Fig. 196.
“The extreme simplicity of these two constructions and the facility with which they may be made with ordinary drawing instruments make them exceedingly convenient, and they should be more widely known than they are. Their application to the present problem is shown in [Fig. 196], which represents a quadrant of an ellipse, the approximate arcs c d, d e, e f, f a having been determined by trial and error. In order to space this off, for the positions of the teeth, a tangent is drawn at d, upon which is constructed the rectification of d c, which is d g, and also that of d e in the opposite direction, that is, d h, by the process just explained. Then, drawing the tangent at f, we set off in the same manner f i = f e, and f k = f a, and then measuring h l = i k, we have finally g l, equal to the whole quadrant of the ellipse.
“Let it now be required to lay out 24 teeth upon this ellipse; that is, 6 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, [Fig. 196]. We therefore divide l g into six equal parts at the points 1, 2, 3, &c., 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 wheel 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 ease 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 workmen. 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 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.
Fig. 197.
“The convenience of the constructions given in [Fig. 194] 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 [Fig. 197], 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, &c.; draw the common tangent at a, upon which rectify the arc a2 by process No. 1, then by process No. 2 set out an equal arc a2 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 a1, a2, a3, &c., of the describing circle, and with centres 1, 2, 3, &c., 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.