Wheels and Pinions
There are few watchmakers who have not some arbitrary rule for determining the size of wheels and pinions by which they can obtain the desired information. There is, however, more to this subject which is shrouded with more or less mystery to the average mechanic and horologist, viz., that after he has selected the diameter of the wheel or pinion, what kind of a tooth he is going to use in such wheel or pinion. There are two general requisites which should be borne in mind and which I am sure every watchmaker does bear in mind: (1) The uniform transmission of power, and (2) the uniform transmission of speed; or, in other words, how to obtain uniform efficiency of the mainspring and how to rotate the pinion at a uniform velocity in proportion to the respective diameters of the wheels and pinions. Starting with the idea that the average watchmaker knows how to compute the outer diameter of his wheel or pinion, we will take into consideration the watch in which the wheel or pinion is to be inserted. The first thing necessary to determine is, as to whether the watch has a fine or a coarse train, a slow or a quick train. When this conclusion is arrived at, there still remains to consider the thickness of the tooth. Assuming that it is a fine quick train watch, in which the center wheel has 80 teeth and the third pinion 10 teeth. In mechanical engineering it is assumed that 6 to 1 is the greatest ratio that can be maintained between an involute wheel and pinion, and 8 to 1 between an epicyclic wheel and pinion; but in watchmaking we brush these considerations aside and provide for larger differences in angular velocity in unique methods, which the size of this paper will not allow me to consider here.
As is well known in mechanical engineering--of which watchmaking is but a finer degree--when considering the driving train of gears, there are what is termed two kinds of friction, viz., an engaging friction, or friction which occurs when the wheel teeth engage before the line of centers; and a disengaging friction which occurs when the wheel teeth leave or recede from the line of centers. The first friction is the most dangerous in a watch, in that it is a kind of dragging movement, which, like the unwinding of a mainspring, is decidedly irregular, and consequently should be avoided as much as possible. The disengaging friction is what might be termed a pushing friction, and has, so far as can be observed by dynamometer tests, a certain ratio of uniformity, and is not so dangerous in the watch. Therefore, the first thing necessary to consider in designing the train is the thickness of the wheel tooth and pinion; and, of course, the outer diameter of the follower or driven pinion.
The 10-leaf pinion is practically the first pinion with which we can with some degree of assurance have engagement between it, and its wheel teeth, or what is termed the "lead" occur on the line of centers. Unfortunately, however, in watches this is not always practicable to obtain, in that it would require that the leaf be made very thin. For instance, dividing the 360 degrees--which is the pitch circumference--into ten equal parts, it allows 36 degrees for the circular pitch. To obtain a tooth of theoretical thickness to insure the "lead" beginning on the line of centers, 11 degrees should be taken for the thickness of the leaf and 25 for the space, making the face of the tooth--or that portion above the pitch line--practically a semi-circle; and as this is merely added for safety, it would insure a pinion in which the lead would take place at the line of centers. But these theoretical conditions could not be fulfilled, in that the tooth would be so thin and weak as to be practically unfit for the transmission of power, and also from the fact that the driving wheel would not drive the pinion one-tenth of a circumference, so that a certain amount of drop would take place and a consequent variation in the time-keeping qualities of the watch. To overcome these objections, however, it is safe, to add on a certain amount--1 1/2 of the circular pitch--taking 13 degrees for the tooth and 23 degrees for the space. Of course some discretion and judgment must be used in these matters, and especially in very small watches even this proportion would be too fine; but the tooth should be made as small as possible consistent with strength, and to prevent drop. This rule, however--or figures which I will give--will be found very satisfactory in watches from 18 to 10 size, inclusive. Below this, circumstances have to be taken into consideration and provided for accordingly. Summing up briefly, a rule which I found most practicable regarding thickness of tooth and pinions: Where the theoretical conditions cannot (always) be complied with for the pinion leaves--11 deg. for the tooth and 25 deg. for the space--practical experiment shows that dividing the circular pitch into 30 parts, and taking 11 for the tooth and 19 for the space, makes a good working pinion. To insure absolute freedom, the circular pitch of the wheel should be divided into 9 parts, taking 4 for the tooth and 5 for the space. Having determined the thickness of the tooth, the next thing to determine is what will be the curve of the tooth used in the wheel and pinion. Here, as in other problems, there are a multitude of conditions to be considered, the principal ones being: (1) Transmission of a substantially uniform power; (2) transmission of a practically uniform speed; and (3) the thickness of the tooth of the gear to which the curve is to be applied. Considering the two principal curves which have been advanced after years of experiment, viz., the involute and the cycloid, it has been demonstrated by large indicating levers attached to a dynamometer that the involute drives with the most uniform transmission of power, for the simple reason that it is a curve developed by the transforming of a circle into a straight line. In other words, it is developed like taking a cord that is wound around a spool and attaching a pencil to the free end, and then unwinding the cord. You will see that the cord becomes a straight line, and the transforming of this cord into a straight line forms a peculiar curve called an "involute." The peculiar feature of this curve is that it is peculiarly adapted to drive the straight flank of a tooth or lever (as such lever is always driven) at right angles to the curve; or, in other words, a tangent. In machine gearing this curve is almost universally used, from the fact that it is very simple to make, and the further fact that the main object required is the uniform transmission of power. In watchmaking, however, we have a further thing to consider, and that is the pinion or wheel must be driven in such a manner that it should travel through equal degrees of its arc or circumference in equal periods of time. For this purpose eminent engineers have found that the epicycloid or cycloid, as the case may be, is the curve best fitted, in that it is a curve that is developed by either rolling a circle on a straight line--cycloid--and tracing a curve from one point only; or it is a curve--epicycloid--developed by rolling a circle upon another circle and describing a curve with one point of such generating circle.
The accompanying diagram, Fig. 314; shows the method of generating an epicycloid curve. The segments of the circle A, represent a portion of the pitch circle of a third wheel. The circle B represents the pitch circle of a fourth pinion. The circles C represent a circle the diameter of a pitch radius, viz., the circle by which the curve 1 in section lines is developed; and the circles B represent circles of the diameter of the pitched radius of the third wheel, rolled by the pitch diameter to develop an epicycloid curve. The peculiar features of this epicycloidal curve is that it will drive the radial line or the flank of a tooth--that is, the portion of the tooth below the pitched circle--with a practically uniform velocity, and, as a consequence, it has been adopted by watchmakers as the proper curve to give to the tooth of a watch. Unfortunately, however, few watchmakers have taken into consideration that there are other conditions to be met in the train of wheels and pinions in a watch. A fixed or arbitrary diameter is selected for the wheel or pinion. An epicycloidal cutter is formed, and the wheel tooth is cut so as to meet in a point at the outer diameter. If it meets in such a point, the operator in charge is satisfied that his tooth is correct, in that he has cut his wheel with what he terms an "epicycloidal cutter." In order to get this form, however, he has varied the angle of his tooth and slightly altered the curve with a diamond lap, in order to get what he terms a proper "epicycloidal tooth," which very much resembles the shape of the tooth 4 in Fig. 315.