Reliability of gears: Determination of statistically validated material strength numbers

One of the most important goals when designing a gearbox is to ensure a sufficient reliability in operation. Depending on the field of application, the acceptable probability of failure may be different, typical values commonly are in the range of 0.01 percent to 10 percent. Material strength values regarding main failure modes such as pitting or tooth root breakage used for designing a gear stage are often obtained from testing. Because of the high effort for such experimental investigations, the usual scope of testing is limited and typically allows for a reliable estimation of the determined strength values for a failure probability of 50 percent only. A conversion to other failure probabilities is possible if sufficient data or literature references for the considered material are available. Furthermore, common standardized methods for rating the load carrying capacity of gears such as ISO 6336 or AGMA 2001 use strength values and S/N-curves specified for a failure probability of 1 percent. Consequently, the calculated safety factors, which are often used as main-design criteria, are strictly correlated to this probability of failure.

This paper is intended to give a review on the statistical reliability behavior of cylindrical gears with regard to pitting and tooth root breakage failures. Initially, the results of different relevant literature references were gathered and compared. It is shown that several previous investigations are not based on extensive research, insufficiently statistically validated, or outdated. Therefore, general applicable distribution parameters for case hardened gears were determined based on a statistical analysis of current and former experimental test data available at FZG. Hence, a mathematical reliability approach was developed and drafted to expand standardized load capacity calculation methods such as ISO 6336. The deduced models and procedures allow for the consideration and conversion of different reliability levels in the design process of cylindrical gears. Finally, the relevance of the approach for industrial applications and the need for further investigations are discussed.

1 Introduction

Gears are designed and optimized in view of different targets such as adequate level of safety, low noise excitation and high efficiency. Furthermore, minimizing costs is one of the most desired requirements in the development process. The material properties and capacities are supposed to be entirely used by the occurring loads. Finally, yet importantly, the desired level of reliability of the gear system in operation must be guaranteed.

The acceptable probability of failure depends on the field of application. The specified permissible level of failure probability is typically between 0.01 percent and 10 percent, but can also be out of this range for special cases. For designing gears, material strength parameters are required. These material values may be determined by empirical testing and are associated with a certain failure probability as determined by the statistical behavior of the test results. Because of the high effort for such experimental investigations, the usual scope of testing is limited and kept to a minimum. A reliable estimation of the determined strength values for a failure probability of 50 percent is usually possible. However, sufficient data or literature references for the considered material have to be available for a calculation of parameters correlated to different failure probability. Strength values and S/N-curves are contained and used in common standardized methods for rating the load carrying capacity of gears such as ISO 6336 [4] or ANSI/AGMA 2101 [2]. These specifications are typically based on a failure probability of 1 percent. In consequence, the calculated safety factors that often serve as main design criteria are strictly associated to this probability of failure. A consideration of the failure probability is included in the calculation method of the bearing capacity according to ISO 281 [9]. However, a comparable standardized and sufficiently validated approach for gears does not exist.

A review on the statistical failure behavior of cylindrical gears is presented in this paper considering pitting and tooth root breakage. The typical characteristics of both failure mechanisms are described initially followed by a short definition of the most relevant statistical probability distributions. Subsequently, the findings of different relevant scientific literature references are gathered. The corresponding statistical parameters are illustrated and compared. The analysis of numerous current and former experimental test data at FZG allows for the determination of general applicable distribution parameters for case hardened gears. Furthermore, a newly developed mathematical calculation approach is proposed for the consideration and conversion of different failure probability levels in the calculation and design process of cylindrical gears. This approach is intended to be an extension to standardized load capacity calculation methods such as ISO 6336 or ANSI/AGMA 2001. The relevance for practical industrial applications is finally discussed.

