Chapter III: THE TEETH OF GEAR-WHEELS (continued) (1)

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.