Integrated Approach for Gear Testing of High-Performance Clean Steels

The power density of gearboxes is continuously increased by different research activities. Besides new material developments, the cleanliness of steels comes to the forefront in order to meet future requirements regarding load carrying capacity of gears. The experimental quantification of the load carrying potentials for high performance steels is the basis for introducing cleanliness as a design parameter in high cycle fatigue applications.

In this paper, an integrated approach for gear testing of high-performance clean steels is presented.  In order to determine the differences between steels of different cleanliness levels, the testing approach for pitting damages has to be improved as a whole. Based on experimental results with a standardized gear geometry for pitting load carrying capacity tests, the limits of the existing test setup are derived. Furthermore, the demands for a new test setup are defined. The main requirement for the gear design is to ensure a clear separation of different damage patterns up to highest loads, which are expected for improved cleanliness levels.

For pitting investigations, a sufficient tooth root strength is obligatory. Analysis of the safety factors according to ISO 6336 parts 2 and 3 confirm that the two damage patterns cannot be separated up to the highest possible load of the back-to-back gear test rig. It is emphasized that a change of face width does not lead to a better separation of the damage patterns. Therefore, the tooth-root strength  is increased by a stepped tooth shape to each side. Furthermore, it is shown that the FZG C-type gear shows better potentials for flank fatigue testing than the 17/18 test gear geometry.

Finally, the tooth mesh is analyzed with regard to premature tooth meshing. In case of a premature tooth meshing, scraper marks influence the pitting strength of the gear. If the test load needs to be increased because of the higher cleanliness of the steel, the effects of a premature tooth meshing become higher keeping the same micro geometry. For the standardized FZG C-type gear, a look-up table for an optimized tip relief is derived for different face widths and torques, which can be easily applied.

As a conclusion of the presented work, it is necessary to guarantee an excellent gear design and a high manufacturing quality for testing of high-performance clean steels with highest load carrying potentials. Only in this case, can the full potential of cleaner steel be outlined.

1 Introduction and Motivation

The optimization of power density is one of the most important criteria during the design process of transmissions. The power density of transmissions has a direct impact on the production costs as well as on the performance of the drivetrain. Furthermore, the carbon footprint depends on the power density because the amount of raw material for manufacturing each gearbox component can be reduced with increased power density on the one hand. On the other hand, less weight of the drivetrain leads to less emission during operation of the transmission in its application.

For a continuous improvement of power density, there are different strategies for transmissions (Figure 1). Besides the optimization of the gear design regarding gear macro and micro geometry or tooth root geometry, the application of surface finishing processes is an established approach to increase the load capacity of gears [20, 17]. Furthermore, the development of new materials as well as the improvement of the material quality shows a high potential for increasing the power density of gearboxes. In this report, the focus is on the improvement of the steel performance concerning a higher cleanliness.

A further increase of the power density of transmissions affects the design process in two ways. On the one hand, the center distance of the gearbox can be reduced transferring the same load (Figure 1). On the other hand, the nominal torque can be increased using the existing design of the gearbox, both using high performance clean steels. Regarding the validation of the strength potential by testing on standardized back-to-back test rigs, the center distance is constant, and a higher nominal torque is applied to force damages on test gears. The design of a test-gear geometry has to ensure that only the desired damage pattern is most critical and the other damage patterns are avoided.

In this report, the focus is on the determination of pitting strength of improved materials. In this case, the safety factor against tooth root breakages should be high enough for the endurance limit of contact fatigue. By continuous increase of pitting load capacity, the safety factor against tooth root breakage decreases for the same gear geometry, if the same allowable stress number for bending fatigue is assumed. For this reason, the well-established test gear geometries for contact fatigue do not guarantee a secure separation of tooth root breakages for improved materials any more. In this report, the well-established FZG C-type and 17/18 test gear geometry are analyzed regarding their potential for pitting tests of high-performance clean steels. Finally, an optimized design of the FZG C-type gear regarding face width and profile correction is introduced for contact fatigue tests.

