Contact fatigue life prediction of a bevel gear under spectrum loading

Rolling contact fatigue (RCF) issues, such as pitting, might occur on bevel gears because load fluctuation induces considerable subsurface stress amplitudes. Such issues can dramatically affect the service life of associated machines. An accurate geometry model of a hypoid gear utilized in the main reducer of a heavy-duty vehicle is developed in this study with the commercial gear design software MASTA. Multiaxial stress–strain states are simulated with the finite element method, and the RCF life is predicted using the Brown–Miller–Morrow fatigue criterion. The patterns of fatigue life on the tooth surface are simulated under various loading levels, and the RCF S–N curve is numerically generated. Moreover, a typical torque–time history on the driven axle is described, followed by the construction of program load spectrum with the rain flow method and the Goodman mean stress equation. The effects of various fatigue damage accumulation rules on fatigue life are compared and discussed in detail. Predicted results reveal that the Miner linear rule provides the most optimistic result among the three selected rules, and the Manson bilinear rule produces the most conservative result.


Introduction
Bevel gears have been extensively used in main reducers and differentials of automobiles, aerospace devices, and printing devices due to their high mechanical efficiency, large loading capacity, low noise behavior, and capability for direct power transmission between two orthogonal shafts. Load fluctuation and considerable subsurface stress amplitude may cause rolling contact fatigue (RCF) issues, such as pitting, on bevel gears; these issues could dramatically affect the service life of associated machines. The prediction of RCF life of a bevel gear pair is imperative for reliability evaluation. Niemann et al. [1] developed the first version of FZG gear test rig and defined relevant experimental standards. Later on, this type of test rig was extensively applied in fatigue and tribological studies on involute parallel gears [2]. He et al. [3] investigated the effect of external load on involute gear contact fatigue life and curve-fitted the stress-life formula on the basis of numerical data. Fernandes and McDuling [4] found that surface contact fatigue is the most common cause of gear failure and can significantly reduce the loadcarrying capacity of components. Liu et al. [5] studied effects of surface roughness and residual stress on the contact fatigue performance of gears by using the modified Dang Van multiaxial fatigue criterion. Carpinteri et al. [6] conducted a comprehensive review of multiaxial fatigue under variable amplitude and random loading. Wang et al. [7,8] predicted the RCF life of a wind turbine carburized gear by using multiaxial fatigue criteria. Wu et al. [9] evaluated six multiaxial fatigue criteria with life data obtained from proportional and non-proportional tensiontorsion fatigue tests on titanium alloy TC4. Zhu et al. [10] conducted a comparative evaluation of typical critical plane criteria by utilizing experimental datasets of four materials under uniaxial tension, torsion, and non-proportional multiaxial loadings.
Compared with involute gears, bevel gears have elicited less research attention. Litvin et al. [11] conducted a pioneering study on loaded tooth contact by using the finite element method; the contact pattern was predicted and proven by tests on spiral bevel gear drives. Sekercioglu and Kovan [12] experimentally studied the fatigue fracture of a bevel gear in an automotive differential and found that excessive contact stress is responsible for tooth pitting and the final fracture. Bhavi et al. [13] performed fatigue testing of spiral bevel gears and obtained the pinion's failure mode. Obtaining the S-N curve through long-term fatigue tests is time consuming. Thus, experimentally acquired RCF S-N curves of bevel gears have rarely been reported. Estimations of fatigue life with numerical methods, such as the finite element method, have become prevalent due to the development of simulation techniques. Ural et al. [14] summarized the applications of the finite element method and fracture mechanics in the prediction of cracks and estimation of the fatigue life of bevel gears in helicopters. Deng et al. [15] investigated contact and bending fatigue by using the cumulative fatigue criterion and stress-life equation.
In most engineering applications, gears experience timevarying loading histories due to the fluctuation of external torque. Liu et al. [16] developed a modular method to investigate the effect of load on the efficiency of a multirange hydromechanical transmission. Liu et al. [17] investigated the influence of work holding equipment errors on the mesh and loading behavior of face-hobbed hypoid gear by using an accurate mesh model established from a generated process. Medepalli and Rao [18] pointed out that the load characteristics of vehicles differ under varying road conditions and illustrated that the prediction of road load via numerical simulation is possible. The fatigue life under a constant load assumption cannot accurately describe damage evolution under a fluctuating load spectrum. Therefore, compilation of the load spectrum and selection of damage accumulation rules are crucial. Liu et al. [19] estimated the fatigue life of fan blades under random loading by considering the combined influence of stress amplitude and mean stress. Shinde et al. [20] utilized a modified rain flow counting method that allows the use of the Miner criterion to evaluate the fatigue life of structures under random loading. Mayer et al. [21] investigated the effect of cyclic loads below the endurance limit on fatigue damage in a series of variable amplitude loads.
According to these previous studies, accurate fatigue life estimation of bevel gears depends on the following prerequisites: Fatigue S-N curve, load spectrum, and proper fatigue damage accumulation rule. Only a few studies on bevel gear fatigue life have considered these factors. To address this shortage, this study used the finite element method to simulate the multiaxial stress-strain states of subsurface material points and the RCF S-N curve of bevel gear. The typical torque-time history on the driven axle was described using a typical truck load spectrum, and the standard load spectrum was constructed with the rain flow method and Goodman mean stress equation. Various fatigue damage accumulation rules were discussed. A diagram of the technique applied in this work is shown in Fig. 1.

