Failure rate is the average time between failures. Mean time between failures is the ratio of the operating time of a restored object to the mathematical expectation of the number of its failures during this operating time. Clear navigation, competent search
The average value of the operating time of products in a batch until the first failure is called the average time to the first failure. This term applies to both repairable and non-repairable products. For non-repairable products, instead of the above, the term mean time to failure can be used.
GOST 13377 - 67 for non-repairable products introduced another reliability indicator, called failure rate.
The failure rate is the probability that a non-repairable product, which worked without failure until moment t, will fail in the next unit of time, if this unit is small.
The failure rate of a product is a function of the time it takes to operate.
Assuming that the failure-free operation of a certain unit in the electronic control system of a vehicle is characterized by a failure rate numerically equal to the calculated one, and this intensity does not change throughout its entire service life, it is necessary to determine the time to failure TB of such a unit.
The control subsystem includes k series-connected electronic units (Fig. 2).
Fig.2 Control subsystem with sequentially connected blocks.
These blocks have the same failure rate, numerically equal to the calculated one. It is required to determine the failure rate of the subsystem λ P and its average time to failure, to plot the dependence of the probability of failure-free operation of one block RB (t) and the subsystem RP (t) on the operating time and to determine the probabilities of failure-free operation of the block RB (t) and the subsystem RP (t) to operating time t= T P.
The failure rate λ(t) is calculated using the formula:
, (5)
Where is the statistical probability of a device failure on an interval, or otherwise the statistical probability of a random variable T falling within a specified interval.
Р(t) – calculated in step 1 – probability of failure-free operation of the device.
Setpoint 10 3 h - 6.5
Interval =
λ(t) = 0.4 / 0.4*3*10 3 h = 0.00033
Let us assume that the failure rate does not change throughout the entire service life of the object, i.e. λ(t) = λ = const, then the time to failure is distributed according to an exponential (exponential) law.
In this case, the probability of failure-free operation of the unit is:
(6)
R B (t) = exp (-0.00033*6.5*10 3) = exp(-2.1666) = 0.1146
And the average operating time of a block to failure is found as:
1/0.00033 = 3030.30 hours.
When k blocks are connected in series, the failure rate of the subsystem they form is:
(8)
Since the failure rates of all blocks are the same, the failure rate of the subsystem is:
λ P = 4*0.00033 = 0.00132 hours,
and the probability of failure-free operation of the system:
(10)
R P (t) = exp (-0.00132*6.5*10 3) = exp (-8.58) = 0.000188
Taking into account (7) and (8), the average time to failure of the subsystem is found as:
(11)
1/0.00132 = 757.58 hours.
Conclusion: As we approach the limit state, the failure rate of objects increases.
Calculation of the probability of failure-free operation.
Exercise: For operating time t = it is necessary to calculate the probability of failure-free operation Рс() of the system (Fig. 3), consisting of two subsystems, one of which is a backup one.
Rice. 3 Scheme of a redundant system.
The calculation is carried out under the assumption that the failures of each of the two subsystems are independent.
The probabilities of failure-free operation of each system are the same and equal to R P (). Then the probability of failure of one subsystem is:
Q P () = 1 – 0.000188 = 0.99812
The probability of failure of the entire system is determined from the condition that both the first and second subsystems have failed, i.e.:
0,99812 2 = 0,99962
Hence the probability of failure-free operation of the system:
,
Р с () = 1 – 0.98 = 0.0037
Conclusion: In this task, the probability of failure-free operation of the system in the event of failure of the first and second subsystems was calculated. Compared to a sequential structure, the probability of failure-free operation of the system is less.
where is the time of proper operation between and m failures of the object; - number of object failures.
With a sufficiently large number of failures, it tends to the average time between two adjacent failures. If several similar objects are tested, then the average time between failures is determined from the expression
number of objects. (1.11)
Failure Rate is the ratio of the number of failed objects per unit of time to the average number of objects that continue to operate properly in a given time interval:
(1.12)
here is the number of failed objects during the time period from to , and 
where is the number of properly working objects at the beginning of the time interval; number of properly functioning objects at the end of the time interval
In reliability theory, a model of the failure rate of an object is adopted, characterized by the curve of the failure rate of an object during operation given below.
Figure 1.3 - Failure rate model of an object
Failure flow parameter is the ratio of the average number of failures of a restored object over an arbitrarily small operating time to the value of this operating time. This indicator is used to assess the reliability of restored objects during operation: in the initial period of time, the object works until failure; after a failure, the object is restored, and the object again works until failure, and so on. It is believed that the restoration of the object occurs instantly. For such objects, failure moments on the total operating time axis (time axis) form a failure flow. As a characteristic of the failure flow, the “leading function” of this flow is used - the mathematical expectation of the number of failures over time t: 
(1.13)
The failure flow parameter characterizes the average number of failures expected over a short time interval
Statistically, the failure flow parameter is determined by the formula
(1.15)
where is the number of failures of the restored object during the time interval from to .
