Reverse Engineering of Bevel Gears


What is Gear Reverse Engineering ?

Reverse Engineering (RE) of gears can take different definitions, depending on the type of gear considered and who is speaking. In some cases, one may consider the module, helix and pressure angles, facewidth, tooth thickness, root and outside diameters as sufficient to define a gear set, which is correct when cylindrical gears are considered [1, 2], that is without profile or lead modification. When topography modifications are introduced, even if they are based on standards, measuring the part would be required to account for the parametric modifications in order to provide a complete definition of the teeth, for which there appears to be no references in the literature.

In [3], Artoni & Al. RE the ease off surface of a hypoid pinion, rather than the teeth per se; since the Fixed Setting is targeted in which each tooth flank is defined by a separate set of machine settings, economically advantageous spiral bevel gear completing cycles such as Duplex Helical are not covered.

In [4], Bae and Schirru provide a RE approach for spiral bevel gears, but the method is based on milling the tooth surfaces on 5Axis CnC machines, and is therefore limited in scope since a parametric definition of the tooth flanks is not obtained that would be compatible with spiral bevel generators. This is Ok if only a few parts are required, but if a significant number of parts are needed, or else if someone requires a definition of the tooth surfaces in terms of machine settings, then no joy.

As 3D transmission components, being able to replicate straight and spiral bevel gears is a common request. It is also quite desirable to be able to analyze existing gears in order to know how they behave in situ. As is shown in the 1st Application Sample below, it can also save the day when the pinion and gear can be RE and either member be modified in order to work correctly with its mate.

In this paper, the HyGEARS [5, 6] approach to bevel gear Reverse Engineering is presented through several examples. The solution is general, is applicable to spiral bevel gears of any type, straight and Coniflex bevel gears, Face Gears, cylindrical and helical gears, shaper and skiving tools. The result is a set of machine settings and cutter dimensions that define the pinion and gear parametrically; spiral bevel and Coniflex gears can then be cut on 5Axis CnC, mechanical or free form machines from Gleason, Klingelnberg and other compatible machines.

Sample Application 1: Saving the Day

A significant lot of spiral bevel gear wheels had been manufactured by Powder Metallurgy (PM) to be used in hand tools. As is usually the case, PM gears can deviate significantly from the design since the mold is obtained by EDM of a copper electrode and a solid block of hard steel. In this case, when the pinion was cut and meshed with a PM gear, the contact patterns shown in Fig. 1 were obtained which are low on the gear tooth and therefore high on the pinion tooth. When such contact patterns are obtained, the Transmission Error (TE) deviates from a parabolic shape and in this case resulted in significant noise issues. Rather than scrapping the bevel gear members, both the pinion and gear were measured, RE (Fig. 2), and the Duplex Helical pinion machine settings were modified to produce good contact patterns and associated TE (Fig. 3) when meshing with the existing PM gear member. This saved the production and eliminated lost time and money waiting for a new series of gear members to arrive – i.e. months in this case.

Source: Involute Simulation Softwares Inc.

Sample Application 2: In Situ Behavior

It is desired to evaluate the in situ behavior of a straight bevel differential gear set. The parts are manufactured by an outside supplier to whom the drawings had been provided, a typical situation in the industry. Since the parts are forged, tooth topography can deviate significantly from the design. In order to model the contact, an initial design is realized using HyGEARS. CMM Nominal inspection files are then generated and the actual parts are measured. Fig. 4 [7] shows the resulting pinion CMM tooth flank measurement; of course, nobody can expect exact agreement from 2 different sources, and as shown differences up to 95 µm are found. The HyGEARS RE algorithm is then applied after which maximum differences are less than 10 µ. Re-measuring using a CMM grid based on the RE part gives the results shown in Fig. 5 [7].

Therefore, the pinion and gear tooth definitions after RE can be used confidently for Loaded Tooth Contact Analysis (LTCA). In this case, the HyGEARS point cloud export function is used to feed the topography to FVA’s [1] BECAL in order to analyze the contact under load, as shown in Fig. 6 [7] below.

Source: ZG Hypoid GmbH

Sample Application 3: Replicate an Existing Part

An existing spiral bevel pinion member is to be replicated on a 5Axis CnC machine. The original machine settings are unknown, but the pinion and gear drawings are available, and the Duplex Helical process is specified for the pinion. Therefore, an initial design is created with HyGEARS, Fig. 7, using data on the drawings. From this initial design, the CMM Nominal inspection files are generated and used to measure the pinion and gear. The pinion CMM result based on the initial design is shown in Fig. 8, where it is clear that the original cutter diameter assumption was wrong, although the pressure angle is correct and the tooth thickness deviation of -0.008 mm is minimal.

Using HyGEARS’ RE algorithm, it is found that the cutter diameter should be 7.5” rather than the assumed initial 6”. Given that the gear set is Formate, i.e. the gear member is non-generated, changing the cutter diameter influences all machine settings and it is therefore preferable here to redesign the gear set from scratch, a 2 minute affair. Using the CMM Nominal inspection files from the 2nd design, the results in Fig. 9 are obtained, where it is clear that the cutter diameter is now correct, but pressure angle, spiral angles and thickness errors are visible.

Applying RE to the result of Fig. 9 reveals that the OB-IB blade angles should be closer to 16.5°-23.5° rather than the assumed 18°-24.5°. In the Duplex Helical process, both flanks being cut simultaneously, the choice of the cutter blade angles has significant influence on the resulting machine settings.

A third design is therefore made using the OB-IB blade angles of 16.5°-23.5°; using the CMM Nominal inspection files from the 3rd design, the results in Fig. 10 are obtained, where it is now obvious that the correct definition of the pinion has been obtained. The final machine settings and cutter definition can then be used to cut the part on any Duplex Helical capable spiral bevel generator, either mechanical, free form or 5Axis CnC.

Source: Involute Simulation Softwares Inc.

[1] „Forschungsvereinigung Antriebstechnik e.V. (FVA)“


  3. Artoni A., Gabaccini M., Guiggiani M., Nonlinear Identification of Machine Settings for Flank Form Modifcations in Hypoid Gears, ASME Journal of Mechanical Design, November 2008, Vol 130
  4. Bae I., Schirru V., An Approach to Pairing Bevel Gears from Conventional Cutting Machine with Gear Produced on 5-Axis Milling Machine, Gear Technology, June 2015, pp. 60-65
  6. Gosselin C., Masseth J., Cutter Interchangeability For Spiral-Bevel and Hypoid Gear Manufacturing, ASME PTG 2003 Conference, Chicago, Sept. 2003.
  7. Thomas J., Gosselin C., Design and rating by means of loaded TCA of straight bevel differential gears, International Conference on Gears, Munich , Sept. 2019


Left to right:

Claude Gosselin, Ph.D., P.Eng., Involute Simulation Softwares Inc., Québec, Canada
J. Thomas, Dr. Ing., ZG Hypoid GmbH, Germany