Analysis of the types and consequences of screen failures. FMEA analysis

An analysis of the types and consequences of failures of the components of the technical and functional structures of the designed system is the first stage of the design study of reliability and safety. The internationally accepted abbreviation for failure mode and effect analysis is FMEA (failure mode and effect analysis). This type of analysis belongs to the class of preliminary qualitative and simplified quantitative analysis at the design stage. If quantitative assessments are carried out, then the term FMECA is used (failure mode, effect and criticality analysis - analysis of the types, consequences and criticality of failures). The first FMEA experiments are related to aerospace projects of the 60s of the USSR and the USA. In the 1980s, FMEA procedures began to be introduced in the US automotive industry at the Ford Motor Company. At present, the analysis of the modes and consequences of failures is a mandatory step. project evaluation reliability and safety of space, aircraft building, nuclear, chemical-technological, gas and oil processing and other industries. In areas where this stage is not mandatory, dangerous incidents occur, leading to large economic and environmental losses and threatening human life and health. Suffice it to recall the dramatic events of the collapse of public Moscow buildings built according to projects where the defect of only one element of the supporting structure (pin, column) led to catastrophic consequences.

There are three main goals for conducting an FMEA

• identifying potential failure modes of system components and determining their impact on the system as a whole and possibly the environment
• classification of failure modes by severity levels or by levels of severity and frequency of occurrence (FMECA)
• issuance of recommendations for the revision of design solutions in order to compensate or eliminate dangerous failure modes

FMEA is the most standardized area of ​​"reliability" research. The procedure for carrying out and the type of input / output documentation is regulated by the relevant standards. Internationally recognized documents are:

· MIL-STD-1629 Style FMECAs - guidance on conducting failure mode and effects analysis, criticality assessment, identifying structural bottlenecks in terms of maintainability and survivability. Initially focused on military applications.

· SAE J1739, AIG-FMEA3, FORD FMEA - a package of documents regulating the analysis of the types and consequences of failures for automotive industry facilities, including the stages of design and manufacture

· SAE ARP5580 - FMEA guidance for both commercial and military projects, integrating MIL-STD-1629 and automotive standards. The concept of groups of equivalent failures is introduced, i.e. failures that generate the same consequences and require the same corrective actions.

Common to all standards is that they regulate only the sequence and interconnection of the stages of analysis, leaving the designer free to act in the specific implementation of each stage. Thus, it is possible to arbitrarily adjust the structure of FMEA tables, determine the scales for the frequency of occurrence of failures and the severity of consequences, the introduction of additional features for classifying failures, etc.

FMEA steps:

construction and analysis of the functional and / or technical structures of the object

analysis of the operating conditions of the facility

analysis of failure mechanisms of elements, criteria and failure modes

Classification (list) of possible consequences of failures

· analysis possible ways prevention (frequency reduction) of isolated failures (failure consequences)

Technical structure the object of analysis usually has a tree-like, hierarchical representation (Fig. 3). Possible failure modes are listed for lower-level components (leaves of the tree), and their consequences are evaluated in terms of their impact on the next-level subsystems (parent nodes of the tree) and the object as a whole.

Fig.3. Hierarchical representation of the object of analysis

In Fig.4. a fragment of the FMEA table is given, containing data on the analysis of the types and consequences of failures of the equipment of a chemical-technological facility.

Fig.4. Fragment of the FMEA table.

When performing quantitative assessments of design solutions for FMEA types Component failures are usually characterized by three parameters: frequency of occurrence, degree of detection, severity of consequences. Since the analysis is preliminary, scores are usually used. expert opinions these options. For example, a number of documents propose the following classifications of failure modes by frequency (Table 2), by detection degree (Table 3), and by severity of consequences (Table 4).

Table 2. Classification of failures by frequency.

