Periodic flank modifications for optimal excitation behavior of practi- cal gear geometries Sebastian Sepp 1 Gear Research Center (FZG), Technical University of Munich Boltzmannstr. 15, 85748 Garching, Germany Michael Otto 2 Gear Research Center (FZG), Technical University of Munich Boltzmannstr. 15, 85748 Garching, Germany Karsten Stahl 3 Gear Research Center (FZG), Technical University of Munich Boltzmannstr. 15, 85748 Garching, Germany

ABSTRACT The design of modern transmissions faces new challenges regarding the main design principles effi- ciency, service life and noise emissions. These challenges are further intensified by the progressing electrification of vehicle drive systems and the associated trend towards higher rotational speeds. In order to achieve the design objectives at the level of flank modifications, the micro geometry has to meet partially competing requirements. Most present gears are designed with combinations of stand- ard modifications to obtain the best possible compromise between the competing goals. The applica- tion of periodic modifications provides a possibility to avoid particularly the trade-off between the two design principles load carrying capacity and excitation. The periodic modifications directly com- pensate the alternating part of the elastic deformations in the mesh without changing the load distri- bution. They can be used to optimize the excitation behavior independently of the load carrying ca- pacity. Theoretical studies show the potential of periodic modifications to optimize the excitation behavior for a special target load or even for broad load ranges. Experimental investigations at two test rigs of the Gear Research Center (FZG) verify the effectiveness of these flank forms to optimize the excitation behavior for different practical gear main geometries.

1. INTRODUCTION

The design of flank modifications of practical gears for modern gear applications is characterized by the trade-off between the objectives load carrying capacity, noise behavior and manufacturing. To ensure a high load carrying capacity, a balanced load distribution in the meshing area without load peaks is desired. For this purpose relieves and crownings are applied to the ideal involute flank form to relieve the meshing start and end as well as the face edges of the gears. The design load for these modifications is usually the nominal load or a specific load spectrum. In contrast, low noise flank

1 sebastian.sepp@tum.de

2 michael.k.otto@tum.de

3 karsten.stahl@tum.de

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designs aim at a constant deflection of the tooth meshing in all variable meshing positions at target load. With a constant deflection, the parameter-excited oscillations in the mesh caused by the loaded gears themselves are minimized and a beneficial noise behavior for the target load is achieved. The realization of constant deflections typically implies micro geometries, which are not in line with ge- ometries for ensuring a high load carrying capacity. Thus, the exclusive use of standard modifications such as tip and root relieves, crownings and end relieves requires compromises between the two target objectives load carrying capacity and noise behavior.

In the literature there are numerous publications on the design of “optimized” flank modifications in order to find the best compromise in the trade-off between the two design goals [1–3]. However, the investigations are not extended to free topological modifications, which provide a method to over- come the trade-off. The fundamental idea is the direct compensation of the loaded transmission error (LTE) by a parallel shifting of all lines of action being active at the same time. Since the LTE develops with a pitch-periodic vibration pattern, the derived topological modification to compensate the LTE is a periodic waveform modification. Griggel and Radev [4] developed this idea and used sine shaped modifications derived from the spectrally analyzed LTE. The periodic modifications do not have an impact on the load distribution because the contact lines are shifted only in parallel direction and thus, there is no load redistribution between and along the contact lines. This allows a separate design of the modifications with regard to the target variables load carrying capacity and noise behavior. How- ever, the investigations revealed that the very small modification amounts down to the sub-microm- eter range represent a major challenge and prevent process-safe manufacturing and practical applica- tion. Based on the pure waveform modifications Kohn [5] derived modified periodic modifications, which are only applied to a partial area of the tooth flanks and thus show locally increased modifica- tion amounts. The effectiveness of the modified modifications for minimizing the excitation of the test gears (ε  = 1.5 and ε  = 1.5) could be confirmed computationally and experimentally. In a project funded by the FVA/AiF [6], the application of periodic modifications to different practical gear main geometries was investigated at the Gear Research Center (FZG).

