Spinning Tops Part 5

9. _Precessional motion of the earth_, 74-91.

Its nature and effects on climate, 75-80; resemblance of the precessing earth to certain models, 80-82; tilting forces exerted by the sun and moon on the {135} earth, 82-84; how the earth's precessional motion is always altering, 85-88; the retrogression of the moon's nodes is itself another example, 88, 89; an exact statement made and a sort of apology for making it, 90, 91.

10. _Influence of possible internal fluidity of the earth on its precessional motion_, 91-98.

Effect of fluids and sand in tumblers, 91-93; three tests of the internal rigidity of an egg, that is, of its being a boiled egg, 93, 94; quasi-rigidity of fluids due to rapid motion, forgotten in original argument, 95; beautiful behaviour of hollow top filled with water, 95; striking contrasts in the behaviour of two tops which are very much alike, 97, 98; fourth test of a boiled egg, 98.

11. Apology for dwelling further upon astronomical matters, and impertinent remarks about astronomers, 99-101.

12. How a gyrostat would enable a person living in subterranean regions to know, _1st, that the earth rotates_; _2nd, the amount of rotation_; _3rd, the direction of true north_; _4th, the lat.i.tude_, 101-111.

Some men's want of faith, 101; disbelief in the earth's rotation, 102; how a free gyrostat behaves, 103, 104; Foucault's laboratory measurement of the earth's rotation, 105-107; to find the true north, 108; all rotating bodies vainly endeavouring to point to the pole star, 108; to find the lat.i.tude, 110; a.n.a.logies between the gyrostat and the mariner's compa.s.s and the dipping needle, 110, 111; dynamical connection between magnetism and gyrostatic phenomena, 111.

13. How the lecturer spun his tops, using electro-motors, 112-114.

14. _Light_, _magnetism_, _and molecular spinning tops_, 115-128.

Light takes time to travel, 115; the electro-magnetic {136} theory of light, 116, 117; signalling through fogs and buildings by means of a new kind of radiation, 117; Faraday's rotation of the plane of polarization by magnetism, with ill.u.s.trations and models, 118-124; chain of gyrostats, 124; gyrostat as a pendulum bob, 126; Thomson's mechanical ill.u.s.tration of Faraday's experiment, 127, 128.

15. _Conclusion_, 129-132.

The necessity for cultivating the observation, 129; future discovery, 130; questions to be asked one hundred years hence, 131; knowledge the thing most to be wished for, 132.

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APPENDIX I.

THE USE OF GYROSTATS.

In 1874 two famous men made a great mistake in endeavouring to prevent or diminish the rolling motion of the saloon of a vessel by using a rapidly rotating wheel. Mr. Macfarlane Gray pointed out their mistake. It is only when the wheel is allowed to _precess_ that it can exercise a steadying effect; the moment which it then exerts is equal to the angular speed of the precession multiplied by the moment of momentum of the spinning wheel.

It is astonis.h.i.+ng how many engineers who know the laws of motion of mere translation, are ignorant of angular motion, and yet the a.n.a.logies between the two sets of laws are perfectly simple. I have set out these a.n.a.logies in my book on _Applied Mechanics_. The last of them between centripetal force on a body moving in a curved path, and torque or moment on a rotating body is the simple key to all gyrostatic or top calculation. When the spin of a top is greatly reduced it is necessary to remember that the total moment of momentum is not about the spinning axis (see my _Applied Mechanics_, page 594); correction for this is, I suppose, what introduces the complexity which scares students from studying the vagaries of tops; but in all cases that are likely to come before an engineer it would be absurd to study {138} such a small correction, and consequently calculation is exceedingly simple.

Inventors using gyrostats have succeeded in doing the following things--

(1) Keeping the platform of a gun level on board s.h.i.+p, however the s.h.i.+p may roll or pitch. Keeping a submarine vessel or a flying machine with any plane exactly horizontal or inclined in any specified way.[14] It is easy to effect such objects without the use of a gyrostat, as by means of spirit levels it is possible to command powerful electric or other motors to keep anything always level. The actual methods employed by Mr. Beauchamp Tower (an hydraulic method), and by myself (an electric method), depend upon the use of a gyrostat, which is really a pendulum, the axis being vertical.