They can be used individually or in combination with each other. If all three types of FMEA - analysis are performed, then their relationship can be represented as follows:

The main application of FMEA analysis is related to the improvement of product design (service characteristics) and processes for its manufacture and operation (service provision). The analysis can be applied both in relation to the newly created products(services) and processes, as well as in relation to existing ones.

FMEA - analysis is performed when a new product, process, service is being developed, or their modernization is being carried out; when a new use is found for an existing product, process or service; when a control plan for a new or changed process is developed. Also, FMEA may be conducted for the purpose of planned improvement of existing processes, products or services, or investigation of emerging nonconformities.

FMEA analysis is performed in the following order:

1. The object of analysis is selected. If the object of analysis is a part of a composite object, then its boundaries must be precisely defined. For example, if you are analyzing a part of a process, you must set a start event and an end event for that part.

2. Options for applying the analysis are determined. FMEA may be part of complex analysis, in which various methods. In this case, the FMEA should be consistent with the overall system analysis.

Key options may include:

• top-down analysis. In this case, the object of analysis is divided into parts and FMEA is started from the largest parts.
• bottom-up analysis. The analysis begins with the smallest elements, successively moving to the elements of a higher level.
• component analysis. FMEA is performed for the physical elements of the system.
• function analysis. In this case, the analysis of the functions and operations of the object is performed. Consideration of functions is carried out from the point of view of the consumer (convenience and safety of execution), and not the designer or manufacturer.

3. The boundaries within which it is necessary to consider inconsistencies are determined. The boundaries can be - time period, type of consumer, geography of application, certain actions, etc. For example, inconsistencies that are only detected during final inspection and testing.

4. A suitable table is developed for recording information. It may vary depending on the factors taken into account. The most commonly used table is the following.

5. Elements are determined in which inconsistencies (failures) may occur. Elements can include various components, assemblies, combinations constituent parts etc. If the list of elements becomes too large and unmanageable, it is necessary to reduce the boundaries of the FMEA.

In the event that potential failures are associated with critical characteristics, additionally, during the FMEA, it is necessary to analyze the criticality of failures. Critical characteristics are standards or indicators that reflect safety or compliance with regulatory requirements and need special control.

6. For each element identified in step 5, a list of the most significant failure modes is compiled. This operation can be simplified by applying a standard list of failures for the considered elements. If a failure criticality analysis is carried out, then it is necessary to determine the probability of failure occurrence for each of the elements. When all possible failure modes for an element are identified, then the total probability of their occurrence should be 100%.

7. For each failure mode identified in step 6, all possible consequences that may occur are determined. This operation can be simplified by using a standard list of consequences. If a failure criticality analysis is carried out, then it is necessary to determine the probability of occurrence of each consequence. When all possible consequences have been identified, the probability of their occurrence should sum to 100% for each element.

8. The rating of the severity of the consequences for the consumer (S) - Severity is determined. The severity rating is usually based on a scale of 1 to 10, where 1 means minor and 10 catastrophic. If a failure mode has more than one consequence, then only the most severe consequence for that failure mode is entered in the FMEA table.

9. For each failure mode, all potential causes are identified. For this, the Ishikawa cause-and-effect diagram can be used. All potential causes for each failure mode are recorded in the FMEA table.

10. For each cause, the probability rating of its occurrence (O) - Occurrence is determined. The probability of occurrence is usually rated on a scale of 1 to 10, where 1 means extremely unlikely and 10 means imminent. The rating value is entered into the FMEA table.

11. For each reason are determined existing methods controls that are currently in place to ensure that failures do not affect the customer. These methods should prevent the occurrence of causes, reduce the likelihood that a failure will occur, or detect a failure after the cause has occurred but before the cause has affected the consumer.

12. For each control method, a detection rating (D) - Detection is determined. The detection rating is usually rated on a scale from 1 to 10, where 1 means that the control method will absolutely detect the problem, and 10 - it will not be able to detect the problem (or there is no control at all). The detection rating is entered into the FMEA table.

