Title: Design for Reliability
1Design for Reliability
Unit 3 Defining Reliability
Mike Robinson CMfgE, CQE Natural Science
Technology Manufacturing Technology
2Introduction
- 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.
3Definition 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.
4Definition 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.
5Example 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
6Definition 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/
7Example 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
8Definition 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.
9Example 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
10Definition 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.
11Example 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
12Failure 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
13Mathematical 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.
14Mathematical 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.
15The 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.
16The 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.
17The 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
18The 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.
19The 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.
20Quantifying 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.
21Quantifying 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
22Mean 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).
23Mean 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.
24Mean 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.
25Summary
- 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.
26Summary
- 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.
27References
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.
28References
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.
29Acknowledgments
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.