Determining tooth thickness of various gear types

How to calculate the nominal values of chordal tooth thickness in all types of gearing.

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In order to determine the tooth size of a gear after taking into account the backlash allowance, you first must determine what the nominal tooth thickness should be. There are three methods for determining this value: chordal tooth thickness measurement, span measurement, and over pin or ball measurement. For this article, we will discuss chordal tooth thickness measurements.

The chordal tooth thickness measurement method uses a tooth caliper that is referenced from the gear’s tip diameter. The thickness is measured at the reference circle as detailed in Figure 1.

Figure 1: Chordal tooth thickness method.

For spur gears, the formulas to calculate the chordal tooth thickness are detailed in Table 1.

For rack, regardless if they are straight tooth or helical tooth, the formulas are simplified because the gear tooth profile is trapezoid. These formulas are detailed in Table 2.

Table 1: Equations for spur gear chordal tooth thickness.
Table 2: Chordal tooth thickness of racks.

Measuring helical gears can be done in either the normal plane or the transverse plane dependent on how the gears are cut. The formulas for measurements in the normal plane are detailed in Table 3 and those for measurements in the transverse plane are detailed in Table 4.

Table 3: Equations for chordal tooth thickness of helical gears in the normal system.
Table 4: Equations for chordal tooth thickness of helical gears in the transverse system.

The measurement of straight tooth bevel gearing is dependent on the tooth system. The formulas for measurements in the standard system are detailed in Table 5 and the formulas for measurements for bevel gears cut using the Gleason system are detailed in Table 6.

Table 5: Equations for chordal tooth thickness of standard straight bevel gears.
Table 6: Equations for chordal tooth thickness of Gleason straight bevel gears.
Figure 2: Chart to determine the tooth thickness factor k for Gleason straight bevel gear.

If a straight bevel gear is cut by a Gleason straight bevel cutter, the tooth angle should be adjusted according to:

This angle is used as a reference in determining the tooth thickness, s, when setting up the gear cutting machine.

For spiral bevel gears cut using the Gleason system, the formulas to calculate the chordal tooth thickness are detailed in Table 7.

Table 7: Equations for chordal tooth thickness of Gleason spiral bevel gears.
Figure 3: Chart to determine the tooth thickness factor k for Gleason spiral bevel gears.

The calculations of chordal tooth thickness of a Gleason spiral bevel gear are so complicated that I will not go any further in this presentation.

The measuring of worm gearing can be done in either the axial plane or the normal plane dependent on how the gears are cut. The formulas for measurements in the axial plane are detailed in Table 8, and those for measurements in the normal plane are detailed in Table 9.

Table 8: Equations for chordal tooth thickness of an axial module worm gear pair.
Table 9: Equations for chordal tooth thickness of a normal module worm gear pair.

Using these tables and formulas, you will be able to determine the proper chordal tooth thickness for your gear. From these values and the measured values, you can determine the tooth thinning or backlash allowance cut into the gear. 

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Brian Dengel
is general manager of KHK USA Inc, a subsidiary of Kohara Gear Industry with a 24-year history of working in the industrial automation industry. He is skilled in assisting engineers with the selection of power-transmission components for use in industrial equipment and automation. Dengel is a member of PTDA and designated as an intern engineer by the state of New York. He is a graduate of Hofstra University with a Bachelor’s of Science in Structural Engineering.