Due to the steadily increasing demands on the power density of mechanical transmissions, gears with special geometries are increasingly coming into focus and, therefore, the need for short-term availability of prototypes. Such special gear designs are, e.g., asymmetrical gears with different normal pressure angles on the drive and coast flank. These are particularly suitable for use in gearboxes with preferred driving direction, whereby the loaded flank can be optimized with regard to load carrying capacity. With symmetrical gears with normal pressure angles in the range of α_{n} = 20°, standardized calculation methods for gear design have been available for decades — mainly theoretical numerical investigations have been carried out on asymmetrical gears so far. For the qualification of any such designed asymmetrical gear geometry with increased load carrying capacity potential for use in industrial practice, however, reliable load carrying capacity values are required. Therefore, according to the current state of the art, prototype tests are indispensable to determine the actual gear strength.

At the Gear Research Centre (FZG), such load capacity investigations are carried out using back-to-back test rigs and pulsator test rigs. The design and procurement of special tools for the production of such prototype gears is often time-consuming and expensive. In this paper, an alternative method for a fast and cost-efficient production of asymmetric gears for prototype tests is presented. The focus is on the grinding process from a full blank test specimen. This process was applied at the FZG in cooperation with Liebherr-Verzahntechnik GmbH in order to produce asymmetrical test gears for experimental investigations of the tooth root bending strength.

Very good results were achieved with regard to gear quality and shape accuracy, especially in the tooth root area, which is then investigated. The results of this paper show therefore a suitable method for the fast, precise, and cost-efficient production of special gears for prototype tests.

### 1 Introduction

The Hertzian contact pressure at the tooth flank and the tooth root bending stresses in the root fillet that occur during operation are decisive factors in determining the gear design. In combination with the properties and characteristic values of the materials and lubricants used, they determine the load carrying capacity of a set of gears and thus the size and weight of the entire gearbox.

Many different approaches are used to increase the load carrying capacity and thus the power density of gearboxes. For example, existing material and heat-treatment concepts can be optimized or high- performance materials and heat-treatment processes adapted to them can be used [4, 7, 15, 18, 25]. Furthermore, an optimized surface treatment, such as vibratory finishing or shot peening for improved residual stress conditions after completion of the heat treatment can further increase the load carrying capacity of gears [5, 8, 14, 26].

The geometric shape of the gear teeth and the tooth root curvature are important influencing factors on the level of the occurring loads. For power-transmitting gear drives, nowadays almost exclusively involute gears with standardized reference profiles (symmetrical tooth form with pressure angle α_{n} ≈ 20°) are used. These usually have a tooth root fillet that corresponds to a trochoid. In the literature, there are various approaches for the optimization of the tooth root geometry regarding an increased tooth root bending strength, compare [6, 13, 20, 21, 22].

Furthermore, in a large number of gear applications with a driving direction that is dominant with regard to the transmitted torque, e.g. in vehicle transmissions as well as in wind power and crane systems, gears with alternative pressure angles or asymmetrical gears are increasingly being considered in order to exploit the resulting advantages in tooth root load carrying capacity. This development is favored by new manufacturing processes. In the following, the design of asymmetric gears as well as the manufacturing possibilities of prototype gears for the experimental investigation of the load carrying capacity of such prototype gears will be discussed in more detail.

### 2 Gears with Asymmetric Tooth Shape

By using alternative special geometries, the load carrying capacity of gears can be improved and thus the power density of gears can be increased. Enlarging the normal pressure angle thus reduces the stresses in the tooth root and, at the same time, reduces the maximum pressure on the tooth flank. Disadvantages from the changed geometry can be compensated by an optimized design of the drive and coast flank, i.e. by an asymmetrical design of the teeth. The potential of such gear geometries, especially for increasing the tooth root load carrying capacity, has already been demonstrated many times on the basis of mainly theoretical work, compare [2, 3, 12].

However, for a reliable application of special involute gears in industrially running transmissions, a comprehensive and confirmed identification and verification of the load carrying capacities are needed for such toothings. This requires an upgrade of existing tooth root load carrying capacity calculation methods as described in [24], so the specific characteristics of special involute gears, as well as experimental test results, must be taken into consideration. At the Gear Research Centre (FZG), a corresponding calculation approach was developed to determine the tooth root stresses occurring on gears with alternative normal pressure angles or asymmetrical tooth profiles [23]. According to FZG/Langheinrich [23], the analytical determination of the tooth root stress for alternative tooth geometries is possible by a relatively simple adaptation of the standardized calculation steps according to ISO 6336 – Part 3 [10]. The following equations from the existing ISO 6336 -Part 3 [10] standard were modified to extend them to alternative gear geometries with a modified pressure angle. The modified form factor Y_{F}’ and the modified stress correction factor Y_{S}’ in Equation 1 are functions of the adjusted relevant bending moment arm h_{Fe}’ and the adjusted tooth root chord at the critical section s_{Fn}’.

where

σ_{F0} is the nominal tooth root stress.

