Analogy test rig for the generation of tooth flank fractures at large gears (WZL-double pulsator)—commissioning and first test results

As a result of the enhanced of load-carrying capacity with regard to pitting and tooth root breakage, generation of tooth flank fractures became more frequent. This type of damage differs from existing fatigue damages on the tooth flank in that crack initiation takes place below the case hardened surface layer and the tooth breaks in the area of the active tooth flank. An isolated investigation of tooth flank fractures is currently not possible because there is no analogy test rig available that can reproduce both, the compressive stress due to Hertzian contact flattening as well as the tensile stress due to tooth bending. For this reason, a test rig concept is being developed, manufactured and put into operation in the DFG research project “Analogy test rig for tooth flank fractures”, which allows the generation of tooth flank fractures in absence of other damage types. This report deals with the commissioning of the WZL-Double Pulsator as well as first test results. The FE model of the analogy test rig is validated by measuring tooth root strains with strain gauges and comparing them with the calculated strains. Finally, the commissioning is carried out and first tests are performed.

a single gear geometry in the running test, since a gear will fail with a critical damage type depending on its design. For the reasons mentioned above, analogy tests enable the investigation of influencing factors on the development of specific gear damages. In contrast to the running test, which comes close to the real application, the damage types can be analyzed separately in the analogy test. In addition, the analogy test is characterized by a more economical operation compared to the running test. The preferred analogy test depends on the investigated damage type, cf. Figure 1.
The tooth flank damages pitting, micropitting and scuffing can be investigated with the aid of the disc-on-disc test rig, Fig. 1 top [5]. The advantages of using the analogy test are the isolated investigation of the tooth flank load-carrying capacity without influences from the tooth root and tooth flank fracture load-carrying capacity, as well as the local reproduction of different contact conditions. The variable slippage from tooth flank contact on the test discs can be realized by using non-circular gears in the transmission gearbox [1].
The investigation of the tooth root load-carrying capacity can take place on pulsators in a time-and cost-efficient manner and detached from other damage types. In addition to the investigation of spur gears, the experimental determination of the tooth root load-carrying capacity of helical gears as well as a qualitative and quantitative transfer to the running test is the focus of research work. For this purpose, a special construction allows clamping of helical gears in the pulsator test rig. [2,3].
An analogy test for the investigation of tooth flank fractures, analogous to the disc-on-disc test and the pulsator test, does not exist. Since tooth flank fractures occur in particular on large gears, which cannot be investigated economically when using the back-to-back test rig, there is great potential for investigating this type of damage by means of an analogy test. For this reason, an analogy test rig for the generation of tooth flank fractures is being developed and tested as part of the DFG research project BR 2905/90 1 "Analogy test rig for tooth flank fractures".

