Transmissions are usually loaded by variable external loads under real operating conditions. The decisive load for a gearbox is in most cases the applied torque. Commonly used allowable stress numbers *σ*_{Hlim}/_{Flim} (ISO) or *σ*_{HP}/_{FP} (AGMA) for calculating the load carrying capacity of cylindrical, bevel, and hypoid gears are usually derived from single stage tests carried out on pulsators or back-to-back test rigs. Variable loads can be considered in the calculation of the load carrying capacity by using application factors, overload factors, or more complex standards such as ISO 6336-6, which was recently revised. In case of variable loads, the calculation of the load carrying capacity of gears is quite different to bearings.

According to ISO 6336-6, a safety factor is determined for gears and, according to ISO 281, a service life is determined for bearings, respectively. Whereas all of these calculation methods only consider a global safety or lifetime, continuously progressing failures such as micropitting or wear can — especially on bevel and hypoid gears — also lead to locally varying stresses even if only a constant external load is applied.

This article is intended to give a brief overview of currently applied methods to consider variable loads in the design process of cylindrical as well as bevel and hypoid gears. Therefore, the scope of application of these methods is shown and critically analyzed for the damage mechanisms pitting, tooth root breakage, and tooth flank fracture. The changes made in the revised version of ISO 6336-6 are shown in detail. Furthermore, the influence of locally changing stresses on the pitting load carrying capacity is explained on bevel and hypoid gears. A method to assess such influence is shown for constant external loads.

### 2 Introduction

Transmissions are usually loaded by variable external loads under real operating conditions. The decisive load for a gearbox is, in most cases, the applied torque. Whereas the calculation of the load carrying capacity for mobile applications is often based on reliability, lifetime, or damage sums, such calculations for industrial gearboxes use standardized safety factors according to ISO 6336 [31], [32], [33], or AGMA 2001/2101 [1]. Basically, all used calculation methods are based on a comparison of occurring loads/stresses and permissible loads/stresses. Commonly used allowable stress numbers are, for example, *σ*_{Hlim}/_{Flim} (ISO) or *σ*_{HP}/_{FP} (AGMA), for calculating the load carrying capacity of cylindrical, bevel, and hypoid gears. If a reliability, lifetime, or damage sum is calculated, S/N-curves and load spectra are compared with special regard to the corresponding statistical distributions. Allowable stress numbers and S/N-curves are usually derived from single stage tests carried out on pulsators or back-to-back test rigs according to Wöhler [12], [14].

If a gearbox is designed for a limited lifetime and suffers variable loads, the calculation of a service life or damage sum is preferred compared to a standardized calculation based on safety factors. This helps to ensure the best possible power-to-weight ratio [13], [19]. Depending on the field of application, a gearbox design for limited life can save significant weight (up to 50%) and space (up to 30%). As the requirements regarding reliability increase more and more, the demand for reliable methods for fatigue life analysis also increases.

Variable loads can be considered in the calculation of the load carrying capacity by using application factors, overload factors, or more complex standards such as ISO 6336-6, which has been revised recently. In case of variable loads, the calculation of the load carrying capacity of gears is quite different to bearings. According to ISO 6336-6, a safety factor is determined for gears and, according to ISO 281, a service life is determined for bearings, respectively. Whereas all of these calculation methods only consider a global safety or lifetime, continuously progressing failures such as micropitting or wear can — especially on bevel and hypoid gears — also lead to locally varying stresses even if only a constant external load is applied.

This article is intended to give a brief overview of currently applied methods to consider variable loads in the design process of cylindrical as well as bevel and hypoid gears. Therefore, the scope of application of these methods is shown and critically analyzed for the damage mechanism’s pitting, tooth root breakage, and tooth flank fracture. Furthermore, the influence of locally changing stresses on the pitting load-carrying capacity is explained on cylindrical as well as bevel and hypoid gears.

### 3 Basic principles

#### 3.1 Damage mechanisms for gears

Transmissions are usually designed for limited life. Depending on the considered gear and damage mechanism, the applied load spectra may contain significant parts above the endurance limit of the gears in contact. The lifetime of vehicle or industrial transmission’s gearings is mainly limited by the following damage mechanisms:

- Tooth root breakage.
- Pitting.
- Tooth flank fracture.
- Scuffing.

These damage mechanisms are described shortly in the following.

#### 3.1.1 Tooth root breakage

Tooth root breakage is a typical fatigue damage on gears. The crack initiation is in the tooth root area, usually located at the 30° tangent (see Figure 1). A decisive criterion for this damage mechanism is the mechanical tooth root bending stress resulting from the attacking tangential force and the tooth geometry. In general, forced rupture and a fatigue-based tooth root breakage are differentiated. Forced tooth root rupture is usually caused by extreme overloads like mechanical blockings. A fatigue-based tooth root breakage is the common failure mechanism for vehicle transmissions as forced rupture can be avoided by a reasonable design. A calculation of the tooth bending strength for spur and helical gears is available within ISO 6336-3 [33]. For single load levels, a safety factor can be determined from the tooth root stress *σ*_{F} and the permissible bending stress *σ*_{FP}.

In any case, tooth root breakage should be avoided as it usually leads to a total breakdown of the whole drivetrain function. For fatigue-based tooth root breakage, a S/N-curve can be used to describe the fatigue life behavior. It can either be obtained by extensive testing or from literature/standards (e.g. ISO 6336-3 [33] and 6336-5 [27]) depending on the material and heat treatment.

