A simulation method for the local pitting load capacity of bevel gears is applied to cylindrical gears where local calculation parameters, such as the local contact pressure, are considered; the method has been successfully applied to cylindrical gears and therefore local safety can be calculated.

In order to carry out an optimal gear design with the aim of cost reduction and the careful handling of resources, load capacity is an important criterion for the evaluation of a gear. Standardized procedures, such as ISO 6336 [1] for cylindrical gears and ISO 10300 [2] for bevel gears, are widely used and established in the design and evaluation of gears. In such procedures, the gear meshing is simplified to a single reference point and a comparative stress is derived. This simplification is based on experience and validation, but the influence of flank modifications is only approximated via factors.

Detailed flank modification design with consideration of all possible types of damage for bevel gears is only possible with the aid of local load capacity analyses [3] [4]. Depending on the load, the location of the maximum stress or damage can occur at different points on the tooth. When calculating load spectra, the minimum safety and the accumulated damage [5] can be determined more precisely with local consideration of stresses, lubrication coefficients such as friction and film thickness, and geometry coefficients such as sliding speeds. In addition, the local load distribution includes the interaction of the overall system with all of its components and their associated loads. For example, bearing stiffness and clearances, shaft bending and torsion, housing deformation, load-dependent center distance, or gravity may be considered. This provides a more accurate picture of the stresses and quickly gives an overview of the possible resulting relevant damage forms.

In this article, the currently available local methods for bevel gears [6] [7] [8] [9] are applied to cylindrical gears. This will demonstrate the benefits of local methods and how they can assist in the design of flank modifications. To validate the accuracy of the calculation methods, a well-documented series of tests from the literature [10] is recalculated and evaluated.

1 Introduction

Power density requirements are constantly increasing in many industries. Optimization based solely on standard calculations is rarely sufficient today. This is why in some cases gear standards are ignored, and gears are designed to meet a contact pressure level or a transmission error and are then tested on a testbench.

Standards give users results that reflect the state of the technology and thus inherently offer a high degree of legal certainty. However, anyone who deviates from the standards must be able to prove, in case of doubt, why the chosen method is more suitable.

Standards in gear load capacity calculations are based on simplified global stress calculations to achieve comparability of results. Contact pressure distribution from tilting of the tooth mesh and flank modifications are considered in a simplified manner using factors. In practice, if complex uneven contact pressure distribution is present in the tooth mesh, it can only be considered by means of a local consideration.

In this article, we discuss the possibility to use a “local method” for gear load carrying capacity. We define the local method as a method that simulates the stresses and safety factors for every point on the gear flank. For this purpose, we have used the available methods for bevel gears [6] [7] [8] [9] and applied them to cylindrical gears. In order to verify the results, we compare the results to available testbench results.

2 Flank load capacity calculation

There are different forms of damage in tooth contact based on different physical phenomena. The forms of damage are:

Pitting: Fatigue damage on the flank, mainly on the profile with negative specific sliding. The crack initiates on the surface and the propagates in the depth direction. Follow up failures are “shell- shaped” material disruption, noise, vibrations, and tooth breakage.

Flank breakage: A sub-surface fatigue failure that initiates between the core and the surface microstructure with a “fish eye” (often on the fragments). The crack starts on the loaded flank and grows toward opposite root fillet. Mostly secondary and tertiary cracks occur, which lead to wedge- shaped material outbreaks.

Scuffing: A sudden local welding due to temperature and contact pressure. It is not a continuous fatigue process, but rather a spontaneous failure that can also occur after a single load cycle. It appears as vertical scratches that cause friction and martensite on the surface.

Micro pitting: Fatigue damage on the surface caused by mixed friction (breakdown of the lubrication film thickness under high pressure with low rotating speed). It results in material removal, profile form deviations, increasing noise, increasing dynamic forces, and decreasing pitting strength. It is visible as gray horizontal stripes, mainly between the start of the active profile and pitch diameter and occurs on all teeth.

In order to determine the flank load capacity, it is necessary to calculate the safety factors of all types of damage with the assumption of minimum safety. The ISO standards for calculating the load capacity on the flank for cylindrical and bevel gears are listed in Table 1.

