1. Definitions Cumulative time to failure (T): Mean life:
When 𝑁 0 components are run for a time t without replacing or repairing failed components:
𝑇=[ 𝑡 1 + 𝑡 2 +…+ 𝑡 𝑘 + 𝑁 0 −𝑘 𝑡]
Mean life:
The average life of the 𝑁 0 components put on test or in service, measured over the entire life curve out to wearout.
Mean time to failure (MTTF):
The sum of the survival time for all of the components divided by the number of failures.
Mean time between failures (MTBF):
The mean time between two successive component failures.
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2. Average Failure Rates For A Variety of Components and Systems
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3. Constant Failure Rate
For the special case of a constant failure rate , ℎ 𝑡 =𝜆,:
𝑅 𝑡 =exp⁡(− 0 𝑡 𝜆 𝑑𝑡 )= 𝑒 −𝜆𝑡
The probability distribution of reliability is a negative exponential distribution.
𝜆= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒 𝑢𝑛𝑖𝑡𝑠 𝑑𝑢𝑟𝑖𝑛𝑔 𝑤ℎ𝑖𝑐ℎ 𝑎𝑙𝑙 𝑖𝑡𝑒𝑚𝑠 𝑤𝑒𝑟𝑒 𝑒𝑥𝑝𝑜𝑠𝑒𝑑 𝑡𝑜 𝑓𝑎𝑖𝑙𝑢𝑟𝑒
The reciprocal of 𝜆 is the mean time between failures (MTBF):
𝑇 = 1 𝜆
𝑅 𝑡 = 𝑒 − 𝑡 𝑇
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4. Example 3 𝑅 𝑡 = 𝑒 − 𝑡 𝑇 = e-100,000/100,000 = e-1= 0.37
If a device has a failure rate of 2 x 10-6 failures/h, what is its reliability for an operating period of 500h? If there are 2000 items in the test, how many failures are expected in 500h? Assume that strict quality control has eliminated premature failures so we can assume a constant failure rate.
𝑅 𝑡 = 𝑒 −𝜆𝑡 = exp(-2 x 10-6 x 500) = e-0.001= 0.999
Ns = N0R(t) = 2000(0.999) =1998
Nf = N0-Ns = 2 failures expected.
If the MTBF for the device is 100,000h, what is the reliability if the operating time equals 100,000 h?
𝑅 𝑡 = 𝑒 − 𝑡 𝑇 = e-100,000/100,000 = e-1= 0.37
If the length of constant failure rate is 50,000h, what is the reliability for operating for that length?
R(50,000) = exp(-2x10-6 x 5 x104) = e-0.1 = 0.905
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5. Weibull Frequency Distribution
The Weibull distribution describes the life of a component for which all values are positive and for which there are occasional long-lived results.
The two-parameter Weibull distribution function is described by:
𝑓 𝑥 = 𝑚 𝜃 𝑥 𝜃 𝑚−1 exp − 𝑥 𝜃 𝑚 𝑥>0
Where f(x)= frequency distribution of the random variable x
m=shape parameter
𝜃=scale parameter
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6. Weibull Frequency Distribution
Probability of x being less than a value q:
𝑃 𝑥≤𝑞 = 0 𝑞 𝑓 𝑥 𝑑𝑥 =1− 𝑒 − 𝑞 𝜃 𝑚
The mean of Weibull distribution:
𝑥 =𝜃−Г(1+ 1 m )
The variance of a Weibull distribution:
𝜎 2 = 𝜃 2 {Г 1+ 2 m − Г 1+ 1 𝑚 2 }
The cumulative frequency distribution of a Weibull distribution:
𝐹 𝑥 =1−exp⁡[− 𝑥 𝜃 𝑚 ]
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7. Reliability With A Variable Failure Rate
Mechanical failures, and some failures of electronic components, do not exhibit a period of constant failure rate.
Since the failure rate is a function of time, the simple exponential relation for reliability no longer applied.
