Modern Machine-Shop Practice Part 8

[Ill.u.s.tration: Fig. 107.]

A cast steel disk is turned in the lathe to the required form and outline. After turning, its circ.u.mference 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 circ.u.mferential 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 circ.u.mference 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 110 show some details of its construction.[5] "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.

[5] From "The Teeth of Spur Wheels," by Professor McCord.

[Ill.u.s.tration: Fig. 108.]

"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.

[Ill.u.s.tration: Fig. 109.]

"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.

[Ill.u.s.tration: Fig. 110.]

"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 106). 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 circ.u.mference will pa.s.s 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 subst.i.tuted for the true cycloid.

[Ill.u.s.tration: Fig. 111.]

THE SYSTEM PRACTICALLY PERFECT.

"The above defects undeniably exist; now, what do they amount to? The diagram is drawn purposely with these sources of error greatly exaggerated, in order to make their nature apparent and their existence sensible. The diameters used in practice, as previously stated, are: describing circle, 7-1/2 inches; cutter for shaping template, 1/8 of an inch; roller used against edge of template, 1-1/8 inches; cutter for shaping a 1-pitch gear cutter, 1 inch.

"With these data the writer has found that the _total length_ of the arc X Y of Fig. 110, which appears instead of the cycloid in the outline of a cutter for a 1-pitch rack, is less than 0.0175 inch; the real _deviation_ from the true form, obviously, must be much less than that.

It need hardly be stated that the effect upon the velocity ratio of an error so minute, and in that part of the contour, is so extremely small as to defy detection. And the best proof of the practical perfection of this system of making epicycloidal teeth is found in the smoothness and precision with which the wheels run; a set of them is shown in gear in Fig. 111, the rack gearing as accurately with the largest as with the smallest. To which is to be added, finally, that objection taken, on whatever grounds, to the epicycloidal form of tooth, has no bearing upon the method above described of producing duplicate cutters for teeth of any form, which the pantagraphic engine will make with the same facility and exactness, if furnished with the proper templates.

"The front faces of the teeth of rotary cutters for gear-cutting are usually radial lines, and are ground square across so as to stand parallel with the axis of the cutter driving spindle, so that to whatever depth the cutter may have entered the wheel, the whole of the cutting edge within the wheel will meet the cut simultaneously. If this is not the case the pressure of the cut will spring the cutter, and also the arbor driving it, to one side. Suppose, for example, that the tooth faces not being square across, one side of the teeth meets the work first, then there will be as each tooth meets its cut an endeavour to crowd away from the cut until such time as the other side of the tooth also takes its cut."

It is obvious that rotating cutters of this cla.s.s cannot be used to cut teeth having the width of the s.p.a.ce wider below than it is at the pitch line. Hence, if such cutters are required to be used upon epicycloidal teeth, the curves to be theoretically correct must be such as are due to a generating circle that will give at least parallel flanks. From this it becomes apparent that involute teeth being always thicker at the root than at the pitch line, and the s.p.a.ces being, therefore, narrower at the root, may be cut with these cutters, no matter what the diameter of the base circle of the involute.

To produce with revolving cutters teeth of absolutely correct theoretical curvature of face and flank, it is essential that the cutter teeth be made of the exact curvature due to the diameter of pitch circle and generating circle of the wheel to be cut; while to produce a tooth thickness and s.p.a.ce width, also theoretically correct, the thickness of the cutter must also be made to exactly answer the requirements of the particular wheel to be cut; hence, for every different number of teeth in wheels of an equal pitch a separate cutter is necessary if theoretical correctness is to be attained.

This requirement of curvature is necessary because it has been shown that the curvatures of the epicycloid and hypocycloid, as also of the involute, vary with every different diameter of base circle, even though, in the case of epicycloidal teeth, the diameter of the generating circle remain the same. The requirement of thickness is necessary because the difference between the arc and the chord pitch is greater in proportion as the diameter of the base or pitch circle is decreased.

But the difference in the curvature on the short portions of the curves used for the teeth of fine pitches (and therefore of but little height) due to a slight variation in the diameter of the base circle is so minute, that it is found in practice that no sensible error is produced if a cutter be used within certain limits upon wheels having a different number of teeth than that for which the cutter is theoretically correct.

