1.3. UNCERTAINTY IN ENGINEERING 9
the component will fail by fracture, even the magnitude of the nominal maximum cyclic
stress is far below the material’s yield strength. is failure is defined as fatigue failure. In
reality, all mechanical components are subjected to cyclic load due to the continuously re-
peated performance or moving components, or mechanical vibrations. So, it is well known
that more than 90% of mechanical metal components fail due to fatigue failure. For exam-
ple, ball bearings or spur gears will undoubtedly fail when the service time is big enough.
1.3 UNCERTAINTY IN ENGINEERING
Uncertainty is the lack of certainty, a situation that is impossible to describe and predict exactly by
one outcome or one numerical value. Uncertainty exists in most phenomena and events observed
worldwide. In practice, the various forms of uncertainty can be classified into two categories:
physical uncertainty and cognitive uncertainty [3, 8, 9].
• Physical uncertainty results from the fact that a system can behave in random ways and is
associated with the state of nature. is type of uncertainty is the inherent randomness that
exists in all physical parameters, so it is an irreducible uncertainty. Physical uncertainty will
be the focus of this book. Uncertainties of the design parameters in the mechanical design
are physical uncertainty. For example, material mechanical properties of the same brand
material such as yield strength, ultimate strength, and Young’s modulus will inevitably vary
due to small variations in chemical composition, the temperature in heat treatment, and
non-homogeneous temperature field during solidification. So, the material mechanical
properties are physical uncertainty. e geometric dimensions of components are physical
uncertainty because no two components can be made identical due to tool wear, errors
in measurement, machine tool vibrations, or the resistance of the shaft materials to cut-
ting. For another example, external load or operation patterns in the mechanical design
are physical uncertainty because the actual external load of the mechanical system in real
service are unpredicted and varies from one working condition to another condition, from
one service to another. e approach in mechanical design to deal with these physical un-
certainties will be reliability-based mechanical design, which is the main topic of this book
and will be discussed and explored in detail later.
• Cognitive uncertainty results from the lack of knowledge about a system and is associated
with our interpretation of the physical world. Cognitive uncertainty describes the inher-
ent vagueness of the system and its parameters. For example, the outcome of the quality
of machined mechanical components by machine operators whose skill and experience in
machining are unknown is cognitive uncertainty. When machine operators’ skills and ex-
perience are fully known, the outcome of the quality of machined mechanical components
by those operators will be a physical uncertainty.
10 1. INTRODUCTION TO RELIABILITY IN MECHANICAL DESIGN
1.4 DEFINITION OF RELIABILITY
When uncertainties of main design variables are taken into consideration during mechanical
design, reliability will be a relative measure of the performance of a product. Although there is a
consensus that reliability is an important attribute of a product, there is no universally accepted
definition of reliability. is book will use the following definition of reliability.
Reliability, denoted by R, is defined as the probability of a component, a device, or a system
performing its intended functions without failure over a specified service life and under specified
operation environments and loading conditions.
Phase Two of the engineering design process, as discussed in Section 1.1, is to determine
and specify the design specifications of the product. e design specifications include the in-
tended functions, service life, operational environments, loading conditions, and the required
reliability. ere are lots of different possible failure, as discussed in Section 1.2. is book will
only focus on three typical mechanical components’ failures: static failure, excessive deflection
failure, and fatigue failure. erefore, the reliability of a component can be expressed by the
following equation:
R D P .S Q/; (1.1)
where S is a component material strength index, which could be material strength such as yield
strength, ultimate strength, fatigue strength, or allowable deflection. Q is a component loading
index, which could be maximum stress, accumulated fatigue damage, or maximum deflection of
a component.
P .S Q/ means the probability of the status that component can perform its intended
functions without failure. R is the reliability and is equal to this probability. e physical mean-
ing of reliability is the percentage of components working properly out of the total of the same
components in service. For example, if 10,000 of the mechanical shafts with the designed relia-
bility 0.99 for the service life of one year are in service, the reliability 0.99 of these shafts indicate
that 0:99 10000 D 9900 of these mechanical shafts are expected to work properly at the end
of one-year service. One hundred of these mechanical shafts might fail at the end of one-year
service.
Reliability is an important attribute of a component and a relative measure of the compo-
nent status through the comparison of materials strength index with component loading index
within the service life. In other words, the reliability of a component is a function of materials
properties, loading conditions, component geometric dimensions, and service life. Since relia-
bility R is expressed by probability, reliability is not an attribute of a specific component, but an
attribute of the batch of same designed components. For example, a company has designed and
sold 10,000 unit of the designed component with a reliability 0.99. Supposed we purchased one
of the components. We cannot claim that the component has a reliability of 0.99 because the
reliability 0.99 is the attribute of the batch of 10,000 units of the same designed components.
1.4. DEFINITION OF RELIABILITY 11
After the definition of reliability is defined, it is easy to define the probability of failure of a
component.
Probability of failure, denoted by F , is defined as the probability of a component, a device, or
a system failing to perform its intended functions over a specified service life and under specified
operation environments and loading conditions.
e probability of failure, F , can be expressed by the following equation:
F D P .S < Q/: (1.2)
e sum of the reliability and the probability of failure should be equal to 1, that is,
F C R D 1: (1.3)
Example 1.1
It is assumed that a company has designed and sold 10,000 units of the designed component
which has a reliability of 0.99 with a service life of 5 years. e record of the service information
of these 10,000 units during 5 years in the service is shown in Table 1.2. Estimate the reliability
of the components at the end of each service year and actual reliability of the components with
the service life of five years.
Table 1.2: e record of the service information of the designed components
Service Years First Year Second Year ird Year Fourth Year Fifth Year
Number of Failures
11 15 13 17 27
Solution:
From Table 1.2, at the end of the first-year service, 11 of the components fails, and 10;000
11 D 9;989 of components still work properly. According to the definition of the reliability and
the probability of failure, we can estimate the reliability R
1
and the probability of failure F
1
of
the components when they are in service of one year:
R
1
D
9;989
10;000
D 0:9989; F
1
D
11
10;000
D 0:0011:
e reliability and the probability of failure of the components at the end of each service year
can be calculated accordingly and is listed in Table 1.3.
From Table 1.3, the actual reliability of the components with a service life of 5 years is
0.9917, which is larger than the required reliability 0.99. is result indicates that the compo-
nents have been designed properly.
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