Demonstrating the construction and optimization of the tooth root fillet while leaving required grinding stock on the involute tooth flanks.

The Direct Gear Design [1] method optimizes various parameters and elements of gear tooth geometry to achieve the required gear-drive performance. One such critical element of the tooth profile is the root fillet. Previous publications had considered asymmetric tooth root fillet optimization assuming the tooth involute flanks and root fillets are processed (machined) simultaneously. However, there are many applications that require high gear tooth flank accuracy and high-load capacity as well as low gear production cost. In order to satisfy these requirements, the tooth involute flanks and root fillets are processed separately. The gear blank is machined by the topping protuberance hob, which finalizes the root fillet, tooth tip diameter, and chamfers but leaves stock for tooth flank grinding (Figure 1). Then, after gear heat treatment (carburizing plus case hardening) — and in some cases shot peening of the tooth root — the tooth flanks are processed by highly productive generating grinding, removing the grinding stock. As a result, the compressive residual stress in the root fillet developed during case hardening is retained. This fabrication sequence is mostly typical for automotive transmission gears.]

Figure 1: Asymmetric gear with tooth root generated with protuberance hob.

Since tooth root load capacity is a major contributor to gear transmission performance, the reduction of tooth bending stress concentration in asymmetric gears generated with a protuberance hob is critically important.

This paper demonstrates the construction and optimization of the tooth root fillet while leaving required grinding stock on the involute tooth flanks. It also describes a reversed generation of the protuberance hob tooth profile using the completely defined gear tooth profile, which includes the optimized root fillet and grinding stock on the tooth flanks.

1 Tooth Root Fillet Construction and Optimization

Construction of the initial root fillet of the asymmetric tooth is shown in Figure 2. The direct gear design method applied to asymmetric gears [2] defines the final (after grinding) involute tooth flanks 1d and 1c and also the trajectory of the mating gear tooth tip in zero backlash mesh 4.

Figure 2: Initial construction of root fillet; 1d and 1c — drive and coast final involute flanks after grinding, 2d and 2c — drive and coast preliminary involute flanks before grinding, 3d and 3c — drive and coast interim involute fillet flanks that will be removed by tooth flank grinding, 4 – trajectory of mating gear tooth tip in zero backlash mesh; 5 — initial tooth root fillet; Dd and Dc — drive and coast flank grinding stocks; gd and gc – drive and coast flank interim fillet profile angles; dlpd and dlpc — drive and coast flank lowest contact point diameters; dfd and dfc — drive and coast flank form diameters.

Then, the preliminary involute flanks 2d and 2c (after hobbing) are constructed equidistant to the final tooth flanks, providing Δd and Δc — drive and coast flank grinding stocks. The interim drive and coast fillet flanks 3d and 3c are for transitioning between the preliminary flanks 2d and 2c and the root fillet. They intersect the preliminary flanks, providing drive and coast flank grinding stocks Δd and Δc at the lowest points of contact with the mating gear tooth tip at the drive and coast flank lowest contact point diameters dlpd and dlpc. These interim fillet profiles are typically involute, formed by the straight protuberance profiles of the protuberance hob. The interim fillet profiles are designed to be machined by hobbing, and later removed by tooth flank grinding after heat treatment.

The initial tooth root fillet 5 is a circular arc that lays below the trajectory of the mating gear tooth tip 4 to exclude root-tip interference. It is tangential to both the interim drive and coast fillet flanks 3d and 3c. For practical purposes, the initial tooth root fillet 5 is also almost tangential to the final involute tooth flanks 1d and 1c, leaving a small amount of undercut (Figure 3) to accommodate manufacturing tolerances and some gear distortion after heat treatment and to guarantee an exit for the grinding wheel without creating a step between the ground tooth flank and hobbed root fillet.

Figure 3: Small undercut Du in the transition point between the final involute flank 1 and initial root fillet 4, 2 — preliminary involute flank before grinding, 3 — interim involute fillet flank; D — flank grinding stock.

The goal of tooth root fillet optimization is to achieve minimal stress concentration in the tooth fillet profile. As a result, the maximum bending stress is evenly distributed along a large portion of the fillet. The root fillet optimization method, developed by Dr. Y.V. Shekhtman, uses three major procedures: defining functions to approximate the fillet profile, FEA for stress calculation, and a random search algorithm to define the optimal set of coefficients for the trigonometric functions, allowing them to reach minimal bending stress.