Average resource is the mathematical expectation of the resource.
Gamma percentage resource% is the operating time during which the object will not reach the limit state with a given probability, expressed as a percentage. The formula for calculation is similar to the formula for gamma percentage time to failure.
Assigned resource is defined as the total operating time of an object, upon reaching which its intended use must be discontinued.
Average service life- mathematical expectation of service life.
Gamma percentage life% is the calendar duration from the start of operation of the object, during which it will not reach the limit state with a given probability, %.
Assigned service life- calendar duration of operation of the object, upon reaching which the intended use of the object must be discontinued.
Assigned resource and assigned service life are established based on subjective or organizational assumptions, and they are indirect indicators of reliability.
The moment of restoration of an object's operability after a failure is a random event. Therefore, the distribution function of this random variable is used as a characteristic of maintainability. Probability of recovery is the probability that the time to restore the operational state of an object will not exceed a given one:
Probability of non-recovery at a given interval, i.e. the probability that is equal
Figure 1.4 - Change in the probabilities of recovery and non-recovery over time
The probability density of the moment of recovery is equal to 
Average recovery time is the 1st order moment (mathematical expectation) of the time of restoration of the operational state of the object.
(1.16)
Statistically, the average recovery time is equal to where is the time of detection and elimination of the th object failure.
An important indicator of the maintainability of an object is recovery intensity, which, following the general methodology, is similar to the reliability indicator - failure rate.
Indicators shelf life – average shelf life and gamma-percentage shelf life– are determined similarly to the corresponding indicators of reliability and durability. The average shelf life is the mathematical expectation of the shelf life; and the gamma-percentage shelf life is the shelf life achieved by an object with a given probability, %.
Since the probabilistic characteristics of individual reliability properties are assumed to be independent, to evaluate several reliability properties they use complex indicators. Let us consider the complex indicators used in reliability theory.
Availability factor is the probability that an object will be in working condition at any point in time, except for planned periods during which the object is not intended to be used for its intended purpose
Operational readiness ratio is defined as the probability that an object will be in working condition at an arbitrary point in time, except for planned periods during which the intended use of the object is not envisaged and, starting from this moment, will work without failure for a given time interval: 
(1.18)
Until this moment, such objects may be on duty, but without performing specified operational functions. In both modes, failures may occur and the object’s functionality may be restored.
Sometimes they use downtime rate
Technical utilization rate is the ratio of the mathematical expectation of the operating time interval of a restored object to the mathematical expectation of the time intervals during which the object remains in downtime conditions due to maintenance and repairs for the same period of operation
(1.20)
where is the mathematical expectation of the operating time of the restored object; mathematical expectation of downtime intervals during maintenance; mathematical expectation of time spent on scheduled and unscheduled repairs. characterizes the proportion of time an object is in working condition relative to the considered duration of operation.
Planned Application Factor is the ratio of the difference between the specified duration of operation and the mathematical expectation of the total duration of scheduled maintenance and repairs for the same period of operation to the value of this period
(1.21)
Efficiency retention factor – the ratio of the value of the efficiency indicator for a certain duration of operation E to the nominal value of the indicator E 0, calculated under the condition that failures of the object do not occur during the same period of operation. This coefficient characterizes the degree of influence of failures of object elements on the efficiency of its intended use.
At the same time, under efficiency of the object's use understand its ability to create a certain useful result (output effect) during a period of operation under certain conditions. An efficiency indicator is a quality indicator that characterizes the performance of an object’s functions. Analytical expressions for calculating the effect of objects of various types are given in GOST 27.003-89. The selection of the range of reliability indicators and their standardization is carried out on the basis of GOST 27.033-83.
1.4 General procedure for ensuring reliability at stages
"life" cycle of an object
In accordance with GOST 27.003-90, we will consider some questions on the given topic.
1.4.1 Composition and general rules for specifying reliability requirements
1 When specifying reliability requirements, the following is determined and agreed upon between the customer and developer:
A typical operating model, in relation to which reliability requirements are set;
Failure criteria based on operating model;
Criteria for limiting states of products, in relation to which requirements for durability and storage are established;
The concept of “output effect” for products, the requirements for which are established by the efficiency retention coefficient K eff . ;
Nomenclature and values of reliability indicators (RI) in accordance with the adopted operating model;
Requirements and restrictions on design, technological and operational methods of ensuring reliability, if necessary, taking into account economic restrictions;
The need to develop a reliability program.
2 A typical product operation model should contain:
Sequence of types, modes of operation (storage, transportation, deployment, waiting for intended use, maintenance and scheduled repairs) with an indication of their duration;
Characteristics of the adopted system of maintenance and repair, provision of spare parts, tools and operating materials;
Levels of external influencing factors and loads for each type and mode of operation;
Number and qualifications of maintenance and repair personnel.