#### 3.1.2 Pitting

Pitting is a typical surface fatigue damage on gears which usually appears at the earliest after 500,000 cycles [37]. In Figure 2, a typical shell-shaped outbreak on the flank surface of a case-carburized helical gearing is shown. The crack initiation is at or near the flank surface mainly in the dedendum flank area of the driving pinion (area of negative specific sliding). The pitting resistance or a safety factor respectively can be calculated according to ISO 6336-2 [32] in a similar way as for tooth root breakage. The decisive parameter for calculation is the contact stress *σ*_{H} in the area of negative specific sliding around the inner point of single tooth contact (B) at the pinion.

Pitting damage typically shows a progressive development and may lead to tooth breakages resulting in a total breakdown of the gearbox. An S/N-curve can be obtained by extensive testing or from literature/standards (e.g. ISO 6336-2 [32] and 6336-5 [27]) depending on the material and heat treatment.

#### 3.1.3 Tooth flank fracture

Tooth flank fracture is a severe fatigue failure on gears with crack initiation below the flank surface [4], [50]. Tooth flank fracture failures are mainly reported from different industrial gear applications, such as spur or helical gears for car, truck, and bus transmissions; wind turbines or turbo transmissions; as well as from bevel gears for water turbines (e.g., [3]). Failures have also occurred from specially designed test gears for gear running tests [6],[49],[55] and typically occur on the driven partner of case-carburized gears but have also been observed on nitrided and induction-hardened gears. Damages may also occur in passenger car’s transmissions due to the high loaded gear flanks.

Tooth flank fracture is characterized by a primary fatigue crack in the region of the active contact area, initiated below the surface due to shear stresses caused by the flank contact. This primary crack is often located at approximately half the height of the tooth. The crack starter is in the material depth, typically in the area of the case-core interface, and is often — but not always — associated with a small non-metallic inclusion. The primary crack grows in both directions, toward the surface, as well as in the material depth in an angle of approximately 40° to 50° to the tooth flank surface. Subsequent cracks growing from the surface may occur (see Figure 3). The final breakage is due to forced rupture. The fractured surface shows typical fatigue characteristics [16], [17], [50]. A calculation of the tooth flank fracture load carrying capacity is possible according to ISO/DTS 6336-4 [16], [17], [30].

Observed tooth flank fractures usually occurred after more than 10^{7} load cycles, which points out the fatigue character of this failure mechanism and is a typical differentiating factor to tooth-root breakage, which usually occurs after <10^{6} load cycles (but can also occur at higher running times in some cases [5]). Standardized S/N-curves for tooth flank fracture are not known yet but may be obtained by extensive testing.

#### 3.1.4 Scuffing

Scuffing is not fatigue damage and is not capable of a fatigue life calculation. Scuffing is an instantaneous welding of the mating surfaces due to a collapse of the separating lubricant. Caused by the relative movement, the tooth flanks get disconnected from each other again instantaneously. Scuffing can occur under critical operating conditions (high circumferential speed, high load, inadequate lubricant temperature) [41] or as a consequence of a momentary overload. The failures occur in the corresponding flank areas of pinion and wheel. Scuffing marks show a pore-like structure and lead to profile form changes with increased dynamics/noise. Additionally, friction martensite of high hardness is formed on the surface, and a temperature influenced zone of low strength can be detected below the surface. A calculation of the scuffing load-carrying capacity can be performed according to ISO/TS 6336-20 and [21], [28], [29].

#### 3.2 Fatigue life calculation based on a local damage accumulation for cylindrical gears

The calculation method presented in the following is based on a local damage accumulation regarding the given load spectrum first introduced by FZG/Ziegler [56] and extended by FZG/Hein [13], [18], [20].

Compared to ISO 6336-6 [34], this approach gives more detailed information about the service life under variable loads. Not only are the analyzed damage mechanisms regarded individually as shown in ISO 6336-6 [34] but also by means of a system analysis, e.g. taking into account more damaged mechanisms in one calculation as well as influences from deformations of the shaft-bearing system. In ISO 6336-6 [34], the damage accumulation is performed globally for the whole flank. Therefore, the calculated service lives can be underrated because effects of an uneven load distribution are not considered completely. The presented method is capable of calculating local damage sums for different points on the tooth flank and in the tooth-root area. Therefore, influences from a varying load distribution can be considered in detail.

In order to compare different damage mechanisms, it is necessary to use one defined parameter to describe the load level. As the gearbox input torque relates to the load and stress parameters for the regarded damage mechanisms, it is a proper criterion for a unified load level description. In Figure 5a, stress/load cycle (S/N) curve (blue, dotted) for pitting is shown compared to the corresponding torque/load cycle (T/N) curve on the right (red, solid). Note the correlation between stress and torque might be nonlinear depending on the regarded damage mechanism. Additionally, a possible basic load spectrum for one application case is shown. This forms the basis for the following considerations.

The main advantage of this way of representing is that the load spectrum is equal for each considered damage mechanism. Only the “Wöhler” curve data need to be converted from stress to torque. Thus, a T/N curve for each fatigue damage mechanism can be described in relation to the load spectrum that should contain all occurring loads also including torque peaks. For example, Figure 6 shows a load spectrum and the corresponding T/N curves for tooth root breakage, pitting, and tooth flank fracture.

