Transcript Document 677577

Slovak University of Technology
Faculty of Material Science and Technology in Trnava
RELIABILITY OF
TECHNICAL
SYSTEMS
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 specialpurpose 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
– 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.
 Serviceability
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;
CLASSIFICATION OF REFUSALS
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;
CLASSIFICATION OF REFUSALS
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 nonfailure 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.
The analysis of a casual operating time to
refusal of system with not loaded reserve
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 ) .
The analysis of a casual operating time to
refusal of system with not loaded reserve
Probability of event A2
Then PNFO the considered system with not loaded reserve
it is equal:
The analysis of a casual operating time to
refusal of system with not loaded reserve
The density of distribution of an operating time to refusal
of system is equal:
The analysis of a casual operating time to
refusal of system with not loaded reserve
IN systems:
DDR systems:
FR systems:
The analysis of a casual operating time to
refusal of system with not loaded reserve
The analysis of a casual operating time to
refusal of system with not loaded reserve
The analysis of a casual operating time to
refusal of system with not loaded reserve
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
The analysis of a casual operating time to
refusal of system with not loaded reserve

Population mean of an operating time to refusal

Dispersion of an operating time to refusal
The analysis of a casual operating time to
refusal of system with not loaded reserve
The analysis of a casual operating time to
refusal of system with not loaded reserve

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.

Reliability of objects at gradual refusals.
The basic concepts and definitions
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.

1, n
Reliability of objects at gradual refusals.
The basic concepts and definitions


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.
The analysis of casual processes of
change DP of objects

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

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.

Models of processes of approach of object to
refusals. Linear stochastic functions.

Real process of change DP X(t) is approximated
by stochastic function of a kind
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 = Dx 0
;
V {v} - the casual normally distributed speed of change DP
in time, possessing expectation mv = M {V} and RMS
Sv = Dv .
Models of processes of approach of object to
refusals. Nonlinear stochastic functions.
The basic types of models.
Fan with nonzero initial dispersion
The basic types of models.
Fan with zero initial dispersion
The basic types of models.
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
Uniform model of change DP
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
Uniform model of change DP
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