Modern Machine-Shop Practice Part 7

Involute teeth possess four great advantages--1st, they are thickest at the roots, where they should be to have a maximum of strength, which is of great importance in pinions transmitting much power; 2nd, the action of the teeth will remain practically perfect, even though the wheels are spread apart so that the pitch circles do not meet on the line of centres; 3rd, they are much easier to mark, and truth in the marking is easier attained; and 4th, they are much easier to cut, because the full depth of the teeth can, on spur-wheels, in all cases be cut with one revolving cutter, and at one pa.s.sage of the cutter, if there is sufficient power to drive it, which is not the case with epicycloidal teeth whenever the flank s.p.a.ce is wider below than it is at the pitch circle. On account of the first-named advantage, they are largely employed upon small gears, having their teeth cut true in a gear-cutting machine; while on account of the second advantage, interchangeable wheels, which are merely required to transmit motion, may be put in gear without a fine adjustment of the pitch circle, in which case the wear of the teeth will not prove destructive to the curves of the teeth. Another advantage is, that a greater number of teeth of equal strength may be given to a wheel than in the epicycloidal form, for with the latter the s.p.a.ce must at least equal the thickness of the tooth, while in involute the s.p.a.ce may be considerably less in width than the tooth, both measured, of course, at the pitch circle. There are also more teeth in contact at the same time; hence, the strain is distributed over more teeth.

[Ill.u.s.tration: Fig. 96.]

These advantages a.s.sume increased value from the following considerations.

In a train of epicycloidal gearing in which the pinion or smallest wheel has radial flanks, the flanks of the teeth will become spread as the diameters of the wheels in the train increase. Coincident with spread at the roots is the thrust shown with reference to Fig. 39, hence under the most favorable conditions the wear on the journals of the wheel axles and the bearings containing them will take place, and the pitch circles will separate. Now so soon as this separation takes place, the motion of the wheels will not be as uniformly equal as when the pitch circles were in contact on the line of centres, because the conditions under which the tooth curves, necessary to produce a uniform velocity of motion, were formed, will have become altered, and the value of those curves to produce constant regularity of motion will have become impaired in proportion as the pitch circles have separated.

[Ill.u.s.tration: Fig. 97.]

In a single pair of epicycloidal wheels in which the flanks of the teeth are radial, the conditions are more favorable, but in this case the pinion teeth will be weaker than if of involute form, while the wear of the journals and bearings (which will take place to some extent) will have the injurious effect already stated, whereas in involute teeth, as has been noted, the separation of the pitch circles does not affect the uniformity of the motion or the correct working of the teeth.

If the teeth of wheels are to be cut to shape in a gear-cutting machine, either the cutters employed determine from their shapes the shapes or curves of the teeth, or else the cutting tool is so guided to the work that the curves are determined by the operations of the machine. In either case nothing is left to the machine operator but to select the proper tools and set them, and the work in proper position in the machine. But when the teeth are to be cast upon the wheel the pattern wherefrom the wheel is to be moulded must have the teeth proportioned and shaped to proper curve and form.

Wheels that require to run without noise or jar, and to have uniformity of motion, must be finished in gear-cutting machines, because it is impracticable to cast true wheels.

When the teeth are to be cast upon the wheels the pattern-maker makes templates of the tooth curves (by some one of the methods to be hereafter described), and carefully cuts the teeth to shape. But the production of these templates is a tedious and costly operation, and one which is very liable to error unless much experience has been had. The Pratt and Whitney Company have, however, produced a machine that will produce templates of far greater accuracy than can be made by hand work.

These templates are in metal, and for epicycloidal teeth from 15 to a rack, and having a diametral pitch ranging from 1-1/2 to 32.

The principles of action of the machine are that a segment of a ring (representing a portion of the pitch circle of the wheel for whose teeth a template is to be produced) is fixed to the frame of the machine. Upon this ring rolls a disk representing the rolling, generating, or describing circle, this disk being carried by a frame mounted upon an arm representing the radius of the wheel, and therefore pivoted at a point central to the ring. The describing disk is rolled upon the ring describing the epicycloidal curve, and by suitable mechanical devices this curve is cut upon a piece of steel, thus producing a template by actually rolling the generating upon the base circle, and the rolling motion being produced by positive mechanical motion, there cannot possibly be any slip, hence the curves so produced are true epicycloids.

The general construction of the machine is shown in the side view, Fig.