The range of these limits, however, must (to avoid sensible error) be more confined as the diameter of the base circle (or what is the same thing, the number of the teeth in the wheel) is decreased, because the error of curvature referred to increases as the diameters of either the base or the generating circles decrease. Thus the difference in the curve struck on a base circle of 20 inches diameter, and one of 40 inches diameter, using the same diameter of generating circle, would be very much less than that between the curves produced by the same diameter of generating circle on base circles respectively 10 and 5 inches diameter.

For these reasons the cutters are limited to fewer wheels according as the number of teeth decreases, or, per contra, are allowed to be used over a greater range of wheels as the number of teeth in the wheels is increased.

Thus in the Brown and Sharpe system for involute teeth there are 8 cutters numbered numerically (for convenience in ordering) from 1 to 8, and in the following table the range of the respective cutters is shown, and the number of teeth for which the cutter is theoretically correct is also given.

BROWN AND SHARPE SYSTEM.

No. of cutter. Involute teeth. Teeth.

1 Used upon all wheels having from 135 teeth to a rack correct for 200 2 " " " " " 55 " to 134 teeth, 68 3 " " " " " 35 " to 54 " 40 4 " " " " " 26 " to 34 " 29 5 " " " " " 21 " to 25 " 22 6 " " " " " 17 " to 20 " 18 7 " " " " " 14 " to 16 " 16 8 " " " " " 12 " to 14 " 13

Suppose that it was required that of a pair of wheels one make twice the revolutions of the other; then, knowing the particular number of teeth for which the cutters are made correct, we may obtain the nearest theoretically true results as follows: If we select cutters Nos. 8 and 4 and cut wheels having respectively 13 and 26 teeth, the 13 wheel will be theoretically correct, and the 26 will contain the minute error due to the fact that the cutter is used upon a wheel having three less teeth than the number it is theoretically correct for. But we may select the cutters that are correct for 16 and 29 teeth respectively, the 16th tooth being theoretically correct, and the 29th cutter (or cutter No. 4 in the table) being used to cut 32 teeth, this wheel will contain the error due to cutting 3 more teeth than the cutter was made correct for.

This will be nearer correct, because the error is in a larger wheel, and, therefore, less in actual amount. The pitch of teeth may be selected so that with the given number of teeth the diameters of the wheels will be that required.

We may now examine the effect of the variation of curvature in combination with that of the thickness, upon a wheel having less and upon one having more teeth than the number in the wheel for which the cutter is correct.

First, then, suppose a cutter to be used upon a wheel having less teeth and it will cut the s.p.a.ces too wide, because of the variation of thickness, and the curves too straight or insufficiently curved because of the error of curvature. Upon a wheel having more teeth it will cut the s.p.a.ces too narrow, and the curvature of the teeth too great; but, as before stated, the number of wheels a.s.signed to each cutter may be so apportioned that the error will be confined to practically unappreciable limits.

If, however, the teeth are epicycloidal, it is apparent that the s.p.a.ces of one wheel must be wide enough to admit the teeth of the other to a depth sufficient to permit the pitch lines to coincide on the line of centres; hence it is necessary in small diameters, in which there is a sensible difference between the arc and the chord pitches, to confine the use of a cutter to the special wheel for which it is designed, that is, having the same number of teeth as the cutter is designed for.

Thus the Pratt and Whitney arrangement of cutters for epicycloidal teeth is as follows:--

PRATT AND WHITNEY SYSTEM.

EPICYCLOIDAL TEETH.

[All wheels having from 12 to 21 teeth have a special cutter for each number of teeth.][6]

Cutter correct for No. of teeth.

23 Used on wheels having from 22 to 24 teeth.

25 " " " " 25 to 26 "

27 " " " " 26 to 29 "

30 " " " " 29 to 32 "

34 " " " " 32 to 36 "

38 " " " " 36 to 40 "

43 " " " " 40 to 46 "

50 " " " " 46 to 55 "

60 " " " " 55 to 67 "

76 " " " " 67 to 87 "