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Summary
Rolling bearing design has benefited from
improved materials technology, especially steel cleanliness and better
understanding of the tribo-physical conditions that prolong operational life.
Users of bearings have put an increasing emphasis on safe
design coupled with lighter, more competitively priced products.
This has put increasing pressure on bearing manufacturers to
offer predictable performance in rolling bearings. In turn, this has encouraged
researchers to review and update standards for bearing life to take into
account the tremendous progress in engineering technology. As a consequence,
SKF has introduced the latest manifestation of its famous formula for
predicting rolling bearing life expectancy. List of symbols:
a |
Contact semi-axis in
transverse direction, [m] |
a1,
a2,a3 |
Life adjustment factors:
1 reliability, 2 material, 3 operating conditions |
a23,aSKF |
Life adjustment factor,
SKF Stress Life Factor |
A |
Constant of proportionality,
scaling factor |
c |
Exponent in the stress-life
equation |
C |
Basic dynamic load rating, [N]
|
dm |
Bearing pitch diameter,
[m] |
e |
Weibull exponent |
h |
Exponent in the stress-life
equation |
l |
Length of raceway contact,
[m] |
L |
Life, [Mrevs] |
L10 |
Basic rating life,
(10 % failure life), [Mrevs] |
L10aa |
Adjusted rating life according
to SKF Life Theory, [Mrevs] |
Ln |
Bearing life at (100-n)%
reliability, [Mrevs] |
Lna |
Adjusted rating life,
[Mrevs] |
N |
Number of load cycles,
number of tests sets |
p |
Exponent in life equation
|
P |
Equivalent dynamic bearing
load, [N] |
Pu |
Fatigue load limit, [N] |
S |
Survival probability [%] |
w |
Exponent in the load-stress
relationship |
zo |
Depth of max. orthogonal shear
stress of Hertzian contact, [m] |
ßcc |
Lubricant cleanliness degree
according to ISO 4406 |
h |
Life factor for added
stress |
hb |
Lubrication factor |
hc |
Contamination factor |
k |
Viscosity ratio |
v,v1 |
Actual and required kinematic
viscosity at the operating
temperature, [m²/s] |
t0 |
Maximum orthogonal shear
stress amplitude in Hertzian contact, [Pa] |
tu |
Fatigue limit shear stress,
[Pa] |
|
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The SKF
formula for rolling bearing life
The demands for safe
design and lighter, competitively priced products have put a new emphasis on
predictability of rolling bearing performance. This has prompted researchers to
update standards for bearing life to take into account progress in engineering
technology.
Present standards for
bearing life calculation are based on work carried out at SKF in 1947 by Gustaf
Lundberg and Arvid Palmgren. The life equation was formulated using the Weibull
probability theory of fatigue developed in 1936. This allowed the calculation
to include bearing reliability. It marked a substantial step forward in the
achievement of predictive methods for the applications of this essential
machine element.
The ability to estimate the life of a bearing and to select,
on a rational basis, a specific bearing suitable for a particular application
was a breakthrough for engineering design. Since then, science and technology,
in particular tribology, have made tremendous strides. The latest theoretical
knowledge and calculation techniques supported by powerful modern computers are
now applied in bearing technology and design work. Also, steel manufacturers
can make cleaner steels to more accurate and consistent formulations. In
addition, greatly improved lubricants allow better separation than before of
moving contacting surfaces in rolling bearings, while modern mass production
manufacturing methods produce high precision bearings to ever-higher levels of
quality.
Such technological progress requires that bearing life
calculations take into account the improved performance of todays
bearings. Over the years, performance improvements were accounted for largely
through increased dynamic load ratings, which yielded longer calculated lives.
Application or adjustment factors were sometimes used to account for
operational conditions, such as special bearing materials and lubricants. Until
recently, these life calculations served industry sufficiently well.
However, observations about the predominant failure
mechanisms in modern rolling bearings have added new knowledge in this field.
It has been found that fatigue failures are more frequently initiated at the
surface rather than from cracks formed beneath the surface.
Consequently, attention has been paid to the influence of
surface finish and contamination on bearing performance. The existence of a
fatigue limit for modern bearing steels has also been observed. Such
considerations have resulted in the publication of improved bearing life
calculations and papers studying the effects of contamination and surface
finish on bearing life. In particular, the rolling contact fatigue
life model published by Eustathios Ioannides and Tedric Harris (1985)
introduced two innovations to the Lundgren and Palmgren theory:
i) a threshold value of the local stress, a fatigue limit, below which no
fatigue is expected to occur in the bearing,
ii) consideration of the stressed volume below a contact as an array of
relatively small volume elements, each experiencing individual local stresses.
