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Layout of the Gear Micro GeometryBy Ulrich Kissling

A three-step methodology can be a successful approach to optimizing flank line and profile modifications.

The last phase in sizing a gear pair is to specify the f lank line and pro-file modifications, also known as the micro geometry. To do so, it is first necessary to select the primary objec-tive 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 fea-tures 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 calcu-lation 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 f lank line modification and then the profile modification.

USE OF MODIFICATIONSTo find the optimum profile and f lank line modifications for a given gear pair a three-step procedure can be imple-mented 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 com-pensated with a nice micro geometry. The choice of the best macro geom-etry [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, design-ing 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 pro-cedure 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 trans-mission 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 light-

er, 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 modifi-cations 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, subsequent-ly, be stopped before the best solution is found.

Confronted with this problem in many engineering projects, the author

Figure 1: Propositions for an optimal flank line modification to get uniform load distribution for a single stage load (input gear stage of the two-stage industrial gearbox)

Figure 2: Propositions used for fHβ / fma and crowning values according to Equation 1 (input gear stage of the two-stage industrial gearbox)

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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 MODIFICATIONSAs the first step in the procedure, the theoretical f lank line is designed. Contrary to profile modifications, where many goals may be reached, f lank line is always designed

Figure 3: Load distribution with different manufacturing deviation values. Left side: fHβ/ fma= 32 / 0 μm (Statistical); Right side: fHβ/ fma= 23 / 28 μm (Maximum)

Figure 4: Gear pair meshing and path of contact calculated with LTCA, showing the prolonged contact at start and end of the mesh

for best uniform load distribution over the face.

So the goal of a f lank 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 f lank line modification in two steps. In Step 1, we specify the ideal f lank line modification using the average position in the tolerance field without taking into account devia-tions 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 accord-ing 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 rea-son, the use of a special method is recom-mended 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 distribu-tion” describes a useful method to get a realistic value for the load distribution and the face load factor KHβ and is much faster than using LTCA. The algorithm is basi-cally a one-dimensional contact analysis that provides good information about the load distribution over the face width. As input, the geometry of both shafts (includ-ing bearings and loads) is needed (same as for LTCA). The current trend in gear soft-ware 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 propo-sition for best f lank line modification needs to be found quickly. Even for complicated duty cycles, it is possible to find the best

Contains all modifications that will not be changed.

Definition of modifications that will be varied.

Figure 5: Input for the modification sizing tool as used in Step 3 (the flank line modifications resulting from Step 2 are fixed, and the profile modifications are varied)

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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 com-posed 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 TOLERANCESOnce the flank line modification for the medi-um 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 fΗβΤ

of the gears (for example, according to ISO 1328 [8])

• Axis alignment tolerances f∑β, f∑δ (par-allelism 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 reduc-tion 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 (Cb value) for a modi-fication of this kind should be, depends on statistical estimates and experience.

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

Cb = fHβΤ Equation 1

Figure 6: Two charts with results (PPTE and efficiency) of 25 modification variants Red: At 100-percent load; Blue: At 75-percent load (input gear stage of the same gearbox as in Figures 2, 3, and 5)

Figure 7: Industrial two-stage gearbox: the housing stiffness is included in the layout of the modifications

Table 1: Face load factors without flank line modifications

Gear Pair KHβ

Without housing deformation

KHβ With housing deformation

Good foundation Extremely bad founda-tion

HSS (High speed stage) 1.166 1.667 2.320

HSS (Low speed stage) 1.299 1.306 2.410

Figure 8: Manually set modifications on both gear pairs based on initial values suggested by the layout function (Figure 1)

Modifications HSS

Modifications LSS

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 Cb 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 KHβ

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 (fHβ

for the lead variation of the gears (fHβΤ1+fHβΤ2) and fma for the axis misalignment in the contact plane). KHβ

has to be calculated five times: without tolerance, then with +fHβ

& +fma, +fHβ

& -fma, -fHβ & +fma, -fHβ

& -fma. For all five combinations, the line load distribu-tion in the operating pitch diameter has to be calculated and checked for edge contact (Figure 3).

