Part 7
On the right of the drawing, (fig. 6) the teeth of the wheel _B_ are angular, (suppose square) and those of the wheel _C_ rounded off by any curve _s_, _within_ an epicycloid. All that is necessary to remark in this case is, that the teeth of the wheel _B_ must not extend _beyond_ its primitive circle, whilst the round parts of those of the wheel _C_, do more or less extend beyond its primitive circle; whence it becomes evident, that the contact of such teeth, (if infinite in number) can _only_ take place in the plane of the common tangent at right angles to _A B_; also that if these teeth are sufficiently hard to withstand ordinary pressure, without indentation in these circumstances, there is no perceptible reason for a sensible change of form; since this contact only takes place where the two motions are alike, both in swiftness and direction. A fact I am going to mention may outweigh this reasoning in the minds of some, but cannot invalidate it. I caused two of these wheels made of brass, to be turned with rapidity under a considerable resistance for several weeks together, keeping them always anointed with _oil_ and _emery_, one of the most destructive mixtures known for rubbing metals; but after this severe trial, the teeth of the wheels, _at their primitive circles_ were found as entire as before the experiment. And why? Certainly for no other reason than that they worked without sensible friction.
Hitherto nothing has been said of wheels in the conical form, usually denominated _mitre and bevel geer_. But my models will prove, that they are both comprehended in the system. The only condition of this unity of principle is, that the axes of two wheels, instead of being _parallel_ to each other, be always found in _the same plane_. With this condition, every property above-mentioned, extends to this class of wheels, which my methods of executing also include, as indeed they do every possible case of geering.
Being afraid of trespassing on the time of the society, I have suppressed a part of this paper, perhaps already too long; but I hope I may be indulged with a few remarks on the application of those wheels to practical purposes. And first, as to what I have myself seen; these wheels have been used in several important machines to which they have given much swiftness, softness or precision of motion as the case required. They have done more; they have given birth to machines of no small importance, that could not have existed without them. In rapid motions they do all that band or cord can perform, with the addition of mathematical exactness, and an important saving of power. In spinning factories these properties must be peculiarly interesting; and in calico-printing, where the various delicate operations require great precision of motion. In clock-making also, this property is of great importance in regulating the action of the weight, and thus giving full scope to the equalizing principle whatever it be. I may add, it almost annuls the cause of anomaly in these machines, since a given clock will go with less than 1/4 of the weight usually employed to move it. Another useful application may be mentioned; in flatting mills, where one roller is driven by a pinion from the other, there is a constant combat between the effort of the plate to pass equally through the rollers, and the action of the common geering, which is more or less convulsive. Whence the plate is _puckered_, and the resistance much increased, both which circumstances these wheels completely obviate; and many similar cases might be adduced.
I shall only add, that my ambition will be highly gratified if, through the approbation of this learned society, I may hope to contribute to the improvement and perfection of the manufactures of this county; and if the invention be found of general utility to my much loved country.”
Subsequently to the reading of the above paper, I had occasion to execute many wheels on this principle; and their appearance, and use, excited on the one hand much interest, and on the other much opposition. I had even to complain of real injury in that contest: against which I defended myself with a warmth that I thought _proportionate_ to the attack.--But all this was local and temporary: and writing now for a more enlarged sphere, and perhaps for a more extended period, I feel inclined to lay aside every consideration, but those immediately connected with the influence of this work on the public prosperity. I shall therefore avoid all reference to the names either of my friends or my opponents. My friends will live in a grateful heart, as long as memory itself shall last; my enemies, if I have any, will be forgiven--or, at worst, forgotten; and my System is henceforward left to wind its way into public notice and usefulness, by its own intrinsic merits.
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CERTAIN OBSERVATIONS which I was induced to make on occasion of a re-print of the above Memoir, may assist in introducing what remains to be said on the subject. They commence thus:
The foregoing little work, which first brought this subject into public notice in this town, was not the only method employed to develope its principles, and urge its adoption. A second paper was read, at the next meeting of the society, and some time after, a third, at the Exchange Dining Room; on both which occasions new modes of reasoning were pursued, and new kinds of proof adduced. On the first, a model was exhibited of two screw-formed teeth (connected with proper centres) exactly like those represented in fig. 6; by the action of which on each other, it became manifest that teeth of this angular shape _do_ work together without inconvenience, and therefore, that all sensible friction is, in this case, done away.
