Methods for calculating basic reliability indicators. Calculation of basic reliability indicators. Purpose and classification of calculation methods
Purpose and classification of calculation methods
Reliability calculations are calculations intended to determine quantitative indicators of reliability. They are carried out at various stages of development, creation and operation of facilities.
At the design stage, reliability calculations are carried out with the aim of forecasting (forecasting) the expected reliability of the system being designed. Such forecasting is necessary to justify the proposed project, as well as to resolve organizational and technical issues:
- choice optimal option structures;
- method of reservation;
- depth and methods of control;
- number of spare elements;
- frequency of prevention.
At the test and operation stage, reliability calculations are carried out to assess quantitative reliability indicators. Such calculations are, as a rule, in the nature of statements. The calculation results in this case show how reliable the objects that were tested or used in certain operating conditions were. Based on these calculations, measures are developed to improve reliability, the weak points of the object are determined, and assessments of its reliability and the influence of individual factors on it are given.
The numerous purposes of calculations have led to their great diversity. In Fig. 4.5.1 shows the main types of calculations.
Elemental calculation- determination of object reliability indicators, determined by the reliability of its components (elements). As a result of this calculation, the technical condition of the object is assessed (the probability that the object will be in working condition, mean time between failures, etc.).
Rice. 4.5.1. Classification of reliability calculations
Calculation of functional reliability - determination of reliability indicators for performing specified functions (for example, the probability that the gas purification system will operate for a given time, in specified operating modes, while maintaining all the necessary parameters for purification indicators). Since such indicators depend on a number of operating factors, then, as a rule, the calculation of functional reliability is more complex than elemental calculation.
By choosing in Fig. 4.5.1 the options for moving along the path indicated by the arrows, each time we get the new kind(case) of calculation.
The simplest calculation- calculation, the characteristics of which are presented in Fig. 4.5.1 on the left: elemental calculation of the hardware reliability of simple products, non-redundant, without taking into account restoration of performance, provided that the operating time to failure is subject to an exponential distribution.
The most difficult calculation- calculation, the characteristics of which are presented in Fig. 4.5.1 on the right: functional reliability of complex redundant systems, taking into account the restoration of their performance and various laws of distribution of operating time and recovery time.
The choice of one or another type of reliability calculation is determined by the task for calculating reliability. Based on the task and subsequent study of the operation of the device (according to its technical description) an algorithm for calculating reliability is compiled, i.e. sequence of calculation stages and calculation formulas.
Sequence of system calculations
The sequence of system calculations is shown in Fig. 4.5.2. Let's consider its main stages.
Rice. 4.5.2. Reliability calculation algorithm
First of all, the task for calculating reliability should be clearly formulated. It must indicate: 1) the purpose of the system, its composition and basic information about its operation; 2) reliability indicators and signs of failures, the purpose of the calculations; 3) the conditions under which the system operates (or will operate); 4) requirements for the accuracy and reliability of calculations, for the completeness of taking into account existing factors.
Based on the study of the task, a conclusion is made about the nature of the upcoming calculations. In the case of calculating functional reliability, the transition is made to stages 4-5-7, in the case of calculating elements (hardware reliability) - to stages 3-6-7.
A structural diagram of reliability is understood as a visual representation (graphical or in the form of logical expressions) of the conditions under which the object under study (system, device, technical complex, etc.) works or does not work. Typical block diagrams are shown in Fig. 4.5.3.
Rice. 4.5.3. Typical structures reliability calculation
The simplest form of a reliability block diagram is a parallel-series structure. It connects elements in parallel, the joint failure of which leads to failure
Such elements are connected in a sequential chain, the failure of any of which leads to the failure of the object.
In Fig. 4.5.3a presents a variant of the parallel-series structure. Based on this structure, the following conclusion can be drawn. The object consists of five parts. Failure of an object occurs when either element 5 or a node consisting of elements 1-4 fails. A node can fail when a chain consisting of elements 3,4 and a node consisting of elements 1,2 fails at the same time. Circuit 3-4 fails if at least one of its constituent elements fails, and node 1,2 - if both elements fail, i.e. elements 1,2. Calculation of reliability in the presence of such structures is characterized by the greatest simplicity and clarity. However, it is not always possible to present the performance condition in the form of a simple parallel-series structure. In such cases, either logical functions are used, or graphs and branching structures are used, according to which systems of performance equations are left.
Based on the reliability block diagram, a set of calculation formulas is compiled. For typical calculation cases, the formulas given in reference books on reliability calculations, standards and guidelines are used. Before applying these formulas, you must first carefully study their essence and areas of use.
Reliability calculation based on the use of parallel-series structures
Let some technical system D be composed of n elements (nodes). Let's say we know the reliability of the elements. The question arises about determining the reliability of the system. It depends on how the elements are combined into the system, what the function of each of them is and to what extent the proper operation of each element is necessary for the operation of the system as a whole.
The parallel-sequential reliability structure of a complex product gives an idea of the relationship between the reliability of the product and the reliability of its elements. Reliability calculations are carried out sequentially - starting from the calculation of elementary nodes of the structure to its increasingly complex nodes. For example, in the structure of Fig. 5.3, and a knot consisting of elements 1-2 is an elementary knot consisting of elements 1-2-3-4, complex. This structure can be reduced to an equivalent one, consisting of elements 1-2-3-4 and element 5 connected in series. Calculation of reliability in this case comes down to the calculation of individual sections of the circuit, consisting of elements connected in parallel and in series.
System with serial connection of elements
The simplest case in a computational sense is the series connection of system elements. In such a system, the failure of any element is equivalent to the failure of the system as a whole. By analogy with a chain of series-connected conductors, the break of each of which is equivalent to opening the entire circuit, we call such a connection “series” (Fig. 4.5.4). It should be clarified that such a connection of elements is “serial” only in the sense of reliability; physically they can be connected in any way.
Rice. 4.5.4. Block diagram of a system with serial connection of elements
From a reliability standpoint, such a connection means that the failure of a device consisting of these elements occurs when element 1 or element 2, or element 3, or element n fails. The operability condition can be formulated as follows: the device is operational if element 1 and element 2, and element 3, and element n are operational.
Let us express the reliability of this system through the reliability of its elements. Let there be a certain period of time (0,t), during which it is required to ensure failure-free operation of the system. Then, if the reliability of the system is characterized by the reliability law P(t), it is important for us to know the value of this reliability at t=t, i.e. Р(t). This is not a function, but a specific number; let's discard the argument t and simply denote the reliability of the system P. Similarly, let's denote the reliability of individual elements P 1, P 2, P 3, ..., P n.
For failure-free operation of a simple system for a period of time t, each of its elements must work without failure. Let us denote S - an event consisting of failure-free operation of the system during time t; s 1, s 2, s 3, ..., s n - events consisting of failure-free operation of the corresponding elements. Event S is the product (combination) of events s 1, s 2, s 3, ..., s n:
S = s 1 × s 2 × s 3 × ... × s n.
Suppose that elements s 1, s 2, s 3, ..., s n fail independently of each other(or, as they say in relation to reliability, “independent of failures”, and very briefly “independent”). Then, according to the rule of multiplication of probabilities for independent events P(S)=P(s 1)× P(s 2)× P(s 3)× ...× P(s n) or in other notations,
P = P 1 × P 2 × P 3 × ... × Р n .,(4.5.1)
and in shortP = ,(4.5.2)
those. The reliability (probability of an operational state) of a simple system composed of failure-independent, series-connected elements is equal to the product of the reliability of its elements.
In the particular case when all elements have the same reliability P 1 =P 2 =P 3 = ... =P n , expression (4.5.2) takes the form
P = Pn.(4.5.3)
Example 4.5.1. The system consists of 10 independent elements, the reliability of each of which is P = 0.95. Determine system reliability.
According to formula (4.5.3) P = 0.95 10 » 0.6.
The example shows how the reliability of the system drops sharply as the number of elements in it increases. If the number of elements n is large, then to ensure at least acceptable reliability P of the system, each element must have very high reliability.
Let us pose the question: what reliability P should an individual element have in order for a system composed of n such elements to have a given reliability P?
From formula (4.5.3) we obtain:
P = .
Example 4.5.2. A simple system consists of 1000 equally reliable, independent elements. What reliability should each of them have in order for the system reliability to be at least 0.9?
According to formula (4.5.4) P = ; logР = log0.9 1/1000; R» 0.9999.
The failure rate of the system under the exponential distribution law of time to failure can be easily determined from the expression
l с = l 1 + l 2 + l 3 + ... + l n ,(4.5.4)
those. as the sum of failure rates of independent elements. This is natural, since for a system in which the elements are connected in series, the failure of an element is equivalent to a failure of the system, which means that all failure flows of individual elements add up to one system failure flow with an intensity equal to the sum of the intensities of the individual flows.
Formula (4.5.4) is obtained from the expression
P = P 1 P 2 P 3 ... P n = exp(-( l 1 + l 2 + l 3 + ... + l n )).(4.5.5)
Average time to failure
T 0 = 1/ l s.(4.5.6)
Example 4.5.3. A simple system S consists of three independent elements, whose failure-free operation time distribution densities are given by the formulas:
at 0< t < 1 (рис. 4.5.5).