2. THEORETICAL FUNDAMENTALS

To design tooth flank modifications, the principal deformation and excitation behavior of the gear pair in contact is modeled in the tooth contact analysis. The vibration excitation in the gear mesh can be explained by the following main excitation mechanisms:

 Variation of the mesh stiffness due to the time-varying meshing positions in the tooth contact  Elastic deformations and displacements of the gears due to load-induced deformations of the

shaft-bearing-housing system  Geometric deviations from the ideal involute shape due to intentionally applied gear modifi-

cations, manufacturing tolerances or assembly inaccuracies  Extended tooth contact due to deformations of loaded teeth The governing excitation mechanism for the noise excitation of a gear pair is the time-varying mesh stiffness. Figure 1 shows an example of the mesh stiffness for spur and helical gears plotted against the path of contact. The mesh stiffness of a gear pair is the superposition of all single tooth pair stiffnesses, which are in contact at the same time. The oscillation of the mesh stiffness results mainly from the variable total length of all simultaneously active contact lines [7]. Thus, the contact ratio ε  and overlap ratio ε  have a significant influence on the variation of the mesh stiffness. For spur gears with contact ratios 1 < ε  < 2 the number of supporting tooth pairs alternates between one and two, resulting in a significant variation of the total length of the contact lines. For helical gears,

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the meshing is carried out gradually and due to the additional overlap ratio, multiple tooth pairs are constantly in contact. For this reason, helical gears generally have a lower excitation level compared to spur gears.

Gears with integer contact ratios - or alternatively with integer overlap ratios in the case of helical gears - always have a constant total length of the contact lines due to the constant number of tooth pairs in the mesh. In order to achieve low-excitation main geometries, integer contact or overlap ratios should be aimed for. Since the effective contact ratio is load-dependent, an integer overlap ratio should be preferred for obtaining low-excitation levels.

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Figure 1: Mesh stiffness of spur and helical gears: mesh stiffness (c s ), single tooth pair stiffness (c), mean value of mesh stiffness (c γ ) (acc. to [8])

Knowing the mesh stiffness of the toothing, the loaded transmission error (LTE) can be calculated. The LTE is a widely used specific value for evaluating the excitation behavior and describes the deflection between pinion and wheel in direction of the path of contact [9]. It is calculated under the boundary condition of very low revolution speeds ( 𝑛→0 ) for several meshing positions, which means that the LTE does not take dynamic effects of the gears or the surrounding system into account. The LTE is calculated according to Equation 1:

𝑥ሺ𝑡ሻ= {𝐹−σ ൣ𝑐 𝑠𝑖 ሺ𝑡ሻ∙𝑥 𝑓𝑖 ሺ𝑡ሻ൧} 𝑖 / {σ 𝑐 𝑠𝑖 ሺ𝑡ሻ 𝑖 } , (1)

where 𝐹 is the load, 𝑐 𝑠𝑖 the local mesh stiffness at the point 𝑖 and 𝑥 𝑓𝑖 the accumulated flank devi- ation at the point 𝑖 . In the flank deviations, the sum of all deviations affecting the tooth contact from flank modifications, manufacturing deviations and elastic deformations and displacements of shafts, bearings and the housing is included. Since the LTE describes the displacement of the tooth contact in relation to the meshing position, this value is particularly suitable for designing modifications and is used in the present investigations as parameter for evaluating the gear excitation behavior.

3. DESIGN OF FLANK MODIFICATIONS

In theoretical and experimental studies, two different practical gear main geometries are used to evaluate the effectiveness of periodic modifications to optimize the excitation behavior. The basic gear data are listed in Table 1.

Table 1: Basic gear data for the investigated meshes

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3.1 DERIVATION OF STANDARD MODIFICATIONS

Two conventional layouts with standard modifications to ensure low noise excitation are designed for the investigated meshes A and B. The mesh A combines a tip and root relief as well as a crowning and a twist to achieve a compromise towards excitation and load carrying capacity. Figure 2 shows the spectral LTE and the load distribution at nominal load for the mesh A. The spectrum presents the amplitudes of the LTE related to the tooth meshing order for several load stages. In the load distribu- tion, the loads acting on the contact lines are plotted over the meshing area. The target load for low noise excitation typically differs from the nominal load in most gear applications with non-stationary operation conditions. For this reason mesh A has a nominal load T T = 2000 Nm and a target load T A = 1200 Nm for excitation. The standard modifications of mesh A are designed to reduce the exci- tation of the first tooth meshing order in the load range around 1200 Nm (see Figure 2a). The ampli- tudes of the higher harmonics are only of secondary significance. However, the load distribution in Figure 2b indicates an uneven load on the tooth flanks with premature and prolonged engagement at nominal load. The reason therefore are the modification amounts, which are too small to compensate the tooth deformation at nominal load. This example demonstrates the trade-off between excitation and load carrying capacity when using only standard modifications. Mesh B shows a comparable behavior with its modifications tip and root relief and crowning.

a) b)

Figure 2: Mesh A with standard modifications (tip / root relief, crowning, twist) to minimize the excitation for target load T A = 1200 Nm. a) Spectral LTE, loads at driving pinion; b) Load distribution at nominal load T T = 2000 Nm