Unlike gears with an optimized ground tooth root where the initial root fillet is the trajectory of the mating gear tooth tip in a zero-backlash mesh (item 4 in the Figure 2), gears with a tooth root generated by the protuberance hob have circular arc initial root fillets (item 5 in the Figure 2). Finite element nodes are evenly distributed along this initial root fillet profile (Figure 4a). The center of the initial root fillet is connected to these finite element nodes. The first and last finite element nodes of the initial fillet profile, located on form diameters dfd and dfc, cannot be moved during the optimization process. The rest of the finite element nodes are moved along straight lines perpendicular to the fillet profile. Bending stresses are calculated for every iteration of the fillet profile configuration. A new position is defined for each finite element node based on the results of the previous iterations in order to reduce stress values. The optimization process stops when the maximum stress value cannot be further reduced. The optimized fillet profile provides even stress distribution along a significant length of the stretched (Figure 4b) and compressed (Figure. 4c) portions of the root fillet.

Figure 4: Tooth root fillet optimization: a — FE node movement, b — tensile stress distribution, c — compressive stress distribution; dfd, dfc — form diameters circles of the drive and coast tooth flanks.
Figure 5: Root fillet optimization: a — tooth profile with the load F applied at the HPSTC, b — stress distribution along tooth profile; dashed line — initial circular root fillet and solid line — optimized root fillet; 1 — tensile stress area, 2 — compressive stress area; Dst — tensile stress reduction, Dsc — compressive stress reduction.

Figure 5 shows a comparison of the root fillet and stress charts along the tooth profile before and after root fillet optimization.

2 Reversed Generation of Tooling (Hob) profile

The fabrication process for gears with ground involute flanks and unground root fillets usually assumes the use of a protuberance hob cutter.

In this case, the reverse generating technique is applied to find the protuberance hob rack profile using the already designed gear profile with the optimized root. This technique assumes that in the gear-rack mesh, every point of the gear tooth profile has a corresponding point on the rack tooth profile. Figure 6 demonstrates how the tooling rack profile point At position is defined from the gear tooth profile point Ag position. In order to find the generating rack profile point At corresponding to the gear tooth profile point Ag, the line AgBg perpendicular to the tooth profile at point Ag is constructed. Point Bg lies at the intersection of the line AgBg with the gear pitch circle. The gear tooth profile and line AgBg are rotated by angle γg relative to the gear center until point Bg reaches its pitch point position Bg‘, where gear pitch circle 3 is tangent to rack pitch line 4. Then point Ag is in position Ag‘, where gear tooth profile 1′ is tangent to rack profile 2’. The line Ag‘Bg‘ is moved parallel to rack pitch line 4 by distance Bg‘Bt, which is equal to the length of arc BgBg‘. This movement puts point Ag‘ in position At along the rack profile, which corresponds to point Ag on the gear tooth profile. This approach allows us to define any generating rack profile point that corresponds a specific gear tooth profile point. For helical gears, this technique is applied in the normal to the gear tooth line section. For topping protuberance hobs, the gear’s outer diameter and chamfer cutting profiles are also generated.

Figure 6: Reverse generating of the protuberance hob rack profile; 1 and 1’ — gear profile positions; 2 and 2’ — rack cutter profile positions; 3 — gear pitch circle in a mesh with the rack; 4 — rack pitch line in a mesh with the gear.

The selection of the radial position of the pitch point Bg‘ (defining the gear pitch circle and rack pitch line) in the reverse generating process is very important. It affects all protuberance hob profile parameters (Figure 7). If the pitch point is located above the nominal gear pitch diameter in the mesh with the mating gear, the hob rack module mn, αnd, and αnc (drive and coast pressure angles) and αprd and αprc (drive and coast protuberance angles) are increased. This leads to a hob protuberance size reduction that may result in increased root fillet surface roughness and excessive hob protuberance wear. If the pitch point is below the nominal gear pitch diameter in the mesh with the mating gear, the hob rack module mn, αnd, and αnc (drive and coast pressure angles) and αprd and αprc (drive and coast protuberance angles) are reduced. This leads to a hob protuberance size increase and a smoother root fillet surface finish. However, the drive and coast protuberance angles should not be less than about 8 degrees, otherwise the back angle of the protuberance cutting edge could be too small for proper machining. 

Figure 7: Protuberance hob profile parameters: mn — normal module, Sn — normal tooth thickness at the generating pitch line, ha — addendum, hd — dedendum, and and anc — drive and coast normal pressure angles, hprd and hprc — drive and coast protuberance heights, aprd and aprc — drive and coast protuberance angles, hchd and hchc — drive and coast chamfer heights, and achd and achc — drive and coast chamfer angles.

