If you work with linear ball or roller bearings, you’re familiar with the L10 bearing life equation, which states that bearing life is equal to the bearing’s rated dynamic load capacity, divided by the applied load, raised to a power of 3 (for ball elements) or 10/3 (for roller elements).
Introduced in 1947 by Gustaf Lundberg and Arvid Palmgren, based on work done at SKF, the bearing life equation represented a significant development in the selection and application of bearings. Prior to the adoption of the Lundberg-Palmgren equation, each manufacturer used its own method to determine bearing life, making comparisons between products difficult, if not impossible. The Lundberg-Palmgren equation represented a significant advancement for the bearing industry, and to this day serves as the foundation of bearing sizing and selection.
In the original paper by Lundberg and Palmgren, the load/life exponents (p) used in the bearing life equation were given as “3” for ball bearings and “4” for roller bearings with pure line contact. However, because the rollers used in bearing applications don’t experience pure line contact with the bearing raceway under some loads, the authors revised the exponent used in the roller bearing equation from “4” to “10/3” in 1952, and this version of the formula is still used today.
The L10 bearing life introduced by Lundberg and Palmgren specifies the distance (or number of revolutions, for radial bearings) that 90% of identical bearings, operating under identical conditions, can travel before fatigue (spalling or flaking) occurs on the rolling elements or raceways.
But in some industries and applications, a reliability greater than 90% is required, especially if the bearing performs a mission-critical function whose failure could result in physical harm to people or loss of an entire system. Case in point: If you’ve ever worked on an application in the aerospace, medical, pharmaceutical, or transport industry, you’ve probably been tasked with selecting a bearing that has 95%, 99%, or even higher reliability.
Fortunately, the most recent version of the ISO standard that deals with bearing life calculations — ISO 281:2007, Rolling bearings – dynamic load ratings and rating life — addresses the need for bearing life reliabilities higher than 90%. The standard now includes a list of factors, denoted a1, that provide adjustments to the bearing life to reflect reliabilities ranging from 95% to 99.95%.
Notice that in this discussion we’ve referenced the ISO 281:2007 standard, which addresses rolling bearings. But if you’ve read some of our other articles on bearing life, you might recall that the dynamic load ratings and life for linear rolling bearings are specified in the ISO 14728-1:2017 standard. We refer to ISO 281:2007 in this article because ISO 14728-1:2017 references the ISO 281:2007 standard for life calculations.
For example, if a linear bearing application requires a reliability of 98% (referred to as L2m life, since it implies that only 2% of bearings are statistically likely to fail), simply multiply the calculated L10 life by 0.37, which is the adjustment factor (a1) for 98% reliability. This gives the distance the bearing will travel with 98% certainty before fatigue occurs.
For a linear ball bearing (p = 3) with an applied load (F) of 1000 N and a load capacity (C) of 1320 N, the L2m life (98% reliability) will be:
Similarly, to determine the required dynamic load capacity that will allow a linear ball bearing with a 1000 N load to travel 85,000 km, with 98% reliability, the equation can be rearranged as:
Regardless of the calculated reliability, it’s important to remember that the bearing life equation assumes bearings are “made with contemporary, commonly used, high quality hardened bearing steel in accordance with good manufacturing practice,” and that the group of bearings are identical and operate under the same conditions.
In other words, bearing life is a theoretical calculation based on probabilities of failure. Calculating bearing life with a higher reliability does not mean that a bearing won’t fail before it reaches the specified travel, but it does reduce the probability of bearing failure.
Controversy in the world of bearing life calculations
Anyone who has worked with radial or linear rolling bearings has probably used the bearing life equation so often that they can recite it in their sleep. But it turns out that this ground-breaking equation — introduced nearly 70 years ago and used by virtually every design engineer in the world — has a history of controversy that exists to this day. A key point of contention regarding bearing life is whether bearing steel actually has a fatigue stress limit — a point of loading under which no material fatigue would occur (assuming elastohydrodynamic lubrication is present between the bearing surfaces).
The most recent ISO standard, ISO 281:2007, incorporates a life adjustment factor (denoted aISO) that includes a fatigue stress limit, but some engineers and researchers argue that there is no evidence for a fatigue limit in bearing steels. In 2010, the Society of Tribologists and Lubrication Engineers (STLE) published an article explaining the evolution of bearing life calculations and how the new ISO 281:2007 standard came to be. One of the issues addressed in the article is the inclusion of the fatigue stress limit. One month later, STLE published a rebuttal to the new ISO standard and its use of the fatigue stress limit, written by Erwin Zaretsky, a distinguished research associate at the NASA Glenn Research Center in Cleveland and a well-respected engineer in the bearing community.
Reading these two articles, I found it interesting that something most of us in the engineering world take for granted — the probability of failure of a bearing under a given load — is still difficult to accurately quantify and whose further development and refinement is being driven by two conflicting theories. Even with incredible advances in material science, testing, and calculation methods, we really don’t know exactly how bearing elements (balls, rollers, and raceways) behave under the extreme loads, pressures, speeds, and lubrication conditions they encounter in even “normal” applications.
If you’re interested in the history of the L10 bearing life equation and the ongoing discussions around its advancement, I highly recommend reading the two articles from STLE.