2 State of the Art and Deficit

2.1 Properties and Potentials of High-Performance Clean Steels

Steel cleanliness as a means to increasing power density or life time — although already proven a valid method in real applications — is still being investigated in many areas [31]. One reason for this is because of the capabilities of available current standards such as the ISO 6336, as well as the limitations of current standardized test methods. These standards, meanwhile, are reliable due to their conservative nature and are perhaps expected to be more of a guideline to manufacturers’ own specifications. The reasoning behind it can partly be because of the weaknesses in how materials are classified from a cleanliness point of view and the impact it has on the performance of the components. Steel making practices have evolved since the standards were put in place, as have the ways of verifying steel cleanliness.

Steel cleanliness can be challenging to define and somewhat depends on application and starting point. However, it can be described as size and frequency of non-metallic inclusions in the steel, such as Al₂O₃-MgO particles. It is then understandable that different applications have different requirements. Steel cleanliness depends on the steel making process, casting format and size, reduction-ratio-to-finished product, and how inclusions are dispersed throughout the material. Large MnS inclusions, for example, located along the tooth root, will likely increase the failure probability at a lower stress level than a smaller inclusion. For high-strength steels, material cleanliness will be a determining factor, especially for high-cycle fatigue.

A recent comparative study clearly reflects the effect of steel cleanliness on material performance [25]. The study was performed on customized gears, not standardized type reference gears. The same gear geometry, manufacturing, and heat treatment was applied to a reference 18CrNiMo7-6 steel and a high cleanliness 18CrNiMo7-6 isotropic property steel, making cleanliness the only significant factor between the two. The results of the study show two very different behaviors. The high cleanliness 18CrNiMo7-6 performs consistently with extremely low scatter compared to the reference material. This is compared to a reference steel with a cleanliness beyond what is commonly used in the automotive industry today and with a reduction ratio of more than 30. For the automotive industry especially, the required reduction ratio is often less than 15:1 and more commonly around 7:1.

Some of the challenges with the ISO 6336-5 standard are how the requirements of the different material classes or performance levels do not deal well with steel cleanliness. One weakness is the demands on O, Ca, and S do not take into account how cleanliness is achieved and how steels can be produced in different ways. Top performing clean steels might not meet all of these requirements, whereas many lower performing steels might adhere to these limits. As mentioned in previous papers, micro-inclusion rating according to the standards only look at a very small area, approximately 100-200 square millimeters per sample, resulting in approximately 1,000 square millimeters  for an entire heat or delivery, which gives a poor indication of material performance [10]. Other challenges using the ISO 6336 standard are the way material cleanliness affect factors such as the size factor. These factors do not take steel cleanliness into account in a relevant way.

Testing of Gear Load Capacity

One challenge of gear testing is the diversity of the different damage types and the multitude of damage mechanisms. According to DIN 50322, there are different levels and test categories for testing of gear load capacity [30]. The back-to-back test rig is an established and standardized testing concept on the level of model tests [32]. During the test run of the so-called running test, a single gear set is tested regarding a specific damage type. The running test is suitable for testing the load capacity of all damage types, because the real meshing of the teeth is ensured. The running test is mostly used for material and lubricant qualification in research using standard reference gears. The center distance is constant and usually a = 91.5 mm. Furthermore, this test principle can also be used to determine the gear load capacity of industry applications using back-to-back test rigs with flexible center distance [16].

Testing of gear load capacity is an obligatory task in order to secure an optimized design of gearboxes. On the one hand, the functionality with regard to load capacity has to be guaranteed over the whole lifetime, while on the other hand, the production costs of each gear set have to be reduced to a minimum. The objective of gear testing regarding load capacity is the derivation of design parameters for the different damage patterns of gears (Figure 2). The design parameters can be used for calculating the gear load capacity according to ISO 6336 [8; 5-7].

The selection of a gear geometry is a crucial point for performing gear load capacity tests on a back- to-back test rig to derive design parameters. There are different damage patterns occurring on gears (see Figure 2). Besides scuffing, the fatigue damages tooth root breakage, contact fatigue strength (pitting), and tooth flank fracture are the most important damage types to avoid in application. The selection of a test gear geometry always has to fulfill the requirement that an appropriate separation of the different damage patterns is guaranteed. Therefore, the safety factor for the intended damage pattern should be below one, and the safety factors for all other damage patterns should be reasonably higher than one. For this reason, the module of the test gear for tooth root bending tests on a back-to-back test rig is always smaller compared to a pitting-critical gear geometry (Figure 2).