2 Failure Description – Pitting and Tooth Root Breakage

Highly loaded gears can suffer different failure modes. An overview is given in ISO 10825 [7] or ANSI/AGMA 1010 [1]. This paper is focused on the main fatigue failure modes pitting and tooth root breakage. Both are caused by material fatigue and may result in a total breakdown of the entire transmission system. Furthermore, validated and well-established calculation methods are available for these two failure mechanisms. Other failure mechanisms such as tooth flank fracture are still part of the contemporary research and therefore not reasonable for a detailed statistical analysis yet [18, 20, 35].

2.1 Pitting

Pitting commonly appears as shell-shaped breakouts from the flank surface (see Figure 1) due to fatigue of the tooth flank under contact pressure. Especially, the region of negative specific sliding in the dedendum flank area between the tooth root and the pitch line is affected. The failure starts with the nucleation of cracks on or near the surface. The crack propagates under repeated contact loading. It eventually grows large enough to become unstable and reaches the tooth surface resulting in the breakout of material. [13] Crack propagation may be reinforced by the hydraulic explosion effect. The lubricant infiltrates the crack because of its low viscosity. The contact pressure squeezes more fluid into the crack under high pressure causing crack propagation and finally breakage. Pitting is substantially influenced by the external load, the material properties, the surface topology, the friction coefficient, and the lubricant [27]. Furthermore, the case hardening depth has a major importance on pitting [34]. Pitting development in most cases has a linear or progressive trend. Secondary damages such as tooth breakage are possible. The occurrence of initial pitting during running-in of the gear can be acceptable in the case of improving the pressure distribution on the flank. Typical effects of pitting are an increase of NVH behavior and additional dynamic forces in the gearbox. [1, 30, 32]

Data for the endurance limit of pitting resistance can be found in standardized load capacity calculation methods considering the influence of material, heat treatment, etc. These parameters determine the allowable contact stress number σ_{H lim} in ISO 6336-5 [6] and the allowable contact stress number σ_HP in ANSI/AGMA 2101 [2]. The effective permissible contact stress limit is calculated by using the life factor Z_NT (ISO, see Figure 2) respectively the stress cycle factor for pitting resistance Z_N (AGMA) that considers the desired number of load cycles. These factors assign higher load capacity to cases with a limited number of load cycles and can be taken from the S-N curves in the referring standards. An overview for the mentioned parameters is given in Table 1.

These material data and S-N curves are based on industrial experience and experimental investigations on test gears and are valid for a failure probability of 1 percent.

2.2 Tooth root breakage

The cyclic bending stress at the tooth root fillet is the determining factor for tooth root breakage (also called: bending fatigue failure). The tooth bending results from high tangential forces with the tooth height acting as a lever. Typically, the damage progress involves three stages: The crack nucleation typically starts on the surface mostly in the critical section at 30° tangent to the tooth-root fillet (see Figure 3). In this area, the highest bending stress occurs and is intensified by notch effects. The crack propagates further and follows a path across the tooth-root section. Ultimately, a final fracture leads to the separation of the tooth. Tooth root breakage absolutely has to be avoided as it leads to an immediate breakdown of the entire transmission system almost without exception. Two types of fractured surfaces usually appear: On the one hand, a fatigue crack growth area can be identified by a smooth crack surface with beach marks. Beach marks may occur when the gear operation is frequently interrupted so the crack propagation process pauses. Beach marks may also appear on gears where both flanks see a reversed and alternating stress cycle. On the other hand, an unstable forced fracture shows a rough and shattered area. It may be caused by abrupt and extreme overloads as blockage in the gearbox. Commonly, tooth root breakage includes both types with an initial fatigue fracture and a final unstable fracture. [1, 13, 32] The case hardening depth has an important influence on the tooth root load carrying capacity [34]. Furthermore, a peening treatment can increase the allowable bending strength creating compressive residual stresses in the surface near area of the tooth root [16, 22, 29].

Current research is also focused on high cycle fatigue of the tooth root. A different failure mechanism with a crack starter in the subsurface area can occur especially after a high number of load cycles. Such cracks most often start at a nonmetallic inclusion in the material [12]. These so-called fish-eyes are not accounted for in this paper as standardized methods do not cover tooth root breakage with a crack initiation below the surface so far.

