New standardized calculation method of the tooth flank fracture load capacity of bevel and hypoid gears

Bevel and hypoid gears are widespread in automotive, industrial, marine and aeronautical applications for transmitting power between crossed axles. Future trends show that the demands on bevel and hypoid gears for higher power transmission and lower weight are continuously increasing. A major aspect in the design process is therefore the load carrying capacity regarding different failure modes. Beside typical fatigue failures like pitting and tooth root breakage, which are the results of cracks initiated at or just below the surface, there are also failures caused by cracks starting in greater material depth in the area of the active flank that can be observed on bevel and hypoid gears. These cracks typically propagate to the tooth root area of the unloaded flank and to the surface of the active flank. The failure mode known as tooth flank fracture occurs particularly frequently on large spiral bevel and hypoid gears because this gear type shows larger equivalent radii of curvature compared to spur and helical gears. As a result of the larger equivalent radius of curvature, the maximum shear stress occurs in a larger material depth, where the material of a case-hardened gear shows a decreased strength. Important parameters influencing the tooth flank fracture load capacity are geometry, operating conditions, material, and heat treatment of the gear set. Tooth flank fracture usually leads to the total breakdown of the gearbox and generally occurs suddenly and unexpectedly since the crack initiation and propagation takes place below the tooth surface and, therefore, cannot be identified within visual inspections.

This paper will give an overview of the subsurface failure mode known as tooth flank fracture on bevel and hypoid gears. Further, a newly developed standardized calculation method for determining the tooth flank fracture load capacity based on the geometry of virtual cylindrical gears according to the standard ISO 10300 (2014) will be explained in detail.

2 Introduction

Case-hardened cylindrical gears, as well as bevel gears, are highly loaded machine elements used for power transmission in a wide variety of applications. Future trends show that the demands on machine elements for higher power transmission and lower weight are continuously increasing. Therefore, a major aspect in the design process of gears is the load carrying capacity regarding different failure modes.

Beside typical fatigue failures such as pitting [27] and tooth root breakage [12], which are the results of cracks initiated at or just below the surface, case-hardened gears frequently show cracks, which start in a larger material depth. These cracks initiate in the area of the active flank below the surface and typically propagate in direction of the tooth root area of the unloaded flank and can progress to the failure mode known as tooth flank fracture. Tooth flank fracture occurs particularly frequently on spiral bevel and hypoid gears because this gear type shows larger radii of relative curvature compared to spur and helical gears [23]. As a result of the larger radii of relative curvature, the maximum shear stress occurs in a larger material depth where the material shows a decreased strength. Tooth flank fracture usually leads to the total breakdown of the gearbox and generally occurs suddenly and unexpectedly since the crack initiation and propagation takes place below the tooth surface and, therefore, cannot be identified within visual inspections. In Figure 1, a typical tooth flank failure at a bevel gear pinion flank is shown.

3 Failure Mode Tooth Flank Fracture

Tooth flank fracture failures are known from different industrial gear applications [8] as well as from specially designed test gears for experimental investigation [13, 25, 29].

Tooth flank fracture is a typical fatigue failure on different gear types with crack initiation below the flank surface due to shear stresses caused by the Hertzian contact pressure. The crack propagates in the direction of the active flank surface as well as in direction of the core area [20]. In Figure 2, an exemplary crack propagation at a cylindrical gear flank is shown. The shown crack propagation corresponds to the behavior of the failure mode occurring on bevel and hypoid gears.

According to FZG/Annast [7] the crack initiation on bevel and hypoid gears is often at the center of the face width and half the height of the tooth. The crack sometimes starts at non-metallic inclusions, which cause an increase of stresses. From there, the crack propagates toward the tooth root fillet of the unloaded flank. In direction of the face width, the crack propagates almost parallel or slightly tilted to the tip of the tooth. [26]

The shape of the emerging breakouts is characteristic for tooth flank fracture and independent of the gear size. They show typical rest lines for fatigue fractures and small areas of final rupture. An exemplary breakout is shown in Figure 1 on the right.

FZG/Hertter [16] detected that a secondary crack may occur in addition to the described primary crack. These result from local changes of the tooth stiffness caused by the internal separation of the material by the primary crack. Starting from the surface, the cracks grow approximately parallel to the tooth tip and with an increasing number of load cycles into the interior of the tooth, where they may hit the primary crack and stop. Especially for gears with a large tooth width, tertiary cracks may occur.

