Produkt

KlappentextThis book is based on a lecture course to students specializing in the safety of technological processes and production.

The author focuses on three main problems in technological risks and safety: elements of reliability theory, the basic notions, models and methods of general risk theory and some aspects of insurance in the context of risk management.

Although the material in this book is aimed at those working towards a bachelor's degree in engineering, it may also be of interest to postgraduate students and specialists dealing with problems related to reliability and risks.

This book is based on a lecture course to students specializing in the safety of technological processes and production.The author focuses on three main problems in technological risks and safety: elements of reliability theory, the basic notions, models and methods of general risk theory and some aspects of insurance in the context of risk management. Although the material in this book is aimed at those working towards a bachelor's degree in engineering, it may also be of interest to postgraduate students and specialists dealing with problems related to reliability and risks.

Details

Weitere ISBN/GTIN9781119347460

ProduktartE-Book

EinbandartE-Book

FormatEPUB

Format Hinweis2 - DRM Adobe / EPUB

FormatFormat mit automatischem Seitenumbruch (reflowable)

Verlag

John Wiley & Sons

Erscheinungsjahr2016

Erscheinungsdatum16.08.2016

Auflage1. Auflage

Seiten228 Seiten

SpracheEnglisch

Dateigrösse9243 Kbytes

Artikel-Nr.3414347

Rubriken

Genre9201

Inhalt/Kritik

Leseprobe

1
Reliability of Engineering Systems
1.1. Basic notions and characteristics of reliability
1.1.1. Basic notions

The notions described below correspond to the usual terminology used in reliability theory and most of the literature sources on reliability. Reliability theory deals with the following basic notions.

An object in reliability theory means a unit (an element or article), an apparatus, an engineering product and any system or its part at all, considering from the point of view of their reliability. Furthermore, the term unit is used for simple objects, which is considered a single entity. For complex objects, the term system is used and the term element means the minimal component of a system.

An exploitation of an object (unit or system) means the collection of all its existence phases (creation, transportation, storage, using, maintenance and repair).

Reliability of an object is its complex property, consisting of its possibility to fulfill assign to it functions under given exploitation conditions1.

According to the definition of Gnedenko [GNE 65], reliability theory is a scientific discipline about the requirements that should be used for projecting, producing, testing and exploitation of an object in order to get the maximal effect from its use. Reliability theory deals with such notions as: reliability, failure (breakdown), longevity, repair, repair-ability, etc.

Reliability means the possibility of an object to maintain its workability during a given time period under a given exploitation condition.

A failure is a partial or full loss of the object s workability. Therefore, we should distinguish full and partial failures.

In addition, failures are divided into sudden, for which the object suddenly (unexpected) loses its workability, gradual, for which the workability of an object is lost gradually (usually as a result of some physical parameters of the object going out of the admissible level) and halting (temporary loss of the workability).

Longevity is the ability of an object to be used for a long time under needed technical service.

Repair is the procedure that renews objects reliability.

Repair-ability is the property of an object to predict, detect and remove its failures.

Safety is the property of an object (system, unit) not to allow situations that could be dangerous for people and the environment.

Further notions and definitions are introduced in the chapter if necessary.

Given the complex property of an object, the reliability is described by many different characteristics and indexes. Furthermore, the term characteristic is used for complex (functional) reliability characteristics, and the term index is usually used for numerical (simple) characteristics.

Among the different reliability characteristics, we first consider those that are used for units and systems which work up to the first failure.
1.1.2. Reliability of non-renewable units

In this section, the reliability of an object is studied independently of the reliability of its components as a single entity, and therefore instead of the term object , here, the term unit is used. Suppose that the unit can be in only two states from the point of view of its reliability: workable (up) and not workable or failure (down). Denote by T the lifetime of the unit. It is a random variable (r.v.) and its basic characteristic is its cumulative distribution function (c.d.f.) that is the probability that this time is not greater than the fixed time t,

[1.1]

Here and later, the symbol P{·} is used for the probability of the event in brackets. In the case of continuous observations for the unit state, this function is a continuous one, but in the case of observations for the unit state in discrete points of time, it is a stepwise one. The function

[1.2]

in reliability theory is known as reliability function2. For continuous distribution, the graphs of these functions are shown in Figure 1.1.