Similar to pitting, data for the endurance limit of bending strength can be found in standardized load capacity calculation methods. There are specified data for the allowable bending stress number σ_{F lim}  in  ISO 6336-5 [6] and for the allowable bending stress number σ_FP in ANSI/AGMA 2101 [2] considering the influence of material, heat treatment, etc. The influence of the number of load cycles on bending stress limit is included in the calculation by the life factor Y_NT (ISO) and the stress cycle factor for tooth root resistance Y_N  (AGMA, see Figure 4). These factors assign a higher load capacity for a limited number of load cycles and can be determined from the S-N curves in the appropriate standards. Table 2 shows an overview for the mentioned parameters.

3 Statistical Definitions and Probability Distributions

First, some basic explanations about statistics are presented to understand the reliability behavior of gears. The probability of failure F(x) and reliability R(x) is the integral of the density function. Both express the same status, but represent the counterpart of each other:

According to Bertsche and Lechner [11], the reliability expresses the probability for functionality (or survival) of a product during a defined time period under certain circumstances. Typically, an S-N curve is determined by experimental tests and is therefore associated to a certain failure probability. The distribution of experimental data for lifetime analysis has to be evaluated in detail. The endurance and static analyses consider the statistical distribution of the load level whereas the limited life analysis examines the variance of the number of load cycles to failure. [11, 17, 19] In Figure 5, S-N curves for different probabilities of failure (F(x) = 1%, 50%, and 99%) are illustrated. The density functions are used to illustrate the appropriate statistical variable.

3.1 Probability Functions

Based on this, the mathematical relations of relevant probability functions for this paper are defined in the following part.

3.1.1 Normal Probability Distribution

The Gaussian distribution or normal distribution is one of the most common distribution functions in statistics. It is defined by the following equation [21, 31, 36]:

The normal probability function can be simplified by the standard normal distribution Φ:

3.1.2 Lognormal Probability Distribution

The lognormal distribution embodies a normal distribution of the logarithmic probability variable x. Hence, the statistical variable log₁₀(x) = lg(x) is used in the normal distribution function. Its mathematical relation is expressed in the following equation [21]:

The lognormal probability function can also be simplified using the standard normal distribution Φ [21]:

3.1.3 Weibull Distribution

The 3-parameter Weibull distribution is described mathematically by the following terms [11, 36]:

The reduced 2-parameter Weibull distribution (x₀ = 0) is frequently used for lifetime analyses. [11, 21, 36]

3.2 Confidence Level

A given set of sample data with a limited number of experimental tests n only represents a part of its entire statistical population. Therefore, an interval of statistical tolerance for a certain confidence level can be considered additionally to the failure probability. This so-called confidence interval gives an estimated range of values. The confidence level represents a certain probability that a sample is part of this tolerance range. The width of the confidence interval gives an idea about how uncertainly a sample describes the statistical population. This allows the examination and comparison of samples with different variations and numbers of tests. [8, 14, 36]

In consequence, the S-N curve for a given failure probability also has a certain range of variation depending on the considered confidence interval. This range of variation disappears for an approximately infinite number of tests (n → ∞) because the total statistical population is described by a defined S-N line. For further  information  about  the  confidence  level  consideration  in  experimental  testing  please  see  ISO 12107 [8].

4 Statistical Parameters Regarding the Reliability of Cylindrical Gears

4.1 Comparison of Relevant Literature References

Initially, the results of different relevant literature references are gathered and compared. Several previous investigations are analyzed regarding their statistical foundation of the experimental test numbers and their being up-to-date.

