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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      In all generality, the expected value or mean of the time to failure T is called the mean time to failure (MTTF), which is defined as

      

(1.7)

      It is equivalent to

      

(1.8)

      The failure rate function or hazard rate function, denoted by h(t), is defined as the conditional probability of failure in the time interval [t, t+Δt] given that it has been working properly up to time t, which is given by

      

(1.9)

      Furthermore, the cumulative failure rate function, or cumulative hazard function, denoted by H(t), is given by

      

(1.10)

      Example 1.2 The failure time of a valve follows the exponential distribution with parameter h(t) (in arbitrary units of time-1). The value is new and functioning at time h(t). Calculate the reliability of the valve at time h(t) (in arbitrary units of time).

       Solution

      The pdf of the failure time of the valve is

      The reliability function of the valve is given by

      At time, the value of the reliability is

      1.2 Component Reliability Modeling

      As mentioned in the previous section, in reliability engineering, the time to failure of an item is a random variable. In this section, we briefly introduce several commonly used discrete and continuous distributions for component reliability modeling.

      1.2.1 Discrete Probability Distributions

      If random variable X can take only a finite number k of different values x1,x2,…,xk or an infinite sequence of different values x1,x2,…, the random variable X has a discrete probability distribution. The probability mass function (pmf) of X is defined as the function f such that for every real number x,

      

(1.11)

      If x is not one of the possible values of X, then f(x)=0. If the sequence x1,x2,… includes all the possible values of X, then ∑if(xi)=1. The cdf is given by

      

(1.12)

      1.2.1.1 Binomial Distribution

      

(1.13)

      Figure 1.1 The pmf of the binomial distribution with n=5, p=0.4.

      The pmf of the binomial distribution is

(1.14)

      For a binomial distribution, the mean, μ, is given by

      

(1.15)

      and the variance, σ2, is given by

      

(1.16)

      1.2.1.2 Poisson Distribution

      

(1.17)

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