On the latter occasion (the lecture at the Exchange) two other methods were brought forward, to corroborate the principles before stated: (see Plate 14, fig. 1.) The first was a kind of transparency, in which _a line of light_ represented the place of contact of two wheels working together; by the partial and _variable_ obscuration of which, the successive action of every portion of the teeth was clearly shewn. The second method consisted of two pair of wheels, _made from loaf sugar_, the teeth of which were cut one pair in the usual form, and the other on the new principle. Here, the difference in the effects of the two methods was so great, that the common teeth were almost immediately worn or broken down, by the very same kind of impulse that the new wheels sustained without injury: and with a loss of matter almost imperceptible, since many thousand revolutions of the wheels took place without detaching so many _grains_ of sugar!
These _Observations_ include likewise the following remarks:
In adverting to a few of the difficulties we have encountered, it will appear curious that one of them should spring from a most useful property of the system: but the paradox is thus explained. As there is no method more effectual for giving the teeth _a perfect form_, than working the wheels together, (covering them with an abrasive substance) we have most frequently chosen to depend on that important property; and have therefore set the wheels at work as they came from the foundry, instead of chipping the teeth, as is usual when common wheels are expected to act well in the first instance. But our wheels being then full of asperities, their action would be of course imperfect and noisy, till time had smoothed and equalized the touching surfaces: a state of things that might well stagger the opinion of a candid observer unacquainted with the system. Happily however we can now appeal to the fact of many wheels having _become silent_, that were once referred to with triumph, as proofs of a radical defect in the principle. It may not be improper to add here, that if highly finished wheels were _particularly desired_, we would engage to cut them in metal on this principle, with all the perfection of surface given to common wheels by the first masters.
In the use of bevel wheels of this description (with singly inclined teeth) there is doubtless a tendency to approach toward or recede from each other; the extent of which (for cylindrical wheels) has been already determined. This tendency goes, so far, to give a _bend_ to the shaft; and, if this be _very weak_, to create a degree of friction on the teeth as the wheels revolve. It is therefore desirable that the shafts should be rather too strong than too weak; since the principle can only exist entire, when the wheels in working, are kept in the same planes which they occupy when at rest. This is too evident to be further insisted on.
But a greater, or at least a more frequent cause of friction in the wheels is the motion, endwise, of the shafts, arising from a want of solidity in the bearers, and especially of connection between them; for whenever these _are strongly connected_, and the shafts well fitted to their steps, all circular commotion is ipso facto destroyed; while the longitudinal tendency produced by the teeth on the shafts _is certainly an advantage_: because it prevents the shaking that often arises from their vibration, endwise, when lying on unsteady bearers, or on bearers between which they have too much liberty.
A few words will make known the process of reasoning by which I arrived at the idea that forms the basis of this invention. I had been conferring with a well-known mechanical character, (to whom the art is greatly indebted)--and hearing his observations on the advantage derived from having two equal cog wheels connected together, with the teeth of the one placed opposite the _spaces_ of the other; so as to reduce the _pitch_ one half, and the friction still more; (since the latter follows the ratio of the double _versed sines_ of the half-angles between the teeth respectively:)--and no sooner had I left that gentleman, than my imagination thus whispered--“What that gentleman says is both true and important.” “But if _two_ wheels thus placed, produce so good an effect, _three_ wheels (_dividing the original pitch into three_), would produce a better: and _four_, a better still: And _five_ a better than _that_. And for the same reason, an indefinite number of such wheels would be indefinitely better! We must then cut off the corners of all those teeth, and we shall have one screw-formed line, that will represent an indefinite number of teeth, and approach indefinitely near to _absolute perfection!_” Thus did this Invention originate: and it soon appeared to me, to be the nearest approach of material exactitude to mathematical precision, that is to be found in the whole circle of practical mechanics. For not only is the _relative motion_ of the touching points of two wheels (that is their _friction_), less than the distance between two of the nearest particles of matter, but it is as many times less than that distance, as that distance is less than the half diameter of any wheel whose teeth are thus formed.
I assert therefore that these teeth, placed in proper circumstances, do work without _sensible friction_ at their pitch lines: as although by means of mathematical abstraction, it may be possible to _assign_ a degree of friction between them, that degree cannot be realized on a material surface: and I fear not the friction on _mathematical surfaces_, if my material surfaces do not suffer from it. I take leave then to repeat, that no _friction_ can justly be said to arise from a motion, too short to carry a _rubbing_ particle from one particle of a _rubbed_ surface to the next! and this is precisely the case in the present instance.
