The last phase in sizing a gear pair is to specify the flank line and profile modifications, also known as the micro geometry. To do so, it is first necessary to select the primary objective for which optimization has to be achieved: noise, service life, scuffing, micropitting, or efficiency. Certainly, it is not possible to achieve all types of optimization simultaneously, and some actions will worsen some features while improving others. It is easy for the design engineer to lose sight of the bigger picture and fail to find the optimum solution because the calculation method for proving the effects achieved by micro geometry and the contact analysis under load (Loaded Tooth Contact Analysis, LTCA) is complex and time-consuming and interpreting the results is challenging.

Today, much more time is needed to optimize the micro geometry than the macro geometry when designing a toothing. When performing a targeted sizing of the micro geometry, a step-by-step approach should be used, first specifying the flank line modification and then the profile modification.

### Use of Modifications

To find the optimum profile and flank line modifications for a given gear pair a three-step procedure can be implemented to perform a targeted sizing. The layout of the modifications is the last step in the gear design process. Therefore, it is extremely important to keep in mind that a bad choice of macro geometry (module, helix angle, profile shift, etc.) can never be compensated with a nice micro geometry. The choice of the best macro geometry [4] is primordial before starting the layout of modifications.

Flank line and profile modifications have been in use in the gear industry for a long time. Nevertheless, designing modifications is not easy.

One problem is that the verification of the effect of modifications can only be made with an LTCA [5]. LTCA is a complex semi-FEM calculation procedure that needs a lot of calculation time. Furthermore, such software was not available or too complicated to use for most gearbox designers. Thus, modifications were designed based on simple rules without checking if the rule used was appropriate for a specific case.

In the last years, it has become easier to use LTCA software. For an LTCA calculation, all gear data together with the geometry and load condition of the shafts is needed. Therefore, the input for a stand-alone program is complicated and time-consuming. In modern system software, such as KISSsys [6] where the complete transmission chain with gears, shafts, and bearings is modeled, all data for an LTCA is available, and the calculation is performed without further input.

Today’s market request for lighter, cheaper, and stronger gearboxes together with the availability of easy-to-use LTCA software changed things considerably in many gearbox design offices. Now the use of LTCA to check and improve the efficiency of modifications is growing fast.

Unfortunately, the interpretation of LTCA results is not easy. All modifications applied on mating gears are interacting, so the decision of which modification to add or to change is difficult. And as the calculation time for a precise LTCA is still in the order of 10-30 seconds, the design process can become tedious and, subsequently, be stopped before the best solution is found.

Confronted with this problem in many engineering projects, the author of this paper developed a strategy to find the optimum combination of modifications with a fast, straightforward procedure.

### Step 1: Layout of the theoretical flank line modifications

As the first step in the procedure, the theoretical flank line is designed. Contrary to profile modifications, where many goals may be reached, flank line is always designed for best uniform load distribution over the face.

So the goal of a flank line modification is to obtain an even load distribution over the face width plus a reduced edge contact. A good strategy is to size the flank line modification in two steps. In Step 1, we specify the ideal flank line modification using the average position in the tolerance field without taking into account deviations due to manufacturing tolerances. The aim is to reach an even load distribution across the full face width. This will achieve the maximum possible service life. As the deformation of the shafts differs according to the load, it is necessary to specify the torque for which the modification is designed.

In the case of a complex load spectrum, this is not a trivial matter. For this reason, the use of a special method is recommended to achieve the maximum service life, while also taking into account the load spectrum. In Annex E in ISO 6336-1 [3], “Analytical determination of load distribution” describes a useful method to get a realistic value for the load distribution and the face load factor K_{H}_{β} and is much faster than using LTCA. The algorithm is basically a one-dimensional contact analysis that provides good information about the load distribution over the face width. As input, the geometry of both shafts (including bearings and loads) is needed (same as for LTCA). The current trend in gear software is to use system programs that are able to handle a complete power transmission chain. In these applications, all data needed to perform a load distribution analysis are available. Thus, the method is easy to use and provides accurate information of the line load distribution over the face width. This information is helpful in the gear design process when a nearly perfect proposition for best flank line modification needs to be found quickly. Even for complicated duty cycles, it is possible to find the best modification, hence improving the overall lifetime considerably [2, 7]. Therefore, using this one-dimensional contact analysis is ideal for the purpose.

