RELIABILITY OF TECHNICAL SYSTEMS Slovak University of Technology Faculty of Material Science and Technology in Trnava

RELIABILITY OF TECHNICAL SYSTEMS

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

THE BASIC CONCEPTS RELIABILITY

Reliability – property of object to carry out the set functions, keeping in time and in the set limits of value of the established operational parameters

Object – the technical product of the certain special-purpose designation considered during the periods of designing, manufacture, tests and operation.

Objects can be various systems and their elements The element – the elementary component of a product, in

problems of reliability can consist of many details. System – the set of in common operating elements intended for

independent performance of set functions.

THE BASIC CONCEPTS RELIABILITY Reliability of object is characterized by following

basic conditions and events Serviceability – a condition of object at which it

corresponds to all requirements established by the specifications and technical documentation (reference document).

Working capacity – a condition of object at which it is capable to carry out the set functions, keeping values of key parameters established by the reference document.

Limiting condition – a condition of object at which its application is to destination inadmissible or inexpedient.

Classification and characteristics of refusals As refusals are subdivided on: Refusals of functioning (performance of the basic

functions by object stops, for example, breakage cogs of cone);

Refusals parametrical (some parameters of object change in inadmissible limits, for example, loss of accuracy of the machine tool)

By the nature refusals can be: Casual, caused by unforeseen overloads, defects of a

material, mistakes of the personnel or failures of a control system and etc.;

Regular, caused by the natural and inevitable phenomena causing gradual accumulation of damages: weariness, deterioration, ageing, corrosion and etc.

CLASSIFICATION OF REFUSALS The basic attributes of classification of refusals: Character of occurrence; The reason of occurrence; Character of elimination; Consequences of refusals; Further use of object; Ease of detection; Time of occurrence.

Character of occurrence: Sudden refusal – the refusal shown in sharp (instant) change of

characteristics of object; Gradual refusal – the refusal occurring as a result of slow, gradual

deterioration of object

CLASSIFICATION OF REFUSALS

The reason of occurrence: The constructional refusal caused by lacks and

a unsuccessful design of object; The industrial refusal connected with mistakes at

manufacturing of object owing to imperfection or infringement of technology;

The operational refusal caused by infringement of service regulations.

Steady refusal;


Character of elimination: An alternating refusal (arising/disappearing).

Consequences of refusal: easy refusal (easily removable);

Average refusal (not causing refusals of adjacent units – secondary refusals);

Heavy refusal (causing secondary refusals or leading threat of a life and health of the person).

The full refusals excluding an opportunity of work of object before their elimination;


Further use of object: Partial refusals at which the object can partially be used. Obvious (obvious) refusals; Ease of detection: The latent (implicit) refusals. Initial the refusals arising in an initial stage of operation;

Time of occurrence: Refusals at normal operation; Refusals of a type a wear caused by irreversible

processes of deterioration of details, ageing of materials and so forth

Components of reliability

Reliability is the complex property including depending on purpose of object:

Non-failure operation Durability Maintainability Retentivity

Non-failure operation – property of object continuously to keep working capacity during some operating time or during some time.

Operating time – duration or volume of work of the object, measured in any not decreasing sizes (a time unit, number of cycles loadings, kilometers of run and etc.).

Components of reliability

Durability – property of object to keep working capacity before a limiting condition at the established system of maintenance service and repairs.

Maintainability – the property of object consisting its fitness to the prevention and detection of the reasons of occurrence of refusals, to maintenance and restoration of working capacity by carrying out of repairs and maintenance service.

Retentivity – property of object continuously to keep demanded operational parameters during (and after) a period of storage and transportations.

The basic parameters of reliability

The parameter of reliability quantitatively characterizes, in what degree the certain properties causing reliability are inherent in the given object.

Technical resource – an operating time of object from the beginning of its operation or renewal of operation after repair before a limiting condition

The appointed resource – a total operating time of object at which achievement operation should be stopped irrespective of its condition.

Service life – calendar duration of operation (including, storage, repair and T. Item) from its beginning before a limiting condition.

