοηβ Ινη Ί ιυ εφίβίίοηβ Οαηοηίοαΐ βςααϋυηβ
409
8βΙεηΐ8 νηίοη ΒΓΘ ηοί ηοΐοηοιηοιιβ
411
ι ιι ι II 1 1 Μ ι κ οεβΜΗΤ ΟΓ ηθ ϋ8ΐΐΒ 1 ΓοΓΠΐ οί ΐΛΓηοκ ευαί ίοιΐ8
413
ι ι Ι ίΐιι ιι ί ιΓ Τουρ οίοοηββίιι ίιιΐιι11Ιι
414
ΙηοΓαϋοη οί οοοπϋηαΙβ3 Ροπηΐίοη οί βφίαϋοηβ οί ηιοΐίοη
415
Κουίηβ ηιΐε ίοΓ ίξηοΓ8ίίοη οί οοοιχίίηβΐββ Οττοβίΐίο ίβπηβ
416
Ογοϋο ββίβηιβ Κίηβΐίο ροίβηίίβΐ
417
ΚβνβΓβίϋΗγ οί ίηβ ιηοΐίοη οί βνβίβιη
418
δίβοίΐίΐγ οί ίηβ ιιηί ϊ π οί α ογοίϊο βΐβπι
419
Υϊβϊβΐθ ηά οοηοβ11 οοοΓάίηαίββ ΝαΙιίΓε οί ροίβηΐίΐ βηετ
420
Κβίαΐίνβ ικίβηΙίίΐ1 βηβΓγ 8ί1ί1ίΙ οί Γβΐίίνβ βηαϊ1ίΙπιιιη
421
Μ 1 1 1 ι πιίϊοηβ οί ΙΗβ ςβηβΓα θΐι1ίοη8 οί πιοίίοη Μοίίοη οί β ραΓίίοΙβ ίη α ρ1β ηβ οιιτνβ
422
Ι Η ι ΜαΙίοηβ οί Ιΐιβ βηβΓα βςιιΙίοηΒ οί ιηοίϊοη ΟοβίΛΐίο ροικίιιΐυιη
423
ΙΜβοιιββίοη οί ίΐιβ 871080 ρβηάιιΐιιιη
424
Α 83τοΒΐίο ρβηάιιΐιιιη οβοίΐΐίίηκ ίΙίΓοπΙι 8ΐη11 ΓΠΘ
426
Μιιΐίοη οί α Ιιοορ ΟΓ Ιίβΐί ΐΓβΛίβθ 1 ιηοΛίββθ ΙιΟταηίαη βςυίίοηβ
427
Ηιηίΐίοηβ ριίηοίρβί ίαιηοΐίοη Ιπίρρ011 οί Ιΐρ οηοηΐοβ1 6ςιιίίοπ8
428
1οοοί3 ΙΙιβοΓβιη οί ίΗβ οοπιρίβίο ίηίβρ οί ίΗβ οαηοηίςΙ ρηιΐίΐίίοη3
430
Οοΐοίά ιηοίίυη
433
Οοΐοίάαΐ βγβίβιηβ οοηίβίηϊη βτηε9
434
Εδβοί οί ΓβρθΛίβά Γοοίβ οί ίηβ άβίβπηίηβηΐβΐ βςιιΐίοη οη βΐΐιίΐίί
435
Μοίίοηΐ ίοΓοββ Βίββίρβίΐοη ίαηοΐίοη
437
Μοίίοηίΐΐ ίοΓοββ ζβΓο θ3θΌ8ίί ίο βρίβπιβ
438
ΟΓΟβΙίίο ρεηάαίιιηι Ιτνο ΪΓββάοιηβ
439
Οτο5ίίίο 95Γ8ίπΐ3 νίΐΗ ΐΗΓββ ίΓββάοηιβ ΕΙβοίΓΐο βηά ιηαβηοίίο αηΛίουθ
440
Οτο8ίίΐΙιο άοπιϊηβίίοη ΙΛΓΘ ιιηά βηαΠ τοοίβ οί Ιηο 1οΙβΓπιίηΐηΙ1 βηυαίίοη
441
Οτο8ΐίίο δίβηΐί ηΙη ίοιίΓ ίΓββιΙοηιβ
442
Μοίίοη οί τίίά οοάγ υηάεΓ ηο ίοΓοεβ
444
Ροίηβοίβ ΓβρΓββεηΙβίίοη οί ίΚο ηιοίίοη οί Ιχκΐ αηάεΓ ηο ίοΓοε ΙηνΓίο1β ρΐβηε ι Γ βηά ΐηνπΗε Ιίηε ι ι
445
ΜεαβιΐΓβπιεηί οί Ιΐιβ ίίηιε οί πιοί ίοη 8ν1νοβίΓ8 ΐΗεοΓβιη
447
Ροϋιοάβ αηά ΗεΓροΙηοάβ Βοθεοπε ηά 8ροεοοηβ
448
Οβββ οί αχϊ1 βγιηιηείΓ
449
Οοηε άββοποεά οη ΐΗβ βούγ ο ίΐιε ρΓθεοΙίοη οί ΐΗο ίηβίβηίβηοοιιβ χΐ8 Ι Α οηΐηβι ρ
450
SECTION PAGI 7 Herpolhodes
451
Variation of the radius vector of the herpolhode with time
453
Radius of curvature of the herpolhode
454
Special case of the herpolhode
455
Spacecone and bodycone according as C is the greatest or the least moment of inertia
456
Illustration of the stability of a body under no forces
457
Stability of a top under no forces Diabolo
458
Stability when the body under no forces is unsymmetrical
459
Extension of Poinsots theory to the motion of a top 460
460
The outer extremity of the i a for a top lies on a fixed spherical surface
461
Reduction of the locus of the extremity of the i a to a plane
462
The locus of the extremities of the i a for the top is a polhode
463
Case of an unspherical top
464
Passage from one Poinsot movement to another
465
Passage back from the second movement to the first
466
Bodycone and spacecone for associated movements
467
The polhode for a top as the intersection of two surfaces of the second degree
468
Relation of the curve of intersection of the two surfaces to a family of confocals
469
Determination of the parameters of the confocals
470
Motion of the axis of a top represented by a deforniable hyperboloid
472
Forces acting on the body carried by tho bodycone
473
Determination of the constants of the surfaces
474
Determination of constants for the associated motion
475
Deformation of the hyperboloid of one sheet as the top moves
476
Summary of results
477
Calculation of the motion of the axis Different forms of the energy equation
478
Calculation of t in terms of cos 0
480
Hodograph for the motion of a top
482
CHAPTER XXII
483
Equations of equilibrium
484
Case of bending in one plane
486
Bending in one plane represented by pendulum motion
487
A thin bar bent into a helix is analogous to a top in steady motion
489
A helix held in equilibrium by a couple or by axial force
492
A gyrostat on an overhanging flexible shaft Equilibrium of the shaft
