workshop 20

*Fundamentals of Reliability Engineering and Applications

Dr. E. A. ElsayedDepartment of Industrial and Systems EngineeringRutgers University ([email protected])

Systems Engineering DepartmentKing Fahd University of Petroleum and MineralsKFUPM, Dhahran, Saudi ArabiaApril 20, 2009

*Reliability Engineering OutlineReliability definitionReliability estimation System reliability calculations*

*Reliability ImportanceOne of the most important characteristics of a product, it is a measure of its performance with time (Transatlantic and Transpacific cables)

Products recalls are common (only after time elapses). In October 2006, the Sony Corporation recalled up to 9.6 million of its personal computer batteries

Products are discontinued because of fatal accidents (Pinto, Concord)

Medical devices and organs (reliability of artificial organs)*

*Reliability ImportanceBusiness data*Warranty costs measured in million dollars for several large American manufacturers in 2006 and 2005. (www.warrantyweek.com)

Some Initial Thoughts Repairable and Non-RepairableAnother measure of reliability is availability (probability that the system provides its functions when needed). *

Some Initial ThoughtsWarrantyWill you buy additional warranty?Burn in and removal of early failures.(Lemon Law).*

*Reliability DefinitionsReliability is a time dependent characteristic.

It can only be determined after an elapsed time but can be predicted at any time.

It is the probability that a product or service will operate properly for a specified period of time (design life) under the design operating conditions without failure.*

*Other Measures of ReliabilityAvailability is used for repairable systems

It is the probability that the system is operational at any random time t.

It can also be specified as a proportion of time that the system is available for use in a given interval (0,T). *

*Other Measures of ReliabilityMean Time To Failure (MTTF): It is the average time that elapses until a failure occurs.It does not provide information about the distributionof the TTF, hence we need to estimate the varianceof the TTF.

Mean Time Between Failure (MTBF): It is the average time between successive failures. It is used for repairable systems.*

*Mean Time to Failure: MTTF2 is better than 1?*

*Mean Time Between Failure: MTBF*

*Other Measures of ReliabilityMean Residual Life (MRL): It is the expected remaining life, T-t, given that the product, component, or a system has survived to time t.

Failure Rate (FITs failures in 109 hours): The failure rate in a time interval [ ] is the probability that a failure per unit time occurs in the interval given that no failure has occurred prior to the beginning of the interval.

Hazard Function: It is the limit of the failure rate as the length of the interval approaches zero.*

*Basic Calculations Suppose n0 identical units are subjected to a test. During the interval (t, t+t), we observed nf(t) failed components. Let ns(t) be the surviving components at time t, then the MTTF, failure density, hazard rate, and reliability at time t are:*

*Basic Definitions Contd*

Time Interval (Hours)Failures in the interval 0-10001001-20002001-30003001-40004001-50005001-60006001-70001004020151087Total200

The unreliability F(t) is

Example: 200 light bulbs were tested and the failures in 1000-hour intervals are

*Calculations*


Time Interval

Failure Density

x

Hazard rate

x

0-1000

1001-2000

2001-3000

6001-7000

..

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_991677941.unknown

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_990614558.unknown

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_990611082.unknown

_990610482.unknown

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_990610449.unknown

*Failure Density vs. Time 1 2 3 4 5 6 7 x 103 Time in hours *10-4

Chart1

5

2

1

0.75

0.5

0.4

0.35

Column 3

Column4

Column5

f(t)

Sheet1

5

2

Column 31

Column40.75

Column50.5

0.4

0.35

*Hazard Rate vs. Time 1 2 3 4 5 6 7 103 Time in Hours *10-4

Chart1

5

4

3.33

3.75

4

5.3

10

h(t)

Sheet1

5

4

3.33

3.75

4

5.3

10

*Calculations*


Time Interval

Reliability

0-1000

1001-2000

2001-3000

6001-7000

200/200=1.0

100/200=0.5

60/200=0.33

0.35/10=.035

_990615208.unknown

*Reliability vs. Time 1 2 3 4 5 6 7 x 103Time in hours*

Chart1

1

0.5

0.3

0.2

0.125

0.075

0.035

column 1

R(t)

