General Theory Of Gears

Fig. 111 represents a pair of disks fastened to shafts A and B, respectively, and touching at the point P. If these disks be pressed tightly against each other, sufficient friction will be produced between them to cause one to drive the other. The number of revolutions B would make in a given time, would be to the number of revolutions made by A as A P is to BP; or, Revolutions B=AP Revolutions A BP.

Such friction disks will transmit but very little power without slipping; and even when required to transmit small power, cannot be depended upon to drive positively, as the least wear or loss of adjustment is liable to make them slip. Hence teeth are provided on each disk, such that they will lock together and make it sure that when one disk is rotated the other must move also, without regard to whether, the disks are pressed tightly together or not. In fact, it is desirable that this side pressure be avoided, in order to prevent excessive friction in the bearings of shafts A and B.

Any shapes whatsoever of teeth would answer, provided they interlocked, so far as positive driving is concerned. But in order that the revolutions of the shafts shall always be inversely as the contact radii, or:

Revolutions B = AP.

Revolutions A BP it can be shown by geometry that the common normal drawn through the point of contact of any pair of teeth must always pass through the point P.

A pair of gears, therefore, may be considered to be based on two disks, touching as in Fig. 111, and provided with teeth such that these two conditions are fulfilled:

1. Positive driving at all times.

2. The common normal through the point of contact of any pair of teeth always passing through the pitch point.

Pitch Circles

The circles corresponding to the disks are known as pitch circles, their diameters pitch diameters, and the point of contact P the pitch point (see Fig. 112). The distance, measured radially, from the pitch circle to the top of the tooth is called the addendum; and the circle through the top of the tooth, the addendum circle. The distance, measured radially, from the pitch circle to the beginning of the fillet at the bottom of the tooth, is called the dedendum; and the circle through this point the dedendum circle. In order that the top of the tooth on one gear shall not strike the surface between the bottoms of the teeth on the other, a further distance is allowed between the dedendum circle and the root circle, known as the clearance. The distance from the center of one tooth to the center of the next, measured on the pitch circle, is called the circular pitch, and is evidently equal to the circumference of the pitch circle divided by the number of teeth.

Diagram of Simple Gear Principle.

Fig. 111. Diagram of Simple Gear Principle.

In order to run together, two gears must have the same circular pitch. The number of teeth in a pair of gears is proportional to the circumference of the pitch circles, and therefore to the pitch diameters, or pitch radii. The speeds of the shafts carrying the gears, being inversely proportional to the diameters of the pitch circles, are also inversely proportional to the numbers of teeth.

Layout for Pair of Gears, Showing Construction Features.

Fig. 112. Layout for Pair of Gears, Showing Construction Features.

Circular And Diametral Pitches

Since the circular pitch is equal to the circumference of the pitch circle divided by the number of teeth, there is a fixed relation, for any given gear, between the pitch diameter and the number of teeth. This relation is known as diametral pitch. Diametral pitch is not a distance, like circular pitch, but is the number of teeth per inch of pitch diameter of the gear. For example, if the diameter of the pitch circle of a gear of 60 teeth were 20", the number of teeth per inch of diameter would be 20 = 3, and the gear would be described as a "60-tooth, 3 diametralpitch gear". The product of the circular pitch times the diametral pitch, is always equal to the constant, 3.1416; that is, if we have one kind of pitch, and wish to change to the other, we divide 3.1416 by the given pitch. For example, 4 diametral pitch is equal to 3.1416 = .7854" circular pitch. Again, 2" circular pitch is equal to 3.1416 = 1.57 diametral pitch. Note carefully that diametral pitch is not "inches", but number of teeth per inch of diameter.

Diametral pitch is very convenient to use, as the calculation is simpler than with circular pitch, and the pitch diameters of the gears come in even figures, or in even fractions of the pitch. For machine-cut gears it is universal practice to use diametral pitch in the specification. For cast gears, where the teeth are fashioned by the pattern maker, it is common to use circular pitch.

The thickness of the tooth LM, Fig. 112, is practically the same as the space TL for machine-cut gears. For cast teeth, however, the tooth must be thinner than the space, to allow for the inaccuracies of the pattern and casting. This allowance measured on the pitch circle is called backlash.

Discussion Of Terms

These terms are illustrated in Fig. 112; also the common normal K P to a pair of teeth in contact. Gear A, being the driver in the direction shown, a pair of teeth are in contact at point K. The curves of the teeth being of the correct shape, if a common tangent be drawn, and a perpendicular erected at the point of tangency K, it will pass through the pitch point P. Now, as the gears move in the direction of the arrows, the teeth slide upon each other, and the point of contact changes, coming closer and closer to point P, then passing through P, and, going on, reaches some point as J, which, in the present example, represents the second pair of teeth in contact. During all this motion of the teeth, the common normal at every point of contact will pass through the pitch point P, thus fulfilling the condition of uniform velocity.

Pressure Line

It will be remembered that the pressure line between two surfaces, as illustrated in the discussion of cams, is the common normal at the point of contact. Now, a pair of gear teeth is like a cam and its follower; and if we wish to find the direction of the pressure between them, we simply draw the common normal.

Hence, knowing that with the teeth of proper outline the common normal will pass through the pitch point, we merely find the point of contact of any pair of teeth and connect it by a straight line to the pitch point, thus giving the direction of pressure between the teeth at the given position.