Numerical methodology
To understand the contact fatigue behavior of a hypoid bevel gear in a heavy-duty vehicle, we investigated fatigue life under a given load spectrum by emphasizing the following aspects: Modeling of gear geometry and finite element contact analysis, fatigue life prediction of bevel gear under constant loading conditions on the basis of the Brown-Miller-Morrow multiaxial fatigue criterion, and compilation of load spectrum and fatigue estimations with various damage accumulation rules.

Bevel gear contact finite element model
Oerlikon and Gleason hypoid gears and spiral bevel gears are the most commonly used gear types in the main gearboxes of modern vehicles [22]. The Gleason-type hypoid gear was selected in this work, and the MC401-AGMA/Gleason hypoid gear design module was obtained from the commercial gear design software MASTA. Table  1 shows the basic design parameters of a hypoid gear pair. Figure 2 shows the geometry of a Gleason hypoid gear pair.
A finite element contact analysis was conducted with the commercial software Abaqus/Standard 6.14. Seven teeth of the wheel were modeled in consideration of axisymmetric characteristics to reduce the computation time. The finite element type C3D10 was selected, and the mesh density at the tooth surface contact zone was sufficiently fine to guarantee convergence. Around 400000 elements were generated in the pinion and wheel. A rotating motion was applied on the driving pinion, and the output torque was applied on the engaged driven wheel, as shown in Fig. 3.

Fatigue life estimation under constant loading conditions
Under repeated gear meshing, the subsurface material points under the contacting surface exhibit complicated time-varying multiaxial stress-strain responses. Normal and shear components co-exist, and the mean values are generally non-zero [23]. Therefore, critical-plane-based multiaxial fatigue criteria are frequently used to determine the complicated stress-strain state and further employed for fatigue behavior analysis during the cycling contact period [24]. A widely accepted multiaxial criterion, the Brown-Miller criterion [25], was used in this work to evaluate the fatigue life under constant contact loading conditions. Given that considerable mean stresses occur at critical material points, the modified Brown-Miller-Morrow criterion that considers the mean stress effect was used as follows [26]: where Δγ max is the maximum shear strain range, Δε n is the normal strain range, # f is the fatigue strength coefficient, m is the mean stress, b is the fatigue strength exponent, c represents the fatigue ductility exponent, ε # f is the fatigue ductility coefficient, and N f is the cycle number to crack initiation. Different loading levels were selected to generate the RCF S-N curve of the bevel gear. Given that this methodology cannot predict the contact fatigue limit, the horizontal segment of a practical S-N curve was not considered. The simulated S-N curve was further used in the following life estimation under a given load spectrum.

