Let it now be required to lay out twenty-four teeth upon this ellipse; that is, six in each quadrant; and for symmetry's sake we will suppose that the centre of one tooth is to be at A, and that of another at C, Figure 253. We, therefore, divide L G into six equal parts at the points 1, 2, 3, etc., which will be the centres of the teeth upon the rectified ellipse. It is practically necessary to make the spaces a little greater than the teeth; but if the greatest attainable exactness in the operation of the wheels is aimed at, it is important to observe that backlash, in elliptical gearing, has an effect quite different from that resulting in the case of circular wheels. When the pitch-curves are circles, they are always in contact; and we may, if we choose, make the tooth only half the breadth of the space, so long as its outline is correct. When the motion of the driver is reversed, the follower will stand still until the backlash is taken up, when the motion will go on with a perfectly constant velocity ratio as before. But in the case of two elliptical wheels, if the follower stand still while the driver moves, which must happen when the motion is reversed if backlash exists, the pitch-curves are thrown out of contact, and, although the continuity of the motion will not be interrupted, the velocity ratio will be affected. If the motion is never to be reversed, the perfect law of the velocity ratio due to the elliptical pitch-curve may be preserved by reducing the thickness of the tooth, not equally on each side, as is done in circular wheels, but wholly on the side not in action. But if the machine must be capable of acting indifferently in both directions, the reduction must be made on both sides of the tooth: evidently the action will be slightly impaired, for which reason the backlash should be reduced to a minimum. Precisely what is the minimum is not so easy to say, as it evidently depends much upon the excellence of the tools and the skill of the workman. In many treatises on constructive mechanism it is variously stated that the backlash should be from one-fifteenth to one-eleventh of the pitch, which would seem to be an ample allowance in reasonably good castings not intended to be finished, and quite excessive if the teeth are to be cut; nor is it very obvious that its amount should depend upon the pitch any more than upon the precession of the equinoxes. On paper, at any rate, we may reduce it to zero, and make the teeth and spaces equal in breadth, as shown in the figure, the teeth being indicated by the double lines. Those upon the portion L H are then laid off upon K I, after which these divisions are transferred to the ellipse by the second of Prof. Rankine's constructions, and we are then ready to draw the teeth.
The outlines of these, as of any other teeth upon pitch-curves which roll together in the same plane, depend upon the general law that they must be such as can be marked out upon the planes of the curves, as they roll by a tracing-point, which is rigidly connected with and carried by a third line, moving in rolling contact with both the pitch-curves. And since under that condition the motion of this third line, relatively to each of the others, is the same as though it rolled along each of them separately while they remained fixed, the process of constructing the generated curves becomes comparatively simple. For the describing line we naturally select a circle, which, in order to fulfil the condition, must be small enough to roll within the pitch ellipse; its diameter is determined by the consideration that if it be equal to A P, the radius of the arc A F, the flanks of the teeth in that region will be radial. We have, therefore, chosen a circle whose diameter, A B, is three-fourths of A P, as shown, so that the teeth, even at the ends of the wheels, will be broader at the base than on the pitch line. This circle ought strictly to roll upon the true elliptical curve; and assuming, as usual, the tracing-point upon the circumference, the generated curves would vary slightly from true epicycloids, and no two of those used in the same quadrant of the ellipse would be exactly alike. Were it possible to divide the ellipse accurately, there would be no difficulty in laying out these curves; but having substituted the circular arcs, we must now roll the generating circle upon these as bases, thus forming true epicycloidal teeth, of which those lying upon the same approximating arc will be exactly alike. Should the junction of two of these arcs fall within the breadth of a tooth, as at D, evidently both the face and the flank on one side of that tooth will be different from those on the other side; should the junction coincide with the edge of a tooth, which is very nearly the case at F, then the face on that side will be the epicycloid belonging to one of the arcs, its flank a hypocycloid belonging to the other; and it is possible that either the face or the flank on one side should be generated by the rolling of the describing circle partly on one arc, partly on the one adjacent, which, upon a large scale, and where the best results are aimed at, may make a sensible change in the form of the curve.
The convenience of the constructions given in Figure 252 is nowhere more apparent than in the drawing of the epicycloids, when, as in the case in hand the base and generating circles may be of incommensurable diameters; for which reason we have, in Figure 254, shown its application in connection with the most rapid and accurate mode yet known of describing those curves. Let C be the centre of the base circle; B, that of the rolling one; A, the point of contact. Divide the semi-circumference of B into six equal parts at 1, 2, 3, etc.; draw the common tangent at A, upon which rectify the arc A 2 by process No. 1; then by process No. 2 set out an equal arc A 2 on the base circle, and stepping it off three times to the right and left, bisect these spaces, thus making subdivisions on the base circle equal in length to those on the rolling one. Take in succession as radii the chords A 1, A 2, A 3, etc., of the describing circle, and with centres 1, 2, 3, etc., on the base circle, strike arcs either externally or internally, as shown respectively on the right and left; the curve tangent to the external arcs is the epicycloid, that tangent to the internal ones the hypocycloid, forming the face and flank of a tooth for the base circle.
Fig. 255.
In the diagram, Figure 253, we have shown a part of an ellipse whose length is ten inches, and breadth six, the figure being half size. In order to give an idea of the actual appearance of the combination when complete, we show in Figure 255 the pair in gear, on a scale of three inches to the foot. The excessive eccentricity was selected merely for the purpose of illustration. Figure 255 will serve also to call attention to another serious circumstance, which is, that although the ellipses are alike, the wheels are not; nor can they be made so if there be an even number of teeth, for the obvious reason that a tooth upon one wheel must fit into a space on the other; and since in the first wheel, Figure 255, we chose to place a tooth at the extremity of each axis, we must in the second one place there a space instead; because at one time the major axes must coincide; at another, the minor axes, as in Figure 255. If, then, we use even numbers, the distribution, and even the forms of the teeth, are not the same in the two wheels of the pair. But this complication may be avoided by using an odd number of teeth, since, placing a tooth at one extremity of the major axes, a space will come at the other.
It is not, however, always necessary to cut teeth all round these wheels, as will be seen by an examination of Figure 256, C and D being the fixed centres of the two ellipses in contact at P. Now P must be on the line C D, whence, considering the free foci, we see that P B is equal to P C, and P A to P D; and the common tangent at P makes equal angles with C P and P A, as is also with P B and P D; therefore, C D being a straight line, A B is also a straight line and equal to C D. If then the wheels be overhung, that is, fixed on the ends of the shafts outside the bearings, leaving the outer faces free, the moving foci may be connected by a rigid link A B, as shown.