4 Calculation Methods

Tooth flank fracture occurs when the shear stress exceeds the material strength in a specific material depth. The shear stresses are caused by the Hertzian stresses [9] on the tooth flank. Therefore, the Hertzian stress is an important influence parameter in the assessment of the risk of tooth flank fracture. Two different types of calculation models for calculating the risk of tooth flank fracture exist, which are applicable to cylindrical and bevel gears.

Material-physically based calculation approach

The material-physically based calculation approach for assessing the risk of tooth flank fracture was developed by FZG/Oster [17] and FZG/Hertter [16] for cylindrical gears and was adapted by FZG/Wirth

[17] for the application on bevel and hypoid gears. The calculation approach is based on the comparison of a local occurring shear stress in an area at or below the surface of the flank and the local material strength at the considered area. A detailed description of the material-physically based calculation approach can be found in [10]. According to FZG/Tobie [25] the material-physically based calculation approach can take into account the following influences on the rolling/sliding contact of two mating gears:

• Normal contact force due to applied torque.
• Modified pressure distribution due to lubricated contact, described by EHL-theory.
• Shear and bending load due to the horizontal component of the normal force.
• Tangential load caused by friction force due to sliding.
• Thermal load due to friction force.
• Stress peaks due to rough surface.
• Residual stresses due to mechanical processes and due to heat treatment.

The local shear stress calculated by use of the material-physically model is determined based on the shear stress intensity hypothesis. For application of the material-physically model, a large number of detailed input values are necessary.

This fact, as well as the necessary integrations and iterations within the calculation approach, leads to a high effort in the application [14]. Therefore, a more practical calculation approach for assessing the risk of tooth flank fracture of cylindrical gears was developed by FZG/Witzig [29].

Practical approach

The practical approach for calculating the risk of tooth flank fracture is based on the material-physically based method described previously and has as a calculation objective the local material exposure, which is defined as the quotient of the local equivalent stress state and the local material shear strength [1]. The aim of this approach is to enable the gear designer to assess the risk of tooth flank fracture on cylindrical gears in an early stage of gear design. Therefore, the calculation method is based on reliable and relatively few input values. FZG/Hein [14] summarizes the practical calculation approach, which also forms the basis of the ISO Technical Specification ISO/TS 6336-4 [1]. In Figure 3, the most important influence factors on the local material exposure are shown. In the following, the corresponding equations are listed and explained [1, 14, 9].

Within the development of the practical calculation approach according to FZG/Witzig [29] the following approximations compared to the material-physically based model have been made [15]:

• Calculation method in closed form solution.
• No consideration of tensile residual stresses.
• No consideration of shear stresses induced by friction, EHD contact, or surface asperities of thermal load.
• Valid for case-carburized gears.

The decisive parameter for assessing the risk of tooth flank fracture by use of the calculation method according to FZG/Witzig [29] is the local material exposure A_FF,Y, which is the quotient of the local equivalent stress state τ_eff,Y and the local material strength τ_per,Y in the material depth y and at the  contact point Y on the path of contact. The material exposure for each considered volume element can be calculated using Equation 1.

The local equivalent stress state τ_eff,Y(y) according to FZG/Witzig [29] considers the local equivalent stress state without consideration of residual stresses τ_eff,L,Y(y), the influence of residual stresses on the local equivalent stress state ∆τ_eff,L,RS,Y(y) and the quasi-stationary residual stress state τ_eff,RS(y).

The local material strength τ_per(y) is a function of the local hardness HV(y) and the material, which is defined by the hardness conversion factor K_τ,per and the material factor K_material.

The hardness depth profile HV(y) is a decisive influence parameter regarding tooth flank fracture. Therefore, it is recommended to use measured values, if they are available (ISO/TS 6336-4 [6], Method B). In the design stage of a gear set, measured data are usually not available. Therefore, method C1 or C2 (ISO/TS 6336-4 [6]) has to be applied if possible. If no experience regarding the hardness depth profile HV(y) is available at all, it is recommended to apply different methods for estimating the hardness depth profile (e.g. [18, 19, 23, 24]) and compare the results. To be on the safe side, the most conservative method should be used for further calculations. [14]

The local equivalent stress state without consideration of residual stress τ_eff,L,Y(y) according to FZG/Witzig [29] is calculated as a function of the radius of relative curvature ρ_rel,Y, the Hertzian stress p_dyn,Y, the reduced modulus of elasticity E_r and the material depth y and does not consider the influence of residual stresses and shear stresses induced by friction, EHD contact, surface asperities, or thermal load [14].