2.2 Deficit of Gear Testing with Standardized Gear Geometry

For testing of pitting load carrying capacity, there are different standardized test gear geometries. The 17/18 test gear geometry is a well-established gear set for a back-to-back test rig with a center distance of a = 91.5 mm and was used in several research projects in the Forschungsvereinigung Antriebstechnik e.V (FVA, German Community for Drivetrain Research) [11, 29, 27]. Depending on the improvement of a surface finishing process of the tooth flank, the pitting load carrying capacity of the 17/18 gear could be improved to a level that tooth root breakage occurred instead of pitting damages [29]. For a center distance of a = 91.5 mm, the C-type gear is another pitting-critical test gear geometry that is mainly used for pitting tests with regard to oil performance [21]. The C-type gear also shows a potential for testing of pitting load carrying capacity of new or improved materials, such as high-performance clean steel. Furthermore, there are pitting-critical test gear geometries for back-to-back test rigs with center distances of a = 112.5 and 200 mm, such as the 21/23 and 24/25 test gear geometry [19, 12]. In order to securely avoid tooth root breakages, the face width of these test gears was improved by a stepped design of the face width [27, 2].

In order to determine the pitting endurance strength of a steel with higher cleanliness, several tests were performed on a back-to-back test rig using the C-type gear geometry (Figure 3). The endurance limit was determined using the stair case method
[15, 9]. Because of the higher material performance, the contact fatigue strength (pitting) could be increased compared to existing test series based on the C-type gear geometry [23, 1, 26]. The allowable stress number according to ISO 6336-5 could be increased up to σ_H,lim ≈1,950 MPa compared to the baseline of σ_H,lim ≈1,500-1,650 MPa for MQ and ME material quality.

With increasing torque, the safety factor for tooth root strength decreases faster than for contact fatigue strength because of a linear relationship between the torque and the bending stresses (Hertzian pressure: square root relationship). Therefore, there was no clear separation of the damage pattern. On the highest load level, tooth root breakages as well as pitting failures occurred (Figure 3). For this reason, the endurance limit of the component can only be determined. Because of this, the full potential regarding pitting strength cannot be derived because tooth root breakages occur before the contact fatigue finite life limit is reached.

Furthermore, the damage patterns were evaluated in terms of micro pitting and scratches in the area of the tooth tip. For the increased torque of the test series, a premature tooth meshing could not be avoided with the applied tip relief of C_a1/2 = 20 µm. Besides the impact caused by the premature tooth meshing, the contact area of the premature tooth meshing is also loaded twice because the contact starts in the direction of the tooth root and moves upwards again. Both effects influence the pitting load capacity and should be avoided [24, 14]. In order to determine the full potential regarding pitting strength of this material in gear applications, an optimization of the gear geometry is necessary to ensure a clear separation of different damage patterns and avoiding premature tooth meshing.

3 Objective and Approach

The continuous improvement of material performance levels leads to a change of the critical damage pattern of existing standard test gear geometries. The aim of this report is the definition of a test gear geometry for investigating contact fatigue strength (pitting) for these types of materials (Figure 4). For increasing endurance limits of high-performance steels, a secure separation of the different gear damage pattern still has to be guaranteed. For testing of contact fatigue, especially tooth root breakage has to be avoided because a tooth root failure directly finishes the test, and the data point cannot be used for the evaluation of the contact fatigue S/N Curve.

The increasing load in order to damage gears of high-strength material also leads to higher elastic deformations and a change of the pitch error under load from the tooth in contact and the following non-loaded tooth. In this case, the resulting premature tooth meshing has a more significant influence compared to smaller loads. The premature tooth meshing should be avoided in order to reduce the occurrence of scratch marks and resulting abrasive wear as well as micro pitting. Furthermore, the maximum torque of the test rig has to be considered in order to avoid damages of bearings and clutches of the back-to-back test rig. In future working packages, the changes of the macro and micro geometry of the test gear geometry are validated by experimental tests for high-performance clean steels.