For a single stage load, it is easy to provide a layout function that gives a proposition for a near optimal flank line modification composed of a helix angle modification combined with crowning. Such a layout functionality is implemented in KISSsoft [6] (Figure 1). Another tool that varies modifications to find the overall highest lifetime is available for duty cycles [2, 7].

### Step 2: Including flank line manufacturing tolerances

Once the flank line modification for the medium tolerance position is determined in Step 1, the manufacturing deviations, respectively, the manufacturing tolerances, must be considered. In gear modification layout, normally two main tolerances are used:

Helix slope tolerance f_{HβΤ} of the gears (for example, according to ISO 1328 [8])

Axis alignment tolerances f_{∑β}, f_{∑δ} (parallelism of the shafts, ISO TR 10064)

(f_{∑β}: Deviation error of axis; f_{∑δ}: Inclination error of axis)

Manufacturing deviations are compensated with an additional modification in Step 2. Deviations cause a random increase or reduction of the gap across the face width. Usually, an additional, symmetrical modification (flank line crowning or end relief) is the only practical solution for preventing edge contact in all possible combinations of allowances. How large the relief (C_{b} value) for a modification of this kind should be, depends on statistical estimates and experience.

When no expertise is available, the following procedure can be applied. In ISO 6336-1, Annex B, for gears having a flank line modification to compensate for deformation, the crowning amount:

_{b}= f

_{HβΤ }Equation 1

for both gears is proposed. If crowning is already used for the compensation of the deformations (Step 1), the actual crowning value has to be increased by C_{b} according to Equation 1.

When such an additional modification is applied, clearly the load distribution over the face width as obtained in Step 1 is not uniformly distributed anymore. Therefore, the face load factor K_{Hβ} will increase. The goal is to avoid edge contact in all possible combinations of deviations. The ISO 6336-1 Annex E procedure is again useful; the procedure advises to take manufacturing tolerances into account (f_{Hβ} for the lead variation of the gears (f_{HβΤ1} +f_{HβΤ2}) and f_{ma} for the axis misalignment in the contact plane). K_{Hβ} has to be calculated five times: without tolerance, then with +f_{Hβ} & +f_{ma}, +f_{Hβ} & -f_{ma}, -f_{Hβ} & +f_{ma}, -f_{Hβ} & -f_{ma}. For all five combinations, the line load distribution in the operating pitch diameter has to be calculated and checked for edge contact (Figure 3).

The axis misalignment in the contact plane can be obtained from f_{∑β}, f_{∑δ} using:

_{ma}= f

_{∑β}* cos(α

_{wt}) + f

_{∑δ}* sin(α

_{wt}) Equation 2

When the calculation of the face load factor according to Annex E with manufacturing tolerances is used, then the tolerances f_{Hβ} and f_{ma} must be introduced, and the crowning values C_{b} set (Figure 2). In KISSsoft software [6], a proposition for the maximum values or realistic values (97-percent probability) is shown. Normally, it is better to use the statistically weighted values.

If the load distribution of all the five +-f_{Hβ}f_{ma} variants are displayed in the same graphic, it is easy to check for edge contact. As shown in Figure 3, for the case with statistically combined tolerances, the load distribution is perfect. Even for the unlikely case with maximum tolerances, edge contact is avoided. Using the suggestion of ISO (Equation 1) is a good choice in this case.

For duty cycles, it is best to normally use the bin with highest torque and then check the result again with the lowest torque.

