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8/2/2019 Class 05 - Failure Analysis Techniques
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Definitions
Failure Density Function f(t)
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Industrial Engineering
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Definitions
Failure Probability F(t)
F(t) = f (t)dt
f (t) = dF(t)/dt
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Industrial Engineering
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Definitions
Survival Probability or Reliability
R(t) =1 F(t)Definitions
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Industrial Engineering
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Definitions
Failure Rate(t )(t)=f(t)/R(t)
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Industrial Engineering
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Definitions
MTTF
(Mean Time to Failure) the lifetime of a non-repairable system
MTTFF
(mean time to first failure)MTBF
(mean time between failure)
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Industrial Engineering
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Definitions
Definitions
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Industrial Engineering
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Analysis Techniques
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Industrial Engineering
Failure function curves of the normal distribution
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Analysis TechniquesIndustrial Engineering
Failure function curves of the normal distribution
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Analysis TechniquesIndustrial Engineering
Equations for the normal distribution
Parameters:
t: Statistical variables (load time, load cycle, number of operations, ...) > 0: Location parameter = tm = tmedian = tmode: scale measurement > 0
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Analysis TechniquesIndustrial Engineering
Equations for the exponential distribution
= 1/tm
tm is
MTTF
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Analysis TechniquesIndustrial Engineering
Equations for the exponential distribution
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Analysis TechniquesIndustrial Engineering
Equations for the exponential distribution
Parameters:
t: Statistical variables (load time, load cycle, number of operations, ...) > 0
: Location of shape parameter= 1/tm > 0
Definitions
Techniques
Examples
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ExamplesIndustrial Engineering
Example:
What is the probability of an item surviving untilt
= 100 units if the item isexponentially distributed with a mean time between failure of 80 units? Given
that the item survived to 200 units, what is the probability of survival until t= 300units? What is the value of the hazard function at 200 units, 300 units?
Definitions
Techniques
Examples
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ExamplesIndustrial Engineering
Example:
What is the probability of an item surviving untilt
= 100 units if the item isexponentially distributed with a mean time between failure of 80 units? Given
that the item survived to 200 units, what is the probability of survival until t= 300units? What is the value of the hazard function at 200 units, 300 units?
The probability of survival until t= 100 units is
The probability of survival until t= 300 units given survival until t= 200 units is
Note that this is equal to the probability of failure in the interval from t=0 to
t=100.
The value of the hazard function is equal to the failure rate and is constanth(t)= 1/80 = 0.125
Definitions
Techniques
Examples
2865.0)100(
80
100
eR
2865.0)200(
)300()200,300(
80/200
80/300
e
e
R
RR
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ExamplesIndustrial Engineering
Example:
The lifetime (in hours) of an electrical component can be described by the
exponential distributionf (t) = exp(
t) t 0; =1/(500h) .
1. What is the probability that the component does not fail before the time t1 =200 h?
2. What is the probability that the component fails before t2 = 100 h?3. What is the probability that the component fails between the times t3 = 200
hand t4 = 300 h?
4. How long, t5, can the component survive with exactly 90% safety and whichrange of time can the component survive with at least 90% safety?5. What value must the parameterhave for a lifetime distribution where the
probability is 90% so that the lifetime of a component is at least 50 h?
Definitions
Techniques
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ExamplesIndustrial Engineering
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ExamplesIndustrial Engineering
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ExamplesIndustrial Engineering
Example:
In a factory, a device that works effectively as good as new during its operating
life, has failure rate of 0.008 failures per day. If the probability of failure for thisdevice is independent of running time, find the following:
1. The probability that this device will fail before 100 days of running time2. The probability that this device will last for more than 80 days3. The probability that this device will not run for 40 days before failing4. The probability that this device will fail before the 10 days that follow the first
100 days of running time5. The probability that this device will last for more than 60 days and less than120 days
6. The probability that this device will fail after 50 days of working and before100 days of running time
7. The probability that this device will fail during the 10 days that follow the first100 days of running time
Definitions
Techniques
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ExamplesIndustrial Engineering
Example:
The reliability of a technical component is given by the equation:
R(t) = exp( (t)2) for t
Calculate the failure density, the failure probability and the failure rate.Show the results graphically
Definitions
Techniques
Examples
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ExamplesIndustrial Engineering
Example:
The reliability of a technical component is given by the equation:
R(t) = exp( (t)2) for t
Calculate the failure density, the failure probability and the failure rate.Show the results graphically
Definitions
Techniques
Examples
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ExamplesIndustrial Engineering
Example:
An electrical meter times to failure are described by the following probability
density function:
(t) = exp (-t)
Where: = 0.0005
Calculate the hazard rate and the motorMTTF.
Definitions
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ExamplesIndustrial Engineering
Example:
A mechanical device times to failure are described by the following probability
density function:
(t) = 2 e(-2t)
Where: = 0.0004
Calculate the failure rate and the device MTTF.
Definitions
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ExamplesIndustrial Engineering
Example:
An item shows a marked wear-out failure pattern. It tends to fail at mean
operating age of 200 days. The dispersion from the mean that is associated withthe times to failure of this item is 40 days measured as standard deviation. Findthe following:1. The probability that this item will last for more than 160 days and less than
300days2. The probability that this item will fail in the first 20 days after 300 of
successful working days
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