3 The PN nomenclature is selected according to GOST 27.002.
4 The total number of selected PNs should be minimal.
5 For restored products, as a rule, a complex PN is specified..., possible combinations of specified indicators K g and T o; K g and T v; T o and T v. Invalid combination K g, T o, T v.
6 Reliability requirements are included in the following documents:
Technical specifications (TOR) for the development or modernization of products;
Technical conditions (TU) for the manufacture of products;
Standards of general technical requirements (GTR), general technical conditions (GTC) and technical conditions (TU).
In passports, forms, instructions and other operational documentation, reliability requirements (RP) are indicated by agreement between the customer and the developer as reference. Reliability requirements may be included in the contract for the development and supply of products.
1.4.2 The procedure for specifying reliability requirements at various
stages of product life cycle
1 Reliability requirements included in the technical specifications are determined at the research and development stage by:
Analysis of customer requirements, operating conditions, restrictions on all types of costs;
Development and agreement with the customer of failure criteria and limit states;
Choosing a rational nomenclature of PN;
Establishing the PN values of the product and its components.
2 At the stages of product development, reliability requirements are specified by:
Consideration of possible options for constructing the product and calculating the PN;
Selecting an option that satisfies the customer in terms of the total cost and costs;
Clarification of the PN values of the product and its components.
3 The specifications for a serial product include those PNs that are supposed to be controlled at the stage of product manufacturing.
4 At the stages of serial production and operation, correction of PN values is allowed based on the results of testing or operation.
5 For complex products during their development, pilot or mass production, it is allowed to set the PN values step by step (subject to their increase) and control plan parameters, taking into account accumulated statistical data on previous analogue products and as agreed between the customer and the developer.
6 If there are prototypes (analogues) with a reliably known level of reliability, the scope of work for setting reliability requirements in paragraphs 1 and 2 can be reduced due to those indicators, information on which is available at the time of formation of the section of the technical specifications, technical specifications “Reliability Requirements”.
1.5 Analytical dependencies between reliability indicators
The relationship between the probability of failure-free operation and the average time to failure:
From here, 
those. the average time to failure is equal to the area under the probability curve of failure-free operation of the object.
Relationship between probability of failure-free operation and failure rate
If put to test N 0 objects, then the number of objects that will work properly at the time t, equals
For a point in time
Number of failed objects
Then 
(1.24)
Since is a positive definite function, then
(1.25)
The relationship between the probability of failure-free operation, failure rate and mean time to failure.
(1.26)
For example, during normal operation
(1.27)
Moreover (1.28)
The relationship between the probability density of failure-free time
work and failure flow parameter.
Let it be tested N 0 number of objects, and failed objects are replaced with new ones (sampling with compensation). If objects are not recoverable, then the failure flow parameter is equal to
(1.29)
The average number of failed objects in a time interval is proportional to the value of , the length of the time interval and .
There are three types of failures:
· caused by hidden errors in design and technological documentation and manufacturing defects in the manufacture of products;
· caused by aging and wear of radio and structural elements;
· caused by random factors of various nature.
To assess the reliability of systems, the concepts of “operability” and “failure” were introduced.
Performance and failures. Performance is the state of a product in which it is capable of performing specified functions with the parameters established by the requirements of technical documentation. Failure is an event leading to the complete or partial loss of functionality of a product. Based on the nature of changes in equipment parameters, failures are divided into sudden and gradual.
Sudden (catastrophic) failures are characterized by an abrupt change in one or more parameters of the equipment and arise as a result of a sudden change in one or more parameters of the elements from which the electronic equipment is built (break or short circuit).
Elimination of a sudden failure is carried out by replacing the failed element with a serviceable one or repairing it.
Gradual (parametric) failures are characterized by a change in one or more hardware parameters over time.
They arise as a result of a gradual change in the parameters of the elements until the value of one of the parameters goes beyond certain limits that determine the normal operation of the elements. This may be a consequence of aging of the elements, exposure to fluctuations in temperature, humidity, pressure, mechanical stress, etc. Elimination of gradual failure is associated either with replacement, repair, adjustment of the parameters of the failed element, or with compensation by changing the parameters of other elements.
Based on their relationship to each other, a distinction is made between independent failures, which are not related to other failures, and dependent ones. Based on the frequency of occurrence, failures can be one-time (failures) or intermittent. A failure is a one-time, self-correcting failure; an intermittent failure is a failure of the same nature that occurs multiple times.
Based on the presence of external signs, a distinction is made between obvious failures, which have external signs of appearance, and implicit (hidden) failures, the detection of which requires certain actions.
Based on their occurrence, failures are divided into structural, production and operational, caused by violation of established norms and rules during the design, production and operation of electronic equipment.
7.2. quantitative characteristics of Reliability
Reliability, as a combination of the properties of reliability, repairability, durability and storage, and these qualities themselves are quantitatively characterized by various functions and numerical parameters. The correct choice of quantitative indicators of the reliability of electronic equipment allows you to objectively compare the technical characteristics of various products both at the design stage and at the operating stage (the correct choice of a system of elements, technical justification for the operation and repair of electronic equipment, the amount of necessary spare equipment, etc.).
