The following excerpt is from the Revised Manual of Gear Design, Section III, covering helical and spiral gears. This section on helical gear mathematics shows the detailed solutions to many general helical gearing problems. In each case, a definite example has been worked out to illustrate the solution. All equations are arranged in their most effective form for use on a computer or calculating machine.

Hard Gear Finishing (HGF), a relatively new technology, represents an advance in gear process engineering. The use of Computer Numerical Controlled (CNC) equipment ensures a high precision synchronous relationship between the tool spindle and the work spindle as well as other motions, thereby eliminating the need for gear trains. A hard gear finishing machine eliminates problems encountered in two conventional methods - gear shaving, which cannot completely correct gear errors in gear teeth, and gear rolling, which lacks the ability to remove stock and also drives the workpiece without a geared relationship to the master rolling gear. Such a machine provides greater accuracy, reducing the need for conventional gear crowning, which results in gears of greater face width than necessary.

A much-used method for checking the tooth thickness of an involute gear tooth is to measure the dimension over two balls placed in most nearly opposite spaces in the case of external gears, and the dimension between the balls in the case of internal gears. This measurement is then checked against a pre-calculated dimension to denote an acceptable part.

The first commandment for gears reads "Gears must have backlash!" When gear teeth are operated without adequate backlash, any of several problems may occur, some of which may lead to disaster. As the teeth try to force their way through mesh, excessive separating forces are created which may cause bearing failures. These same forces also produce a wedging action between the teeth with resulting high loads on the teeth. Such loads often lead to pitting and to other failures related to surface fatigue, and in some cases, bending failures.

The last decade has been a period of far-reaching change for the metal working industry. The effect of higher lubricant costs, technical advances in machine design and increasing competition are making it essential that manufacturers of gears pay more attention to testing, selecting and controlling cutting fluid systems. Lubricant costs are not a large percentage of the process cost relative to items such as raw materials, equipment and labor, and this small relative cost has tended to reduce the economic incentive to evaluate and to change cutting fluids.

When specifying a complete gear design, the novice designer is confronted with an overwhelming and frequently confusing group of options which must be specified. This array of specifications range from the rather vague to the very specific.

Rotary gear honing is a hard gear finishing process that was developed to improve the sound characteristics of hardened gears by: Removing nicks and burrs; improving surface finish; and making minor corrections in tooth irregularities caused by heat-treat distortion.

The forming of gear teeth has traditionally been a time-consuming heavy stock removal operation in which close tooth size, shape, runout and spacing accuracy are required. This is true whether the teeth are finished by a second forming operation or a shaving operation.

Below are listed a variety of commonly used constants arranged numerically to permit ease of reference. Wherever an asterisk (*) is shown, the constant is exact as given, it being generally a mathematical constant or one fixed by definition. In cases where the first constant listed is followed by another in parenthesis, the first is the round number generally used, while the second is the more exact value.

The most conclusive test of bevel and hypoid gears is their operation under normal running conditions in their final mountings. Testing not only maintains quality and uniformity during manufacture, but also determines if the gears will be satisfactory for their intended applications.