Hence, a damage accumulation can be performed for each failure mechanism in order to identify the specific cause of damage. As the choice of a suitable damage accumulation hypothesis (DAH) for each failure is a mandatory requirement, recommendations on that can be found in [20].

The local damage accumulation on cylindrical gears has been roughly described. More detailed descriptions can be found in the given literature. Chapter 6 shows parts of this method applied on bevel gears and extended to consider the influence of locally changing stresses on the service life of bevel gears.

### 4 Common methods for calculating the service life of gears

When designing gears under variable loads, two different approaches can be distinguished. Both are treated separately in the following:

- Optimized application-based design.
- Calculation of safety factors according to standards such as ISO 6336 or AGMA 2001.

An optimized application-based design is often used for vehicle transmissions, where maximizing the power-to-weight ratio is one of the main goals. Thus, it is necessary to consider all relevant damage mechanisms simultaneously. This approach requires extensive knowledge regarding relevant loads (load spectrum, torque-time-traces), component strength (S/N-curves), and damage behavior under variable loads (relevant damage accumulation hypothesis (DAH) and permissible damage sums) for the considered material. This approach was presented and successfully applied for the first time by Renius in 1976 [42], [43], [44].

The standardized design of gears under variable loads according to ISO 6336 [34] or AGMA 2101 [1] is often used for industrial gearboxes or wind turbines, which have to fulfill certain classification requirements [2]. This method is also preferred when only a little experience is available or a rough estimation of the service life is sufficient. By using standardized S/N-curves and simplified load assumptions, for example, by using application or overload factors, a safety factor can be calculated for both pitting and tooth-root breakage.

#### 4.1 Calculation of safety factors by using the application factor K_{A}

A calculation of safety factors according to ISO 6336-2/3 [32], [33] by using application factors is state-of-the-art for industrial gearboxes under variable loads. The applied nominal torque T_{N} is adjusted by multiplication with the application factor K_{A} to represent the underlying load spectrum. K_{A} can be interpreted in two ways:

**• K**_{A}** as shock or overload factor: **If the application factor is interpreted as shock or overload factor, K* _{A}* is the ratio of some kind of maximum-occurring load to the nominal load. Guideline values can be found in ISO 6336-1 [31]. The overload factor K

*used in AGMA 2101 [1] is also defined in such a way. According to the authors’ experience, this approach should be mainly used for gearbox designs for unlimited life.*

_{O}**• K**_{A}** as operational factor: **If K* _{A}* is considered as operational factor, it is intended to represent the load spectrum as damage equivalent single stage load F

*∙K*

_{t}*. K*

_{A}*is determined for every considered damage mechanism separately [31] according to ISO 6336-6 [34] based on a decisive design load spectrum. According to the new revision of ISO 6336 series, application factors for pitting damages are designated as K*

_{A}*and for bending damages as K*

_{HA}*followed by the used method for determination as additional index (for example K*

_{HF}*for an application factor for pitting damages based on method A acc. ISO 6336-1). According to method A, also application factors determined with more complex methods such as [7], [8] can be used.*

_{HA-A}Using application or overload factors in combination with well-known standardized calculation methods such as AGMA 2101 or ISO 6336, is state-of-the-art for many fields of application. Nevertheless, it should not be forgotten that this approach reduces a complex problem to a single factor and is only valid for a single damage mechanism.

#### 4.2 Calculation of service life according ISO 6336-6

ISO 6336-6 [34] describes the calculation of service life for cylindrical gears and is also referred in some AGMA standards such as AGMA 6006 [2]. In summary, this method is the application of the linear DAH according to Palmgren and Miner with a permissible damage sum of D_{per} = 1.0 to cylindrical gears. In the 2019 revision of this standard, the possibility to use other DAHs (for example with continuously decreasing endurance limit) and also other permissible damage sums has been implemented. In addition to the calculation of a damage sum, the calculation of a safety factor is described in ISO 6336-6 [34]. The calculation is done iteratively by multiplying the load spectrum with the safety factor determined in each step until the calculated damage sum for the load spectrum is 0.99 < D < 1.0.

For practical application, it is important to mention that load dependent factors such as K_{H}_{β} or K_{v} have to be determined separately for every single load level of the basic load spectrum. When calculating a safety factor, calculated load levels during the iteration process may also be above the static strength. Safety factors calculated according to ISO 6336-6 [34] should be treated carefully as they are not fully comparable to safety factors calculated according to parts 2 (pitting) or 3 (tooth root breakage).

### 5 Optimized application-based design of gears under variable loads

The optimized application-based design of gears under variable loads can be structured in the following three steps which are described in the following:

- Evaluation of load spectra.
- Evaluation and definition of S/N-curves.
- Choice of adequate DAH and permissible damage sums.

Basically, the application of this approach needs detailed knowledge which can be gained from systematic experimental investigations or long-time experiences with comparable gearboxes of the same field of application.