Table 1: ISO standards for gear analysis.

In the following, the calculation of the pitting load capacity will be considered as an example.

2.1 ISO 6336-2: Calculation of surface durability (pitting) on cylindrical gears

ISO 6336-2 [1] is an internationally recognized standard based on experience and extensive testing.

The method is based on the calculation of the nominal contact stress at the pitch point σH0 (Equation 2-1). For this purpose, the nominal tangential load Ft on the reference diameter is converted to the normal load on the pitch diameter using the zone factor ZH with the flank curvature of the pitch circle. The material properties are considered via the elasticity factor ZE, the profile contact ratio determines the contact ratio factor Zε, and the helix angle is used for the helix angle factor Zβ.

Since this nominal contact stress does not correspond to the critical location and load on the tooth flank for pitting damage, it must be converted to the relevant contact stress for the pinion and gear (Equation 2-2). For this purpose, the force factors and the contact factors for the pinion ZB and wheel ZD are necessary.

The force factors are described in more detail in ISO 6336-1. They are the application factor KA, mesh load factor Kγ, dynamic factor KV, face load factor KHβ, and the transverse load factor KHα.

The only factor that can consider the system’s stiffnesses and the resulting flank line deviations in tooth contact is the face load factor KHβ. This also considers the face width modifications, such as lead crowning and helix angle. It is defined as the quotient of the highest line load and the mean line load across the tooth width.

The only factor that considers a modification in the profile direction is the dynamic factor KV. However, this only considers the tip or root relief, and only with minor effects.

To consider the uncertainty of the influence of unevenly distributed contact pressure, there is the auxiliary factor fZCa, which affects ZB and ZD. The auxiliary factor provides a qualitative statement on whether the flank has been optimally corrected and carries completely and uniformly fZCa = 1.0 up to an uncorrected and uneven contact pressure distribution with fZCa = 1.2.

In order to be able to calculate the safety factors SH1, SH2, the maximum permissible contact stresses σHP1, σHP2 are required in addition to the maximum loads σH1, σH2.

The permissible contact stress σHP is calculated from the allowable stress σH lim depending on the material, the heat treatment, and the surface roughness of a standard reference test gear, along with the following factors: the life factor ZNT, lubricant factor ZL, velocity factor ZV, roughness factor ZR, work hardening factor ZW, and size factor ZX (Equation 2-3) and (Equation 2-4).

The safety factors for surface durability (against pitting) SH1 and SH2 are now calculated by the quotient of the permissible contact stress σHP and maximum load σH (Equation 2-5) and should be greater than 1.

2.2 ISO 10300-2: Calculation of surface durability (pitting) on bevel gears

The calculation of bevel gears according to ISO 10300 [2] is largely based on the methodology of ISO 6336 [1]. First, the bevel gear is converted to a virtual cylindrical gear according to ISO 10300 part 1 [2]. This is used to calculate the corresponding load, gear mesh sizes, force factors, and Z-factors. This, too, is similar to ISO 6336.

Once again, the physical output variable for the load capacity calculation is the Hertzian contact stress at the tooth flanks, which is also referred to as contact pressure.

The contact pressure σH is calculated according to Equation 2-6 from the maximum contact stress in the line contact σH0, the application factor KA, dynamic factor KV, the face load factor KHβ, and the transverse load factor KHα. The respective application factors are specified in ISO 10300-1.

The maximum contact stress in the line contact σH0 can be calculated according to Equation 2-7 from the tooth normal force Fn of the length of the contact line Ibm and the radius of the relative curvature ρrel determined vertically to the contact line.

Other influences are considered via the mid-zone factor ZM-B, the load sharing factor (flank) ZLS, the elasticity factor ZE, and the bevel gear factor ZK.

The nominal normal force Fn of the virtual cylindrical gear (Equation 2-8) is calculated from the tangential force Fmt1, the normal pressure angle of the tooth αn, and the mean helix angle βm1.

