The line r p q is at a right angle to the curves p e and p f, at their point of contact, and, therefore, fills the conditions referred to in Fig. 41. Now the line r p q denotes the path of contact of tooth upon tooth as the wheels revolve; or, in other words, the point of contact between the side of a tooth on one wheel, and the side of a tooth on the other wheel, will always move along the line q r, or upon a similar line passing through d, but meeting the base circles upon the opposite sides of the line of centres, and since line q r always cuts the line of centres at the point of contact of the pitch circles, the conditions necessary to obtain a correct angular velocity are completely fulfilled. The velocity ratio is, therefore, as the length of b q is to that of a r, or, what is the same thing, as the radius of the base circle of one wheel is to that of the other. It is to be observed that the line q r will vary in its angle to the line of centres a b, according to the diameter of the base circle from which it is struck, and it becomes a consideration as to what is its most desirable angle to produce the least possible amount of thrust tending to separate the wheels, because this thrust (described in Fig. 39) tends to wear the journals and bearings carrying the wheel shafts, and thus to permit the pitch circles to separate. To avoid, as far as possible, this thrust the proportions between the diameters of the base circles d and e, Fig. 91, must be such that the line d e passes through the point of contact on the line of centres, as at c, while the angles of the straight line d e should be as nearly 90° to a radial line, meeting it from the centres of the wheels (as shown in the figure, by the lines b e and d e), as is consistent with the length of d e, which in order to impart continuous motion must at least equal the pitch of the teeth. It is obvious, also, that, to give continuous motion, the length of d e must be more than the pitch in proportion, as the points of the teeth come short of passing through the base circles at d and e, as denoted by the dotted arcs, which should therefore represent the addendum circles. The least possible obliquity, or angle of d e, will be when the construction under any given conditions be made such by trial, that the base circles d and e coincide with the addendum circles on the line of centres, and thus, with a given depth of both beyond, the pitch circle, or addenda as it is termed, will cause the tooth contacts to extend over the greatest attainable length of line between the limits of the addendum circles, thus giving a maximum number of teeth in contact at any instant of time. These conditions are fulfilled in Fig. 92,[3] the addendum on the small wheel being longer than the depth below pitch line, while the faces of the teeth are the narrowest.
[3] From an article by Prof. Robinson.
In seeking the minimum obliquity or angle of d e in the figure, it is to be observed that the less it is, the nearer the base circle approaches the pitch circle; hence, the shorter the operative length of tooth flank and the greater its wear.
In comparing the merits of involute with those of epicycloidal teeth, the direction of the line of pressure at each point of contact must always be the common perpendicular to the surfaces at the point of contact, and these perpendiculars or normals must pass through the pitch circles on the line of centres, as was shown in Fig. 41, and it follows that a line drawn from c (Fig. 91) to any point of contact, is in the direction of the pressure on the surfaces at that point of contact. In involute teeth, the contact will always be on the line d e (Fig. 92), but in epicycloidal, on the line of the generating circle, when that circle is tangent at the line of centres; hence, the direction of pressure will be a chord of the circle drawn from the pitch circle at the line of centres to the position of contact considered. Comparing involute with radial flanked epicycloidal teeth, let c d a (Fig. 91) represent the rolling circle for the latter, and d c will be the direction of pressure for the contact at d; but for point of contact nearer c, the direction will be much nearer 90°, reaching that angle as the point of contact approaches c. Now, d is the most remote legitimate contact for involute teeth (and considering it so far as epicycloidal struck with a generating circle of infinite diameter), we find that the aggregate directions of the pressures of the teeth upon each other is much nearer perpendicular in epicycloidal, than in involute gearing; hence, the latter exert a greater pressure, tending to force the wheels apart. Hence, the former are, in this respect, preferable.
It is to be observed, however, that in some experiments made by Mr. Hawkins, he states that he found “no tendency to press the wheels apart, which tendency would exist if the angle of the line d e (Fig. 92) deviated more than 20° from the line of centres a b of the two wheels.”
A method commonly employed in practice to strike the curves of involute teeth, is as follows:—
In Fig. 93 let c represent the centre of a wheel, d d the full diameter, p p the pitch circle, and e the circle of the roots of the teeth, while r is a radial line. Divide on r, the distance between the pitch circle and the wheel centre, into four equal parts, by 1, 2, 3, &c. From point or division 2, as a centre, describe the semicircle s, cutting the wheel centre and the pitch circle at its junction with r (as at a). From a, with compasses set to the length of one of the parts, as a 3, describe the arc b, cutting s at f, and f will be the centre from which one side of the tooth may be struck; hence from f as a centre, with the compasses set to the radius a b, mark the curve g. From the centre c strike, through f, a circle t t, and the centres wherefrom to strike all the teeth curves will fall on t t. Thus, to strike the other curve of the tooth, mark off from a the thickness of the tooth on the pitch circle p p, producing the point h. From h as a centre (with the same radius as before,) mark on t t the point i, and from i, as a centre, mark the curve j, forming the other side of the tooth.