Nevertheless, residual stresses in the carburized layer can influence the total stress state. Therefore, FZG/Witzig [29] developed formulae for calculating the influence of residual stresses on the local equivalent stress state ∆τ_eff,L,RS,Y(y), which is influenced by the residual stress depth profile σ_RS(y), the case hardening depth CHD at 550 HV, and the maximum value of the residual stresses σ_RS,max. These equations are based on the approach of FZG/Oster [20] and FZG/Hertter [16]. By use of the adjustment factors K₁, K₂, K_pH,σRS,max and K_CHD, which are given in Equations 6 to 9, a closed form description of the influence of the residual stresses on the local equivalent stress state ∆τ_eff,L,RS,Y(y)v is possible.

The quasi-stationary residual stress state τ_eff,RS can be calculated according to Equation 10.

As shown in the equations, the risk of tooth flank fracture can be assessed by comparing the local occurring stress with the local material strength. From experimental investigation on case carburized gears [22], it is known that a maximum material exposure A_FF,max ≤ 0.8 may lead to tooth flank fracture for gear materials characterized by typical quality and cleanness. FZG/Boiadjiev [11] has shown that this critical value can be also applied for bevel and hypoid gears. FZG/Witzig [29] gives the following limitations concerning the practical calculation approach, which were also confirmed for bevel and hypoid gears in [11]:

500 N/mm² ≤ p_H ≤ 3,000 N/mm².

mm ≤ ρ_rel ≤ 150 mm.

0.3 mm ≤ CHD ≤ 4.5 mm.

The practical calculation approach for assessing the risk of tooth flank fracture is validated for material depths y ≥ 1 · b_H and therefore not suitable for evaluation of the tooth flank fracture risk at or just below the surface [29].

The aim of this paper is to present a practical calculation approach for assessing the risk of tooth flank fracture on bevel and hypoid gears. Therefore, the presented equations according to ISO/TS 6636-4 [1] should be used. As FZG/Hein [14] has shown, the basic input parameters for calculating the material exposure according to FZG/Witzig [29] contain information about external loads, material, and tooth-flank geometry. As external loads as well as parameters regarding the material of the gearing are largely similar for cylindrical and bevel or rather hypoid gears, the modification of the calculation approach affects the input parameters regarding the geometry and load, as shown in Figure 4.

The geometry data needed for assessing the risk of tooth flank fracture on bevel and hypoid gears can be determined by use of a local calculation method (loaded tooth contact analysis, Method A) as well as by use of a standardized method (e.g. ISO 10300-series [1, 2, 3], Method B). Within the local calculation approach, the values for local Hertzian stress can be directly determined by a loaded tooth contact analysis. Figure 5 shows an exemplary contact stress distribution of a bevel gear flank used for experimental investigations at FZG calculated by use of the program BECAL [21] and evaluated by an internal software. The calculated stresses can serve as input data for calculating the local material exposure A_FF,Y.

In comparison to a loaded tooth contact analysis, the standardized calculation approach uses the geometry of a virtual cylindrical gear derived from the real bevel or hypoid gear geometry as input data. This procedure allows assessing the tooth flank fracture risk in an early stage of gear design at a time when the final microgeometry of the gear set is not yet available. In the following, the simplified, practical calculation approach for determining the necessary geometry and load input data for calculating the tooth flank fracture risk of bevel and hypoid gears is explained.

5 New practical calculation method for assessing the risk of tooth flank fracture on bevel and hypoid gears

5.1 General

The described procedure for the calculation of the tooth flank fracture load capacity of bevel and hypoid gears is based on theoretical investigations [11] on different test gears and gears from industrial applications. The calculation method was developed within the research project FVA 556 III [9] by one of the co-authors. The presented calculation method is the basis for a new ISO technical specification for calculating the tooth flank fracture load capacity of bevel and hypoid gears (ISO/DTS 10300-4 [4]) similar to ISO/TS 6336-4 [6] for spur and helical gears. A comparison of the results of the practical calculation method and calculation results based on a loaded tooth contact analysis assesses the quality of the practical method within Section 4.

The simplified calculation approach allows an estimation of the tooth flank fracture risk at a very early stage of designing a bevel or hypoid gear. The calculation method is based on the standard ISO 10300- series [1, 2, 3] in terms of geometry, stresses, and sliding velocities.

5.2 Virtual Cylindrical Gear

The calculation method is based on a virtual cylindrical gear geometry [17], which is structured according to the schematic construction in Figure 6 according to FZG/Wirth [28]. Detailed information regarding the representation of a hypoid gear as a virtual cylindrical gear are available in Annex A of ISO 10300-1 [1].