Stresses in tooth flank contact and tooth flank fracture
The stress of a volume element, which lies on or below the tooth flank surface, can be characterized by three phases, see Fig. 2. Without external load, a constant residual stress state prevails in the volume element, phase 1. For case hardened gears, the residual stresses are resulting from the phase transformation of the carburized surface layer during quenching as well as the grinding process. This results in residual compressive stresses near the surface and residual tensile stresses below the hardened surface layer or in the interior volume [6]. In the active tooth flank contact, predominantly compressive stresses build up in the volume element as a result of the Hertzian contact flattening, which are superimposed on the residual stress state from phase 1, see Fig. 2 "Primary stress (2)" [7]. The stress state below the Hertzian contact zone is characterized by normal, tangential and temperature stresses due to friction, slippage, micro and macro Hertzian contact [NIEM03]. At higher material depths, the influence of the macro Hertzian contact dominates and the tribological influencing factors are negligible small [8,9]. For large-modulus cylindrical gears, the radii of curvature ρ1/2 of the contacting tooth  [9,12] flanks increase. As a result, the Hertzian flattening width bH increases and the equivalent stress maximum according to the von Mises hypothesis shifts to higher material depths [6]. Downstream of the Hertzian contact, the volume element is stressed by tooth deformation. Tensile and shear stresses build up, which increase until upper transition of single tooth contact or until the end of tooth contact due to the increasing lever arm, see Fig. 2 "Secondary stress (3)" [10,11].
The crack initiation point of the tooth flank fracture is below the surface at a depth of t 2.5 CHD550, preferably at a defect in the microstructure (e.g. void, non-metallic inclusion) [9]. According to the current state of the art, the cause of crack initiation is assumed an excessive stress due to the difference in Young's modulus between the defect and the surrounding microstructure as well as the notch effect of the defect [9,13,14]. In the crack propagation phase, a primary crack develops, which propagates with an angle of approx. 45°to the active tooth flank, see Fig. 2 right [15]. In addition, secondary and tertiary cracks can develop which occur due to the changed stiffness properties after the formation of the primary crack and run parallel to the tooth tip [9]. The damage type tooth flank fracture is significantly influenced by the contact pressure in combination with the mechanical secondary stress (bending, shear) and by the superimposed residual stress state [4,9,14,16]. Compared to the stress state of the tooth flank near the surface (0 mm ≤ t ≤ tvon Mises,max), lower stresses are present in the tooth flank fracture critical region and the influence of the local strength on the tooth flank fracture load capacity predominates [16]. The reduced strength in the tooth flank fracture critical region is due to the transition between hardened surface layer and unhardened core structure, resulting in a locally low hardness in combination with low compressive residual stresses or tensile residual stresses [4].

Reproduction of the TFF critical stress with two pulsators
In order to reproduce the stress sequence of the running test by a double pulsator concept previous work focused on an automatic, iterative calculation method to determine the required pulsator forces for a defined torque. The running test rsp. the real tooth contact was realized by a quasistatic rolling-sliding simulation in Abaqus CAE. The target variable was the stress tensor sequence of the simulated tooth contact in the point of interest (POI). The POI is the position in the discrete volume element next to the crack initiation location of most tooth flank fractures in the running test. In the first calculation step, the Hertzian contact, which is represented by the lower actuator (primary actuator), was optimized. The contact between primary actuator and tooth flank is initially loaded with a force ramp within the power limits of the actuator. The upper actuator (secondary actuator) is load-free during the optimization of the Hertzian contact. The result of the initial loading are stress tensors in the POI depending on the simulated forces. From the relationship between pulsator force and resulting stress tensor at the POI, all stresses at the POI can be interpolated to any desired force within the simulated force ramp. The optimization of the force sequence is based on a quadratic minimization between the required stress amount of the running test and the stress amounts in the analogy test. The optimization algorithm uses the determined interpolation functions and iterates the pulsator force until the quadratic minimization function reaches the smallest possible value.

Double pulsator Running test
Stress curves at point of interest (POI)

Fig. 3 Comparison of Stress Curves between Running Test and Analogy Test
The first optimization algorithm yields pulsator force sequences and their associated stress tensors for the primary actuator. The procedure for optimizing the pulsator force of the secondary actuator is analogous to the optimization of the primary actuator. In the last step, merging of the pulsator forces and stress tensor sequences of both optimization steps is carried out. A subsequent FE simulation with the calculated pulsator force sequences and a comparison to the running test simulation is done in order to verify the determined stress tensor sequences. Figure 3 shows the comparison of the stress sequences between analogy and running test at the POI for the test gear geometry in the DFG research project at a pinion torque of M1 = 3500 Nm [17,18]. The comparison of the stress sequences between the running and analogy test shows a high correlation after optimization of the the pulsator force sequences. However, the stationary force application of the primary actuator means that the maximum stress magnitudes of all stress components will not match, since they are phase-shifted in rollingsliding-contacts. If the primary actuator was moved further towards the tooth tip, a higher agreement of the stress component σtan could be achieved at the expense of the stress components σrad and τrad/tan. In the chosen actuator position an overshoot in the shear stress peak τrad/tan compared to the running test is inevitable in order to have a better fit for the normal radial stress σrad as well as the normal tangential stress σtan. A match of all stress components shown will not be possible due to the fixed actuator position. This is a disadvantage of the concept compared to a rolling-sliding contact based test. However, Fig. 3 bottom right shows the stress sequence of the von Mises equivalent stress, where a high correlation between the running and analogy tests can be observed. This is an indicator for a high stress state in the volume at the POI, which is assumed to be a requirement for damage occurrence. The von Mises equivalent stress is even higher in the analogy concept so it is assumed, that the TFF risk increases compared to the running test [17].