4.1.1 ANSI/AGMA 2101 [2]

A model for the statistics of failures is included in the standardized load carrying capacity calculation method ANSI/AGMA 2101 [2] by the reliability factor Y_Z for pitting (see equation (8)) and tooth root breakage (see equation (9)):

The reliability factor Y_Z is used to modify the allowable stress in terms of the failure probability. Y_Z can be extracted from Table 3 and accounts for the effect of failures found in materials testing by the normal statistical distribution. According to the explanations in ANSI/AGMA 2101, the values are based on data developed for pitting and bending failure by the U.S. Navy. The standard also permits using other values if specific data are available. The factor Y_Z = 1 for a failure probability of 1 percent again clarifies that the given values for the allowable contact stress number σ_HP and the allowable bending stress number σ_FP are associated with this probability level. [2]

The analysis of the values for Y_Z by the related standard deviation shows that the calculation is founded on more than one normal probability distribution. A consistent standard deviation (in percentage, related to the allowable endurance limit for a failure probability of 50 percent) cannot be identified.

Furthermore, the model of the reliability factor Y_Z does not differentiate between long-life and limited-life analysis. As it is shown in Figure 5, a different statistical variable usually determines the statistical failure behavior for each fatigue area. Therefore, a detailed distinction is reasonable and mandatory. Also, Beermann [10] considers the values for Y_Z in the limited-life range as questionable.

Moreover, an identical approach for pitting and bending failure mechanisms is questionable. There is indeed a note added that a greater hazard is sometimes considered for tooth breakage than for pitting [2]. However, the standard does not contain recommendations on how to adapt the value of Y_Z to a reliable and appropriate level.

Because of the classified data, it isn’t possible to retrace the statistical coverage and the actual distribution properties. In summary, the given values for the reliability factor Y_Z in ANSI/AGMA 2101 can be questioned. The basic idea of the integration in a standardized load carrying capacity calculation method is incorporated and refined in the framework of this paper (see Part 5).

4.1.2 ISO 6336 [4]

In ISO 6336-5 [6], the values of σ_{H lim} (pitting resistance) and σ_{F lim} (tooth root bending resistance) are given for a probability of failure of 1 percent. The standard permits an adjustment of the values by a statistical analysis in order to correspond to other failure probabilities if available. Though, there are no recommended parameters regarding the statistical behavior specified or integrated in the load carrying capacity calculation of ISO 6336 [4].

4.1.3 Research of Bertsche and Lechner [11]

The investigations of Bertsche and Lechner [11] are focused on the statistical failure behavior in the limited- life range. To this end, the variance of the numbers of load cycle to failure N is considered. A comparison between the statistical parameters of gears, shafts, and bearings is illustrated in Table 4. For this purpose, Bertsche and Lechner assume a Weibull distribution function with three parameters. Hence, the Weibull factor b and the starting time without failure x₀ (related to the number of load cycles N_10% for a failure probability of 10%) are given.

For gears, it is differentiated between two failure mechanisms: The statistical behavior of pitting damages is characterized by a Weibull factor b = 1.1 … 1.5 and a starting time without failure x₀ = 0.4 … 0.8 · N_10%. The determined values for tooth root breakage failure are b = 1.2 … 2.2 and x₀ = 0.8 … 0.95 · N_10%. Bertsche and Lechner describe that the Weibull parameter b increases with the load level. That means a reduced deviation of the number of load cycles to failure for higher fatigue strength levels in the limited-life range. [11]

However, the analysis for gears is based on a low number of test runs (n = 5 … 20) per test series. There is no comment about the total number of tests or the analysis method. Information about the test conditions, the gear dimensions, or other properties also is not available.

Several paper authors such as Kissling and Stangl [26] or Beermann [10] reveal the determined parameters of Bertsche and Lechner are integrated in commercial gear designing software. Therefore, further verifications are necessary to ensure the applicability of these numbers for the calculation of reliability levels.

4.1.4 FVA Research Project 304 by Stahl [33]

Detailed information about the statistical failure behavior regarding pitting and tooth root breakage of case hardened cylindrical gears can be extracted from the research of Stahl [33] (see Table 5). He specifies the statistical probability distribution and the appropriate parameters for the long-life and limited-life range.