Continuing to reflect on this important subject, I soon perceived that the _screw-formed_ line would give the teeth a tendency to slide out of each other; and to drive the shafts of the wheels endwise in opposite directions; but even that evil is not great: for, confining the obliquity within 15 degrees, that tendency is only about one quarter of the useful effort; and a _stop_ acting on the central points of the axes, will annul this tendency _without any sensible loss of power_. We need not even have recourse to this expedient when any good reason opposes it: for this tendency can be destroyed altogether by using _two opposite inclinations_: giving the teeth the form of a V on the surface of the wheels--a method which I actually followed on the very first pair I ever executed, which I believe are now in the Conservatory of Arts at Paris.
A circumstance somewhat remarkable deserves to be here noticed. In the specification of a Patent which I have seen in a periodical work since my return from Paris, _for things respecting steam engines_, and dated, if I recollect right, in 1804 or 5, this V formed tooth is introduced--as an article of the specification, yet having no connection whatever with its other subjects; nor being attended with the most distant allusion to the _principle_ of this _geering_. The fact is that I had these V wheels in my _Portique_, in 1801, when that exhibition took place in which my Parallel motion appeared and was rewarded by a Medal from Bonaparte: so that _two_ of my countrymen at least, engineers like myself, appear to have taken occasion from that exhibition, to draw my inventions from France to England--a thing by no means wrong in itself nor displeasing to me: who was then totally precluded from holding any communication of that kind with my native country.
It would be repeating the statements contained in the foregoing memoir, to say more on the general principles of this System. I request therefore, my readers to give that paper an attentive perusal; and to accept the following recapitulation of its contents:
1. To cut teeth of this form in any wheel is, virtually, to divide it into a number of teeth as near to _infinite_, as the smallness of a material point is to that of a mathematical one.
2. By the use of these teeth, and the _multitude_ of contacts succeeding each other thence arising, all perceptible noise or commotion is prevented. (This of course supposes _good execution_, or long-continued previous working.)
3. For the same reasons, all sensible abrasion is avoided: for we have proved that the passage of any point of one wheel, over the corresponding point of another, is indefinitely _less_ than the distance between the nearest particles of matter. (This supposes the action confined to the pitch line of the wheel; and this it will be in all common cases--since the teeth wear each other in preference, within and without that line; _which therefore must remain prominent_.)
4. From the foregoing it appears that the teeth of two wheels working together tend constantly to assume a form more and more perfect: as they abrade each other _while imperfect_, and cannot wear themselves _beyond perfection_.
5. For a similar reason the division of the teeth cannot remain unequal: for those that are too far distant from a given tooth will be _attacked behind_, and those that are too near before; so that the division also will finally become perfect.
But it must be remembered that these _recoveries_ of form are in their nature _very slow_; since the nearer the teeth come to perfection the slower is their approach to it: so that in thus dwelling on these properties, we do not advise the making of _bad_ wheels that they may become _good_; but only wish to destroy an _honest prejudice_ that has already much impeded the progress of the System; namely, that it requires great nicety to adjust them so as to work together at all: which is--(to say the least) a very great error.
In Plate 14, fig. 1, I have shewn the apparatus presented at the Exchange, as mentioned in page 110 preceding. _A B_ is the stand; _C D_ is a disk turning on the centre _E_; _b a_ is the transparent line cut through the stand, and representing the place of contact of two wheels geering together. It is there seen, (supposing the disk to turn in the direction of the arrow) that the action of the teeth, is always progressive _along_ the transparent line _a b_; whether the single or double obliquity _G_ or _F_ be used. In reality, the lower end of any tooth _c_, does not uncover the line _a b_, till the upper point of the succeeding tooth _d_ has begun to cover it; whereas, observing a few of the common teeth represented at _H_, as directed to the centre of the disk, _they_ would be seen to pass the line _a b_ all at once; and thus to represent, with a certain exaggeration, the transient manner of acting of the common geering.
Some knowledge of the nature of this geering may be gathered from its very appearance: see fig. 5 Plate 14. To represent these teeth properly, no light must appear between them. The tops of the teeth offer a continued circular line, similar to what it would be if there were no teeth at all: and the latter are distinguished only by a different shading of their front and lateral surfaces. The reason (as has been already observed) is, that they are necessarily so placed, as that the _last_ end of any tooth shall not quit the plane of the centres, until the _first_ end of the succeeding tooth arrives at it; which principle precludes the possibility of any space remaining between the teeth, that an eye directed _parallelly_ to the axes could penetrate. Such a space indeed would introduce a portion of the properties of the old geering, which it is the object of this System to avoid. As this wheel then _appears_ in fig. 5, _so it acts_: that is equally and perpetually.