THE BASIC DATA FROM PROBABILITY THEORY

The most important parameters of reliability of nonrestorable objects – parameters of non-failure operation to which concern: Probability of non-failure operation; Density of distribution of refusals; Failure rate; Average operating time to refusal.

Parameters of reliability are represented in two forms (definitions): Statistical (selective estimations); Likelihood.

Axioms of probability theory

The probability of event A is designated P(A) or P{A}. Probability chooses so that it satisfied to following conditions or axioms:

If Ai and Aj not joint events

Frequency definition of probability of any event A:

The theorem of addition of probabilities If A1, A2, …, An - Not joint events and A – the sum of

these events the probability of event A is equal to the sum of probabilities of events A1, A2, …, An:

As opposite events A and

are incompatible also form full group, the sum of their probabilities

The theorem of multiplication of probabilities

If events A1 and A2 are independent, corresponding conditional probabilities

Therefore the theorem of multiplication of probabilities (8) becomes

The formula of full probability

PARAMETERS OF NON-FAILURE OPERATION

Statistical estimation PNFO (empirical function of reliability) is defined:

Density of distribution of refusals The statistical estimation

Density of distribution of refusals

Failure rate

Statistical estimation FR is defined

Failure rate

THE EQUATION OF COMMUNICATION OF

PARAMETERS OF RELIABILITY

The size (t) dt – is probability of that the element which has trouble-free worked in an interval of an operating time [0, t], will give up in an interval [t, t + dt].

Numerical characteristics of non-failure operation of nonrestorable objects

Statistical estimation of an average operating time to refusal

Where ti – An operating time to refusal of i-th object.

Numerical characteristics of non-failure operation of nonrestorable objects

Dispersion of a random variable of an

operating time

STATISTICAL PROCESSING OF RESULTS OF TESTS

Calculation of empirical functions

Calculation of empirical functions

THE NORMAL LAW OF DISTRIBUTION OF THE OPERATING TIME TO REFUSAL

THE NORMAL LAW OF DISTRIBUTION OF THE OPERATING TIME TO REFUSAL

The truncated normal distribution

Exponential distribution

Logarithmic normal (logarithmically normal)

distribution

Scale-distribution

Bases of calculation of reliability of

systems Problem of calculation of reliability: definition of

parameters of non-failure operation of the system consisting of nonrestorable elements, according to about reliability of elements and communications between them.

The purpose of calculation of reliability: To prove a choice of this or that constructive decision; To find out an opportunity and expediency of

reservation; To find out, whether demanded reliability is

achievable at existing technology of development and manufacture.

Bases of calculation of reliability of systems

Calculation of reliability consists of following stages: 1. Definition of structure of counted parameters of

reliability. 2. Drawing up (synthesis) of the structural logic scheme

of reliability (structure of system), based on the analysis of functioning of system (what blocks are included in what their work consists, the list of properties of serviceable system and т. Item), and a choice of a method of calculation of reliability.

3. Drawing up of the mathematical model connecting counted parameters of system with parameters of reliability of elements.

4. Realization calculation, the analysis of the received results, updating of settlement model.

The mathematical model of reliability

Models can be realized by means of: Method of the integrated and differential

equations; On the basis of the column of possible

conditions of system; On the basis of logic - likelihood methods; On the basis of a deductive method (a tree of

refusals).

Systems with reservation

Working capacity of systems without reservation demands working capacity of all elements of system.

In complex technical devices without reservation never it is possible to reach high reliability even if to use elements with high parameters of non-failure operation.

The system with reservation

The system with reservation is a system with redundancy of elements, with reserve components, superfluous in relation to minimally necessary (basic) structure and carrying out the same functions, as basic elements.

In systems with reservation working capacity is provided until for replacement of the given up basic elements are available reserve.

Structural reservation can be

Examples of not loaded reservation

RELIABILITY OF THE BASIC SYSTEM

Probability of non-failure operation (PNFO) OS:

Probability of refusal (IN) OS:

RELIABILITY OF SYSTEMS WITH THE LOADED

RESERVATION

Probability of refusal (IN)

Probability of non-failure operation (PNFO):

Population mean (expectation) operating time to refusal

Reliability of systems with restriction on loading

Number of necessary working elements – r, reserve – (n - r).

