Worm gear pairs are a common gearing technology. They consist of two components, the first being the worm and the second being the worm wheel. The worm is a cylindrical piece with gear teeth produced in a manner that resembles a screw. Worms are produced either as an integral part of a shaft or they are produced with a bore in order to be fixed to a shaft.

The worm wheel is similar to a helical gear; however, the tooth surface is produced in a concave shape in order to improve the surface contact area with the worm.

The calculations for the geometry of a worm gear pair are dependent on several key parameters. These include the module, the pressure angle, the number of teeth on the worm wheel, the number of threads on the worm, the pitch diameter of the worm, and the center distance between the two axes. When these factors are known, the other dimensions can be calculated as detailed in Table 1.

In the case in Table 1, the module was set to three, the shaft angle was set to 90°, the number of teeth on the worm wheel was set to thirty, the number of threads was set to two. The pitch diameter of the worm was set to 44 millimeters and the center distance was set to 67 millimeters. Based on these parameters, a coefficient of profile shift was necessary.

The most important value that is determined by the calculations in Table 1 is the reference cylinder lead angle. The lead angle of the worm is so important because it becomes the effective helix angle for the worm wheel. The unique feature of a worm gear pair is that you can maintain the diameter of the worm, and the number of teeth of the worm wheel, and the center distance of the pair, but change the number of threads on the worm and achieve a different reduction ratio.

An example of this would be if you choose a Module 3 worm gear pair where the worm pitch diameter was 44 millimeters and the center distance was 67 millimeters, as noted in the table, and set the number of threads on the worm to one — then the lead angle would be 3.90956°. If the number of threads on the worm is two, then the lead angle would be 7.83748°. If the number of threads on the worm is three, then the lead angle would be 11.80289°. If the number of threads on the worm is four, then the lead angle would be 15.82662°. With each of these values, the profile shift coefficient can be adjusted to maintain the center distance of 67 millimeters. However, the reduction ratio for each pair would be different. With a single thread on the worm and 30 teeth on the worm wheel, the reduction ratio would be 30:1. With two threads on the worm, the reduction ratio would be 30:2 or 15:1. If there are three threads on the worm, then the reduction ratio would be 30:3 or 10:1. And if the number of threads on the worm is four, then the reduction ratio would be 30:4 or 7.5:1.

The downside to worms with a high number of threads is that the increased lead angle value limits the ability of the worm gear pair to be self-locking. The self-locking phenomena of worm gear pairs as detailed in Figure 1 is the ability of a worm gear pair to prohibit the rotation of the worm if driven by the worm wheel. This phenomenon exists due to the forces of friction in the gear pair exceeding the forces of the worm wheel being back driven.

Although worm gear pairs can have a self-locking feature, it does not prohibit the worm from being driven in either a clockwise or counterclockwise direction. Figure 2 shows how the forces act on the teeth of a worm gear pair mesh with a shaft angle Σ = 90 °. Since the power transmission of the worm gear pair mesh has a sliding contact nature, the coefficient of friction on the tooth surface has a great effect on the transmission efficiency ηR and the force acting on the gear mesh.

The calculation of these forces is detailed in Table 2. The unique features of worm gear pairs make them an excellent choice for right-angle gearboxes. The same gearbox housing can be used to produce gearboxes with different output speeds and the self-locking feature can limit damage to the drivetrain if the system experiences any back driving. The input can be clockwise or counterclockwise and the torque is always increased as an inverse proportion of the reduction ratio.