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with statistically combined tolerances, the load distribution is perfect. Even for the unlikely case with maximum tolerances, edge contact is avoided. Using the sugges-tion 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 MODIFICATIONSWhen the flank line modification is defined, the third step is to specify the profile modi-fications. 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 opti-mal. 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 meth-od, 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 modifi-cation 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 NOISELow-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

Table 2: Face load factors with modifications from Step 1

Gear Pair KHβ

HSS 1.002

LSS 1.003

Figure 9: Statistically evaluated and maximum values for manufacturing tolerances

HSS LSS

Figure 10: Load distribution with different manufacturing deviation values for both gear pairs

Gear pair KHβ max (statistical deviations) KHβ max (max. deviations)

HSS 1.14 1.21LSS 1.10 1.16

HSS, statistical deviations (left) and maximum deviations (right)

LSS, statistical deviations (left) and maximum deviations (right)

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

fma = f∑β * cos(αwt) + f∑δ * sin(αwt) Equation 2

When the calculation of the face load factor according to Annex E with manufacturing tolerances is used, then the tolerances fHβ

and fma must be introduced, and the crown-ing values Cb 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 +-fHβfma variants are displayed in the same

graphic, it is easy to check for edge contact. As shown in Figure 3, for the case

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if LTCA also documents the real transverse contact ratio εαeff. If εαeff is bigger than the theoretical transverse contact ratio εα, then the path of contact is elongated and contact shock appears. Therefore, when a low PPTE is obtained, εαeff 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 propo-sition for the tip relief Ca, 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 DESIGNOptimization of profile modifications in a case-by-case manner is extremely time-con-suming and demanding. Results of an LTCA are not easy to evaluate. Comparing results of different LTCA calculations with slightly changed modifications is even more chal-lenging.

Knowing this problem, KISSsoft developed a concept for a so-called “modification siz-ing” 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 cer-tain number of variants with different modi-fications 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 dis-play 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 KHβ

resulted identical for all variants. Therefore, there was no need to change the flank line modifica-tions.

CONSIDERING HOUSING AND/OR PLANET CARRIER STIFFNESSA 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

Figure 11: Input for the modification sizing tool and resulting radar charts (HSS)

Definition of modifications that will be varied for HSS

Radar charts for PPTE and Efficiency of the HSS

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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 pro-cess described here can be applied to cylin-drical gear or bevel gear pairs. If required, it can be combined with the housing defor-mation. In the case of planetary stages, the optimization is performed for all the mesh-

ings in the system, including the deforma-tion of the planet carrier from an integrated FEM calculation.

INDUSTRIAL GEARBOX EXAMPLEFor 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 with-out considering housing stiffness, to get an indication on the influence of the housing.

Before starting with Step 1, the load distri-butions of the two gear pairs without modi-fications 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 KHβ

, 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 inter-medium shaft so that the load distribution in the meshing becomes bad with KHβ values above 2.

Without housing stiffnessSTEP 1In Step 1, flank modifications are evaluated without considering manufacturing toleranc-es. The first suggestions for crowning and helix angle modifications proposed by the layout tool result in KHβ

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 distribu-tion. The final modifications (Figure 8) result in a perfect uniform load distribution with KHβ

values near to 1 (Table 2).

STEP 2In Step 2, manufacturing tolerances are con-sidered 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. KHβ

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.

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Figure 12: Results from the LTCA for HSS without (left) and with (right) profile modifications

SS with modifications from Step 2:PPTE: 5.96 μmEfficiency: 98.7%

HSS with additional profile modifications:PPTE: 2.94 μmEfficiency: 99.2%

Figure 13: Final modifications including profile modifications

HSS

LSS

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Figure 14: A stiffness matrix created by FEM can be included in a KISSsys model, and the housing stiffness is considered in the load distribution analysis

Gearbox with extremely bad foundation

Modifications HSS

Gearbox with good foundations

Modifications LSS

Figure 15: Bearing outer ring displacements in mm (x,y: horizontal, z: vertical)

