Crossed helical gearing a.k.a. screw gears

A common refrain from my childhood was “if you keep crossing your eyes, they are going to get stuck like that!” It never did happen, but it was a real-life example of what nonparallel meant. In gearing, the axis of the pair is usually parallel, or they are intersecting. Crossed helical gears, also known as screw gears, are both nonparallel and nonintersecting.  They allow for some very unique designs.

The term screw gearing includes various types of gears used to drive nonparallel and nonintersecting shafts where the teeth of one or both members of the pair are helical in nature. The Figure 1 below shows the meshing of screw gears where the pitch and pressure angles are the same but the helix angle of the teeth on each gear is different.

Two screw gears can only mesh together under the conditions that normal modules (m_n1) and (m_n2) and normal pressure angles (α_n1, α_n2) are the same. 

If we set a pair of screw gears to have the shaft angle Σ and helix angles β₁ and β₂:

If they are the same hand (both left hand or both right hand), then

Σ  =  β₁ + β₂

If they are the opposite hand (one left hand and the other right hand), then

Σ  = β₁ – β₂      or     Σ  =  β₂ – β1

If these screw gears are profile shifted, then the meshing would become a little more complex. If we let β_w1 and β_w2 represent the working pitch cylinders:

If they are the same hand (both left hand or both right hand), then

Σ  =  β_w1 + β_w2

If they are the opposite hand (one left hand and the other right hand), then

Σ  =  β_w1 – ββ_w2     or     Σ  =  β_w2 – ββ_w1

The unique situation of crossed axis helical gears is that if you set the helix angles β₁ and β₂ to 45 degrees and both gears are of the same hand, then the resulting shaft angle Σ becomes 90 degrees and the gear pair will operate as a right angle drive. However, if the same gears are opposite in hand and the helix angles β₁ and β₂ are 45 degrees, then the resulting shaft angle Σ becomes 0 degrees and the gear pair will operate as a set of regular helical gears. For all configurations of crossed helical gears, the speed ratio of the gear pair is equal to z₂/z₁. Thus the speed ratio of these gear pairs is limited by the size of the larger gear and practically speaking are usually less than 6:1.

Table 1 presents the equations for a profile shifted screw gear pair. When the normal profile shift coefficients x_n1 =  x_n2 = 0, the equations and calculations are the same as for standard helical gears and the following apply:

d_w1  =  d₁ d_w2  =  d₂

β_w1  = β₁ β_w2  =  β₂

For all screw gears, the proper thrust bearings must be selected so that they absorb the thrust loads imparted by the helix angle.

Although the most popular helix angle for screw gears is 45 degrees, any helix angle greater than zero degrees is possible. The ability to mix helix angles allows for screw gears to be used in unique applications where the input and output shafts are neither intersecting nor parallel and when combined with profile shifting, they also allow for a variety of center distances. These features are not found in any other form of gearing.