### 5.1 Evaluation of load spectra

Basically, application-specific load spectra can be determined by measurement or simulation of torque- time-curves. A meaningful load spectrum is highly relevant to design gearboxes with good power-to- weight ratio. The determined load spectra often show a large scattering, which is often based on different customer usage behavior [15], [51] or different environmental conditions [9], [23], [35], [36]. Hence, it is necessary to always link the design load spectrum to a specified probability of failure. For example, it is common practice to use a “99% customer” for designing vehicle transmissions. For some applications such as industrial gearboxes, racing applications, marine transmissions, or test rigs, relevant load spectra can be specified without significant scattering.

Foulard [11] shows a method for an online determination of load spectra for vehicle transmissions. Based on these load spectra, damage sums can be calculated online in order to allow predictive maintenance of the gearbox. The possibility to use machine learning and big data analysis for the service life prediction of gears is a topic of interest of the current research in the area of fatigue life analysis [52].

#### 5.2 Evaluation and definition of S/N-curves

The determination of S/N-curves is usually performed specific for each damage mechanism and each component in “Wöhler”-tests. For the relevant damage mechanisms pitting and tooth-root breakage, quasi-standardized approaches are available to gain S/N-curves with characteristic endurance strength number, slope, number of load cycles at the transition point from limited life area to infinite life, and static-strength number. These approaches are based on experimental investigations on standardized test gears. Especially for the calculation of the service life, the area of limited life has to be investigated in detail [38]. Decisive S/N-curves used for the calculation of damage sums and service life always have to be linked to a probability of failure. Based on experimental investigations, usually S/N-curves for a probability of failure of PA = 50% can be determined reliably. The conversion to other probabilities of failure (commonly used are 1% or 10%) can be done for example with conversion factors shown in [14] or [37].

#### 5.3 Choice of adequate DAH and permissible damage sums

If decisive load spectrum and S/N-curve have been determined, they have to be linked by a damage accumulation hypothesis (DAH) in order to calculate damage sums and consequently the service life. The choice of an appropriate DAH has to be based on experience for the considered field of application. For typical applications with loads in the area of limited life, the application of the DAH Miner-Haibach [12] has shown reasonable results in several research works on the service life on gears [10], [13], [45], [47], [48].

To determine the service life, it is necessary to specify a permissible damage sum D_{per}. If the calculated damage sum is D > D_{per}, damages are expected with the given probability of failure of load spectrum and S/N-curve. Permissible damage sums can be chosen according to experiences based on recalculations of “old” gearbox designs or according to findings of FZG/Schaller [45], FZG/Eberspächer [10], FZG/Stahl [47], or FZG/Suchandt [48].

#### 5.4 Summary of the optimized application-based design of gears under variable loads

The optimized application-based design of gears under variable loads needs extensive knowledge about occurring loads and the strength of all considered components in the gearbox. Load as well as strength are subject to scattering and uncertainties, which have to be considered. Figure 7 shows this relation graphically. Load and strength are shown in form of a normal distribution function. The probability of failure can be graphically determined as overlap area of both curves.

### 6 Influence of locally changing stresses on the service life of bevel gears

#### 6.1 General

Case-hardened bevel and hypoid gears are highly loaded machine elements used for power transmission in a wide variety of applications. Therefore, a major aspect in the design process of bevel and hypoid gears is the load carrying capacity regarding different failure modes. Besides typical fatigue failure modes such as pitting [21], [53] and tooth root breakage [37], the failure mode tooth flank fracture occurs on highly loaded tooth flanks also. This failure mode is the result of cracks initiated in a larger material depth compared to the failure mode pitting [40]. Furthermore, spontaneous failures such as scuffing also are observed on bevel and hypoid gears if the load carrying capacity of the tribological system consisting of gears and lubricant is exceeded [39].

In general, the calculation of the load-carrying capacity of bevel and hypoid gears can be performed by use of standardized methods as well as by the use of a tooth-contact analysis. In the following, a short overview of currently available calculation methods regarding the fatigue failure modes pitting and tooth root breakage is given and a possibility for calculating the service life under changing local stresses is shown.

A typical pitting failure on a pinion flank of a bevel gear set is shown in Figure 8 [54]. Pitting failures may lead to higher dynamic forces and can cause other failure modes such as tooth breakage.

The failure mode tooth root breakage shows an initial crack in the 30°-tangent area on the tooth root fillet and normally starts at the surface. Due to tooth-root breakage, the gear set is damaged substantially and a total breakdown of the gearbox is likely to occur. A typical tooth root breakage failure on a tooth of a bevel gear is shown in Figure 9.

#### 6.2 Standardized calculation

A widely used approach for a standardized calculation of the load carrying capacity regarding the failure modes pitting and tooth root breakage of bevel and hypoid gears is the international standard

ISO 10300:2014 [24]. The calculation of the load-carrying capacity according to the international standard ISO 10300:2014 [24] is based on a virtual cylindrical gear. The virtual cylindrical gear geometry is structured according to the schematic construction in Figure 10 according to FZG/Wirth [54] and is derived from the reference cone of the bevel gear or rather the hypoid gear. Detailed information regarding the representation of a hypoid gear as a virtual cylindrical gear is available in Annex A of ISO 10300-1 [24].

#### 6.2.1 Failure mode pitting

For determination of the load carrying capacity regarding the failure mode pitting according to ISO 10300-2 [25] the occurring contact stress *σ*_{H} is compared to the permissible contact stress *σ*_{HP}. The nominal contact stress *σ*_{H0} can be calculated with Equation 2, where F_{n} is the nominal force of the virtual cylindrical gear at mean point P.

