Class 05 - Failure Analysis Techniques

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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Survival Probability or Reliability

R(t) =1 F(t)Definitions

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Definitions

Failure Rate(t )(t)=f(t)/R(t)

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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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Analysis Techniques

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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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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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Equations for the exponential distribution

= 1/tm

tm is

MTTF

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Parameters:

t: Statistical variables (load time, load cycle, number of operations, ...) > 0

: Location of shape parameter= 1/tm > 0

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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?

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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

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2865.0)100(

80

100

eR

2865.0)200(

)300()200,300(

80/200

80/300

e

e

R

RR


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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?

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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

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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

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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

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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.

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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.

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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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Class 05 - Failure Analysis Techniques

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