The permissible contact stress σHP is calculated according to Equation 2-9 from the flank endurance limit σH,lim and includes the influence of the life factor ZNT, size factor ZX, lubricant factor ZL, velocity factor ZV, roughness factor ZR, work hardening factor ZW, and the hypoid factor ZHyp.

The flank safety factor SH for pitting can now be determined (Equation 2-10) from the existing contact stress σH and the permissible contact stress σHP and should be greater than the desired minimum value for SH,min = 1.0. For a more detailed description of the influencing factors, refer to ISO 10300-2.

2.3 FVA 411: Local pitting load capacity

The local pitting safety factor is determined using findings from Wirth [3] according to Equations 2-11 and 2-12.  Accordingly, the existing local contact stress σH is decisive for the occurrence of pitting damage.

The permissible contact stress σHG is calculated from the value of the material endurance limit σHlim, considering other influences such as material fatigue due to an increasing number of load cycles, material and lubricant influences, and the kinematics on the flank.

Based on pitting load capacity tests, a critical pitting safety factor of SH,krit = 1 has been determined with a failure probability of 1%, below which the tooth flank may fail.

The contact stress σH is determined locally using gear contact analysis.

The lubricant influence factor ZL considers the influence of the type of lubricant and its viscosity on the durability of the surface. It is therefore a global factor and is calculated from the lubricant properties. Other global factors are the size factor ZX, work hardening factor ZW, and life factor ZNT.

The roughness factor ZR describes the influence of the surface condition of the tooth flanks on the flank load capacity. It is calculated from the surface roughness Rz1, Rz2 and the radius of relative curvature ρ (Equation 2-13). The radius of curvature can also be determined locally after analyzing the flank geometry. The influence of the surface roughness is thus also included locally in the calculation.

The velocity factor ZV considers the influence of the tangential sliding velocity on the load capacity of the tooth flank against pitting damage. Once the gear contact analysis has been performed, the sliding speed can be calculated from the contact kinematics and the velocity factor can thus be used locally (Equation 2-14).

The strength values according to ISO 10300 were mainly determined on cylindrical gears with straight teeth. The decisive strength is relative to the inner single point of engagement of the pinion in the area of push sliding. Due to negative specific sliding, the strength in this area is lower than in the area of positive specific sliding. The local slip factor ZS measures the slipping influence. This was determined empirically from test results and is described by Equation 2-15. Therefore, the relative slippage perpendicular to the contact line must be determined in a gear contact analysis for the local calculation.

The hypoid factor Zhyp considers the influence of longitudinal sliding of hypoid gears compared to bevel gears. Since no longitudinal sliding occurs in the case of the cylindrical gear calculation, the hypoid factor is set to Zhyp = 1.0.

Application of the formulas to cylindrical gears enables calculation of the local pitting load capacity. Figure 1 shows an example of this for the gearing from Section 5.

Figure 1: Local pitting load capacity according to FVA 411 for GV REF example.

The procedure presented here is related to Schaefer [11]. In his investigations, he successfully applied the analytical load distribution according to Weber/Banaschek [12] to the methodology of Wirth [3].

3 Stress calculation

While the load capacity calculation according to the standard method is performed at the design point and can be carried out using the available formulae from ISO 6336 or ISO 10300, the starting point for the local load capacity calculation at the tooth flank is a calculation of the local load and pressure distribution.

Calculation of the local load distribution requires a tooth contact analysis program. The starting point of the calculation is the geometry of the gearing. Based on the gear geometry, a deformation model can be generated for the linear and non-linear deformation components of the contact problem. An efficient deformation calculation can be created using deformation influence numbers [13]. These allow the linearized deformation results to be reused after the deformation behavior has been calculated once. For this purpose, the calculation of the total deformation is divided into two influence number models (Figure 2). The non-linear deformation component is in the area near the contact zone and the linear deformation component is further away. Deformation from the flattening of both contact partners is considered as the near range. The far range includes all deformation components outside of this area. When solving the contact problem, only the non-linear component has to be determined iteratively. The linear deformation influence can be included directly in the solution. The prerequisite for this is linear elastic material behavior according to Hook’s law. Both the advantages and disadvantages of this calculation method result from this principle. Compared to a numerical solution using a universal finite element contact solver, the calculation time can be significantly reduced, but the contact position remains unchanged. Various models have been established for the calculation of the linear deformation component, including beam and disk models [14], two-dimensional numerical methods [13], and three- dimensional numerical methods [6]. BEM and FEM can be used as numerical methods.