Where

F_{n} is the nominal force.

l_{bm} is the length of contact line.

*ρ*_{rel} is the radius of relative curvature.

Z_{M-B} is the mid-zone factor.

Z_{LS} is the load sharing factor.

Z_{E} is the elasticity factor.

Z_{K} is the bevel gear factor.

The actual contact stress *σ*_{H} can be calculated by Equation 3.

Where

K_{A} is the application factor.

K_{v} is the dynamic factor.

K_{h}_{β} is the face load factor.

K_{H}_{α} is the transverse load factor.

*σ*_{HP} is the permissible contact stress.

To avoid pitting, the occurring contact stress *σ*_{H} should be lower than the permissible contact stress *σ*_{HP}, which can be calculated according to Equation 4.

Where

*σ*_{H,lim} is the allowable stress number.

Z_{NT} is the life factor.

Z_{X} is the size factor.

S_{Hmin} is the minimum safety factor.

Z_{L}, Z_{V}, Z_{R} are the lubricant factors.

Z_{W} is the work hardening factor.

Z_{Hyp} is the hypoid factor.

#### 6.2.2 Failure mode tooth root breakage

The calculation of the tooth root load carrying capacity according to ISO 10300-3 [26]) is also based on a virtual cylindrical gear geometry (see Figure 10) comparing the occurring tooth root stress *σ*_{F} with the permissible tooth root stress *σ*_{FP}.

The nominal tooth root stress *σ*_{F0} can be calculated according to Equation 5, where F_{vmt} is the nominal tangential force of the virtual cylindrical gear.

Where

F_{vmt} is the nominal force.

b_{V} is the face width.

m_{mn} is the mean normal module.

Y_{Fa} is the tooth form factor.

Y_{Sa} is the stress correction factor.

Y_{ε} is the contact ratio factor.

Y_{BS} is the bevel spiral angle factor.

Y_{LS} is the load sharing factor.

The actual tooth root stress *σ*_{F} can be calculated by Equation 6.

Where

K_{A} is the application factor.

K_{v} is the dynamic factor.

K_{F}_{β} is the face load factor.

K_{F}_{α} is the transverse load factor.

To avoid tooth root breakage, the tooth root stress *σ*_{F} should be lower than the permissible tooth root stress *σ*_{FP}, which can be calculated with Equation 7.

Where

*σ*_{FE} is the allowable stress number.

Y_{NT} is the life factor.

Y_{δ }_{rel T} is the relative notch sensitivity factor.

Y_{R rel T} is the relative surface condition factor.

Y_{X} is the size factor.

S_{Fmin} is the minimum safety factor.

#### 6.3 Local calculation

The local calculation method is based on the real bevel or hypoid gear geometry and uses local stresses to determine the safety factors against different failure modes. The local stresses are determined by use of a loaded tooth contact analysis (LTCA), which can be performed by applying a software tool such as BECAL [46].

#### 6.3.1 Failure mode pitting

The local contact stresses *σ*_{H,Y} can be calculated by the multiplication of the calculated Hertzian stresses and the square root of the dynamic factor K_{V} according to ISO 10300-1 [24] in order to take load increments due to internal dynamic effects into account.

Where

*σ*_{H,lok,i} is the local Hertzian stress.

K_{v} is the dynamic factor.

The local permissible contact stress *σ*_{HP,Y} is calculated on the basis of the strength values according to ISO 6336-5 [27] using the corresponding equations according to ISO 10300-2 [25], whereas the bevel specific influence on the permissible contact stress is taken into account by the additional slip factor Z_{S,Y}.

Where

*σ*_{H,lim} is the allowable stress number.

Z_{NT} is the life factor.

Z_{X} is the size factor.

Z_{L}, Z_{V}, Z_{R} are lubricant factors.

Z_{W} is the work hardening factor.

Z_{Hyp} is the hypoid factor.

Z_{S,Y} is the slip factor.

For the pinion and wheel, the local safety factors are calculated separately, whereas it is assumed that the minimum factor is decisive.

#### 6.3.2 Failure mode tooth root breakage

The local tooth root stress *σ*_{F,Y} can be calculated by the multiplication of the calculated tooth root stress *σ*_{F,lok,Y} and the dynamic factor K_{v} according to ISO 10300-1 [24] in order to take load increments due to internal dynamic effects into account.

Where

*σ*_{F,lok,Y} is the local stress.

K_{v} is the dynamic factor.

The local permissible contact stress *σ*_{FP,Y} is calculated on the basis of the strength values according to ISO 6336-5 [27] using the corresponding equations given in ISO 10300-3 [26].

Where

*σ*_{FE} is the allowable stress number.

Y_{NT} is the life factor.

Y_{δ}_{rel T} is the local relative notch sensitivity factor.

Y_{R rel T} is relative surface condition factor.

Y_{X} is the size factor.

#### 6.4 Local load spectrum

For a more precise consideration of the variable loads occurring during the operation of a bevel or hypoid gear set, the standardized methods according to the ISO 10300-series [24], [25], [26] described earlier also can be applied in accordance with ISO 6336-6 [27] in form of a service life calculation. This procedure allows a more precise consideration of the variable load occurring during operation but also provides only global safety factors.