It were well also to observe the appearance of these wheels on their edges; or in the planes which, as wheels they occupy. The 4th. figure of this Plate is outlined with some care, in order to shew the varying, and seemingly anomalous form which the teeth assume as they approach the boundaries of the figure. Although cut as obliquely to the axis there, as any where else, the receding cylindrical surface, thus seen, appears to take this obliquity away; and the _very_ outward teeth seem nearly parallel to the axis of the wheel. But this is only appearance: and we give here _one_ example of it, that we may not be obliged to lose much time hereafter, in drawing correctly, wheels on this principle--a process indeed which in many cases, would be found very difficult, if not impossible.
We have already adverted to the oblique tendencies of these wheels, when used with a single inclination of the teeth; from which, among other things, it follows that, in the act of urging the shafts endwise, they tend also to bend these shafts: for which reason the shafts require to be stronger than those of common wheels--that is, when the effort bears any proportion to their stiffness--a circumstance which, in light rapid movements, is of small moment. And in heavier works, when it is desirable to get rid of these tendencies altogether, we have peremptory means of avoiding the very appearance of this evil.
Suppose then (fig. 2 and 3, plate 14) _a b_ to be a straight rack on this principle; driven by the wheel or pinion _c_. The motion, backward, of the pinion, tends, clearly, to urge the pinion endwise towards _d_, and the rack sideways towards _a b_. But either of these motions is prevented by fixing to the pinion, or the rack, a _cheek_ _e f_, to support them against this lateral pressure. But then, exclaims a doubting friend, you introduce _friction_: and it is true: there is now a real rubbing of the _ends_ of the teeth against this cheek; but the pressure there being only one quarter of what it would be on the front of straight teeth, we avoid (on a rough estimate) three quarters of the friction; while preserving _all_ the constancy and smoothness of motion which the system gives; and which after all, is the most important part of the business.
This idea then applies among other things to the racks of slide-lathes; giving a regular motion to the _rest and cutting tool_, thereby adding to the perfection of the turning process: and many other cases might be adduced.
But instead of using a rack and pinion, as thus described, _two wheels_, of any desired proportions might have been thus treated, and the result would have been the same. _They_ would have worked with perfect smoothness, under about one quarter of the friction attendant upon common wheels in similar circumstances. There are cases therefore, in which it would be expedient thus to employ the System. I cannot but observe likewise, that this method of using _cheeks_ to prevent any side motion in spur wheels, might also be applied to bevel-wheels, to prevent the _angular_ tendency which the obliquity of their teeth gives them: and that I prefer such a method of obviating this evil (where it is one) to any attempt at using teeth in the V form, on bevel wheels. Still however, as before observed, this counteraction of the oblique tendencies is not always necessary. It may be dispensed with in all light and rapid movements; especially in the use of perpendicular shafts; and where the _driven_ wheels are small and distributed round a central wheel in positions nearly opposite each other: of all which cases we shall see examples in the spinning machinery to be described hereafter.
OF THE CUTTING ENGINE, _To form Spur-wheels, on my late Patent principle_.
The figures of this Engine (see Plates 15 and 16,) are drawn to a scale, from the Machine itself, now before me. The scale of the objects on Plate 15, is one inch and three quarters to the foot; and that of the objects on Plate 16, one inch and one third. These were convenient proportions for introducing this object into the present work; but the size itself of the Machine is arbitrary. I did not make it according to my ideas of the _best_ dimensions: but bought it as a common cutting Engine, and gave it those other properties that my System required.
The first remarkable deviation from the usual form is in the shaft or axis of the dividing plate. See fig. 1 and 2 of the Plates 15 and 16. The dividing plate _a b_, is concentric with, and fixed to an axis _A B_ made as perfectly cylindrical as possible, so as _both to slide and turn_ in the bars _C D_, and _E F_ composing the frame. These bars are _bushed_, to fit the axis _A B_, either with a contracting ring of brass, as usual in some mathematical instruments; or with _type metal_, cast around the axis into _rough_ holes in those bars:--which metal, closing upon the axis makes a good centre; and will last a long time. My Engine is made in this manner; and has been renewed in this part only twice in several years. This frame _C D E F_ of the Engine, is strongly connected with the feet _G G H H_, by means of the nuts _E F_ in the plan: and by these feet it is fixed to its bench or table, as will be seen in Plate 16.