Sizing the gear micro geometry Step by Stepkadkraft.com/wp-content/uploads/2015/12/PRESENTATI… · · 2015-12-07Sizing the gear micro geometry – Step by Step Dr.-Eng. Ulrich

Ulrich Kissling – KISSsoft User Conference India 2015 1

Sizing the gear micro geometry – Step by Step

Dr.-Eng. Ulrich Kissling, KISSsoft AG, Switzerland

Abstract

The last phase in sizing a gear is to specify the flank line and profile modifications (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. One thing is certain: 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, the contact analysis under load ("Loaded Tooth Contact Analysis", or LTCA), is complex and time-consuming, and interpreting the results is complex.

Today, we need much more time to optimize the micro geometry than the macro geometry, when designing a toothing. This makes it all the more surprising that the technical literature barely mentions the topic of micro geometry. In Niemann [1], for example, the topic of profile shift is discussed over 5 pages, while only 3 pages are devoted to flank line and profile modifications!

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. This paper describes how a 3-step process can be implemented to perform a targeted sizing.

Usually, there is only one lay-out criterion for specifying the optimum flank line modification: to achieve an as evenly distributed load across the face width as possible, and, in particular, to avoid edge contact (highest load on the end of the face). The course of the gap in the meshing is caused by the elastic deformation of the shafts, generated by the operating forces and manufacturing allowances (tolerances).

It is best 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 face width. This will enable the maximum possible service life to be achieved. As the deformation of the shafts differs according to the load, it is necessary to specify the torque for which the modification has to be sized. In the case of a complex load spectra this is not a trivial matter. For this reason, a special method has been developed, which can be used to achieve the maximum service life while also taking into account the load spectrum. Using the "one-dimensional contact analysis" [2] (according to ISO 6336-1, Appendix E [3]) is ideal for this purpose.

Once the flank line modification for the medium tolerance position is determined in step 1, the manufacturing tolerances are compensated with an additional modification in step 2. Tolerances (manufacturing allowances) cause a randomly increase/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 (Cb value) for a modification of this kind should be depends on statistical estimates and experience.

When the flank line modification is defined, the third step is to specify the profile modifications. Now the primary aim (sizing criterion such as noise, service life, etc.) is very important. LTCA has to be used as calculation method, and this may require a lot of time if several variants are to be checked. A program module has been developed specially for this purpose. It generates a list of variants, processes them, and then displays a clear summary of the results. The LTCA calculation runs completely automatically, as it may run for hours in extreme cases, if hundreds of profile modification combinations are calculated. A typical application is minimizing the transmission error by systematically varying the value and length of the pinion and gear tip relief, independently of one another.

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

The micro geometry optimization process described here can be applied to cylindrical gear or bevel gear pairs. If required, it can also be combined with an analysis of the housing deformation from an FEM calculation. 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. The methodology has proven highly successful since it was introduced two years ago, in applications such as wind power or ship transmission systems, in which it is very demanding to define the modifications, due to the extreme load spectra.

1 Introduction: Use of Modifications

This paper explains how to find straight forward the optimum profile and flank line modifications for a given gear pair using a 3-step-procedure. The layout of the modifications is the last step in the gear design process. Therefore, it is extremely important to remind, that a bad choice of macro geometry (as module, helix angle, profile shift, etc.) never can 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 are in use in gear industry since long time. Nevertheless, designing modifications is not easy. In literature, astonishing few information about the topic can be found. In the Niemann book [1] just few generic hints are given – compared to the detailed discussion of much simpler problems as for example profile shift layout!

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 much calculation time. Furthermore such software was or not available or too complicated to use for most gearbox designers. Therefore, modifications were designed based on simple rules without a check, if the rule used was appropriate for a specific case.

In the last years is 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 modelled, 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 developed a strategy to find the optimum combination of modifications with a fast, straightforward procedure.

2 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 here, a straightforward technique can be used.

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. Annex E in ISO6336-1 [3], "Analytical determination of load distribution" describes a very useful method to get a realistic value for the load distribution

and the face load factor KH 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 an 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, which gives a proposition for a near to optimum flank line modification composed of a helix angle modification combined with a crowning. Such a layout functionality is implemented in KISSsoft [6] (fig. 1). Another tool


that varies modifications to find the overall highest life time is available for duty cycles. This method is described in earlier publications [2, 7].