The occurrence of failures is random. The process of failure occurrence in electronic equipment is described by complex probabilistic laws. In engineering practice, to assess the reliability of REA, quantitative characteristics are introduced based on the processing of experimental data.
Reliability of products characterized
Probability of failure-free operation P(t) (characterizes the rate of decrease in reliability over time),
Failure rate F(t),
Failure rate l(t),
Mean time between failures T avg.
The reliability of REA can also be assessed by the probability of failure q(t) = 1 - P(t).
Let's consider assessing the reliability of non-repairable systems. The given characteristics are also true for repaired systems, if they are considered for the case before the first failure.
Let a batch containing N(0) products be delivered for testing. During the testing process, by time t n items failed. Remained intact:
N(t) = N(0) – n.
The ratio Q(t) = n/N(0) is an estimate of the probability of product failure during time t. The greater the number of products, the more accurate the assessment of the reliability of the results, the strict expression for which is as follows:
The value P(t), equal to
P(t) = 1 – Q(t)
is called the theoretical probability of failure-free operation and characterizes the probability that a failure will not occur by time t.
The probability of failure-free operation P(t) is the probability that within a specified period of time t, an object failure will not occur. This indicator is determined by the ratio of the number of object elements that worked without failure until time t to the total number of object elements that were operational at the initial moment.
The probability of failure-free operation of the product can be determined for an arbitrary time interval (t 1 ; t 2) from the moment of start of operation. In this case, we talk about the conditional probability P(t 1 ; t 2) in the period (t 1 ; t 2) in the operating state at time t 1 . The conditional probability P(t 1 ; t 2) is determined by the relation:
P(t 1 ; t 2) = P(t 2)/ P(t 1),
where P(t 1) and P(t 2) are the probability values at the beginning (t 1) and end (t 2) of operating time, respectively.
Failure rate. The value of the failure rate over time t in a given experiment is determined by the relation f(t) = Q(t)/t = n/(N(0)*t). As an indicator of the reliability of non-repairable systems, the time derivative of the failure function Q(t) is often used, which characterizes the distribution density of the product’s time to failure f(t):
f(t) = dQ(t)/dt = - dP(t)/dt.
The value f(t)dt characterizes the probability that the system will fail in the time interval (t; t+dt) provided that at time t it was in working condition.
Failure rate. A criterion that more fully determines the reliability of non-repairable electronic equipment and its modules is the failure rate l(t). The failure rate l(t) represents the conditional probability of a failure occurring in the system at some point in the operating time, provided that there were no failures in the system before that moment. The value l(t) is determined by the relation
l (t) = f(t)/P(t) = (1/P(t)) dQ/dt.
Failure rate l (t) is the number of failures n(t) of object elements per unit of time, divided by the average number of object elements N(t) operational at time t:
l (t)=n(t)/(N(t)*t), where
t - a specified period of time.
For example: 1000 object elements worked for 500 hours. During this time, 2 elements failed. Hence, l(t)=n(t)/(N*t)=2/(1000*500)=4*10-6 1/h, i.e. 4 out of a million elements can fail in 1 hour.
The reliability of an object as a system is characterized by a failure flow l, numerically equal to the sum of the failure rates of individual devices:
The formula calculates the flow of failures and individual devices of an object, which, in turn, consist of various nodes and elements, characterized by their failure rate. The formula is valid for calculating the failure rate of a system of 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. The relationship between the probability of failure-free operation P(t) and the failure flow l is determined:
P(t)=exp(-lt), it is obvious that 0 Indicators of component failure rates are taken based on reference data [1, 6, 8]. For example in table. Figure 1 shows the failure rate l(t) of some elements. It follows that the value l(t)dt characterizes the conditional probability that the system will fail in the time interval (t; t+dt) provided that at time t it was in working condition. This indicator characterizes the reliability of the electronic equipment at any time and for the interval Δt i can be calculated using the formula: l = Δn i /(N avg Δt i), where Δn i = N i - N i+1 - number of failures; N c p = (N i + N i +1)/2 - average number of serviceable products; N i, and N i+1 - the number of workable products at the beginning and end of the time period Δt i. The probability of failure-free operation is related to the values of l(t) and f(t) by the following expressions: P(t) = exp(- l(t) dt), P(t) = exp(- f(t) dt) Knowing one of the reliability characteristics P(t), l(t) or f(t), you can find the other two. If you need to estimate the conditional probability, you can use the following expression: P(t 1 ; t 2) = exp(- l(t) dt). If the REA contains N series-connected elements of the same type, then l N (t) = Nl (t). Mean time between failures