3 Comparing Gears with Conventional Trochoidal and Optimized Tooth Roots

Table 1 presents a comparison of two asymmetric gear pairs with identical flank geometry and different root fillets — a conventional trochoidal fillet generated by a protuberance hob with a full radius tooth tip, and an optimized fillet generated by a protuberance hob with a special shape tooth tip.

Under the same load, the asymmetric gear pair with the optimized root has its root tensile stress reduced by about 10 percent compared to the asymmetric gear pair with the conventional trochoidal root.

Table 1

4 Protuberance Hob Root Generation Limitations

In some cases, a combination of the asymmetric gear’s geometric parameters does not allow the use of a protuberance hob for preliminary machining of the tooth flanks and final machining of the root fillet. This is typical for gears with a low number of teeth and also a low coast profile pressure angle. In such cases (see Figure 8), the sweep of the protuberance hob tooth tip undercuts the tooth flanks near the drive and coast flank form diameters of the gear tooth with the grinding stock.

Figure 8: Gear profile undercut by protuberance hob tooth tip: 1 — machined gear profile with the grinding stock, 2 —final gear flanks profiles after grinding, 3 —protuberance hob profile, 4 —hobbing generating pitch line, 5 —hobbing generating pitch diameter, 6 —hob tooth tip trajectories, 7 and 8 — drive and coast flank undercut areas, dfmd and dfmc — drive and coast flank form diameters of the gear tooth with the grinding stock.

In this case, the gear machining process should be altered, replacing protuberance hobbing with conventional non-protuberance hobbing (Figure 9a) and optimized root milling using a mill cutter shaped as the optimized root fillet (Figure 9b). After heat treatment, the gear flanks are ground with a helical grinding wheel. Asymmetric gears have different drive and coast flank form diameters, and the helical grinding wheel thread tip should be dressed with a chamfer angle to avoid interference with the root fillet near the coast flank form diameter (Figure 9c).

Figure 9: Processing of asymmetric gear with low number of teeth: a — preliminary gear hobbing, 1 — preliminary machined gear profile, 2 — roughing non-protuberance hob profile; b — optimized root milling, 3 — gear with machined root fillet, 4 — form disk mill cutter; c — tooth flank grinding, 5 — gear with ground tooth flanks, 6 — generating grinding wheel; f — grinding wheel thread tip angle.

Generating grinding of the asymmetric gear flank is described in [3].

5 Inspection of Optimized Tooth Root

The optimized tooth fillet of an asymmetric gear generated with a protuberance hob must be inspected by CMM in the same way as the involute flanks (Figure 10). Deviation of the actual root fillet profile is relative to the designed (CAD) profile at average material conditions. Root profile tolerance should be about 2-3 accuracy grades lower than the ground involute flank profile tolerance.

Figure 10: CMM measurement of asymmetric gear.

Summary

The article presents the direct gear design tooth root fillet optimization technique for asymmetric gears generated with a protuberance hob.

Optimization of the tooth root fillet of asymmetric gears generated with a protuberance hob allows for a reduction in root tensile stress by about 10 percent compared to an asymmetric gear pair with conventional trochoidal roots generated by a full tip radius profile protuberance hob. Combined with other benefits of asymmetric gears [2], this root stress reduction makes asymmetric gears highly desirable for automotive transmissions.

The optimized root fillet of asymmetric gears generated with a protuberance hob must be CMM inspected as thoroughly as the involute tooth flanks. 

References

  1. A.L. Kapelevich, Direct Gear Design, 2013, CRC Press, Taylor & Francis Group, Boca Raton, London, New York.
  2. A.L. Kapelevich, Asymmetric Gearing, 2018, CRC Press, Taylor & Francis Group, Boca Raton, London, New York.
  3. A. Mehr, K. Iguchi, 2017. Hard Finishing of Asymmetric Tooth Profiles – Solutions for Series Production. Proceedings of MPT2017-Kyoto, the JSME International Conference on Motion and Power Transmissions. February 28 – March 3, Kyoto, Japan.
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operates the gear-design consulting firm AKGears, LLC in Shoreview, Minnesota. He has a master’s degree from Moscow Aviation Institute and a Ph.D from Moscow State Technical University and 40 years of experience with custom gear drive development. He is author of the book, “Direct Gear Design,” and many technical articles.
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is an expert in mathematical modeling and stress analysis and is a software developer for AKGears, LLC, in Shoreview, Minnesota. He can be reached at ys@akgears.com. Go online to www.akgears.com.