Within the design of bevel and hypoid gears, the gears microgeometry can be designed freely within certain limits. Therefore, a gear set with the same macro-geometry can show very different micro-geometries, which can significantly influence the load carrying capacity and the operating behavior of the gear set. The local calculation methods (loaded tooth contact analysis) described earlier were developed to take this micro-geometry into account within the load-carrying calculation of bevel and hypoid gears. In addition to the theoretical microgeometry, measured flank topographies also can be considered within the local calculation and thus removals of the gear flank by wear or micropitting as well as possible manufacturing deviations can be included. In order to also take the influence of variable loads occurring during operation regarding the failure mode pitting in the local calculation method into account, FZG/Hombauer [22] proposes the procedure shown in Figure 11.

The locally occurring internal load spectrum can result from variable external loads as well as from an ongoing change of the gear flank’s microgeometry (for example caused by wear or micropitting). Both are expressed by varying local stresses on the tooth flank; therefore, an internal load spectrum also can be derived from experimental investigations under a constant load where the microgeometry is locally changing by wear or micropitting. These changes of the microgeometry can be documented by regular inspections and can be transferred to an internal load spectrum by performing successive local calculations.]

The local safety factors S_{H}(Y) against pitting cannot be derived directly from the local damage sum U_{Y}(Y) of a point. The local safety factor of a local point has to be determined by use of an iteration. Starting from the internal locally occurring load spectrum Lk(Y), the local safety factor S_{H}(Y) is adjusted until the damage sum is nearly equal to 1. This procedure is used for all points of contact.

Figure 12 shows an exemplary safety factor distribution of a bevel gear flank used for experimental investigations at FZG created by use of the program BECAL [46] and evaluated by an internal software.

Within the recalculation of experimental investigations made at FZG, a satisfactory agreement between the calculated damage and the flank states documented within the tests could be found. If a local safety factor S_{H}(Y) smaller than 1 is calculated, a pitting failure is most likely to occur corresponding the 1% failure probability.

### 7 Conclusion

In the scope of this article, a brief overview of currently applied methods to consider variable loads in the design process of cylindrical as well as bevel and hypoid gears was given. Furthermore, an approach for a local damage accumulation regarding pitting damages on cylindrical as well as bevel and hypoid gears has been shown. All of the presented methods have their intended use case and differ in complexity and required input. It is up to the designer to select the appropriate calculation method for every phase of the design process. The use of a particular method is always a tradeoff. On the one hand, there are methods requiring little input but have limited information of the results. On the other hand, methods with very accurate and detailed output require significantly more input and calculation effort. This article presented an overview of this spectrum of methods.

Industrial gearboxes under variable loads are usually designed by using standardized calculation methods such as ISO 6336 or AGMA 2101 in combination with an appropriate application factor K_{A} or overload factor K_{O}. More complex calculation methods such as ISO 6336-6 or the shown optimized application-based design approach should be used for applications where power-to-weight ratio is important. But these methods need a lot more experience and knowledge. As the power-to-weight ratio is becoming more important, it is suggested to also implement more complex methods for calculating the service life of gears in the well-known standards as soon as there is more knowledge available. At the moment, there are only few systematic experimental investigations available dealing with the service life of gears under variable load. This work should be significantly intensified in the near future to form a sustainable basis for a future extension of the standards.

Locally changing stresses on the tooth flank of bevel and hypoid or cylindrical gears lead to an internal load spectrum. This can be treated in the same way as a global load spectrum but has to be applied within a local damage accumulation.

All in all, modern methods for calculating the service life of gears offer a lot of potential for a further increase of the power-to-weight ratio if they are applied appropriate.