Refusal of system comes under condition of refusal (n – r + 1) elements.

PNFO such system it is defined by means of binomial distribution.

Reliability of systems with restriction on loading

Where

Dependence of reliability of system on frequency rate of reservation

RELIABILITY OF SYSTEM WITH NOT LOADED RESERVATION

Assumptions: 1. Time of replacement of the given up

element reserve is equal 0 (t3 0). 2. The switching device of connection of a

reserve element instead of the given up core – is absolutely reliable.

The analysis of a casual operating time to

refusal of system with not loaded reserve

Where T0i = M (Ti ) – expectation operating time to refusal of i-th element of system.

The analysis of a casual operating time to refusal of system with not loaded reserve

The events corresponding working capacity of system for an operating time (0, t):

A = {non-failure operation (БР) systems for an operating time (0, t)};

A1 = {БР ОЭ for an operating time (0, t)};A2 = {Refusal ОЭ during the moment t>, inclusion (t3 = 0) РЭ

and БР РЭ on an interval (t–)}.Event A = A1 A2, Therefore PNFO systems to an operating

time t (for an operating time (0, t)), it is defined:P (A) = P (A1 ) + P (A2 ) ,

Where P (A) = PWith(t); P (A1 ) – PNFO ОЭ to an operating time t, P (A1 ) = P1 (t);P (A2 ) = Pр (t) – probability of refusal ОЭ and БР РЭ after

refusal ОЭ.At the known law of distribution an operating time to refusal ОЭ

calculation P1 (t) does not represent complexity.


Event A2 Is the "complex" event including simple:A21 = {Refusal ОЭ at < i>t (near to the considered

moment)};A22 = {БР РЭ from the moment of up to t, т. е. In

an interval (t-)}.Event A2 It is carried out at simultaneous

performance of events A21 And A22:A2 = A21 A22 .

Events A21 And A22 Are dependent, therefore probability of event A2

P (A2 ) = P (A21 ) P (A22| A21 ) .


Probability of event A2

Then PNFO the considered system with not loaded reserve it is equal:


The density of distribution of an operating time to refusal of system is equal:

IN systems:


DDR systems:

FR systems:




At not loaded reserve with fractional frequency rate (at m> 1) and exponential distribution of an operating time

where k* = n – m.

The casual operating time to refusal of elements of system submits to normal distribution with DDR are considered

where

- number of elements of system


Population mean of an operating time to refusal

Dispersion of an operating time to refusal



For system with elements which time between failures submits exponential to distribution Pi (t) = exp (-i t), it is possible to accept Pi(t) 1-i t, therefore expressions IN and PNFO:

Reliability of systems with the facilitated

reserve The events providing non-failure operation (NFO) of system

for an operating time (0, t):A = {NFO systems for an operating time (0, t)};A1 = {NFO BE for an operating time (0, t)};

A2 ={Refusal BE during the moment , inclusion RE and NFO

it on an interval (t-)} A = A1V A2

Reliability of systems with the facilitated reserve

PNFO systems for an operating time (0, t), i.e. to an operating time t it is equal to the sum of probabilities of events A1 and A2: P (A) = P (A1 ) + P (A2 ) ,

P (A2 ) = P (A21 ) P (A22 ) P (A23 ) .

PNFO reserved system with the facilitated reserve

PNFO the system consisting from n identical reliability of elements:

Reliability of systems with the facilitated reserve

At presence of one BE and one RE (n = 2), PNFO it is defined:

For system from n elements with exponential an operating time to refusal

Sliding reservation

Work of a reserve element

RELIABILITY OF RESTORED OBJECTS AND SYSTEMS

t <t0 t> t0

Key rules of drawing up of model

Reliability of objects at gradual refusals.

The basic concepts and definitions

1. The basic technical parameters describing working capacity of object and a quality being by its measure, we shall name defining parameters (DP).

