cles c and d will be the pitch circles. Such gears, called knuckle gears, are sometimes employed on slow-moving work where no special accuracy is required. They will not transmit speed uniformly. If the driver of such a pair of gears rotated at a uniform rate, the driven gear would have a more or less jerky movement as the successive teeth came into contact, and if run at high speed they would be noisy. Various curves may be employed to give to gear teeth such an outline that the driver of a pair of gears will impart a uniform speed to the driven one, but in common practice only two kinds are used, the cycloidal, or, as it is sometimes called, epicycloidal, and the involute.

Epicycloidal Gearing.-Let the circles a, b and c, Fig. 158, having their centers on the same straight line, be made to rotate so that their circumferences roll upon each other without slipping. If the circle c has tracing points 1, 2, 3 upon its circumference, and when we start to rotate the circles point 1 is half way around from the position in which it is shown, then in rotating the circles sufficiently to bring the tracing points to the position in which they are shown, point 1 will trace the line 1' inwardly from the circle a, and the line 1" outwardly from the circle b. Point 2 will trace the two lines which are shown meeting at that point, one inwardly from the circle a, and one outwardly from the circle b. Point 3 will similarly trace the two lines which met at that point. Inasmuch as these lines were traced simultaneously by points at a fixed distance apart, it is evident that if the circle c were to be removed, and the circles

a and b were rolled back upon each other, these lines would work smoothly together, being in contact and tangent to each other at all times upon the line of the circle c. If the circle c is now placed beneath the circle b in the position shown, and the three circles are rolled together as before, the tracing points would trace lines inwardly from b, and

outwardly from a, which would also work together smoothly if the circle c were removed and the circles a and b were rolled back upon each other. It is evident that as the three circles are rolled together the lines formed by the tracing points are the same as though either a or b were taken by itself, and the circle c were rolled either within or upon it, hence the lines formed by the tracing points are either epicycloids or hypocycloids as the case may be, and so could be formed by the

plotting method described in the geometrical problems.

POINT

FACE

ADDENDUM

If these two sets of lines are now joined together so that the lines which extend inwardly from a or b form a continuation of those which extend outwardly and reverse curves are made at a distance from the first set equal to the thickness of a gear tooth, and they are then cut off at such a distance both outside and inside of the circles a and b as to give to the teeth the proper length, it is evident that we will have a pair of perfectly working gears. The circles a and b would roll upon each other without slipping and hence would be true pitch circles. The teeth would work smoothly together in constant contact, the point of contact being always on the line of the generating circle.

ROOTS

PITCH
LINE

DEDENDUM

CLEARANCE

FIG. 160.-Definitions of
Gear Tooth Terms.

The length of the point of the gear tooth, that is the portion lying outside of the pitch line, is usually made one-third of the circular pitch-the latter being the distance between the teeth measured from center to center on the pitch line. The distance below the pitch line is made somewhat greater for the sake of clearance. For the names of the various parts of a gear tooth see Fig. 160. Cast gears have some backlash between the teeth to allow for the roughness of the castings, as shown in Figs. 161 and 163.

It is evident that if another circle, either larger or smaller, were substituted for b in Fig. 158, the

lines formed by the generating circle c either within or upon the circle a would remain unchanged. Or if a different circle were substituted for a, the curves formed within or upon b would remain unchanged. Hence it follows that all gears in the epicycloidal system, having their teeth formed by the same generating circle and made of the same

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FIG. 162.-Rack with
Epicycloidal Teeth.

FIG. 161.-Gears with Epicycloidal

Teeth.

size, will work together correctly, or, as it is commonly expressed, are interchangeable.

In standard interchangeable gears the generating circle is made one-half the diameter of the smallest gear of the set, which has twelve teeth. This smallest gear will have radial flanks, as that part of the working surface lying within the pitch line is called, because the hypocycloid of a circle formed by a generating circle of half its size will be a straight line passing through its center.

Fig. 161 shows a portion of a pair of such gears, Fig. 162 showing the rack.

Gears with Strengthened Flanks. A further examination of Fig. 158 will show that the curves formed by the generating circle when it is in the upper of the two positions in which it appears, work together by themselves, and those formed when it is in the lower position work similarly, so that it is not necessary that the same sized gener

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FIG. 164.-Rack with
Involute Teeth.

FIG. 163.-Gears with Involute Teeth.

ating circle should be used in both positions, unless the gears are to be members of an interchangeable set of gears. Advantage may be taken of this fact to strengthen the roots of the teeth in a pinion.

If, for instance, in Fig. 161, a smaller generating circle were used in the upper position, the effect would be to broaden out the roots of the teeth in the pinion, and to correspondingly round off the points of the teeth of the other gear.

Gears with Radial Flanks.-Another modification which may be made is to have the teeth of both gears with radial flanks. If, for instance, in Fig. 161 a generating circle were to be used in the