Compilation of the load spectrum
Heavy-duty vehicles often operate under rough and  complicated road conditions. The average value of the load-time history is T 2 = 2153 N•m, as depicted in Fig. 4. When the vehicle is driving on a complicated road, the load fluctuates significantly due to the unevenness of the road surface.
Statistical counting was conducted using the widely accepted rain flow method to derive the amplitude-mean rain flow matrix. The resultant torque amplitude-mean frequency histogram is shown in Fig. 5. Subsequently, the amplitude-mean probability distribution function was determined. Then, the two-dimension load spectrum was compiled, based on which the one-dimensional load spectrum was generated through the Goodman equivalent damage rule [27]. Finally, the one-dimensional standard load spectrum was used in fatigue life estimation.
The least squares method was used to test the distribution of the mean and amplitude frequency. Logarithmic normal, three-parameter Weibull, Weibull, and normal distributions were fitted. The results showed that the torque amplitude follows the Weibull distribution, and the mean torque follows the normal distribution. Figure 6(a) shows that the statistical amplitude frequency follows the Weibull distribution. Figure 6(b) depicts the fitted amplitude probability density curve, and Fig. 6(c) shows that the statistical mean frequency follows the normal distribution. Figure 6(d) illustrates the fitted mean probability density curve.
The mean value and amplitude of torque are independent variables, and the two-dimension load spectrum relates to both variables. Therefore, the joint probability density function of the mean and amplitude is defined as where x stands for the mean and y refers to the amplitude.
The joint probability density function was then used to calculate the frequency of each load interval. The formula is expressed as where N is the cycle number, S a1 and S a2 are the integral lower and upper limits of amplitude, respectively, S m1 and S m2 are the integral lower and upper limits of the mean, respectively, and N # is the cumulative frequency. The 600-second data of torque shown in Fig. 4 are insufficient to represent the entire loading history. Therefore, cumulative loading cycles should be extended to the well-recognized threshold of 10 6 cycles. To reflect a more realistic load history experienced during the entire life cycle, the two-dimension load spectrum is shown in a tabular form in Table 2.
The German company LBF [28] developed an eightlevel program load spectrum, which has been widely used for fatigue life estimation of truck gears. The main advantage of such a standard load spectrum is that it accurately reflects the actual loading history conditions of vehicles. On the basis of this method and the Goodman equivalent damage rule, the one-dimensional standard load spectrum with zero mean stress for the bevel gear was generated in this work, as shown in Table 3.

Damage accumulation rules
Many fatigue damage accumulation rules [29,30] have been proposed, and they can be classified into three main types, namely, linear, bilinear, and nonlinear. The Miner  linear, Manson bilinear, and Corten-Dolan nonlinear rules are generally accepted in engineering practice [31].
The fatigue damage variable D of the Miner linear rule is the ratio between cycle number n under service stress and fatigue life N of the material under this stress, and it is expressed as Under multiple stress amplitudes, the Miner damage is expressed as where D i means the damage suffered during a loading block, n i is the cycle number under the ith class of load, and N i is the fatigue life under the ith load class. The Manson bilinear rule assumes that the fatigue life consists of two stages for a two-level cyclic loading condition. As shown in Fig. 7, the fatigue damage for each where N i,f is the fatigue life under the ith load class, N 1 represents the fatigue life at the highest stress level in the load spectrum, and N 2 is the fatigue life at the lowest stress level in the load spectrum.
According to the Corten-Dolan nonlinear rule under the action of multistage stress, high-level stress not only causes damage to the structure but also affects the damage induced by low stress. The formula for fatigue life estimation under multistage stresses has the form where N is the total fatigue life under multistage stresses, 1 represents the stress value of maximum load with a unit of MPa, N 1 is the fatigue life under stress level 1 , i is the stress value for the ith load class, a i represents the ratio of cycle number of the ith load class stress to the total cycle number, and d is the material constant. In the absence of test data, Corten assumed that d/m = 0.85, where m is the slope of the S-N curve.  Figure 9 shows the calculated fatigue life of the bevel gear under various constant loading conditions with the torque T 2 increasing from 1000 to 5000 N•m. As the load level increased, the minimum fatigue life of the bevel gear decreased significantly from 7:1 Â 10 8 to 1:45 Â 10 6 .

