Determining tooth thickness of various gear types – Part III

How to calculate the nominal values of over-pin or ball measurement of teeth for various 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 measurement over rollers, which is more commonly known as over-pin or ball measurement.

The over-pin measurement, M, is made over the outside of two pins that are inserted in diametrically opposite tooth spaces for even tooth number gears and as close as possible for odd tooth number gears. See Figure 1a for details.

Figure 1a: Over-pin (ball) measurement.

In measuring a standard spur gear, the size of the pin must meet the condition that its surface should have a tangent point at the standard pitch circle. When measuring a profile shifted gear, the surface of the pin should have a tangent point at the d + 2xm circle. Under the conditions mentioned earlier, Table 1 details the formulas that determine the diameter of the pin (ball) for the spur gear in Figure 1b.

Table 1: Equations for calculating ideal pin diameters.
Figure 1b: Over-pins measurement of spur gear (far right).
Figure 2: Over-pins measurement for a rack using a pin or a ball.

An ideal diameter of pins when calculated from the equations of Table 1 may not be practical. So, in practice, you should select a standard pin diameter close to the ideal value. After the actual diameter of pin dp is determined, the over-pin measurement M can be calculated from Table 2.

Table 2: Equations for over-pins measurement of spur gears.
Table 3: The size of pin which has the tangent point at d = 2xm circle for spur gears.

In Table 3 the calculated values for pin size under the conditions of module m = 1 and pressure angle α = 20° wherein the pin has the tangent point at d + 2xm circle.

If you are measuring a straight tooth rack, the pin is ideally tangent with the tooth flank at the pitch line. The equations in Table 4 can, thus, be derived. In the case of a helical rack with module m and pressure angle α, in Table 4, can be substituted by normal module mn , and normal pressure angle αn , resulting in Table 5.

Table 4: Equations for over-pins measurement of straight racks.
Table 5: Equations for over-pins measurement of helical racks.

As shown in Figure 3, measuring an internal gear needs a proper pin that has its tangent point at d + 2xm circle. The equations are in Table 6 for obtaining the ideal pin diameter. The equations for calculating the between-pin measurement, M, are in Table 7.

Figure 3: Between pin dimension of internal gears.
Table 6: Equations for calculating pin diameter for internal gears.
Table 7: Equations for between pin measurement of internal gears (above).

The Table 8 lists ideal pin diameters for standard and profile shifted internal gears under the conditions of module m = 1 and pressure angle α = 20°, which makes the pin tangent to the reference circle d + 2xm.]

Table 8: The size of pin that is tangent at reference circle d + 2xm for internal gears (left).
Table 9: Equations for calculating pin diameter for helical gears in the normal system. (below)
Table 10: Equations for calculating over-pins measurement for helical gears in the normal system.

Another gear type to consider is the helical gear. The ideal pin that makes contact at the d + 2xnmn reference circle of a helical gear can be obtained from the same above equations but with the teeth number z substituted by the equivalent (virtual) teeth number zv. Table 9 presents equations for deriving over-pin diameters. Table 10 presents equations for calculating over-pin measurements for helical gears in the normal system.

Tables 11 and 12 present equations for calculating the pin measurements for helical gears in the transverse (perpendicular to axis) system.

Table 11: Equations for calculating pin diameter for helical gears in the transverse system.
Table 12: Equations for calculating over-pins measurement for helical gears in the transverse system.
Figure 4: Three wire method of a worm.

As noted in Figure 4, worms can be measured using the three-wire method. The tooth profile of type III worms, which are the most popular, are cut by standard cutters with a pressure angle α0 = 20°. This results in the normal pressure angle of the worm being a bit smaller than 20°. Equation 1 shows how to calculate the normal pressure angle of a type III worm in the AGMA system.

Where r: Worm reference radius

r0:  Cutter radius

z1: Number of threads

γ: Lead angle of worm

The exact equation for measuring type III worms using the three-wire method is not only difficult to comprehend but also hard to calculate precisely. As such, there are two approximate calculation methods that you can use:

a) Regard the tooth profile of the worm as a straight tooth profile of a rack and apply its equations. Using this system, the three-wire method of a worm can be calculated as detailed in Table 13.

Table 13: Equations for three wire method of worm measurement.
Table 14: Equations for three wire method of worm measurement.

These equations presume the worm lead angle to be very small and can be neglected. Of course, as the lead angle gets larger, the equations’ error gets correspondingly larger. If the lead angle is considered as a factor, the proper equations are detailed in Table 14.

b) Consider a worm to be a helical gear. This means applying the equations for calculating over pins measurement of a helical gear to the case of three wire method for a worm. Because the tooth profile of Type III worm is not an involute curve, this method yields an approximation. However, the accuracy is quite adequate in practice.

Table 15: Equations for calculating pin diameter for worms in the axial system.
Table 16: Equations for three wire method for worms in the axial system.

Tables 15 and 16 contain equations based on the axial system. Tables 17 and 18 are based on the normal system.

Table 17: Equations for calculating pin diameter for worms in the normal system.
Table 18: Equations for three wire method for worms in the normal system.

Tables 17 and 18 show the calculations of a worm in the normal module system. Basically, the normal module system and the axial module system have the same form of equations. Only the notations of module make them different.

Due to the tooth form of bevel gearing, whether it is straight or spiral tooth, this measurement over pins method is not possible and cannot be used.                

Using these tables and formulas, you will be able to determine the proper over-rollers measurement for your spur gear, helical gear, internal ring gear, gear rack, or worm. From these calculated values and the actual 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.