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, 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 class cannot be used to cut teeth having the width of the space 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 spaces 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 space 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 spaces 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 spaces too narrow, and the curvature of the teeth too great; but, as before stated, the number of wheels assigned 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 spaces 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 " 100 " " " " 87 to 123 " 150 " " " " 123 to 200 " 300 " " " " 200 to 600 " Rack " " " " 600 to rack.

[6] For wheels having less than 12 teeth the Pratt and Whitney Co. use involute cutters.

Here it will be observed that by a judicious selection of pitch and cutters, almost theoretically perfect results may be obtained for almost any conditions, while at the same time the cutters are so numerous that there is no necessity for making any selection with a view to taking into consideration for what particular number of teeth the cutter is made correct.

For epicycloidal cutters made on the Brown and Sharpe system so as to enable the grinding of the face of the tooth to sharpen it, the Brown and Sharpe company make a separate cutter for wheels from 12 to 20 teeth, as is shown in the accompanying table, in which the cutters are for convenience of designation denoted by an alphabetical letter.

24 CUTTERS IN EACH SET.

Letter A cuts 12 teeth. B " 13 " C " 14 " D " 15 " E " 16 " F " 17 " G " 18 " H " 19 " I " 20 " J " 21 to 22 " K " 23 " 24 " L " 25 " 26 " M " 27 " 29 " N " 30 " 33 " O " 34 " 37 " P " 38 " 42 " Q " 43 " 49 " R " 50 " 59 " S " 60 " 74 " T " 75 " 99 " U " 100 " 149 " V " 150 " 249 " W " 250 " Rack. X " Rack.

In these cutters a shoulder having no clearance is placed on each side of the cutter, so that when the cutter has entered the wheel until the shoulder meets the circumference of the wheel, the tooth is of the correct depth to make the pitch circles coincide.

In both the Brown and Sharpe and Pratt and Whitney systems, no side clearance is given other than that quite sufficient to prevent the teeth of one wheel from jambing into the spaces of the other. Pratt and Whitney allow 1/8 of the pitch for top and bottom clearance, while Brown and Sharpe allow 1/10 of the thickness of the tooth for top and bottom clearance.

It may be explained now, why the thickness of the cutter if employed upon a wheel having more teeth than the cutter is correct for, interferes with theoretical exactitude.

First, then, with regard to the thickness of tooth and width of space. Suppose, then, Fig. 112 to represent a section of a wheel having 12 teeth, then the pitch circle of the cutter will be represented by line A, and there will be the same difference between the arc and chord pitch on the cutter as there is on the wheel; but suppose that this same cutter be used on a wheel having 24 teeth, as in Fig. 113, then the pitch circle on the cutter will be more curved than that on the wheel as denoted at C, and there will be more difference between the arc and chord pitches on the cutter than there is on the wheel, and as a result the cutter will cut a groove too narrow.

The amount of error thus induced diminishes as the diameter of the pitch circle of the cutter is increased.

But to illustrate the amount. Suppose that a cutter is made to be theoretically correct in thickness at the pitch line for a wheel to contain 12 teeth, and having a pitch circle diameter of 8 inches, then we have

3.1416 = ratio of circumference to diameter. 8 = diameter. ------- Number of teeth = 12 ) 25.1328 = circumference. ------- 2.0944 = arc pitch of wheel.

If now we subtract the chord pitch from the arc pitch, we shall obtain the difference between the arc and the chord pitches of the wheel; here

2.0944 = arc pitch. 2.0706 = chord pitch. ------ .0238 = difference between the arc and the chord pitch.

Now suppose this cutter to be used upon a wheel having the same pitch, but containing 18 teeth; then we have

2.0944 = arc pitch. 2.0836 = chord pitch. ------ .0108 = difference between the arc and the chord pitch.

Then

.0238 = difference on wheel with 12 teeth. .0108 = " " " 18 " ----- .0130 = variation between the differences.

And the thickness of the tooth equalling the width of the space, it becomes obvious that the thickness of the cutter at the pitch line being correct for the 12 teeth, is one half of .013 of an inch too thin for the 18 teeth, making the spaces too narrow and the teeth too thick by that amount.

Now let us suppose that a cutter is made correct for a wheel having 96 teeth of 2.0944 arc pitch, and that it be used upon a wheel having 144 teeth. The proportion of the wheels one to the other remains as before (for 96 bears the proportion to 144 as 12 does to 18).

Then we have for the 96 teeth

2.0944 = arc pitch. 2.0934 = chord pitch. ------ .0010 = difference.

For the 144 teeth we have

2.0944 = arc pitch. 2.0937 = chord pitch. ------ .0007 = difference.

We find, then, that the variation decreases as the size of the wheels increases, and is so small as to be of no practical consequence.

If our examples were to be put into practice, and it were actually required to make one cutter serve for wheels having, say, from 12 to 18 teeth, a greater degree of correctness would be obtained if the cutter were made to some other wheel than the smallest. But it should be made for a wheel having less than the mean diameter (within the range of 12 and 18), that is, having less than 15 teeth; because the difference between the arc and chord pitch increases as the diameter of the pitch circle increases, as already shown.

A rule for calculating the number of wheels to be cut by each cutter when the number of cutters in the set and the number of teeth in the smallest and largest wheel in the train are given is as follows:--

Rule.--Multiply the number of teeth in the smallest wheel of the train by the number of cutters it is proposed to have in the set, and divide the amount so obtained by a sum obtained as follows:--

From the number of cutters in the set subtract the number of the cutter, and to the remainder add the sum obtained by multiplying the number of the teeth in the smallest wheel of the set or train by the number of the cutter and dividing the product by the number of teeth in the largest wheel of the set or train.

Example.--I require to find how many wheels each cutter should cut, there being 8 cutters and the smallest wheel having 12 teeth, while the largest has 300.

Number of teeth in Number of cutters smallest wheel. in the set. 12 × 8 = 96

Then

Number of cutters Number of in set. cutter. 8 - 7 = 1

Then

Number of teeth in The number of the The number of the teeth smallest wheel. cutter. in largest wheel. 12 × 8 ÷ 300

12 8 --- 300 ) 960 ( 0.32 900 --- 600 600

Now add the 1 to the .32 and we have 1.32, which we must divide into the 96 first obtained.

Thus

1.32 ) 96.00 ( 72 924 ---- 360 264 --- 96

Hence No. 8 cutter may be used for all wheels that have between 72 teeth and 300 teeth.

To find the range of wheels to be cut by the next cutter, which we will call No. 7, proceed again as before, but using 7 instead of 8 as the number of the cutter.

Thus

Number of teeth in Number of cutters in smallest wheel. the set. 12 × 8 = 96

Then

Number of cutters Number of in the set. cutters. 8 - 6 = 2

And

Number of teeth in The number of the The number of teeth smallest wheel. cutter. in the largest wheel. 12 × 8 ÷ 300

Here

12 8 --- 300 ) 960 ( 0.32 900 --- 600 600

Add the 2 to the .32 and we have 2.32 to divide into the 96.

Thus

2.32 ) 96.00 ( 41 928 --- 320 232 --- 88

Hence this cutter will cut all wheels having not less than the 41 teeth, and up to the 72 teeth where the other cutter begins. For the range of the next cutter proceed the same, using 6 as the number of the cutter, and so on.

By this rule we obtain the lowest number of teeth in a wheel for which the cutter should be used, and it follows that its range will continue upwards to the smallest wheel cut by the cutter above it.

