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TEST

Fundamentals of reliability theory and diagnostics

Exercise

Based on the results of testing products for reliability according to plan, the following initial data were obtained for assessing reliability indicators:

5 sample values of time to failure (unit: thousand hours): 4.5; 5.1; 6.3; 7.5; 9.7.

5 sample values of operating time before censoring (i.e. 5 products remained in working condition at the end of testing): 4.0; 5.0; 6.0; 8.0; 10.0.

Define:

Point estimate of mean time to failure;

With confidence probability, lower confidence limits and;

Draw the following graphs to scale:

distribution function;

probability of failure-free operation;

upper confidence limit;

lower confidence limit.

Introduction

The calculation part of the practical work contains an assessment of reliability indicators based on given statistical data.

Reliability indicator assessments are numerical values of indicators determined based on the results of observations of objects under operating conditions or special reliability tests.

When determining reliability indicators, two options are possible:

- the type of operating time distribution law is known;

- the type of operating time distribution law is not known.

In the first case, parametric assessment methods are used, in which the parameters of the distribution law included in the calculation formula of the indicator are first assessed, and then the reliability indicator is determined as a function of the estimated parameters of the distribution law.

In the second case, nonparametric methods are used, in which reliability indicators are assessed directly from experimental data.

1. Brief theoretical information

failsafe trust distribution point

Quantitative indicators of the reliability of rolling stock can be determined from representative statistical data on failures obtained during operation or as a result of special tests carried out taking into account the operating characteristics of the structure, the presence or absence of repairs and other factors.

The initial set of observation objects is called the general population. Based on the coverage of the population, there are 2 types of statistical observations: continuous and sample. Continuous observation, when every element of the population is studied, is associated with significant costs and time, and sometimes is not physically feasible at all. In such cases, they resort to selective observation, which is based on the selection from the general population of a certain representative part of it - a sample population, which is also called a sample. Based on the results of studying the characteristic in the sample population, a conclusion is made about the properties of the characteristic in the general population.

The sampling method can be used in two ways:

- simple random selection;

- random selection according to typical groups.

Dividing the sample population into typical groups (for example, by gondola car models, by years of construction, etc.) gives an increase in accuracy when estimating the characteristics of the entire population.

No matter how thoroughly the sample observation is carried out, the number of objects is always finite, and therefore the volume of experimental (statistical) data is always limited. With a limited amount of statistical material, only some estimates of reliability indicators can be obtained. Despite the fact that the true values of reliability indicators are not random, their estimates are always random (stochastic), which is associated with the randomness of the sample of objects from the general population.

When calculating an estimate, one usually tries to choose a method so that it is consistent, unbiased, and efficient. A consistent estimate is one that, with an increase in the number of observation objects, converges in probability to the true value of the indicator (condition 1).

An unbiased estimate is one whose mathematical expectation is equal to the true value of the reliability indicator (condition 2).

An estimate is called effective, the variance of which, compared to the dispersions of all other estimates, is the smallest (condition 3).

If conditions (2) and (3) are satisfied only when N tends to zero, then such estimates are called asymptotically unbiased and asymptotically efficient, respectively.

Consistency, unbiasedness and efficiency are qualitative characteristics of assessments. Conditions (1) - (3) allow us to write down only an approximate equality for a finite number of observation objects N

a~b(N)

Thus, the estimate of the reliability indicator in (N), calculated from a sample population of objects of volume N, is used as an approximate value of the reliability indicator for the entire population. This estimate is called a point estimate.

Given the probabilistic nature of reliability indicators and the significant scatter of statistical data on failures, when using point estimates of indicators instead of their true values, it is important to know what the limits of possible error are and what its probability is, that is, it is important to determine the accuracy and reliability of the estimates used. It is known that the quality of a point estimate is higher, the more statistical material it is obtained from. Meanwhile, the point estimate itself does not carry any information about the volume of data on which it was obtained. This determines the need for interval estimates of reliability indicators.

The initial data for assessing reliability indicators are determined by the observation plan. The initial data for the plan (N V Z) are:

- sample values of time to failure;

- sample values of operating time of machines that remained operational during the observation period.

The operating time of machines (products) that remained operational during testing is called the operating time before censoring.

Censoring (cutting off) on the right is an event leading to the termination of testing or operational observations of an object before the onset of failure (limit state).

Reasons for censoring are:

- different times of the beginning and (or) end of testing or operation of products;

- removal from testing or operation of some products for organizational reasons or due to failures of components, the reliability of which is not investigated;

- transfer of products from one mode of use to another during testing or operation;

- the need to assess the reliability before failure of all tested products.

