Teeth of Wheels. Oil for Clocks.
Teeth of Wheels
xBefore explaining the construction of the largest clock in the world it is necessary to consider the shape of wheel teeth suitable for different purposes, and also of the cams requisite to raise heavy hammers, which had been too much neglected by clockmakers previously. At the same time we are not going to wrote a treatise on all the branches of the important subject of wheel-cutting; but assuming a knowledge of the general principles of it, to apply them to the points chiefly involved in clock-making. The most comprehensive mathematical view of it is perhaps to be found in a paper by the astronomer royal in the Cambridge Transactions many years ago, which is further expanded in Professor Willis Principles of Mechanism. Respecting the latter book, however, we should advise the reader to be content with the mathematical rules there given which are very simple, without attending much to those of the odontograph, which seem to give not less but more trouble than the mathematical, and are only approximate after all, and also do not explain themselves, or convey any knowledge of the principle to those who use them.
For all wheels that are to work together, the first thing to do is to fix the geometrical, or primitive, or pitch circles of the two wheels, i.e., the two circles which, if they rolled perfectly together, would give the velocity-ratio you want. Draw a straight line joining the two centres; then the action which takes place between any two teeth as they are approaching that line of centres. Now, with a view to reduce the friction, it is essential to have as little action before the line of centres as you can; for if you make any rude sketch, on a large scale, of a pair of wheels acting together, and serrate the edges of the teeth (which is an exaggeration of the roughness which produces friction), you will see that the further the contact begins before the line of centres, the more the serration will interfere with the motion, and that at a certain distance no force whatever could drive the wheels, but would only jam the teeth faster; and you will see also that this cannot happen after the line of centres. But with pinions of the numbers generally use in clocks you cannot always get rid of action before the line of centres; for it may be proved (but the proof is too long to give here), that if a pinion has less than 11 leaves, no wheel of any number of teeth can drive it without some action before the line of centres. And generally it may be stated that the greater the number of teeth the less friction there will be, as indeed be evident enough from considering that if the teeth were infinite in number, and infinitesimal in size, there would be no friction at all, but simple rolling of one pitch circle on the other. And since in clock-work the wheels always drive the pinions, except the hour pinion in the dial work, and the winding pinions in large clocks, it has long been recognized as important to have high numbered pinions, except where there is a train remontoire, or gravity escapement, to obviate that necessity.
And with regard to this matter, the art of clock-making has in one sense retrograded; for the pinions which are now almost university used in English and French clocks are of a worse form than those of several centuries ago, to which we have several times alluded under the name of lantern pinions, so called from their resembling a lantern with upright ribs. A sketch of one, with a cross section on a large scale, is given at fig. 24. Now it is a property of these pinions, that when they are driven, the action begins juts when the centre of the pin is on the line of centres, however few the pins may be; and thus the action of a lantern pinion of 6 is about equal to that of a leaved pinion of 10; and indeed, for some reason or other, it appears in practice to be even better, possibly from the teeth of the wheel not requiring to be cut so accurately, and from the pinion never getting clogged with dirt. Certainly the running of the American clocks, which all have these pinions, is remarkably smooth, and they require a much smaller going weight than English clocks; and the same may be said of common "Dutch," i.e., German clocks. It should be understood, however, that as the action upon these pinions is all after the line of centres when they are driven, it will be all before the line of centres if they drive, and therefore they are not suitable for that purpose. In some of the French clocks in the 1851 Exhibition they were wrongly used, not only for train, but for winding pinions; and some of them also had the pins not fixed in the lantern, but rolling,a very useless refinement, and considerably diminishing the strength of the pinion. For it is one of the advantages of lantern pinions with fixed pins, that they are very strong, and there is no risk of their being broken in hardening, as there is with common pinions.
The fundamental rule for the tracing of teeth, though very simple, is not so well known as it ought to be, and therefore we will give it, premising that so much of a tooth as lies within the pitch circle of the wheel is called its root or flank, and the part beyond the pitch circle is called the point, or the curve, or the addendum; and moreover, that before the line of centres the action is always between the flanks of the driver and the points of the driven wheel or runner (as it may be called, more appropriately than the usual term follower); and after the line of centres, the action is always between the points of the driver and the flanks of the runner. VConsequently the points of the driver and the flanks of the runner. Consequently, if there is no action before the line of centres, no points are required for the teeth of the runner.
In fig. 23, let AOK be the pitch circle of runner, and ARY that of the driver; and let GAP be any curve whatever of smaller curvature than AQX (of course a circle is always the kind of curve used); and QP the curve which is traced out by any point P in the generating circle GAP, as it rolls in the pitch circle AQX; and again let RP be the curve traced by the point P, as the generating circle GAP is rolled on the pitch circle ARY; then RP will be the form of the point of a tooth on the driver ARY, which will drive with uniform and proper motion the flank QP of the runner; though not without some friction, because that can only be done with involute teeth, which are traced a different way, and are subjec6t to other conditions, rendering them practically useless for machinery , as may be seen in Professor Williss book. If the motion is reversed, so that the runner become the driver, then the flank QP is of the proper form to drive the point RP, if any action has to take place before the line of centres.