Having by this means found the range of wheels for each cutter, it remains to find for what particular number of teeth within that range the cutter teeth should be made correct, in order to have whatever error there may be equal in amount on the largest and smallest wheel of its range. This is done by using precisely the same rule, but supposing there to be twice as many cutters as there actually are, and then taking the intermediate numbers as those to be used.

Applying this plan to the first of the two previous examples we have--

Number of teeth in the Number of cutters in smallest wheel. the set. 12 × 16 = 192

Then

Number of cutters Number of the in the set. cutter. 16 - 15 = 1

And

Number of teeth in The number of the The number of the teeth in smallest wheel. cutter. the largest wheel. 12 × 15 ÷ 300

12 15 --- 60 12 ----- 300 ) 180.0 ( 0.6 1800

Then add the 1 to the .6 = 1.6, and this divided into 192 = 120.

By continuing this process for each of the 16 cutters we obtain the following table:--

Number of Number of Cutter. Teeth. 1 12 *2 13 3 14 *4 15 5 17 *6 18 7 20.61 *8 23 9 26 *10 30 11 35 *12 42 13 54 *14 75 15 120 *16 300

Suppose now we take for our 8 cutters those marked by an asterisk, and use cutter 2 for all wheels having either 12, 13, or 14 teeth, then the next cutter would be that numbered 4, cutting 14, 15, or 16 toothed wheels, and so on.

A similar table in which 8 cutters are required, but 16 are used in the calculation, the largest wheel having 200 teeth in the set, is given below.

Number of Number of Cutter. Teeth. 1 12.7 2 13.5 3 14.5 4 15.6 5 16.9 6 18 7 21 8 23.5 9 26.5 10 29 11 35 12 40.6 13 52.9 14 67.6 15 101 16 200

To assist in the selections as to what wheels in a given set the determined number of cutters should be made correct for, so as to obtain the least limit of error, Professor Willis has calculated the following table, by means of which cutters may be selected that will give the same difference of form between any two consecutive numbers, and this table he terms the table of equidistant value of cutters.

TABLE OF EQUIDISTANT VALUE OF CUTTERS.

Number of Teeth.

Rack--300, 150, 100, 76, 60, 50, 43, 38, 34, 30, 27, 25, 23, 21, 20, 19, 17, 16, 15, 14, 13, 12.

The method of using the table is as follows:--Suppose it is required to make a set of wheels, the smallest of which is to contain 50 teeth and the largest 150, and it is determined to use but one cutter, then that cutter should be made correct for a wheel containing 76; because in the table 76 is midway between 50 and 150.

But suppose it were determined to employ two cutters, then one of them should be made correct for a wheel having 60 teeth, and used on all the wheels having between 50 and 76 teeth, while the other should be made correct for a wheel containing 100 teeth, and used on all wheels containing between 76 and 150 teeth.

In the following table, also arranged by Professor Willis, the most desirable selection of cutters for different circumstances is given, it being supposed that the set of wheels contains from 12 teeth to a rack.

+-----------+------------------------------------------------------+ |Number of | | |cutters in | Number of Teeth in Wheel for which the Cutter is to | |the set. | be made correct. | +-----------+----+----+--------------------------------------------+ | 2 | 50 | 16 | | | --+----+----+----+ | | 3 | 75 | 25 | 15 | | | --+----+----+----+----+ | | 4 | 100| 34 | 20 | 14 | | | --+----+----+----+----+----+----+ | | 6 | 150| 50 | 30 | 21 | 16 | 13 | | | --+----+----+----+----+----+----+----+----+ | | 8 | 200| 67 | 40 | 29 | 22 | 18 | 15 | 13 | | | --+----+----+----+----+----+----+----+----+----+----+ | | 10 | 200| 77 | 50 | 35 | 27 | 22 | 19 | 16 | 14 | 13 | | | --+----+----+----+----+----+----+----+----+----+----+----+ | | 300| 100| 60 | 43 | 34 |27 | 23 | 20 | 17 | 15 | 14 | | 12 +----+----+----+----+----+----+----+----+----+----+----+ | | 13 | | | --+----+----+----+----+----+----+----+----+----+----+----+ | | 300| 150| 100| 70 | 50 | 40 | 30 | 26 | 24 | 22 | 20 | | 18 +----+----+----+----+----+----+----+----+----+----+----+ | | 18 | 16 | 15 | 14 | 13 | 12 | | | --+----+----+----+----+----+----+----+----+----+----+----+ | |Rack| 300| 150| 100| 76 | 60 | 50 | 43 | 38 | 34 | 30 | | +----+----+----+----+----+----+----+----+----+----+----+ | 24 | 27 | 25 | 23 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | | +----+----+----+----+----+----+----+----+----+----+----+ | | 13 | 12 | | +-----------+----+----+--------------------------------------------+

Suppose now we take the cutters, of a given pitch, necessary to cut all the wheels from 12 teeth to a rack, then the thickness of the teeth at the pitch line will for the purposes of designation be the thickness of the teeth of all the wheels, which thickness may be a certain proportion of the pitch.

But in involute teeth while the depth of tooth on the cutter may be taken as the standard for all the wheels in the range, and the actual depth for the wheel for which the cutter is correct, yet the depth of the teeth in the other wheels in the range may be varied sufficiently on each wheel to make the thickness of the teeth equal the width of the spaces (notwithstanding the variation between the arc and chord pitches), so that by a variation in the tooth depth the error induced by that variation may be corrected. The following table gives the proportions in the Brown and Sharpe system.

+------------+-----------------+-----------------+ | Arc Pitch. | Depth of Tooth. | Depth in terms | | | |of the arc pitch.| +------------+-----------------+-----------------+ | inches. | inches. | inches. | | 1.570 | 1.078 | .686 | | 1.394 | .958 | .687 | | 1.256 | .863 | .686 | | 1.140 | .784 | .697 | | 1.046 | .719 | .687 | | .896 | .616 | .686 | | .786 | .539 | .685 | | .628 | .431 | .686 | | .524 | .359 | .685 | | .448 | .307 | .685 | | .392 | .270 | .686 | | .350 | .240 | .686 | | .314 | .216 | .687 | +------------+-----------------+-----------------+

To avoid the trouble of measuring, and to assist in obtaining accuracy of depth, a gauge is employed to mark on the wheel face a line denoting the depth to which the cutter should be entered.

Suppose now that it be required to make a set of cutters for a certain range of wheels, and it be determined that the cutters be so constructed that the greatest permissible amount of error in any wheel of the set be 1/100 inch. Then the curves for the smallest wheel, and those for the largest in the set, and the amount of difference between them ascertained, and assuming this difference to amount to 1/16 inch, which is about 6/100, then it is evident that 6 cutters must be employed for the set.

It has been shown that on bevel-wheels the tooth curves vary at every point in the tooth breadth; hence it is obvious that the cutter being of a fixed curve will make the tooth to that curve. Again, the thickness of the teeth and breadth of the spaces vary at every point in the breadth, while with a cutter of fixed thickness the space cut will be parallel from end to end. To overcome these difficulties it is usual to give to the cutter a curve corresponding to the curve required at the middle of the wheel face and a thickness equal to the required width of space at its smallest end, which is at the smallest face diameter.