98 (Plate I.), and top view, Fig. 99 (Plate I.), details of construction being shown in Figs. 100, 101 (Plate I.), 102, 103, 104, 105, and 106. A A is the segment of a ring whose outer edge represents a part of the pitch circle. B is a disk representing the rolling or generating circle carried by the frame C, which is attached to a rod pivoted at D. The axis of pivot D represents the axis of the base circle or pitch circle of the wheel, and D is adjustable along the rod to suit the radius of A A, or what is the same thing, to equal the radius of the wheel for whose teeth a template is to be produced.

When the frame C is moved its centre or axis of motion is therefore at D and its path of motion is around the circ.u.mference of A A, upon the edge of which it rolls. To prevent B from slipping instead of rolling upon A A, a flexible steel ribbon is fastened at one end upon A A, pa.s.ses around the edge of A A and thence around the circ.u.mference of B, where its other end is fastened; due allowance for the thickness of this ribbon being made in adjusting the radii of A A and of B.

E' is a tubular pivot or stud fixed on the centre line of pivots E and D, and distant from the edge of A A to the same amount that E is. These two studs E and E' carry two worm-wheels F and F' in Fig. 102, which stand above A and B, so that the axis of the worm G is vertically over the common tangent of the pitch and describing circles.

[Ill.u.s.tration: _VOL. I._ =TEMPLATE-CUTTING MACHINES FOR GEAR TEETH.= _PLATE I._

Fig. 98.

Fig. 99.

Fig. 100.

Fig. 101.]

The relative positions of these and other parts will be most clearly seen by a study of the vertical section, Fig. 102.[4] The worm G is supported in bearings secured to the carrier C and is driven by another small worm turned by the pulley I, as seen in Fig. 101 (Plate I.); the driving cord, pa.s.sing through suitable guiding pulleys, is kept at uniform tension by a weight, however C moves; this is shown in Figs. 98 and 99 (Plate I.).

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

[Ill.u.s.tration: Fig. 102.]

Upon the same studs, in a plane still higher than the worm-wheels turn the two disks H, H', Figs. 103, 104, 105. The diameters of these are equal, and precisely the same as those of the describing circles which they represent, with due allowance, again, for the thickness of a steel ribbon, by which these also are connected. It will be understood that each of these disks is secured to the worm-wheel below it, and the outer one of these, to the disk B, so that as the worm G turns, H and H' are rotated in opposite directions, the motion of H being identical with that of B; this last is a rolling one upon the edge of A, the carrier C with all its attached mechanism moving around D at the same time.

Ultimately, then, the motions of H, H', are those of two equal describing circles rolling in external and internal contact with a fixed pitch circle.

[Ill.u.s.tration: Fig. 103.]

[Ill.u.s.tration: Fig. 104.]

In the edge of each disk a semicircular recess is formed, into which is accurately fitted a cylinder J, provided with f.l.a.n.g.es, between which the disks fit so as to prevent end play. This cylinder is perforated for the pa.s.sage of the steel ribbon, the sides of the opening, as shown in Fig.

103, having the same curvature as the rims of the disks. Thus when these recesses are opposite each other, as in Fig. 104, the cylinder J fills them both, and the tendency of the steel ribbon is to carry it along with H when C moves to one side of this position, as in Fig. 105, and along with H' when C moves to the other side, as in Fig. 103.

This action is made positively certain by means of the hooks K, K', which catch into recesses formed in the upper f.l.a.n.g.e of J, as seen in Fig. 104. The spindles, with which these hooks turn, extend through the hollow studs, and the coiled springs attached to their lower ends, as seen in Fig. 102, urge the hooks in the directions of their points; their motions being limited by stops _o_, _o'_, fixed, not in the disks H, H', but in projecting collars on the upper ends of the tubular studs.

The action will be readily traced by comparing Fig. 104 with Fig. 105; as C goes to the left, the hook K' is left behind, but the other one, K, cannot escape from its engagement with the f.l.a.n.g.e of J; which, accordingly, is carried along with H by the combined action of the hook and the steel ribbon.

On the top of the upper f.l.a.n.g.e of J, is secured a bracket, carrying the bearing of a vertical spindle L, whose centre line is a prolongation of that of J itself. This spindle is driven by the spur-wheel N, keyed on its upper end, through a flexible train of gearing seen in Fig. 99; at its lower end it carries a small milling cutter M, which shapes the edge of the template T, Fig. 105, firmly clamped to the framing.