13. The risk priority number is calculated ( consumer risk - RPN) which is equal to the product

S*O*D. This number allows you to rank potential failures in terms of significance.

14. Recommended actions are identified, which may include design or process modifications to reduce the severity or likelihood of failures. They may also take additional measures control to increase the probability of failure detection.

Tests of technological processes for completeness.

Structural testing for completion.

These tests are carried out on the first prototypes of the product. Their purpose is to show that the design of the product satisfies the requirements for reliability.

It does not matter how the prototype was built and what efforts went into its debugging. If the required level of product reliability is not achieved, the design must be improved. Testing continues until the product meets all specified requirements.

During these tests, failures are recorded in initial period operation of the product. With this data, full consistency is achieved between the design of the product and the processes required for its manufacture, and determines the amount of testing necessary to achieve the required reliability in delivery [ of the product to consumers.

Tests are also carried out on the first samples of products. These I samples work for a given period (run-in period). The characteristics of their work are carefully monitored, decreasing failure rate is measured. After a run-in period, experience data is collected to measure and verify the performance of the product and compare it with the results. tatami, obtained during testing of the product for completeness. I Observations made during these tests, allow you to set the value of the period of the product's running-in.

Durability tests. During these tests, wear failures of product elements are recorded and their distribution is built. The data obtained is used for elimination. causes of those failures, the occurrence of which leads to an unacceptable reduction in the expected life of the product. Durability tests are carried out on a number of samples of this product. During these tests, it is necessary to determine the boundary of the transition from a constant to an increasing failure rate and construct a distribution for each observed failure mode.

One of the effective means of improving the quality of technical objects is the analysis of the types and consequences of potential failures (Potential Failure Mode and Effects Analysis - FMEA). The analysis is carried out at the stage of designing a structure or a technological process (the corresponding stages life cycle products - development and preparation for production), as well as when finalizing and improving products already put into production. It is advisable to divide this analysis into two stages: a separate analysis at the design development stage and at the development stage technological process.

The standard (GOST R 51814.2-2001. Quality systems in the automotive industry. A method for analyzing the types and consequences of potential defects) also provides for the possibility of using the FMEA method in the development and analysis of other processes, such as sales, service, and marketing processes.

The main objectives of the analysis of the types and consequences of potential failures:

Identification of critical failures associated with a danger to human life and environment and development of activities
to reduce the likelihood of their occurrence and the severity of possible consequences;

Identification and elimination of the causes of any possible failures of the product to improve its reliability.

During the analysis, the following tasks are solved:

Identification of possible failures of an object (product or process) and its elements (this takes into account the experience of manufacturing and operating similar objects),

Studying the causes of failures, quantifying the frequency of their occurrence,

Classification of failures according to the severity of consequences and quantitative assessment of the significance of these consequences,

Evaluation of the sufficiency of monitoring and diagnostic tools Evaluation of the possibility of detecting a failure, the possibility of preventing a failure in the practical use of these tools,

Development of proposals for changing the design and manufacturing technology in order to reduce the likelihood of failures and their criticality,

Development of rules for personnel behavior in the event of critical failures,

analysis of possible personnel errors.

To conduct the analysis, a group of specialists with practical experience and high professional level in the field of designing similar objects, knowing the processes of manufacturing components and assembling an object, "technology for monitoring and diagnosing the state of an object, methods" of maintenance and repair. The brainstorming method is used. At the same time, at the stage of qualitative analysis, a structural scheme object: the object is considered as a system consisting of subsystems different levels, which in turn consist of separate elements.

Possible types of failures and their consequences are analyzed from the bottom up, i.e. from elements to subsystems, and then to the object as a whole. The analysis takes into account that each failure can have several causes and several different consequences.

At the stage of quantitative analysis, the criticality of the failure is assessed expertly, in points, taking into account the probability of its occurrence, the probability of its detection and assessment of the severity of possible consequences. Failure risk (priority risk number) can be found using the formula: I

where the value of O is determined in points depending on the probability of failure, - on the probability of detecting (detecting) a failure, "depends on the severity of the consequences of the failure.