Rice. 4.5.5. Distribution densities of failure-free operation time
Find the failure rate of the system.
Solution. We determine the unreliability of each element: 
at 0< t < 1.
Hence the reliability of the elements: 
at 0< t < 1.
Failure rates of elements (conditional failure probability density) - ratio f(t) to p(t): 
at 0< t < 1.
Adding, we have: l c = l 1 (t) + l 2 (t) + l 3 (t).
Example 4.5.4. Let us assume that for the operation of a system with a series connection of elements at full load, two pumps of different types are required, and the pumps have constant failure rates equal to l 1 =0.0001h -1 and l 2 =0.0002h -1 , respectively. It is required to calculate the average failure-free operation of this system and the probability of its failure-free operation for 100 hours. It is assumed that both pumps start working at time t =0.
Using formula (4.5.5), we find the probability of failure-free operation P s of a given system for 100 hours:
P s (t)= .
P s (100)=е -(0.0001+0.0002)× 100 =0.97045.
Using formula (4.5.6), we obtain
h.
In Fig. 4.5.6 shows a parallel connection of elements 1, 2, 3. This means that a device consisting of these elements goes into a failure state after the failure of all elements, provided that all elements of the system are under load, and the failures of the elements are statistically independent.
Rice. 4. 5.6. Block diagram of a system with parallel connection of elements
The condition for the operability of a device can be formulated as follows: the device is operable if element 1 or element 2, or element 3, or elements 1 and 2, 1 are operational; and 3, 2; and 3, 1; and 2; and 3.
The probability of a failure-free state of a device consisting of n parallel-connected elements is determined by the theorem of addition of probabilities of joint random events as
Р=(р 1 +р 2 +...р n)-(р 1 р 2 +р 1 р 3 +...)-(р 1 р 2 р 3 +р 1 р 2 р n +... )-...±
(р 1 р 2 р 3 ...р n).(4.5.7)
For the given block diagram (Fig. 4.5.6), consisting of three elements, expression (4.5.7) can be written:
R = r 1 + r 2 + r 3 - (r 1 r 2 + r 1 r 3 + r 2 r 3) + r 1 r 2 r 3 .
With regard to reliability problems, according to the rule of multiplying the probabilities of independent (together) events, the reliability of a device of n elements is calculated by the formula
Р = 1- ,(4.5.8)
those. when connecting independent (in terms of reliability) elements in parallel, their unreliability (1-p i =q i) is multiplied.
In the particular case when the reliabilities of all elements are the same, formula (4.5.8) takes the form
Р = 1 - (1-р) n.(4.5.9)
Example 4.5.5. The safety device, which ensures the safety of the system under pressure, consists of three valves that duplicate each other. The reliability of each of them is p=0.9. The valves are independent in terms of reliability. Find device reliability.
Solution. According to formula (4.5.9) P = 1-(1-0.9) 3 = 0.999.
The failure rate of a device consisting of n parallel-connected elements with a constant failure rate l 0 is defined as
.(4.5.10)
From (4.5.10) it is clear that the failure rate of the device for n>1 depends on t: at t=0 it is equal to zero, and as t increases, it monotonically increases to l 0.
If the failure rates of elements are constant and subject to the exponential distribution law, then expression (4.5.8) can be written
Р(t) =
.(4.5.11)
We find the average failure-free operation time of the system T 0 by integrating equation (4.5.11) in the interval:
T 0 =
=(1/
l 1 +1/ l 2 +…+1/ l n )-(1/(l 1 + l 2 )+ 1/(l 1 + l 3 )+…)+(4.5.12)
+(1/(l 1 + l 2 + l 3 )+1/(l 1 + l 2 + l 4 )+…)+(-1) n+1 ´ .
In the case when the failure rates of all elements are the same, expression (4.5.12) takes the form
T 0 = .(4.5.13)
The average time to failure can also be obtained by integrating equation (4.5.7) in the interval
Example 4.5.6. Let us assume that two identical fans in an exhaust gas purification system operate in parallel, and if one of them fails, the other is capable of operating at full system load without changing its reliability characteristics.
It is required to find the failure-free operation of the system for 400 hours (the duration of the task) provided that the failure rates of the fan motors are constant and equal to l = 0.0005 h -1 , the motor failures are statistically independent and both fans start working at time t = 0.
Solution. In the case of identical elements, formula (4.5.11) takes the form
P(t) = 2exp(- l t) - exp(-2 l t).
Since l = 0.0005 h -1 and t = 400 h, then
P (400) = 2exp(-0.0005 ´ 400) - exp(-2 ´ 0.0005 ´ 400) = 0.9671.
We find the mean time between failures using (4.5.13):
T 0 = 1/l (1/1 + 1/2) = 1/l ´ 3/2 = 1.5/0.0005 = 3000 hours.
Let's consider the simplest example of a redundant system - a parallel connection of the system's backup equipment. Everything in this diagram n identical pieces of equipment operate simultaneously, and each piece of equipment has the same failure rate. This picture is observed, for example, if all equipment samples are kept at operating voltage (the so-called “hot reserve”), and for the system to function properly, at least one of the equipment must be in working order. n equipment samples.
In this redundancy option, the rule for determining the reliability of parallel-connected independent elements is applicable. In our case, when the reliability of all elements is the same, the reliability of the block is determined by the formula (4.5.9)
P = 1 - (1-p) n.
If the system consists of n samples of backup equipment with different failure rates, then
P(t) = 1-(1-p 1) (1-p 2)... (1-p n).(4.5.21)
Expression (4.5.21) is represented as a binomial distribution. It is therefore clear that when a system requires at least k serviceable ones n equipment samples, then
P(t) = p i (1-p) n-i , where
.(4.5.22)
At a constant failure rate of l elements, this expression takes the form
P(t) =
,(4.5.22.1)
where p = exp(-l t).
Enabling backup system equipment by replacement
In this connection diagram n Of identical equipment samples, only one is in operation all the time (Fig. 4.5.11). When a working sample fails, it is certainly turned off, and one of the ( n-1) reserve (spare) elements. This process continues until everything ( n-1) Reserve samples will not be exhausted.
Rice. 4.5.11. Block diagram of the system for switching on backup equipment of the system by replacement
Let us accept the following assumptions for this system:
1. System failure occurs if everyone fails n elements.
2. The probability of failure of each piece of equipment does not depend on the condition of the others ( n-1) samples (failures are statistically independent).
3. Only equipment in operation can fail, and the conditional probability of failure in the interval t, t+dt is equal to l dt; spare equipment cannot fail before it is put into operation.
4. Switching devices are considered absolutely reliable.
5. All elements are identical. The spare parts have the same characteristics as new.
The system is capable of performing the functions required of it if at least one of the n equipment samples. So in this case the reliability is simply the sum of the probabilities system states, excluding the failure condition, i.e.
P(t) = exp(- l t) .(4.5.23)
As an example, consider a system consisting of two backup equipment samples switched on by replacement. In order for this system to work at time t, it is necessary that by time t either both samples or one of the two are operational. That's why
P(t) = exp(- l t) =(exp(- l t))(1+ l t).(4.5.24)
In Fig. 4.5.12 shows a graph of the function P(t) and for comparison a similar graph for a non-redundant system is shown.
Rice. 4.5. 12. Reliability functions for a redundant system with the inclusion of a reserve by replacement (1) and a non-redundant system (2)
Example 4.5.11. The system consists of two identical devices, one of which is operational, and the other is in unloaded reserve mode. The failure rates of both devices are constant. In addition, it is assumed that the backup device has the same characteristics as the new one at the beginning of operation. It is required to calculate the probability of failure-free operation of the system for 100 hours, provided that the failure rate of devices l = 0.001 h -1 .
Solution. Using formula (4.5.23) we obtain Р(t) = (exp(- l t))(1+ l t).
For given values of t and l, the probability of failure-free operation of the system is
P(t) = e -0.1 (1+0.1) = 0.9953.
In many cases, it cannot be assumed that spare equipment will not fail until it is put into service. Let l 1 be the failure rate of working samples, and l 2 - backup or spare (l 2 > 0). In the case of a duplicated system, the reliability function has the form:
P(t) = exp(-(l 1 + l 2 )t) + exp(- l 1 t) - exp(-(l 1 + l 2 )t).
This result for k=2 can be extended to the case k=n. Really
P(t) = exp(- l 1 (1+ a (n-1))t)
(4.5.25)
, where a = l 2 / l 1 > 0.
Reliability of a redundant system in case of combinations of failures and external influences
In some cases, system failure occurs due to certain combinations of failures of equipment samples included in the system and (or) due to external influences on this system. Consider, for example, a weather satellite with two information transmitters, one of which is a backup or spare. System failure (loss of communication with the satellite) occurs when two transmitters fail or in cases where solar activity creates continuous interference with radio communications. If the failure rate of a working transmitter is equal to l, and j is the expected intensity of radio interference, then the system reliability function
P(t) = exp(-(l + j )t) + l t exp(-(l + j )t).(4.5.26)
This type of model is also applicable in cases where there is no reserve under the replacement scheme. For example, suppose that an oil pipeline is subject to hydraulic shocks, and the impact of minor hydraulic shocks occurs with intensity l, and significant ones - with intensity j. For the break welds(due to the accumulation of damage), the pipeline should receive n small water hammers or one significant one.
