Failure takes place when a unit, which can be a machine, or a sub-part of a machine, or a component stop fulfilling its purpose. We mostly make use of failure statistics from the past to understand exactly what the failure entails. This then enables us to take the appropriate decision regarding the strategy for managing a specific failure. This video explores the various ways in which failure is described quantitatively for (1) systems (machines), and (2) components.

1. Quantitative Descriptions of Failure
Refer to Coetzee, J.L., RCM ProAktiv, pp. 62-71

2. 1. ROCOF
2. MTTF
3. Failure Density
4. Cumulative Failure Distribution
5. Survival Function
6. Hazard Function (Force of Mortality)
Quantitative descriptions of failure

3. 1. ROCOF
Quantitative descriptions of failure

4. • The ROCOF (Rate of OCcurrence Of Failures) is an indication of how fast
failures are following one another.
• It is calculated by dividing the number of failures by some quantitative
measure of the cumulative use of the system or component to accumulate
all these failures.
• The ROCOF is a very useful measure in determining whether the reliability
of a system or sub-system is busy improving (ROCOF decreasing), constant
or decreasing (ROCOF increasing) over time.
ROCOF (failure rate)

5. ROCOF
ρ( )
T =
Number of failures in period
Length of p
period
Typical units: Failures/hou
r
Failures/to
n
Failures/unit produced

6. Typical ROCOF for system

7. 1. ROCOF
2. MTTF
Quantitative descriptions of failure

8. • The Mean Time To Failure (MTTF) is the reciprocal of the
ROCOF (i.e. 1/ROCOF).
• Thus where the ROCOF represents the number of failures per
time (or other measuring) unit, the MTTF gives an indication
of the average life between successive failures in operating
hours (or other measuring unit).
Mean Time To Failure
Textbook p. 59

9. • It thus serves the same purpose as the ROCOF, and is
sometimes preferred by maintenance people.
• As is the case with a decreasing ROCOF, an increasing MTTF
will be an indication that the system’s reliability is improving,
while a decreasing MTTF (like an increasing ROCOF) will
indicate reliability degradation.
Mean Time To Failure

10. Typical MTTF for system

11. 1. ROCOF
2. MTTF
3. Failure Density
Quantitative descriptions of failure

12. • The failure density gives a picture of the probability of failure
of a component over its own life.
• It thus gives, at any particular point in the life of the
component, the probability of failure at exactly that point.
Failure Density
Textbook p. 60

13. Failure Density
f(t)
0,000
0,225
0,450
0,675
0,900
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Textbook p. 60

14. 1. ROCOF
2. MTTF
3. Failure Density
4. Cumulative Failure Distribution
Quantitative descriptions of failure

15. • By cumulative summation of the probabilities of failure as
depicted in the failure density above, the Cumulative Failure
Distribution results.
• It thus gives the probability that the component will have failed
before or at any point in time (or other measuring unit) of a
component’s life.
Cumulative Failure Distribution
∫
=
−∞
F t f t dt
( ) ( )
t

16. Textbook p. 60

17. • The Cumulative Failure Distribution always starts at a
probability of zero, which is equivalent to stating that a new
component will not be in the failed state.
• It will always eventually have a value of one (or one hundred
percent), indicating that the component will eventually fail
with one hundred percent certainty.
Cumulative Failure Distribution

18. Cumulative Failure Distribution
F(t)
0,00
0,25
0,50
0,75
1,00
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2

19. 1. ROCOF
2. MTTF
3. Failure Density
4. Cumulative Failure Distribution
5. Survival Function
Quantitative descriptions of failure

20. • The Cumulative Failure Distribution gives the probability of
failure before or at a certain age.
• The difference between that and a level of one hundred percent
gives the probability that the component will survive up to that
point.
• This is a very useful measure, as it gives the percentage of
units that will survive up to that age.
Survival Function
R t F t
( ) ( )
= −
1
Textbook p. 60

21. Textbook p. 61

22. Survival Function
Textbook p. 61

23. 1. ROCOF
2. MTTF
3. Failure Density
4. Cumulative Failure Distribution
5. Survival Function
6. Hazard Function (Force of Mortality)
Quantitative descriptions of failure

24. • The most useful of the last four quantitative measures (also
called the four reliability functions) is the Hazard Function {or
Force of Mortality (f.o.m.)}.
• It is also called the conditional probability of failure or the
momentary probability of failure.
Hazard Function (f.o.m.)
Textbook p. 61

25. • It gives the probability that the component will fail at a certain life, given
that it has survived up to that age.
which leads to:
• It is a measure of the risk of failure of the component at that specific age.
Hazard Function
z t f o m t
P t t dt t t
dt
( ) . . .( )
{ | }
* *
= =
< + >
z t
f t
R t
( )
( )
( )
=
Textbook p. 61

26. • If the Hazard Function is decreasing with age or constant, there
will be no incentive towards Use Based Preventive
Maintenance as the risk after such maintenance action will not
be lower than before it.
• In such a case the only preventive strategy available will be
that of Condition Based Preventive Maintenance.
Hazard Function
Textbook p. 61

27. • But, in the case of an increasing Hazard Function,
• a situation often encountered in components,
• Use Based Preventive Maintenance will certainly be an option,
as that will lower the risk of failure.
• See next two slides.
Hazard Function
Textbook p. 61

28. Hazard Function
3000
0 500 1000 1500 2000 2500
0.009
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Life Units
f.o.m.
Textbook p. 62

29. The effect of Use Based Prevention
3000
0 500 1000 1500 2000 2500
0.009
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Life Units
f.o.m.

30. The effect of Use Based Prevention
3000
0 500 1000 1500 2000 2500
0.0014
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
Life Units
f.o.m.
β=1, η=1500
3000
0 500 1000 1500 2000 2500
0.0014
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
Life Units
f.o.m.
β=0.5, η=1500