The found value for each element for each cause and for each possible consequence is compared with the critical one. The critical value is set in advance and is selected from 100 to 125. Reducing the critical value corresponds to the development of more reliable products and processes.

For each failure, for which the value of R exceeds the critical one, measures are developed to reduce it by improving the design and manufacturing technology. For a new version of the object, the criticality of the object R is recalculated. If necessary, the refinement procedure is repeated again.

FMEA methodology, examples

FMEA (Failure Mode and Effects Analysis) is an analysis of the modes and effects of failures. Originally developed and published by the US military-industrial complex (in the form of MIL-STD-1629), failure mode analysis is so popular today because specialized FMEA standards have been developed and published in some industries.

A few examples of such standards are:

• MIL-STD-1629. Developed in the USA and is the ancestor of all modern FMEA standards.
• SAE-ARP-5580 is a modified MIL-STD-1629, supplemented by a library of some elements for the automotive industry. Used in many industries.
• SAE J1739 is an FMEA standard describing Potential Failure Mode and Effects Analysis in Design (DFMEA) and Potential Failure Mode and Effects Analysis in Manufacturing and Assembly Processes, PFMEA). The standard helps to identify and reduce risk by providing relevant conditions, requirements, rating charts and worksheets. As a standard, this document contains requirements and guidelines to guide the user through the implementation of the FMEA.
• AIAG FMEA-3 is a specialized standard used in the automotive industry.
• Internal FMEA standards of large car manufacturers.
• Historically developed in many companies and industries, procedures similar to failure modes and effects analysis. Perhaps today these are the "standards" of the FMEA with the widest coverage.

All failure mode and effects analysis standards (whether published or developed historically) are generally very similar to each other. Below general description gives a general idea of ​​FMEA as a methodology. It is intentionally not too deep and covers most of the current FMEA approaches.

First of all, the boundaries of the analyzed system must be clearly defined. The system may be technical device, process or anything else subject to FME analysis.

Next, the types of possible failures, their consequences and possible reasons occurrence. Depending on the size, nature, and complexity of the system, the determination of possible failure modes can be performed for the entire system as a whole or for each of its subsystems individually. In the latter case, the consequences of failures at the subsystem level will manifest themselves as failure modes at the level above. Identification of failure modes and consequences should be carried out in a bottom-up manner, until top level systems. To characterize the types and consequences of failures defined at the top level of the system, parameters such as intensity, criticality of failures, probability of occurrence, etc. are used. These parameters can either be calculated "bottom-up" from the lower levels of the system, or explicitly set at its upper level. These parameters can be both quantitative and qualitative. As a result, for each element of the top-level system, its own unique measure is calculated, calculated from these parameters according to the corresponding algorithm. In most cases, this measure is referred to as "risk priority ratio", "criticality", "risk level" or similar. The ways in which such a measure can be used and how it is calculated can be unique in each case and are a good starting point for making the manifold modern approaches to conduct a failure modes and effects analysis (FMEA).

An example of the application of FMEA in the military-industrial complex

The purpose of the "Criticality" parameter is to demonstrate that the system safety requirements are fully met (in the simplest case, this means that all criticality indicators are below a predetermined level.

The acronym FMECA stands for Failure Mode, Effects and Criticality Analysis.

The main indicators used to calculate the Severity value are:

• failure rate (determined by calculating the time between failures - MTBF),
• probability of failure (as a percentage of the failure rate indicator),
• working time.

Thus, it is obvious that the criticality parameter has a real exact value for each specific system (or its component).