Instead, reliability is expressed by the Weibull distribution:
𝑅 𝑡 =1−𝐹 𝑡 = 𝑒 − 𝑡 𝜃 𝑚
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8. Example 4
The failure of a group of mechanical components follows a Weibull distribution, where m = 1.5 and θ = 6 x 105 cycles. What is the probability that one of the component will have a life of 5 x 105h. 𝑅 𝑡 =1−𝐹 𝑡 = 𝑒 − 𝑡 𝜃 𝑚 F(t) = 1-exp[(-5x105/6x105)1.5] = 1-e = =53% R(t) = = 0.468
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9. System Reliability
The overall reliability of the system depends on how the individual components with their individual failure rates are arranged.
If the components are arranged so that the failure of any component causes the system to fail, it is said to be arranged in series:
𝑅 𝑠𝑦𝑠𝑡𝑒𝑚 = 𝑅 𝐴 × 𝑅 𝐵 ×…× 𝑅 𝑛
A much better arrangement of components is one in which it is necessary for all components in the system to fail in order for the system to fail. This is called parallel reliability:
𝑅 𝑠𝑦𝑠𝑡𝑒𝑚 =1− 1− 𝑅 𝐴 1− 𝑅 𝐵 …(1− 𝑅 𝑛 )
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10. System Reliability (2)
If we are dealing with a constant-failure-rate system:
Series:
𝑅 𝑠𝑦𝑠𝑡𝑒𝑚 = 𝑅 𝐴 × 𝑅 𝐵 = 𝑒 𝜆 𝐴 𝑡 × 𝑒 𝜆 𝐵 𝑡 = 𝑒 −( 𝜆 𝐴 + 𝜆 𝐵 )𝑡
Parallel:
𝑅 𝑠𝑦𝑠𝑡𝑒𝑚 = 1−(1−𝑅 𝐴 ) 1− 𝑅 𝐵 = 𝑒 − 𝜆 𝐴 𝑡 + 𝑒 − 𝜆 𝐴 𝑡 − 𝑒 − (𝜆 𝐴 + 𝜆 𝐵 )𝑡
The reliability of an n-out-of-m system is given by a binomial distribution:
𝑅 𝑛 𝑚 = 𝑖=𝑛 𝑚 𝑚 𝑖 𝑅 𝑖 1−𝑅 𝑚−𝑖
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11. Maintenance and Repair
An important category of reliability problems deals with maintenance and repair of systems.
If a failed component can be repaired while a redundant component has replaced it in service, then the overall reliability of the system is improved.
If components subject to wear can be replaced before they have failed, then the system reliability will be improved.
Preventive maintenance is aimed at minimizing system failure.
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12. Maintainability
A redundant system continues to operate when a component has failed, but it may become vulnerable to shutdown unless the component is repaired and placed back in service.
𝑀𝑇𝐵𝐹=𝑀𝑇𝑇𝐹+𝑀𝑇𝑇𝑅
Where MTBF= mean time between failures
MTTF=mean time to fail
MTTF=mean time to repair
Maintainabiltiy is the probability that a component or system that has failed will be restored to service within a given time.
𝑀 𝑡 =1− 𝑒 −𝑟𝑡 =1− 𝑒 − 𝑡 𝑀𝑇𝑇𝑅
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13. Availability
Availability is the concept that combines both reliability and maintainability; it is the proportion of time the system is working “on line” to the total time, when that is determined over a long working period.
𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑖𝑙𝑖𝑡𝑦= 𝑡𝑜𝑡𝑎𝑙 𝑜 𝑛 𝑙𝑖𝑛𝑒 𝑡𝑖𝑚𝑒 𝑡𝑜𝑡𝑎𝑙 𝑜 𝑛 𝑙𝑖𝑛𝑒 +𝑡𝑜𝑡𝑎𝑙 𝑑𝑜𝑤𝑛𝑡𝑖𝑚𝑒
= 1 1+𝜆𝑀𝑇𝑇𝑅
If 𝑀𝑇𝑇𝐹= 1 𝜆
𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑖𝑙𝑖𝑡𝑦= 𝑀𝑇𝑇𝐹 𝑀𝑇𝑇𝐹+𝑀𝑇𝑇𝑅
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14. Causes of Unreliability
Design mistakes:
Among the common design errors are failure to include all important operator factors, incomplete information on loads and environmental conditions, erroneous calculations, and poor selection of materials.