There is a two-step approach for determining a pitting-critical gear geometry for high strength materials. In the first step, the macro geometry for a back-to-back test rig with a center distance of a = 91.5 mm is selected considering the potential regarding maximum contact pressure and a sufficient tooth root load carrying capacity. Therefore a comparison between the well-established, pitting-critical test gear geometries 17/18 and C-type performed  using  a  diagram for defining  an optimum face width depending on the test rig limits and the tooth root strength. The chosen macro geometry is optimized in terms of tooth root strength as well as the maximum Hertzian pressure considering the torque limit of the test rig. In order to fulfill both requirements, the face width is designed in a stepped way guaranteeing a sufficient tooth root strength as well as high contact pressure in the area of the smaller face width. The lead crowning is not changed to increase the contact pressure, since a point contact leads to a change of the lubrication conditions [13].

In the second step, the micro geometry regarding profile corrections is optimized for high loads. The premature tooth meshing resulting from the increased elastic deformation of the tooth is avoided by a circular tip relief. The tip relief is short according to NIEMANN/WINTER [18]. The definition of the amount of the tip relief is based on a FE-based calculation approach according to BRECHER ET. AL. [3, 4]. The other micro geometry corrections are taken over from the original design of the C-type gear [21].

4 Selection and Optimization of Test Gear Macro Geometry

For the selection of the test gear macro geometry, the well-established gear sets 17/18 and C-type are compared to each other in the following chapter [11, 21]. Both gear geometries are pitting-critical for the contact fatigue strength of existing steels from standard gear applications, but showed tooth root breakages as the torque increases for high-performance steels or special surface finishing processes, see section 2.2 and [27]. In order to guarantee a high comparability to existing test results of these gear geometries, the macro geometry is only optimized with regard to face width. All other gear parameters are kept constant compared to the original design of the 17/18 and C-type gear set (Table 1).

The aim of the adaption of the face width is the increase of the maximum Hertzian pressure and likewise a sufficient tooth root strength. The contact pressure and the safety factor against tooth root breakage are calculated according to ISO 6336 [5-7]. At the beginning of the calculation, the constant input parameters are defined (Figure 5). Besides the material properties of the gears (Young’s Module: E_1/2 = 206,000 MPa; Poisson’s number: v_1/2 = 0.3) the allowable stress number for tooth root strength is defined for a reference gear material (16MnCr5 case hardened:
σ_F,lim = 430 MPa).

The tool geometry of the soft machining process (hobbing) is given in Table 2. The grinding stock is q = 120 µm with a tooth span of the finished part for the C-type gear of W_{k1/2 ,3 teeth} = 34.68/48.44 mm and for the 17/18 gear of W_{k1/2 ,3 teeth} = 39.72/39.69 mm. Based on the gear geometry and the input parameters of the material properties and the manufacturing process, the Y- and Z-factors according to ISO 6336 are calculated (Table 3). According to the standard, all factors are constant for a certain gear geometry and testing condition. An increasing load does not affect the Y- and Z-factors according to the overall calculation approach of ISO 6336. The test rig limitations of the back-to-back test rig with a center distance of a = 91.5 mm is a minimum torque of T_min = 150 Nm and a maximum torque of T_max = 715 Nm [22]. The maximum torque refers to the gear and the minimum torque to the pinion. The K-factors according to ISO 6336 for the back-to-back test rig are given in Table 4.

Finally, the calculation of the contact pressure and the safety factor for bending strength are calculated according to ISO 6336 depending on the face width and the torque. Based on the results, the maximum contact pressure can be derived considering the test rig limitations (maximum torque) and the safety against tooth root breakage (Figure 5).

The results of the calculation are shown for the 17/18 gear geometry in Figure 6. Based on the face width on the x-axis and the limitations of the test rig, the field of the contact pressure can be determined for different face widths. The maximum contact pressure is also limited by the safety factor for bending strength of the gear. The safety factor for bending strength S_F is a horizontal line because the ratio between the normal force and the face width F/b is always constant to achieve the same contact pressure. For small face widths, the maximum contact pressure is limited by the safety factor for bending strength. This type of diagram can be used to derive the optimum face width for contact fatigue tests depending on the expected range of contact pressures of the S/N curve. The derivation of the correct face width is important for high strength steel, but also for weaker materials, because the expected torque of the endurance limit should not fall below the limit of test rig.