Figure 1.1. C.d.f. of lifetime and reliability function of some unit

In the case of continuous observation, the r.v. T can also be characterized by its probability density function (p.d.f.) f(t) = Fâ²(t). At that lifetime c.d.f. connected with p.d.f. by the equality,

[1.3]

For small values Ît, the quantity f(t)Ît is the probability of a unit s failure in time interval (t, t + Ît). Because in practice, the probability is measured with frequency, this value is also called frequency of failures.

In reliability practice the time is usually measured in discrete units. Therefore, the discrete distributions are more appropriate models for the lifetime s description. However, for theoretical study, the continuous distributions are more convenient. Therefore, according to these reasons, mostly continuous distributions will be used for the units lifetime distribution description. By the way, when the time is measured in discrete units, the discrete distribution can be obtained from the continuous one by discretization of time,

[1.4]

where Î means the unit of time (in minutes, hours, months or years).

Besides lifetime distribution of a new unit, an important reliability of its characteristic is its residual lifetime. Conditional distribution, after its reliable working time t, represents conditional failure probability in time interval (t, t + x] given up to time t a failure does not occur,

[1.5]

For small values of x, we have:

where the function Î»(t) represents a conditional probability density of residual lifetime of a unit under condition that is used without failure during time t. More precisely, this function is determined by the equality,

[1.6]

and in reliability literature, it is also known as hazard rate function (h.r.f.). This function allows us to evaluate the failure probability of a unit during a small time interval Ît after time t as follows:

as an area under the curve, as is shown in Figure 1.2.

Figure 1.2. Typical hazard rate function

Equality [1.6] allows us to represent the c.p.f. of a unit lifetime and its reliability function in terms of its h.r.f. In fact, it can be represented as

which after integration gives

Supposing that there are no instant failures, which means that F(+0) = 0, it gives

or

[1.7]

Analogously, for conditional lifetime probability in interval (t, t + x], we can find

[1.8]

Besides functional characteristics in practice, the lifetime of units is also measured with some numerical indexes such as:
- mean lifetime, i.e. expectation of lifetime,

[1.9]
- variance of lifetime, which shows the variation of the lifetime around its mean value,

[1.10]

Here and later, symbols E[·] and Var[·] indicate expectation and variance, respectively.

One of the main problems of reliability theory is elements and unit lifetime distribution modeling. Some parametric families of continuous distributions of non-negative random variables that are usually used for the unit lifetime modeling are presented in the next section. Some of these distributions will also be used later in section 2.3 for modeling of the damage value distributions.
1.1.3. Some parametric families of continuous distributions of non-negative random variables

Consider some parametric families of continuous distributions of non-negative random variables along with their indexes.
1.1.3.1. Exponential distribution
Exponential lifetime is used for modeling the reliability of units, subject to instantaneous (sudden, unexpected) failures. Its p.d.f. and c.d.f. are

[1.11]

where Î» > 0 is its parameter. The reliability function of these units is

[1.12]

and their h.r.f. is constant and coincides with the distribution parameter Î»,

[1.13]

The graphs of these functions are represented in Figure 1.3.

Moreover, the property of h.r.f. to be constant is a characteristic property of the exponential reliability law. From relation [1.7] we have

[1.14]

Another characteristic property of an exponential distribution is its memoryless property, which is presented in the following theorem.

THEOREM 1.1.- A unit has an exponential reliability law iff the distribution of its residual lifetime does not depend on the elapsed working time (its age),

[1.15]

Figure 1.3....
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