Here, the state of the destruction process is represented by the number of impacts (or damage), and one powerful hydraulic shock is equivalent to n small ones. Reliability or the probability that the pipeline will not be destroyed by microshocks at time t is equal to:
P(t) = exp(-(l + j )t) .(4.5.27)
Analysis of system reliability under multiple failures
Let us consider a method for analyzing the reliability of loaded elements in the case of statistically independent and dependent (multiple) failures. It should be noted that this method can be applied to other models and probability distributions. When developing this method, it is assumed that for each element of the system there is some probability of multiple failures occurring.
As is known, multiple failures do exist, and to take them into account, the parameter is introduced into the corresponding formulas a . This parameter can be determined based on experience in operating redundant systems or equipment and representsproportion of failures caused by a common cause. In other words, parameter a can be considered as a point estimate of the probability that the failure of some element is one of multiple failures. In this case, we can assume that the failure rate of an element has two mutually exclusive components, i.e. e. l = l 1 + l 2, where l 1 - constant rate of statistically independent element failures, l 2 - the rate of multiple failures of a redundant system or element. Because thea= l 2 / l, then l 2 = a/l, and therefore l 1 =(1- a ) l .
We present formulas and dependencies for the probability of failure-free operation, failure rate and average time between failures in the case of systems with parallel and serial connection of elements, as well as systems with k serviceable elements from P and systems whose elements are connected via a bridge circuit.
System with parallel connection of elements(Fig. 4.5.13) - a conventional parallel circuit to which one element is connected in series. The parallel part (I) of the diagram displays independent failures in any system from n elements, and the series-connected element (II) - all multiple system failures.
Rice. 4.5.13. Modified system with parallel connection of identical elements
A hypothetical element, characterized by a certain probability of occurrence of multiple failures, is connected in series with elements that are characterized by independent failures. Failure of a hypothetical series-connected element (i.e., multiple failure) results in failure of the entire system. It is assumed that all multiple failures are completely interrelated. The probability of failure-free operation of such a system is determined as R р =(1-(1-R 1) n) R 2, where n - number of identical elements; R 1 - probability of failure-free operation of elements due to independent failures; R 2 is the probability of failure-free operation of the system due to multiple failures.
l 1 and l 2 the expression for the probability of failure-free operation takes the form
R р (t)=(1-(1-e -(1- a )
l t ) n ) e - al t ,(4.5.28)
where t is time.
The effect of multiple failures on the reliability of a system with parallel connection of elements is clearly demonstrated in Fig. 4.5.14 – 4.5.16; when increasing the parameter value a the probability of failure-free operation of such a system decreases.
Parameter a takes values from 0 to 1. When a = 0 the modified parallel circuit behaves like a regular parallel circuit, and when a =1 it acts as one element, i.e. all system failures are multiple.
Since the failure rate and mean time between failures of any system can be determined using(4.3.7) and formulas 
,
,
taking into account the expression forR p(t ) we find that the failure rate (Fig. 4.5.17) and the average time between failures of the modified system are respectively equal
,(4.5.29)
,Where
.(4.5.30)
Rice. 4.5.14. Dependence of the probability of failure-free operation of a system with a parallel connection of two elements on the parameter a
Rice. 4.5.15. Dependence of the probability of failure-free operation of a system with a parallel connection of three elements on the parameter a
Rice. 4.5.16. Dependence of the probability of failure-free operation of a system with a parallel connection of four elements on the parameter a
Rice. 4.5.17. Dependence of the failure rate of a system with a parallel connection of four elements on the parameter a
Example 4.5.12. It is required to determine the probability of failure-free operation of a system consisting of two identical parallel-connected elements, if l =0.001 h -1; a =0.071; t=200 h.
The probability of failure-free operation of a system consisting of two identical parallel-connected elements, which is characterized by multiple failures, is 0.95769. The probability of failure-free operation of a system consisting of two parallel-connected elements and characterized only by independent failures is 0.96714.
System with k serviceable elements from n identical elementsincludes a hypothetical element corresponding to multiple failures and connected in series with a conventional system of the type k from n, which is characterized by independent failures. The failure represented by this hypothetical element causes the entire system to fail. Probability of failure-free operation of a modified system with k serviceable elements from n can be calculated using the formula
,(4.5.31)
where R 1 - probability of failure-free operation of an element characterized by independent failures; R 2 - probability of failure-free operation of the system with k serviceable elements from n , which is characterized by multiple failures.
At constant intensities l 1 and l 2 the resulting expression takes the form
.(4.5.32)
Dependence of the probability of failure-free operation on the parameter a for systems with two serviceable elements out of three and two and three serviceable elements out of four are shown in Fig. 4.5.18 - 4.5.20. When increasing the parameter a the probability of failure-free operation of the system decreases by a small amount(l t).
Rice. 4.5.18. The probability of failure-free operation of a system that remains operational when two of them fail n elements
Rice. 4.5.19. The probability of failure-free operation of a system that remains operational if two of the four elements fail
Rice. 4.5.20. Probability of failure-free operation of a system that remains operational when three out of four elements fail
System failure rate with k serviceable elements from n and mean time between failures can be determined as follows:
,(4.5.33)
where h = (1-e -(1-b )l t ),
q = e (r a -r- a ) l t
.(4.5.34)
Example 4.5.13. It is required to determine the probability of failure-free operation of a system with two serviceable elements out of three, if l =0.0005 h - 1; a =0.3; t =200 h.
Using the expression for R kn we find that the probability of failure-free operation of a system in which multiple failures have occurred is 0.95772. Note that for a system with independent failures this probability is equal to 0.97455.
System with parallel-series connection of elementscorresponds to a system consisting of identical elements, which are characterized by independent failures, and a number of branches containing imaginary elements, which are characterized by multiple failures. The probability of failure-free operation of a modified system with a parallel-series (mixed) connection of elements can be determined using the formula R ps =(1 - (1-) n ) R 2 , where m - number of identical elements in a branch, n- number of identical branches.
At constant failure rates l 1 and l 2 this expression takes the form
R рs (t) = e - bl t . (4.5.39)
(here A=(1- a ) l ). Dependency of system failure-free operation Rb (t) for various parameters a shown in Fig. 4.5.21. At small values l t the probability of failure-free operation of a system with elements connected via a bridge circuit decreases with increasing parameter a.
Rice. 4.5.21. Dependence of the probability of failure-free operation of a system, the elements of which are connected via a bridge circuit, on the parameter a
The failure rate of the system under consideration and the mean time between failures can be determined as follows:
l + .(4.5.41)
Example 4.5.14. It is required to calculate the probability of failure-free operation for 200h for a system with identical elements connected via a bridge circuit, if l =0.0005 h - 1 and a =0.3.
Using the expression for Rb(t), we find that the probability of failure-free operation of a system with elements connected using a bridge circuit is approximately 0.96; for a system with independent failures (i.e. when a =0) this probability is 0.984.
Reliability model for a system with multiple failures
To analyze the reliability of a system consisting of two unequal elements, which are characterized by multiple failures, consider a model in the construction of which the following assumptions were made and the following notations were adopted:
Assumptions (1) multiple failures and other failure types are statistically independent; (2) multiple failures are associated with the failure of at least two elements; (3) if one of the loaded redundant elements fails, the failed element is restored; if both elements fail, the entire system is restored; (4) the rate of multiple failures and the rate of recovery are constant.
Designations
P 0 (t) - the probability that at time t both elements are functioning;
P 1 (t) - the probability that at time t element 1 is out of order and element 2 is functioning;
P 2 (t) - the probability that at time t element 2 is out of order, and element 1 is functioning;
P 3 (t) - the probability that at time t elements 1 and 2 are out of order;
P 4 (t) - the probability that at time t there are specialists and spare elements to restore both elements;
a- a constant coefficient characterizing the availability of specialists and spare parts;
b- constant intensity of multiple failures;
t - time.
Let's consider three possible cases of restoration of elements when they fail simultaneously:
Case 1. Spare elements, repair tools and qualified technicians are available to refurbish both elements, i.e. elements can be refurbished simultaneously.
Case 2. Spare parts, repair tools and qualified personnel are only available to refurbish one item, i.e. only one item can be rebuilt.
Happening 3 . Spare parts, repair tools and qualified personnel are not available, and there may be a waiting list for repair services.
Mathematical system model, shown in Fig. 4.5.22, is the following system of first order differential equations:
P" 0 (t) = -
,
P" 1 (t) = -( l 2 + m 1 )P 1 (t)+P 3 (t)
Rice. 4.5.22. Model of system readiness in case of multiple failures
Equating the time derivatives in the resulting equations to zero, for the steady state we obtain
-
,
-( l 2 + m 1 )P 1 +P 3 m 2 +P 0 l 1 = 0,
-(l 1 + m 2 )P 2 +P 0 l 2 +P 3 m 1 = 0,
P 2 =
P 3 =
P 4 =
The stationary availability factor can be calculated using the formula
,
,
.