Manufacturing defects:
Poor surface finish, decarburization crack in heat-treated steel.
Maintenance
Exceeding design limits:
Exceeding limits of temperature, speed, etc.
Environmental factors:
Subjecting equipment to environmental conditions for which it was not designed.
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15. Minimizing Failure
A variety of methods are used in engineering design practice to improve reliability:
Margin of safety
Derating
Redundancy
Durability
Damage tolerance
Ease of Inspection
Specificity
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16. 14.5 Failure Mode and Effects Analysis (FMEA)
What is FMEA?
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17. FMEA
Failure mode and effects analysis (FMEA) is a team-based methodology for identifying potential problems with new or existing designs.
FMEA was first used to identify and correct safety hazards.
FMEA identifies the mode of failure of every component in a system and determines the effect on the system of each potential failure.
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18. Factors in Developing FMEA
Three factors are considered in developing a FMEA:
The severity of a failure
The probability of occurrence of the failure.
The likelihood of detecting the failure in either design or manufacturing, before the product is used by the customer.
Risk Priority Number (RPN):
𝑅𝑃𝑁 = 𝑠𝑒𝑣𝑒𝑟𝑖𝑡𝑦 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 × 𝑜𝑐𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 × 𝑑𝑒𝑡𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑛𝑔
Value of RPN can vary from a maximum of 1000, the greatest risk, to a minimum of 1.
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19. Rating for Severity of Failure
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20. Rating of Occurrence of Failure
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21. Rating of Detection of Failure
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22. Results Of A FMEA Analysis
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23. Creating a FMEA Chart
The design is reviewed to determine the interrelations of assemblies and the interrelations of the components of each subassembly.
Now look more broadly, and ask what are the consequences to the system of each failure identified in step1.
For each of the functions, list the potential failure modes.
For each of the failure modes identifies, describe the consequences or effect of the failure.
Using the severity of failure table, enter the numerical value.
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24. Creating a FMEA Chart (2)
Identify the possible causes of the failure mode.
Using the occurrence of failure table, enter a value for the occurrence of the cause of each failure.
Determine how the potential failure will be detected.
Using Table 14.14, enter a rating that reflects the ability to detect the cause of the failure identified in step 8.
Calculate the risk priority number (RPN).
For each potential failure, determine the corrective action to remove a potential design, manufacturing, or operational failure.
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25. What is fault tree analysis?
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26. Fault Tree Analysis
Fault tree analysis (FTA) is a systematic method to identify undesired events (faults) in a system.
A fault is when a system does something it is not supposed to do or does not do something it is supposed to do.
Often these faults are reliability or safety issues.
Fault tree analysis starts with the top undesired event and develops in a tree-like fashion all potential causes for that event.
FTA is the ability to identify combinations of events that can affect the top undesired event.
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27. Fault Tree For The Failure Of A Lawn Mower Engine to Start
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28. 14.7 Defects and Failure Modes
What are typical defects and failure modes in engineering design?
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29. Defects and Failure Modes
Failures of engineering designs and systems are a result of deficiencies in four broad categories:
Hardware failure:
Failure of a component to function as designed
Software failure:
Failure of the computer software to function as designed
Human failure:
Failure of human operators to follow instructions or respond adequately to emergency situations.
Organizational failure:
Failure of the organization to properly support the system.
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30. Causes of Hardware Failure
Design deficiencies
Deficiency in selection of material
Imperfection in material due to manufacturing
Improper testing or inspection
Overload and other abuses in service
Inadequate maintenance and repair
Environmental factors
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31. Failure Modes
The specific modes of failure of engineering components can usually be grouped into four general classes:
Excessive elastic deformation
Excessive plastic deformation
Fracture
Loss of required part geometry through corrosion or wear
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32. Failure Modes for Mechanical Components
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33. Examples of Failure Modes in Components
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