(2) Greatly reducing the rolling (or pitching) of a s.h.i.+p, or the saloon of a s.h.i.+p. This is the problem which Mr. Schlick has solved with great success, at any rate in the case of torpedo boats.

(3) In Mr. Brennan's Mono-rail railway, keeping the resultant force due to weight, wind pressure, centrifugal force, etc., exactly in line with the rail, so that, however the load on a wagon may alter in position, and although the wagon may be going round a curve, it is quickly brought to a position such that there are no forces tending to alter its angular position. The wagon leans over towards the wind or towards the centre of the curve of the rail so as to be in equilibrium.

(4) I need not refer to such matters as the use of gyrostats for the correction of compa.s.ses on board s.h.i.+p, referred to in page 111.

{139}

[Ill.u.s.tration: FIG. 1.]

{140} Problems (2) and (3) are those to which I wish to refer. For a s.h.i.+p of 6,000 tons Mr. Schlick would use a large wheel of 10 to 20 tons, revolving about an axis E F (fig. 1) whose mean position is vertical. Its bearings are in a frame E C F D which can move about a thwart-s.h.i.+p axis C D with a precessional motion. Its centre of gravity is below this axis, so that like the s.h.i.+p itself the frame is in stable equilibrium. Let the s.h.i.+p have rolled through an angle R from its upright position, and suppose the axis E F to have precessed through the angle P from a vertical position.

Let the angular velocity of rolling be called [.R], and the angular velocity of precession [.P]; let the moment of momentum of the wheel be m.

For any vibrating body like a s.h.i.+p it is easy to write out the equation of motion; into this equation we have merely to introduce the moment m [.P]

diminis.h.i.+ng R; into the equation for P we merely introduce the moment m [.R] increasing P. As usual we introduce frictional terms; in the first place F [.R] (F being a constant co-efficient) stilling the roll of the s.h.i.+p; in the second case f [.P] a fluid friction introduced by a pair of dash pots applied at the pins A and B to still the precessional vibrations of the frame. It will be found that the angular motion P is very much greater than the roll R. Indeed, so great is P that there are stops to prevent its exceeding a certain amount. Of course so long as a stop acts, preventing precession, the roll of the s.h.i.+p proceeds as if the gyrostat wheel were not rotating. Mr. Schlick drives his wheels by steam; he will probably in future do as Mr. Brennan does, drive them by electromotors, and keep them in air-tight cases in good vacuums, because the loss of energy by friction against an atmosphere is proportional to the density of the atmosphere. The solution of the equations to find the nature of the R and P motions is sometimes tedious, but requires no great amount of mathematical knowledge. In a case considered by me of {141} a 6,000 ton s.h.i.+p, the period of a roll was increased from 14 to 20 seconds by the use of the gyrostat, and the roll rapidly diminished in amount. There was accompanying this slow periodic motion, one of a two seconds' period, but if it did appear it was damped out with great rapidity. Of course it is a.s.sumed that, by the use of bilge keels and rolling chambers, and as low a metacentre as is allowable, we have already lengthened the time of vibration, and damped the roll R as much as possible, before applying the gyrostat. I take it that everybody knows the importance of lengthening the period of the natural roll of a s.h.i.+p, although he may not know the reason. The reason why modern s.h.i.+ps of great tonnage are so steady is because their natural periodic times of rolling vibration are so much greater than the probable periods of any waves of the sea, for if a series of waves acts upon a s.h.i.+p tending to make it roll, if the periodic time of each wave is not very different from the natural periodic time of vibration of the s.h.i.+p, the rolling motion may become dangerously great.

If we try to apply Mr. Schlick's method to Mr. Brennan's car it is easy to show that there is instability of motion, whether there is or is not friction. If there is no friction, and we make the gyrostat frame unstable by keeping its centre of gravity above the axis C D, there will be vibrations, but the smallest amount of friction will cause these vibrations to get greater and greater. Even without friction there will be instability if m, the moment of momentum of the wheel, is less than a certain amount.

We see, then, that no form of the Schlick method, or modification of it, can be applied to solve the Brennan problem.

{142}

[Ill.u.s.tration: FIG. 2.]