GOST 27.301-95
Group T51
INTERSTATE STANDARD
RELIABILITY IN TECHNOLOGY
RELIABILITY CALCULATION
Basic provisions
Dependability in technology.
Dependability prediction. Basic principles
ISS 21.020
OKSTU 0027
Date of introduction 1997-01-01
Preface
1 DEVELOPED MTK 119 "Reliability in technology"
INTRODUCED by Gosstandart of Russia
2 ADOPTED by the Interstate Council for Standardization, Metrology and Certification (Protocol No. 7 of April 26, 1995)
The following voted for adoption:
State name | Name of the national standardization body |
Republic of Belarus | State Standard of the Republic of Belarus |
The Republic of Kazakhstan | Gosstandart of the Republic of Kazakhstan |
The Republic of Moldova | Moldovastandard |
Russian Federation | Gosstandart of Russia |
The Republic of Uzbekistan | Uzgosstandart |
Ukraine | State Standard of Ukraine |
3 The standard was developed taking into account the provisions and requirements of international standards IEC 300-3-1 (1991), IEC 863 (1986) and IEC 706-2 (1990)
4 Resolution of the Committee Russian Federation on standardization, metrology and certification dated June 26, 1996 N 430, the interstate standard GOST 27.301-95 was put into effect directly as state standard Russian Federation January 1, 1997
5 INSTEAD GOST 27.410-87 (in part 2)
6 REISSUE
1 area of use
1 area of use
This standard specifies general rules calculation of the reliability of technical objects, requirements for methods and the procedure for presenting the results of reliability calculations.
2 Normative references
This standard uses references to the following standards:
GOST 2.102-68 Unified system of design documentation. Types and completeness of design documents
GOST 27.002-89 Reliability in technology. Basic concepts. Terms and Definitions
GOST 27.003-90 Reliability in technology. Composition and general rules for specifying reliability requirements
3 Definitions
This standard uses general terms in the field of reliability, the definitions of which are established by GOST 27.002. Additionally, the standard uses the following terms related to reliability calculations.
3.1. reliability calculation: The procedure for determining the values of reliability indicators of an object using methods based on their calculation from reference data on the reliability of object elements, from data on the reliability of analogue objects, data on the properties of materials and other information available at the time of calculation.
3.2 reliability prediction: A special case of calculating the reliability of an object based on statistical models reflecting trends in the reliability of analogue objects and/or expert assessments.
3.3 element: A component of an object, considered when calculating reliability as a single whole, not subject to further disaggregation.
4 Basic provisions
4.1 Procedure for calculating reliability
The reliability of an object is calculated at stages life cycle and corresponding to these stages the stages of the types of work established by the reliability assurance program (REP) of the facility or documents replacing it.
The PON must establish the goals of the calculation at each stage of the types of work, the regulatory documents and methods used in the calculation, the timing of the calculation and performers, the procedure for registration, presentation and control of the calculation results.
4.2 Goals of reliability calculations
Calculation of the reliability of an object at a certain stage of types of work corresponding to a certain stage of its life cycle may have as its goals:
justification of quantitative reliability requirements for the object or its components;
checking the feasibility of the established requirements and/or assessing the likelihood of achieving the required level of reliability of the facility within the established time frame and with allocated resources, justifying the necessary adjustments to the established requirements;
comparative analysis of the reliability of options for the circuit design of an object and justification for choosing a rational option;
determination of the achieved (expected) level of reliability of the object and/or its components, including the calculated determination of reliability indicators or distribution parameters of reliability characteristics of the component parts of an object as initial data for calculating the reliability of the object as a whole;
justification and verification of the effectiveness of the proposed (implemented) measures to improve the design, manufacturing technology, system Maintenance and repairs of the facility aimed at increasing its reliability;
solving various optimization problems in which reliability indicators act as target functions, controlled parameters or boundary conditions, including such as optimization of the structure of an object, distribution of reliability requirements between indicators of individual reliability components (for example, reliability and maintainability), calculation of spare parts kits , optimization of maintenance and repair systems, justification warranty periods and assigned service life (resource) of the object, etc.;
checking compliance with the expected (achieved) level of object reliability established requirements(reliability control), if direct experimental confirmation of their level of reliability is technically impossible or economically impractical.
4.3 General scheme calculation
4.3.1 Calculation of the reliability of objects in the general case is a procedure for sequential step-by-step refinement of estimates of reliability indicators as the design and manufacturing technology of the object, algorithms for its functioning, operating rules, maintenance and repair systems, failure criteria and limit states are developed, accumulation of more complete and reliable information about all factors that determine reliability, and the use of more adequate and accurate calculation methods and calculation models.
4.3.2 Calculation of reliability at any stage of the types of work provided for by the operational plan includes:
identification of the object to be calculated;
determination of the goals and objectives of the calculation at this stage, the nomenclature and required values of the calculated reliability indicators;
selection of calculation method(s) adequate to the characteristics of the object, the purposes of the calculation, the availability of the necessary information about the object and the initial data for the calculation;
drawing up calculation models for each reliability indicator;
obtaining and preliminary processing of initial data for calculations, calculating the values of object reliability indicators and, if necessary, comparing them with the required ones;
registration, presentation and protection of calculation results.
4.4 Object identification
4.4.1 Identification of an object to calculate its reliability includes obtaining and analyzing the following information about the object, its operating conditions and other factors determining its reliability:
purpose, scope and functions of the object;
criteria for quality of functioning, failures and limit states, possible consequences failures (the object reaches a limit state) of the object;
the structure of the object, composition, interaction and load levels of its elements, the possibility of restructuring the structure and/or algorithms for the operation of the object in the event of failures of its individual elements;
availability, types and methods of reservation used in the facility;
a standard model for the operation of an object, establishing a list of possible operating modes and the functions performed during this, the rules and frequency of alternating modes, the duration of the object’s stay in each mode and the corresponding operating hours, the nomenclature and parameters of loads and external influences on the object in each mode;
planned system of technical maintenance (TO) and repair of an object, characterized by types, frequency, organizational levels, methods of implementation, technical equipment and logistical support for its maintenance and repair work;
distribution of functions between operators and means of automatic diagnostics (monitoring) and management of the object, types and characteristics of human-machine interfaces that determine the parameters of the operability and reliability of the operators;
personnel qualification level;
quality of software used in the facility;
planned technology and production organization for the manufacture of the object.
4.4.2 The completeness of identification of an object at the considered stage of calculating its reliability determines the choice of the appropriate calculation method that provides acceptable accuracy at this stage in the absence or impossibility of obtaining part of the information provided for in 4.4.1.
4.4.3 Sources of information for identifying an object are design, technological, operational and repair documentation for the object as a whole, its components and components in composition and kits corresponding at this stage reliability calculations.
4.5 Calculation methods
4.5.1 Reliability calculation methods are divided into:
by the composition of the calculated reliability indicators (RI);
according to the basic principles of calculation.
4.5.2 Based on the composition of the calculated indicators, calculation methods are distinguished:
reliability,
maintainability,
durability,
preservation,
complex reliability indicators (methods for calculating availability factors, technical use, maintaining efficiency, etc.).
4.5.3 According to the basic principles for calculating the properties that make up reliability, or complex indicators of the reliability of objects, the following are distinguished:
forecasting methods,
structural calculation methods,
physical calculation methods.
Forecasting methods are based on the use of data on the achieved values and identified trends in changes in the PN of objects that are similar or close to the one being considered in terms of purpose, principles of operation, circuit design and manufacturing technology, element base and materials used, conditions and modes to assess the expected level of reliability of an object. operation, principles and methods of reliability management (hereinafter referred to as analogous objects).
Structural calculation methods are based on representing an object in the form of a logical (structural-functional) diagram that describes the dependence of the states and transitions of the object on the states and transitions of its elements, taking into account their interaction and the functions they perform in the object, with subsequent descriptions of the constructed structural model with an adequate mathematical model and calculation PN of an object according to the known reliability characteristics of its elements.
Physical calculation methods are based on the use of mathematical models that describe physical, chemical and other processes leading to failures of objects (to objects reaching a limit state), and calculation of the load factor based on the known parameters of the object’s load, the characteristics of the substances and materials used in the object, taking into account the features of its design and manufacturing technologies.
Characteristics of the listed methods and recommendations for their use are given in Appendix A.
4.5.4 The method for calculating the reliability of a specific object is selected depending on:
calculation purposes and requirements for the accuracy of determining the PN of an object;
availability and/or possibility of obtaining the initial information necessary to apply a certain calculation method;
the level of sophistication of the design and manufacturing technology of the object, its maintenance and repair system, which allows the use of appropriate reliability calculation models.
4.5.5 When calculating the reliability of specific objects, it is possible to simultaneously use various methods, for example, methods for predicting the reliability of electronic and electrical elements with subsequent use of the results obtained as initial data for calculating the reliability of the object as a whole or its components using various structural methods.