T av and the probability of failure-free operation P(t) are related by the dependence T av = P(t) dt. According to statistical data T av = Dn i t av i, t av i = (t i +t i +1)/2, m = t/Dt where Δn i is the number of failed products during the time interval Δt av i = (t i +1 -t i); t i , t i +1 - respectively, the time at the beginning and end of the test interval (t 1 =0); t is the time interval during which all products failed; m is the number of test time intervals. Mean time to failure To is the mathematical expectation of the operating time of an object before the first failure: To=1/l=1/(N*li), or, from here: l=1/To Failure-free operation time is equal to the reciprocal of the failure rate. For example: the technology of the elements provides an average failure rate of li=1*10 -5 1/h. When using N=1*10 4 elementary parts in an object, the total failure rate is lо= N*li=10 -1 1/h. Then the average failure-free operation time of the object is To=1/lо=10 hours. If the object is built on the basis of 4 large integrated circuits (LSI), then the average failure-free operation time of the object will increase by N/4=2500 times and will be 25000 hours or 34 months or about 3 years. Example. Out of 20 non-repairable products, 10 failed in the first year of operation, 5 in the second, and 5 in the third. Determine the probability of failure-free operation, failure rate, failure rate in the first year of operation, as well as the average time to first failure. P(1)=(20-10)/20 = 0.5, P(2)=(20-15)/20 = 0.25, P(1;2)= P(2)/ P(1) = 0.25/0.5 = 0.5, P(3)=(20-20)/20 = 0, P(2;3)= P(3)/ P(2) = 0/0.25 = 0, f(1)=10/(20·1) = 0.5 g -1 , f(2)=5/(20·1) = 0.25 g -1 , f(3)=5/(20·1) = 0.25 g -1 , l(1)=10/[(20*1] = 0.5 g -1 , l(2)=5/[(10*1] = 0.5 g -1 , l(3)=5/[(5*1] = 1 g -1 , T av = (10·0.5+5·1.5+5·2.5)/20 = 1.25 g. Correctly understanding the physical nature and essence of failures is very important for a reasonable assessment of the reliability of technical devices. In operating practice, three characteristic types of failures are distinguished: running-in, sudden and failures due to wear. They differ in physical nature, methods of prevention and elimination, and appear during different periods of operation of technical devices. Failures can be conveniently characterized by the “life curve” of a product, which illustrates the dependence of the intensity of failures occurring in it l(t) on time t. Such an idealized curve for REA is shown in Figure 7.2.1. It has three distinct periods: running-in I, normal operation II, and wear III. Run-in failures
are observed during the first period (0 - t 1) of operation of the REA and arise when some of the elements included in the REA are defective or have hidden defects. The physical meaning of running-in failures can be explained by the fact that the electrical and mechanical loads placed on the electronic components during the running-in period exceed their electrical and mechanical strength. Since the duration of the running-in period of the electronic equipment is determined mainly by the failure rate of the low-quality elements included in its composition, the duration of failure-free operation of such elements is usually relatively low, therefore it is possible to identify and replace them in a relatively short time. Depending on the purpose of the REA, the running-in period can last from several to hundreds of hours. The more critical the product, the longer the duration of this period. The running-in period is usually fractions and units of percent of the time of normal operation of the REA in the second period. As can be seen from the figure, the section of the “life curve” of the REA, corresponding to the running-in period I, is a monotonically decreasing function l(t), the steepness of which and the length in time are smaller, the more perfect the design, the higher the quality of its manufacture and the more carefully the running-in regimes are observed . The running-in period is considered completed when the failure rate of the electronic equipment approaches the minimum achievable (for a given design) value l min at point t 1 . Run-in failures can be the result of design (for example, unsuccessful layout), technological (poor quality assembly) and operational (violation of run-in modes) errors. Taking this into account, when manufacturing products, enterprises are recommended to carry out run
products for several tens of hours of operation (up to 2-5 days) using specially developed methods that provide for operation under the influence of various destabilizing factors (cycles of continuous operation, on-off cycles, changes in temperature, supply voltage, etc.). Period of normal operation.
Sudden failures are observed during the second period (t 1 -t 2) of operation of the REA. They arise unexpectedly due to the action of a number of random factors, and it is practically impossible to prevent their approach, especially since by this time only full-fledged components remain in the REA. However, such failures are still subject to certain patterns. In particular, the frequency of their appearance over a fairly large period of time is the same in the same types of CEA classes. The physical meaning of sudden failures can be explained by the fact that with a rapid quantitative change (usually a sharp increase) of any parameter, qualitative changes occur in the electronic components, as a result of which they lose completely or partially their properties necessary for normal functioning. Sudden failures of electronic equipment include, for example, breakdown of dielectrics, short circuits of conductors, unexpected mechanical damage to structural elements, etc. The period of normal operation of REA is characterized by the fact that the intensity of its failures in the time interval (t 1 -t 2) is minimal and has an almost constant value l min » const. The value of l min is smaller, and the interval (t 1 – t 2) is larger, the more perfect the design of the electronic equipment, the higher the quality of its manufacture and the more carefully observed operating conditions. The period of normal operation of REA for general technical purposes can last tens of thousands of hours. It may even exceed the obsolescence time of the equipment. Wear period.