Sheet1

column 11

0.5

0.3

0.2

0.125

0.075

0.035

*Exponential DistributionDefinition*

*Exponential Model ContdStatistical Properties*MTTF=200,000 hrs or 20 yearsMedian life =138,626 hrs or 14 years

*Empirical Estimate of F(t) and R(t)When the exact failure times of units is known, weuse an empirical approach to estimate the reliabilitymetrics. The most common approach is the RankEstimator. Order the failure time observations (failuretimes) in an ascending order:

*Empirical Estimate of F(t) and R(t) is obtained by several methods

Uniform naive estimator

Mean rank estimator

Median rank estimator (Bernard)

Median rank estimator (Blom)

*Empirical Estimate of F(t) and R(t)Assume that we use the mean rank estimator *Since f(t) is the derivative of F(t), then

*Empirical Estimate of F(t) and R(t)*Example:

Recorded failure times for a sample of 9 units are observed at t=70, 150, 250, 360, 485, 650, 855, 1130, 1540. Determine F(t), R(t), f(t), ,H(t)

*Calculations *

it (i)t(i+1)F=i/10R=(10-i)/10f=0.1/Dtl =1/(Dt.(10-i))H(t)0070010.0014290.00142901701500.10.90.0012500.0013890.1053605221502500.20.80.0010000.0012500.2231435532503600.30.70.0009090.0012990.3566749443604850.40.60.0008000.0013330.5108256254856500.50.50.0006060.0012120.6931471866508550.60.40.0004880.0012200.91629073785511300.70.30.0003640.0012121.20397288113015400.80.20.0002440.0012201.6094379191540-0.90.12.30258509

*Reliability Function*

*Probability Density Function*

*Failure RateConstant *

*Exponential Distribution: Another ExampleGiven failure data:

Plot the hazard rate, if constant then use the exponential distribution with f(t), R(t) and h(t) as defined before.

We use a software to demonstrate these steps.*

*

Input Data*

*Plot of the Data*

*Exponential Fit*

Exponential Analysis

*

Go Beyond Constant Failure Rate Weibull Distribution (Model) and Others*

*The General Failure CurveTime tFailure Rate0*

*Related Topics (1)Time tFailure Rate0 Burn-in:According to MIL-STD-883C, burn-in is a test performed to screen or eliminate marginal components with inherent defects or defects resulting from manufacturing process.*

*21Motivation Simple ExampleSuppose the life times (in hours) of several units are:1 2 3 5 10 15 22 28 3-2=1 5-2=3 10-2=8 15-2=13 22-2=20 28-2=26After 2 hours of burn-in

*Motivation - Use of Burn-inImprove reliability using cull eliminatorAfter burn-inBefore burn-in*

*Related Topics (2)Time tHazard Rate0 Maintenance:An important assumption for effective maintenance is that component has an increasing failure rate.

Why?

*

*Weibull ModelDefinition*

*Weibull Model Cont. Statistical properties*

*Weibull Model*

*Weibull Analysis: Shape Parameter*

*Normal Distribution*

Weibull Model

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Input Data

Plots of the Data

Weibull Fit

Test for Weibull Fit

Parameters for Weibull

Weibull Analysis

Example 2: Input Data

Example 2: Plots of the Data

Example 2: Weibull Fit

Example 2:Test for Weibull Fit

Example 2: Parameters for Weibull

Weibull Analysis

*Versatility of Weibull ModelHazard rate: Time tHazard Rate0*

*Graphical Model ValidationWeibull Plotis linear function of ln(time).Estimate at ti using Bernards Formula For n observed failure time data*

*Example - Weibull Plot T~Weibull(1, 4000) Generate 50 dataIf the straight line fits the data, Weibull distribution is a good model for the data*

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workshop 20

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