4.6 Initial data
4.6.1 The initial data for calculating the reliability of an object can be:
a priori data on the reliability of analogue objects, components and components of the object in question based on the experience of their use in similar or similar conditions;
assessments of reliability indicators (parameters of the laws of distribution of reliability characteristics) of the component parts of the object and the parameters of the materials used in the object, obtained experimentally or by calculation directly during the development (manufacturing, operation) of the object in question and its components;
calculated and/or experimental assessments of the loading parameters of the components and structural elements used in the object.
4.6.2 Sources of initial data for calculating the reliability of an object can be:
standards and technical specifications for the component parts of the facility, components used in it for cross-industry use, substances and materials;
reference books on the reliability of elements, properties of substances and materials, standards for the duration (labor intensity, cost) of typical maintenance and repair operations and other information materials;
statistical data (data banks) on the reliability of analogue objects, their constituent elements, the properties of the substances and materials used in them, the parameters of maintenance and repair operations, collected during the process of their development, manufacturing, testing and operation;
results of strength, electrical, thermal and other calculations of the object and its components, including calculations of reliability indicators of the component parts of the object.
4.6.3 If there are several sources of initial data for calculating the reliability of an object, priorities in their use or methods for combining data from different sources must be established in the calculation methodology. In the reliability calculation included in the set of working documentation for the facility, it should be preferable to use the initial data from the standards and technical specifications for components, elements and materials.
4.7.1 The adequacy of the selected calculation method and the constructed calculation models for the purposes and tasks of calculating the reliability of an object is characterized by:
complete use in the calculation of all available information about the object, its operating conditions, maintenance and repair system, reliability characteristics of its components, properties of substances and materials used in the object;
the validity of the assumptions and assumptions adopted when constructing the models, their impact on the accuracy and reliability of PN estimates;
the degree of correspondence of the level of complexity and accuracy of the calculation models of the reliability of the object to the available accuracy of the initial data for the calculation.
4.7.2 The degree of adequacy of models and methods for calculating reliability is assessed by:
comparison of calculation results and experimental assessment of the PT of analogue objects, for which similar models and calculation methods were used;
studies of the sensitivity of models to possible violations of the assumptions and assumptions adopted during their construction, as well as to errors in the initial data for calculation;
examination and testing of applied models and methods, carried out in accordance with the established procedure.
4.8 Requirements for calculation methods
4.8.1 To calculate the reliability of objects, use:
standard calculation methods developed for a group (type, type) of objects that are homogeneous in purpose and principles of ensuring reliability, formalized in the form of appropriate regulatory documents(state and industry standards, enterprise standards, etc.);
calculation methods developed for specific objects, the design features and/or conditions of use of which do not allow the use of standard reliability calculation methods. These methods usually include directly accounting documents according to reliability calculations or are drawn up in the form of separate documents included in the set of documentation for the corresponding stage of object development.
4.8.2 The standard methodology for calculating reliability should contain:
characteristics of the objects to which the methodology applies, in accordance with the rules for their identification established by this standard;
a list of calculated PN of the object as a whole and its components, methods used to calculate each indicator;
standard models for calculating PN and rules for their adaptation for calculating the reliability of specific objects, calculation algorithms corresponding to these models and, if available, software;
methods and corresponding techniques for assessing the load parameters of component parts of objects taken into account in reliability calculations;
requirements for source data for calculating reliability (sources, composition, accuracy, reliability, form of presentation) or the source data themselves, methods for combining heterogeneous source data for calculating reliability, obtained from different sources;
decisive rules for comparing the calculated PN values with the required ones, if the calculation results are used to monitor the reliability of objects;
methods for assessing errors in the calculation of PT, introduced by the assumptions and assumptions adopted for the models and calculation methods used;
methods for assessing the sensitivity of calculation results to violations of accepted assumptions and/or to errors in source data;
requirements for the form of presentation of the results of calculation of the PN and rules for protecting the results of the calculation at the corresponding control points of the PN and during examinations of facility designs.
4.8.3 The methodology for calculating the reliability of a specific object must contain:
information about the object, ensuring its identification for reliability calculations in accordance with the requirements of this standard;
the range of calculated PNs and their required values;
models for calculating each PT, assumptions and assumptions adopted during their construction, corresponding algorithms for calculating PT and the software used, estimates of errors and sensitivity of the selected (built) models;
initial data for calculation and sources of their receipt;
methods for assessing the loading parameters of an object and its components or directly assessing these parameters with references to the corresponding results and methods of strength, thermal, electrical and other calculations of the object.
4.9 Presentation of calculation results
4.9.1 The results of calculating the reliability of an object are presented in the form of a section explanatory note to the corresponding project (draft, technical) or in the form of an independent document (RR according to GOST 2.102, report, etc.) containing:
goals and methodology (link to the corresponding standard methodology) of calculation;
calculated values of all PNs and conclusions on their compliance with the established reliability requirements of the facility;
identified deficiencies in the design of the facility and recommendations for their elimination with assessments of the effectiveness of the proposed measures in terms of their impact on the level of reliability;
a list of components and elements that limit the reliability of an object or for which there is no necessary data for calculating the PN, proposals for including in the PN additional measures to improve (in-depth study) their reliability or to replace them with more reliable ones (tested and proven);
conclusion on the possibility of moving to the next stage of development of the object when the calculated level of its reliability has been achieved.
4.9.3 Calculated estimates of PN, conclusions about their compliance with established requirements and the possibility of moving to the next stage of types of work on the development (putting into production) of an object, recommendations for modifications in order to increase its reliability are included in the acceptance test report if a decision is made to control reliability object by calculation method.
APPENDIX A (for reference). METHODS FOR CALCULATING RELIABILITY AND GENERAL RECOMMENDATIONS FOR THEIR APPLICATION
APPENDIX A
(informative)
1 Reliability prediction methods
1.1 Forecasting methods are used:
to justify the required level of reliability of objects during development technical assignments and/or assessing the likelihood of achieving the specified PN when developing technical proposals and analyzing the requirements of the technical specifications (contract). An example of appropriate methods for predicting the maintainability of objects is contained in MP 252-87;
for an approximate assessment of the expected level of reliability of objects at the early stages of their design, when the necessary information for applying other methods of reliability calculation is not available. An example of a methodology for predicting block reliability indicators radio-electronic equipment depending on its purpose and the number of elements used in it (groups of active elements), it is contained in the American military standard MIL-STD-756A;
to calculate the failure rates of serially produced and new electronic and electrical elements of various types, taking into account their load level, manufacturing quality, and areas of application of the equipment in which the elements are used. Examples of relevant techniques are contained in the American military reference book MIL-HDBK-217 and domestic reference books on the reliability of IET for general industrial and special purposes;
to calculate the parameters of typical tasks and operations of maintenance and repair of objects, taking into account the structural characteristics of the object, which determine its maintainability. Examples of relevant techniques are contained in MP 252-87 and the American military reference book MIL-HDBK-472.
1.2 To predict the reliability of objects, use:
methods of heuristic forecasting (expert assessment);
forecasting methods using statistical models;
combined methods.
Heuristic forecasting methods are based on statistical processing independent assessments values of the expected operating conditions of the object being developed (individual forecasts), given by a group of qualified specialists (experts) based on the information provided to them about the object, its operating conditions, the planned manufacturing technology and other data available at the time of the assessment. A survey of experts and statistical processing of individual PI forecasts are carried out using methods generally accepted for expert assessment of any quality indicators (for example, the Delphi method).
Forecasting methods using statistical models are based on extra- or interpolation of dependencies that describe identified trends in changes in the PN of analogue objects, taking into account their design and technological features and other factors, information about which for the object being developed is known or can be obtained at the time of the assessment. Models for forecasting are built based on data on PN and parameters of analogue objects using well-known statistical methods (multivariate regression or factor analysis, methods of statistical classification and pattern recognition).
Combined methods are based on the joint use of forecasting methods based on statistical models and heuristic methods to predict the reliability of objects, followed by comparison of the results. In this case, heuristic methods are used to assess the possibility of extrapolation of the statistical models used and to refine the forecast of PN based on them. The use of combined methods is advisable in cases where there is reason to expect qualitative changes in the level of reliability of objects that are not reflected by the corresponding statistical models, or when the number of analogue objects is insufficient to apply only statistical methods.
2 Structural methods for calculating reliability
2.1 Structural methods are the main methods for calculating reliability indicators, maintainability and complex PN in the process of designing objects that can be disaggregated into elements, the reliability characteristics of which at the time of calculations are known or can be determined by other methods (forecasting, physical, from statistical data collected in the process their use in similar conditions). These methods are also used to calculate the durability and storability of objects, the limit state criteria of which are expressed through the parameters of the durability (stability) of their elements.
2.2 Calculation of PN by structural methods in the general case includes:
representation of an object in the form of a structural diagram describing the logical relationships between the states of the elements and the object as a whole, taking into account the structural and functional connections and interaction of elements, the adopted maintenance strategy, types and methods of reservation and other factors;
description of the constructed structural reliability diagram (SSN) of the object with an adequate mathematical model that allows, within the framework of the introduced assumptions and assumptions, to calculate the PN of the object based on data on the reliability of its elements under the considered conditions of their use.