F_{t} is the (nominal) transverse tangential load at reference cylinder per mesh.

b is the face width.

m_{n} is the normal module.

Y_{F}’ is the tooth form factor for the influence on nominal tooth root stress with load applied at the outer point of single pair tooth contact, according to Langheinrich [16].

Y_{S}’ is the stress correction factor for the conversion of the nominal tooth root stress, determined for application of load at the outer point of single pair tooth contact, to the local tooth root stress, according to Langheinrich [16].

Y_{β} is the helix angle factor (tooth root).

h_{Fe}’ is the bending moment arm for tooth root stress relevant to load application at the outer point of single pair tooth contact, according to Langheinrich [16].

s_{Fn}’ is the tooth root chord at the critical section, according to Langheinrich [16].

α_{Fen} is the load direction angle relevant to direction of application of load at the outer point of single pair tooth contact.

Y_{S}_{α} is the pressure angle factor (tooth root).

The required tooth root thickness of gears with changed pressure angles s_{Fn}’ in the calculation of the nominal tooth root stress can be determined by the mean value of the tooth root thicknesses. This approach for asymmetric teeth is already described in ISO 10300 – Part 3 [11] for bevel gears and is shown at right in Figure 1.

The bending moment around the point N in the tooth root of asymmetric gears can be determined by shifting the acting force along its line of action. As basis for the relevant bending moment arm h_{Fe}’ in the tooth root area, the tooth root thickness is used for asymmetric as well as for symmetric gears. It is determined with the tooth root fillet of the loaded flank. This principle is shown in Figure 1 on the left for symmetric gears and in the middle for asymmetric gears respectively. The diameter d_{Fn,bel} is determined according to ISO 6336 – Part3 [10].

In addition, the stress correction factor Y_{S}’ in Equation 3 has been extended by the pressure angle factor Y_{S}_{α}. This factor takes into account the effects on the tooth root stresses, which are caused by teeth with normal pressure angles α_{n} ≠ 20° at the loaded (α_{n,bel}) and unloaded (α_{n,unb}) tooth flank. The approach for calculating the pressure angle factor Y_{S}_{α} is shown in the analytical simplified Equation 4.

where

Y_{S}_{α} is the pressure angle factor (tooth root).

α_{n,bel} is the pressure angle at the loaded tooth flank.

α_{n,unb} is the pressure angle at the unloaded tooth flank.

F_{1}/F_{2} are auxiliary variables.

For external involute gears, the auxiliary variables F_{1} and F_{2} have the values 78 and 852 respectively. It can be seen that both calculating methods according to Langheinrich [17] and ISO 6336 – Part3 [10] give the same result for symmetric involute gears with pressure angle α_{n} = 20°. The determination of the auxiliary variables F_{1} and F_{2} in this calculation approach according to Langheinrich [16] is based on variational calculations on more than 2,000 external spur gear geometries of different normal pressure angle combinations, number of teeth, profile modification factors, and tool parameters. These were evaluated with regard to the tooth root stresses occurring in each case. For each variant, the nominal tooth root stress was determined using both the numerically Finite Elements (FE) Method and the analytical calculation approach according to ISO 6336 – Part 3 [10]. For asymmetric pressure angle combinations, the modified calculation method (see Equations 1-4) according to Langheinrich [16] was used. The results of the analytical calculations were then compared with the FE calculations.

The test results of first experimental investigations on an asymmetric gear geometry with pressure angle combination α_{n} = 28°/18° show a good agreement with this new analytical calculation approach [24].

Currently at the FZG, further extensive experimental investigations are conducted to validate the standard-compliant calculation approach according to Langheinrich [16]. Three different, asymmetrical gear geometries are considered in comparison to a symmetrical reference variant. The main geometry data of these gears are shown in Table 1.

The production of these asymmetric test gears faced challenges discussed in the following.

### 3 Experimental Investigation of The Tooth Root Bending Strength

#### 3.1 Test Rig and Test Method

Experimental investigations to determine the tooth root bending strength are usually time-consuming and costly. However, this is often indispensable for the validation of theoretical investigations. At the FZG, the experimental investigation of the tooth root bending strength of involute external spur gears is usually carried out in pulsator test rigs. In Figure 2, an example of the schematic design of an electromagnetic resonance pulsator (system Roell-Amsler) and the typical test configuration are shown.

