One issue that periodically arises in gear discussions is independent profile shift in bevel gears. Standard practice for such gears uses zero-sum shifts, meaning the gear profile shift coefficient x2 equals and is opposite in sign to the pinion profile shift coefficient x1, i.e., x2 = –x1. Cylindrical gears use independent coefficients, raising the question of where this zero-sum rule comes from and whether it’s still justified. To examine this, we’ll first use the fact that cylindrical gears are a special case of bevel gears where the shaft angle approaches zero, allowing analysis of meshing in the transverse plane.
Cylindrical meshing
In Figure 1, black shows the involute meshing of two cylindrical gears with numbers of teeth z1 = 23 and z2 = 32, profile shift coefficients x1 = +0.7 and x2 = +0.4, and zero backlash. Pitch diameters d1 and d2 do not touch because positive profile shifts on both gears cause them to spread apart. Due to the unchanged gear ratio (numbers of teeth remain the same), working pitch diameters dw1 and dw2 exist in this setup that touch and roll without slipping. Pink shows a basic rack profile associated with these working pitch diameters, forming zero-sum profile shifts, where pressure angle is equal to its profile angle.
Cylindrical meshing parameters
Table 1 shows cylindrical pinion tooth thickness calculated at five different diameters. The values match across sets, indicating identical tooth shapes. Thus, we have two parameter sets describing the same involute gearing. Note that in Set I, module and profile angle are independent of profile shift coefficients, unlike Set II. For cylindrical gear manufacturing, methods using tools with fixed pitch have become widespread, so using Set I parameters allows straightforward verification if a tool matches the gearing’s pitch, supporting standardization. Describing cylindrical gearing with Set II parameters would often necessitate recalculating them to Set I parameters, which are tool-centric and are usually needed for manufacturing. Therefore, Set I parameters have become standard for cylindrical gearing.
Bevel meshing
In bevel gear meshing, the elements rolling without slipping are pitch cones δ1 and δ2, drawn black in Figure 2. The associated profile shift coefficients are x1 = +0.2 and x2 = –0.2, summing to zero. Pink shows independent profile shifts for each gearing with coefficients x1 = +0.7 and x2 = +0.7. These shifts increase face and root cone angles, so meshing can only occur with an increased shaft angle Σ. However, since this angle is fixed by design, it needs to be modified to alter the resulting pitch cones so that after applying the independent profile shifts the shaft angle ends up with its nominal design value.
Bevel meshing parameters
Table 2 shows bevel pinion tooth thickness calculated at five different measuring cone angles. These thicknesses are distances between points on the tooth profile, computed by solving mathematical equations derived for the actual generated octoid gearing case (as octoid tooth shape is predominantly used in bevel gears), without simplifications like the Tredgold’s approximation. Again, the values match across sets, indicating Sets I and II describe the same gearing. Thus, introducing independent profile shifts in bevel gears, which requires additional operations and calculations, yields the same effect as changing the profile angle α.
The standard Set I parameters allow simpler description of bevel gears than Set II. In Set I, pitch cones and working pitch cones are the same, tooth thicknesses can be defined on rolling pitch cones without conversions or corrections, back cones can be perpendicular to pitch cones and mutually aligned, and the sum of pitch cone angles gives the shaft angle Σ. Therefore, using Set II parameters, i.e. independent profile shifts in bevel gears, violates Occam’s Razor, which favors the simplest explanation that fits the facts. For bevel gearings, which are commonly generated with tools with no fixed pitch, Set I parameters serve as the standard description.
Indeterminacy in reverse engineering
From the above considerations, an involute or octoid gearing can be described by an infinite number of parameter variants. One special variant has zero-sum profile shifts. This means parameters cannot be uniquely determined in reverse engineering.
Although determined parameters will match measurements 100 percent, the original drawing may still show a different set. Hints on original parameters may lie in the transition curve at the tooth root, as detailed more broadly in the article “Transition Curve: Much more than a radius at the root fillet of a tooth”, published in the September 2025 issue of Gear Solutions.