Figure 2: Addition of the deformation models from the linear and non-linear components gives the total deformation.

Another important aspect of the load distribution calculation is the interaction with the gear environment. This includes the shaft deflection as well as the bearing deflection and the housing deformation. The deformation components can be linear or non-linear, depending on the physical problem. Two approaches have been established for consideration of the displacements of the gear environment. One is calculation of the rigid-body motion of the gearing with subsequent load-free tooth contact analysis to identify the contact line position. For example, this method is used in the calculation of local load distribution on bevel gears with BECAL [13] and is described in the following as bevel gear method (BGM). For this purpose, the relative position of the gears must first be determined in a program for deformation analysis of the entire gear. In addition, the iterative calculation of the contact line requires the most accurate possible mathematical representation of the flank geometry as a fitting surface using Non-Uniform Rational B-Splines (NURBS). Therefore, the geometric deviation of the fitting surface quality from the underlying data must be checked in each case. The deviation of the fitting surface for the example pinion from Section 5 can be derived from Figure 3. The maximum flank deviation is 0.06 µm at the starting edge of the tip relief. The geometric deviation can thus be neglected for further investigations.

Figure 3: Geometric deviation of the fitting surface from the underlying data.

Another way to consider the interaction with the gear environment is to analyze the gear in a mechanical model of the overall transmission. By mapping the gearing by springs distributed along the tooth width in a mechanical system, the twist and deflection of the gearing can be discretized for each width section and then considered in the tooth contact analysis as a change in the local flank gap. This method has become established, especially in cylindrical gear calculation. For example, it is used in the FVA-Workbench to calculate the load distribution on cylindrical gears [14] and is described in the following as cylindrical gear method (CGM). The formation of fitting surfaces is not necessary. The curvatures are calculated directly on the tooth contour, discontinuous transitions of tip reliefs are not included in the curvature.

4 Verification calculation (CGM/BGM)

In the following, both previously described variants of the local load distribution calculation (CGM/BGM) will be compared. Figure 4 shows the local pressure distribution on an example model with straight toothing with normal modulus mn = 2 mm, number of teeth z2/z1 = 40/25, and a tooth width of b = 26 mm. The calculation is performed using both the analytical deformation approach according to Weber/Banaschek [12] in the FVA-Workbench [14] (CGM) (Figure 4, top) and with a numerical two-dimensional BEM approach according to Schaefer [13] (BGM) (Figure 4, bottom). The shafts and bearings are modeled rigidly. Comparable compression patterns are shown, with both the single engagement and double engagement regions easily identifiable. The deviation of the maximum Hertzian pressure is only ΔσH,max′ < ca. 2%. Accordingly, both calculation models provide comparable results with respect to the local pressure distribution.

Figure 4: Pressure distribution comparison of analytical (top) and BEM (bottom) calculation models with rigid environment.

The influence of an elastic environment on the local pressure distribution is investigated in the following example. Thus, the shafts and bearings in the model are elastically constrained. The shafts are modeled using Timoshenko beam elements, and the bearing stiffnesses are calculated non-linearly. The deformation effect can be seen in Figure 5 (left). The torsion and lateral deformation are summed and plotted along the tooth width. Figure 5 (right) shows the resulting modification proposal with respect to the flank line deviation. It can be seen that the environmental interaction leads to an increase in the contact distance in the center of the tooth as well as skewing along the tooth width. However, the resulting modification proposal is only 1.1 µm.

Figure 5: Transverse deformation and torsion (left) and proposed modification (right).

In Figure 6, the influence of the elastic environment on the pressure distribution can be examined. In the analytical model (CGM) (Figure 6, top), a trough-shaped pressure distribution is formed. The torsion and lateral deflection of the gears leads to a slight increase in pressure at the tooth ends. In addition, the helical position leads to a lateral increase in pressure. In contrast, the calculation with the BEM model (BGM) (Figure 6, bottom) only shows the skewing of the contact and thus a slight lateral pressure increase. In both models, the maximum pressure is formed on one end face and is slightly increased compared to the previous model with a rigid toothing environment.