The pulsator test rig essentially consists of a machine frame to accommodate the testing device, a load cell, and the test gear. As shown schematically in Figure 2, the test gear is clamped symmetrically between two clamping jaws. A special device ensures the force application point on the tooth flank can be set precisely and be reproduced for every test of a series when clamping the test wheels. Deviations in the flank angle can be compensated by fine adjustment of the clamping jaws so an even load distribution over the tooth width can be ensured. By means of the mid-load actuator and the mid-load springs, the gear to be tested is preloaded with a static load. This prevents the gear from falling out during the test run. The dynamic load is then mechanically generated by an electronically controlled excitation magnet and transmitted via the pole springs to the vibrating traverse and the test gear. The test conditions for each test are continuously recorded and logged by PC.

The main advantage of this test method is that several test points can be determined for each test gear. Depending on the gear geometry to be tested, only between three and 10 test gears are required to determine a complete S-N-line with up to 25 test points for characterizing the tooth root load carrying capacity. Furthermore, the running times per test point are relatively short due to the high achievable test frequencies. Altogether, this test method makes it possible to carry out the experimental verification of the tooth root bending strength of a gear geometry comparatively quickly and cost-effectively.

#### 3.2 Requirements for Test Gears for Tooth Root Bending Tests

The tooth root load carrying capacity of spur gears is significantly influenced by the geometry of the tooth root. Therefore, it is necessary to consider this geometry as accurately as possible when carrying out experimental investigations to determine the tooth root bending strength. As described by Hyatt et al. [9], in contrast to the tooth flank, the exact geometry in the area of the tooth root is typically not tolerated and is generally not subject to standardized tolerance. For this reason, the actual tooth root geometry is determined by means of a contour scan on a Klingelnberg P40 gear measuring machine at the FZG for all experimental investigations and compared with the designed nominal geometry. This is illustrated in Figure 3 for the hobbed reference test variant P-m4-2020 from Table 1.

As can be seen in Figure 3, there are slight deviations both between the two measured test gears and between the gear geometry actually manufactured and originally designed. As described by Nigade and Wink [19], such deviations between designed test gear geometry and test gear geometry produced by gear hobbing can lead to considerable deviations of up to 22 percent between actual tooth root stresses occurring in operation and those assumed during the design. For this reason, good contour accuracy in the tooth root area must be ensured, especially when manufacturing test gears for the experimental validation of the tooth root load carrying capacity of new tooth respectively tooth root geometries.

### 4 Manufacturing of Prototype Gears for Tooth Root Bending Tests

As described in Section 3.1, only a few test gears per test variant are required for the experimental determination of the tooth root load carrying capacity in the pulsator test rig, depending on the test gear geometry. However, even this can be a problem if, as shown in Table 1, several different, new gear geometries are to be investigated experimentally for which no gear hobbing tools have been available so far. In this case, the purchasing of new gear hobbing tools can be disproportionately expensive and also time-consuming, so an alternative manufacturing option should be considered.

As already mentioned, almost all spur gears produced in series production today are manufactured by gear hobbing. This manufacturing process has economic advantages when manufacturing many gears of the same geometry. But the design and procurement of special tools for the production of only a few prototype gears may be time-consuming and expensive. Therefore, alternative manufacturing processes come to focus. Bouquet et al. [1] give an overview of alternative manufacturing methods of test gears for prototype tests and evaluate these with regard to manufacturing effort, manufacturing quality, and process time. The following manufacturing processes are considered:

- Milling (manufacturing on a 5-axis milling machine with standard end and ball mills).
- Wire-EDM.
- Selective laser melting.

The authors came to the conclusion that of these, milling is the most promising process for the rapid production of prototype gears [1]. The reasons given for this are the high flexibility and good surface quality of the gears produced. Contour accuracy is described as good for the milling manufacturing process. The two other manufacturing methods are described as not practical due to the current state of the art.

Since no 5-axis milling machine was available on the FZG at the time the asymmetrical gears listed in Table 1 were manufactured, the gears were instead ground from a full blank test specimen on the Liebherr LGG 280 gear grinding machine available at the Institute. This was done in close cooperation with Liebherr-Verzahntechnik GmbH, which provided software updates for asymmetric gears as well as for full grinding and technical support. The grinding-setup for a test gear of the variant P-m4-3015 (see Table 1) from a full blank test specimen is shown in Figure 4.

