SKF Bearing Reliability
SKF Bearing Reliability
SKF Bearing Reliability
Summary
This article provides an introduction to reliability engineering
terminology and concepts, such as reliability functions, failure
rates, typical failure patterns, Weibull and exponential failure
distributions, and Mean-Time-Between-Failure (MTBF). Life
concepts of rolling element bearings are explained. Examples
and references for further reading are also provided.
GS02005
Gerard Schram and René Klerx
SKF Engineering & Research Center
14 pages
April 2003
Mean Time Between Failures (MTBF): A Often a distinction is made between inherent
basic measure of reliability for repairable and operational reliability. Inherent reliability
items. Mean time between failures is relates to a product reliability from design
calculated from the total accumulated point-of-view. Operational reliability stands
operating time divided by the number of for the product being operational in the field.
failures during the same period. In the context of maintenance, we generally
speak in terms of operational reliability.
Total Accumulated Operating Time
MTBF = Failure rate: The total number of failures
Number of Failures
within an item population, divided by the total
Mean Time Between Critical Failure time expended by that population during a
(MTBCF): A measure of system reliability, particular measurement interval under stated
which includes the effects of any fault conditions. Other formats include the number
tolerance that exist. The average time between of failures per year, and in some cases, it is
failures that cause a loss of a system function common to express failure rate as the number
defined as “critical” by the customer. of failures per hour, or the number of failures
of an item per unit time.
Mean Time to Failure (MTTF): A basic
measure of reliability for non-repairable items. Pump example (1): Assume 2 pumps failed
Mean time to failure is calculated from the out of a population of 16 pumps during three
total elapsed time or operating time divided by months. The failure rate is (2/16) / (0.25 year)
the total number of failures (in case of more = 0.5/year.
than one item).
Hazard rate: Hazard rate is the instantaneous
speed of failure - being the instantaneous
Total Elapsed Time
MTTF = failure rate. Hazard rate is the ratio of failures
Number Of Failures that occur in an interval to the size of the
population at the start of the interval, divided
While MTBF excludes time that the
by the length of time. Of course, in case of
equipment is not available or in use, MTTF
© 2004 SKF Reliability Systems All Rights Reserved 2
Reliability and Life
constant failure rate, the hazard rate is properties. These failures are due to old
constant. age.
• Wear out: constant hazard rate with a • Infant mortality: high failure rates that
distinct wear out region. Failure rate show up early in usage followed by a
increases with operating time. Process constant hazard rate.
indicates a deterioration of material
Figure 1. Different Failure Rate Patterns. The Percentages Refer to the Typical Spread of Equipment that Show
These Curves.
Infant mortality is clearly the primary failure Infant - Inadequate stresses during
pattern. Scheduled maintenance activities do mortality running-in / burn-in (e.g. due to
little or nothing to defend against the portion bad installation, procedures)
of infant mortality failures. Assets where - Quality problems components
failures are a significant problem should be - Rework / refurbishment
examined using root cause failure analysis to problems
identify the root causes and prevent them from
occurring again. Random - Maintenance / human errors
- Inherent failures, not induced
In order to give some idea of possible failure - Mixtures of failure modes
reasons, consider the table below.
Wear out - low cycle fatigue In other words, the probability of a failure
- constant corrosion / erosion occurring can be expressed as a number in the
failure modes range 0 to 1, where zero means that it will not
- Scheduled replacement may be occur, and 1 means that it will occur.
cost effective
By way of an example consider a quantity of
100 components that all fail within a year of
purchase, as shown at table 1 below.
Of the bearings that fail in service, past SKF
research shows that approximately 16% fail Month 1 2 3 4 5 6 7 8 9 10 11 12
due to faulty mounting, 36% due to
Failures 4 10 15 17 15 13 10 8 6 2 0 0
inadequate lubrication, 14% due to ingress of
contamination, and 34% from fatigue. Fatigue Table 1Example - component failures
failure often occurs because of abuse in
service (e.g. result of overloading). Therefore, The data shown above can be represented
in practice, bearings fail prematurely graphically in the form of a Relative
according to a combination of infant and Frequency Histogram, as at figure 1 below:
random failure patterns. In principle, pure
fatigue or wear under normal conditions
follow a wear out curve.
Reliability Modeling
Failure Distributions, Failure and
Survival Probabilities
First we introduce failure distribution f(t): the
probability density function of failure
distributed over time.
t
The total number of failures occurring in the
F (t ) = ∫
o
f (t )dt
first six months is 74. Dividing this number by
the total population (in this case 100) gives
Notice that the probability of failure starts 0.74, a number that represents the cumulative
from zero and converges to one over time. probability of failure F(t) occurring within the
first half year.
The reliability (or survival) function R(t),
which means the probability of survival is the By transforming the number of failures by
complement of the failure function: month into these probabilities the data shown
above can be represented in a different form
R (t ) = 1 − F (t )
known as the probability density function F(t).
This can be plotted as represented at figure 2
Being the complement of failure probability,
below, such that the area under the curve
the reliability starts from one and converges to
between time 0 and any time t represents the
zero over time.
cumulative probability of failure.
exp( λ .t )
[MTBF ]system ≈ 1 /[h(t )]system 0.5
λ
• β is the shape parameter, which makes 1< β < 4 Implies early wear out; the failure
the shape of the distribution, when the rate is increasing over time.
Weibull function reduces to the
exponential distribution function. When β >4 Implies old age and rapid wear
β ≈ 3.4 , the distribution comes close to a out; the failure rate is increasing
quickly.
normal distribution.
1.9 0.96177 2.9 1.82736 Finally, assume a case with increasing hazard
rate, β = 1.2, γ = 0, θ = 1 , which
2.0 1.00 3.0 2.00
represents wear-out failure behavior. The
MTBF is calculated as 0.94 (Figure 7).