Figure 6: Pressure distribution comparison for analytical (top) and BEM (bottom) calculation models with elastic environment.

By extending the deformation model of the BGM with an elastic FE approach according to [8], the torsion and lateral deflection of the gearing can be included to the calculation approach. The meshing used for the calculation example can be seen in Figure 7. The shaft constraints are also considered here. As a result, this FEM calculation variant of the (BGM) also produces a trough-shaped pressure distribution. The maximum pressure increases slightly but is still located on the tooth end.

Figure 7: FE wheel body (blue) and FE shaft constraint (red), pressure distribution with consideration of the elastic wheel bodies (BGM).

Comparison of the results for the load distribution calculations with CGM and BGM on the cylindrical gear example shows the models based on the analytical deformation calculation according to Weber/Banaschek, the numerical BEM deformation calculation, and the FEM deformation calculation provide comparable results with regard to the local pressure distribution. There are only slight deviations in the calculation results with regard to the tooth torsion and wheel body deflection. Since this influence was not observed with the rigid connection, this can be explained by the different approach to the interaction with the gear environment. Since the FEM-based deformation calculation is also suitable for consideration of complex wheel body structures, the calculation method has an advantage in terms of geometric flexibility. However, due to the significant increase in the required computing time and computing power associated with FEM, the analytical method is preferable for efficient variation calculations.

5 Validation

In the following, the local method from FVA 411 [3] for bevel gears (BGM) is applied and validated based on the results of the FVA 284 V research project [10] on cylindrical gears (CGM).

In project FVA 284 V, the influence of load distribution on the pitting load capacity of case-hardened cylindrical gears was investigated. For this purpose, standardized flank load capacity calculation methods according to DIN3990 and ISO6336 were applied for spur and helical gears with different face modifications and compared with running tests.

Six different variants were investigated and compared, based on two main geometries for the straight- toothed variants GV and helical-toothed variants SV. The geometries are described in Table 2 and shown in 2D tooth mesh and in 3D in Figure 8.

Table 2: Basic macro geometry data of the gears.
Figure 8: Gear mesh and 3D model.

Table 3 describes the micro geometry of the variants in detail. For each main geometry, there is a reference variant REF, a variant with significant lead crowning BB, and a variant with helix angle modification SWK.

Table 3: Micro modifications of the variants.

The designations “short” and “long” refer to the beginning of the tip relief. With “short,” the tip relief is made between the end of the tooth contact point E on the line of action and half the distance to the upper point of single tooth contact point D, see Figure 8 gear mesh. With “long,” the tip relief is between the end of the tooth contact and the upper single point of tooth contact.

The gears are made of 18CrNiMo7-6 and are case-hardened. The gear quality is level 5 according to DIN 3962 [15] with an arithmetic mean roughness of RaH = 0.3 µm.

The experimental running tests were performed using an FZG back-to-back gear test rig [16]. The lubricant used was FVA reference oil no. 3 with an additive of 4% Anglamol 99 (i.e., FVA 3 A [17]) with a kinematic viscosity at ν40°C = 85-95 mm2/s and at ν100°C = 10-11 mm2/s. Injection lubrication was used with a lubricant temperature of υE = 60°C. To determine the fatigue strength against pitting, the test loads were selected according to the modified Probit method [18] at a speed of n1 = 3,000 min-1 at the pinion.

The derived endurance limits for the torque T for 99% survival probability, which represents a pitting safety factor of 1, are shown in Table 4.

Table 4: Pitting load capacity results comparison.

ISO 6336:2019 flank load capacity with endurance limit torque calculations have also been performed and are included in Table 4 for comparison purposes. For the calculations, an ideally stiff system environment is assumed, similar to the test bench conditions. The flank modifications are included in the calculation of the dynamic factor KV to a limited extent, to a significant degree in the calculation of the face load factor KHβ, and potentially in the auxiliary factor fZCa, depending on the selection. The modifications in the profile direction are virtually ignored in the calculation. The calculations were performed in the FVA-Workbench [14], whose calculation approach for the load distribution of the gear contact is based on the Weber/Banaschek [12] methodology.