The statistical long-life behavior (that means the deviation of the endurance limit) for pitting and bending is described by a normal probability function of the load level. For the analysis, the modified Probit method [23] as well as the enhanced staircase method acc. to Hück [24] are applied. The normal distribution of pitting failures is characterized by a standard deviation s = 3.5% (related to the allowable endurance limit for a failure probability of 50 percent). Regarding the bending load carrying capacity, the surface treatment in the tooth root area has a decisive influence on the standard deviation according to Stahl [33]. Hence, the peened specimen shows a lower standard deviation (s = 3.4%) than the unpeened variants (s = 6.0%). [33]

The Weibull distribution with two parameters is the most suitable function to describe the statistical failure behavior in the pitting limited-life range according the data of Stahl [33]. The spreading of the numbers of load cycles to failure is characterized by a Weibull parameter b = 3.2 in all load levels. For tooth root breakage, the lognormal distribution constitutes the most appropriate approximation. The logarithmic standard deviation varies between s_log = 0.06 … 0.13 depending on the load level. Here, the spreading of the numbers of load cycles to failure for lower and higher loads is increased compared to intermediate loads. [33]

The evaluation by Stahl [33] considers different test gears and testing conditions from several research projects and experimental investigations. Especially, the influence of the following factors is exemplified:

Pitting tests

•  Material (e.g. 16MnCr5, 18CrNiMo7-6, …), all case-carburized
•  Normal module m_n = 4.5 … 8mm
•  Case hardening depth CH D₅₅₀ = 0.06 … 0.2 · m_n

Bending tests

• Material (e.g. 16MnCr5, 18CrNiMo7-6, …), all case-carburized
•  Surface treatment in the tooth root (peened, unpeened)
•  Normal module m_n = 2 … 20mm
•  Case hardening depth CH D₅₅₀ = 0.04 … 0.33 · m_n

The data used in [33] are statistically sufficiently founded on the analysis of 509 pitting tests on the FZG back-to-back gear test rig [28] and 2,219 bending tests on a pulsating test rig [15]. It has to be mentioned that the deduced distribution parameters are just as valid under the applied controlled and simplified test conditions of single-stage loads.

4.1.5 Further References

Detailed research on the statistical failure behavior regarding the pitting long-life and limited-life range was performed by Joachim [25]. He specifies the normal distribution for long-life and the 2-parameter Weibull distribution for limited-life as the most suitable approach.

Niemann and Winter [32] list a compilation of several research projects considering the pitting and tooth root load carrying capacity in the long-life range. It allows for a comparison of the statistical failure behavior based on the normal distribution for different materials, heat, and surface treatments as well as gear tooth qualities.

Furthermore, standardized S-N curves (pitting and tooth root breakage) of common case hardened steels (16MnCr5, 20MnCr5, 25MnCr4 and 20MoCr4) for a failure probability of 1 percent and 50 percent can be found  in DIN 3990-41 [3]. Based on this standard, conclusions about the statistical failure behavior are also possible.

For more information, please see the reference list.

4.2 Statistical Analysis of Experimental Data at FZG

Numerous test data regarding the pitting and bending load carrying capacity of cylindrical gears are available at FZG. The tests were typically run on standardized test rigs under controlled and documented test conditions. This allows for an appropriate comparability of the test runs.

4.2.1 Analysis of Current Test Data

In the framework of this research, as many current test data as possible are gathered and analyzed according to the methods used by Stahl [33]. This permits a further validation of the results of Stahl. Furthermore, it shows whether a change of the statistical behavior of case hardened gears after a certain time can be identified. Standardized analyzation methods are also explained in ISO 12107 [8].