At the end of the equipment service life, the number of failures begins to increase again. In most cases, they are a natural consequence of gradual wear and natural aging of the materials and elements used in the equipment. They depend mainly on the duration of operation and the “age” of the REA. The average service life of a component before wear is a more definite value than the time of occurrence of run-in and sudden failures. Their appearance can be predicted on the basis of experimental data obtained from testing specific equipment. Physical meaning of failures due to wear can be explained by the fact that in as a result of a gradual and relatively slow quantitative change in some parameter REA component, this parameter goes beyond the established tolerance, completely or partially loses its properties necessary for normal functioning. With wear, partial destruction of materials occurs, and with aging, a change in their internal physical and chemical properties occurs. Failures as a result of wear include loss of sensitivity, accuracy, mechanical wear of parts, etc. The section (t 2 -t 3) of the “life curve” of the REA, corresponding to the period of wear, is a monotonically increasing function, the steeper of which is the smaller (and the length in time the more), the higher quality materials and components used in the equipment. Operation of the equipment stops when the failure rate of the electronic equipment approaches the maximum permissible for a given design. Probability of failure-free operation of REA.
The occurrence of failures in electronic equipment is random. Consequently, failure-free operation time is a random variable, which is described using different distributions: Weibull, exponential, Poisson. Failures in electronic equipment containing a large number of similar non-repairable elements obey the Weibull distribution quite well. The exponential distribution is based on the assumption of a constant failure rate over time and can be successfully used in calculating the reliability of disposable equipment containing a large number of non-repairable components. When operating a radio electronic equipment for a long time, in order to plan its repair, it is important to know not the probability of failures, but their number over a certain period of operation. In this case, the Poisson distribution is used, which allows one to calculate the probability of the occurrence of any number of random events over a certain period of time. The Poisson distribution is applicable to assess the reliability of a repaired electronic equipment with the simplest failure flow. The probability of no failure during time t is P 0 = exp(-t), and the probability of i failures occurring during the same time is P i = i t i exp(-t)/i!, where i = 0, 1, 2, ..., n - number of failures. 7.3. Structural reliability of the equipment The structural reliability of any radio-electronic device, including electronic equipment, is its resulting reliability with a known structural diagram and known reliability values of all elements that make up the structural diagram. In this case, elements are understood as integrated circuits, resistors, capacitors, etc., performing certain functions and included in the general electrical circuit of the REA, as well as auxiliary elements that are not included in the structural diagram of the REA: soldered connections, plug-in connections, fastening elements, etc. .d. The reliability of these elements is described in sufficient detail in the specialized literature. When further considering issues of reliability of REA, we will proceed from the fact that the reliability of the elements that make up the structural (electrical) circuit of REA is uniquely specified. Quantitative characteristics
structural reliability of REA. To find them, they draw up a block diagram of the electronic equipment and indicate the elements of the device (blocks, nodes) and the connections between them. Then the circuit is analyzed and elements and connections are identified that determine the performance of the main function of this device. From the identified main elements and connections, a functional (reliability) diagram is made, and in it the elements are distinguished not according to their design, but according to their functional characteristics in such a way that each functional element is ensured independence, i.e., so that the failure of one functional element does not cause a change in probability occurrence of a failure in another adjacent functional element. When drawing up separate reliability diagrams (devices of units, blocks), it is sometimes necessary to combine those structural elements whose failures are interrelated, but do not affect the failures of other elements. Determining quantitative indicators of the reliability of REA using block diagrams makes it possible to solve the issues of choosing the most reliable functional elements, assemblies, blocks that make up the REA, the most reliable structures, panels, racks, consoles, rational operating procedures, prevention and repair of REA, composition and quantity Spare parts Related information. Failure Rate-
conditional probability density of the occurrence of a failure of a non-repairable object, determined for the considered moment in time, provided that the failure did not occur before this moment. Thus, statistically, the failure rate is equal to the number of failures that occurred per unit of time, divided by the number of objects that have not failed at a given moment. A typical change in failure rate over time is shown in Fig. 5. Experience in operating complex systems shows that the change in failure rate λ( t) the majority of objects are described U- shaped curve. Time can be divided into three characteristic sections: 1. Run-in period. 2. Period of normal operation. 3. The aging period of the object. Rice. 5. Typical change in failure rate The period of running-in of an object has an increased failure rate, caused by running-in failures caused by defects in production, installation and adjustment. Sometimes the end of this period is associated with warranty service of the object, when the elimination of failures is carried out by the manufacturer. During normal operation, the failure rate practically remains constant, while failures are random in nature and appear suddenly, primarily due to random load changes, non-compliance with operating conditions, unfavorable external factors, etc. It is this period that corresponds to the main operating time of the facility. An increase in failure rate refers to the aging period of an object and is caused by an increase in the number of failures due to wear, aging and other reasons associated with long-term operation. That is, the probability of failure of an element that has survived for the moment t in some subsequent period of time depends on the values of λ( u) only over this period, and therefore the failure rate is a local indicator of the reliability of the element over a given period of time. Topic 1.3. Reliability of restored systems