And again, any generating curve, not even necessarily the same as before, may be sued to trace the flanks of the driver an the points of the runner, by being rolled within the circle ARY, and on the circle AQX.
Now the, to apply this rule is particular cases. Suppose the generating circle is the same as the pitch circle of the driven pinion itself, it evidently cannot roll at all; and the tooth of the pinion is represented by the mere point P on the circumference of the pitch circle; and the tooth to drive it will be simply an epicycloids traced by rolling the pitch circle of the pinion on that of the wheel. And we know that in that case there is no action before the line of centres, and no necessity for any flanks on the teeth of the driver. But inasmuch as the pins of a lantern pinion must have some thickness, and cannot be mere lines, a further process is necessary to get the exact form of the teeth; thus if RP, fig. 24, is the tooth that would drive a pinion with pins of no sensible thickness, the tooth to drive a pin of the thickness 2 Pp must have the width Pp or Rr ganged off it all round. This, in fact, brings it very nearly to a smaller tooth traced with the same generating circle; and therefore in practice this mode of construction is not much adhered to, and the teeth are made of the same shape, only thinner, as if the pins of the pinion had no thickness. Of course they should be thin enough to allow a little shake, or "back-lash," but in clock-work the backs of the teeth never come in contact at all.
Next suppose the generating circle to be half the size pf the pitch circle of the pinion. The curve, or hypocycloid, traced by rolling this within the pinion, is no other than the diameter of the pinion; and constantly the flanks of the pinion teeth will be merely radii of it, and such teeth or leaves are called radical teeth; and they are far the most common; indeed, no others are ever made (except lanterns) for clock-work. The corresponding epicycloidal points of the teeth of the driver are more curved, or a less pointed arc, than those required for a lantern pinion of the same size and number. The teeth in fig. 25 are made of a different from on the opposite sides of the line of centres CA, in order to show the difference between driving and driven or running teeth, when the number of the pinion happens to be as much as 12, so that no points are required to its teeth when driven, since with that number all the action may be after the line of centres. The great Westminster clock affords a very good illustration of this. In both the striking parts the great wheel of the train and the great winding-wheel on the other end of the barrel are about the same size; but in the train the wheel drives, and in winding the pinion drives. And therefore in the train the pinion-teeth have their points cut off, and wheel-teeth have their points on, as on the right side of fig. 25, and in the winding-wheels the converse; and thus in both cases the action is made to take place in the way in which there is the least friction. Willis gives the following table, "derived organically" (i.e., by actual trial with large models), of the least numbers which will work together without any action before the line of centres, provided there are no points to the teeth of the runner, assuming them to be radial teeth, as usual:
54 30 24 20 17 15 14 13 12 11 10 9 8 7 6
Runner..11 12 13 14 151 16 171 18 19 21 23 26 35 32 176
In practice it is hardly safe to leave the driven teeth without points, unless the numbers slightly exceed these; because, if there is any irregularity in them, the square edges of those teeth would not work smoothly with the teeth of the driver. Sometimes it happens that the same wheel has to drive two pinions of different numbers. It is evident that, if both are lanterns, or both pinions with radial teeth, they cannot properly be driven by the same wheel, because they would require teeth of a different shape. It is true that on account of the greater indifference of lantern pinions to the accuracy of the teeth which are to drive them, the same wheel will drive two pinions of that kind, differing in the numbers in the ratio of even 2 to 1, with hardly any sensible shake; but that would not be so with radial pinions, and of course it is not correct. Accordingly, in clocks with the spring remontoire, as in fig. 21, where the scape-wheel or remontoire pinion is double the size of the fly pinion, the larger one is made with radial teeth and the smaller a lantern, which makes the same wheel teeth exactly right for both. In clocks of the same construction as fig. 22, and in the Westminster clock, there is a case of a different kind, which cannot be so accommodated; for there the great wheel has to drive both the second wheels pinion of 10 or 12, and the hour-wheel of 40 or 48; the teeth of the great wheel were therefore made to suit the lantern pinion, and those of the hour-wheel (i.e., their flanks) then depend on those of the great wheel, and they were accordingly traced by rolling a generating circle of the size of the lantern pinion on the inside of the pitch circle of the hour-wheel; the results is a tooth thicker at the bottom than usual. These are by no means unnecessary refinements; for if the teeth of a set of wheels are not properly shaped so as to work smoothly and regularly into each other, it increases their tendency to wear out in proportion to their inaccuracy, besides increasing the inequalities of force in the train. Sometimes turrets clocks are worn out in a few years from the defects in their teeth, especially when they are made of brass or soft gun-metal.