The cutter thus formed produces, when passed through the wheel once, and to the required depth, a tooth of one curve from end to end, having its thickness and width of space correct at the smaller face diameter only, the teeth being too thick and the spaces too narrow as the outer diameter of the wheel is approached. But the position and line of traverse of the cutter may be altered so as to take a second cut, widening the space and reducing the tooth thickness at the outer diameter.

By moving the cutter's position two or three times the points of contact between the teeth may be made to occur at two or three points across the breadth of the teeth and their points of contact; the wear will soon spread out so that the teeth bear all the way across.

Another plan is to employ two or three cutters, one having the correct curve for the inner diameter, and of the correct thickness for that diameter, another having the correct curve for the pitch circle, and another having the correct curve at the largest diameter of the teeth.

The thickness of the first and second cutters must not exceed the required width of space at the small end, while that for the third may be the same as the others, or equal to the thickness of the smallest space breadth that it will encounter in its traverse along the teeth.

The second cutter must be so set that it will leave the inner end of the teeth intact, but cut the space to the required width in the middle of the wheel face. The third cutter must be so set as to leave the middle of the tooth breadth intact, and cut the teeth to the required thickness at the outer or largest diameter.

CUTTING WORM-WHEELS.

The most correct method of cutting the teeth of a worm-wheel is by means of a worm-cutter, which is a worm of the pitch and form of tooth that the working worm is intended to be, but of hardened steel, and having grooves cut lengthways of the worm so as to provide cutting edges similar to those on the cutter shown in Fig. 107.

The wheel is mounted on an arbor or mandril free to rotate on its axis and at a right angle to the cutter worm, which is rotated and brought to bear upon the perimeter of the worm-wheel in the same manner as the working worm-wheel when in action. The worm-cutter will thus cut out the spaces in the wheel, and must therefore be of a thickness equal to those spaces. The cutter worm acting as a screw causes the worm-wheel to rotate upon its axis, and therefore to feed to the cutter.

In wheels of fine pitch and small diameter this mode of procedure is a simple matter, especially if the form of tooth be such that it is thicker, as the root of the tooth is approached from the pitch line, because in that case the cutter worm may be entered a part of the depth in the worm-wheel and a cut be taken around the wheel. The cutter may then be moved farther into the wheel and a second cut taken around the wheel, so that by continuing the process until the pitch line of the cutter worm coincides with that of the worm-cutter, the worm-wheel may be cut with a number of light cuts, instead of at one heavy cut.

But in the case of large wheels the strain due to such a long line of cutting edge as is possessed by the cutter worm-teeth springs or bends the worm-wheel, and on account of the circular form of the breadth of the teeth this bending or spring causes that part of the tooth arc above the centre of the wheel thickness to lock against the cutter.

To prevent this, several means may be employed. Thus the grooves forming the cutting edges of the worm-cutter may wind spirally along instead of being parallel to the axis of the cutter.

The distance apart of these grooves may be greater than the breadth of tooth a width of worm-wheel face, in which case the cutting edge of one tooth only will meet the work at one time. In addition to this two stationary supports may be placed beneath the worm-wheel (one on each side of the cutter). But on coarse pitches with their corresponding depth of tooth, the difficulty presents itself, that the arbor driving the worm-cutter will spring, causing the cutter to lift and lock as before; hence it is necessary to operate on part of the space at a time, and shape it out to so nearly the correct form that the finishing cut may be a very light one indeed, in which case the worm-cutter will answer for the final cut.

The removal of the surplus metal preparatory to the introduction of the worm-cutter to finish, may be made with a cutter-worm that will cut out a narrow groove being of the thickness equal to the bottom of the tooth space and cutting on its circumference only. This cutter may be fed into the wheel to the permissible depth of cut, and after the cut is taken all around the wheel, it may be entered deeper and a second cut taken, and so on until it has entered the wheel to the necessary depth of tooth. A second cutter-worm may then be used, it being so shaped as to cut the face curve only of the teeth. A third may cut the flank curve only, and finally a worm-cutter of correct form may take a finishing cut over both the faces and the flanks. In this manner teeth of any pitch and depth may be cut. Another method is to use a revolving cutter such as shown in Fig. 107, and to set it at the required angle to the wheel, and then take a succession of cuts around the wheel, the first cut forming a certain part of the tooth depth, the second increasing this depth, and so on until the final cut forms the tooth to the requisite depth. In this case the cutter operates on each space separately, or on one space only at a time, and the angle at which to set the cutter may be obtained as follows in Fig. 114. Let the length of the line A A equal the diameter of the worm at the pitch circle, and B B (a line at a right angle to A A) represent the axial line of the worm. Let the distance C equal the pitch of the teeth, and the angle of the line D with A A or B B according to circumstances, will be that to which the cutter must be set with reference to the tooth.

If then a piece of sheet metal be cut to the lines A, D, and the cutter so set that with the edge D of the piece held against the side face of the cutter (which must be flat or straight across), the edge A will stand truly vertical, and the cutter will be at the correct angle supposing the wheel to be horizontal.

In making patterns wherefrom gear-wheels may be cast in a mould, the true curves are frequently represented by arcs of circles struck from the requisite centres and of the most desirable radius with compasses, and this will be treated after explaining the pattern maker's method of obtaining true curves by rolling segments by hand. If, then, the wheels are of small diameter, as say, less than 12 inches in diameter, and precision is required, it is best to turn in the lathe wooden disks representing in their diameters the base and generating circles. But otherwise, wooden segments to answer the same purpose may be made as from a piece of soft wood, such as pine or cedar, about three-eighths inch thick, make two pieces A and B, in Fig. 115, and trim the edges C and D to the circle of the pitch line of the required wheel. If the diameter of the pitch circle is marked on a drawing, the pieces may be laid on the drawing and sighted for curvature by the eye. In the absence of a drawing, strike a portion of the pitch circle with a pair of sharp-pointed compasses on a piece of zinc, which will show a very fine line quite clear. After the pieces are filed to the circle, try them together by laying them flat on a piece of board, bringing the curves in contact and sweeping A against B, and the places of contact will plainly show, and may be filed until continuous contact along the curves is obtained. Take another similar piece of wood and form it as shown in Fig. 116, the edge E representing a portion of the rolling circle. In preparing these segments it is an excellent plan to file the convex edges, as shown in Fig. 117, in which P is a piece of iron or wood having its surface S trued; F is a file held firmly to S, while its surface stands vertical, and T is the template laid flat on S, while swept against the file. This insures that the edge shall be square across or at least at the same angle all around, which is all that is absolutely necessary. It is better, however, that the edges be square. So likewise in fitting A and B (Fig. 115) together, they should be laid flat on a piece of board. This will insure that they will have contact clear across the edge, which will give more grip and make slip less likely when using the segments. Now take a piece of stiff drawing paper or of sheet zinc, lay segment A upon it, and mark a line coincident with the curved edge. Place the segment representing the generating circle flat on the paper or zinc, hold its edge against segment A, and roll it around a sufficient distance to give as much of the curve as may be required; the operation being illustrated in Fig. 118, in which A is the segment representing the pitch or base circle, E is the segment representing the generating circle, P is the paper, C the curve struck by the tracing point or pencil O.

This tracing point should be, if paper be used to trace on, a piece of the _hardest_ pencil obtainable, and should be filed so that its edge, if flat, shall stand as near as may be in the line of motion when rolled, thus marking a fine line. If sheet zinc be used instead of paper a needle makes an excellent tracing point. Several of the curves, C, should be struck, moving the position of the generating segment a little each time.