Modern automation systems are complex, restorable systems. Such systems are repaired during operation; if some elements fail, they are repaired and continue to operate. The ability of systems to be restored during operation is “laid in” during their design and ensured during manufacturing, and repair and restoration operations are provided for in regulatory and technical documentation. Carrying out repair and restoration activities is essentially another method aimed at increasing the reliability of the system. 1.3.1. Reliability indicators of restored systems
On the quantitative side, such systems, in addition to the previously discussed reliability indicators, are also characterized by complex reliability indicators. A complex reliability indicator is a reliability indicator that characterizes several properties that make up the reliability of an object. Complex reliability indicators that are most widely used to characterize the reliability of restored systems are: Availability factor; Operational readiness ratio; Technical utilization rate. Availability factor-
the probability that the object will be in working condition at any point in time, except for planned breaks, during which the object is not intended to be used for its intended purpose. Thus, the availability factor simultaneously characterizes two different properties of an object - reliability and maintainability. Availability factor is an important parameter, however, it is not universal. Operational readiness ratio-
the probability that the object will be in working condition at an arbitrary point in time, except for planned breaks, during which the use of the object for its intended purpose is not intended, and, starting from this moment, will work without failure for a given time interval. The coefficient characterizes the reliability of objects, the need for use of which arises at an arbitrary point in time, after which a certain failure-free operation is required. Until this moment, the equipment can be in standby mode, the mode of use in other operating functions. Technical utilization rate-
the ratio of the mathematical expectation of time intervals for objects to remain in working condition for a certain period of operation to the sum of mathematical expectations of time intervals for an object to remain in working condition, downtime due to maintenance, and repairs for the same period of operation. There are probabilistic (mathematical) and statistical indicators of reliability. Mathematical reliability indicators are derived from theoretical distribution functions of failure probabilities. Statistical reliability indicators are determined empirically when testing objects on the basis of statistical data from the operation of equipment. Reliability is a function of many factors, most of which are random. It is clear from this that a large number of criteria are needed to assess the reliability of an object. The reliability criterion is a sign by which the reliability of an object is assessed. Reliability criteria and characteristics are probabilistic in nature, since the factors influencing the object are random in nature and require statistical assessment. Quantitative characteristics of reliability can be: Serves as one of the main indicators when calculating reliability. In the future, we assume that the operation of the object occurs continuously, the duration of operation of the object is expressed in time units t and operation began at time t=0. It is clear from the graph that: In practice, sometimes a more convenient characteristic is the probability of malfunction of an object or the probability of failure: The failure rate is the ratio of the number of failed objects to their total number before the test, provided that the failed objects are not repaired or replaced with new ones, i.e. a*(t) = n(t)/(NΔt) The failure rate is the distribution density of the operating time of a product before it fails. Probabilistic determination of failure rate a(t) = -P(t) or a(t) = Q(t). Thus, there is a unique relationship between the frequency of failures, the probability of failure-free operation and the probability of failures under any failure time distribution law: Q(t) = ∫ a(t)dt. Failure is treated in reliability theory as a random event. The theory is based on the statistical interpretation of probability. Elements and systems formed from them are considered as mass objects belonging to the same general population and operating under statistically homogeneous conditions. When we talk about an object, we essentially mean an object taken at random from a population, a representative sample from this population, and often the entire population. For mass objects, a statistical estimate of the probability of failure-free operation P(t) can be obtained by processing the results of reliability tests of sufficiently large samples. How the score is calculated depends on the test design. Let tests of a sample of N objects be carried out without replacement or restoration until the failure of the last object. Let us denote the duration of time until failure of each of the objects t 1, ..., t N. Then the statistical estimate is: P*(t) = 1 - 1/N ∑η(t-t k) where η is the Heaviside unit function. For the probability of failure-free operation on a certain segment, the estimate P*(t) = /N is convenient, The failure rate, determined by replacing failed products with serviceable ones, is sometimes called the average failure rate and is denoted ω(t). The failure rate λ(t) is the ratio of the number of failed objects per unit of time to the average number of objects operating in a given period of time, provided that the failed objects are not restored or replaced by serviceable ones: λ(t) = n(t)/ Lifetime tests and observations