2.3 As block diagrams reliability can be used:
structural block diagrams of reliability, representing an object as a set of elements connected in a certain way (in terms of reliability) (IEC 1078 standard);
object failure trees, representing a graphical display of the cause-and-effect relationships that cause certain types of its failures (IEC 1025 standard);
graphs (diagrams) of states and transitions that describe the possible states of an object and its transitions from one state to another in the form of a set of states and transitions of its elements.
2.4 Mathematical models used to describe the corresponding SSN are determined by the types and complexity of the specified structures, the accepted assumptions regarding the types of laws of distribution of reliability characteristics of elements, the accuracy and reliability of the initial data for calculation and other factors.
The most commonly used mathematical methods for calculating PN are discussed below, which does not exclude the possibility of developing and using other methods that are more adequate to the structure and other features of the object.
2.5 Methods for calculating the reliability of non-repairable objects of type I (according to the classification of objects in accordance with GOST 27.003).
As a rule, to describe the reliability of such objects, fail-safe block diagrams are used, the rules for the compilation and mathematical description of which are established by IEC 1078. In particular, this standard establishes:
methods for direct calculation of the probability of failure-free operation of an object (FBO) based on the corresponding parameters of failure-free operation of elements for the simplest parallel-series structures;
FBG calculation methods for more complex structures, belonging to the class of monotone, including the method of direct enumeration of states, the method of minimal paths and sections, the method of expansion with respect to any element.
To calculate indicators such as the average time to failure of an object, the specified methods use the method of direct or numerical integration of the distribution of time to failure of an object, which represents a composition of the corresponding distributions of time to failure of its elements. If information on the distribution of time to failure of elements is incomplete or unreliable, then various boundary estimates of the object’s load capacity, known from reliability theory, are used.
In the special case of a non-recoverable system with different ways reservation and with the exponential distribution of time to failure of elements, its structural mapping is used in the form of a transition graph and its mathematical description using the Markov process.
When used to structurally describe the reliability of fault trees in accordance with IEC 1025, the probabilities of the corresponding failures are calculated using a Boolean representation of the fault tree and the method of minimum cuts.
2.6 Methods for calculating the reliability and complex PN of restored objects of type I
A universal calculation method for objects of any structure and for any cross-section of distributions of operating time between failures and recovery times of elements, for any strategies and methods of restoration and prevention is the method of statistical modeling, which in general includes:
synthesis of a formal model (algorithm) for the formation of a sequence of random events occurring during the operation of an object (failures, restorations, switching to reserve, beginning and end of maintenance);
development software for implementation on a computer of the compiled algorithm and calculation of the object’s PN;
conducting a simulation experiment on a computer through repeated implementation of a formal model that ensures the required accuracy and reliability of the calculation of PN.
The statistical modeling method for calculating reliability is used in the absence of adequate analytical models from those discussed below.
For redundant sequential structures with restoration and arbitrary methods of reserving elements, Markov models are used to describe the corresponding state graphs (diagrams).
In some cases, for objects with non-exponential distributions of operating time and recovery time, the non-Markov problem of calculating the operational load can be reduced to a Markov one by introducing fictitious states of the object into its transition graph in a certain way.
Another effective method calculation of the PT of objects with reserve is based on the representation of their operating time between failures in the form of the sum of a random number of random terms and the direct calculation of the PT of objects without involving methods of the theory of random processes.
2.7 Methods for calculating maintainability indicators
Methods for calculating maintainability indicators in the general case are based on representing the maintenance or repair process of a certain type as a set of individual tasks (operations), the probabilities and goals of which are determined by the reliability (durability) indicators of the objects and the adopted maintenance and repair strategy, and the duration (labor intensity, cost) The completion of each task depends on the structural adaptability of the object to maintenance (repair) of this type.
In particular, when calculating the maintainability indicators of objects during current unscheduled repairs, the distribution of time (labor intensity, cost) of its restoration represents a composition of cost distributions for individual restoration tasks, taking into account the expected probability of completing each task for a certain period of operation of the object. These probabilities can be calculated, for example, using fault trees, and the cost distribution parameters for performing individual tasks are calculated using one of the methods established, for example, MP 252-87 (normative coefficients, regression models, etc.).
The general calculation scheme includes:
compiling (for example, by AVPKO methods according to GOST 27.310) a list of possible object failures and assessing their probabilities (intensities);
selection from the compiled list using the stratified random sampling method of a certain fairly representative number of tasks and calculation of the parameters of their duration distributions (labor intensity, cost). The truncated normal or alpha distribution is usually used as such distribution;
constructing an empirical distribution of costs for Maintenance object by adding, taking into account the probabilities of failures, the cost distributions for individual tasks and smoothing it using the appropriate theoretical distribution (log-normal or gamma distribution);
calculation of object maintainability indicators based on the parameters of the selected distribution law.
2.8 Methods for calculating reliability indicators of objects of type II (according to GOST 27.003 classification)
For objects of this type, a PN of the “efficiency retention coefficient” () type is used, during the calculation of which the general principles calculation of the reliability of objects of type I, but each state of an object, determined by the set of states of its elements or each possible trajectory in the space of states of elements, must be associated with a certain value of the share of retained nominal efficiency from 0 to 1 (for objects of type I, efficiency in any state can only take two possible values: 0 or 1).
There are two main calculation methods:
the method of averaging over states (analogous to the method of direct enumeration of states), used for short-term objects performing tasks whose duration is such that the probability of a change in the object’s state during the task can be neglected and only its initial state can be taken into account;
method of averaging along trajectories, used for long-term objects, the duration of tasks of which is such that the probability of a change in the object’s states during their execution cannot be neglected due to failures and restorations of elements. In this case, the process of object functioning is described by the implementation of one of the possible trajectories in the state space.
There are also some special cases of calculation schemes for determining , used for systems with certain types of efficiency functions, for example:
systems with an additive efficiency indicator, each element of which makes a certain independent contribution to the output effect from the use of the system;
systems with a multiplicative efficiency indicator obtained as the product of the corresponding efficiency indicators of subsystems;
systems with redundant functions;
systems that perform a task in several possible ways using different combinations of elements involved in the execution of the task by each of them;
symmetrical branching systems;
systems with overlapping coverage areas, etc.
In all the above schemes, the system is represented as a function of its subsystems or PN elements.
The most important point in the calculations is the assessment of the efficiency of the system in various states or when implementing various trajectories in the space of states, carried out analytically, or by modeling, or experimentally directly on the object itself or its full-scale models (models).
3 Physical methods for calculating reliability
3.1 Physical methods are used to calculate the reliability, durability and storage of objects for which the mechanisms of their degradation under the influence of various external and internal factors are known, leading to failures (limit states) during operation (storage).
3.2 The methods are based on the description of the corresponding degradation processes using adequate mathematical models that make it possible to calculate the PT taking into account the design, manufacturing technology, modes and operating conditions of the object based on reference or experimentally determined physical and other properties of substances and materials used in the object.
In the general case, these models, with one leading degradation process, can be represented by a model of emissions of some random process beyond the boundaries of the permissible region of its existence, and the boundaries of this region can also be random and correlated with the specified process (non-exceedance model).
In the presence of several independent degradation processes, each of which generates its own resource distribution (time to failure), the resulting resource distribution (object time to failure) is found using the “weakest link” model (distribution of the minimum of independent random variables).
3.3 Components of non-exceedance models may have different physical natures and, accordingly, be described different types distributions of random variables (random processes), and can also be in damage accumulation models. This explains the wide variety of non-exceedance models used in practice, and only in relatively rare cases do these models allow a direct analytical solution. Therefore, the main method for calculating reliability using non-exceedance models is statistical modeling.
APPENDIX B (for reference). List of reference books, normative and methodological documents on reliability calculations
APPENDIX B
(informative)
1 B.A. Kozlov, I.A. Ushakov. Handbook for calculating the reliability of radio electronics and automation equipment. M.: Soviet radio, 1975. 472 p.
2 Reliability technical systems. Handbook ed. I.A.Ushakova. M.: Radio and communication, 1985. 608 p.
3 Reliability and efficiency in technology. Directory in 10 volumes.
T.2 ed. B.V. Gnedenko. M.: Mechanical Engineering, 1987. 280 p.;
T. 5, ed. V.I.Patrushev and A.I.Rembeza. M.: Mechanical Engineering, 1988. 224 p.
4 B.F. Khazov, B.A. Didusev. Handbook for calculating machine reliability at the design stage. M.: Mechanical Engineering, 1986. 224 p.
5 IEC Standard 300-3-1 (1991) Reliability management. Part 3. Guides. Section 1. Review of reliability analysis methods.
6 Standard IEC 706-2 (1991) Guidelines for ensuring the maintainability of equipment. Part 2, section 5. Maintainability analysis at the design stage.
7 IEC Standard 863 (1986) Presentation of prediction results for reliability, maintainability and availability.
8 IEC Standard 1025 (1990) Fault tree analysis.
9 IEC Standard 1078 (1991) Methods for reliability analysis. Method for calculating reliability using block diagrams.
10 RD 50-476-84 Guidelines. Reliability in technology. Interval assessment of the reliability of a technical object based on the test results of its components. General provisions.
11 RD 50-518-84 Guidelines. Reliability in technology. General requirements to the content and forms of presentation of reference data on the reliability of components for cross-industry use.
