Reliability Analysis, Safety Assessment and Optimization. Enrico Zio. Читать онлайн. Newlib. NEWLIB.NET

Автор: Enrico Zio
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Техническая литература
Год издания: 0
isbn: 9781119265863
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alt="upper F left-parenthesis t right-parenthesis equals StartFraction lamda Superscript beta Baseline Over upper Gamma left-parenthesis beta right-parenthesis EndFraction integral Subscript 0 Superscript t Baseline x Superscript beta minus 1 Baseline e Superscript minus lamda x Baseline d x comma t greater-than-or-equal-to 0 period"/>(1.34)

      If T denotes the failure time of an item with gamma distribution, the reliability function will be

      upper R left-parenthesis t right-parenthesis equals StartFraction lamda Superscript beta Baseline Over upper Gamma left-parenthesis beta right-parenthesis EndFraction integral Subscript t Superscript infinity Baseline x Superscript beta minus 1 Baseline e Superscript minus lamda x Baseline d x comma t greater-than-or-equal-to 0 period(1.35)

      The hazard rate function is

      h left-parenthesis t right-parenthesis equals StartFraction t Superscript beta minus 1 Baseline e Superscript minus lamda t Baseline Over integral Subscript t Superscript infinity Baseline x Superscript beta minus 1 Baseline e Superscript minus lamda x Baseline d x EndFraction comma t greater-than-or-equal-to 0 period(1.36)

      The mean, μ, and variance, σ2, are

      sigma squared equals StartFraction beta Over lamda squared EndFraction period(1.37)

      1.2.2.4 Lognormal Distribution

      A random variable T follows the lognormal distribution if and only if the pdf (shown in Figure 1.6) of T is

      f left-parenthesis t right-parenthesis equals StartFraction 1 Over sigma t StartRoot 2 pi EndRoot EndFraction exp left-bracket minus StartFraction 1 Over 2 sigma squared EndFraction left-parenthesis ln t minus mu right-parenthesis squared right-bracket comma t ampersand gt semicolon 0 comma(1.38)

      where σ>0 is the shape parameter and μ>0 is the scale parameter of the distribution. Note that the lognormal variable is developed from the normal distribution. The random variable X=lnT is a normal random variable with parameters μ and σ. The cdf of the lognormal distribution is

      upper F left-parenthesis t right-parenthesis equals upper Phi left-parenthesis StartFraction ln t minus mu Over sigma EndFraction right-parenthesis comma t ampersand gt semicolon 0 comma(1.39)

      where Φ(x) is the cdf of a standard normal random variable. If T denotes the failure time of an item with lognormal distribution, the reliability function of T will be

      upper R left-parenthesis t right-parenthesis equals 1 minus upper Phi left-parenthesis StartFraction ln t minus mu Over sigma EndFraction right-parenthesis comma t ampersand gt semicolon 0 period(1.40)

      The hazard rate function is

      h left-parenthesis t right-parenthesis equals StartStartFraction f left-parenthesis t right-parenthesis OverOver 1 minus upper Phi left-parenthesis StartFraction ln t minus mu Over sigma EndFraction right-parenthesis EndEndFraction comma t ampersand gt semicolon 0 period(1.41)

      The mean, μ, and variance, σ2, are

      sigma squared equals e Superscript 2 mu plus sigma squared Baseline left-parenthesis e Superscript sigma squared Baseline minus 1 right-parenthesis period(1.42)

       Example 1.3

      The random variable of the time to failure of an item, T, follows the following pdf:

f left-parenthesis t right-parenthesis equals StartLayout Enlarged left-brace 1st Row StartFraction 1 Over 6000 EndFraction comma 0 less-than-or-equal-to t less-than-or-equal-to 6000 comma 2nd Row 0 comma o t h e r w i s e period EndLayout

      where t is in days and t≥0.

      1 What is the probability of failure of the item in the first 100 days?

      2 Find the MTTF of the item.

       Solution

      1 The cdf of the random variable isThe probability of failure in the first 100 days is

      2 The MTTF of the item is

      1.2.3 Physics-of-Failure Equations

      Different from the traditional reliability assessment approach, the Physics-of-Failure (P-o-F) represents an approach to reliability assessment based on modeling and simulation of the physical processes leading to the occurrence of failures in an item [2]. The P-o-F approach begins within the first stages of the design of the item. A model is constructed based on the customer’s requirements, service environment, and stress analysis [1]. Once the models are established, a reliability assessment can be conducted on the item.

      1.2.3.1 Paris’ Law for Crack Propagation

      Figure 1.7 Illustration of Paris Law.

      StartFraction normal d a Over normal d upper N EndFraction equals upper C left-parenthesis upper Delta upper K right-parenthesis Superscript m Baseline comma(1.43)

      upper Delta upper K equals upper K Subscript m a x Baseline minus upper K Subscript m i n Baseline period(1.44)

      1.2.3.2 Archard’s Law for Wear

      The Archard’s wear equation is a simple model used to describe sliding wear, which is based on the theory of asperity contact [4]. The volume of the removed debris due to wear is proportional to the work done by friction forces. The Archard’s wear equation is given by

      where Q is the total volume of the