Design of the WZL-double pulsator
The design of the analogy test rig is based on VDI 2221 and the overall concept, based on the preliminary work of Konowalczyk, consists of a hydraulic system with two connected actuators, which are operated with a phase shift [4,19]. The test system is arranged horizontally on a machine table, see Fig. 4. The size of the machine table is L × D = 3000 mm × 1000 mm. The primary actuator is designed as a compression actuator with a dynamic maximum force of Fprim,dyn,max = -80 kN and the secondary actuator as a tension actuator with a dynamic maximum force of Fsec,dyn,max = 53 kN. Both actuators have one translational degree of freedom and two rotational degrees of freedom at the rear guide, which can be clamped after an initial positioning for the dynamic operation. The force is applied translationally via the piston stroke (primary actuator: t2, secondary actuator: t4). The dynamic force is applied stationary. The primary pulsator jaw, which is positioned on the tooth flank to apply the Hertzian contact, is made of quenched and tempered tool steel. The secondary pulsator jaw consists of a C-shaped mounting bracket and a pulsator tip, which is attached to the mounting bracket with screws and is also made of quenched and tempered tool steel [20].
To set the correct position, the rear guides are equipped with magnetic tape sensors. After initial positioning and locking of the rear translational guides, the actuator is freely movable on a circular path around the clamping axis perpendicular to the machine table (primary actuator: r1, secondary actuator: r4). This clamping axis defines the angle of the actuator and is adjusted by a laser distance sensor at the front guidance. To ensure that a large number of test segments of a gear geometry are tested under the same loading conditions, the clamping situation must remain unchanged. The actuators or the pulsator jaws are locked after initial positioning at the beginning of a test series (examination of a single gear geometry) and are not realigned in the further course of the test series. The change of a test segment is necessary for each new test. Each test segment, including a cuboid gear body segment, is separated from the gear by electronic discharge machining (EDM), cf. Figure 4 gear segment in the middle. Figure 5 provides a general overview of the analogy test rig [20].

Objective and approach
An analogy test rig is necessary for the economical investigation of tooth flank fracture load-carrying capacity in isolation from other damage types. Chapter 1 shows that analogy test rigs are a proven concept for the isolated investigation of individual damage types and that they already  exist for the common gear damage types such as pitting, scuffing, micropitting and tooth root breakage. Tooth flank fracture is influenced by both the stress due to Hertzian contact and the downstream stress due to tooth bending. For this reason, an analogy test rig is required that represents both loading scenarios. In preliminary work, a test rig concept (WZL-Double pulsator) was developed which reproduces the local stress at the point of crack initiation (POI: point of interest) appropriate in FE-simulations [17,18,20]. Based on the preliminary work, the following objective is derived for the successful use of a tooth flank fracture analogy test rig:

Validation of the FE-simulation as well as commissioning of the analogy test rig and test execution
In order to validate the simulated stresses, a strain measurement in the tooth root using strain gauges is conducted and compared to the FE-simulation. In a second step, the commissioning of the test rig is done by analyzing the measured pulsator force profiles under dynamic loading with different nominal pulsator force profiles. The analysis includes the quality of imaging the nominal pulsator force profile. The quality of imaging the nominal profile is an indicator of the dynamic stability of the test rig, since the measured pulsator force profile gets worse if the controllability of the test rig is struggling with the required nominal pulsator force profile. In the last step, endurance tests are carried out in order to generate tooth flank fractures with the analogy test concept.