There is a fairly wide range of available catalogs (libraries) containing failure probabilities different types for various electronic components:

• FMD97
• MIL-HDBK-338B
• NPRD3

The library descriptor for a specific component, in general, looks like this:

Since in order to calculate the failure criticality parameter it is necessary to know the values ​​of the failure rate index, in the military-industrial complex, before applying the FME[C]A methodology, the MTBF calculation is performed, the results of which are used by FME[C]A. For system elements whose failure criticality index exceeds the tolerances established by safety requirements, an appropriate Fault Tree Analysis (FTA, Fault Tree Analysis) should also be carried out. In most cases, Failure Modes, Effects, and Criticality Analysis (FMEA) for the needs of the defense industry is performed by a single person (either an electronic circuit design expert or a quality control specialist) or a very small group of such experts.

FMEA in the automotive industry

For each Risk Priority Number (RPN) of a failure that exceeds a predetermined level (often 60 or 125), corrective actions are identified and implemented. As a rule, responsible for the implementation of such measures, the timing of their implementation and the way to subsequently demonstrate the effectiveness of the corrective actions taken are determined. After the implementation of corrective measures, the value of the Failure Risk Priority Factor is re-evaluated and compared with the limit value set.

The main indicators used to calculate the value of the Risk Priority Ratio are:

• probability of failure
• criticality,
• failure detection probability.

In most cases, the Risk Priority Ratio is derived based on the values ​​of the above three indicators (whose dimensionless values ​​range from 1 to 10), i.e. is a calculated value that varies within similar limits. However, in cases where there are actual (retrospective) exact values ​​of the failure rate for a particular system, the boundaries for finding the Risk Priority Coefficient can be expanded many times, for example:

In most cases, FMEA analysis in the automotive industry is carried out internally. working group representatives of different departments (R & D, production, service, quality control).

Features of FMEA, FMECA and FMEDA analysis methods

The FMEA (Failure Modes and Effects Analysis), FMECA (Failure Modes, Effects and Criticality Analysis) and FMEDA (Failure Modes, Effects and Diagnosability Analysis) reliability analysis methods, while having much in common, contain several notable differences.

Whereas FMEA is a methodology that allows you to determine scenarios (methods) in which a product (equipment), emergency protection device (ESD), technological process or system can fail (see IEC 60812 "Analysis techniques for system reliability - Procedure for failure mode and effects analysis (FMEA)"),

FMECA, in addition to FMEA, ranks the identified failure modes in order of their importance (criticality) by calculating one of two indicators - the risk priority number (Risk Priority Number) or failure criticality,

and the goal of FMEDA is to calculate the failure rate (failure rate) of the final system, which can be considered a device or group of devices that performs a more complex function. The FMEDA failure modes, effects and diagnosability analysis methodology was first developed for the analysis of electronic devices and subsequently extended to mechanical and electromechanical systems.

General concepts and approaches of FMEA, FMECA and FMEDA

FMEA, FMECA and FMEDA use common basic concepts components, devices and their arrangement (interactions). The safety instrumented function (SIF) consists of several devices that must ensure that the necessary operation is performed to protect the machine, equipment or process from the consequences of a hazard, failure. Examples of SIS devices are a converter, an insulator, a contact group, etc.

Each device is made up of components. For example, a transducer may consist of components such as gaskets, bolts, diaphragm, electronic circuit etc.

An assembly of devices can be considered as one combined device that implements the SIS function. For example, an actuator-positioner-valve is an assembly of devices that together can be considered as the ultimate safety element of an ESD. Components, devices, and assemblies may be part of an end system for the purposes of FMEA, FMECA, or FMEDA evaluation.

The basic methodology underlying FMEA, FMECA and FMEDA can be applied before or during the design, manufacture or final installation of the final system. The basic methodology considers and analyzes the failure modes of each component that is part of each device in order to estimate the chance of failure of all components.

In cases where FME analysis is performed for an assembly, in addition to identifying failure modes and effects, a reliability block diagram (diagram) of this assembly should be developed to assess the interaction of devices with each other (see IEC 61078:2006 "Analysis techniques for dependability - Reliability block diagram and boolean methods").

