Gearing basics

July 1, 2002
Here to answer many often-encountered gear questions is a quick reference that includes formulas and definitions.

Gears are toothed wheels that transmit motion from one shaft to another and determine the speed, torque, and direction of rotation of machine elements. Gears can be grouped into two major categories: those that operate in pairs to connect parallel shafts and those that connect nonparallel shafts. Parallel types include spur and helical. Nonparallel types include bevel, hypoid, and worm.

Types of gears

Spur gears. Cylindrical gears with teeth that are straight and parallel to the axis of rotation.

Helical gears. Cylindrical gears with teeth at an angle to the axis of rotation.

External gear. A gear with teeth on the outside of a cylinder or cone.

Internal gear. A gear with teeth on the inside of a hollow cylinder. Both spur and helical gears can be made as internal gears. The mating gear for an internal gear must be an external gear.

Bevel gears. Gears with teeth on the outside of a cone-shaped body. Teeth may be straight or spiral. Normally used at right angles (perpendicular to each other).

Worm gears. Gearsets in which one member of the pair has teeth wrapped around a cylinder like screw threads. Normally, this gear, called the worm, is at a right angle to the mating gear. Worm gears may be cylindrical, single-enveloping, or double-enveloping. Enveloping designs have curved worm or gear-tooth shapes to obtain more tooth contact area.

Face gears. Gears with teeth on the end of the cylinder.

Hypoid gears. Similar to bevel gears, but they operate on nonintersecting axes.

Pinion. Where two gears run together, the one with the smaller number of teeth is called the pinion.

Rack. A gear with teeth spaced along a straight line, and suitable for straightline motion.

Elements of gear teeth

Tooth surface. The side of a gear tooth.

Tooth profile. One side of a tooth in a cross section between the outside circle and the root circle.

Involute. A tooth profile generated from the involute of a circle. A common tooth shape for spur gears.

Base circle. The circle from which involute tooth profiles are derived.

Flank. The working or contacting side of a tooth. Usually has an involute profile in a transverse section.

Top land. The top surface of a gear tooth.

Bottom land. The surface at the bottom of the space between adjacent teeth.

Crown. A modification consisting of a slight outward bulge in the center of the tooth flank. The tooth becomes gradually thinner toward each end. A fully crowned tooth has a little material removed at the tip and root areas also. The purpose of crowning is to ensure that the center of the flank carries its full share of the load even if the gears are slightly misaligned or distorted.

Root circle. A tangent to the bottom of the tooth spaces in a cross section.

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Pitch circle. A circle that contains the pitch point. Pitch circles are tangent in mating gears. A circle at which gear teeth theoretically roll without slipping.

Gear center. The center of the pitch circle.

Line of centers. A line connecting the centers of the pitch circles of two engaging gears. It is also the common perpendicular of the axes in crossed helical gears and worm gears.

Pitch point. The point of a tooth profile which lies on the pitch circle of the gear. As the pitch point of one gear contacts its mating gear, the contact occurs at the pitch point of the mating gear. This common pitch point lies on a line connecting the two gear centers.

Path of action. A curve along which contact occurs during the engagement of two tooth profiles.

Line of action. The path of action for involute gears. It is the straight line passing through the pitch point and tangent to the base circle.

Line of contact. The line or curve along which two tooth surfaces are tangent to each other.

Point of contact. Any point at which two tooth profiles touch each other.

Linear and circular dimensions

Center distance. The distance between parallel axes of spur gears or parallel helical gears, or the crossed axes of crossed helical gears or of worms and worm gears. Also, it is the distance between the centers of the pitch circles.

Offset. The perpendicular distance between the axes of hypoid gears or offset face gears.

Pitch. The distance between similar, equally spaced tooth surfaces along a given line or curve.

Axial pitch. Linear pitch in an axial plane and in a pitch surface. In helical gears and worms, axial pitch has the same value at all diameters. In other gears, axial pitch may be confined to the pitch surface and may be a circular measurement.