The test results and the standard load capacity according to ISO 6336 provide the basis for comparison for the local load capacity calculation according to FVA 411, which is the focus of this article. Local load capacity calculations also were performed for the six variants and are included in Table 4.

Calculating the pitting safety factors according to ISO 6336 without taking the modifications into account tends to result in higher safety factors, which should be evaluated critically since there is an increased risk due to uncertain contact pressure distribution.

ISO 6336 with modifications considers the contact pressure distribution to a better degree via the face load factor KHβ. In the past, however, tests [10] have shown the calculation according to ISO 6336:2008 generates unreasonably high safety factors for helical gears. This was counteracted with the introduction of the auxiliary factor fZCa in ISO 6336 in 2019. In this example, the flanks have been modified in such a way that the contact pressure distribution is clearly uneven. Thus, the auxiliary factor for the unmodified case was adopted, fZCa = 1.2. Nevertheless, the safety factors are still slightly above 1.

The local safety factor according to FVA 411 I shows a good correlation with values close to 1, especially for the helical toothed variants, but these values are still somewhat exaggerated. Furthermore, it should be noted the local safety factors for GV BB and GV SWK are slightly too low, below 1, and are therefore too conservative. What this comparison does not show are the effects when the profile modifications differ significantly. In reality, this leads to different stresses, but these only have an effect in the local calculation.

The FVA 411 I on edges calculation represents an evaluation at the beginning of the tip relief. Here, numerical singularities, clearly excessive stresses, and lower pitting safeties occur in the contact calculation due to the edge. This is a fundamental issue in contact analyses: The stresses at edges and margins of the contact are exaggerated and often smoothed by the respective programs used. Therefore, evaluation at these points should be avoided or at least critically analyzed.

A comparison of the pitting damage test results for the six variants with the locally calculated pitting safety factors is shown in Figure 9 and Figure 10. In principle, the damage locations and wear marks on the surface match the calculated safety factors quite well, although the damage is in an advanced state and the starting point of the pitting cannot clearly be determined. In the case of the straight-toothed variant REF, the most damage is seen in the area of the lowest safety, with secondary damage in the upper area of the tooth flank. In the case of GV BB, not only does the damage location match, but the wear marks also align very well. In the case of the helical-toothed variants, the local calculation also corresponds with the test results. A certain degree of uncertainty is always necessary, since the test results are scattered, and the damage can develop differently during the course of the test.

Figure 9: Local pitting safety factors compared with test results GV variants.
Figure 10: Local pitting safety factors compared with test results SV variants.

6 Conclusion

In this study, a simulation method for the local pitting load capacity of bevel gears (BGM) [3] is applied to cylindrical gears (CGM). In this method, local calculation parameters, such as the local contact pressure, are considered. The method has been successfully applied to cylindrical gears and therefore local safety can be calculated.

The comparison of different methods for calculating deformation for tooth contact analysis and interaction with the environment (CGM/BGM) has shown, that the investigated analytical method, as well as the BEM and FEM provide similar results for pressure distribution.

In summary, it can be stated the local method FVA 411 and the modified ISO 6336 represents the test results [10] well. However, it must be critically noted that almost all calculation results ISO 6336, with the exception of the modified spur gear variant GV BB, tend to be on the uncertain side. For the results of FVA 411, this is especially noted for the helical toothing variants.

The local method has advantages compared to the global standard methods [1] [2], because the influences of the profile modifications are considered, no auxiliary factor fZCa is required and the damage-critical location in the tooth contact can be determined. This makes it possible to determine a suitable flank modification in order to specifically increase the critical local safety.

However, the load capacity calculated here at the transition edge of the tooth contour and the tooth relief must be questioned, since the calculation method is not suitable for this contact behavior. Further studies and research on this topic should therefore be considered desirable. 

Bibliography

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Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2023 at the AGMA Fall Technical Meeting. 23FTM18