The analysis of 614 pitting tests in the FZG back-to-back gear test rig [28] and 2,119 mechanical pulsating bending tests [15] considers the following specimen properties:

Pitting tests

•  Materials (e.g. 16MnCr5, 18CrNiMo7-6, …), all case-carburized
•  Normal module m_n = 0.3 … 5mm
•  Case hardening depth CH D₅₅₀ = 0.15 … 0.83 · m_n

Bending tests

•  Materials (e.g. 16MnCr5, 18CrNiMo7-6, …), all case-carburized
•  Surface treatment in the tooth root (peened, unpeened)
•  Normal module m_n = 0.45 … 8mm
•  Case hardening depth CH D₅₅₀ = 0.06 … 0.51 · m_n

All associated research projects were performed after 2002 and were not available at the time of the work performed by Stahl [33], most of them were performed in the last five years. Consequently, the analyzed test data represent up-to-date properties of commonly used case hardened steels for gears.

The performed analysis of the pitting endurance limit shows a result similar to the investigations of Stahl. In consideration of a normal distribution, a standard deviation
s = 3.5% (related to the allowable endurance limit for a failure probability of 50 percent) can be deduced from the contemporary pitting test series. Regarding the bending endurance limit, lower levels for the standard deviation are observed for the latest data. The peened specimens reveal a standard deviation s = 2.5%, the unpeened variants s = 2.9%. The influence of the surface treatment in the tooth root area on the standard deviation is less distinct but also apparent matching the conclusions of Stahl [33].

The pitting failure behavior in the area of limited-life of the applied tests shows comparable results as Stahl [33]. A Weibull factor b = 2.8 can be identified meaning a slight increase of the variation of numbers of load cycles for pitting damage as compared to Stahl’s results. For an example, Figure 6 illustrates the appropriate experimental data in a Weibull plot to prove the applicability of the Weibull distribution for the pitting load carrying capacity in the limited-life range. To commonly analyze the data, the load cycle numbers are referred to the load cycle number N_50% for a failure probability of 50 percent of every test series. The good match between the regression line and the data favors a reasonable approximation by the Weibull distribution. Consequently, the line in the Weibull plot allows specification of the Weibull factor by the gradient as well as the recommended factor to calculate another failure probability: Here, for example, the load cycle number N_1% for a failure probability of 1 percent is determined by multiplication of N_50% with the factor 0.22 (see Figure 6). A dependency of the variance of the number of load cycles on the load level could not be detected analogically to Stahl [33]. Regarding the load carrying capacity for bending, the statistical analysis shows similar logarithmic standard deviations s_log = 0.05 … 0.15 depending on the load level in the limited-life range. The trend of the research of Stahl [33] can be confirmed.

In general, the current results match well with the analyses of Stahl. However, there are some discrepancies especially regarding the bending endurance limit. The statistical failure behavior of gears may have changed over the last decades with respect to improvements in material quality, heat treatment, and gear production. Common manufacturing processes became more reproducible and stable, and new production methods were developed. Quality assurance now ensures high product standards. Furthermore, measurement technologies also have gotten more exact and affordable.

4.2.2 Recommendations for Distribution Parameters

A combined analysis of the current test data and the former test series by Stahl [33] allows for the determination of general applicable distribution parameters for case hardened gears. In total, there is a sufficient and significant statistical foundation with 1,123 pitting tests and 4,338 bending tests available at FZG. In Table 6, the results of this analysis are given as best approach to describe the statistics of the failure behavior.

The normal distribution with a standard deviation s = 3.5% is confirmed to be the most suitable function for the statistical description of the pitting endurance limit. A significant correlation between s and different typical case hardening steels, the gear module m_n, the case hardening depth CH D₅₅₀ or the surface hardness cannot be identified. Regarding failures by bending, a standard deviation of s = 3.2% is recommended for gears with a peening treatment in the tooth root to characterize the variance of the endurance limit. An increased value of the standard deviation s = 5.3% is suggested for unpeened gears. A higher chemical content of nickel in the material, a lower gear module m_n , as well as an increased case hardening depth CH D₅₅₀ in the tooth root show a trend to reduced standard deviations.

