Rolling bearings and gears may suffer from surface and sub-surface initiated fatigue, termed “spalling” for bearings and “pitting” for gears. Life estimation for rolling bearings assumes the Palmgren-Miner linear damage law to account for steady and dynamic loads. Bearing manufacturers’ catalogues and international standards such as IEC 61400-4 for design of wind turbine gearboxes present methods to calculate the equivalent bearing dynamic load from the various applied loads and their fraction of operating time. Because of the assumption of linear damage, the Palmgren-Miner law cannot take account of load level dependence, load sequence dependence and load interaction.
Bearing life estimation methods such as ISO 281 implicitly assume a basic stress-cycles fatigue curve with the application of factors to the basic life estimate to account for reliability level, material cleanliness, lubricant cleanliness and lubricant viscosity. Bearing manufacturers may choose to use a different fatigue curve based on practical experience in particular applications, but in general, the original Lundberg and Palmgren fatigue curve is used.
It accounts for both surface and sub-surface initiated fatigue. In theory, bearings can achieve infinite life if contact stresses are held below about 1500MPa providing high quality steels and manufacturing techniques are used. However, poorer quality may result in a lower endurance limit.
ISO 281 uses the concepts of a fatigue limit load Cu below which life becomes infinite. This is reflected in the life modification factor aISO that is limited to a maximum value of 50. So in effect, infinite life is not predicted but can lead to long estimated lives.
In principle, it is possible to re-calculate aISO for different values of endurance limit, with, for example, a reduction in fatigue life of a factor of 3 if the endurance limit is 1200MPa.
Turning now to pitting strength of gears, ISO 6336-5 uses allowable contact stress numbers σHlim for a wide range of gear materials and qualities relating to a probability of failure of 1% at 2 million cycles. For example, the stress number varies from 1300 to 1650MPa for case hardened wrought steels of qualities ML, MQ and ME, with MQ corresponding to 1500MPa. This applies to hardness levels similar to bearing steels. Whilst not clearly defined, it would appear that this corresponds to a maximum contact stress as opposed to a mean contact stress so direct comparison can be made with ISO 281.
ISO 6336-2 then calculates the nominal contact stress σH0 from the tooth loading, gear dimensions and a number of Z factors covering zone and contact ratio amongst others. The zone factor ZH transforms the tangential load to normal load at the pitch point, whilst the contact ratio Zε may be taken as 1 in this instance.
These can be categorized according to geometry, materials, lubrication and general influence factors.
The calculated contact stress σH has further factors applied to the nominal stress σH0 by means of general influence factors K. KA and KV relate to the external load sources and internal manufacturing accuracy. KA may reach values in excess of 2 where excessive torque fluctuations may arise from driven and driving machinery, whilst KV may approach 1 for high quality manufacture. The face factor KHβ accounts for non-uniform distribution of load over the gear face whilst KHα accounts for non-uniform distribution of transverse load between pairs of simultaneously contacting teeth. For purposes of comparison, these are assumed to be 1. ZB is also applied to σH0 and here is assumed to be 1.
The resulting value of σH should be less than the value of the permissible value of stress σHP that is calculated from σHlim with modification factors for the effects of the lubricant film, geometry and materials. However, a safety factor SHmin is introduced that is agreed between designer and customer. The life factor ZNT accounts for higher contact stresses that may be tolerated for a limited life. It may typically be in the range 1 to 1.4 and increases the permissible contact stress.
The size factor ZX accounts for the greater likelihood of encountering a material defect in a larger stressed volume. Interestingly, ISO 6336-2 does not provide any guidance on a suitable value for larger gears.
The lubricant factors ZL, ZV and ZR generally amount to less than 10% so are set to 1 here as well. So in summary, the nominal stress σHo should be enhanced by a factor of up to 2 due to KA and the result should be less than σHP. The permissible contact stress σHP is calculated by enhancing the contact stress number σHlim by factor ZNT with an additional factor ZX (presumably less than 1) to account for size. The permissible contact stress is reduced by the safety factor SHmin that is recommended to be a minimum of 1.25 in IEC 61400-4.
In comparison with ISO 281, therefore, gear design allows for higher reliability through its use of 1% failure probability (normally 10% for bearings), introduces a safety factor SHmin, has a factor KA that accounts for transient torques, has a size factor ZX and different contact stress numbers σHlim related to material quality.
Whilst bearing manufacturers may have recommendations for safety factors (especially for shock loadings), they are not explicitly stated in ISO 281.
Turning to variable loadings, ISO 6336-6 uses fatigue curves for different material types that may or may not show an endurance limit. In the former case, stresses below the endurance limit do not contribute towards the fatigue damage summation. In order to calculate the damage summation, contact stresses are determined for each bin in a histogram of torques. Then the cumulative ratio of cycles to cycles to failure for each stress level is formed according to Miner’s summation.
So the real issue here is the construction of the fatigue curves, which can be related to different levels of failure probability like 1, 5 or 10%. ISO 6336-6 states that pitting strength may be determined by experiment or by the rules of ISO 6336-2. It is possible to incorporate a factor of safety S by repeated iterations of the Miner’s summation until a value of 1 is reached.
McVittie and Errichello conclude that Miner’s summation seems to work for assessment of pitting strength under variable loading from analysis of a range of industrial gearboxes. Furthermore, peak loadings cannot be ignored in gear life calculations as they frequently cause the most damage.
The apparent success of Miner’s summation may therefore in part be due to the safety factors, reliability level in the fatigue curve and material-specific fatigue curves. Gear fatigue curves have different shapes to those assumed in ISO 281 for bearings, possibly with lower permissible contact stresses at lower cycles. Specific comparison with bearing fatigue curves may prove enlightening.
In conclusion, bearing life estimate methods such as ISO 281 may benefit from fatigue curves more appropriate to the material in use, endurance limits related to the material quality, factors for size and an appropriate safety factor. Should life estimates still deviate significantly from actual performance under variable loading conditions, then alternatives to Miner’s summation may have to be considered.