The virtual cylindrical gear is derived from the reference cone of the bevel gear or rather the hypoid gear. The transverse path of contact between the pinion and wheel of the virtual cylindrical gear is divided in a number of section (e.g. n = 10) to determine the local safety factors regarding tooth flank fracture for n+1 points of contact. To locate the contact points, the coordinate g_Y is established, which originates in the pitch point C (= design point P of the bevel gear set). Toward the pinion tip g_Y is defined as positive; toward the pinion root it is defined as negative. Figure 7 shows an exemplary path of contact including the coordinate g_Y.

In the boundary points A and E on the transverse path of contact, g_Y is determined as follows [5]:

The length of transverse path of contact can be subdivided in a number of sections i. For a contact point Y on the transverse path of contact, the corresponding coordinate g_Y(Y) can be calculated as follows [5]:

In a next step, the length of contact lines l_b,Y can be calculated by use of Equation 14. Figure 8 shows the general definition of length of contact lines l_b,Y according to [5].

The theoretical length of contact line at contact point l_b0,Y can be calculated by use of the following equation:

The correction factor C_lb,Y can be determined with the following equation [5]:

The local equivalent radius of curvature ρ_rel,Y plays a decisive role within the formation of tooth flank fracture and can be calculated for each considered contact point with following equations [5]:

The conversion of the maximum line load at point M into the contact point Y is calculated by use of the local face load factor K_Hβ,Y according to Equations 19 to 21 [5].

The modified Hertzian contact stress at contact point Y_pdyn,Y,mod plays a decisive role within the assessment of the tooth flank fracture risk and can be calculated based on the nominal Hertzian contact stress p_H,Y,B. The nominal Hertzian contact stress p_H,Y,B is calculated  using the described load sharing and curvature factors according to Equation 23. Within the determination of the modified Hertzian contact stress p_dyn,Y,mod, the profile crowning of the gear set is taken into account by use of the exponent e according to ISO 10300-2 [2].

6 Comparison of the Results of the Practical Method B and A Local Calculation Method A Regarding the Risk of Tooth Flank Fracture

6.1 Procedure

To evaluate the presented standardized calculation approach, a comparison of calculation results to the results of a local calculation method was conducted as shown in Figure 10.

Several systematic investigations [15, 29, 25] have demonstrated that the following parameter shows a remarkable influence on the load carrying capacity regarding tooth flank fracture on gears:

Relative radius of curvature ρ_rel,Y.

Hertzian stress p_dyn,Y,mod.

Case-hardening depth CHD.

Therefore, for validation of the presented calculation approach, the relative radius of curvature ρ_rel,Y and the Hertzian stress p_dyn,Y,mod determined according to the calculation approach presented in Section 3 (Method B) are compared to values derived from a local tooth contact analysis based on the software BECAL (Method A) [21]. Because of the independence of the case-hardening depth CHD and gear geometry, the case hardening depth will not be taken into account within this comparison. The influence of the hardness depth profile on the calculated risk of tooth flank fracture has already been investigated by FZG/Hein [15].

Within the evaluation, three different gear geometries were examined. The basic geometry data of the gear sets are shown in Table 1.

6.2 Comparison of Radius of Relative Curvature ρ_rel,Y

The relative radius of curvature ρ_rel,Y is calculated using the approach according to ISO 10300-1 [1]. The calculation uses the real geometry of the bevel gear with the exception of the inclination of the contact line, which is determined using a virtual cylindrical gear as described in ISO 10300-1 [1].

Exemplary values of the local radius of relative curvature over the path of contact are shown in Figure 11. The dashed lines in Figure 11 represent the values determined within a tooth contact analysis using the software BECAL [21]. The solid lines show the values calculated using the standardized approach according to ISO 10300 [1], which is based on the approach of Shtipelman [22].

The results of the standardized calculation approach (Method B) correspond well to the results of the loaded tooth contact analysis (Method A). The small deviations can be explained by the fact that the microgeometry of the bevel and hypoid gears cannot be represented in detail by the virtual cylindrical gear geometry. However, the presented standardized calculation approach aims at determining the risk of tooth flank fracture in an early design stage of the gear set without using a complex tooth contact analysis. From this point of view, the shown accuracy is sufficient for a reliable prediction of the tooth flank fracture risk of bevel and hypoid gears in an early design stage.

6.3 Comparison of Hertzian Stresses p_dyn,Y,mod

The Hertzian contact stress p_dyn,Y,mod represents a significant factor on the load carrying capacity of gears regarding tooth flank fracture. Therefore, it was investigated to what extent the standard calculation approach is able to map the real occurring stresses. In Figure 12, the results of the described comparison are shown. The dashed lines show the values determined within a loaded tooth contact analysis using the software BECAL [21]. The solid lines show the values for Hertzian stresses according to the standardized calculation approach.