In the construction of clocks which have to raise heavy hammers it is important to obtain the best form for the cams, as pins are quite unfit for the purpose. The conditions which are most important arethat the action should begin at the greatest advantage, and therefore at the end of the lever, that when it ceases the face of the lever should be a tangent to the cam at both their points, and that in no part of the motion should the end of the lever scrape on the cam. In the common construction of clocks the first condition is deviated from as far as possible, by the striking pins beginning to act at some distance from the end of the lever; and consequently, at the time when the most force is required to lift the hammer there is the least given, and a great deal is wasted afterwards.
The construction of curve for the cams, which is the most perfect mathematically, is that which is described in mathematical books under the name of the tractrix. But there are such practical difficulties in describing it that it is of no use. It should be observed that, in a well-known book with an approximate name (Camus on the Teeth of Wheels), a rule for drawing cams has been inserted by some translator, which is quite wrong. It may be proved that epicycloidal came described as follows are so nearly of the proper mathematical form that they may be used without any sensible error. Let r be the radius of the circle or barrel on which the cams are to be set theoretically, i.e., allowing nothing for the clearance which must be cut out afterwards, for fear the lever should scrape the back of the cams in falling; in other words, r is the radius of the pitch circle of the cams. Call the length of the lever. Then the epicycloidal cams may be traced by rolling on the pitch circle a smaller one whose diameter is &Mac195;¯¯r2 + l2 r. Thus, if l is 4 inches and r 8 inches (which is about the proper size for an 18-inch striking wheel with 20 cams), the radius of the tracing circle from the cams will be 0·9 inch. The advantage of cams of this kind is that they waste as little force as possible in the lift, and keep the lever acting upon them as a tangent at its point the whole way; and the came themselves may be of any length, according to the angle through which you want the lever to move.
More people however prefer dealing with circles, when they can, instead of epicycloids; and drawing by compasses is safer than calculating in most hands. We therefore give another rule, suggested by Mr E. J. Lawrence, a member of the horological jury in the 1851 Exhibition, which is easier to work, and satisfies the principal conditions states just now, though it wastes rather more in lift than the epicycloidal curve; an the cams must not have their points cur off, as epicycloidal ones may, to make the lever drop off sooner; because a short cam has to be drawn with a different radius from a long one, to work a lever of any given length. But, on the other hand, the same curve for the cams will suit a lever of any length, whereas with epicycloidal cams you must take care to put the centre or axis of the lever at the exact distance from the centre of the wheel for which the curve was calculatedan easy enough thing to do, of course, but for the usual disposition of workmen to deviate from your plans, apparently for the mere pleasure of doing wrong. It is astonishing how, by continually making one machine after another, with a little deviation each time, the thing gradually assumes a form in which you can hardly recognize your original design at all. The prevention of this kind of blundering is one of the many advantages of making machines by machinery, for which no machine offers more facilities than clocks, and yet there is none to which it is less applied.
In fig. 26 let CA be a radius of the wheel, L in the same straight line the centre of the lever, and AB the space of one cam on the pitch circle of the cams, A being a little below the line of centres; AP is the arc of the lever. Draw a tangent to the two circles at A, and a tangent to the cam circle at B; then T, their point of intersection, will be the centre of the circle which is the face of the cam BP; and TB also=TA, which is a convenient test of the tangents being rightly drawn. The action begins at the point of the lever, and advances a little way up, but receded again to the point and ends with the lever as a tangent to the cam of P. The backs of the cams must be cut out rather deeper than the circle AP, but retaining the point P, to allow enough for clearance of the lever, which should fall against some fixed stop or banking on the clock-frame, before the next cam reaches it. The point of the lever must not be left quite sharp, for if it is, it will in time cut off the points of the cast-iron cams.
Oil for Clocks
We will add a few words in the subject of oil for clocks. Olive-oil is most commonly used, sometimes purified in various ways, and sometimes not purified at all. We believe, however, that purified animal oil is better than any of the vegetable oils, as some of them are too thin, while others soon get thick and viscid. For turret clocks and common houses clocks, good sperm oil is fine enough, and is probably the best. For finer work the oil requires some purification. Even common neats foot oil may be made fine and clear by the following method. Mix it with about the same quantity of water, and shake it in a large bottle, not full, until it becomes like a white soup; then let it stand till fine oil appears at the top, which may be skimmed off; it will take several months before it has all separatedinto water at the bottom, dirt in the middle, and fine oil at the top. And it should be done in cold weather, because heat makes some oil come out as fine, which is cold would remain among the dirty oil in the middle, and in cold weather that fine oil of hot weather will become muddy. There are various vegetable oils sold at tool-shops as oil for watches, including some for which a prize medal was awarded in the Exhibition, but not by any of the mechanical juries; we have no information as to the test which was applied to it, and none but actual use for a considerable time would be of much value.
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