The usual proportions are for cast gears: Addendum =.3 X circular pitch. Root = .4 X circular pitch. Thickness of tooth = .48 x circular pitch.

The gears most often met with are the cut gears of small and medium size like those, for example, on machine tools, which are almost invariably diametral-pitch gears. The teeth are cut from the solid with standard milling cutters, proportioned with the diametral pitch as a basis. This system is also coming into use for cast gearing. In all diametral-pitch gears, the addendum, in inches, is made equal to 1 divided by the diametral pitch, and the working depth to twice the addendum. The end clearance is usually taken equal to the addendum for cut gears, though The Brown & Sharpe Manufacturing Company use the thickness of the tooth on the pitch line as the clearance. The side clearance, or "backlash," is barely enough to give a good working fit, and seldom exceeds the pitch.

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Using the above proportions, a 4-pitch gear will have the addendum 14 inch; the working depth will be 2x inch; and the clearance, if made the addendum, Xinch. The whole length of the tooth will be + inch. The thickness of the tooth will be onehalf the circular pitch, nearly. In a 10-pitch wheel, the addendum will be inch and the length of the tooth 1 inch; in a 2-pitch, it will be 12 inch and the length 1 inch.

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FORMS OF GEAR-TEETH.

50. The forms of teeth used in ordinary practice form part of certain curves known as the epicycloid, hypocycloid, and involute.

51. The Epicycloidal Tooth.-In the so-called epicycloidal tooth, which more properly is called a cycloidal tooth, the face of the tooth is part of an epicycloid, and the flank, part of a hypocycloid.

An epicycloidal curve is the path described by any point of a circle rolling, without slipping, on the outside of another circle. A hypocycloid is the path described by any point of a circle rolling, without slipping, on the inside of another circle.

Epicycloidal teeth can always be recognized by their appearance; they are formed by two curves that, commencing at the pitch circle, curve in opposite directions. Fig. 25 clearly exhibits the characteristic tooth form.

The use of epicycloidal teeth is being gradually abandoned, as they possess some practical defects, the chief defect being that the center-to-center distance of the two gears must be practically perfect in order to insure a uniform velocity of the driven gear.

52. Involute Teeth.-In Fig. 26 is shown the involute form of tooth, which is composed of but one curve.

The involute is the path described by any point of a string that is being wound on or off a cylinder, the cylinder being stationary. In the involute system, the sides of the teeth of the rack are straight lines, as shown in Fig. 26.

Involute teeth have two great advantages over epicycloidal teeth: (1) They are stronger for the same pitch, as they are thicker at the root. (2) (2) The gears may be spread

slightly apart so that their pitch circles do not run tangent to

FIG. 26.

one another, without affecting the perfect action of the teeth to an appreciable extent.

GEAR CALCULATIONS.

53. The Circular-Pitch System.-For calculating the pitch diameter, number of teeth, etc. of gear-wheels, we have, for the circular-pitch system, the following rules, where

P

circular pitch in inches;

T= number of teeth;

D=

pitch diameter of the gear in inches.

54. To find the pitch diameter of a gear-wheel in inches, when the pitch and number of teeth are given:

Rule 13.-The pitch diameter equals the product of the pitch and the number of teeth divided by 3.1416.

EXAMPLE.-What is the diameter of the pitch circle of a gear-wheel that has 75 teeth and whose pitch is 1.675 inches?

SOLUTION.-Applying rule 13, we have

D=

1.675 X 75

55. To find the number of teeth in a gear-wheel when the diameter and pitch are given:

Rule 14.-The number of teeth equals the product of 3.1416 and the diameter divided by the pitch.

EXAMPLE. The diameter of a gear-wheel is 40 inches and the pitch of the teeth is 1.675 inches; how many teeth are there in the wheel? SOLUTION.-Applying the rule just given, we have

56. To find the pitch of a gear-wheel when the diameter and the number of teeth are given:

Rule 15.-The pitch of the teeth equals the product of 3.1416 and the diameter divided by the number of teeth.

EXAMPLE. The diameter of a gear-wheel is 40 inches and it has 75 teeth; what is the pitch of the teeth?

57. The Diametral-Pitch System. The diameter of the gear-wheel, the number of teeth, etc. are given by the following rules, where Pa = diametral pitch; D, outside diameter; N = number of teeth; and the other letters have the same meaning as in the three preceding rules.

d

58. To find the pitch diameter of the gear-wheel when the number of teeth and the pitch are given:

Rule 16.-Divide the number of teeth by the diametral

EXAMPLE.-A wheel is to have 40 teeth, 4 pitch; what is its pitch diameter ?

SOLUTION.-By applying rule 16, we have

59. To find the diameter over all, that is, the diameter of the blank from which the gear-wheel is cut, the number of teeth and the diametral pitch being given:

Rule 17.-Add 2 to the number of teeth and divide by the diametral pitch.

EXAMPLE. In the last example, what is the diameter over all the blank?

SOLUTION.--Applying the rule just given, we get

60. The number of teeth and the outside diameter of the gear-wheel being known, to find the diametral pitch: Rule 18.-Add 2 to the number of the teeth and divide by the outside diameter.

EXAMPLE. A gear-wheel has 60 teeth and is 6 inches in diameter over all; what is the diametral pitch?

SOLUTION.-By applying rule 18, we get

61. To find the diametral pitch, the number of teeth and the pitch diameter being known:

Rule 19.-Divide the number of teeth by the pitch diam

EXAMPLE.-A wheel has 90 teeth and its pitch diameter is 30 inches; what is the diametral pitch?