Determining a critical safety factor for root strength S_F = 1.3, the maximum contact pressure limited by the bending strength is σ_H,max,SF ≈ 1,900 MPa (Figure 6). Regarding the test rig limitations, there is a higher potential of the maximum contact pressure for small face widths, because with the same maximum torque of the test rig, the contact pressure progressively increases if the face width is lowered. In comparison to the 17/18 gear set, the C-type test gear geometry shows the same pattern in this type of diagram (Figure 7). The maximum contact pressure is also limited by the bending strength for low-face widths instead of the test-rig limitations. Considering the bending strength of the C-type  gear  geometry,  the  maximum  contact  pressure  is   similar   to   the   17/18   gear   (σ_H,max,SF ≈ 1,900 MPa).

For high performance materials and based on the test series in section 2.2, the contact pressure needs to be higher than the maximum contact pressure limited by the bending strength (σ_H,max,SF ≈ 1,900 MPa). Therefore, the face width can be designed in a stepped way, determining a different face width in two sections. In this case, the face width below the pitch diameter b_diff of pinion and gear is wider in order to strengthen the tooth root. At the same time, the contact width b is small for the whole path of contact because the face width of either the pinion or the gear is small. At no point in the path of contact are the wider face width b_diff of the pinion and the gear in contact at the same time. This design approach has already been applied on a test gear with module m_n = 8 mm by TOBIE [28].

Applying a stepped tooth shape to both gear geometries, the maximum contact pressure increases and is therefore limited by the test rig limitations. In this case, the general potential with regard to the maximum contact pressure is higher for the C-type than for the 17/18 gear set. Choosing a face width of b = 10 mm, the maximum contact pressure for the C-type gear geometry is σ_H,max,C-type ≈ 2,450 MPa compared to the 17/18 gear set with σH,max,17/18 ≈ 2,300 MPa (Figure 6 and Figure 7). Based on this analysis, the macro geometry of the C-type gear should be chosen as a test gear geometry for high strength materials. It has to be noticed, that pinion and gear of the C-type gear set have an integral common divisor, which might lead to higher loaded teeth depending on pitch errors (hunting teeth).

Considering the high manufacturing quality of modern grinding machines, this hunting effect is not the main criterion for the choice of a test gear geometry.

For the final design of the optimized C-type gear, a smaller face width of b = 10 mm is chosen in order to achieve the largest contact pressures. The face width below the pitch diameter b_diff is increased by factor 2 (b_diff = 20 mm) in order to strengthen the tooth root (stepped tooth shape). The highest diameter of the wider section is d_stepped,1/2 = 69/105 mm with a radius of 2 mm between the wide and small face width. The design of a stepped tooth root guarantees a secure separation of the different damage pattern and high contact pressures.

5 Optimization of Micro Geometry

The elastic deformation of the tooth increases due to the higher endurance limit and, thus, required torque for testing of high-performance steels. The additional elastic deformation may cause premature tooth meshing. This irregular tooth contact is the reason for an increase of the stresses on the tooth flank. The purpose of a tip relief is to prevent premature tooth meshing and corresponding negative effects on the contact fatigue strength. The tip relief for the chosen C-type gear geometry in this report is designed based on a FE-based calculation approach according to BRECHER ET. AL. [3, 4]. In the first step, the elastic deformation is calculated for the point of contact, where the single tooth contact ends (D) (Figure 8, left). Based on the amount of deformation, the theoretical penetration of the following tooth is calculated. In the following step, further influences on the penetration caused by geometrical deviations are considered. Finally, the resulting theoretical penetration at the following tooth is calculated with regard to the local stiffness and the geometrical deviations caused by modifications and manufacturing tolerances.

Figure 8 shows the result of the calculation of the tip relief for the C-type gear geometry depending on the torque of the pinion and the face width of the contact area. In the upper part of the diagram, the contact pressure according to ISO 6336 is shown, whereas in the lower part, the amount of a circular tip relief is given to avoid premature tooth meshing. The tip relief is short according to NIEMANN/WINTER with diameters d_Ca1/2 = 79.43/115.63 mm [18]. In order to define the amount of the tip relief, the maximum contact pressure of the test series has to be estimated. From this point on the upper y-axis the intersection with the curve of the face width in both parts of the diagram leads to the amount of the tip relief in the lower part, compare Figure 8. Based on the test series in section 2.2, the expected maximum contact pressure for the endurance limit is approximately σ_H,max ≈ 2,100 MPa. With a face width of the contact area of b = 10 mm, the premature tooth meshing is securely prevented with a tip relief of C_a = 85 µm. The wider face width below the pitch diameter increased the stiffness of the contact. Therefore, the design of the tip relief is on the safe side and a premature tooth meshing can also be avoided for higher loads. The test rig torque of the pinion for the contact pressure of σ_H = 2,100 MPa is T_pinion ≈ 350 Nm.