To approximate the variance of the load cycle number to pitting failure, a Weibull distribution with a Weibull factor b = 3.0 is recommended for all load levels in the limited-life range. For gears with a higher chemical content of nickel or an increased case hardening depth CH D₅₅₀ at the tooth flank, higher values of b can be assumed meaning less spreading. Regarding the limited-life region of tooth root breakage failures, a distinct influence of the load level is identified: The best approximation to describe the variance of the load cycle number is the lognormal distribution. The appropriate logarithmic standard deviation can be taken from Figure 7 depending on the load level.

These recommended statistical parameters are not supposed to be applied without a previous test for statistical significance. Hence, the results of a statistical analysis for an experimental sample can be proofed considering a certain confidence level. Certain confidence intervals are specified in many common test methods as for example the staircase method according to Hück [24]. Furthermore, ISO 12107 [8] contains a general approach for the consideration of confidence levels. The determination of confidence intervals is also part of current research at FZG. Additional publications are planned to be published in the near future.

5 Reliability Model for the Calculation of Load Capacity of Spur and Helical Gears

In the framework of this paper, a mathematical reliability approach was developed and drafted to expand standardized load capacity calculation methods such as ISO 6336 [4] or ANSI/AGMA 2101 [2]. It integrates the results of the previous analyses and allows for the consideration and conversion of different reliability levels in the design process of cylindrical gears.

5.1 Reliability Factor

To account for levels of failure probability other than 1 percent, a reliability factor Z_Z for pitting and a reliability factor Y_Z for tooth root breakage is suggested to be included in the calculation methods of the referring standards. The adapted calculation of ISO 6336 is presented in the following equations where Z_Z and Y_Z are added:

For ANSI/AGMA 2101, the former reliability factor in the calculation could be replaced by Z_Z and Y_Z (positioned in the numerator). Unlike in the actual method, it is distinguished between the failure mechanisms pitting and tooth root breakage:

In summary, the reliability factor is intended to modify the allowable stress to the considered reliability level.

5.2 Mathematical Calculation of Reliability Levels

As shown in Figure 5, the pertinent statistical variable characterizing the failure behavior is different for endurance and limited-life range. Therefore, a translation model is necessary to determine the appropriate modification of the load level depending on the reliability level.

For that purpose, the relation between a life and load consideration (see Figure 8) according to Haibach [17] is applied:

Here, the material specific S-N curve gradient k is calculated by the following equation:

The mathematical relations to determine the reliability factors are based on this assumption as well as on the results of the statistical analysis (see Part 4.2.2).

Based on this, the equations to calculate the reliability factor Z_Z for pitting are developed. A normal probability distribution is assumed for the endurance range. The decisive value is represented by the related standard deviation s_H:

For example, a recommended standard deviation s_H = 3.5% (see Table 6) for pitting of case hardened gears leads to a reliability factor Z_Z,50% = 1.09 to consider a failure probability of 50 percent. A reliability factor Z_Z,1% = 1.0 documents that the endurance limit parameters of the standard are based a failure probability of 1 percent.

In the limited-life range for pitting, the relevant parameters for the calculation of Z_Z are the S-N curve gradient k (to be derived from the appropriate S-N curve in the standard) and the Weibull factor b:

The reliability factor Y_Z for tooth root breakage is calculated analogically to Equation 16 as a normal probability distribution of the endurance limit is assumed. The associated distribution parameter is the related standard deviation s_F:

The calculation of Y_Z in the limited-life range of tooth root breakage is based on the lognormal distribution including the S-N curve gradient k (to be derived from the appropriate S-N curve in the standard) and the logarithmic standard deviation s_log:

These mathematical relations for the reliability factors Z_Z (pitting) and Y_Z (tooth root breakage) allow for a simple conversion of the reliability level. Every equation only depends on one statistical parameter that has to be specified for the considered gear type (material, heat and surface treatment, etc.). The presented relations are only based on the statistical behavior of case hardened gears so far. Therefore, further analyses, especially with other materials or heat treatments and practical validation need to be performed before the reliability factors should be adopted to the referring standards.

