Design for Reliability

1
Design for Reliability
Unit 3 Defining Reliability
Mike Robinson CMfgE, CQE Natural Science
Technology Manufacturing Technology
2
Introduction

• While there is no universally accepted definition
  of reliability, there are key elements,
  including probability, performance, operating
  conditions, and time.
• First and foremost, reliability is the
  probability that a part or system will
  successfully complete its mission.
• For most products, the time-to-failure in the
  constant failure rate portion of the bathtub
  curve can be represented by the exponential
  distribution.
• Generalized formulas exist for the calculation of
  quantitative reliability metrics.

3
Definition of Reliability

• There is no universal definition of
  reliability.
• But all definitions in use share common
  elements.
• One definition (based on IEEE-90)

The probability that a system or component will
perform its required function under stated
conditions for a specified period of time.
4
Definition of Reliability - Probability

• Reliability is first and foremost defined as a
  probability
• A numerical value between 0 and 1 (unity).
• The ability to express reliability in numerical
  terms allows for the direct comparison between
  different design alternatives for products.
• For example, a reliability of 0.97 indicates
  that, on average, 97 of 100 items will perform
  their intended function for a given period of
  time under given operating conditions.

5
Example Reliability as a Probability
Applying this definition, the reliability of a
standard equipment automobile tire is essentially
1 for 10,000 miles of normal operation on a
typical passenger car, but it virtually 0 for
use at the Daytona 500 stock car race.
Reliability 0
Reliability 1
6
Definition of Reliability - Performance

• Performance is the second element of the
  definition and refers to the object for which the
  product or system was designed.
• The term failure is used when the expectations
  of performance are not met.

A failure can be defined as any incident that
prevents the mission from being accomplished, or
any incident requires unscheduled maintenance.
http//todayinspacehistory.wordpress.com/category/
tragedy/
7
Example Hard Disk Drive Performance
Maxtor Atlas 10K V 75GB HDD
http//en.wikipedia.org/wiki/ImageHard_disk_platt
er_reflection.jpg
1.4 Million Hour Mean Time to Failure (MTTF)
http//www.darklab.rutgers.edu/MERCURY/t15/disk.pd
f
8
Definition of Reliability Operating Conditions

• The third aspect of the definition is operating
  conditions, which involves the type and amount
  of usage and the environment in which the product
  or system is used.
• Operating conditions include such considerations
  as operating temperatures, storage temperatures,
  altitude, humidity, chemical exposure, shock,
  vibration, transportation/handling.
• Reliability often must include extreme conditions
  in addition to ambient environments.

9
Example Hard Disk Drive Operating Conditions
Maxtor Atlas 10K V 75GB HDD
http//en.wikipedia.org/wiki/ImageHard_disk_platt
er_reflection.jpg
1.4 Million Hour Mean Time to Failure (MTTF)
http//www.darklab.rutgers.edu/MERCURY/t15/disk.pd
f
10
Definition of Reliability - Time

• The final aspect of the definition of reliability
  is time.
• All products must operate for a given lifetime.
• While long life is typically a design
  requirement, some products only need to operate
  for a relatively short time frame (ie., Mars
  rovers)
• Clearly, a device having a reliability of 0.97
  for 1,000 hours of operations is inferior to one
  having the same reliability for 5,000 hours,
  assuming that the mission of the device is long
  operating life.

11
Example Time
Mars Rover
Cardiac Pacemaker
http//www.techshout.com/science
http//www.heartzine.com/
High Reliability over Long Time Period
High Reliability over Relatively Short Time
Period
12
Failure Distribution

• In most cases, the distribution of failure times
  follows the classic bathtub curve.

• Frequently in the reliability evaluation of an
  item, only the middle portion of the bathtub
  curve is taken into consideration. It is in this
  portion that the failure rate is essentially
  constant.

http//en.wikipedia.org/wiki/Bathtub_curve
13
Mathematical Basis of Reliability

• The mathematics of reliability is based in
  probability theory, which deals with the
  uncertainty of random events.
• In most reliability applications, we are dealing
  with quantitative measures such the
  time-to-failure of a part, where the random
  variable is time ( t ).
• Because a product can be found failed at any time
  after time 0 (i.e. at 12.4 hours or at 100.12
  hours, etc.), the random variable (time) can take
  on any value in this range. In this case, the
  random variable is said to be a continuous random
  variable.

14
Mathematical Basis of Reliability

• All random events (including failures) have an
  underlying probability function.
• Given a continuous random variable, we denote
• the probability density function (pdf) as f(t),
  and
• the cumulative distribution function (cdf) as
  F(t).
• The pdf and cdf give a complete description of
  the probability distribution of the random
  variable.

15
The Exponential Function

• The majority of reliability (failure rate)
  prediction standards assume that the underlying
  distribution of failures is exponential.
• Even if individual components fail according to
  other distributions, in a long run, the
  successive times of a repairable series system
  follows the exponential distribution.
• The exponential function is NOT the normal
  distribution (bell curve) that most people are
  familiar with.