6 Summary and Discussion

Increase of power density is a continuous challenge for the development of modern transmissions. The power density influences the production cost as well as the performance within the application because of weight reduction. During the design phase of transmissions, the fulfilment of load carrying capacity is guaranteed by the application of different calculation methods. For the calculation of pitting load capacity according to ISO 6336, the allowable stress number for contact fatigue σ_H,lim is required. In order to determine the allowable stress number, experimental tests for different materials and process chains are performed on back-to-back test rigs. There are different, well-established test gear geometries, which are usually used for these standard tests in order to guarantee a comparability between different test series. The test gear geometries are designed in a way to force pitting damages and to avoid all other damage types such as tooth root breakage. With regard to an increasing power density, the endurance limit of the test gear geometry in terms of maximum torque increases. In this case, the safety factors for the different damage patterns (pitting and tooth root breakage) merge and a secure separation of the different damage patterns cannot be ensured.

In this report, a test gear geometry for investigating contact fatigue strength (pitting) of high-performance clean steels is determined. Based on two well-established test gear geometries for contact fatigue test, a detailed comparison with regard to their potential of maximum contact pressure is performed. The C-type test gear geometry is finally chosen and optimized regarding face width and amount of tip relief. The further gear macro geometry is not changed in order to guarantee a high comparability to existing test results. The face width is designed in a stepped way with a high face width below the pitch diameter and a small face width above the pitch diameter. In this way, the tooth root load capacity can be increased and at the same time, the contact pressure can be maximized because of the small contact area on the entire path of contact. The contacting face width is defined  b = 10 mm, and the face width below the pitch diameter is b_diff = 20 mm. Furthermore, the amount of the tip relief is optimized in order to avoid premature tooth meshing for the increased endurance limit. Based on a FE-based calculation method, a look-up table for the maximum contact pressure and different face widths are developed. The design of the tip relief is circular with an amount of C_a = 85 µm.

Based on the optimized test, gear geometry pitting load carrying capacity can be investigated for high strength material without the risk of tooth root breakage and premature tooth meshing. In the next work package, this will be investigated for high performance clean steel. The allowable stress number for contact fatigue can be securely determined for future material improvements regarding pitting load capacity. 