Figure 16: Manually set modifications on both gear pairs based on initial values suggested by the layout function (Figure 1)

Gear Pair KHβ

HSS 1.001LSS 1.002 Table 3: Face load factors with modifications from Step 1

Gear pair KHβ max (statistical deviations) KHβ max (max. deviations)HSS 1.14 1.22LSS 1.10 1.15 Table 4: Face load factors with modifications from Step 2

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

The f lank line modifications are fixed while a suitable profile crowning modifica-tion is found using the modification sizing tool as described earlier. The crowning value, Ca, 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 Ca, which is implemented at KISSsoft. For HSS, a Ca-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 improve-ment (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 repeat-ed, but the results are not documented here (see Step 3, Gearbox with Bad Foundations, for example).

With housing stiffnessIn any KISSsys model [6], the housing stiff-ness can be considered using a stiffness matrix imported from an FEM software (Figure 14). The resulting housing deforma-tion 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 gear-box 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 modifica-tions 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).

Definition of modifications that will be varied for LSS

Radar charts for PPTE and Efficiency of the LSS

Figure 17: Input for the modification sizing tool and resulting radar charts (LSS)

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Gearbox with bad foundations STEP 1As before, the f lank line modifications are evaluated without considering manufacturing tolerances. Compared with the analy-sis earlier, it is evident that the helix angle modifications are increased to compensate the housing deformation.

STEP 2The 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 3The 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 improve-ment (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 docu-mented in Figure 19.

Summary of industrial gearbox exampleIn typical industrial gearboxes, the housing def lections have a

Figure 18: Results from the LTCA for LSS without (left) and with (right) profile modification

LSS with modifications from Step 2:PPTE: 35.11 μmEfficiency: 98.72%

LSS with additional profile modifications:PPTE: 13.74 μmEfficiency: 99.02%

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negligible influence on the gear mesh if the foundation is accurate. However, if strong hous-ing 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 manufac-turing 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.

CONCLUSIONOptimization of flank line and profile modifi-cations 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.

REFERENCES1. Niemann, “Maschinenelemente,” Band II,

Springer Verlag, 1985.2. U. Kissling, “Auslegung optimaler

Flankenkorrekturen für Stirnradpaare und Planetenstufen mit komplex-en Lastkollektiven,” DMK 2013, S.67, ISBN978-3-944331-33-1. (Or: “Flankenlinienkorrekturen per Software –

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ABOUT THE AUTHOR: Dr. Ulrich Kissling studied mechanical engineering at the Swiss Federal Institute of Zurich (ETH). From 1981 to 2001, he worked as a calculation engineer, technical director, and then as managing director of Kissling Co., a Swiss gearbox company located in Zurich, focusing on planetary, turbo, and bevel-helical gearboxes for industrial applications and in the ski business. In 1998, he founded KISSsoft AG and acts as CEO. He is chairman of the Gear committee of the Swiss Standards Association (SNV) and voting member for Switzerland in the ISO TC 60 committee. He has been involved in numerous engineering projects ranging from micro plastic gears to large open gears and has held presentations at the major international gearing conferences. He has published over 50 publications on calculation procedures for machine design.

eine Fallstudie”; antriebstechnik 11/2013, antriebstechnik 12/2013.)3. ISO 6336, Part 1, “Calculation of load capacity of spur and helical gears,” ISO Geneva, 2006.4. Bae, I; Kissling, U.; An Advanced Design Concept of Incorporating Transmission Error

Calculation into a Gear Pair Optimization Procedure; International VDI conference, Munich, 2010.

5. Mahr, B.; Kontaktanalyse; Antriebstechnik 12/2011, 2011.6. KISSsoft; Calculation software for machine design, www.KISSsoft.AG.7. Kissling, U.; Application and Improvement of Face Load Factor Determination based on

AGMA 927, AGMA Fall Technical Meeting 2013.8. ISO 1328-1, Cylindrical gears — ISO system of f lank tolerance classification,

Geneva, 2013.