Validation of the stress in the specimen with strain gauge measurement
The FE model is validated with the aid of strain gauge measurements. Since a strain measurement was not possible due to the risk of collision between the pulsator jaw and the strain gauge directly at or below the force application point, strain measurements were carried out in the tooth root. For this purpose, a strain gauge was applied to the tensile stress side of a cut tooth segment. The strain was measured separately for the primary and secondary actuators as a function of the applied pulsator force, cf. Figure 6. Afterwards, the measured strain was compared to the calculated primary strains at the surface of the hexahedron-meshed tooth. The comparison between measurement and simulation is shown in Fig. 6 on the right. The comparison shows that the measured strain for both actuators is higher than the calculated strain. The maximum deviation between measurement and calculation is εTension,primary = 1.25 10 -4 ( O = 26.3 MPa) for the primary actuator and εTension,secondary = 2.15 10 -4 ( O = 45.2 MPa) for the secondary actuator. The relative deviation for both actuators is in the range 7% ≤ (εmeasurement -εcalculation)/εcalculation ≤ 12%, cf. Figure 6, bottom right. It can be assumed that the deviations are attributed to neglected components in the simulation which, due to their deformation under load, lead to an angular deviation between the pulsator jaw and tooth segment and consequently cause a higher circumferential force (e.g. load cells, hydrostatic piston bearings).

Dynamic test rig commissioning
The implementation of an automatic test procedure as part of the research project allows the operation of the test rig. During a test, manually preset force and displacement parameters serve as limit values for a test stop. Furthermore, a peak-valley detector monitors the change of the displacement signal for both actuators and stops the test run when the displacement delta exceeds a limit value. After completion of the test by premature termination or reaching the limiting load cycle number, the actuators automatically return to their starting position. To determine the test rig limits with the elaborated concept and the optimized control parameters, tests were carried out with the individual loading profile and the phase-shifted double sinusoidal profile, cf. Figure 7. The individual loading profile is the optimized pulsator force sequence from the FE optimization. In the phase-shifted double sinusoidal profile, the load of the primary and secondary actuator is applied as a sinus function containing a phase-shift with the maximum force of one actuator acting when the clamping force is applied to the other actuator. Figure 7 shows the comparison of the nominal and actual profiles of both actuators during a load cycle with the individual loading profile and the phase-shifted double sinusoidal profile. The imaging quality of the nominal signal by the actual signal is evaluated with the coefficient of determination R 2 . In addition, the phase-shift deviation t is plotted for different test frequencies. The phase-shift deviation t describes the percentage offset between the time of the measured, maximum actual force on the secondary actuator and the time of the maximum nominal force on the secondary actuator.
The analysis of the imaging quality of the individual loading profile was carried out for the test frequencies f = 0.1 Hz, f = 1 Hz and f = 10 Hz, see Fig. 7 top. The pulsator force profile corresponds to a torque equivalent at the pinion of Mpinion 600 Nm in the running test and thus relatively low loads. Tests were also carried out with varying loads and the same test frequency. Due to the similar control behavior for varying loads, the results are not shown separately. For the pulsator force profile investigated, the nominal signal can be reproduced well for the test frequencies f = 0.1 Hz as well as f = 1 Hz. For the test frequency f = 10 Hz, a worse quality of imaging can be observed. On the one hand, there is an undershooting of the clamping force at the secondary actuator. Falling below the clamping force can lead to hammering, since the pulsator jaw lifts off the tooth flank as soon as the force exceeds the zero crossing. Furthermore, the required maximum force on the secondary actuator is not reached and there is an increase in the phase-shift deviation t = 5.5%. If the phase-shift deviation t is too high, the secondary actuator reaches its maximum force while the primary actuator is already actively applying load. There is a correlation between the coefficient of determination R 2 and the phase-shift deviation t. The general phase-shift between nominal and actual signal is filtered in advance for the primary and secondary actuator and only the phase-shift leading to an interference between primary and secondary load is analyzed. This is equivalent to a poor quality of imaging for the secondary actuator with respect to the temporal offset between nominal and actual signal. For the individual loading profile, the test rig limit was accordingly reached at a test frequency of f < 10 Hz and a load equivalent of Mpinion = 600 Nm, so that the individual loading profile is not suitable for long- term tests. Assuming that the type of loading is negligible compared to the maximum loads applied, the pulsator forces of primary and secondary actuator can be applied by a phase-shifted double sinusoidal profile [21]. With respect to the calculation of the tooth flank fracture load-carrying capacity, the approximation does not result in any loadcarrying capacity differences. Here, the load on primary and secondary actuator is imposed by a sinusoidal profile with the force limits clamping force and maximum force, and the nominal phase-shift between primary and secondary pulsator force corresponds to φ = 0.5 π. Hydraulic controllability of a phase-shifted double sinusoidal profile is better compared to the individual loading profile. Accordingly, the analysis could be performed at higher loads and test frequencies, see Fig. 7 bottom. The maximum forces analyzed were Fprim,max = -80 kN for the primary actuator and Fsec,max = +40 kN for the secondary actuator. The test frequencies analyzed were f = 1/10/20/30 Hz. For the test frequencies f = 1 Hz and f = 10 Hz, a good reproduction of the nominal signal was demonstrated. With increased test frequencies f, the coefficient of determination R 2 decreases and the phase-shift deviation t increases. For a test frequency f = 30 Hz, a similarly poor control behavior could be observed as for the individual loading profile at f = 10 Hz. For the phase-shifted double sinusoidal profile, the test rig limit was accordingly reached at a test frequency of f = 20 Hz and a running test load equivalent of Mpinion = 4800 Nm.