Input data, results and evaluation of the results of the implementation of FMEA, FMECA, FMEDA shown schematically in the picture (right). Enlarge picture.

The general approach defines the following main steps of FME analysis:

• definition of the final system and its structure;
• identification of possible scenarios for performing the analysis;
• assessment of possible situations of combinations of scenarios;
• performing FME analysis;
• evaluation of the results of FME analysis (including FMECA, FMEDA).

The application of the FMECA methodology to the results of the failure mode and effects analysis (FMEA) makes it possible to assess the risks associated with failures, and the FMEDA methods - the ability to assess the reliability.

For everybody simple device an FME table is developed, which is then applied to each specific analysis scenario. The structure of the FME table may vary for FMEA, FMECA or FMEDA, and also depending on the nature of the final system being analyzed.

The result of the failure modes and effects analysis is a report containing all verified (if necessary, adjusted by the working group of experts) FME tables and conclusions / judgments / decisions regarding the final system. If the target system is modified after performing an FME analysis, the FMEA procedure must be repeated.

Differences in assessments and results of FME-, FMEC- and FMED-analysis

Although the basic steps in performing an FME analysis are generally the same for FMEA, FMECA, and FMEDA, the evaluation and results differ.

The results of the FMECA analysis include the results of the FMEA, as well as the ranking of all failure modes and effects. This ranking is used to identify components (or devices) with more a high degree influence on the reliability of the final (target) system, characterized by such safety indicators as the average probability of failure on demand (PFDavg), the average dangerous failure frequency (PFHavg.), the mean time between failures (MTTFs) or the mean time to dangerous failure ( MTTFd).

FMECA results can be used for qualitative or quantitative evaluation, and in both cases they should be presented with an end system criticality matrix showing in graphical form which components (or devices) have a greater / lesser impact on the reliability of the final (target) system.

FMEDA results include FMEA results and final system reliability data. They can be used to verify that a system meets a target SIL, to certify a SIL, or as a basis for calculating the target SIL of an SIS device.

FMEDA provides quantitative assessments of reliability indicators such as:

• Safe detected failure rate (rate of diagnosed / detected safe failures) - the frequency (rate) of failures of the final system, transferring its operating state from normal to safe. The ESD system or operator is notified, the target plant or equipment is protected;
• Safe undetected failure rate (rate of undiagnosed / undetected safe failures) - the frequency (rate) of failures of the final system, transferring its operating state from normal to safe. The ESD system or operator is not notified, the target plant or equipment is protected;
• Dangerous detected failure rate (rate of diagnosable / detected dangerous failures) - the frequency (rate) of failures of the end system, at which it will remain in a normal state when necessary, but the system or operator of the ESD is notified to correct the problem or perform Maintenance. The target plant or equipment is not protected, but the problem is identified and there is a chance to correct the problem before the need arises;
• Dangerous undetected failure rate - The rate (rate) of failures of an end system at which it will remain in a normal state when the need arises, but the system or the ESD operator is not notified. The target plant or equipment is not protected, the problem is hidden, and the only way identifying and eliminating a malfunction is to perform a control test (verification). If necessary, the FMEDA evaluation can reveal how much of the undiagnosed dangerous failures can be identified using a control test. In other words, the FMEDA score helps to ensure that Test Test Efficiency (Et) or Control Test Coverage (PTC) measures are taken when performing proof testing (validation) of the end system;
• Annunciation failure rate (rate of failure-alerts) - the frequency (rate) of failures of the final system, which will not affect the safety performance when its operating state is transferred from a normal to a safe state;
• No effect failure rate - The rate (rate) of any other failures that will not cause the end system's operating state to go from normal to safe or dangerous.

KConsult C.I.S. offers professional services certified European practicing engineers for performing FMEA, FMECA, FMEDA analysis, as well as implementing the FMEA methodology in the daily activities of industrial enterprises.