β 1.2 γ 0 θ 1
A few examples follow. First assume
β = 1, γ = 0, θ = 1 . This means the
2
β
MTBF becomes 1. The functions are plotted exp
(t
θ
γ)
1
in Figure 5. β 1
β. t γ
θ θ 0.5
β 1 γ 0 θ 1
2 0
0 2 4
t
β. t γ
β 1
.exp (t γ)
β
1.5
Figure 7. Example Weibull Distribution, Reliability
θ θ θ Function, and Hazard Function
exp
(t γ)
β
1
β = 1.2, γ = 0,θ = 1 .
θ
β 1
β. t γ
θ θ 0.5
For the sake of completeness, the Weibull
failure distribution is shown for
0
0 2 4
β = 3.4, γ = 0, θ = 1 . In this specific case,
t the Weibull distribution approximates a
Figure 5. Example Weibull Distribution, Reliability normal distribution. The MTBF is calculated
Function, and Hazard Function
as 0.90 (Figure 8).
for β = 1, γ = 0,θ = 1 .
β 3.4 γ 0 θ 1
Next, assume a case with decreasing hazard 2
rate, β = 0.8, γ = 0, θ = 1 , which
represents infant mortality failure behavior. In β. t γ
β 1
.exp (t γ)
β
1.5
θ θ θ
this case, the MTBF is calculated as 1.13. The β
functions are plotted in Figure 6. exp
(t
θ
γ)
1
β 1
β. t γ
β 0.8 γ 0 θ 1 θ θ 0.5
0
0 2 4
β 1 β
β. t γ .exp (t γ) 1.5 t
θ θ θ
Figure 8. Example Weibull Distribution, Reliability
exp
(t γ)
β
1
Function, and Hazard Function
θ
β = 3.4, γ = 0,θ = 1 .
β 1
β. t γ
θ θ 0.5
The Weibull distribution can be used to
0
0 2 4
determine points on a bathtub curve where the
t failure rate is changing from decreasing, to
Figure 6. Example Weibull Distribution, Reliability constant, to increasing. The bathtub curve
Function, and Hazard Function accounts for failure patterns according to
β = 0.8, γ = 0,θ = 1 . Weibull distributions ß <1, ß = 1 and ß >1.
Based on experience, a test is designed and Raw 2.31 4.4 1.65 3.73 6.27
Est
performed on 7 hub units over 3 months.
Within that test, 84 wheels are used. For both Bias 2.29 4.4 1.63 3.72 6.29
corr
wheels and hub units, a 2 parameter Weibull 50%
failure distribution is estimated ( β ,θ ) , from
Bias 1.97 4.0 1.34 3.39
which L10 , L50 , L90 are distillated (R(t)=0.9, Corr
R(t)=0.5, R(t)=0.1, respectively). 5%
Figure 9. Weibull Plot for Wheels. Vertical Axis Show the Failed Percentage of the Population Wheels, Horizontal
Axis Show the Hours that Were Run (Life). The Blocks Indicate the Individual Wheel Failures. The Solid Line is the
Estimated Failure Function.
Figure 10. Weibull Plot for Hub Units. Vertical Axis Show the Ailed Percentage of the Population Hub Units,
Horizontal Axis Show the Hours that Were Run (Life). The Stars Indicate the Individual Hub Units that Failed: 4
Out of 7. The Solid Line is the Estimated Failure Function.
data), relevant information may be found. A typical failure modes for common
few example databases: components.
• OREDA for Offshore Reliability Data, Another list of typical life numbers for various
with turbines, compressors, etc. equipment and components can be found at
(http://www.oreda.com) this website [2]:
http://www.barringer1.com/wdbase.htm
• Process Equipment Reliability Database
(PERD) of the American Institute of Rolling Element Bearing Life
Chemical Engineers
(http://www.AIChe.org) Generally speaking, suppliers provide L10
values for their bearings, but what does that
• Reliability Analysis Center (RAC), see mean in the context of reliability? And what
also [1]: http://rac.iitri.org/
is the relation between L10 and MTBF?
Typical MTBF numbers are listed at the
Reliability Analysis Center’s site. To give an Generally, MTTF is used instead of MTBF, as
idea, some data is provided below. bearings are often replaced after failure. Only
large and relatively expensive bearings are
MTBF (Hours) worth repairing.
(Adapted from [1])
Reliability of rolling element bearings are
Air handling unit 24,000 - 70,000
modeled with Wiebull failure distributions:
Boiler 18,000 - 31,000
β
Chiller 18,000 - 82,000 1 L − L0
Compressor 34,000 - 66,000 R ( L) = exp − ln
1 − P L p − L0
Condensor 26,000 - 200,000
Control Panel 465,000 - 780,000 Notice that β , L0 , L p are known as parameters,
Diesel Engine 7,000 - 14,000
and P stands for the probability of survival
Generator
corresponding with L p . Compared to the
Gas Turbine 5,000 - 30,000
Generator general Wiebull function, L p corresponds to
Motor Generator Set 71,000 - 145,000 the scale parameter, and L0 is the location
Pump 31,000 - 143,000 parameter. Generally, a value β = 10 / 9 is
Computer Controller 19,000 - 22,000 used for rolling element bearings. The
assumption of minimum life may hold (for
Switchgear 240,000 -
example L0 = 0.05L10 ), but usually a value of
(insulated bus) 2,5000,000
zero is used L0 = 0 .
Transformer, High 179,000 -
Voltage 12,000,000
Uninterruptible 38,000 - 785,000
Power Supply
Moreover, various (charged) databases with Applied Mechanics, Vol. 18, No. 3, September
RAM data exist. 1951.