### Step 3: Profile modifications

When the flank line modification is defined, the third step is to specify the profile modifications. Important features such as noise, losses, micropitting, scoring, and wear can be improved by profile modifications. Therefore, the layout criteria must be defined. Then, the corresponding strategy is used.

Additionally, the designer must decide at which torque level (or at which bin if a duty cycle is used) the modification should be optimal. This is not always obvious. For scoring, it would be the peak torque, but for noise, it is better to use the most frequent driving situation. For example, the aim for a truck transmission is to have the lowest noise at 80 km/h when driving on the highway in the fifth gear. In that case, the corresponding torque will be used for the layout.

LTCA has to be applied as calculation method, which may require a lot of time if several variants must be checked. A special tool has been developed specially for this purpose. It generates a list of variants, processes them, and then displays a summary of the results.

Clearly, a profile modification has a certain influence on the face load distribution as well, so the previously specified flank line modification may be varied slightly along with the profile modification. The results will then be displayed both as a graph and in a table. For interesting individual variants, a report is generated that contains all the detailed results from the LTCA.

### Layout for low noise

Low-noise design is one of the most important criteria in the layout procedures. For low-noise behavior, the peak-to-peak transmission error (PPTE) must become as low as possible and contact shock (due to deflection, the contact between the teeth starts too early) must be avoided. In KISSsoft, the contact shock is visualized in the meshing diagram where the real path of contact (Figure 4) is displayed. The transmission error is a direct result of the LTCA analysis. Unfortunately, a low PPTE value does not automatically mean that the contact shock is reduced as well. The contact shock can indirectly be controlled if LTCA also documents the real transverse contact ratio ε_{αeff}. If ε_{αeff} is bigger than the theoretical transverse contact ratio ε_{α}, then the path of contact is elongated and contact shock appears. Therefore, when a low PPTE is obtained, ε_{αeff} must also be controlled.

Good practice for reducing the PPTE is to use long tip relief for spur gears and profile crowning for helical gears. As a first proposition for the tip relief C_{a}, the simple rule according to Niemann [1] may be used. The proposition must be checked by performing a first LTCA calculation and then be slightly adapted after verifying the resulting PPTE and length of the effective contact path.

### Use of a modification sizing tool to find the optimal design

Optimization of profile modifications in a case-by-case manner is extremely time-consuming and demanding. Results of an LTCA are not easy to evaluate. Comparing results of different LTCA calculations with slightly changed modifications is even more challenging.

Knowing this problem, KISSsoft developed a concept for a so-called “modification sizing” tool in partnership with a German gear company. The basic idea is to systematically vary the properties of an unlimited number of modifications. Also, the possibility to cross-vary properties of individual modifications (e.g., tip relief and length of modification) must be available (Figure 5). With this, a certain number of variants with different modifications is defined. Then, for every variant a full LTCA is performed, and all relevant data is stored. This can be time-consuming if hundreds of variants are analyzed, but the process is fully automatic.

A major problem was to find a way to display the results. The data is displayed in a table (with the possibility to export into Microsoft Excel), but with so many numbers in a table, it is difficult to maintain a good overview. Principally, if the PPTE, losses, and lifetime of different variants should be represented in the same graphic, a 5D- or even 10D-diagram would be needed. As this was not an issue, an unlimited number of radar charts displayed in parallel was used (Figure 6). In the example shown, compared to no profile modifications (variant in Figure 6), the PPTE can be reduced from 6.3 to 1.3 μm and the losses from 1.1 to 0.7 percent. The face load factor K_{H}β resulted identical for all variants. Therefore, there was no need to change the flank line modifications.

### Considering housing and/or planet carrier stiffness

A clever combination of an FE-application (gearbox housing) with a gearbox design software is currently the most efficient approach. With KISSsoft’s KISSsys, it is possible to easily import a stiffness matrix from any commercial FEM, consider the effect of the housing deformation on the bearing and shaft displacement, and then relay to the load distribution in the gear mesh.

