Design Principles of Bevel Gears: Rationale for Zero Sum Profile Shifts and Generating Gear Choices

A figure from “Gear Geometry—Terms and Definitions” adopted by the AGMA as a recommended practice in April 1936. (Image: MPMA)

Bevel gears are typically designed with a zero sum of profile shifts, meaning that the amount of pinion profile shift is equal and opposed in sign to that of the wheel (Ref. 1). This design constraint stands in contrast to cylindrical gears, which often feature independently adjusted profile shifts to optimize performance parameters such as load capacity and center distance. Despite this apparent limitation, the practice of maintaining a zero sum of profile shifts in bevel gears is well-founded. In the following sections, this article will demonstrate why such an approach is both practical and beneficial for bevel gear design.

In bevel gear technology, the planar generating gear serves as the counterpart to the generating rack used in cylindrical gears. However, in practical applications, conical generating gears are often employed instead. This practice is also well-founded, and its rationale will be illustrated through a detailed example.

Bevel Gear Generation

Figure 1 shows an example of a bevel gearset with the number of teeth z₁ = 9 and z₂ = 13, a shaft angle R = 90 degrees, a tooth height coefficient h = 0.8, a pressure angle a = 22.5 degrees, and profile shift coefficients x₁ = 0.2 and x₂ = −0.2. The pitch cones, drawn in black, have angles d₁ = 34.695 degrees and d₂ = 55.305 degrees (Ref. 2) and roll on each other without sliding.

If profile shift coefficients of x₁ = 0.7 and x₂ = 0.7 are introduced, the shaft angle Σ must be increased to achieve proper meshing. However, since this angle is fixed, it must first be preliminarily reduced so that after increasing it by applying two positive profile shifts, it ultimately reaches the specified value of 90 degrees. As a result, the pitch cone angles decrease and will no longer be in contact. Using the proprietary bevel gear calculation software at Oktoida, the pitch cone angles for these gearings with double positive profile shifts were calculated to be d₁ = 31.534 degrees and d₂ = 49.267 degrees, and are illustrated in pink in Figure 1. Additionally, tooth thickness must be adjusted to enable proper meshing, since those defined on the decreased pitch cones no longer apply due to the lack of rolling contact between these cones.

Figure 2 illustrates the original pinion, with a profile shift coefficient of x₁ = 0.2, meshing with its planar generating gear (depicted in black). It also shows the pinion with a reduced pitch cone angle and a profile shift coefficient of x₁ = 0.7, along with its planar generating gear (depicted in pink). Note that in the latter case, the rolling radius r₁ has decreased while the generating gear radius r₀ remains unchanged, resulting in a change in the generating ratio i = r₁/r₀. This effect is even clearer in Figure 5, where, in the shown kinematic setup of the generating process, only the value of this generating ratio has changed. Consequently, due to the change in the pitch cone angle, the generating motion is modified, causing changes in the generated profile angle (Ref. 1).

Using the same software, the octoid geometry of the original gear tooth flank was calculated in the form of a point cloud with corresponding normal vectors. This cloud extends from the heel (R_e) to the toe (R_i) and from the outer cone (d_a) to the start of the generated profile at the root (d_gf) and will serve as the baseline (zero geometry) for comparisons with geometries obtained through subsequent modifications. Next, in the same manner, the tooth flank geometry of the gear with a reduced pitch cone angle was determined. A comparison of these geometries is presented in Figure 3.

Subsequently, the function fitting the tooth flank geometry from the same software was used, yielding a pressure angle value of a = 31.910 degrees. The original geometry with zero sum of profile shifts and a modified pressure angle corresponds to the geometry with a double positive profile shift, as shown in Figure 4. It should be noted that, in the original case, both during gear generation and its operation in the transmission, the rolling cone is identical to the pitch cone; therefore, in both instances, the term “pressure angle” can be applied.

Figure 2 shows a planar generating gear; however, in industrial practice, conical gears are often used, such as the one shown in Figure 5. Figure 6 illustrates the effect of using such a conical gear—a crowning on the tooth profile is created.

The use of a conical generating gear causes a slight deviation of the pressure angle, since it is measured relative to the pitch cone of the generating gear rather than its plane of rotation. Therefore, Figure 7 shows the tooth geometry with the appropriate roll ratio correction and adjusted tooth thickness applied.

In the final step of the flank topography development, lead crowning was applied, and the effect is shown in Figure 8. The resulting tooth geometry is of the ease-off type, which counteracts the concentration of stresses at a single location on the tooth surface, which may result from the displacement of mating gears from their nominal mounting dimensions due to housing deflection under load, as well as from elastic deformations of the gears themselves under operational forces.

Conclusion

The performed calculations for bevel gear transmissions confirm that the original geometry with zero sum of profile shifts and a modified pressure angle corresponds to the geometry with a double positive profile shift. This implies that the use of independent profile shifts, which entails providing additional parameters and complicating as well as increasing the amount of required calculations, is not warranted.

If tools with no fixed pitch are used for gear cutting (typical for bevel gears), nominal parameters can be used to describe the generating gear, so the standard set of parameters with a zero sum of profile shifts is correct and sufficient.

If tools with a fixed pitch are used (typical for cylindrical gears), the gearing description is performed using the generating rack parameters, and the use of independent profile shifts is warranted.

The use of a conical generating gear brings benefits in the form of tooth profile crowning. The resulting modification is well proven in industrial practice, is based on the gear’s roll angle, and ensures backward compatibility with older designs. Therefore, its replacement with other types of modifications should be preceded by appropriate theoretical and industrial studies.

References

1. Klingelnberg, J. Editor (2016). Bevel Gear: Fundamentals and Applications, Springer-Verlag Berlin Heidelberg.
2. Zarębski, I. (2025), “The Intrinsic Pitch Cones in Hypoid Gears,” Gear Technology, September/October 2025.