Base pitch. In an involute gear, the pitch on the base circle or along the line of action. Corresponding sides of involute gear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them measured along the base circle.

Axial base pitch. The base pitch of helical involute tooth surfaces in an axial plane.

Lead. The axial advance of a thread or a helical spiral in 360 deg (one turn about the shaft axis).

Circular pitch. Distance along the pitch circle between corresponding profiles of adjacent teeth.

Outside diameter. The diameter of the outer circle of a gear. In a bevel gear, it is the diameter of the crown circle. In a throated worm gear, it is the maximum diameter of the blank.

Face width. Length of the tooth in an axial plane.

Circular tooth thickness. Length of arc between the two sides of a gear tooth on the pitch circle.

Chordal tooth thickness. Length of the chord subtending a circular tooth thickness arc.

Backlash. Amount by which the width of a tooth space exceeds the thickness of a mating gear tooth on the operating pitch circles. Normally thought of as the freedom of one gear to move while the mating gear is held stationary.

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Angular dimensions

Helix angle. The inclination of the tooth in a lengthwise direction. If the helix angle is 0 deg, the tooth is parallel to the axis of the gear and is really a spur gear tooth.

Lead angle. The inclination of a thread at the pitch line from a Iine 90 deg to the shaft axis. Shaft angle. The angle between the axes of two nonparallel gear shafts.

Pitch angle. In bevel gears, the angle between an element of a pitch cone and its axis.

Angular pitch. The angle subtended by the circular pitch, usually expressed in radians.

Pressure angle. The angle at a pitch point between the line of pressure which is normal to the tooth surface, and the plane tangent to the pitch surface.

Profile angle. The angle at a specified pitch point between a line tangent to a tooth surface and the line normal to the pitch surface.


Gear tooth ratio. The ratio of the larger to the smaller number of teeth in a pair of gears.

Diametral pitch. The ratio of the number of teeth to the pitch diameter in inches. As tooth size increases, diametral pitch decreases.

Contact ratio. To assure smooth, continuous tooth action as one pair of teeth passes out of action, a succeeding pair must have already started action. It is desired to have as much overlap as possible. The contact ratio is a measure of this overlapping action and can be thought of as the average number of teeth in contact.

Hunting ratio. A ratio of numbers of gear and pinion teeth which ensures that each tooth in the pinion will contact every tooth in the gear before it contacts any gear tooth a second time. (13 to 48 is a hunting ratio; 12 to 48 is not.)

General terms

Runout. A measure of eccentricity relative to the axis of rotation. Runout is measured in a radial direction and the amount is the difference between the highest and lowest reading in 360 deg, or one turn. For gear teeth, runout is usually checked by measurement over pins put between the teeth, by a ball probe, or using a master gear. Cylindrical surfaces are checked for runout by a measuring probe that reads in a radial direction as the part is turned on its axis.

Undercut. When part of the involute profile of a tooth is cut away near its base, the tooth is said to be undercut. Undercutting becomes a problem when the number of teeth is small.

Flash temperature. The temperature at which a tooth surface is calculated to be hot enough to destroy the oil film and allow instantaneous welding at the contact point.

Working depth. The depth of engagement of two gears. Full depth tee

th. Those in which the working depth equals 2.000 divided by normal diametral pitch.

Tip relief. A modification of a tooth profile, whereby a small amount of material is removed near the tip of the tooth.

Technical references

Gear Nomenclature, Definitions of Terms with Symbols (ANSI/AGMA 1012-F90), American Gear Manufacturers Association, Alexandria, Va., 1990.

Darle W. Dudley, Hand book of Practical Gear Design, McGraw-Hill Inc., New York, 1984.

Robert O. Parmley, Mechanical Components Handbook, McGraw-Hill Inc., New York, 1985.

Raymond J. Drago, Fundamentals of Gear Design, Butterworths, Stoneham, Mass., 1988.

P. M. Dean, Jr., Gear Manufacturing and Performance, American Society for Metals, Metals Park, Ohio, 1974

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