12 MR 159-85 Reliability in technology. Selection of types of distributions of random variables. Guidelines.
13 MR 252-87 Reliability in technology. Calculation of maintainability indicators during product development. Guidelines.
14 R 50-54-82-88 Reliability in technology. Selection of reservation methods and methods.
15 GOST 27.310-95 Reliability in technology. Analysis of types, consequences and criticality of failures. Basic provisions.
16 US military standard MIL-STD-756A. Modeling and forecasting of failure-free operation.
17 US Military Standards Handbook MIL-HDBK-217E. Forecasting the reliability of electronic equipment elements.
18 US Military Standards Handbook MIL-HDBK-472. Predicting maintainability.
The text of the document is verified according to:
official publication
Reliability in technology: Sat. GOST. -
M.: IPK Standards Publishing House, 2002
Calculation of reliability indicators of non-recoverable non-redundant systems
As an object whose reliability needs to be determined, consider some complex system S, consisting of individual elements (blocks). The task of calculating the reliability of a complex system is to determine its reliability indicators if the reliability indicators of individual elements and the structure of the system are known, i.e. the nature of connections between elements from the point of view of reliability.
The simplest structure is a non-redundant system consisting of n elements, in which the failure of one of the elements leads to the failure of the entire system. In this case, system S has a logically sequential connection of elements (Fig. 4).
Figure 4. Diagram of logical connection of elements of a non-redundant system
Calculation methods
Depending on the completeness of taking into account the factors influencing the operation of the product, a distinction is made between an approximate and a complete calculation of reliability indicators.
At approximate When calculating reliability indicators, it is necessary to know the structure of the system, the range of elements used and their quantity. The approximate calculation takes into account the impact on reliability only of the number and types of elements included in the system, and is based on the following assumptions:
All elements of this type are equally reliable, i.e. the failure rate values () for these elements are the same;
All elements operate in the nominal (normal) mode provided technical specifications;
The failure rates of all elements do not depend on time, i.e. During the service life, the elements included in the product do not experience aging or wear, therefore;
Failures of product elements are random and independent events;
All elements of the product work simultaneously.
The approximate calculation method is used at the preliminary design stage after the development of fundamental electrical diagrams products and allows you to outline ways to improve the reliability of the product.
Let element failures be events independent of each other. Since a system is operational if all its elements are operational, then according to the theorem on the multiplication of probabilities, the probability of failure-free operation of the system P c (t) is equal to the product of the probabilities of failure-free operation of its elements:
,
where is the probability of failure-free operation of the i-th element.
Let the exponential distribution of reliability be valid for the elements and their failure rates are known. Then the exponential law of reliability distribution is valid for the system:
,
where is the system failure rate.
The failure rate of a non-redundant system is equal to the sum of the failure rates of its elements:
If all elements of this type are equally reliable, then the failure rate of the system will be
where: - number of elements of the i-th type; r – number of element types.
The selection for each type of element is made according to the corresponding tables.
The mean time to failure and the system failure rate are respectively equal to:
, 
.
In practice, it is very often necessary to calculate the probability of failure-free operation of highly reliable systems. In this case, the product is significantly less than one, and the probability of failure-free operation P(t) is close to one. In this case, the quantitative characteristics of reliability can be calculated with sufficient accuracy for practice using the following approximate formulas:
, , , 
.
When calculating the reliability of systems, it is often necessary to multiply the probabilities of failure-free operation of individual calculation elements and raise them to a power. For probability values P(t) close to unity, these calculations can be performed with sufficient accuracy for practice using the following approximate formulas:
, ,
where is the probability of failure of the i-th block.
Full calculation of product reliability indicators is carried out when the actual operating modes of the elements after testing in laboratory conditions product layouts.
Product elements are usually in different operating modes, very different from the nominal value. This affects the reliability of both the product as a whole and its individual components. Performing a final calculation of reliability parameters is possible only if there is data on the load factors of individual elements and if there are graphs of the dependence of the failure rate of elements on their electrical load and temperature environment and other factors, i.e. for the final calculation it is necessary to know the dependencies
.
These dependencies are presented in the form of graphs or they can be calculated using the so-called failure rate correction factors.
When developing and manufacturing elements, certain, so-called “normal” operating conditions are usually provided for. The failure rate of elements in the “normal” operating mode is called rated failure rate .
The failure rate of elements during operation under real conditions is equal to the nominal failure rate multiplied by correction factors, i.e.
,
where: - failure rate of an element operating under normal conditions at a rated electrical load; - correction factors depending on various influencing factors.
A full reliability calculation is applied at the stage technical design products.
Example 1. The system consists of two devices. The probabilities of failure-free operation of each of them during time t = 100 hours are equal to: p 1 (100) = 0.95; p 2 (100) = 0.97. The exponential law of reliability distribution is valid. It is necessary to find the average time until the first failure of the system.
Solution. Let's find the probability of failure-free operation of the system using the formula:
Let's find the failure rate of the system. To do this we use the formula:
Then . From this expression we find .
Or 
(1/h).
Average time to first failure
(h).
Example 2. Only elements with a failure rate of 1/hour can be used in systems. The systems have a number of elements N 1 = 500, N 2 = 2500. It is required to determine the average time to first failure and the probability of failure-free operation at the end of the first hour P c (t)
Part 1.
Introduction
The development of modern equipment is characterized by a significant increase in its complexity. Increasing complexity leads to an increase in the guarantee of timeliness and correctness of problem solving.
The problem of reliability arose in the 50s, when the process of rapid complication of systems began, and new objects began to be put into operation. At this time, the first publications appeared defining concepts and definitions related to reliability [1] and a methodology for assessing and calculating the reliability of devices using probabilistic and statistical methods was created.
Studying the behavior of equipment (object) during operation and assessing its quality determines its reliability. The term "exploitation" comes from the French word "exploitation", which means to gain benefit or benefit from something.
Reliability is the property of an object to perform specified functions, maintaining over time the values of established operational indicators within specified limits.
To quantify the reliability of an object and for planning operation, special characteristics are used - reliability indicators. They make it possible to assess the reliability of an object or its elements in various conditions and at different stages of operation.
More detailed information on reliability indicators can be found in GOST 16503-70 - "Industrial products. Nomenclature and characteristics of main reliability indicators.", GOST 18322-73 - "Equipment maintenance and repair systems. Terms and definitions.", GOST 13377-75 - "Reliability in technology. Terms and definitions."
Definitions
Reliability- the property [hereinafter - (its)] of an object [hereinafter - (OB)] to perform the required functions, maintaining its performance indicators for a given period of time.
Reliability is a complex property that combines the concepts of operability, reliability, durability, maintainability and safety.
Performance- represents the state of the OB in which it is able to perform its functions.
Reliability- the ability of the OB to maintain its functionality for a certain time. An event that disrupts the operation of the OB is called a failure. A failure that resolves itself is called a failure.
Durability- the freedom of the OB to maintain its operability to the limit state, when its operation becomes impossible for technical, economic reasons, safety conditions or the need for major repairs.
Maintainability- determines the adaptability of the equipment to prevent and detect malfunctions and failures and eliminate them through repairs and maintenance.
Storability- the ability of the OB to continuously maintain its performance during and after storage and maintenance.
Main reliability indicators
The main qualitative indicators of reliability are the probability of failure-free operation, failure rate and mean time to failure.
Probability of failure-free operation
P(t) represents the probability that within a specified period of time t, OB failure will not occur. This indicator is determined by the ratio of the number of OB elements that have worked without failure up to the point in time t to the total number of OB elements operational at the initial moment.
Failure Rate l(t) is the number of failures n(t) OB elements per unit of time, related to the average number of elements Nt OB operational at the moment of time Dt:
l (t )= n (t )/(Nt * D t )
, Where
D
t- a specified period of time.
For example: 1000 OB elements worked for 500 hours. During this time, 2 elements failed. From here, l (t )= n (t )/(Nt * D t )=2/(1000*500)=4*10 -6
1/h, i.e. 4 out of a million elements can fail in 1 hour.
Indicators of component failure rates are taken based on reference data [1, 6, 8]. For example, the failure rate is given l(t) some elements.
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Item name |
Failure rate, *10 -5, 1/h |
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Resistors |
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Capacitors |
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Transformers |
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Inductors |
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Switching devices |
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Solder connections |
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Wires, cables |
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The reliability of the OB as a system is characterized by a flow of failures L, numerically equal to the sum of the failure rates of individual devices: L = ål i The formula calculates the flow of failures and individual OB devices, which in turn consist of various units and elements, characterized by their failure rate. The formula is valid for calculating the failure rate of a system from n elements in the case when the failure of any of them leads to the failure of the entire system as a whole. This connection of elements is called logically consistent or basic. In addition, there is a logically parallel connection of elements, when the failure of one of them does not lead to failure of the system as a whole. Relationship between the probability of failure-free operation P(t) and failure rate L defined: P (t )= exp (- D t ) , it's obvious that 0 Mean time to failure To is the mathematical expectation of the operating time of the OB before the first failure: To=1/ L =1/(ål i) , or, from here: L =1/To Failure-free operation time is equal to the reciprocal of the failure rate. For example : element technology ensures medium failure rate l i =1*10 -5 1/h . When used in OB N=1*10 4 elementary parts total failure rate l o= N * l i =10 -1 1/h . Then the average non-failure time of the OB To =1/ l o=10 h. If you perform an OB based on 4 large-scale integrated circuits (LSI), then the average time between failures of the OB will increase by N/4=2500 times and amount to 25,000 hours or 34 months or about 3 years. Reliability calculation
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RELIABILITY INDICATOR. Quantitative characteristics of one or more properties that make up reliability object.