Figure 3: Mesh B with standard modifications (tip / root relief, crowning) to minimize the excitation for target load T A = 150 Nm. a) Spectral LTE, loads at driving pinion; b) Load distribution at nominal load T T = 300 Nm

a) b)

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3.2 DERIVATION OF PERIODIC MODIFICATIONS

The major advantage of using periodic modifications is the separate design process of the modifi- cations in terms of load carrying capacity and noise behavior. In the first step, modifications are designed to achieve a balanced load distribution at nominal load without load peaks or engagement prolongations. Figure 4 shows the mesh topography of mesh A with standard modifications at the pinion to ensure a high load carrying capacity. The spectral LTE indicates high excitation amplitudes for lower load levels and an excitation minimum in the load range around 1400 Nm. In order to pre- cisely modify the excitation behavior, the load carrying capacity modifications serve as an input var- iable for the subsequent design of the periodic modifications, which are specifically adapted to the operating meshing conditions. The adapted periodic modification is shown in Figure 5. It is based on a pure waveform modification with an additional scaling of the sinusoidal amplitude. The local mod- ification amount 𝑧 can be calculated according to Equation 2, where C SIN describes the sinusoidal amplitude,  the mesh coordinates along the path of contact, 𝑏 the coordinates along the face width and 𝜆 𝑆𝐼𝑁 the wavelength. The phase shift 𝜙 of the modification is adapted to the phase of the LTE 𝜙 𝐿𝑇𝐸 , the face width coordinate and the base helix angle  𝑏 . The linear scaling function of the sinusoidal amplitude is given in Equation 3, where 𝑔  describes the total length of the path of contact. With the periodic modification, it is possible to minimize the excitation for the target load T A = 1200 Nm (see Figure 5b) without negatively affecting the load distribution to a significant ex- tent. However, the calculated spectrum indicates that the periodic effect is only beneficial in a narrow load range, while the excitation of some load levels is even negatively affected.

𝑧ሺ  , 𝑏ሻ= 𝐴ሺ  ሻ∙𝐶 𝑆𝐼𝑁 ∙sin ൜ 2𝜋

∙  + 𝜙൫𝑏,  𝑏 , 𝜙 𝐿𝑇𝐸 ൯ൠ (2)

𝜆 𝑆𝐼𝑁

𝐴ሺ  ሻ= 1 − 

= 0 … 1 (3)

𝑔 

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a) b)

Figure 4: Mesh A with standard modifications (tip / root relief, crowning) at the pinion to ensure a high load carrying capacity. a) Pinion topography; b) Spectral LTE, loads at driving pinion

C SIN = 0.7 µm

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Figure 5: Mesh A with standard modifications at the pinion to ensure a high load carrying capacity (see Figure 4) and additional adapted periodic modification at the wheel to minimize the excitation for target load T A = 1200 Nm. a) Wheel topography; b) Spectral LTE, loads at driving pinion

a) b)

If standard and periodic modifications are specifically adjusted to each other, it is possible to op- timize the excitation over broad load ranges. This is achieved by utilizing the load-dependent increase in the contact ratio. Figure 6 shows a mesh topography with standard modifications combined with an applied periodic modification to reduce the excitation for partial loads.

o 8 $e unt /qunowe pour

C SIN = 0.3 µm

\ th il 1 2 3 annnints=

a) b)

Figure 6: Mesh B with standard modifications (profile crowning, crowning, twist) to ensure a high load carrying capacity and additional adapted periodic modification to minimize the excitation for partial loads. a) Mesh topography; b) Spectral LTE, loads at driving pinion

Analogous to the modifications of mesh A in Figure 4 and 5, the standard modifications ensure a high load carrying capacity but have an unfavorable noise behavior on their own. The complex periodic modification is composed of a pure waveform modification with C SIN = 0.3 µm and an additionally applied periodic modification in the flank areas of the load-dependent contact ratio

increase. The combination of all modifications provides a high load carrying capacity and at the same time an advantageous noise behavior for partial load stages.

4. EXPERIMENTAL INVESTIGATIONS

The experimental investigations were performed at two FZG back-to-back test rigs with center distances of 140 mm (mesh A) and 91.5 mm (mesh B). Due to the mechanical power circulation principle it is possible to examine the gears over broad load ranges (mesh A: 400 – 2000 Nm and mesh B: 50 – 300 Nm). The LTE is measured using high-resolution angular position sensors at the pinion and wheel. The measurements are executed at low revolution speeds, which allows the detec- tion of the excitation behavior without dynamic effects.