On removing the segments from the paper, there will appear the lines shown in Fig. 119; A representing the pitch circle, and O O O the curves struck by the tracing point.

Cut out a piece of sheet zinc so that its edge will coincide with the curve A and the epicycloid O, trying it with all four of the epicycloids to see that no slip has occurred when marking them; shape a template as shown in Fig. 120. Cutting the notches at _a_ _b_, acts to let the file clear well when filing the template, and to allow the scriber to go clear into the corner. Now take the segment A in Fig. 118, and use it as a guide to carry the pitch circle across the template as at P, in Fig. 120. A zinc template for the flank curve is made after the same manner, using the rolling segment in conjunction with the segment B in Fig. 115.

But the form of template for the flank should be such as shown in Fig. 121, the curve P representing, and being of the same radius as the pitch circle, and the curve F being that of the hypocycloid. Both these templates are set to the pitch circles and to coincide with the marks made on the wheel teeth to denote the thickness, and with a hardened steel point a line is traced on the tooth showing the correct curve for the same.

An experienced hand will find no difficulty in producing true templates by this method, but to avoid all possibility of the segments slipping on coarse pitches, and with large segments, the segments may be connected, as shown in Fig. 122, in which O represents a strip of steel fastened at one end into one segment and at the other end to the other segment. Sometimes, indeed, where great accuracy is requisite, two pieces of steel are thus employed, the second one being shown at P P, in the figure. The surfaces of these pieces should exactly coincide with the edge of the segments.

The curve templates thus produced being shaped to apply to the pitch circle may be correctly applied to that circle independently of its concentricity to the wheel axis or of the points of the teeth, but if the points of the teeth are turned in the lathe so as to be true (that is, concentric to the wheel axis) the form of the template may be such as shown in Fig. 123, the radius of the arc A A equalling that of the addendum circle or circumference at the points of the teeth, and the width at B (the pitch circle) equaling the width of a space instead of the thickness of a tooth. The curves on each side of the template may in this case be filed for the full side of a tooth on each side of the template so that it will completely fill the finished space, or the sides of two contiguous teeth may be marked at one operation. This template may be set to the marks made on the teeth at the pitch circle to denote their requisite thickness, or for greater accuracy, a similar template made double so as to fill two finished tooth spaces may be employed, the advantage being that in this case the template also serves to mark or test the thickness of the teeth. Since, however, a double template is difficult to make, a more simple method is to provide for the thickness of a tooth, the template shown in Fig. 124, the width from A to B being either the thickness of tooth required or twice the thickness of a tooth plus the width of a space, so that it may be applied to the outsides of two contiguous teeth. The arc C may be made both in its radius and distance from the pitch circle D D to equal that of the addendum circle, so as to serve as a gauge for the tooth points, if the latter are not turned true in the lathe, or to rest on the addendum circle (if the teeth points are turned true), and adjust the pitch circle D D to the pitch circle on the wheel.

The curves for the template must be very carefully filed to the lines produced by the rolling segments, because any error in the template is copied on every tooth marked from it. Furthermore, instead of drawing the pitch circle only, the addendum circle and circle for the roots of the teeth or spaces should also be drawn, so that the template may be first filed to them, and then adjusted to them while filing the edges to the curves.

Another form of template much used is shown in Fig. 125. The curves A and B are filed to the curve produced by rolling segments as before, and the holes C, D, E, are for fastening the template to an arm, such as shown in Fig. 126, which represents a section of a wheel W, with a plug P, fitting tightly into the hub H of the wheel. This plug carries at its centre a cylindrical pin on which pivots the arm A. The template T is fastened to the arm by screws, and set so that its pitch circle coincides with the pitch circle P on the wheel, when the curves for one side of all the teeth may be marked. The template must then be turned over to mark the other side of the teeth.

The objection to this form of template is that the length of arc representing the pitch circle is too short, for it is absolutely essential that the pitch line on the template (or line representing the arc of the addendum if that be used) be greater than the width of a single tooth, because an error of the thickness of a line (in the thickness of a tooth), in the coincidence of the pitch line of the template with that of the tooth, would throw the tooth curves out to an extent altogether inadmissible where true work is essential.

To overcome this objection the template may be made to equal half the thickness of a tooth and its edge filed to represent a radial line on the wheel. But there are other objections, as, for example, that the template can only be applied to the wheel when adjusted on the arm shown in Fig. 126, unless, indeed, a radial line be struck on every tooth of the wheel. Again, to produce the template a radial line representing the radius of the wheel must be produced, which is difficult where segments only are used to produce the curves. It is better, therefore, to form the template as shown in Fig. 127, the projections at A B having their edges filed to coincide with the pitch circle P, so that they may be applied to a length of one arc of pitch circle at least equal to the pitch of the teeth.

The templates for the tooth curves being obtained, the wheel must be divided off on the pitch circle for the thickness of the teeth and the width of the spaces, and the templates applied to the marks or points of division to serve as guides to mark the tooth curves. Since, however, as already stated, the tooth curves are as often struck by arcs of circles as by templates, the application of such arcs and their suitability may be discussed.

MARKING THE CURVES BY HAND.

In the employment of arcs of circles several methods of finding the necessary radius are found in practice.

In the best practice the true curve is marked by the rolling segments already described, and the compass points are set by trial to that radius which gives an arc nearest approaching to the true face and flank curves respectively. The degree of curve error thus induced is sufficient that the form of tooth produced cannot with propriety be termed epicycloidal teeth, except in the case of fine pitches in which the arc of a circle may be employed to so nearly approach the true curve as to be permissible as a substitute. But in coarse pitches the error is of much importance. Thus in Fig. 128 is shown the curve of the _former_ or _template_ attachment used on the celebrated Corliss Bevel Gear Cutting Machine, to cut the teeth on the bevel-wheels employed upon the line shafting at the Centennial Exhibition. These gears, it may be remarked, were marvels of smooth and noiseless running, and attracted wide attention both at home and abroad. The engraving is made from a drawing marked direct from the _former_ itself, and kindly furnished me by Mr. George H. Corliss. A A is the face and B B the flank of the tooth, C C is the arc of a circle nearest approaching to the face curve, and D D the arc of a circle nearest approaching the flank curve. In the face curve, there are but two points where the circle coincides with the true curve, while in the flank there are three such points; a circle of smaller radius than C C would increase the error at _b_, but decrease it at _a_; one of a greater radius would decrease it at _b_, and increase it at _a_. Again, a circle larger in radius than D D would decrease the error at _e_ and increase it at _f_; while one smaller would increase it at _e_ and decrease it at _f_. Only the working part of the tooth is given in the illustration, and it will be noted that the error is greatest in the flank, although the circle has three points of coincidence.

In this case the depth of the _former_ tooth is about three and three-quarter times greater than the depth of tooth cut on the bevel-wheels; hence, in the figure the actual error is magnified three and three-quarter times. It demonstrates, however, the impropriety of calling coarsely pitched teeth that are found by arcs of circles "epicycloidal" teeth.

When, however, the pitches of the teeth are fine as, say an inch or less, the coincidence of an arc of a circle with the true curve is sufficiently near for nearly all practical purposes, and in the case of cast gear the amount of variation in a pitch of 2 inches would be practically inappreciable.

To obtain the necessary set of the compasses to mark the curves, the following methods may be employed.