### Bibliography

- 2101-D04: Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth (2004).
- 6006-B20: Standard for Design and Specifications of Gearboxes for Wind Turbines (2020).
- Bauer, E.; Böhl, A.: Flank Breakage on Gears for Energy Systems. Gear Technology 28. Heft: 8, S. 36–42 (2011).
- Boiadjiev, I.; Witzig, J.; Tobie, T.; Stahl, K.: Tooth flank fracture – basic principles and calculation model for a sub surface initiated fatigue failure mode of case hardened gears. International Gear Conference, Lyon Villeurbanne, France. Woodhead Publishing (2014).
- Bretl, N.; Schurer, S.; Tobie, T.; Stahl, K.; Höhn, B.-R.: Investigations on Tooth Root Bending Strength of Case Hardened Gears in the Range of High Cycle Fatigue. 2013 Fall Technical Meeting (FTM), Indianapolis, USA. American Gear Manufacturers Association (2013).
- Bruckmeier, S.: Flankenbruch bei Stirnradgetrieben. Dissertation, Technische Universität München (2006).
- Curà, F.: Method for Calculating the Application Factor KA in Gears Subjected to Variable Working Conditions. Procedia Engineering 101, S. 109–116 (2015).
- Curà, F.: ISO Standard based method for calculating the in-operation application factor KA in gears subjected to variable working conditions. International Journal of Fatigue 91, S. 459–465 (2016).
- Dąbrowski, D.; Natarajan, A.: Assessment of Gearbox Operational Loads and Reliability under High Mean Wind Speeds. Energy Procedia 80, S. 38–46 (2015).
- Eberspächer, C.; Höhn, B.-R.: FVA-Nr. 125/ III- Heft 457- Zahnflankenlebensdauer Abschlussbericht – Untersuchungen zur nicht-linearen Schädigung bei der Grübchenbildung an Zahnflanken. Forschungsvereinigung Antriebstechnik e.V. (1995).
- Foulard, S.; Ichchou, M.; Rinderknecht, S.; Perret-Liaudet, J.: Online and real-time monitoring system for remaining service life estimation of automotive transmissions – Application to a manual transmission. Mechatronics 30, S. 140–157 (2015).
- Haibach, E.: Betriebsfestigkeit – Verfahren und Daten zur Bauteilberechnung. Springer, Berlin [u.a.], 3. Auflage (2006).
- Hein, M.: Zur ganzheitlichen betriebsfesten Auslegung und Prüfung von Getriebezahnrädern. Dissertation, Technische Universität München (2018).
- Hein, M.; Geitner, M.; Tobie, T.; Stahl, K.; Pinnekamp, B.: Reliability of Gears – Determination of Statistically Validated Material Strength Numbers – 18FTM26. 2018 Fall Technical Meeting (FTM), Chicago, USA. American Gear Manufacturers Association (2018).
- Hein, M.; Kunze, C.; Lee, J.-K.; Tobie, T.; Stahl, K.: Customer-focused modular test procedure for driveline components. Getriebe in Fahrzeugen 2017 – VDI-Bericht 2313, Bonn. VDI – Verein Deutscher Ingenieure (2017).
- Hein, M.; Tobie, T.; Stahl, K.: Calculation of tooth flank fracture load capacity – Practical applicability and main influence parameters – 17FTM21. 2017 Fall Technical Meeting (FTM), Columbus, USA. American Gear Manufacturers Association (2017).
- Hein, M.; Tobie, T.; Stahl, K.: Parameter study on the calculated risk of tooth flank fracture of case hardened gears. Journal of Advanced Mechanical Design, Systems, and Manufacturing 11. Heft: 6 (2017).
- Hein, M.; Tobie, T.; Stahl, K.: Test method for time-scaled fatigue tests of gear transmission systems. Forschung im Ingenieurwesen 81. Heft: 2-3, S. 291–297 (2017).
- Hein, M.; Tobie, T.; Stahl, K.: Holistic approach for fatigue life design of vehicle transmissions. Dritev 2018 – Drivetrain for Vehicles – VDI-Berichte 2328, Bonn. VDI – Verein Deutscher Ingenieure (2018).
- Hein, M.; Ziegler, A.; Tobie, T.; Stahl, K.: System service life testing: Shortened test method for vehicle transmissions – Systemlebensdauerprüfung: verkürztes testverfahren für Getriebe in Kraftfahrzeugen. Drivetrain for Vehicles / Getriebe in Fahrzeugen – VDI-Bericht 2218, Friedrichshafen. VDI – Verein Deutscher Ingenieure (2014).
- Höhn, B.-R.; Michaelis, K.; Otto, H.-P.: Pitting load carrying capacity under increased thermal conditions. Industrial Lubrication and Tribology 63. Heft: 1, S. 11–16 (2011).
- Hombauer, M.: Grauflecken an Kegelrad- und Hypoidverzahnungen und deren Einfluss auf die Grübchentragfähigkeit. Dissertation, Technische Universität München (2013).
- Igba, J.; Alemzadeh, K.; Durugbo, C.; Henningsen, K.: Performance assessment of wind turbine gearboxes using in-service data – Current approaches and future trends. Renewable and Sustainable Energy Reviews 50, S. 144–159 (2015).
- 10300-2: Calcualtion of load capacity of bevel gears – Part 2: Calculation of surface durability (pitting) (2014).
- 10300-3: Calcualtion of load capacity of bevel gears – Part 3: Calculation of tooth root strength (2014).
- 10300-1: Calculation of load capacity of bevel gars – Part 1: Introduction and general influence factors (2014).
- 6336-5: Calculation of load capacity of spur and helical gears — Part 5: Strength and quality of materials (2016).
- 6336-20: Calculation of load capacity of spur and helical gears — Part 20: Calculation of scuffing load capacity (also applicable to bevel and hypoid gears) — Flash temperature method (2017).
- 6336-21: Calculation of load capacity of spur and helical gears — Part 21: Calculation of scuffing load capacity (also applicable to bevel and hypoid gears) — Integral temperature method (2017).