493
Determination of the gyrostatic couple
494
A rotor carried midway between the two bearings ot a flexible shaft
495
The free period of an oscillatory disturbance of a rotating shaft
497
Quasirigidity of a moving chain Equations of motion
498
A bend in a moving chain is not carried along the chain by the moving links
499
A revolving chain under no forces General oase 600
500
A revolving chain in a plane containing the axis of rotation
502
Calculation of the vectorial angle for the case of 16
505
CHAPTER XXIII
507
A gyroscope on gimbal rings Equation of energy
508
Differential equations of motion for Example 3
509
A gyroscope in a spherical case hung by a string
510
Pseudoelliptic case of the motion of a top
511
A cylinder rolling on the circular edge of ono end
512
motion of a Bphere on the surface
513
A sphere rolling on a vertical plane which turns about a vertical axis
514
The vertical plane and sphere of 13 when the plane turns also about a normal axis
515
A rigid body turning about a principal axis while another principal axis lies in a plane through the former and a fixed line
516
Stability of a ring of wire spinning on the top of a sphere
517
A body supported at its centroid and under the action of a constant couple
518
The ordinary problem of a rapidly spinning top started by unwinding a string
519
The arc described by a point on the axis of a rapidly spinning top
520
Motion relative to the earth
521
Analogy of Foucaults pendulum to a gyrostatic pendulum
522
Revolving balance showing the earths rotation Experiment of Eotviis
523
Index
535
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Page 413 - ... energy as part of their thermal energy. The view therefore is plausible that the emission and absorption of radiant energy in band spectra represent changes in this rotational energy. If molecules were really rigid bodies, the application of Bohr's principles to the suggestion would be simple. If / is the moment of inertia of the body about its axis of rotation, to the angular velocity of rotation, the energy of rotation is given by W =\Ia?
Page 33 - ... of a very large one. In the second mode of operating the gyrostat, friction is introduced at the bearings on which the frame of the gyrostat is mounted. With this addition the ship is forcibly prevented from excessive rolling. In the trials of the device it was found that, with the control in operation, the angle of roll of the ship did not exceed 1 in a cross-sea which produced a total swing of 35 when the control was out of action. It is interesting to notice that, contrary to the opinions...
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Page 33 - ... a ship does not result in the waves breaking over her ; a ship controlled by a gyrostat is, I believe, a dry one. I have here a motor-gyrostat fitted within a skeleton frame representing a ship (Fig. 12). The frame is mounted on two bearings arranged on wooden uprights, and may be made to oscillate on these bearings, so as to imitate the rolling of a ship in a cross-sea. The frame of the gyrostat is mounted on two bearings placed athwart the frame, and a weight is attached to the outside of the...
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Page vi - Yet some, from subtle hints, have got Mysterious light it was a trot But let that pass : they now begun To spur their living engines on : For as whipp'd tops and bandy'd balls, The learned hold, are animals ; So horses they affirm to be Mere engines made by geometry, And were invented first from engines, As Indian Britains were from penguins. So let them be, and, as I was saying, They their live engines ply'd, not staying Until they reach'd the fatal champaign Which th...
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