First by rolling the true curves with segments as already described, and the setting the compass points (by trial) to that radius which gives an arc nearest approaching the true curves. In this operation it is not found that the location for the centre from which the curve must be struck always falls on the pitch circle, and since that location will for every tooth curve lie at the same radius from the wheel centre it is obvious that after the proper location for one of the curves, as for the first tooth face or tooth flank as the case may be, is found, a circle may be struck denoting the radius of the location for all the teeth. In Fig. 129, for example, P P represents the pitch circle, A B the radius that will produce an arc nearest approaching the true curve produced by rolling segments, and A the location of the centre from which the face arc B should be struck. The point A being found by trial with the compasses applied to the curve B, the circle A C may be struck, and the location for the centres from which the face arcs of each tooth must be struck will also fall on this circle, and all that is necessary is to rest one point of the compasses on the side of the tooth as, say at E, and mark on the second circle A C the point C, which is the location wherefrom to mark the face arc D.

If the teeth flanks are not radial, the locations of the centre wherefrom to strike the flank curves are found in like manner by trial of the compasses with the true curves, and a third circle, as I in Fig. 130, is struck to intersect the first point found, as at G in the figure. Thus there will be upon the wheel face three circles, P P the pitch circle, J J wherefrom to mark the face curves, and I wherefrom to mark the flank curves.

When this method is pursued a little time may be saved, when dividing off the wheel, by dividing it into as many divisions as there are teeth in the wheel, and then find the locations for the curves as in Fig. 131, in which 1, 2, 3 are points of divisions on the pitch circle P P, while A, B, struck from point 2, are centres wherefrom to strike the arcs E, F; C, D, struck also from point 2 are centres wherefrom to strike the flank curves G, H.

It will be noted that all the points serving as centres for the face curves, in Fig. 130, fall within a space; hence if the teeth were rudely cast in the wheel, and were to be subsequently cut or trimmed to the lines, some provision would have to be made to receive the compass points.

To obviate the necessity of finding the necessary radius from rolling segments various forms of construction are sometimes employed.

Thus Rankine gives that shown in Fig. 132, which is obtained as follows. Draw the generating circle D, and A D the line of centres. From the point of contact at C, mark on circle D, a point distance from C one-half the amount of the pitch, as at P, and draw the line P C of indefinite length beyond C. Draw a line from P, passing through the line of centres at E, which is equidistant between C and A. Then multiply the length from P to C by the distance from A to D, and divide by the distance between D and E. Take the length and radius so found, and mark it upon P C, as at F, and the latter will be the location of centre for compasses to strike the face curve.

Another method of finding the face curve, with compasses, is as follows: In Fig. 133, let P P represent the pitch circle of the wheel to be marked, and B C the path of the centre of the generating or describing circle as it rolls outside of P P. Let the point B represent the centre of the generating circle when that circle is in contact with the pitch circle at A. Then from B, mark off on B C any number of equidistant points, as D, E, F, G, H, and from A, mark on the pitch circle, points of division, as 1, 2, 3, 4, 5, at the intersection of radial lines from D, E, F, G, and H. With the radius of the generating circle, that is, A B, from B, as a centre, mark the arc I, from D the arc J, from E the arc K, &c., to M, marking as many arcs as there are points of division on B C. With the compasses set to the radius of divisions 1, 2, step off on arc M the five divisions, N, O, S, T, V, and V will be a point in the epicycloidal curves. From point of division 4, step off on L four points of division, as _a_, _b_, _c_, _d_, and _d_ will be another point in the epicycloidal curve. From point 3 set off three divisions on K, from point 2 two dimensions on L, and so on, and through the points so obtained, draw by hand or with a scroll the curve represented in the cut by curve A V.

Hypocycloids for the flanks of the teeth may be traced in a similar manner. Thus in Fig. 134 P P is the pitch circle, and B C the line of motion of the centre of the generating circle to be rolled within P P, and R a radial line. From 1 to 6 are points of equal division on the pitch circle, and D to I are arc locations for the centre of the generating circle. Starting from A, which represents the supposed location for the centre of the generating circle, the point of contact between the generating and base circles will be at B. Then from 1 to 6 are points of equal division on the pitch circle, and from D to I are the corresponding locations for the centres of the generating circle. From these centres the arcs J, K, L, M, N, O, are struck. From 6 mark the six points of division from _a_ to _f_, and _f_ is a point in the curve. Five divisions on N, four on M, and so on, give respectively points in the curve which is marked in the figure from A to _f_.

There is this, however, to be noted concerning the constructions of the last two figures. Since the circle described by the centre of the generating circle is of different arc or curve to that of the pitch circle, the chord of an arc having an equal length on each will be different. The amount is so small as to be practically correct. The direction of the error is to give to the curves a less curvature, as though they had been produced by a generating circle of larger diameter. Suppose, for example, that the difference between the arc N 5 (Fig. 133) and its chord is .1, and that the difference between the arc 4 5, and its chord is .01, then the error in one step is .09, and, as the point V is formed in 5 steps, it will contain this error multiplied five times. Point _d_ would contain it multiplied four times, because it has 4 steps, and so on.

The error will increase in proportion as the diameter of the generating is less than that of the pitch circle, and though in large wheels, working with large wheels (so that the difference between the radius of the generating circle and that of the smallest wheel is not excessive), it is so small as to be practically inappreciable, yet in small wheels, working with large ones, it may form a sensible error.

An instrument much employed in the best practice to find the radius which will strike an arc of a circle approximating the true epicycloidal curve, _and for finding at the same time_ the location of the centre wherefrom that curve should be struck, is found in the Willis' odontograph. This is, in reality, a scale of centres or radii for different and various diameters of wheels and generating circles. It consists of a scale, shown in Fig. 135, and is formed of a piece of sheet metal, one edge of which is marked or graduated in divisions of one-twentieth of an inch. The edge meeting the graduated edge at O is at angle of 75° to the graduated edge.

On one side of the odontograph is a table (as shown in the cut), for the flanks of the teeth, while on the other is the following table for the faces of the teeth:

TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE.

CENTRES FOR THE FACES OF THE TEETH.

Pitch in Inches and Parts.

+------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ |No. of|1/4|3/8|1/2|5/8|3/4| 1|1- |1- |1- | 2|2- |2- | 3|3- | |Teeth | | | | | | |1/4|1/2|3/4| |1/4|1/2| |1/2| |------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 12 | 1| 2| 2| 3| 4| 5| 6| 7| 9| 10| 11| 12| 15| 17| | 15 | ..| ..| 3| ..| ..| ..| 7| 8| 10| 11| 12| 14| 17| 19| | 20 | 2| ..| ..| 4| 5| 6| 8| 9| 11| 12| 14| 15| 18| 21| | 30 | ..| 3| 4| ..| ..| 7| 9| 10| 12| 14| 16| 18| 21| 25| | 40 | ..| ..| ..| ..| 6| 8| ..| 11| 13| 15| 17| 19| 23| 26| | | | | | | | | | | | | | | | | | 60 | ..| ..| ..| 5| ..| ..| 10| 12| 14| 16| 18| 20| 25| 29| | 80 | ..| ..| ..| ..| ..| 9| 11| 13| 15| 17| 19| 21| 26| 30| | 100 | ..| ..| ..| ..| 7| ..| ..| ..| ..| 18| 20| 22| ..| 31| | 150 | ..| ..| 5| 6| ..| ..| ..| 14| 16| 19| 21| 23| 27| 32| |Rack. | ..| 4| ..| ..| ..| 10| 12| 15| 17| 20| 22| 25| 30| 34| +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+

