This article mainly introduces the definitions and impacts of root undercut and steps in the manufacturing of transmission gears, discusses the step problems caused by incorrect design and manufacturing of the convex angle amount of roughing tools before and after heat treatment, and provides judgment conditions and methods for correcting the convex angle amount. It analyzes the principle that the mismatch between the overtravel calculation of finishing tools (before and after heat treatment) and roughing tools leads to steps in the tooth root transition area, and presents common failure cases and solutions. In view of the state of parts before and after finishing, it examines the impact of errors in tooth profile, helix, pitch, and radial runout on gear root steps, and proposes relevant solutions from the perspectives of tool modification, process selection, and thermal deformation in gear manufacturing.

With the continuous tightening of fuel and emission regulations and the implementation of the dual-credit policy, the process of new energy, lightweight, and electrification of automobiles is accelerating. Transmission gears are increasingly required to withstand high torque loads and achieve high-speed precision. While meeting NVH requirements, avoiding gear root steps has become a key issue. This article analyzes the shape characteristics of different steps, discusses their causes, and provides solutions.

Gear Undercut and Steps

As shown in Figure 1, the undercut of a gear is the area cut by the convex edge of the roughing tool, which reserves retraction space for subsequent finishing of the tooth surface to avoid tool joint steps. If the shape and size of the undercut before finishing are problematic, steps or undercut issues may occur in subsequent processing.

Steps reduce the bending strength of the gear root. Without steps, only the shape and size of the dangerous section at the maximum undercut position need to be considered; with steps, the shape and size of the steps must also be considered (see Figure 2). For details, refer to the relevant chapters of the international standard ISO-6336.

Estimation formula for strength reduction:

tg: Maximum step amount along the normal direction at the gear root step on the end face (mm);

ρg: Curvature radius at the position of maximum step (mm);

Ys: Gear bending strength coefficient without steps;

Ysg: Estimated gear bending strength coefficient with steps.

In addition, if the step is located in the normal meshing area of the gear, it will cause abnormal transmission noise, which not only affects the user experience of the transmission but also may pose functional risks.

Convex Angle Amount Issues of Roughing Tools

1. Definition of Convex Angle Amount of Roughing Tools

The convex angle amount of a roughing tool refers to the raised area of the tool tip along the main cutting edge, which extends from the secondary cutting edge to the end of the tip arc. This is most common in hobs and shaper cutters, as shown in Figures 3 and 4.

Note that the convex angle amount cannot be directly equated with the processed undercut amount. The convex angle amount is the normal size marked on the tool tooth profile, while the undercut amount measured by a gear measuring instrument is the end face size of the part. The convex angle amount can only be converted to the undercut amount through the axial intersection angle between the tool and the part.

2. Problems and Solutions for Convex Angle Amount of Roughing Tools

Gear finishing processes are generally divided into two types: before and after heat treatment. Processes before heat treatment typically involve combinations of gear shaping/ Hobbing with shaving; processes after heat treatment involve combinations of gear shaping/ Hobbing with honing and grinding. In terms of convex angle amount issues of roughing tools, their manifestations are basically the same. However, finishing before heat treatment is not affected by thermal deformation, making it easier to control undercut and steps. If step problems occur, the first step is to check whether the design and manufacturing of the convex angle amount of the roughing tool are correct. The following analysis takes the hobbing-shaving process as an example.

(1) The step is located near the maximum undercut position. This type of step is difficult to identify in reports and usually requires marking the maximum undercut position before shaving for identification, as shown in Figure 5.

(2) The step is located above the maximum undercut position. This type of step is easy to identify, with the characteristic of a raised shape between the undercut arc cut by the hob and that cut by the shaving cutter, as shown in Figure 6.

For the above two types of problems, attempting to increase the addendum height of the finishing tool to remove the step will result in both the step and undercut being cut off by the finishing tool tip, leaving a new step. Therefore, the convex angle amount of the roughing tool should be checked and corrected.

​Method 1: Check for manufacturing errors in the convex angle amount of the hob. The measured report of parts processed by the hob shows that the undercut amount ranges from 0.043 to 0.048 mm. Figure 7 is a simulation diagram of the undercut area on the part's end face obtained through simulated hobbing generation, with the undercut amount being 0.056±0.006 mm. It can be found that the upper limit of the measured undercut amount (0.048 mm) is smaller than the lower limit of the simulation result (0.050 mm), indicating that the processing of the tool's convex angle amount does not match the design. The tool tooth profile should be regrinded until a qualified undercut can be processed.

​Method 2: Adjust the nominal value of the M parameter in the process to adapt the allowance to the tool's convex angle amount. However, this method has limitations. If adjusting the M parameter still fails to eliminate the step, the tool tooth profile must be regrinded.

