Physics of Failures
In theory failures can be understood using principles of physics
and chemistry
Constructing mathematical models to correlate causal variables with
failures
Can demonstrate how failures occur and when they will occur
Only a very lim
Exercise 4.5
Problem:
Determine goodness of the least square parameters obtained in example 4.7,
4.8 and 4.9 using the coefficient of determination. Which one has a better fit?
Solution:
For example 4.7
With a1 = 0.261 = from example 4.7, fi = a1 ti
Exercise 5.1
Problem:
The strength of a product manufactured in a factor is measured to have an
exponential distribution with a mean S. A buyer bought N of such products
selected randomly from a lot to be used in an environment with a load L
which is ap
Exercise 6.1
Products believed to have 0 =1.0 in example 6.2 are tested against the null
hypothesis that R0 = exp(0t) = 0.61 at t0 = 0.5. The alternate hypothesis is
that R1 = exp(1t) = 0.37 at t0 = 0.5. The desired producers risk and
customers risk ar
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Chapter 5 Other models
5.1 Covariate Models
Up to now, we have considered reliability models as a function of failure time only. Many other
variables such as temperature, humidity, current or voltage, are also associated with fai
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Chapter 6 Test planning
Tests are expensive. Not only units under tests can be damaged in tests, but also the environmental set
up, labor and facility costs for the tests can be quite high. Since damages and costs are all directl
Exercise 7.2
Problem:
(a) Find the system reliability of the below configuration in terms of the
independent component reliabilities R1, R2, R3, R4, and R5. (b) Find system
MTTF assuming components are independent, identical, and have a constant
IDLOXUH
Exercise 7.1
Problem:
A system consists of 6 identical and independent components connected as
system consists of identical and independent components connected as
in the diagram below. The component failures are exponentially distributed
with a constan
Exercise 7.3
Problem
Consider a system with one active unit and two standby units. When the
online unit fails, the first backup unit is placed online. When it fails the third
unit is placed online. Units have constant failure rates 1, 2 and 3
respect
Exercise 8.1
Problem:
A system has two independent failure modes with constant failure rates 1
and 2. Let the constant repair rates for the two independent modes be r1
and r2 respectively. Find the system availability as a function of time
8C
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Ex
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Chapter 8 Maintenance Measures of Simple Systems
There are two types of maintenance: reactive and proactive. A reactive maintenance is performed after
a failure has occurred. A proactive maintenance on the other hand is performed
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Chapter 9 Multilevel Maintenance of Complex Systems
Repairs of a complex system are normally accomplished at subsystem or component levels. As in
reliability study, understanding the relationships between different levels of mai
Exercise 4.2
Problem:
Following failure times were observed in a reliability testing of 50 identical units.
Plot failure and failure rate histograms and determine the model that can best
describe units reliability.
0.623
0.889
1.829
2.056
2.575
2.966
3.4
Exercise 4.3
Problem:
Determine the failure rate of exercise 4.2 by MLE assuming Weibull failure
distribution
4F
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Exercise 4.3
(cont)
Solution:
1 ln( )
=
Since the data are not censored, the equation for is
1
=
1 ( ) ln ( )
=
1 ( )
=

1
Exercise 4.4
Problem
Make Weibull plots for failure data in example 4.3, exercise 4.2 and example
4.6 to determine if Weibulll is a good model for them. Determine the Weibull
model parameters for the data sets Weibull is a good model.
4F
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Exercise
Probability of Independent Events
Definition:
Events A and B are independent if the occurrence of event B does not
alter the occurrence of event A
P(AIB) = P(A)
P(A B) = P(A)P(B)
Proof:
U
P(AIB)
( )
()
= P(A)
Brief review:
Consider a system with two f
Characteristics of a probability distribution
(cont)
Skewness (cont)
For positive 1
The distribution is more concentrated on the left of the mean
Has a longer tail on the right of the mean
Referred to as positively skewed or rightskewed
Mean is general
Engineering 202
Reliability,Maintainability
& Supportability
Lecture 3
Reliability Models
Dr. Beidwo Chang
beidwo@gmail.com
(Cell) 310 9861858
page1
Theoretical Models for Failure Distribution
Introduced four functions of Time to first failure
Probabil
Acceptance Testing with Variable Number of Units
This is a sequential test with the number of units under test as a
variable
Each unit is tested to time t0 unless it fails before t0
After each unit is tested, the hypotheses are evaluated against criter
Exercise 1.1
A detector has a fault detection probability of 98% (If a fault is present, 98%
of time the detector can detect it). But this detector also has a false alarm
rate of 1% (There is 1% of the probability that when the detector detects a
failure
Exercise 2.2
Problem:
For a system having the following hazard rate (t) (A) determine its reliability
function (B) What is the reliability of the system after it operates without failure
for a period of time [0, a]? (C) What is the reliability of the sys
Exercise 2.3
Problem:
Assume the three modes of failures in the bathtub example occur in all
phases. Namely, there are three independent modes with failure rates as
shown below. (A) What is (t)? (B) Determine its reliability function (C) What is
the reli
Exercise 2.1
Problem:
A products reliability is described by the function R(t) = exp ( t). Plot the
reliability function, the CDF and the failure distribution function with = 0.10.
Determine the mean time to failure using the reliability function. Dete
Exercise 3.1
Problem
A products failure distribution is normal. If at t1 the products reliability is 95%
and at t2 the products reliability is 90%. Find its mean, median, mode and
variance in terms of t1 and t2
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Exercise 3.1 (Cont)
Solution
R(
Exercise 3.2
Problem:
A product has two independent Weibull failure modes, one with 1 = 0.5, 1 =
1.0 and the other with 2 = 1.5, 2 =1.0. (a) How much does burn in of duration
tb improve the reliability? Express the improvement as a ratio between
reliabil
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Preface
These are my lecture notes on Reliability, Maintainability and Supportability (RMS) prepared for a
Systems Engineering online course at UCLA. I studied RMS in the 90s when my company was tasked to
improve a radar systems
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Chapter 2 Reliability Measures
In this and next several chapters we will consider a sample space resulted from operating a large
number of identical new products until they fail. A random time variable which tells when each failu
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Chapter 3 Theoretical Models
In chapter 2 we have introduced four functions, f(t), F(t), R(t), and (t), each represents a different
aspect of a failure distribution. We have also shown that given any one of the four, the other th