The method of using the instrument is as follows: In Fig. 136, let C represent the centre, and P the pitch circle of a wheel to contain 30 teeth of 3 inch arc pitch. Draw the radial line L, meeting the pitch circle at A. From A mark on the pitch circle, as at B, a radius equal to the pitch of the teeth, and the thickness of the tooth as A _k_. Draw from B to C the radial line E. Then for the flanks place the slant edge of the odontograph coincident and parallel with E, and let its corners coincide with the pitch circle as shown. In the table headed _centres for the flanks of the teeth_, look down the column of 3 inch pitch, and opposite to the 30 in the column of numbers of teeth, will be found the number 49, which indicates that the centre from which to draw an arc for the flank is at 49 on the graduated edge of the odontograph, as denoted in the cut by _r_. Thus from _r_ to the side _k_ of the tooth is the radius for the compasses, and at _r_, or 49, is the location for the centre to strike the flank curve _f_. For the face curve set the slant edge of the odontograph coincident with the radial line L, and in the table of centres for the faces of teeth, look down the column of 3-inch pitch, and opposite to 30 in the number of teeth column will be found the number 21, indicating that at 21 on the graduated edge of the odontograph, is the location of the centre wherefrom to strike the curve _d_ for the face of the tooth, this location being denoted in the cut at R.

The requisite number on the graduated edge for pitches beyond 3-1/2 (the greatest given in the tables), may be obtained by direct proportion from those given in the tables. Thus for 4 inch pitch, by doubling the numbers given for a 2 inch pitch, containing the same number of teeth, for 4-1/2 inch pitch by doubling the numbers given for a 2-1/4 inch pitch. If the pitch be a fraction that cannot be so obtained, no serious error will be induced if the nearest number marked be taken.

An improved form of template odontograph, designed by Professor Robinson of the Illinois School of Industry, is shown in Fig. 137.

In this instrument the curved edge, having graduated lines, approaches more nearly to the curves produced by rolling circles than can be obtained from any system in which an arc of a circle is taken to represent the curve; hence, that edge is applied direct to the teeth and used as a template wherefrom to mark the curve. The curve is a logarithmic spiral, and the use of the instrument involves no other labor than that of setting it in position. The applicability of this curve, for the purpose, arises from two of its properties: first, that the involute of the logarithmic spiral is another like spiral with poles in common; and, second, that the obliquity or angle between a normal and radius sector is constant, the latter property being possessed by this curve only. By the first property it is known that a line, lying tangent to the curve C E H, will be normal or perpendicular to the curve C D B; so that when the line D E F is tangent to the pitch line, the curve A D B will coincide very closely with the true epicycloidal curve, or, rather, with that portion of it which is applied to the tooth curve of the wheel. By the second quality, all sectors of the spiral, with given angle at the poles, are similar figures which admit of the same degree of coincidence for all similar epicycloids, whether great or small, and nearly the same for epicycloids in general; thus enabling the application of the instrument to epicycloids in general.

To set the instrument in position for drawing a tooth face a table which accompanies the instrument is used. From this table a numerical value is taken, which value depends upon the diameters of the wheels, and the number of teeth in the wheel for which the curve is sought. This tabular value, when multiplied by the pitch of the teeth, is to be found on the graduated edge on the instrument A D B in Fig. 137. This done, draw the line D E F tangent to the pitch line at the middle of the tooth, and mark off the half thickness of the tooth, as E, D, either on the tangent line or the pitch line. Then place the graduated edge of the odontograph at D, and in such a position that the number and division found as already stated shall come precisely on the tangent line at D, and at the same time so set the curved edge H F C so that it shall be tangent to the tangent line, that is to say, the curved edge C H must just meet the tangent line at some one point, as at F in the figure. A line drawn coincident with the graduated edge will then mark the face curve required, and the odontograph may be turned over, and the face on the other side of the tooth marked from a similar setting and process.

For the flanks of the teeth setting numbers are obtained from a separate table, and the instrument is turned upside down, and the tangent line D F, Fig. 137, is drawn from the side of the tooth (instead of from the centre), as shown in Fig. 138.

It is obvious that this odontograph may be set upon a radial arm and used as a template, as shown in Fig. 126, in which case the instrument would require but four settings for the whole wheel, while rolling segments and the making of templates are entirely dispensed with, and the degree of accuracy is greater than is obtainable by means of the employment of arcs of circles.

The tables wherefrom to find the number or mark on the graduated edge, which is to be placed coincident with the tangent line in each case, are as follows:--

TABLE OF TABULAR VALUES WHICH, MULTIPLIED BY THE ARC PITCH OF THE TEETH, GIVES THE SETTING NUMBER ON THE GRADUATED EDGE OF THE INSTRUMENT.