The micro geometry optimization process described here can be applied to cylindrical gear or bevel gear pairs. If required, it can be combined with the housing deformation. In the case of planetary stages, the optimization is performed for all the meshings in the system, including the deformation of the planet carrier from an integrated FEM calculation.

### Industrial Gearbox Example

For a typical industrial two-stage parallel shaft reducer (Figure 7), the modifications are optimized using the three-step method. The process is repeated twice, with and without considering housing stiffness, to get an indication on the influence of the housing.

Before starting with Step 1, the load distributions of the two gear pairs without modifications are calculated. The face load factors are calculated according to Annex E in ISO 6336-1, using the axis deformations from the shaft calculation (Table 1).

The housing is 1,400 mm long, 400 wide, and 750 mm high. The wall thickness is 20 mm, which is moderate. The elastic yielding in the bearing supports is about 0.1 mm, but as the yielding is similar in both bearings of every shaft, the gap in the meshing is only minimally changed. As displayed in Table 1, the face load factor K_{H}β, calculated based on the shaft deformation including housing deformation, is unchanged compared to the same factor without housing deformation.

To test the three-step procedure, a weak foundation was simulated under the intermedium shaft so that the load distribution in the meshing becomes bad with K_{H}_{β} values above 2.

#### Without housing stiffness

**Step 1**

In Step 1, flank modifications are evaluated without considering manufacturing tolerances. The first suggestions for crowning and helix angle modifications proposed by the layout tool result in K_{Hβ} values of 1.016 for high-speed stage (HSS) and 1.012 for low-speed stage (LSS).

These modifications are then manually adjusted to reach a more even load distribution. The final modifications (Figure 8) result in a perfect uniform load distribution with K_{Hβ} values near to 1 (Table 2).

**Step 2**

In Step 2, manufacturing tolerances are considered as explained earlier. The proposed statistical and the maximum values for the helix slope deviation and the misalignment of axes are shown in Figure 9.

The perfectly uniform load distribution resulting in Step 1 changes significantly if the tolerances are considered. K_{Hβ} increases up to 1.23 (statistically evaluated tolerance) or 1.36 (maximum tolerance), and the highest load is now on the left or right end of the face width (edge contact).

To avoid edge contact in all tolerance combinations, the crowning values must be increased according to Equation 1. Then, an initial check suggested that acceptable load distribution without edge contact resulted, as shown in Figure 10. The crowning of the HSS was increased from 4 to 13 μm (both gears) and of the LSS from 8 to 18 μm.

**Step 3**

In Step 3, profile modifications are added to reduce the transmission error and gear losses at 90 percent of the nominal load.

The flank line modifications are fixed while a suitable profile crowning modification is found using the modification sizing tool as described earlier. The crowning value, C_{a}, must be defined carefully, so that the contact shock (Figure 4) can be eliminated. Niemann [1] proposes a simple rule to obtain an approximate value for C_{a}, which is implemented at KISSsoft. For HSS, a C_{a}-value of 25 μm is suggested for LSS 38 μm. Therefore, the input for the sizing tool can be deduced; for HSS, the profile crowning values are varied from 20 to 60 μm in 10 μm steps (Figure 11). The modifications are cross-varied between gears 1 and 2, therefore, 25 variants are checked.

The next figures show for HSS, the input and output of the modification sizing tool (Figure 11) and the obtained improvement (Figure 12) in noise behavior (PPTE reduced by 50 percent and contact shock eliminated) and in power loss (reduction of the losses by 40 percent). The resulting modifications are documented in Figure 13. For LSS, the same procedure is repeated, but the results are not documented here (see Step 3, Gearbox with Bad Foundations, for example).