In Fig. 94 the process is shown carried out for several teeth. On the pitch circle p p, divisions 1, 2, 3, 4, &c., for the thickness of teeth and the width of the spaces are marked. The compasses are set to the radius by the construction shown in Fig. 93, then from a, the point b on t is marked, and from b the curve c is struck.
In like manner, from d, g, j, the centres e, h, k, wherefrom to strike the respective curves, f, i, l, are obtained.
Then from m the point n, on t t, is marked, giving the centre wherefrom to strike the curve at h m, and from o is obtained the point p, on t t, serving as a centre for the curve e o.
A more simple method of finding point f is to make a sheet metal template, c, as in Fig. 95, its edges being at an angle one to the other of 75° and 30′. One of its edges is marked off in quarters of an inch, as 1, 2, 3, 4, &c. Place one of its edges coincident with the line r, its point touching the pitch circle at the side of a tooth, as at a, and the centre for marking the curve on that side of the tooth will be found on the graduated edge at a distance from a equal to one-fourth the length of r.
The result obtained in this process is precisely the same as that by the construction in Fig. 93, as will be plainly seen, because there are marked on Fig. 93 all the circles by which point f was arrived at in Fig. 95; and line 3, which in Fig. 95 gives the centre wherefrom to strike curve o, is coincident with point f, as is shown in Fig. 95. By marking the graduated edge of c in quarter-inch divisions, as 1, 2, 3, &c., then every division will represent the distance from a for the centre for every inch of wheel radius. Suppose, for example, that a wheel has 3 inches radius, then with the scale c set to the radial line r, the centre therefrom to strike the curve o will be at 3; were the radius of the wheel 4 inches, then the scale being set the same as before (one edge coincident with r), the centre for the curve o would be at 4, and arc t would require to meet the edge of c at 4. Having found the radius from the centre of the wheel of point f for one tooth, we may mark circle t, cutting point f, and mark off all the teeth by setting one point of the compasses (set to radius a f) on one side of the tooth and marking on circle t the centre wherefrom to mark the curve (as o), continuing the process all around the wheel and on both sides of the tooth.
This operation of finding the location for the centre wherefrom to strike the tooth curves, must be performed separately for each wheel, because the distance or radius of the tooth curves varies with the radius of each wheel.
In Fig. 96 this template is shown with all the lines necessary to set it, those shown in Fig. 95 to show the identity of its results with those given in Fig. 93 being omitted.
The principles involved in the construction of a rack to work correctly with a wheel or pinion, having involute teeth, are as in Fig. 97, in which the pitch circle is shown by a dotted circle and the base circle by a full line circle. Now the diameter of the base circle has been shown to be arbitrary, but being assumed the radius b q will be determined (since it extends from the centre b to the point of contact of d q, with the base circle); b d is a straight line from the centre b of the pinion to the pitch line of the rack, and (whatever the angle of q d to b d) the sides of the rack teeth must be straight lines inclined to the pitch line of the rack at an angle equal to that of b d q.
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 passage of the cutter, if there is sufficient power to drive it, which is not the case with epicycloidal teeth whenever the flank space 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 space must at least equal the thickness of the tooth, while in involute the space 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.
These advantages assume 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.
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 11⁄2 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.
| VOL. I. | TEMPLATE‑CUTTING MACHINES FOR GEAR TEETH. | PLATE I. | |
| Fig. 98. | |||
| Fig. 99. | |||
| Fig. 100. | Fig. 101. | ||
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 circumference 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, passes around the edge of a a and thence around the circumference 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.
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, passing 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.
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.
In the edge of each disk a semicircular recess is formed, into which is accurately fitted a cylinder j, provided with flanges, between which the disks fit so as to prevent end play. This cylinder is perforated for the passage 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 flange 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 flange 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 flange 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.
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.
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 71⁄2 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 space; also its relation to the side of the proposed tooth should be as here shown. As previously explained, the depth of the space depends upon the pitch. In the system adopted by the Pratt & Whitney Company, the whole height of the tooth is 21⁄8 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 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 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.
“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 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 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.
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, 71⁄2 inches; cutter for shaping template, 1⁄8 of an inch; roller used against edge of template, 11⁄8 inches; cutter for shaping a 1-pitch gear cutter, 1 inch.