Method 3: If the above methods fail, collect processing reports of parts before heat treatment, after heat treatment, and finished products after stable heat treatment deformation. Statistically analyze the undercut variation caused by thermal deformation to obtain the convex angle compensation value, thereby correcting the hob design.

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Overtravel Issues of Finishing Tools

1. Definition of Overtravel of Finishing Tools

There are many types of finishing tools, including shaving cutters, honing tools, and grinding tools. To avoid steps, these tools must be designed with positive overtravel. Overtravel is defined as the difference between the length of the involute extension to the lowest point that the finishing tool can process and the length of the extension to the actual involute starting point. Refer to Figure 8 for a schematic diagram of the tooth profile.

It can be seen that when the scraping starting point is below the actual involute starting point, the overtravel is positive, indicating that the actual involute starting point can be processed without steps. When the scraping starting point is above the actual involute starting point, the overtravel is negative, indicating that the actual involute starting point cannot be processed, resulting in steps.

2. Overtravel Problems and Solutions

(1) Before heat treatment: Processing before heat treatment is mainly represented by the shaving process. Overtravel issues of shaving cutters are shown in Figure 9.

The main causes are as follows:

• First, problems with the regrinding of the shaving cutter, where the outer diameter and tooth thickness do not match. A too-small outer diameter leads to insufficient overtravel and steps; a too-large outer diameter causes interference between the tool tip and the tooth root transition arc of the part. This can be controlled through the regrinding parameter table of the shaving cutter to ensure correct matching of outer diameter and tooth thickness, avoiding such problems.
• Second, design issues with the shaving cutter, where the overtravel does not cover the position of the actual involute starting point (referred to as TIF point) under the maximum allowance limit. This position is much lower (approximately 0.2–0.5 mm) than the TIF point under medium allowance. If the shaving cutter is designed only considering overtravel for medium allowance, it may fail to meet requirements when the allowance is close to the maximum, leading to steps.

(2) After heat treatment: Processing after heat treatment mainly involves honing and grinding, with tools including diamond wheels and grinding wheels. Overtravel issues are mainly related to the involute starting point of the diamond wheel; an excessively high position of this point leads to small overtravel during tool cutting of parts.

Overtravel issues of honing tools are basically the same as those of shaving cutters, and the problem diagnosis mode and solutions for shaving can be fully referenced. Overtravel issues of grinding tools are somewhat different, but the step manifestation is similar, as shown in Figure 10.

Grinding methods are divided into generating grinding and form grinding, with different principles. The calculation of the scraping involute starting point in generating grinding is similar to that in hobbing, and the trajectory of the tool tip scraping is also a cycloid, but smaller and denser than that in hobbing. The scraping involute starting point in form grinding is not a cycloid but is ground by diamond tools according to coordinates.

The main causes of overtravel issues are:

• First, problems with grinding wheel regrinding. In generating grinding, improper matching of the addendum height and tooth thickness of the grinding wheel dressed by the diamond wheel leads to overtravel issues. For example, dressing the grinding wheel with an excessively small outer diameter, or offset in the joint between the grinding wheel addendum arc and the main cutting edge; in form grinding, wear of the diamond wheel or diamond pen results in incomplete dressing of the grinding wheel tooth profile, causing steps at the part root.
• Second, issues with diamond wheel design or dressing programs. In generating grinding, the involute starting point of the diamond wheel is designed too high (see Figure 10); in form grinding, the scraping starting point parameter in the diamond wheel dressing program for the grinding wheel is too large. Both cause the scraping involute starting point of the grinding wheel to be higher than the actual involute starting point of the part, resulting in insufficient overtravel and steps.

(3) Solutions: In the initial stage of designing the generating grinding diamond wheel or setting the form grinding dressing program, it is necessary to simulate and verify the position of the tool tooth profile's scraping starting point to ensure sufficient overtravel under the maximum allowance limit. In addition, during the manufacturing of diamond wheels and dressing of grinding wheels, the dimensional tolerances of addendum height and tooth thickness must be strictly controlled. Dressing tools should be replaced compulsorily according to their service life, and inspection should be strengthened (100% inspection of key tooth profile dimensions is recommended).

Issues with Tooth Profile, Helix, Pitch, and Radial Runout of Parts

In addition to the impact of roughing and finishing tools, the precision of parts before and after finishing is also closely related to the generation of steps (for the meanings of each precision parameter, refer to standards such as GB, ISO, or VDI).

1. Step Problems Caused by Tooth Profile and Solutions

(1) Problem analysis: From the perspective of part tooth profile, if the tooth profile angle error fHa and tooth profile crowning Ca have inconsistent values or directions before and after finishing, steps may occur at the root. The cutting allowance at the root in the gear's mid-section under fHa and Ca is shown in Figure 11.