SINGLE RELIABILITY INDICATOR. Index reliability, characterizing one of the properties that make up reliability object.
COMPLEX RELIABILITY INDICATOR. Index reliability, characterizing several properties that make up reliability object.
ESTIMATED RELIABILITY INDICATOR. Index reliability, the values of which are determined by the calculation method.
EXPERIMENTAL RELIABILITY INDICATOR. Reliability indicator
OPERATIONAL RELIABILITY INDICATOR. Reliability indicator, the point or interval estimate of which is determined from operating data.
PROBABILITY OF FAILURE-FAILURE OPERATION –P(t) 0 before t ) object failure does not occur:
P(t)=N(t)/N 0 ,
Where N(t) t ;
N 0– number of operational devices at a time t=0
The probability of failure-free operation is expressed as a number from zero to one (or as a percentage). The higher the probability of failure-free operation of a device, the more reliable it is.
Example. During the operation of 1000 power transformers of the OM type, 15 failed in a year. We have N 0 = 1000 pcs., N(t) = 985 PC. P(t)=N(t)/N 0 = 985/1000 = 0 ,985.
PROBABILITY OF FAILURE –q(t) . The probability that within a given operating time (or within the time interval from 0 before t ) a failure will occur:
q(t)=n(t)/N 0 ,
Where n(t) – number of devices that failed at the time t ;
N 0– number of operable device elements at a time t=0 (number of monitored devices).
q(t) = 1 - P(t).
AVERAGE TIME TO FAILURE. Expected value developments object to the first refusal T avg (average value of the duration of operation of the device being repaired until the first failure):
Where t i – duration of operation (running time) until failure i -th device;
N 0– number of monitored devices.
Example. When operating 10 starters, it was revealed that the first failed after 800 switchings, the second - 1200, then 900, 1400, 700, 950, 750, 1300, 850, 1150, respectively.
T av = (800 + 1200 + 900 + 1400 + 700 + 950 + 750 + 1300 + 850 + 1150)/10 = 1000 switchings
AVERAGE TIME TO FAILURE. T - total ratio operating time of the restored object to the mathematical expectation of its number failures during this developments(mean time between failures).
FAILURE RATE. Conditional probability density of occurrence refusal object, determined under the condition that before the considered moment in time refusal did not occur (average number of failures per unit of time):
l(t) = n(Dt) / N Dt ,
Where n(Dt) - number of devices that failed during a period of time Dt ;
N- number of monitored devices;
Dt– observation period.
Example. When operating 1000 transformers for 10 years, 20 failures occurred (and each time a new transformer failed). We have: N = 1000pcs., n(Dt) = 20 pcs., Dt = 10 years.
l(t)= 20/(1000 × 10) = 0.002 (1/year).
AVERAGE RECOVERY TIME. Mathematical expectation of time restoration of working condition object after refusalT avg (average time of forced or routine downtime of a device caused by detection and elimination of a failure).
Where i – serial number of the failure;
t i– average time of detection and elimination i-th refusal.
READINESS RATIO. K G - the probability that the object will be in in working condition at an arbitrary point in time, except for planned periods during which the intended use of the object is not envisaged.
It is defined as the ratio of the device's time between failures in units of time to the sum of this time between failures and the recovery time.
K G = T / (T + T V).
Reliability calculation
The main method for calculating reliability is based on an exponential mathematical model of failure-free operation of elements (most often encountered in studying the reliability of control systems and assuming a constant failure rate over time):
probability of failure-free operation per operating time t :
,
mean time between failures (to failure) is equal to the reciprocal of the failure rate:
,
Assumptions predetermined by this method:
failures of component elements are random independent events;
two or more elements cannot fail at the same time;
the failure rate of elements during their service life in the same operating modes and operating conditions is constant;
There are two types of element failures: open (O) and short circuit (SC).
The probability of failure-free operation of a system containing N elements (blocks):
,
Where P i (t) - probability of failure-free operation of the element (unit).
Failure rate of a block consisting of M components:
.
Failure rate of elements operating in variable modes for a given period of time:
,
Where l 1, l 2- failure rates at intervals t 1, t 2 respectively.
Relationship between failure rate and operating time and probability of failure-free operation:
.
Before starting the calculation, based on a logical analysis of schematic and structural diagrams and functional purposes, the structure of the object is determined from the point of view of reliability ( sequential And parallel connection of elements).
Parallel from the point of view of reliability, the connection of elements is when the device fails if all elements fail.
Sequential from the point of view of reliability, the connection of elements is when the device fails if at least one element fails.
Moreover, elements connected electrically in series (parallel) can, from the point of view of reliability, be, on the contrary, parallel (series).
For different types of failures (short circuit or open), elements may, from a reliability point of view, be consistent for one type of failure and consistent for another. For example, a string of insulators electrically connected in series for a short-circuit type failure, from the point of view of reliability, has a parallel connection, and for a break-type failure, it has a serial connection.
Maintenance (MRO) and repair (R) strategies
STRATEGY. Any rule that prescribes certain actions in each situation of a decision-making process. Formally, a strategy is a function of currently available information that takes values on the set of alternatives available at the moment.
MAINTENANCE (REPAIR) STRATEGY. Management rules system technical condition in progress Maintenance (repairs).
MAINTENANCE. A set of operations or an operation to maintain the functionality or serviceability of a product when used for its intended purpose, waiting, storing and transporting.
RECOVERY. The process of transferring an object to operational state from inoperative state.
REPAIR. Complex of operations on restoration of serviceability or performance products and resource recovery products or their components.
MAINTENANCE AND REPAIR SYSTEM OF EQUIPMENT. A set of interconnected tools and documentation maintenance and repair and performers necessary to maintain and restore the quality of products included in this system.
PERIODICITY OF MAINTENANCE (REPAIR). Time interval or operating time between this type maintenance (repair) and subsequent ones of the same type or others of greater complexity. Under the guise Maintenance(repair) understand maintenance (repair), allocated (allocated) according to one of the characteristics: stage of existence, frequency, volume of work, operating conditions, regulation, etc.
PERIODIC MAINTENANCE. Maintenance, carried out through the values established in the operational documentation developments or time intervals.
REGULATED MAINTENANCE. Maintenance, provided for in the regulatory, technical or operational documentation and carried out with the frequency and to the extent established therein, regardless of technical condition products at the start Maintenance.
MAINTENANCE WITH PERIODIC CONTROLS. Maintenance, in which control technical condition is carried out with the frequency and volume established in the regulatory, technical or operational documentation, and the volume of other operations is determined technical condition products at the start Maintenance.
MAINTENANCE WITH CONTINUOUS MONITORING. Maintenance, provided for in the regulatory, technical or operational documentation and carried out based on the results continuous monitoring of technical condition products .
Selecting the optimal maintenance and repair strategy
The solution to this problem should include the development of a procedure for assigning one or another type of maintenance and repair, ensuring maximum efficiency in using the power supply system.
Three main maintenance and repair strategies are possible:
1) recovery after a failure;
2) preventive restoration based on operating time - after completing a certain amount of work or duration of use;
3) preventive restoration based on technical condition (TS) (with parameter control). In relation to the aggregate-node method, one more strategy can be called - restoration by TS with control of reliability indicators.
For such complex technical systems as the power supply system, it is inappropriate to prescribe the same strategy for carrying out maintenance and repair - for each element, device, unit, its own strategy must be chosen, taking into account their role in ensuring performance indicators of machine operation using economic and mathematical models. In this case, the following information is used as initial information:
Reliability indicators of equipment and its elements, assessed at the development stage and determined during operation;
Costs of planned and unscheduled maintenance and repairs;
Values of damage from equipment downtime;
The influence of the technical condition of elements on power quality indicators;
Cost of technical diagnostics;
Existing maintenance and repair system;
Ensuring traffic safety, electrical safety and environmental safety requirements.
Recovery effects after failure are used for elements whose failures do not lead to loss of functionality of the power supply system and violations of safety requirements.
For elements whose failure is simultaneously a system failure, with this maintenance and repair strategy, any actions that control the reliability and level of specific losses are impossible. The level of failure-free operation and the lower limit of losses from failure are predetermined only by the reliability of the element and cannot be reduced without increasing it, i.e., without changing the design.
recovery based on operating hours There are two types of losses - failures of some elements and underutilization of others. It is impossible to reduce one type of loss without simultaneously increasing another; it is only possible to minimize the total specific losses (with optimal frequency of maintenance and repair).
With a preventive strategy restoration based on the results of parameter monitoring(technical diagnostics) it becomes possible to reduce losses from failure and losses from underutilization of a resource, and to a greater extent, the lower the level of diagnostic costs.