{143} Mr. Brennan's method of working is quite different from that of Mr.

Schlick. Fig. 2 shows his model car (about six feet long); it is driven by electric acc.u.mulators carried by the car. His gyrostat wheels are driven by electromotors (not shown in fig. 3); as they are revolving in nearly vacuous s.p.a.ces they consume but little power, and even if the current were stopped they would continue running at sufficiently high speeds to be effective for a length of time. Still it must not be forgotten that energy is wasted in friction, and work has to be done in bringing the car to a new position of equilibrium, and this energy is supplied by the electromotors.

Should the gyrostats really stop, or fall to a certain low speed, two supports are automatically dropped, one on either side of the car; each of them drops till it reaches the ground; one of them dropping, perhaps, much farther than the other.

The real full-size car, which he is now constructing, may be pulled with other cars by any kind of locomotive using electricity or petrol or steam, or each of the wheels may be a driving wheel. He would prefer to generate electropower on his train, and to drive every wheel with an electric motor.

His wheels are so independent of one another that they can take very quick curves and vertical inequalities of the rail. The rail is fastened to sleepers lying on ground that may have sidelong slope. The model car is supported by a mono-rail bogie at each end; each bogie has two wheels pivoted both vertically and horizontally; it runs on a round iron gas pipe, and sometimes on steel wire rope; the ground is nowhere levelled or cut, and at one place the rail is a steel wire rope spanning a gorge, as shown in fig. 2. It is interesting to stop the car in the middle of this rope and to swing the rope sideways to see the automatic balancing of the car. The car may be left here or elsewhere balancing itself with n.o.body in charge of it. If the load on the car--great lead weights--be dumped about into new positions, the car adjusts itself to the new conditions with great {144} quickness. When the car is stopped, if a person standing on the ground pushes the car sidewise, the car of course pushes in opposition, like an indignant animal, and by judicious pus.h.i.+ng and yielding it is possible to cause a considerable tilt. Left now to itself the car rights itself very quickly.

[Ill.u.s.tration: FIG. 3.]

{145}

[Ill.u.s.tration: FIG. _3^b_ (showing the ground-plan of Fig. 3).]

{146} Fig. 3 is a diagrammatic representation of Mr. Brennan's pair of gyrostats in sectional elevation and plan. The cases G and G', inside which the wheels F and F' are rotating _in vacuo_ at the same speed and in opposite directions (driven by electromotors not shown in the figure), are pivoted about vertical axes E J and E' J'. They are connected by spur-toothed segments J J and J' J', so that their precessional motions are equal and opposite. The whole system is pivoted about C, a longitudinal axis. Thus when precessing so that H comes out of the paper, so will H', and when H goes into the paper, so does H'. When the car is in equilibrium the axes K H and K' H' are in line N O O' N' across the car in the plane of the paper. They are also in a line which is at right angles to the total resultant (vertical or nearly vertical) force on the car. I will call N O O' N' the mid position. Let m be the moment of momentum of either wheel. Let us suppose that suddenly the car finds that it is not in equilibrium because of a gust of wind, or centrifugal force, or an alteration of loading, so that the shelf D comes up against H, the spinning axis (or a roller revolving with the spinning axis) of the gyrostat. H begins to roll away from me, and if no slipping occurred (but there always is slipping, and, indeed, slipping is a necessary condition) it would roll, that is, the gyrostats would precess with a constant angular velocity [alpha], and exert the moment m[alpha] upon the shelf D, and therefore on the car. It is to be observed that this is greater as the diameter of the rolling part is greater. This precession continues until the roller and the shelf cease to touch. At first H lifts with the shelf, and afterwards the shelf moving downwards is followed for some distance by the roller. If the tilt had been in the opposite direction the shelf D' would have acted upwards upon the roller H', and caused just the opposite kind of precession, and a moment of the opposite kind.

We now have the spindles out of their mid position; how are they brought back from O Q and O' Q' to O N and O' N', {147} but with H permanently lowered just the right amount? It is the essence of Mr. Brennan's invention that after a restoring moment has been applied to the car the spindles shall go back to the position N O O' N' (with H permanently lowered), so as to be ready to act again. He effects this object in various ways. Some ways described in his patents are quite different from what is used on the model, and the method to be used on the full-size wagon will again be quite different. I will describe one of the methods. Mr. Brennan tells me that he considers this old method to be crude, but he is naturally unwilling to allow me to publish his latest method.