With an exponential law of distribution of recovery time and time between failures, the mathematical apparatus of Markov random processes is used to calculate the reliability indicators of systems with recovery. In this case, the functioning of systems is described by the process of changing states. The system is depicted as a graph called a state-to-state transition graph.

Random process in any physical system S , is called Markovian, if it has the following property : for any moment t 0 probability of the state of the system in the future (t > t 0 ) depends only on the current state

(t = t 0 ) and does not depend on when and how the system came to this state (in other words: with a fixed present, the future does not depend on the prehistory of the process - the past).

t< t 0

t > t 0

For a Markov process, the "future" depends on the "past" only through the "present", i.e., the future course of the process depends only on those past events that affected the state of the process at the present moment.

The Markov process, as a process without aftereffect, does not mean complete independence from the past, since it manifests itself in the present.

When using the method, in the general case, for the system S , it is necessary to have mathematical model as a set of system states S 1 , S 2 , … , S n , in which it can be during failures and restoration of elements.

When compiling the model, the following assumptions were introduced:

The failed elements of the system (or the object itself) are immediately restored (the beginning of the restoration coincides with the moment of failure);

There are no restrictions on the number of restorations;

If all flows of events that transfer the system (object) from state to state are Poisson (the simplest), then random process transitions will be a Markov process with continuous time and discrete states S 1 , S 2 , … , S n .

Basic rules for compiling a model:

1. The mathematical model is depicted as a state graph, in which

a) circles (vertices of the graphS 1 , S 2 , … , S n ) – possible states of the system S , arising from element failures;

b) arrows– possible directions of transitions from one state S i to another S j .

Above/below arrows indicate transition intensities.

Graph examples:

S0 - working condition;

S1 – failure state.

"Loop" denotes delays in a particular state S0 and S1 relevant:

Good condition continues;

The failure state continues.

The state graph reflects a finite (discrete) number of possible system states S 1 , S 2 , … , S n . Each of the vertices of the graph corresponds to one of the states.

2. To describe the random process of state transition (failure / recovery), state probabilities are used

P1(t), P2(t), … , P i (t), … , Pn(t) ,

where P i (t) is the probability of finding the system at the moment t v i-th state.

Obviously, for any t

(normalization condition, since other states, except for S 1 , S 2 , … , S n No).

3. According to the graph of states, a system of ordinary differential equations of the first order (Kolmogorov-Chapman equations) is compiled.

Let's consider an installation element or installation itself without redundancy, which can be in two states: S 0 - trouble-free (workable),S 1 - state of failure (restoration).

Let us determine the corresponding probabilities of element states R 0 (t): P 1 (t) at an arbitrary point in time t under different initial conditions. We will solve this problem under the condition, as already noted, that the flow of failures is the simplest with λ = const and restorations μ = const, the law of distribution of time between failures and recovery time is exponential.

For any moment of time, the sum of the probabilities P 0 (t) + P 1 (t) = 1 is the probability of a certain event. Let us fix the moment of time t and find the probability P (t + ∆ t) that at the moment of time t + ∆ t item is in progress. This event is possible when two conditions are met.

At time t the element was in the state S 0 and for time t there was no failure. The probability of the element operation is determined by the rule of multiplying the probabilities of independent events. The probability that at the moment t item was and condition S 0 , is equal to P 0 (t). The probability that in time t he did not refuse e -λ∆ t . Up to a higher order of smallness, we can write

Therefore, the probability of this hypothesis is equal to the product P 0 (t) (1- λ ∆ t).

2. At the point in time t element is in state S 1 (in a state of recovery), during the time ∆ t restoration has ended and the element has entered the state S 0 . This probability is also determined by the rule of multiplying the probabilities of independent events. The probability that at the time t the element was in the state S 1 , is equal to R 1 (t). The probability that the recovery has ended is determined through the probability of the opposite event, i.e.