[Ill.u.s.tration: Fig. 105.]

When the machine is in operation, a heavy weight, seen in Fig. 98 (Plate I.), acts to move C about the pivot D, being attached to the carrier by a cord guided by suitably arranged pulleys; this keeps the cutter M up to its work, while the spindle L is independently driven, and the duty left for the worm G to perform is merely that of controlling the motions of the cutter by the means above described, and regulating their speed.

The centre line of the cutter is thus automatically compelled to travel in the path R S, Fig. 105, composed of an epicycloid and a hypocycloid if A A be the segment of a circle as here shown; or of two cycloids, if A A be a straight bar. The radius of the cutter being constant, the edge of the template T is cut to an outline also composed of two curves; since the radius M is small, this outline closely resembles R S, but particular attention is called to the fact that it is _not identical with it, nor yet composed of truly epicycloidal curves of any generation whatever:_ the result of which will be subsequently explained.

NUMBER AND SIZES OF TEMPLATES.

With a given pitch every additional tooth increases the diameter of the wheel, and changes the form of the epicycloid; so that it would appear necessary to have as many different cutters, as there are wheels to be made, of any one pitch.

But the proportional increment, and the actual change of form, due to the addition of one tooth, becomes less as the wheel becomes larger; and the alteration in the outline soon becomes imperceptible. Going still farther, we can presently add more teeth without producing a sensible variation in the contour. That is to say, several wheels can be cut with the same cutter, without introducing a perceptible error. It is obvious that this variation in the form is least near the pitch circle, which is the only part of the epicycloid made use of; and Prof. Willis many years ago deduced theoretically, what has since been abundantly proved by practice, that instead of an infinite number of cutters, 24 are sufficient of one pitch, for making all wheels, from one with 12 teeth up to a rack.

[Ill.u.s.tration: Fig. 106.]

Accordingly, in using the epicycloidal milling engine, for forming the template, segments of pitch circles are provided of the following diameters (in inches):

12, 16, 20, 27, 43, 100, 13, 17, 21, 30, 50, 150, 14, 18, 23, 34, 60, 300.

15, 19, 25, 38, 75,

In Fig. 106, the edge T T is shaped by the cutter T T, whose centre travels in the path R S, therefore these two lines are at a constant normal distance from each other. Let a roller P, of any reasonable diameter, be run along T T, its centre will trace the line U V, which is at a constant normal distance from T T, and therefore from R S. Let the normal distance between U V and R S be the radius of another milling cutter N, having the same axis as the roller P, and carried by it, but in a different plane as shown in the side view; then whatever N cuts will have R S for its contour, if it lie upon the same side of the cutter as the template.

The diameter of the disks which act as describing circles is 7-1/2 inches, and that of the milling cutter which shapes the edge of the template is 1/8 of an inch.

Now if we make a set of 1-pitch wheels with the diameters above given, the smallest will have twelve teeth, and the one with fifteen teeth will have radial flanks. The curves will be the same whatever the pitch; but as shown in Fig. 106, the blank should be adjusted in the epicycloidal engine, so that its lower edge shall be 1/16th of an inch (the radius of the cutter M) above the bottom of the s.p.a.ce; also its relation to the side of the proposed tooth should be as here shown. As previously explained, the depth of the s.p.a.ce depends upon the pitch. In the system adopted by the Pratt & Whitney Company, the whole height of the tooth is 2-1/8 times the diametral pitch, the projection outside the pitch circle being just equal to the pitch, so that diameter of blank = diameter of pitch circle + 2 diametral pitch.

We have now to show how, from a single set of what may be called 1-pitch templates, complete sets of cutters of the true epicycloidal contour may be made of the same or any less pitch.

Now if T T be a 1-pitch template as above mentioned, it is clear that N will correctly shape a cutting edge of a gear cutter for a 1-pitch wheel. The same figure, reduced to half size, would correctly represent the formation of a cutter for a 2-pitch wheel of the same number of teeth; if to quarter size, that of a cutter for a 4-pitch wheel, and so on.

But since the actual size and curvature of the contour thus determined depend upon the dimensions and motion of the cutter N, it will be seen that the same result will practically be accomplished, if these only be reduced; the size of the template, the diameter and the path of the roller remaining unchanged.

The nature of the mechanism by which this is effected in the Pratt & Whitney system of producing epicycloidal cutters will be hereafter explained in connection with cutters.

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 cla.s.s of cutters is made as follows:--