Load distribution before sizing

Load distribution after sizing

Figure 1: Proposition 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)


3 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 fHof the gears (for example according to ISO 1328 [7])

- Axis alignment tolerances f,f (parallelism of the shafts, ISO TR 10064)

(f: Deviation error of axis; f: Inclination error of axis)

Manufacturing deviations are compensated with an additional modification in Step 2. Deviations cause a randomly 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 (Cb 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

Cb = fH (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 Cb according eq. 1.

When such an additional modification is applied, clearly the load distribution over the face width as obtained in step 1 is not anymore uniform distributed. Therefore the face load factor KH will increase. The goal is to avoid edge contact in all possible combination of deviations.

The ISO6336-1 Annex E procedure is again very useful; the procedure advises to take manufacturing tolerances into account (fH for the lead variation of the gears (fHT1+fHT2) and

fma for the axis misalignment in the contact plane). KH has to be calculated five times: Without

tolerance, than with +fH & +fma, +fH & -fma, -fH & +fma, -fH & -fma. The highest KH–value found will be used in the load capacity calculations. For all five combinations, the line load distribution in the operating pitch diameter has to be calculated and checked for edge contact (fig. 3).

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

fma = f * cos(wt) + f * sin(wt) (2)

In KISSsoft [6] this task is implemented, when the calculation of the face load factor according Annex E with manufacturing tolerances is used. Then the tolerances fH and fma can be

introduced, the crowning values Cb set (fig. 2). 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 +-fHfma variants are displayed in the same graphic, it is easy to check for edge contact. As shown in fig. 3, for the statistically combined tolerances case the load distribution is perfect. Even for the unlikely case with maximum tolerances, edge contact is avoided. Using the suggestion of ISO (eq. 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.

Figure 2: Propositions used for fH / fma and crowning values according eq. 1

(Input gear stage of the two-stage-industrial gearbox)

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

4 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 used 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 layout procedure. For low-noise behavior, the peak-to-peak transmission error (PPTE) must become lowest 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 (fig. 4) is displayed. The transmission error is a direct result of the LTCA analysis. Unfortunately a low PPTE value does not automatically mean that also the contact shock is reduced. The contact shock can indirectly be controlled if LTCA also documents the

real transverse contact ratio eff. If eff is bigger than the theoretical transverse contact

ratio, then the path of contact is prolonged and contact shock appears. Therefore, when a low

PPTE is obtained, eff must be controlled.

Figure 4: Gear pair meshing and path of contact calculated with LTCA, showing the

prolonged contact at start and end of the mesh.

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 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 optimum 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 vary systematically properties


of an unlimited number of modifications. Also the possibility to cross-vary properties of individual modifications (as tip relief to length of modification) must be available (fig. 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’s are stored. This can be time-consuming if hundreds of variants are analysed, but the process is fully automatic.

Tab Base: Contains all modifications which will not be changed

Tab Conditions: Definition of modifications which 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).

Figure 6: Two charts with results (PPTE and efficiency) of 25 modification variants

Red: At 100 precent load; Blue: At 75 precent load (Input gear stage of the same gearbox as in fig. 2, 3, 5)


A major problem was to find a way to display the results. The data is displayed in a table (with 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 is used (fig. 6). In the example shown, compared to no profile modifications (variant – in fig. 6), the PPTE

can be reduced from 6.3 to 1.3 m and the losses from 1.1 to 0.7 percent. The face load factor

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

modifications.

If more variants with finer resolution of the parameters are checked, then another graphical display of the results is preferable as shown in fig. 7. It is amazing how big influence of the modification on some failures as Micropitting (safety varies from 3.1 down to 1.2), on PPTE or on the efficiency (losses varies from 0.62 % up to 1.12 %) is.

