Chapter III.—The Teeth of Gear-Wheels (continued)

The revolving cutters employed in gear-cutting machines, gear-cutters, or cutting engines (as the machines for cutting the teeth of gear-wheels to shape are promiscuously termed), are of the form shown in Fig. 107, which represents what is known as a Brown and Sharpe patent cutter, whose peculiarities will be explained presently. This class of cutters is made as follows:—

A cast steel disk is turned in the lathe to the required form and outline. After turning, its circumference is serrated as shown, so as to provide protuberances, or teeth, on the face of which the cutting edges may be formed. To produce a cutting edge it is necessary that the metal behind that edge should slope or slant away leaving the cutting edge to project. Two methods of accomplishing this are employed: in the first, which is that embodied in the Brown and Sharpe system, each tooth has the curved outline, forming what may be termed its circumferential outline, of the same curvature and shape from end to end, and from front to back, as it may more properly be termed, the clearance being given by the back of the tooth approaching the centre of the cutter, so that if a line be traced along the circumference of a tooth, from the cutting edge to the back, it will approach the centre of the cutter as the back is approached, but the form of the tooth will be the same at every point in the line. It follows then that the radial faces of the teeth may be ground away to sharpen the teeth without affecting the shape of the tooth, which being made correct will remain correct.

This not only saves a great deal of labor in sharpening the teeth, but also saves the softening and rehardening process, otherwise necessary at each resharpening.

The ordinary method of producing the cutting edges after turning the cutter and serrating it, is to cut away the metal with a file or rotary cutter of some kind forming the cutting edge to correct shape, but paying no regard to the shape of the back of the tooth more than to give it the necessary amount of clearance. In this case the cutter must be softened and reset to sharpen it. To bring the cutting edge up to a sharp edge all around its profile, while still preserving the shape to which it was turned, the pantagraphic engine, shown in Fig. 108, has been made by the Pratt and Whitney Company. Figs. 109 and show some details of its construction. “The milling cutter n is driven by a flexible train acting upon the wheel o, whose spindle is carried by the bracket b, which can slide from right to left upon the piece b, and this again is free to slide in the frame f. These two motions are in horizontal planes, and perpendicular to each other.

“The upper end of the long lever p c is formed into a ball, working in a socket which is fixed to p c. Over the cylindrical upper part of this lever slides an accurately fitted sleeve d, partly spherical externally, and working in a socket which can be clamped at any height on the frame f. The lower end p of this lever being accurately turned, corresponds to the roller p in Fig. 109, and is moved along the edge of the template t, which is fastened in the frame in an invariable position.

“By clamping d at various heights, the ratio of the lever arms p d, p d, may be varied at will, and the axis of n made to travel in a path similar to that of the axis of p, but as many times smaller as we choose; and the diameter of n must be made less than that of p in the same proportion.

“The template being on the left of the roller, the cutter to be shaped is placed on the right of n, as shown in the plan view at z, because the lever reverses the movement.

“This arrangement is not mathematically perfect, by reason of the angular vibration of the lever. This is, however, very small, owing to the length of the lever; it might have been compensated for by the introduction of another universal joint, which would practically have introduced an error greater than the one to be obviated, and it has, with good judgment, been omitted.

“The gear-cutter is turned nearly to the required form, the notches are cut in it, and the duty of the pantagraphic engine is merely to give the finishing touch to each cutting edge, and give it the correct outline. It is obvious that this machine is in no way connected with, or dependent upon, the epicycloidal engine; but by the use of proper templates it will make cutters for any desired form of tooth; and by its aid exact duplicates may be made in any numbers with the greatest facility.

“It forms no part of our plan to represent as perfect that which is not so, and there are one or two facts, which at first thought might seem serious objections to the adoption of the epicycloidal system. These are:

“1. It is physically impossible to mill out a concave cycloid, by any means whatever, because at the pitch line its radius of curvature is zero, and a milling cutter must have a sensible diameter.

“2. It is impossible to mill out even a convex cycloid or epicycloid, by the means and in the manner above described.

“This is on account of a hitherto unnoticed peculiarity of the curve at a constant normal distance from the cycloid. In order to show this clearly, we have, in Fig. 110, enormously exaggerated the radius c d, of the milling cutter (m of Figs. 105 and ). The outer curve h l, evidently, could be milled out by the cutter, whose centre travels in the cycloid c a; it resembles the cycloid somewhat in form, and presents no remarkable features. But the inner one is quite different; it starts at d, and at first goes down, inside the circle whose radius is c d, forms a cusp at e, then begins to rise, crossing this circle at g, and the base line at f. It will be seen, then, that if the centre of the cutter travel in the cycloid a c, its edge will cut away the part g e d, leaving the template of the form o g i. Now if a roller of the same radius c d, be rolled along this edge, its centre will travel in the cycloid from a, to the point p, where a normal from g, cuts it; then the roller will turn upon g as a fulcrum, and its centre will travel from p to c, in a circular arc whose radius g p = c d.

“That is to say even a roller of the same size as the original milling cutter, will not retrace completely the cycloidal path in which the cutter travelled.

“Now in making a rack template, the cutter, after reaching c, travels in the reversed cycloid c r, its left-hand edge, therefore, milling out a curve d k, similar to h l. This curve lies wholly outside the circle d i, and therefore cuts o g at a point between f and g, but very near to g. This point of intersection is marked s in Fig. 110, where the actual form of the template o s k is shown. The roller which is run along this template is larger, as has been explained, than the milling cutter. When the point of contact reaches s (which so nearly corresponds to g that they practically coincide), this roller cannot now swing about s through an angle so great as p g c of Fig. 110; because at the root d, the radius of curvature of d k is only equal to that of the cutter, and g and s are so near the root that the curvature of s k, near the latter point, is greater than that of the roller. Consequently there must be some point u in the path of the centre of the roller, such, that when the centre reaches it, the circumference will pass through s, and be also tangent to s k. Let t be the point of tangency; draw s u and t u, cutting the cycloidal path a r in x and y. Then, u y being the radius of the new milling cutter (corresponding to n of Fig. 109), it is clear that in the outline of the gear cutter shaped by it, the circular arc x y will be substituted for the true cycloid.