1 - e -μ∆ t = μ· ∆ t

Therefore, the probability of the second hypothesis is P 1 (t) ·μ· ∆ t/

Probability of the operating state of the system at a point in time (t + ∆ t) is determined by the probability of the sum of independent incompatible events when both hypotheses are fulfilled:

P 0 (t+∆ t)= P 0 (t) (1- λ ∆ t)+ P 1 (t) ·μ ∆ t

Dividing the resulting expression by ∆ t and taking the limit at ∆ t → 0 , we obtain the equation for the first state

dP 0 (t)/ dt=- λP 0 (t)+ µP 1 (t)

Carrying out similar reasoning for the second state of the element - the state of failure (restoration), we can obtain the second equation of state

dP 1 (t)/ dt=- µP 1 (t)+λ P 0 (t)

Thus, to describe the probabilities of the state of the element, a system of two differential equations was obtained, the state graph of which is shown in Fig. 2

d P 0 (t)/ dt = - λ P 0 (t)+ µP 1 (t)

dP 1 (t)/ dt = λ P 0 (t) - µP 1 (t)

If there is a directed state graph, then the system of differential equations for state probabilities R TO (k = 0, 1, 2,…) can be written immediately using the following rule: on the left side of each equation is the derivativedP TO (t)/ dt, and in the right one there are as many components as there are edges connected directly with the given state; if the edge ends in this state, then the component has a plus sign, if it starts from given state, then the component has a minus sign. Each component is equal to the product of the intensity of the flow of events that transfers an element or system along a given edge to another state, by the probability of the state from which the edge begins.

The system of differential equations can be used to determine the PBR of electrical systems, the function and availability factor, the probability of being under repair (restoration) of several elements of the system, the average time the system is in any state, the failure rate of the system, taking into account the initial conditions (states of the elements).

Under initial conditions R 0 (0)=1; R 1 (0)=0 and (P 0 +P 1 =1), the solution of the system of equations describing the state of one element has the form

P 0 (t) = μ / (λ+ μ )+ λ/(λ+ μ )* e^ -(λ+ μ ) t

Failure state probability P 1 (t)=1- P 0 (t)= λ/(λ+ μ )- λ/ (λ+ μ )* e^ -(λ+ μ ) t

If at the initial moment of time the element was in the state of failure (restoration), i.e. R 0 (0)=0, P 1 (0)=1 , then

P 0 (t) = μ/ (λ +μ)+ μ/(λ +μ)*e^ -(λ +μ)t

P 1 (t) = λ /(λ +μ)- μ/ (λ +μ)*e^ -(λ +μ)t

Usually in the calculations of reliability indicators for sufficiently long time intervals (t ≥ (7-8) t v ) without a large error, the probabilities of states can be determined by the established average probabilities -

R 0 (∞) = K G = P 0 and

R 1 (∞) = TO P =P 1 .

For steady state (t→∞) P i (t) = P i = const a system of algebraic equations with zero left-hand sides is compiled, since in this case dP i (t)/dt = 0. Then the system of algebraic equations has the form:

Because Kg there is a probability that the system will be operational at the moment t at t , then from the resulting system of equations is determined P 0 = kg., i.e. the probability of the element operation is equal to the stationary availability factor, and the probability of failure is equal to the forced downtime factor:

limP 0 (t) = Kg =μ /(λ+ μ ) = T/(T+ t v )

limP 1 (t) = Кп = λ /(λ+μ ) = t v /(T+ t v )

i.e., the same result was obtained as in the analysis of limit states using differential equations.

The method of differential equations can be used to calculate reliability indicators and non-recoverable objects (systems).

In this case, the inoperable states of the system are "absorbing" and the intensities μ exits from these states are excluded.

For a non-restorable object, the state graph looks like:

System of differential equations:

Under initial conditions: P 0 (0) = 1; P 1 (0) = 0 , using the Laplace transform of the probability of being in a working state, i.e., FBG to operating time t will be .