2. Generally DP can be a vector.

3. The limiting values established on everyone DP of object, are admissible values DP, which limit working area (a floor of the admission).

Reliability of objects at gradual refusals. The basic concepts and definitions

While values vector DP object are inside of multivariate working area, the object is considered efficient.

However eventually under influence of the factors connected with ageing, wear process or violation of regulating the end of vector x(t) can reach border of working area.


In the general statement of a problem the border of working area can be considered as system of random variables or vector casual process.

Process of change DP of identical objects at operation we shall consider as stochastic function X(t) time.


If from the moment of inclusion in work (for t = 0) by measurements with identical t = ti +1 - ti = ti - ti-1 or various periodicity (interval) t to supervise values DPj objects it is possible to predict the further changes DP and the moment approach of refusal.

n,1

Casual process of change DP X(t) can be generally presented:

Stationary casual process (t) is converted changes of parameters at change of external conditions, leads alternating (appear / disappear) to refusals.

Non-stationary casual process (t), characterizes long-term irreversible changes of parameters as a result of wear process, ageing or violation of regulating.

The analysis of casual processes of

change DP of objects

The analysis of casual processes of change DP of objects

Casual process X (t) can be considered wear processes

X0 - initial (factory) value DP; B(t) - half random process of change of speed of wear process.

The analysis of casual processes of change DP of objects

Integral characterizes accumulation of irreversible changes is as a result of ageing, wear processes or violation of regulating.

Change DP depending on time or can be presented operating time generally three periods.

Models of processes of approach of object to refusals

The first period - extra earnings of object. The second period characterizes the basic period of

operation.

The third period - the period of "ageing" of object.

Real process of change DP X(t) is approximated by stochastic function of a kind

Models of processes of approach of object to

refusals. Linear stochastic functions.

where X0 = X (t=0) = {x}0 - casual initial value DP ( t = 0), a having population mean (expectation) mxo = M {X0} and an average square-law deviation RMS) Sxo = ;

V {v} - the casual normally distributed speed of change DP in time, possessing expectation mv = M {V} and RMS

Sv = .

0xD

vD

Models of processes of approach of object to refusals. Nonlinear stochastic functions.

The basic types of models.

Fan with nonzero initial dispersion


Fan with zero initial dispersion


Uniform

RELIABILITY OF OBJECTS AT GRADUAL

REFUSALS

Where f(X)i - density of distribution of values DP at t = ti , that is in i-m section of casual process X(t).

RELIABILITY OF OBJECTS AT GRADUAL REFUSALS

Density of distribution of an operating time to refusal

And in view of function Laplas Ф(z) at normal distribution DP in ti, sections

Q (ti) = 0.5 - Ф(z).

The general models is calculation of

distribution density of an operating time Casual process X(t) is distinct from linear

For calculation [fi]ср, corresponding an interval ti, it is necessary to know the law of distribution DP in the beginning (ti) And the end ti+1 = ti + ti this interval

The fan models of change DP

For objects, casual process of change DP which can be presented fan models, a random variable of time of achievement DP X (t) borders Xp working area

Casual process X(t) is linear

The fan models of change DP

Density of distribution of time of achievement DP of border Xp working area it is defined by a rule of reception of laws of distribution of functions of casual arguments known from probability theory:

The density of distribution f [X (t)], certain on expression, looks like

Uniform model of change DP

Time of preservation of working capacity tс after transformation becomes

tс = mt - St

.


For fan models with nonzero initial dispersion we express speed of change DP under condition of achievement by process X (t) borders Xp working area, i.e. X(t) = Xp


The density of distribution of time of crossing DP

Private questions of an estimation of

parametrical reliability of objects

R (t) = R0 + Qt

Q - casual speed violation of regulating;

t - time counted from the moment of carrying out last maintenance service.

Private questions of an estimation of parametrical reliability of objects

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RELIABILITY OF TECHNICAL SYSTEMS Slovak University of Technology Faculty of Material Science and Technology in Trnava