Results and discussion
The material point with the minimum fatigue life was selected as a representative, and the evolutions of various stress components experienced by this material point under the loading condition T 2 = 3000 N•m are displayed in Fig. 10. The shear stress components presented different tendencies during a meshing cycle compared with the normal components. The normal stress components remained compressive, whereas the shear components changed their signs during the loading cycle, indicating a typical multiaxial state with considerable mean stress.
When lives at several constant loading levels are Table 3 Eight-level standard load spectrum estimated, the RCF S-N curve can indicate the relation between failure cycle N f and the maximum contact pressure, as shown in Fig. 11. For the classical S-N curve of contact fatigue, the relationship between stress and cycle number can be expressed by many forms of empirical equations and can be approximated linearly on a log-log graph. The most common empirical equation is the Basquin formula [32]: Use the logarithm of both sides to obtain where S is the stress amplitude in MPa, C is the constant, and N f is the cycle number to failure. The contact fatigue S-N curve of the driven gear can be curve-fitted as follows: lgS ¼ -10lgN f þ 23:9:     If the simple Miner linear rule is applied, since the damage is expressed as If the Manson bilinear rule is adopted, given that we have Considering that D II ¼ 1, N 2 ¼ 3:55 Â 10 7 , from which we can readily see N ¼ N f ¼ 7:1 Â 10 7 .
If the Corten-Dolan nonlinear rule is applied, then In a constant loading case, because no load sequence is involved, the selection of damage accumulation rule has no effect on fatigue life. This section illustrates the basic procedure for life estimation based on a damage accumulation law.

Load spectrum case
The aforementioned fluctuating load spectrum consisting of eight levels of torque was applied in this section.
With regard to the Miner linear rule, ( The following relationship should be satisfied: The Manson bilinear rule gives rise to the minimum fatigue life, and the Miner linear rule provides the maxima because the Miner linear rule assumes fatigue damage as a stationary process independent of the loading level and sequence. The Manson bilinear rule divides fatigue damage into two stages, which can be determined according to the loading sequence of high and low loads, as shown in Fig. 12. When a multistage load case is assumed in which more than two loading levels are involved, the total damage estimated by the Manson bilinear rule is mainly determined by the highest and lowest levels. Final failure is expected to occur when the total damage is less than unity, which indicates a shorter life compared with the Miner linear rule. According to the Corten-Dolan nonlinear rule, all damages are related to the maximum stress and the relationship between the maximum stress and each stress level. Owing to the consideration of the relationship between the current stress level and the maximum one, the damage rate at each stress level increases compared with that in the Miner linear rule, resulting in a reduced fatigue life. Notably, the life magnitude predicted by the Corten-Dolan rule significantly depends on the selection of parameter d, which should be determined experimentally.

Conclusions
A systematic numerical model that considers multiaxial fatigue and the load spectrum was developed to thoroughly understand the contact failure process of bevel gears. Various fatigue damage accumulation rules were also discussed, and the following conclusions were obtained.
1) The proposed numerical methodology can be utilized in the contact fatigue life estimation of bevel gears under given load spectrums because the multiaxial stress response is correctly captured in the repeated contact loading process. This methodology can be a promising approach for fatigue analysis of other types of gears or bearings because it allows a flexible selection of any other damage accumulation rule.
2) In a constant loading case, because no load sequence is involved, the selection of the damage accumulation rule has no effect on fatigue life. Under a fluctuating loading spectrum, the Miner rule provides the most optimistic result among three rules, and the Manson bilinear rule produces the most conservative one.