16
The Probability Density Function

• The probability density function (pdf) of the
  exponential function is given generically as
• f(x) ?e -?x x 0
• where ? is the parameter.
• For reliability applications, the parameter is
  the constant failure rate of the item (?) and the
  random variable ( x ) is operating time.

17
The Probability Density Function
The Probability Density Function (pdf) of the
Exponential Function
f(x) ?e - ?x x 0
Probability Density (frequency of occurrence)
Continuous Variable
18
The Cumulative Density Function

• The cumulative density function (cdf) of the
  exponential function is given as
• F(x) 1 - e -?x x 0
• where ? is the parameter.
• For reliability applications, the parameter is
  the constant failure rate of the item (?) and the
  random variable ( x ) is operating time.

19
The Cumulative Density Function
The Cumulative Density Function (cdf) of the
Exponential Function
F(x) 1 - e - ?x x 0
Probability Of Failure
Continuous Variable
http//en.wikipedia.org/wiki/Exponential_distribut
ion.
20
Quantifying Reliability

• The reliability (probability of mission success
  for a given operating time) is mathematically
  derived from the exponential distributions pdf
  and cdf
• Reliability R(t) e -?t
  t 0
• Using this simple formula, the numerical
  reliability for any amount of operating time can
  be calculated if the constant failure rate (? )
  is known.

21
Quantifying Failure Probability (Unreliability)

• Similarly, the unreliability (the probability of
  failure during a given operating time) can be
  expressed as
• Unreliability P(t) 1 - e -?t
  t 0

22
Mean Time Between Failures

• The expected value of the time to failure of a
  non-repairable system is known as the Mean Time
  to Failure (MTTF).
• If the equipment is repairable, the expected
  value of the time between repaired failures is
  commonly described as Mean Time Between Failure
  (MTBF).

23
Mean Time Between Failures

• Thus, the mean waiting time between successive
  failures can be found by the fraction
• or the reciprocal of the constant failure
  rate.
• Thus, reliability can be defined in terms of the
  average or mean time a device or item will
  operate without failure, or the average time
  between failures for a repairable item.

24
Mean Time Between Failures

• Like any mean or average value, the MTTF or MTBF
  of a particular system is an average and it is
  unlikely that the actual time before a failure
  occurs will exactly equal the MTTF.
• Over a very long time or for a very large number
  of systems, the times to failure will average out
  to the MTTF. The MTTF is neither a minimum value
  nor a simple arithmetic average.
• If we impose the condition that every time a
  repairable item fails we restore it completely
  (back to the t 0 state), MTTF and MTBF can be
  used interchangeably.

25
Summary

• While there is no universally accepted definition
  of reliability, there are key elements,
  including
• probability,
• performance,
• operating conditions, and
• time
• First and foremost, reliability is the
  probability that a part or system will
  successfully complete its mission.
• The mathematics of reliability is based in
  probability theory, which deals with the
  uncertainty of random events.

26
Summary

• Constant failure rates are assumed to exist
  during the useful life portion of the bathtub
  curve.
• The majority of reliability (failure rate)
  prediction standards assume that the underlying
  distribution of failures during this period of
  time is exponential.
• Reliability is quantified using the exponential
  distribution function and is a function of the
  constant failure rate and time.
• The mean waiting time between successive failures
  is equivalent to the reciprocal of the constant
  failure rate.

27
References

•  
•  

National Institute of Standards and Technology
(NIST) Engineering Statistics Handbook.
Retrieved in December 2007 from
http//www.itl.nist.gov/div898/handbook/ IEEE
90 Institute of Electrical and Electronics
Engineers. IEEE Standard Computer Dictionary A
Compilation of IEEE Standard Computer Glossaries.
New York, NY 1990. Operating specification
for Maxtor Atlas 10KV Ultra 320 Hard Disk Drive.
Retrieved in January 2008 from
http//www.darklab.rutgers.edu/MERCURY/t15/disk.pd
f. US Army TM 5-698-1, Reliability/Availability
of Electrical Mechanical Systems for Command,
Control, Communications, Computer, Intelligence,
Surveillance, and Reconnaissance (C4ISR)
Facilities. US Army Corps of Engineers -
Technical Manuals). Retrieved in January 2008
from http//www.usace.army.mil/publications/armyt
m/tm5-698-1/a-b.pdf.
28
References

•  
•  

Exponential Distribution. Retrieved in January
2008 from http//en.wikipedia.org/wiki/Exponentia
l_distribution. Kales, Paul. Reliability for
Technology, Engineering and Management. Upper
Saddle River Prentice Hall, 1998. Neubeck,
Ken. Practical Reliability Analysis. Upper
Saddle River Pearson Prentice Hall, 2004.
29
Acknowledgments
The author wishes to acknowledge the support from
the National Science Foundation Advanced
Technology Education Program, NSF Grant 0603362
for Midwest Coalition for Comprehensive Design
Education.