# Cylindrical worm gear pair

One distinguishes two types of cylindrical worm and wormwheel:

• A set “worm and helical gear”
• A set “worm and wormwheel”

The first set is a special case of the ‘helical gears for crossed shafts’ where the pinion for only counts l, 2, 3 (or a small number of) teeth and is characterised by a large helix angle of the tooth.
The calculation is the same as for the helical gears.
The choice of the module should be limited to the normalized set of normal modules. As with helical gears for crossed shafts there is only point contact between the tooth flanks so this kind of transmissions is not applicable for high loads.
For the set ‘worm and wormwheel’ the teeth of the wormwheel are being milled with a worm mill with the same basic features as the worm. In this case the gear contact is characterized by line touch, through which higher tooth forces are admissible.

## Important notes

• The lead angle of the worm toothing (thread) is the angle formed by the tooth trace which pulls the tooth trace to the reference cylinder; in other words: a surface perpendicular to the axis of the worm, while the helix angle of the wormwheel teeth represents the angle formed by a tangent to the tooth flank on the height of the pitch circle and a diametrically surface cutting through the middle of the same tooth flank. That results in identical helix angles for worm and worm wheel.
• The dimensions for the wormwheel gear pair are generally calculated on basis of an apparent (or transverse) module. The corresponding normal module is often then not an integer.
• Furthermore, there is also something like globoïd worm pairs which allow a larger transfer capability than the standard cylindrical worm gear pairs. This globoïd worms are difficult to produce and not easy to assemble. Internal frictions cause a negative effect on the efficiency of this transmission and often cause minimum contacts at the ends of the worms.
• The formulas for standard worms pairs:
Axial module mx = mn/cos(γm) and sin(γm) = mn·z1/dm1
Axial pitch of the worm px = mn·π/cos(γm)
Tipdiameter Ø worm da1 = dm1 + 2·mx
Pitch circle Ø of the wheel d2 = z2·mn/cos(γm)
Throat circle Ø of the wheel da2 = d2 + 2·mx
The other dimensions are derived from these elements.

## Pressure angle for worm and worm wheels

As pressure angle normally is chosen for 20° and sometimes for 14°30′, as long as one respects the critical numbers, indicated for cylindrical gears. For tooth numbers below 20 the pressure angle will always be choosen to be 20° or more, depending on the existing tools. If one has only a worm mill of 14°30′ then one can avoid the undercut and its effects by an adjusted tooth correction. (See chapter Tooth correction).
The axial and radial components of the on the worm working forces are, however, different for 14°30′ or 20° pressure angle and can therefore affect the choice of the bearings of the worm shaft.

## Direction of the Helix angle

This will be chosen in function of the existing tools. When the direction is not determined by any need, one chooses one preferably for “right”, because the most hobs have a right handed lead.

## Efficiency of a worm gear

The efficiency of worm and worm gear pair transmissions is often worse than those of transmissions with parallel shafts with the same proportion. This is the reason why this type of transmission is often not adopted, however, recent calculations and studies have seriously improved the efficiency. The efficiency depends of several factors, including:

• the lead angle of the worm
• the coefficient of friction of the used materials
• the smoothness and hardness of the tooth flanks
• the tooth shape
• the quality of the used lubricants
• The friction also changes with the peripheral velocity of the worm. The simplified formulas for calculating the efficiency are:
 in which: η = tan(γm)/tan(γm+f)·100% η = efficiency in % γm = lead angle of the tooth trace on the eference cylinder f = friction angle

The angle of friction “f” can vary from 1° to 8°. The worm gear transmissions in our standardized wormwheel gearboxes normally have a friction angle between 1° and 2°.