References

1. Bagh, A., 2015, “Auslegung PVD-beschichteter Stirnräder,” Diss., Werkzeugmaschinenlabor (WZL) der RWTH Aachen; Fraunhofer-Institut für Produktionstechnologie (IPT), RWTH Aachen University.
2. Brecher, C. and Löpenhaus, C. and Goergen, F. and Mevissen, D., 2017, Crack Propagation Analysis of Pitting Damages of High-Strength Material Systems in Gear Applications, VDI, Düsseldorf.
3. Brecher, C. and Löpenhaus, C. and Konowalczyk, P., 2015, FE-based Design Method for Pressure Optimized Profile Corrections, VDI Wissensforum GmbH, Düsseldorf.
4. Brecher, C. and Löpenhaus, C. and Konowalczyk, P., 2016, FE-based Approaches for Tip Relief Design, Brecher C.; Klocke F., Aachen.
5. ISO, September 2006, “Calculation of load capacity of spur and helical gears. Basic principles, introduction and general influence factors,” ISO 6336 Teil 1.
6. ISO, September 2006, “Calculation of load capacity of spur and helical gears. Calculation of surface durability (pitting),” ISO 6336 Teil 2.
7. ISO, September 2006, “Calculation of load capacity of spur and helical gears. Calculation of tooth bending strength,” ISO 6336 Teil 3.
8. ISO, Juli 2003, “Calculation of load capacity of spur and helical gears. Strength and quality materials,” ISO 6336 Teil 5.
9. Dixon, W.J. and Mood, A.M., 1948, “A Method for Obtaining and Analyzing Sensitivity Data,” Journal of the American Statistical Association.
10. Fagerlund, J. and Kamjou, L., 2015, Fatigue Performance and Cleanliness of Carburizing Steels for Gears, AGMA.
11. 1976, Grundlagenversuche zur Ermittlung der richtigen Härtetiefe bei Wälz- und Biegebeanspruchung, Frankfurt a.M.
12. 1986, Grundlagenversuche zur Ermittlung der Richtigen Härtetiefe bei Wälz- und Biegebeanspruchung – Ergänzungsversuche zum Größeneinfluss an einsatzgehärteten Rädern aus 16MnCr5, Frankfurt a.M.
13. Hamrock, B.J. and Schmid, S.R. and Jacobson, B.O., 2004, Fundamentals of fluid film lubrication, Dekker, New York.
14. Hergesell, M., 2013, “Grauflecken- und Grübchenbildung an einsatzgehärteten Zahnrädern mittlerer und kleiner Baugröße,” Diss., Institut für Maschinen- und Fahrzeugtechnik, TU München.
15. Hück, M., 1983, “Ein verbessertes Verfahren für die Auswertung von Treppenstufenversuchen,” Werkstofftechnik.
16. Klocke, F. and Brecher, C., 2017, Zahnrad- und Getriebetechnik, Carl Hanser, München.
17. Klocke, F. and Schröder, T. and Bugiel, C., 2005, Influence of the PVD Process on the Load Carrying Capacity of Case Hardened Components, Brecher C.; Klocke F.
18. Niemann, G. and Winter, H., 2003, Maschinenelemente, Springer, Berlin.
19. 1988, Oberflächenstruktur, Frankfurt a.M.
20. Oberg, E. and MacCauley, C.J., 2012, Machinery’s handbook, Industrial Press, New York, NY.
21. 2010, Pittingtest – Einfluss des Schmierstoffes auf die Grübchenlebensdauer einsatzgehärteter Zahnräder im Einstufen- und im Lastkollektivversuch, Frankfurt a.M.
22. DIN, April 1990, “Prüfung von Schmierstoffen. FZG-Zahnrad-Verspannungs-Prüfmaschine. Allgemeine Arbeitsgrundlagen,” DIN 51354 Teil 1.
23. Schedl, U., 1998, “Einfluß des Schmierstoffs auf die Grübchenlebensdauereinsatzgehärteter Zahnräder,” Diss., Forschungsstelle für Zahnräder und Getriebebau (FZG) der TU München, TU München.
24. Schrade, U., 2002, “Einfluss von Verzahnungsgeometrie und Betriebsbedingungen auf die Graufeckentragfähigkeit von Zahnradgetrieben,” Diss., Forschungsstelle für Zahnräder und Getriebebau (FZG) der TU München, TU München.
25. Stahl, K., 2018, “Determination of the pitting and tooth bending capacity, preliminary report,” Forschungsstelle für Zahnräder und Getriebebau (FZG) der TU München, TU München .
26. Staudt, J., 2016, “Funktionsgerechte Bearbeitung von Verzahnungen durch Freiformfräsen,” Diss., RWTH Aachen University.
27. 2010, Steigerung der Zahnflankentragfähigkeit durch Kombination von Strahlbehandlung und Finishingprozess, Frankfurt a.M.
28. Tobie, T., 2001, “Zur Grübchen- und Zahnfusstragfähigkeit einsatzgehärteter Zahnräder,” Diss., Forschungsstelle für Zahnräder und Getriebebau (FZG) der TU München, TU München.
29. 2015, Tragfähigkeit gestrahlter und gleitgeschliffener Zahnflanken unter besonderer Berücksichtigung des Randzonen- und des Schmierfilmzustands, Frankfurt a.M.
30. DIN, März 1986, “Verschleiß; Kategorien der Verschleißprüfung,” DIN 50322.
31. Wallin, K., 2014, Internal defects in case hardened gears and its influence on fatigue life, Sweden.
32. DIN ISO, Mai 2006, “Zahnräder. FZG-Prüfverfahren. FZG-Prüfverfahren A/8,3/90 zur Bestimmung der relativen Fresstragfähigkeit von Schmierölen,” DIN ISO 14635 Teil 1.