In this way, real local stresses and many effects stemming from surface
stress concentrations, such as edge stresses and contamination dents, can be
introduced consistently into fatigue life predictions. Such a comprehensive
approach requires access to complex computer programs for bearing life
calculations, as integration of the failure risk over complicated stress fields
is required. Many bearing users do not have access to such resources. Instead
they usually apply simple formulae readily available in bearing
manufacturers catalogues or standards.
To support these needs, and reap the benefits provided by
advances in technology and knowledge, the SKF Life Equation* was made
available. This equation was developed by introducing an effective real
weighted stress that is averaged throughout the risk volume. The fatigue limit
and the local stress were implemented into equations similar to those used by
Lundberg-Palmgren. In practice, this is done by introducing a stress factor
that acts on the fatigue limit of the bearing. This factor accounts for the
actual stress condition of real bearings but is not included with the ideal
Hertzian stress field, as used in the Lundberg-Palmgren classical analysis.
This approach led to a simple expression that can be readily calculated in a
similar way to the ISO 281 (1990) formulation. This equation retains much of
the original simplicity of the load-life equation but also describes the
performance of modern bearings more accurately than before.
Standardised life equations
Fatigue life modelling is central to all bearing life prediction. The
traditional crack initiation or cumulative damage model invokes a stress power
law to account for the portion of the life spent in the initiation of a crack,
which dominates the complete lifetime of rolling contacts. Probabilistic
methods were applied early in bearing endurance prediction because of the
natural dispersion of bearing life. Lundberg and Palmgren (1947) used the
Weibull (1939) probability distribution of metal fatigue to establish the basic
theory of the stochastic dispersion of bearing lives, as shown by the equation
below:
(1)
By substituting the Hertzian contact parameters (in terms of the applied
load and contact geometry), they obtained the load-life relationship for
rolling bearings. This can be expressed in its final form in a very simple way:
(2)
where C is the bearing basic dynamic load rating, a factor that
depends on the bearing geometry, and P is the equivalent bearing load.
The exponent p is 3 for ball bearings and 10/3 for roller bearings.
Equation 2 was adopted by ISO in the recommendation R281-(1962). In 1977, the
multiplicative constants a1,
a2 and a3 were introduced to account for different
levels of reliability, material fatigue properties and lubrication, resulting
in the standard used today, ISO 281 (1990):
(3)
Many bearing manufacturers, who have recognised the interrelationship of
material and lubrication effects, use it as
(4)
SKF life formula
A simple analytical formulation for bearing life that includes consideration of
the fatigue strength of the material evolved from the rolling contact fatigue
model set forth by Ioannides and Harris (1985). This was initially applied in
the numerical solutions of the fatigue risk of 3-D stress fields of rolling
contacts. Due to the self-similarity of Hertzian stress fields, it was found
that in the case of strictly Hertzian contacts, simplification was possible.
This could be introduced by applying stress criteria based on the maximum
alternating shear stress amplitude tO
above a threshold value and of a fatigue limit stress tu of the rolling contact*. Thus the
introduction of the fatigue limit stress tu can be effected through the replacement of
the shear stress amplitudetO of the
Hertzian stress field of equation 1, by the difference tO tu . Then equation 1 becomes:
(5)
In the above formulation, Macauley bracket notation is used; in other words,
the term is set to zero if the quantity enclosed is negative. To obtain the
Life Equation for rolling bearings in the presence of a material fatigue limit,
one may use equation 5. This equation differs from the original Lundberg and
Palmgren form (equation 1) only by the Macauley bracket on the right side of
equation 5. It is, therefore, possible to work through the bearing capacity
formulation very much as in Lundberg and Palmgrens original work.
Applying the well-known relationship between induced stresses and applied load
for Hertzian contacts, either point or line, as done by Lundberg and Palmgren
(1947), an equation corresponding to equation 5, but written in terms of the
equivalent load and the basic dynamic load rating, is obtained. For 10 percent
probability of failure, the corresponding bearing life is expressed as*:
(6)
where the h parameter is a factor (corresponding
to a stress factor) that is introduced to account for the actual stresses
present in real bearing contacts. This parameter is not included with the ideal
Hertzian stress field used in the derivation of the original Lundberg and
Palmgren (1947) life formula. It can be shown that, besides the bulk stresses
due to the heat treatment and mounting, the stress factor h
depends to a large extent on the lubrication condition of the contact
and on micro-scale stress concentrations due to denting or imperfections.
Consequently, h is given as a function of the
environmental conditions, (lubrication and contamination), and in relation to
the bearing size.