The production of the test gears by grinding into solid material took place in a discontinuous profile form grinding process before the heat treatment of the material (18CrNiMo7-6). Due to the geometrically indeterminate cutting edge of the grinding wheel, many possibilities are realizable with regard to the geometric design of the gears. This applies both to the tooth flank geometry and to the tooth root geometry. However, due to the different pressure angles on the drive and coast flank and the associated different cutting speeds on both flanks, the correct adjustment of the grinding parameters such as speed rate and sharpening cycle must be ensured. Otherwise, a significant change in the flank surface can occur, particularly on the flank with a smaller normal contact angle, as a result of an increased temperature input. The pure machining time on the grinding machine of approximately 240 minutes per test gear is significantly higher than the cycle time of a comparable test gear in the hobbing process. However, production can be initiated at short notice with a short lead time to machine setup. Long procurement times for specially designed production tools are eliminated.

### 5 Results of the Test Gears Grinded into Full Material

In the following, contour scans of the manufactured test gears as described in section 3.2 are shown. First of all, Figures 5-7 show the comparison between the originally designed test gear geometry and the actual test gear geometry manufactured by grinding into full material for each asymmetric variant from Table 1.

It can be seen that there is very good contour accuracy between the designed and manufactured gear geometry for all of the three asymmetric test variants.

Furthermore, for each of the three test gear geometries produced using the manufacturing process “grinding into solid material,” a comparison was carried out between three of the test gears produced for each test variant. Figures 8-10 show this comparison graphically by means of the contour scans determined on three different test gears for each variant.

Here, too, there is very good consistency within the individual test gear geometries with regard to contour accuracy. It can therefore be assumed that the manufacturing process is very reproducible within one production batch.

Finally, the manufacturing accuracy on a single test gear was also examined. For this purpose, three contour scans evenly distributed over the circumference were made on a test gear of the test geometry P-m4-3015 and compared with each other. This is shown in Figure 11.

Here, too, there is very good contour accuracy, which is an indicator for a favorable choice of the specified grinding parameters.

Altogether, the production process “grinding into solid material” can be assumed to have a very high production quality with a reasonable choice of process parameters and thus a good suitability for the production of test gears for prototype tests.

### 6 Comparison to Other Manufacturing Processes for Gear Prototypes

Since the gears considered in this paper (see Table 1) were produced exclusively by grinding into the solid material, a comparison with other methods is made on the basis of the test results of Bouquet et al. [1], see Section 4. It should be mentioned here that Bouquet et al. [1] consider a different gear geometry. This is a spur gear of size m_{n} = 2.12 mm with a number of teeth of z = 36 and a tooth width of b = 31.5 mm. Since the tooth width is about twice as large as that of the gears ground to the solid material, only half the production time is assumed for the manufacturing processes described by Bouquet et al. [1] in the comparative analysis. Furthermore, only the machining steps defining the tooth geometry with a comparable surface roughness are used to determine a comparative machining time. The results of this comparison are shown in Table 2.

It can be seen that the production of gear prototypes by grinding into full material at better contour accuracy requires a machining time comparable to that of milling. However, it should also be noted that according to Bouquet et al. [1] the machining times refer to a gear geometry with 25 percent fewer teeth and only half the size regarding the normal module m_{n}. Especially for the manufacturing processes milling and selective laser melting, the production time for the gear geometry that has been ground into full material would be increased even further.

Also as mentioned in Section 4, the two manufacturing processes wire-EDM and selective laser melting are described as not practical due to the current state of the art [1].

### 7 Conclusion

Three different gear geometries with asymmetric tooth shapes were manufactured by grinding into solid material (18CrNiMo7-6) for future investigations on the tooth root strength on a gear grinding machine of the type LGG 280 available at the Institute. The extensive geometrical comparisons show a very good contour accuracy for these gears compared to the originally designed gear geometry as well as within one test gear series. Therefore, this manufacturing process is suitable for small prototype test series.

Especially when focusing on the tooth root bending strength where manufacturing deviations in the tooth root geometry can lead to a misinterpretation of the bending strength of the designed geometry, a high contour accuracy is necessary.

The relatively high production times per test gear compared to gear hobbing can, however, be compensated by the elimination of long design and procurement times for special production tools as well as the costs of such tools.

Therefore, the results of this paper show a suitable method for the fast, precise and cost-efficient production of special gears for prototype tests.

Furthermore, due to the acquisition of a 5-axis machining center at the research institute in the meantime, a direct comparison can also be made with regard to machining time and production costs for gears of identical geometry in future research work.

### Acknowledgments

The authors thank Liebherr-Verzahntechnik GmbH for their support.

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