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Figure 7 shows the measured 1st Fourier coefficient derived from the LTE time signals for the two layouts of mesh A and mesh B. With the combination of standard and periodic modifications a sig- nificant reduction of the transmission error compared with the standalone standard modifications is assessed for both geometries. The average excitation reduction over all load stages is approx. 28 % (-2.9 dB) for the mesh A and approx. 30 % (-3.0 dB) for the mesh B.

a) b)

‘transmission error x / jim 04 (= Wiest A wih standard medications | —— Mesh A with standard and periodic maifcations | 400 700 1000 1200 1400 1600 load /Nm 2000

Figure 7: Measured 1st Fourier coefficient of LTE, loads at driving pinion. a) Mesh A with standard modi- fications according to Figure 2; Mesh A with standard and periodic modifications according to Figure 5; b) Mesh B with standard modifications according to Figure 3; mesh B with standard and periodic modifi- cations according to Figure 6

5. CONCLUSION

In this study, the application of periodic flank modifications for two practical gear main geometries is investigated. For this purpose, one low noise modification with only standard modifications and one combined layout with standard and periodic modifications are designed for each of the two main geometries. In the theoretical investigations, the additional application of periodic modifications achieved a clearly advantageous excitation behavior. Either the excitation of individual load stages can be specifically reduced to a minimum or the excitation can be lowered over broad load ranges. However, the low-excitation gear main geometries generally result in very low excitation levels, which has a direct impact on the necessary modification amounts of the periodic modifications. The experimental results demonstrate the current possibilities to manufacture even very complex micro

02 ‘transmission error x / jim = Niesh 6 with standard medications | —— Mesh 8 with standard and periodic maifeations| 100150200 load /Nm 250 300

geometry specifications with modification amounts in the sub-micrometer range with a very high accuracy. The LTE measurements gathered at two FZG test rigs verify the effectiveness of periodic modifications to optimize the excitation behavior for practical gear main geometries.

The key potential in the use of periodic modifications for practical gear main geometries is the separation of the two design principles load carrying capacity and excitation behavior. The periodic modifications can be superimposed on the previously applied standard load carrying modifications without significantly affecting the load distribution. In this way, periodic modifications increase the degree of freedom in the trade-off between the load carrying capacity and the excitation behavior and allow flexible and time-delayed excitation optimization for given load carrying modifications.

ACKNOWLEDGEMENTS

The presented results are based on the research project IGF no. 19869 N/1 (FVA 338/VIII) under- taken by the Research Association for Drive Technology e.V. (FVA); supported partly by the FVA and through the German Federation of Industrial Research Associations e.V. (AiF) in the framework of the Industrial Collective Research Programme (IGF) by the Federal Ministry for Economic Affairs and Energy (BMWi) based on a decision taken by the German Bundestag. The authors would like to thank for the sponsorship and support received from the FVA, AiF and the members of the project committee.

6. REFERENCES

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3. Zhang, S., Zheng, J., Cheng, L.: Theoretical and Experimental Study on Optimum Design of Gear Profile Modification Curves. Applied Mechanics and Materials, 471-472, 269–272 (2004).

4. Radev, S., Griggel, T., Oster, P., Höhn, B.-R.: FVA-Heft 739: Forschungsvorhaben 338/IIb, Anre- gungsoptimierte Flankenkorrektur für konstante und veränderliche Last. Abschlussbericht, Forschungs- vereinigung Antriebstechnik e.V. (FVA), Frankfurt/Main (2004).

5. Kohn, B., Utakapan, T., Fromberger, M., Otto, M., Stahl, K.: Flank modifications for optimal excitation behaviour. Forschung im Ingenieurwesen, 81, 65-71 (2017). 6. Sepp, S., Otto, M., Stahl, K.: FVA-Heft 1455: Forschungsvorhaben 338/VIII, Optimale Flan- kenkorrektur praxisnahe Verzahnung - Entwicklung von Auslegungsstrategien für anregungsop- timale Flankenkorrekturen bei praxisnahen Verzahnungshauptgeometrien und Tragfähigkeits- modifikationen. Abschlussbericht, Forschungsvereinigung Antriebstechnik e.V. (FVA), Frank- furt/Main (2021). 7. Velex, P., Bruyère, J., Houser, D.R.: Some Analytical Results on Transmission Errors in Narrow- Faced Spur and Helical Gears: Influence of Profile Modifications. Journal of Mechanical Design, 133 (3), (2011). 8. Höhn, B.-R.: Improvements on noise reduction and efficiency of gears. Meccanica, 45, 425–437 (2010). 9. Gregory, R., Harris, S., Murno, R.: Dynamic behaviour of spur gears. Proceedings of the Insti- tution of Mechanical Engineers, 178 (1), 207–218 (1963).

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