(7)
introducing equation 7 in equation 6 we get:
(8)
To simplify the use of the above equation, in 1989 SKF introduced a Stress
Life Factor aSKF for each specific
bearing (brg) class: radial ball bearings, radial roller bearings, thrust ball
bearings, thrust roller bearings. This factor can be calculated and plotted as
shown in figure 1:
(9)
Furthermore curves of the function hc(k,dm,
bcc) were also derived, as
explained by Bergling and Ioannides (1994) and Ioannides and his colleagues
(1999). Examples of these parametric curves are given in figure 2. With
this information, the life equation 8 can be written in a simple way:
(10)
This is the formulation of the life equation used in the SKF General
Catalogue since 1989. This equation, in conjunction with the diagrams of
figures 1 and 2, can be used to calculate the life of rolling bearings
in a straightforward manner that is similar to the previous life calculation,
ISO 281:1990. However, equation 10 can account for specific lubrication and
contamination conditions of the bearing and for the increased life that is
experienced in case of lightly loaded, clean and well lubricated bearing
applications.
Experimental verification
To verify the accuracy of the SKF Bearing Life Equation (equation 10),
extensive bearing endurance tests were performed on more than 8,000 bearings
under a variety of lubrication and contamination test conditions. An overview
of the mean L10 measured in these tests is shown in figure
3.
A comparison was made between predicted lives and the actual endurance test
data of figure 3. This was performed by plotting the ratio between the
experimental bearing life and the basic rating life of the bearing, i.e.
aexp=L10exp/
L10 versus the corresponding stress factor parameter,
hc
Pu/P.
In figure 4, a group of experimental points of ball
bearing operating with a full oil film is shown. The data points of the test
results mostly overlap with the corresponding life-ratio curve, i.e.
aSKF , calculated according to equation
9. Figure 4 shows very good agreement between the relative life
calculated with the SKF Life Equation and the endurance test results. A more
comprehensive evaluation of the experimental data also indicated that the total
bias in life calculation could be reduced by half using the SKF Life Equation
in comparison to the results of life calculations based only on bearing
standard ratings. This improved accuracy provides a better match between theory
and test data, validating the choice of the model constants used and the
present Life Equation. Furthermore, the asymptotic trend displayed by the
function aSKF (k, hc Pu/P),
figures 1 and 4 as the stress factor parameter hc
Pu/Ptends towards lower stress conditions for the
bearing, and is a further indication of the postulated fatigue stress limit in
rolling contact fatigue. The behaviour of the curve aSKF(relative stress vs. relative life) is
indeed similar to the familiar Wöhler (stress vs. number of stress cycles)
curves used for plotting the survival probability of specimens subjected to
fatigue testing at different stress level.
Conclusions
Rolling bearings have made it possible to support heavy loads at high
rotational speeds with good reliability and a minimum of friction. This basic
technology has allowed the massive mechanisation process that has characterised
the past century and has profoundly revolutionised our way of life. Practical
tools for better selection and use of rolling bearings have a significant
effect on the way machines operate, their efficiency and cost. This affects
each of us through the energy consumption and running costs of society at
large. The SKF Life Equation introduces a new and higher standard in life
calculations to help in the prediction of bearing performance and to respond to
the continual quest for better ways to design, select and use bearings.
The SKF Life Equation manages to describe the complex
tribo-system in which the bearing operates using a few key parameters. This is
an important feature of the model that concentrates on the effect of a few
important factors that have significant consequences for the bearing life.
Furthermore, the SKF Life Equation recognises that the fatigue effect arising
from these factors cannot be written as a linear superposition of the risks
induced from individual stress components. Rather than attempting to derive
independent life modifying factors for lubrication, contamination etc., a
single multidimensional factor, aSKF =
f (k,
hc Pu/P), is introduced depending on
those relevant factors describing the tribo-physical state of the system. By
this approach, the best use is made of the available input data, to the benefit
of both the designer and the user of a machine. The good agreement with the
experimental results and improved accuracy achieved using the SKF Life Equation
support the use of this approach for the calculation of the life expectancy of
modern rolling bearings.
* Ioannides E., Bergling G. and Gabelli A. An analytical
formulation for the life of rolling bearings. Acta Polytechnica
Scandinavica, ME 137, Espoo (1999).
by Eustathios
Ioannides,SKF Engineering and Research Centre (ERC),
Nieuwegein,
the Netherlands & Imperial College of Science, Technology and Medicine,
London, UK;
Gunnar Bergling, AB SKF, Gothenburg, Sweden, and
Antonio Gabelli, SKF Engineering and Research Centre (ERC), Nieuwegein,
the Netherlands.
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