Figure 11a shows that inconsistent fHa before and after processing leads to excessive cutting allowance at the root; Figure 11b shows that inconsistent Ca before and after processing leads to excessive cutting allowance at the root; Figure 11c considers the combined effect of inconsistent fHa and Ca before and after processing, where the increased allowance at the root 叠加，increasing the risk of steps.

Of course, the above figures only describe the scenario where "fHa = 0 before processing and fHa > 0 after processing". In actual production, there is also the scenario where "fHa < 0 before processing and fHa > 0 after processing", which further increases the root allowance, making steps almost unavoidable.

Another common scenario in grinding processes is that when the helix crowning of the ground tooth is large, the tooth profile shows natural distortion in three sections, as shown in Figures 12 and 13.

On the cross-sections at both ends of the part's tooth width, the tooth profile angle fHa changes before and after grinding. The originally vertical tooth profile angle (fHa = 0) becomes inclined in opposite directions (fHa > 0 at the upper end face and fHa < 0 at the lower end face). At this time, the root allowance at the upper end face increases, easily causing steps.

(2) Solutions: The ideal condition for addressing the first problem is that the tooth profile fHa and Ca before and after processing are very close. After obtaining stable data on the tooth profile thermal deformation of the part, adjust the tooth profile requirements of the previous process. This allows for corresponding tooth profile modifications in tool design to align with the finished product requirements after processing.

Solutions for step problems in grinding processes:

• First, the natural distortion can be compensated in the previous process, but this is challenging due to the difficulty in controlling differences between roughing and finishing tools and thermal deformation trends.
• Second, an anti-twist program function can be added to the grinding machine, which is relatively easier and particularly helpful for reducing the development cycle and cost of process trial production.

2. Step Problems Caused by Helix and Solutions

(1) Problem analysis: The helix angle error fHb and helix crowning Cb also affect the generation of gear root steps, as shown in Figure 14.

It can be seen that if the helix angle deviation before and after finishing is consistent, the tooth thickness processing allowance along the helix is uniform; if the difference is large and the allowance varies significantly from one end to the other, the end with more allowance is at risk of generating steps.

(2) Solutions: Similar to tooth profile correction methods, obtain stable data on helix thermal deformation through experiments and compensate for it in the previous process. Note that once helix modification requirements are added, tooth profile modification will also be affected. Therefore, tool design must consider both tooth profile and helix modifications to achieve correct compensation and adjustment, ensuring consistent modification states before and after processing.

3. Step Problems Caused by Pitch and Radial Runout and Solutions

(1) Step problems caused by excessive individual pitch error Fp and adjacent pitch error fu:

Analysis: Such problems usually occur in roughing processes, most typically in hobbing. When the number of teeth of the part is small and the hob has multiple starts, especially when the number of teeth of the part is divisible by the number of hob starts (e.g., 14 teeth for the part and 2 starts for the hob), the probability of problems is very high. As shown in Figure 15, in the case study, the center of each tooth of the part deviates from the center of its theoretical tooth profile. During subsequent finishing, the tooth thickness processing allowance on the side far from the centerline increases, leading to gear root steps.

Solutions: Start with hob design. If processing cycle allows, use as few starts as possible; if high cycle requirements necessitate multiple starts, improve the manufacturing precision of the hob or adopt the more costly full-ground tooth profile process.

(2) Step problems caused by excessive cumulative error and radial runout:

Analysis: Such problems usually occur in finishing processes after heat treatment (grinding, honing), with reports showing both undercut and steps on the same tooth flank, as shown in Figure 16.

The main cause is significant changes in the cumulative pitch error Fp and radial runout error of the part after heat treatment, as shown in Figure 17. The manifestation of cumulative errors in step problems is similar to that of individual pitch errors. After one full circle of pitch accumulation, combined with the impact of radial runout, the centerline of some teeth deviates significantly from the centerline of their theoretical tooth profile, as shown in Figure 18.

Finished parts after finishing (grinding) also have comprehensive cumulative errors and radial runout errors. If the curve trends of these errors are opposite to those before processing, the possibility of steps in some teeth will be greatly increased.

Solutions: Optimize the gear cutting parameters in the process based on the form of part heat treatment deformation; optimize the positioning and clamping methods related to the processing datum hole and end face; or optimize heat treatment process parameters and fixture positioning methods; if necessary, even optimize the gear material type and structural design.

In summary, only by identifying the causes of irregular thermal deformation can we targetedly control the form of thermal deformation, minimize the thermal deformation amount, and reduce or even eliminate tooth root step problems as thermal deformation decreases.