First test results and review of test concept
The geometry investigated for the research project was tested experimentally with respect to the tooth flank frac-ture load-carrying capacity both on the analogy test rig as well as on the back-to-back test rig according to DIN ISO 14635 with center distance a = 200 mm [22]. The comparison of the results is shown in Fig. 8. The blue crosses and circles represent tooth flank fractures and run-outs on the back-to-back test rig, where the limiting load cycle number was defined as NL = 50 10 6 load cycles. As an example, two damage patterns at a pinion torque Mpinion = 2600 Nm and Mpinion = 3000 Nm are shown in Fig. 8. With the geometry, tooth flank fractures could be reproducibly generated in the running test. The tooth segments selected for the analogy test came from gears of the identical batch. The yellow (or unfilled, light) circles show test points where a previously unloaded tooth flank was examined in the analogy test. The first tests were performed at moderate forces (primary actuator: -34 kN ≤ Fprim ≤ 43 kN, secondary actuator: 11 ≤ Fsec ≤ 13 kN), which after recalculation corresponds to an equivalent torque in the running test of 2000 Nm ≤ M1 ≤ 2540 Nm. A majority of the tests were subsequently carried out at pulsator forces Fprim = -70 kN and Fsec = 30 kN, which corresponds to an equivalent torque Mpinion = 4130 Nm and is thus within the time yield strength of the gear stage with respect to tooth flank fracture. One test was performed at the test rig limit with pulsator forces Fprim = -80 kN and Fsec = 40 kN, which corresponds to an equivalent torque Mpinion = 4720 Nm. With the unloaded tooth segments, no tooth flank fracture could be generated in the analogy test. Furthermore, tests were carried out with tooth segments that had previously been subjected to a load in the back-to-back test rig. The gears selected for this purpose have previously experienced a pinion torque in the range 2600 Nm ≤ Mpinion ≤ 3000 Nm at a load cycle Since the test results did not meet expectations, the calculations and tests were reflected with a view to differences between the two test methods. Since tooth flank fracture is usually initiated at a non-metallic inclusion, the first step was to compare the total number of teeth tested. It became clear that the number of teeth tested in the analogy test and thus the volume tested in the analogy test corresponded to only 3.7% of the running test, cf. Figure 8 "tested teeth". For future investigations, consideration must be given to analyzing gears in advance of the tests by means of ultrasonic measurement for large, non-metallic inclusions in individual teeth and using those teeth for the analogy test.
In the second step, the highly stressed volumes were compared by FE calculations in Abaqus CAE. The highly stressed volume HSV90 is the volume, which experiences 90% or more of the maximum material exposure in the component. The comparison in Fig. 9 on the left between the running and analogy tests shows that the highly stressed volume is 10% lower in the analogy test than in the running test (HSV90 = 7.0%). This is due to the stationary load application in the analogy test, which causes a lower volume stressed by Hertzian contact.
In the last step, the stress tensors at the POI were evaluated in detail. The evaluation is shown in Fig. 9 on the right. It can be seen that the maximum magnitudes of the different stress components in the running test are phaseshifted and in the analogy test they are simultaneous. This circumstance cannot be considered by the currently applied equivalent stress hypotheses [4,9,[23][24][25][26]. For this reason, focus in future work should lie on the extension of existing equivalent stress hypotheses for the representation of the equivalent stress in rolling-sliding contacts.