- 6336-4: Calculation of load capacity of spur and helical gears — Part 4: Calculation of tooth flank fracture load capacity (2019).
- 6336-1: Calculation of load capacity of spur and helical gears — Part 1: Basic principles, introduction and general influence factors (2019).
- 6336-2: Calculation of load capacity of spur and helical gears — Part 2: Calculation of surface durability (pitting) (2019).
- 6336-3: Calculation of load capacity of spur and helical gears — Part 3: Calculation of tooth bending strength (2019).
- 6336-6: Calculation of load capacity of spur and helical gears — Part 6: Calculation of service life under variable load (2019).
- Nejad, A. R.; Gao, Z.; Moan, T.: Fatigue Reliability-based Inspection and Maintenance Planning of Gearbox Components in Wind Turbine Drivetrains. Energy Procedia 53, S. 248–257 (2014).
- Nejad, A. R.; Gao, Z.; Moan, T.: On long-term fatigue damage and reliability analysis of gears under wind loads in offshore wind turbine drivetrains. International Journal of Fatigue 61, S. 116–128 (2014).
- Niemann, G.; Winter, H.: Maschinenelemente – Band 2: Getriebe allgemein, Zahnradgetriebe- Grundlagen, Stirnradgetriebe. Springer-Verlag, Berlin Heidelberg New York, 2. Auflage (2003).
- Olsson, E.; Olander, A.; Öberg, M.: Fatigue of gears in the finite life regime — Experiments and probabilistic modelling. Engineering Failure Analysis 62, S. 276–286 (2016).
- Pellkofer, J.; Boiadjiev, I.; Kadach, D.; Klein, M.; Stahl, K.: New calculation method for the scuffing load-carrying capacity of bevel and hypoid gears. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 233. Heft: 21-22, S. 7328–7337 (2019).
- Pellkofer, J.; Hein, M.; Stahl, K.; Reimann, T.; Boiadjiev, I.: New standardized calculation method of the tooth flank fracture load capacity of bevel and hypoid gears – 19FTM23 (2019).
- Reimann, T.; Stemplinger, J.-P.; Stahl, K.: The scuffing test A/44/Cr – A method for testing the scuffing and wear behavior of hypoid oils [Der Fresstest A/44/Cr – Eine Methode zur Prüfung des Fress- und Verschleißverhaltens von Hypoidölen]. Tribologie und Schmierungstechnik 62. Heft: 2, S. 46–53 (2015).
- Renius, K.-T.: European Tractor Transmission Design Concepts. American Society of Agricultural Engineers. ASAE-Paper 76-1526, St. Joseph, MI (1976).
- Renius, K.-T.: Betriebsfestigkeitsberechnungen von Maschinenelementen in Ackerschleppern mit Hilfe von Lastkollektiven. Konstruktion 29. Heft: 3, S. 85–93 (1977).
- Renius, K.-T.: Betriebsfestigkeitsberechnung und Laborerprobung von Zahnrädern in Ackerschleppergetrieben. VDI-Bericht 332. VDI – Verein Deutscher Ingenieure (1979).
- Schaller, K. V.: IGF Nr. 7609, FVA-Nr. 125/ II- Heft 320- Zahnradlebensdauerprüfung Abschlussbericht – Betriebsfestigkeitsuntersuchung zur Grübchenbildung an einsatzgehärteten Stirnflanken. Forschungsvereinigung Antriebstechnik e.V. (1991).
- Schlecht, B.; Schaefer, S.; Hutschenreiter, B.: FVA-Forschungshaft 1037 – BECAL – Programm zur Berechnung der Zahnflanken- und Zahnfußbeanspruchung an Kegel- und Hypoidgetrieben bei Berücksichtigung von Relativlage und Flankenmodifikationen (Version 4.1.0) – Benutzeranleitung und Programmdokumentation. Forschungsvereinigung Antriebstechnik e.V., Frankfurt/Main (2012).
- Stahl, K.: IGF Nr. 9889, FVA-Nr. 188/II- Heft 502- Zahnfuß-Betriebsfestigkeit Abschlussbericht – Zahnfuß-Betriebsfestigkeitsuntersuchungen an einsatzgehärteten Zahnrädern. Forschungsvereinigung Antriebstechnik e.V. (1996).
- Suchandt, T.: IGF Nr. 8387, FVA-Nr. 188/I- Heft 408- Zahnfuß-Betriebsfestigkeit Abschlussbericht – Zahnfuß-Betriebsfestigkeitsuntersuchungen an einsatzgehärteten Zahnrädern. Forschungsvereinigung Antriebstechnik e.V. (1993).
- Tobie, T.: Zur Grübchen- und Zahnfußtragfähigkeit einsatzgehärteter Zahnräder – Einflüsse aus Einsatzhärtungstiefe, Wärmebehandlung und Fertigung bei unterschiedlicher Baugröße. Dissertation, Technische Universität München (2001).
- Tobie, T.; Höhn, B.-R.; Stahl, K.: Tooth flank Breakage – Influences on subsurface initiated fatigue failures of case hardened gears. ASME 2013 DETC, Portland, Oregon. ASME (2013).
- Trencséni, B.; Palkovics, L.: Driveline torque observer for heavy duty vehicle. Periodica Polytechnica Transportation Engineering 39. Heft: 2, S. 91 (2011).
- Vietze, D.; Hein, M.; Stahl, K.: Method for a cloud based Remaining-Service-Life-Prediction for Vehicle-Gearboxes based on Big-Data-Analysis and Machine Learning. Forschung im Ingenieurwesen (2020 (accepted; not yet published)).
- Weber, C.; Tobie, T.; Stahl, K.: Investigation on the flank surface durability of gears with increased pressure angle. Forschung im Ingenieurwesen 81. Heft: 2-3, S. 207–213 (2017).
- Wirth, C.: Zur Tragfähigkeit von Kegelrad- und Hypoidgetrieben. Dissertation, Technische Universität München (2008).
- Witzig, J.: Flankenbruch – Eine Grenze der Zahnradtragfähigkeit in der Werkstofftiefe. Dissertation, Technische Universität München (2012).
- Ziegler, A. G.: Zur verkürzten Systemlebensdauerprüfung von Zahnradgetrieben. Dissertation, Technische Universität München (2011).