Machinery's Handbook, 31st Edition
Gears and Gearing 2209 Properties of the Involute Curve.— The involute curve is used almost exclusively for gear- tooth profiles, because of the following important properties. 1) The form or shape of an involute curve depends upon the diameter of the base circle from which it is derived. (If a taut line were unwound from the circumference of a circle— the base circle of the involute—the end of that line or any point on the unwound portion would describe an involute curve.) 2) If a gear tooth of involute curvature acts against the involute tooth of a mating gear while rotating at a uniform rate, the angular motion of the driven gear will also be uniform, even though the center-to-center distance is varied. 3) The relative rate of motion between driving and driven gears having involute tooth curves is established by the diameters of their base circles. 4) Contact between intermeshing involute teeth on a driving and driven gear is along a straight line that is tangent to the two base circles of these gears. This is the line of action . 5) The point where the line of action intersects the common centerline of the mating involute gears establishes the radii of the pitch circles of these gears; hence true pitch cir cle diameters are affected by a change in the center distance. (Pitch diameters obtained by dividing the number of teeth by the diametral pitch apply when the center distance equals the total number of teeth on both gears divided by twice the diametral pitch.) 6) The pitch diameters of mating involute gears are directly proportional to the diameters of their respective base circles; thus, if the base circle of one mating gear is three times as large as the other, the pitch circle diameters will be in the same ratio. 7) The angle between the line of action and a line perpendicular to the common centerline of mating gears is the pressure angle ; hence the pressure angle is affected by any change in the center distance. 8) When an involute curve acts against a straight line (as in the case of an involute pinion acting against straight-sided rack teeth), the straight line is tangent to the involute and per pendicular to its line of action. 9) The pressure angle, in the case of an involute pinion acting against straight-sided rack teeth, is the angle between the line of action and the line of the rack’s motion. If the involute pinion rotates at a uniform rate, movement of the rack will also be uniform. Nomenclature: φ = Pressure Angle a = Addendum a G = Addendum of Gear a P = Addendum of Pinion
b = Dedendum c = Clearance C = Center Distance D = Pitch Diameter
D G = Pitch Diameter of Gear
D P = Pitch Diameter of Pinion D R = Root Diameter
D B = Base Circle Diameter F = Face Width h k = Working Depth of Tooth
D O = Outside Diameter
h t = Whole Depth of Tooth
m G = Gear Ratio N = Number of Teeth p = Circular Pitch
N G = Number of Teeth in Gear
N P = Number of Teeth in Pinion
P = Diametral Pitch Diametral and Circular Pitch Systems.— Gear tooth system standards are established by specifying the tooth proportions of the basic rack. The diametral pitch system is applied to most of the gearing produced in the United States. If gear teeth are larger than about one diametral pitch, it is common practice to use the circular pitch system. The circular pitch system is also applied to cast gearing and it is commonly used in connection with the design and manufacture of worm gearing. Pitch Diameters Obtained with Diametral Pitch System.— The diametral pitch system is arranged to provide a series of standard tooth sizes, the principle being similar to the standardization of screw thread pitches. Inasmuch as there must be a whole number of
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