Tab Conditions: Definition of modifications which will be varied


Figure 7: PPTE, maximum Hertzian pressure, Efficiency, Micropitting safety and Tooth

Fracture Safety when the parameters (Tip relief (“Value m”) and Length (“Factor 1”) of an arc-like tip relief are varied (225 solutions, all at 100% load)


5 Considering housing and/or planet carrier stiffness

A clever combination of a FE-application (gearbox housing) with a gearbox design software is today a most efficient approach. In KISSsys it is possible to import easily 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, 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.

Figure 8: Industrial two-stage gearbox; the housing stiffness is included in the layout of the modifications. Additional Force on the input shaft (yellow circle) and the resulting bearing

forces (red arrows).

6 Example: Use of the 3-step-procedure with a industrial 2-stage gearbox

For a typical industrial two-stage parallel shaft reducer (fig. 8) the modifications are optimized using the 3-step method. The process is repeated twice, with and without considering housing stiffness, to get an indication on the influence of the housing. To further increase the housing deformation, the housing made in aluminium with relatively thin walls is used and a high radial force is applied on the input shaft.

First the load distributions of the two gear pairs without modifications are calculated. The face load factors are calculated according to Annex E in ISO6336-1, using the axis deformations from the shaft calculation (table 1).


The housing is 1400 mm long, 400 mm wide and 750 mm high. The wall thickness is 20 mm, which is moderate. The elastic yielding in the bearing supports is about 0.2 mm, but as the yielding is similar in the bearings on the same side of every shaft, the gap in the meshing is

only minimally changed. As displayed in table 1, the face load factor KH, calculated based on the shaft deformation including housing deformation, is only slightly changed compared to the same factor without housing deformation. This indicates that the housing stiffness has a small impact.

Gear Pair KH

Without housing deformation

KH

With housing deformation

HSS (High speed stage) 1.17 1.16

HSS (Low speed stage) 1.30 1.32

Table 1: Face load factors without flank line modifications

In any KISSsys model [6] the housing stiffness can be considered using a stiffness matrix imported from a FEM software (fig. 9). The resulting housing deformation at the bearing positions are shown in a results table (fig. 10). The deformations are assigned to the bearings (typically outer ring) in the shaft calculation and considered in the gear contact analysis.

Fig. 9: A stiffness matrix, created by FEM, is included in a KISSsys model. Thus, the housing stiffness is considered in the load distribution analysis.


Figure 10: Bearing outer ring displacements in mm (x, y: horizontal; z: vertical)

The documentation and discussion of the proceeding step by step with this example would be too long at this point. All the details of this example can be found on the homepage [6]. The important results were as follows:

- The time used by the design engineer to find optimum modifications for both stages was 15 minutes.

- The optimum flank line modifications as defined in Step 1 are only slightly different when housing stiffness is considered (only 10% change in the helix angle modification value).

- The additional modifications in Step 2 and the profile modifications in Step 3 are identical with and without consideration of housing stiffness.

- The additional crowning added in Step 2 to compensate for manufacturing tolerances is much bigger (5 times) than the difference between modifications in Step 1 used to compensate shaft deflection with and without considering housing stiffness. Therefore, for practice-oriented solutions the influence of the housing stiffness is so small that it is negligible.

The conclusion, that the housing stiffness in most gearbox applications has negligible influence on the load distribution in the gear meshing may be surprising if we consider that the displacement of the housing in the bearing locations is quite important (about 0.2 mm in the discussed example – with high external load on the input shaft and aluminium housing). But for a reasonable designed housing the deflections in the bearing locations are usually symmetric, the shafts are all changing inclination in similar way, so the gap in the gear mesh and the load distribution is practically unchanged.

7 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 for a gearbox with parallel shafts including external forces acting on it, the housing deformations have an insignificant influence on the resulting gap in the meshing of the gears.

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.


8 References

[1] Niemann, „Maschinenelemente“, Band II, Springer Verlag, 1985.

[2] Kissling, U.; Application and Improvement of Face Load Factor Determination based on AGMA 927, AGMA Fall Technical Meeting 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] ISO 1328-1, Cylindrical gears — ISO system of flank tolerance classification, Geneva, 2013.

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Sizing the gear micro geometry Step by Stepkadkraft.com/wp-content/uploads/2015/12/PRESENTATI… · · 2015-12-07Sizing the gear micro geometry – Step by Step Dr.-Eng. Ulrich