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 circ.u.mference 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 s.p.a.ces 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 exact.i.tude.

[Ill.u.s.tration: Fig. 112.]

[Ill.u.s.tration: Fig. 113.]

First, then, with regard to the thickness of tooth and width of s.p.a.ce.

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 ill.u.s.trate 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 circ.u.mference to diameter.

8 = diameter.

------- Number of teeth = 12 ) 25.1328 = circ.u.mference.

------- 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 s.p.a.ce, 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 s.p.a.ces 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:--