Summary and outlook
Tooth flank fractures occur more frequently due to the continuous optimization with regard to pitting and tooth root load-carrying capacity. The characteristic of tooth flank fracture is a crack initiation in the material volume, usually at a non-metallic inclusion. Since the probability of large, non-metallic inclusions increases with component size, large-modulus gears in particular are at risk of tooth flank fracture. An economical and statistically proven load capacity test of such gears is currently not possible. For this reason, an analogy test rig was developed in the DFG research project BR 2905/90-1 "Analogy test rig for tooth flank fractures" where a tooth segment of a gear is loaded with two hydraulic actuators in such a way that the stress sequence corresponds as closely as possible to the real tooth flank contact. Previous work was carried out on the implementation of a method for the determination of the pulsator forces as well as on the design and manufacturing of the analogy test rig [17,18,20].
In order to verify the FE-simulations, measurement of the principal strain in the tooth root by means of strain gauges was carried out and provided appropriate agreement with the calculated principal strains in the FE simulation. In the next step, the dynamic commissioning was carried out and the tests were performed. To this end, tooth segments were cut out of gears by EDM and clamped in the analogy test rig. Since the load-optimized pulsator force profile could only be controlled with limited dynamic stability, a sim-plified phase-shifted double-sinusoidal profile was successfully implemented and tested. This pulsator force profile leads computationally to an identical stress state and can be controlled in a dynamically stable manner so that pulsator forces up to the test rig limit can be applied with a test frequency of f = 20 Hz. In the test series, no tooth flank fracture could be generated with the new test concept, whereby both previously unloaded tooth flanks as well as tooth flanks previously loaded in the running test were tested in the area of time yield strength with regard to tooth flank fracture. For this reason, a critical reflection of the test rig concept was carried out. Possible causes could be the reduced number of tested teeth and the reduced highly stressed volume HSV90 compared to the running test. In addition, due to the fixed positioning, the analogy concept is not able to reproduce the phase-shift between the maxima of the individual stress tensor components, which is characteristic for the rollingsliding contact.
For future work, consideration must be given to ultrasonic measurement techniques to examine gears for large, non-metallic inclusions in order to identify teeth at risk of tooth flank fracture for the analogy test. Furthermore, the current equivalent stress hypotheses for evaluating rollingsliding strength problems need to be questioned and extended with focus on phase-shifted stress maxima. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4. 0/.