D' is a circular shelf extending from the mid position in my direction; D is a similar shelf extending from the mid position into the paper, or away from me. It is on these shelves that H' and H roll, causing precession away from N O O' N', as I have just described. When H' is inside the paper, or when H is outside the paper, they find no shelf to roll upon. There are, however, two other shelves L and L', for two other rollers M and M', which are attached to the frames concentric with the spindles; they are free to rotate, but are not rotated by the spindles. When they are pressed by their shelves L or L' this causes negative precession, and they roll towards the N O O' N' position. There is, of course, friction at their supports, r.e.t.a.r.ding their rotation, and therefore the precession. The important thing to remember is that H and H', when they touch their shelves (when one is touching the other is not touching) cause a precession away from the mid position N O O' N' at a rate [alpha], which produces a restoring moment m[alpha] of nearly constant amount (except for slipping), whereas where M or M' touches its shelf L or L' (when one is touching the other is not touching) the pressure on the shelf and friction determine the rate of the precession towards the mid position N O O' N', {148} as well as the small vertical motion. The friction at the supports of M and M' is necessary.

Suppose that the tilt from the equilibrium position to be corrected is R, when D presses H upward. The moment m[alpha], and its time of action (the total momental impulse) are too great, and R is over-corrected; this causes the roller M' to act on L', and the spindles return to the mid position; they go beyond the mid position, and now the roller H' acts on D', and there is a return to the mid position, and beyond it a little, and so it goes on, the swings of the gyrostats out of and into the mid position, and the vibrations of the car about its position of equilibrium getting rapidly less and less until again neither H nor H', nor M nor M' is touching a shelf. It is indeed marvellous to see how rapidly the swings decay.

Friction accelerates the precession away from N O O' N'. Friction r.e.t.a.r.ds the precession towards the middle position.

It will be seen that by using the two gyrostats instead of one when there is a curve on the line, although the plane N O O' N' rotates, and we may say that the gyrostats precess, the tilting couples which they might exercise are equal and opposite. I do not know if Mr. Brennan has tried a single gyrostat, the mid position of the axis of the wheel being vertical, but even in this case a change of slope, or inequalities in the line, might make it necessary to have a pair.

It is evident that this method of Mr. Brennan is altogether different in character from that of Mr. Schlick. Work is here actually done which must be supplied by the electromotors.

One of the most important things to know is this: the Brennan model is wonderfully successful; the weight of the apparatus is not a large fraction of the weight of the wagon; will this also be the case with a car weighing 1,000 times as {149} much? The calculation is not difficult, but I may not give it here. If we a.s.sume that suddenly the wagon finds itself at the angle R from its position of equilibrium, it may be taken that if the size of each dimension of the wagon be multiplied by n, and the size of each dimension of the apparatus be multiplied by p, then for a sudden gust of wind, or suddenly coming on a curve, or a sudden s.h.i.+ft of position of part of the cargo, R may be taken as inversely proportional to n. I need not state the reasonable a.s.sumption which underlies this calculation, but the result is that if n is 10, p is 7.5. That is, if the weight of the wagon is multiplied by 1,000, the weight of the apparatus is only multiplied by 420.

In fact, if, in the model, the weight of the apparatus is 10 per cent. of that of the wagon, in the large wagon the weight of the apparatus is only about 4 per cent. of that of the wagon. This is a very satisfactory result.[15]

My calculations seem to show that Mr. Schlick's apparatus will form a larger fraction of the whole weight of a s.h.i.+p, as the s.h.i.+p is larger, but in the present experimental stage of the subject it is unfair to say more than that this seems probable. My own opinion is that large s.h.i.+ps are sufficiently steady already.

In both cases it has to be remembered that if the _diameter_ of the wheel can be increased in greater proportion than the dimensions of s.h.i.+p or wagon, the proportional weight of the apparatus may be diminished. A wheel of twice the diameter, but of the same weight, may have twice the moment of momentum, and may therefore be twice as effective. I a.s.sume the stresses in the material to be the same.

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