+--------------------+-----------------------------------------------------+ | | Number of Teeth in Wheel Sought; or, Wheel for | | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | 8 | 12 | 16 | 20 | 30 | 40 | 50 | 60 | 70 | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Faces: Flanks Radial or Curved._ | | RATIOS.[7] | Draw Setting Tangent at Middle of Tooth.-- | | | Epicycloidal Spur or Bevel Gearing. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | 1/12 = .083 | .32 | .39 | .46 | .51 | | | | | | | 1/4 = .250 | .31 | .37 | .44 | .49 | .61 | .70 | .78 | .85 | .92 | | 1/2 = .500 | .28 | .34 | .41 | .46 | .57 | .66 | .73 | .80 | .87 | | 2/3 = .667 | .27 | .32 | .38 | .43 | .54 | .62 | .70 | .77 | .83 | | 1 | .23 | .28 | .34 | .39 | .49 | .58 | .65 | .72 | .78 | | 3/2 = 1.50 | .19 | .25 | .29 | .34 | .44 | .51 | .58 | .64 | .69 | | 2 | .17 | .22 | .26 | .30 | .38 | .46 | .53 | .59 | .63 | | 3 | | .16 | .19 | .23 | .31 | .38 | .44 | .49 | .53 | | 4 | | .14 | .17 | .20 | .26 | .33 | .38 | .42 | .46 | | 6 | | | | | .22 | .26 | .30 | .34 | .37 | | 12 | | | | | | .20 | .23 | .25 | .28 | | 24 | | | | | | | | | | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | Number of Teeth in Wheel Sought; or, Wheel for| | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | 80 | 90 | 100 | 120 | 150 | 200 | 300 | 500 | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Faces: Flanks Radial or Curved._ | | RATIOS.[7] | Draw Setting Tangent at Middle of Tooth.-- | | | Epicycloidal Spur or Bevel Gearing. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ | 1/12 = .083 | | | | | | | | | | 1/4 = .250 | .99 | 1.05| 1.11| 1.22| 1.36| 1.55| 1.94| 2.54| | 1/2 = .500 | .93 | 1.00| 1.06| 1.15| 1.29| 1.50| 1.86| 2.41| | 2/3 = .667 | .89 | .95| 1.01| 1.11| 1.24| 1.45| 1.79| 2.32| | 1 | .83 | .89| .94| 1.03| 1.15| 1.36| 1.65| 2.10| | 3/2 = 1.50 | .74 | .79| .84| .93| 1.05| 1.25| 1.53| 1.94| | 2 | .68 | .72| .76| .84| .95| 1.13| 1.40| 1.81| | 3 | .57 | .60| .63| .71| .82| .97| 1.23| 1.60| | 4 | .49 | .53| .56| .63| .73| .87| 1.08| 1.42| | 6 | .41 | .44| .47| .53| .61| .71| .90| 1.20| | 12 | .30 | .32| .34| .37| .42| .49| .60| .82| | 24 | | .19| .21| .23| .26| .31| .40| .57| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ +--------------------+-----------------------------------------------------+ | | Number of Teeth in Wheel Sought; or, Wheel for | | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | 8 | 12 | 16 | 20 | 30 | 40 | 50 | 60 | 70 | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Flanks, when Curved._ | | | Draw Setting Tangent at Side of Tooth.-- | | | Epicycloidal Spur and Bevel Gearing. | |D C | Faces of Internal, and Flanks of Pinion Teeth. | |e u +-----+-----+-----+-----+-----+-----+-----+-----+-----+ |g F r { 1.5 slight.| .77 | .98 | 1.18| 1.36| 1.75| 2.05| 2.31| 2.56| 2.75| |r l v { 2 good. | .44 | .54 | .63| .72 | .92| 1.09| 1.24| 1.38| 1.49| |e a a { 3 more. | .20 | .28 | .35| .40 | .54| .65| .76| .86| .95| |e n t { 4 much. | | .20 | .23| .25 | .34| .42| .51| .59| .66| | k u { 6 | | | .16| .17 | .26| .32| .38| .43| .48| |o r {12 | | | | | .19| .24| .28| .31| .34| |f e {24 | | | | | | | | | | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | Number of Teeth in Wheel Sought; or, Wheel for| | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | 80 | 90 | 100 | 120 | 150 | 200 | 300 | 500 | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Flanks, when Curved._ | | | Draw Setting Tangent at Side of Tooth.-- | | | Epicycloidal Spur and Bevel Gearing. | |D C | Faces of Internal, and Flanks of Pinion Teeth.| |e u +-----+-----+-----+-----+-----+-----+-----+-----+ |g F r { 1.5 slight.| 2.92| 3.08| 3.24| 3.52| 3.87| 4.51| 5.50| 7.20| |r l v { 2 good. | 1.59| 1.79| 1.79| 1.98| 2.23| 2.67| 3.22| 4.50| |e a a { 3 more. | 1.02| 1.10| 1.18| 1.31| 1.46| 1.67| 2.08| 2.76| |e n t { 4 much. | .71| .77| .82| .92| 1.06| 1.25| 1.64| 2.15| | k u { 6 | .52| .56| .60| .66| .76| .93| 1.20| 1.54| |o r {12 | .36| .38| .40| .45| .52| .63| .80| .98| |f e {24 | | | .22| .25| .28| .33| .47| .60| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ +--------------------+-----------------------------------------------------+ | | Number of Teeth in Wheel Sought; or, Wheel for | | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | 8 | 12 | 16 | 20 | 30 | 40 | 50 | 60 | 70 | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | _For Faces of Racks; and of Pinions for Racks and Internal Gears; for | | Flanks of Internal and Sides of Involute Teeth._ | | Draw Setting Tangent at Middle of Tooth, regarding Space as Tooth in | | Internal Teeth. For Rack use Number of Teeth in Pinion. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | Pinion. | .31 | .39 | .48 | .57 | .73 | .88 | 1.00| 1.10| 1.20| | Rack. | .32 | .38 | .44 | .50 | .62 | .72 | .80| .87| .93| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | Number of Teeth in Wheel Sought; or, Wheel for| | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | 80 | 90 | 100 | 120 | 150 | 200 | 300 | 500 | | +-----+-----+-----+-----+-----+-----+-----+-----+ | _For Faces of Racks; and of Pinions for Racks and Internal Gears; | | for Flanks of Internal and Sides of Involute Teeth._ | |Draw Setting Tangent at Middle of Tooth, regarding Space as Tooth in| | Internal Teeth. For Rack use Number of Teeth in Pinion. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ | Pinion. | 1.30| 1.40| 1.48| 1.65| 1.85| 2.15| 2.65| 3.50| | Rack. | .99| 1.03| 1.08| 1.16| 1.27| 1.49| 1.86| 2.44| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+

[7] These ratios are obtained by dividing the radius of the wheel sought by the diameter of the generating circle.

From these tables may be found a tabular value which, multiplied by the pitch of the wheel to be marked (as stated at the head of the table), will give the setting number on the graduated edge of the instrument, the procedure being as follows:--

For the teeth of a pair of wheels intended to gear together only (and not with other wheels having a different number of teeth).

For the face of such teeth where the flanks are to be radial lines.

Rule.--Divide the pitch circle radius of the wheel to have its teeth marked by the pitch circle radius of the wheel with which it is to gear: or, what is the same thing, divide the number of teeth in the wheel to have its teeth marked by the number of teeth in the wheel with which it is to gear, and the quotient is the "ratio." In the ratio column find this number, and look along that line, and in the column at the head of which is the number of teeth contained in the wheel to be marked, is a number termed the tabular value, which, multiplied by the arc pitch of the teeth, will give the number on the graduated edge by which to set the instrument to the tangent line.

Example.--What is the setting number for the face curves of a wheel to contain 12 teeth, of 3-inch arc pitch, and to gear with a wheel having 24 teeth?

Here number of teeth in wheel to be marked = 12, divided by the number of teeth (24) with which it gears; 12 ÷ 24 = .5. Now in column of ratios may be found 1/2 = .500 (which is the same thing as .5), and along the same horizontal line in the table, and in the column headed 12 (the number of teeth in the wheel) is found .34. This is the tabular value, which, multiplied by 3 (the arc pitch of the teeth), gives 1.02, which is the setting number on the graduated edge. It will be noted, however, that the graduated edge is marked 1, 2, 3, &c., and that between each consecutive division are ten subdivisions; hence, for the decimal .02 an allowance may be made by setting the line 1 a proportionate amount below the tangent line marked on the wheel to set the instrument by.

Required now the setting number for the wheel to have the 24 teeth.

Here number of teeth on the wheel = 24, divided by the number of teeth (12) on the wheel with which it gears; 24 ÷ 12 = 2. Now, there is no column in the "number of teeth sought" for 24 teeth; but we may find the necessary tabular value from the columns given for 20 teeth and 30 teeth, thus:--opposite ratio 2, and under 20 teeth is given .30, and under 30 teeth is given .38--the difference between the two being .08. Now the difference between 20 teeth and 24 teeth is 4/10; hence, we take 4/10 of the .08 and add it to the tabular value given for 20 teeth, thus: .08 × 4 ÷ 10 = .032, and this added to .30 (the tabular value given for 20 teeth = .33, which is the tabular value for 24 teeth). The .33 multiplied by arc pitch (3) gives .99. This, therefore, is the setting number for the instrument, being sufficiently near to the 1 on the graduated edge to allow that 1 to be used instead of .99.

It is to be noted here that the pinion, having radial lines, the other wheel must have curved flanks; the rule for which is as follows:--

CURVED FLANKS FOR A PAIR OF WHEELS.

Note.--When the flanks are desired to be curved instead of radial, it is necessary to the use of the instrument to select and assume a value for the degree of curve, as is done in the table in the column marked "Degree for flank curving;" in which

1.5 slight--a slight curvature of flank. 2 good--an increased curvature of flank. 3 more--a degree of pronounced spread at root. 4 much--spread at root is a distinguishing feature of tooth form. 6--still increased spread in cases where the strength at root of pinion is of much importance to give strength. 12--as above, under aggravated conditions. 24--undesirable (unless requirement of strength compels this degree), because of excessive strain on pinion.