#### With housing stiffness

In any KISSsys model [6], the housing stiffness can be considered using a stiffness matrix imported from an FEM software (Figure 14). The resulting housing deformation at the bearing positions are shown in a results table (Figure 15). The deformations are assigned to the bearings (typically, outer ring) in the shaft calculation and considered in the gear contact analysis.

As explained earlier, the industrial gearbox with a good, stiff foundation has a small change in the meshing gap when gearbox stiffness is considered. As shown in Figure 15, the displacements at the bearing positions are similar in the two bearings of the same shaft. Because the meshing gap is almost unchanged, the resulting modifications are all identical to the previous section.

In this section, the gearbox with bad foundations is used. The displacement of the bearings of the intermedium shaft is unbalanced due to a weak foundation under the intermedium shaft (Figure 15).

#### Gearbox with bad foundations

**Step 1**

As before, the flank line modifications are evaluated without considering manufacturing tolerances. Compared with the analysis earlier, it is evident that the helix angle modifications are increased to compensate the housing deformation.

**Step 2**

The manufacturing tolerances are the same as before (Figure 9). Changing the crowning values according to Equation 1 results in load distributions without edge contact. The crowning values used are exactly the same as before (crowning of the HSS was increased from 4 to 13 μm, and of the LSS, from 8 to 18 μm).

**Step 3**

The profile modifications are designed to reduce the transmission error and losses at 90 percent of the nominal load. Because the proceeding for the optimization is very similar to the example without Housing Stiffness, only the LSS stage is documented here.

The next figures show for LSS, the input and output of the modification sizing tool (Figure 17) and the obtained improvement (Figure 18) in noise behavior (PPTE reduced by 60 percent and contact shock eliminated) and in power loss (reduction of the losses by 25 percent). The resulting modifications are documented in %%0915-KISSsoft-19%%.

#### Summary of industrial gearbox example

In typical industrial gearboxes, the housing deflections have a negligible influence on the gear mesh if the foundation is accurate. However, if strong housing deflections occur due to bad foundation or extremely lightweight design, then housing deformation must be considered in the first layout step. For the compensation of manufacturing deviations (Step 2), it does not matter if housing stiffness is considered or not. This is also valid for the profile modifications (Step 3). If the flank line modifications designed provide a uniform load distribution, then the optimum profile modifications are mostly identical with and without housing stiffness consideration.

### Conclusion

Optimization of flank line and profile modifications for a specific application is not an easy task. The three-step methodology has proven highly successful since it was introduced two years ago. The layout of the modifications for an industrial gearbox shows that the housing deformations have an insignificant influence on the resulting gap in the meshing of the gears. When the housing is further deformed due to a bad foundation, then the deformations must be considered.

This method can also be used in applications such as wind power, ship transmission systems, or helicopters in which it is demanding to define the modifications due to the extreme load spectrum or high housing deflections.

### References

- Niemann, “Maschinenelemente,” Band II, Springer Verlag, 1985.
- U. Kissling, “Auslegung optimaler Flankenkorrekturen für Stirnradpaare und Planetenstufen mit komplexen Lastkollektiven,” DMK 2013, S.67, ISBN978-3-944331-33-1. (Or: “Flankenlinienkorrekturen per Software – eine Fallstudie”; antriebstechnik 11/2013, antriebstechnik 12/2013.)
- ISO 6336, Part 1, “Calculation of load capacity of spur and helical gears,” ISO Geneva, 2006.
- Bae, I; Kissling, U.; An Advanced Design Concept of Incorporating Transmission Error Calculation into a Gear Pair Optimization Procedure; International VDI conference, Munich, 2010.
- Mahr, B.; Kontaktanalyse; Antriebstechnik 12/2011, 2011.
- KISSsoft; Calculation software for machine design, www.KISSsoft.AG.
- Kissling, U.; Application and Improvement of Face Load Factor Determination based on AGMA 927, AGMA Fall Technical Meeting 2013.
- ISO 1328-1, Cylindrical gears — ISO system of flank tolerance classification, Geneva, 2013.