6 Relevance for the Design of Industrial Applications

Although reliability is very important in all industrial applications, little focus is directed to its impact on gear calculating and/or required safety factors. If a very high reliability is required, e.g. 99.99 percent, reliability factors Z_Z and Y_Z of less than 1.0 are calculated (e.g. Z_Z = 0.95) according to the equations proposed in this paper. In other words, the actual safety margin is less than calculated with the current standards. As for other improvements to the calculation methods, a validated approach to consider the impact of reliability on gear design and safety factors is important to write clear specifications and to compare different designs. Detailed knowledge of all influencing factors such as load distribution, dynamic loads, external loads, and, last but not least, desired and realized reliability allow for an optimum gear design combining cost efficiency, performance, and availability. The proposals in this paper are a good starting point for case hardened gears.

Most industrial gear applications are designed for unlimited service life. Only in a few cases, e.g. for power generation or extruder drives, the operating conditions may be considered steady. In most applications, operation is under variable load and speed. To what extent the proposed methods may be used for variable load conditions needs to be evaluated.

Especially for high speed gears, e.g. gears for power generation or compressor drives, very high numbers of load cycles apply. The limit of the S-N curves in the pertinent standards [2, 4] is typically at 10¹⁰ load cycles, a value that in many applications is reached after less than three years. The desired service life for these applications is usually more than 20 years. This begs the question: How does the endurance limit and the probability of failure, especially the standard deviation s_H behave in the range of very high numbers of load cycles reaching or even exceeding 10¹¹? Test data can hardly be produced in this range due to time limitation. Industry and research institutes are therefore encouraged to provide data and experience with respect to high cycle operation and failures.

7 Conclusions

Due to increased requirements on transmission systems, the design of gears includes many different targets such as adequate level of safety, low noise excitation and high efficiency. The reliability level of the gear system is an important parameter considered during the development process. The operational reliability of a gearbox depends on many factors, one of them is the scattering of the material properties.

This paper is intended to analyze the statistical behavior of gears regarding pitting and tooth root breakage failures. Different information about failure statistics is available in several literature references. The results and parameters in these references were gathered and compared. An analysis of the appropriate experimental data gives an impression of the statistical basis and the applicability. Based on the available information, several simplifications and assumptions about the statistical behavior are obvious. Furthermore, information about the database is often not transparent, making conclusions about the statistical validation even more difficult. Moreover, the analysis of current test data shows that a change in the statistical behavior is possible because of improvements in material, heat treatment, and gear manufacturing over the last several years. Consequently, an expansion, a verification, or an adaption of the given statistical parameters of previous investigations could be reasonable where appropriate.

A number of test data is necessary to identify the failure statistics of gears. Therefore, current experimental data on case hardened gears available at FZG were analyzed regarding pitting and tooth root breakage. This analysis allows for the determination of generally applicable distribution parameters with a sufficient data base to describe the appropriate failure statistics. The standardized strength properties commonly are based on a failure probability of 1 percent. Based on the evaluation and equations presented, conversion to different reliability levels for strength values and S-N curves can be performed.

Furthermore, a mathematical reliability approach was developed to expand or modify standardized load carrying capacity calculation methods such as ISO 6336 and ANSI/AGMA 2101 [2, 4] for pitting resistance and bending strength. The suggested addition integrates the statistical behavior in the calculation process of gears and allows for a detailed and target-oriented development.

The method presented only applies to case hardened gears. The knowledge about the statistical failure behavior of many further materials and heat treatments (e.g. through-hardened steels, nitrided steels, cast materials, fiber-reinforced plastics) is limited. The influence of load spectrum applications on the failure statistics is also largely unexplored. Real components often are exposed to various composed load levels, which is not comparable to single-stage tests considered in this paper. Finally, the research can be expanded to further failure mechanisms as tooth flank fracture or scuffing and applied to other gear types such as bevel and hypoid gears if there is enough data available. 

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