Rule.--For faces of teeth to have curved flanks.

Divide the number of teeth in the wheel to be marked by the number of teeth in the wheel with which it gears, and multiply by the degree of flank curve selected for the wheel with which that to be marked is to gear, and this will give the ratio. Find this number in ratio column, and the tabular number under the column of number of teeth of wheel to be marked; multiply tabular number so found by arc pitch of wheel to be marked, and the product will be the setting number for the instrument.

Example.--What is the setting number on the graduated edge of the odontograph for the faces of a wheel (of a pair) to contain 12 teeth of 2-inch arc pitch, and to gear with a wheel having 24 teeth and a flank curvature represented by 3 in "Degree of flank curving" column?

Here teeth in wheel to be marked (12) divided by number of teeth in the wheel it is to gear with (24), 12 ÷ 24 = .5, which multiplied by 3 (degree of curvature selected for flanks of 24-teeth wheel), .5 × 3 = 1.5. In column of ratio numbers find 1.5, and in 12-teeth column is .25, which multiplied by pitch (2) gives .5 as the setting number for the instrument; this being the fifth line on the instrument, and half way between the end and mark 1.

FOR CURVED FLANKS.

Rule.--Assume the degree of curve desired for the flanks to be marked, select the corresponding value in the column of "Degrees of flank curving," and find the tabular value under the number of teeth column.

Multiply tabular value so found by the arc pitch of the teeth, and the product is the setting number on the instrument.

Example.--What is the setting number on the odontograph for the flanks of a wheel to contain 12 teeth and gear with one having 24 teeth, the degree of curvature for the flanks being represented by 4 in the column of "Degree of flank curvature?"

Here in column of degrees of flank curvature on the 3 line and under 12 teeth is .20, which multiplied by pitch of teeth (2) is .20 × 2 = 40, or 4/10; hence, the fourth line of division on the curved corner is the setting line, it representing 4/10 of 1.

FOR INTERCHANGEABLE GEARING (THAT IS, A TRAIN OF GEARS ANY ONE OF WHICH WILL WORK CORRECTLY WITH ANY OTHER OF THE SAME SET).

Rule--both for the faces and for the flanks. For each respective wheel divide the number of teeth in that wheel by some one number not greater than the number of teeth in the smallest wheel in the set, which gives the ratio number for the wheel to be marked. On that line of ratio numbers, and in the column of numbers of teeth, find the tabular value number; multiply this by the arc pitch of the wheel to be marked, and the product is the setting number of the instrument.

Example.--A set of wheels is to contain 10 wheels; the smallest is to contain 12 teeth; the arc pitch of the wheels is four inches. What is the setting number for the smallest wheel?

Here number of teeth in smallest wheel of set is 10; divide this by any number smaller than itself (as say 5), 10 ÷ 5 = 2 = the ratio number on ratio line for 2; and under column for 12 is .17, which is the tabular value, which multiplied by pitch (4) is .17 × 4 = 68, or 6/10 and 8/100; hence, the instrument must be set with its seventh line of division just above the tangent line marked on the wheel. It will be noted that, if the seventh line were used as the setting, the adjustment would be only the 2/100 of a division out, an amount scarcely practically appreciable.

Both for the faces and flanks, the second number is obtained in _precisely_ the same manner for every wheel in the set, except that instead of 10 the number of teeth in each wheel must be substituted.

RACK AND PINION.--_For radial flanks_ use for faces the two lower lines of table. _For curved flanks_ find tabular value for pinion faces in lowest line. For flanks of pinion choose degree of curving, and find tabular value under "flanks," as for other wheels. For faces of rack divide number of teeth in pinion by degree of curving, which take for number of teeth in looking opposite "rack." Flanks of rack are still parallel, but may be arbitrarily curved beyond half way below pitch line.

INTERNAL GEARS.--For tooth curves within the pitch lines, divide radius of each wheel by any number not greater than radius of pinion, and look in the table under "flanks." For curves outside pitch line use lower line of table; or, divide radii by any number and look under "faces." In applying instrument draw tangents at middle and side of _space_, for internal teeth.

INVOLUTE TEETH.--For tabular values look opposite "Pinion," under proper number of teeth, for each wheel. Draw setting tangent from "base circle" of involute, at middle of tooth. For this the instrument gives the whole side of tooth at once.

In all cases multiply the tabular value by the pitch in inches.

BEVEL-WHEELS.--Apply above rules, using the developed normal cone bases as pitch lines. For right-angled axes this is done by using in place of the actual ratio of radii, or of teeth numbers, the square of that ratio; and for number of teeth, the actual number multiplied by the square root of one plus square of ratio or radii; the numerator of ratio, and number of teeth, belonging to wheel sought.

When the first column ratio and teeth numbers fall between those given in the table, the tabular values are found by interpolating as seen in the following examples:

EXAMPLES OF TABULAR VALUES AND SETTING NUMBERS.

_Take a pair of 16 and 56 teeth; radii 5.09 and 17.82 inches respectively; and 2 inches pitch._

+----------------+------+----------------+------+---------------+-------+ | |Number} | | First Column | Tab. | |Kind of Gearing.| of } Kind of Flank. |Ratio | Ratio. | Val. | | |Teeth.} |Radii.+--------+------+---+---+ | | | | | Flank. |Face. |[A]|[B]| +----------------+------+----------------+------+--------+------+---+---+ |Epicycloidal, }|Small |Radial | .29 |Radial | .29 |.. |.44| |Radial Flanks }|Large |Radial | 3.5 |Radial | 3.5 |.. |.44| |Epicycloidal, } |Small |Curved 2 deg. | .29 | 2 | .87 |.63|.36| |Curved Flanks.} |Large |Curved 3 deg. } | 3.5 | 3 | 7. |.82|.30| |Epicycloidal, }|Small |"Sets," Divide} | 2. | 2 | 2. |.63|.26| |Interchange'bl.}|Large |Radii by 2.55 } | 7. | 7 | 7. |.40|.30| |Epicycloidal, } |Pinion|Curved 2 deg. | | 2 |Pinion|.63|.44| |Internal. } |Wheel |Int. face 7 deg.| 3.5 |Pinion | 7[8] |.84|.39| |Epicycloidal, }|Pinion|Curved 2 deg. | | 2 |Pinion|.63|.44| |Rack & Pinion. }|Rack |Parallel | |Parallel|Rack |.. |.31| |Involute } |Small |Face and Flank | | Pinion. | .44 | |Gearing. } |Large |One Curve | | Pinion. | .84 | +----------------+------+----------------+------+--------+------+-------+

Legend: A = Flank. B = Face.

[8] The face being here internal, the tabular value is to be found under "flanks." If bevels, use ratio radii .082 and 12.25; and teeth numbers 16.6 and 203.8 respectively.

WALKER'S PATENT WHEEL SCALE.--This scale is used in many manufactories in the United States to mark off the teeth for patterns, wherefrom to mould cast gears, and consists of a diagram from which the compasses may be set to the required radius to strike the curves of the teeth.

The general form of this diagram is shown in Fig. 139. From the portion A the length of the teeth, according to the pitch, is obtained. From the portion B half the thickness of the tooth at the pitch line is obtained. From the part C half the thickness at the root is obtained, and from the