of large samples of objects show that in most cases the failure rate varies non-monotonically over time. From the curve of failures versus time it can be seen that the entire period of operation of the facility can be conditionally divided into 3 periods. Run-in failures are, as a rule, the result of the presence of defects and defective elements in an object, the reliability of which is significantly lower than the required level. As the number of elements in a product increases, even with the most strict control, it is not possible to completely eliminate the possibility of elements having certain hidden defects getting into the assembly. In addition, failures during this period can also be caused by errors during assembly and installation, as well as insufficient mastery of the facility by maintenance personnel. The physical nature of such failures is random in nature and differs from sudden failures during the normal period of operation in that here failures can occur not under increased, but also under insignificant loads (“burning out of defective elements”). To increase the reliability of an object, taking into account the possibility of running-in failures, you need to: The average running-in time is determined during testing. For particularly important cases, it is necessary to increase the running-in period several times compared to the average. II – nd period – normal operation Maintaining the level of failure intensity at this stage is characterized by the fact that the failed element is replaced by the same one with the same probability of failure, and not by a better one, as happened at the running-in stage. Rejection and preliminary running-in of elements used to replace failed ones is even more important for this stage. III period – wear The likelihood of failures due to wear increases as the service life approaches. From a probabilistic point of view, system failure in a given time period Δt = t 2 – t 1 is defined as the probability of failure: ∫a(t) = Q 2 (t) - Q 1 (t) Failure rate is the conditional probability that a failure will occur in a time interval Δt, provided that it has not occurred before λ(t) = /[ΔtP(t)] These expressions establish the relationship between the probability of failure-free operation and the frequency and intensity of failures. If a(t) is a non-increasing function, then the following relation holds: The mean time between failures is the mathematical expectation of the time between failures. Probabilistic definition: MTBF is equal to the area under the MTBF curve. Statistical definition: T* = ∑θ i /N 0 It is obvious that the parameter T* cannot fully and satisfactorily characterize the reliability of durable systems, since it is a characteristic of reliability only up to the first failure. Therefore, the reliability of long-term use systems is characterized by the average time between two adjacent failures or time between failures t av: MTBF is the average time between adjacent failures, provided that the failed element is restored.
№
Item name Failure rate, *10 -5, 1/h
Resistors 0,0001…1,5
Capacitors 0,001…16,4
Transformers 0,002…6,4
Inductors 0,002…4,4
Relay 0,05…101
Diodes 0,012…50
Triodes 0,01…90
Switching devices 0,0003…2,8
Connectors 0,001…9,1
Solder connections 0,01…1
Wires, cables 0,01…1
Electric motors 100…600
Rice. 7.2.1.
probability of failure-free operation;
mean time between failures;
failure rate;
failure rate;
various reliability coefficients.1. Probability of failure-free operation
The probability of failure-free operation of an object is the probability that it will maintain its parameters within specified limits for a certain period of time under certain operating conditions.
Let us denote P(t) the probability of failure-free operation of an object over a period of time. The probability, considered as a function of the upper bound of the time interval, is also called the reliability function.
Probabilistic assessment: P(t) = 1 – Q(t), where Q(t) is the probability of failure.
1. P(t) – non-increasing function of time;
2. 0 ≤ P(t) ≤ 1;
3. P(0)=1; P(∞)=0.
Q(t) = 1 – P(t).
Statistical characteristic of failure probability: Q*(t) = n(t)/N2. Failure rate
where a*(t) is the failure rate;
n(t) – number of failed objects in the time interval from t – t/2 to t+ t/2;
Δt – time interval;
N – number of objects participating in the test.
where n(t) is the number of objects that failed at time t.3. Failure rate
where N av = /2 is the average number of objects that worked properly in the time interval Δt;
N i – number of products operating at the beginning of the interval Δt;
N i+1 – the number of objects that were working properly at the end of the time interval Δt.
1st period – running-in.
A decrease in the failure rate of an object as a whole, with a constant value of this parameter for each of the elements separately, is precisely explained by the “burning out” of weak links and their replacement with the most reliable ones. The steeper the curve in this area, the better: fewer defective elements will remain in the product in a short time.
carry out more stringent screening of elements;
carry out tests of the object in conditions close to operational ones and use only elements that have passed the tests during assembly;
improve the quality of assembly and installation.
This period is characterized by the fact that running-in failures have already ended, and wear-related failures have not yet occurred. This period is characterized exclusively by sudden failures of normal elements, the time between failures of which is very high.
The designer has the greatest capabilities in solving this problem. Often, changing the design or facilitating the operating modes of just one or two elements provides a sharp increase in the reliability of the entire facility. The second way is to improve the quality of production and even the cleanliness of production and operation.
The period of normal operation ends when wear failures begin to occur. The third period in the life of the product begins - the period of wear.
λ(t) = lim /[ΔtP(t)] = / = Q"(t)/P(t) = -P"(t)/P(t)
since a(t) = -P"(t), then λ(t) = a(t)/P(t).
ω(t) ≥ λ(t) ≥ a(t).4. MTBF
where θ I is the operating time of the i-th object until failure;
N 0 – initial number of objects.
t av = ∑θ i /n = 1/ω(t),
where n is the number of failures during time t;
θ i is the operating time of the object between the (i-1)th and i-th failures.