All in all, the Hertzian stresses calculated by Method A and Method B respectively are in good accordance. However, the determination of the stresses again shows that the geometry of the test gears can only be represented roughly, but sufficiently accurate, by use of a virtual cylindrical gear geometry. In these cases, the standardized calculation method shows slightly higher stresses than the local calculation method. In summary, it can by stated that the assessment of the tooth flank fracture risk is conservative if method B is used for the herein investigated examples. For a more accurate consideration of the real flank geometry, the described local calculation approach, Method A, should be used.

6.4 Comparison of Material Exposure A_FF,Y

As described, the standardized calculation method, Method B, is carried out for the virtual cylindrical gear geometry. The virtual cylindrical gear geometry is calculated in the mean section of the bevel or hypoid gear. In Figure 13, the calculation points along the path of contact are illustrated as dots for the standardized calculation method. Unlike the standardized calculation (Method B) the local calculation (Method A) considers the whole flank of the gear within the determination of the material exposure A_FF,Y by use of tooth sections, which are distributed over the whole flank. The tooth sections are shown as dashed lines in Figure 13. For comparing the standardized (Method B) and the local calculation method (Method A), material exposures A_FF,Y for three different tooth sections (Method A) have been compared to the material exposure A_FF,Y at midsection (Method B).

• Tooth section 11.
• Tooth section 12.
• Tooth section 13.

Subsequently, the calculated material exposures at the three tooth sections are compared to the material exposure determined according to the standardized calculation approach at the midface of the virtual cylindrical gear. Figure 14 shows exemplary comparisons of values for material exposure for all three tooth sections for the test gear set G31,75. The black bars reflect the material exposure derived from the virtual cylindrical gear geometry; the values for material exposure based on a loaded tooth contact analysis are shown as grey bars. The material exposure at tooth section 12 shows the best correlation with the values determined by use of the standardized calculation approach.

Figure 14 shows the material exposure calculated by method A strongly depends on the considered tooth section within the calculation. Therefore, the accuracy of the practical calculation approach (Method B) is limited in comparison to the local method (Method A).

Nevertheless, to ensure the area of the midface is reliable regarding the occurrence of the tooth flank fracture on bevel and hypoid gears, different tooth sections near the toe and heel have been investigated regarding the occurring material exposure. Figure 14 shows a comparison of tooth sections near the toe and heel for an exemplary gear out of a rear axle drive of a passenger car. The black bars reflect the material exposure derived from the virtual cylindrical gear geometry. The values for material exposure based on a loaded tooth contact analysis are shown as gray bars.

As shown in Figure 15, the sections near the toe and heel show smaller material exposures in comparison to the material exposures determined according to the standardized calculation method. This result proves the midsection of a bevel gear tooth is suitable to assess the risk of tooth flank fracture sufficiently within a practical and efficient calculation for typical bevel gear designs with a contact pattern under load in the middle of the tooth flank. A more accurate and detailed information about the risk of tooth flank fracture of bevel or hypoid gears can be reached using the described local calculation (Method A) based on a loaded tooth contact analysis as soon as the microgeometry of the gearing is available within the design process.

7 Conclusion

The presented calculation method provides a new approach for the determination of the tooth flank fracture risk for bevel and hypoid gears. This method is based on the calculation method according to ISO 6336-4 [6] regarding the calculation of the material exposures and uses geometry data based on the virtual cylindrical gear geometry according to ISO 10300-1 [1]. For a more accurate determination of the tooth flank fracture risk, a calculation based on a loaded tooth contact analysis can be performed for determining the local values regarding Hertzian stress and geometry as input data.

The presented investigations show the real flank geometry of bevel and hypoid gears can be mapped by the virtual cylindrical gear geometry according to ISO 10300-1 [1] in a sufficiently accurate way. Furthermore, the studies have shown the material exposure strongly depends on the considered tooth section within the calculation. Therefore, the accuracy of the practical calculation approach (Method B) is limited in comparison to the local calculation approach (Method A).

Nevertheless, the studies proved the midsection of a bevel gear tooth is suitable to assess the risk of the tooth flank fracture sufficiently, and, therefore, the practical calculation approach is able to predict the tooth flank fracture risk of bevel and hypoid gears in an accurate way for typical gear designs. For a detailed prediction of the tooth flank fracture load carrying capacity, the application of the local calculation approach (Method A) is recommended as soon as the micro geometry is defined within the design process. It is intended to publish the described calculation method as ISO/TS 10300-4 [4] in the near future to gain more practical experience with the results obtained by this approach. 

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