Operating time before censoring is the operating time of the object from the start of testing to the onset of censoring. A sample whose elements are the values of time to failure and before censoring is called a censored sample.

A once censored sample is a censored sample in which the values of all times before censoring are equal to each other and not less than the longest time before failure. If the values of the operating time before censoring in the sample are not equal, then such a sample is repeatedly censored.

2. Estimation of reliability indicators using a nonparametric method

1 . We arrange the time to failure and the time to censoring in a general variation series in the order of non-decreasing operating time (the time before censoring is marked ): 4.0; 4.5; 5.0; 5.1; 6.0; 6.3; 7.5; 8.0; 9.7; 10.0.

2 . We calculate point estimates of the distribution function for operating time using the formula:

; ,

where is the number of serviceable products of the j-th failure in the variation series.

;

3. We calculate the point estimate of the average time to failure using the formula:

,

Where;

;

.

;

thousand hours

4. The point estimate of failure-free operation per thousand hours is determined using the formula:

,

Where;

.

;

5. We calculate point estimates using the formula:

.

;

.

6. Based on the calculated values, we construct graphs of the operating time distribution functions and reliability functions.

7. The lower confidence limit for the average time to failure is calculated using the formula:

,

where is the quantile of the normal distribution corresponding to the probability. Accepted according to the table depending on the confidence level.

According to the conditions of the task, confidence probability. We select the corresponding value from the table.

thousand hours

8 . We calculate the values of the upper confidence limit for the distribution function using the formula:

,

where is the quantile of the chi-squared distribution with the number of degrees of freedom. Accepted according to the table depending on the confidence level q.

.

The curly brackets in the last formula mean taking the integer part of the number enclosed in these brackets.

For;

For.

;

.

9. The values of the lower confidence limit of the probability of failure-free operation are determined by the formula:

.

;

.

10. The lower confidence limit of the probability of failure-free operation at a given operating time, thousand hours, is determined by the formula:

,

Where; .

.

Respectively

11 . Based on the calculated values, we construct graphs of the functions of the upper confidence limit and lower confidence limit as previously constructed models of point estimates and

Conclusion on the work done

When studying the results of reliability testing of products according to plan, the following reliability indicators were obtained:

- point estimate of average time to failure, thousand hours;

- point estimate of the probability of failure-free operation per thousand hours of operation;

- with confidence probability lower confidence limits thousand hours and;

Using the found values of the distribution function, the probability of failure-free operation, the upper confidence limit and the lower confidence limit, graphs were constructed.

Based on the calculations performed, it is possible to solve similar problems that engineers face in production (for example, when operating cars on the railway).

Bibliography

1. Chetyrkin E.M., Kalikhman I.L. Probability and statistics. M.: Finance and Statistics, 2012. - 320 p.

2. Reliability of technical systems: Handbook / Ed. I.A. Ushakova. - M.: Radio and communication, 2005. - 608 p.

3. Reliability of engineering products. A practical guide to standardization, confirmation and provision. M.: Publishing house of standards, 2012. - 328 p.

4. Guidelines. Reliability in technology. Methods for assessing reliability indicators based on experimental data. RD 50-690-89. Enter. P. 01.01.91, M.: Standards Publishing House, 2009. - 134 p. Group T51.

5. Bolyshev L.N., Smirnov N.V. Tables of mathematical statistics. M.: Nauka, 1983. - 416 p.

6. Kiselev S.N., Savoskin A.N., Ustich P.A., Zainetdinov R.I., Burchak G.P. Reliability of mechanical systems of railway transport. Tutorial. M.: MIIT, 2008-119 p.

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FUNDAMENTALS OF RELIABILITY THEORY

AND DIAGNOSTICS

Lecture course

Omsk – 2012

Ministry of Education and Science of the Russian Federation

Federal state budget educational

institution of higher professional education

"Siberian State Automobile and Highway Academy

(SibADI)"

A.N. Cheboksary

FUNDAMENTALS OF RELIABILITY THEORY

AND DIAGNOSTICS

Course of lectures Omsk SibADI 2012 UDC 629.113.004 BBK 39.311-06-5 Ch 34 Reviewer Ph.D. tech. Sciences, Associate Professor THEM. Knyazev The work was approved at a meeting of the department “Operation and Repair of Automobiles” of the Federal State Budgetary Educational Institution of Higher Professional Education SibADI as a course of lectures for students of all forms of study in specialties 190601 “Automobiles and Automotive Industry”, 190700 “Organization and Traffic Safety”, training areas 190600 “Operation of Transport and Technological Machines” and complexes."

Cheboksarov A.N. Fundamentals of reliability theory and diagnostics: a course of lectures / A.N. Cheboksarov. – Omsk: SibADI, 2012. – 76 p.

The basic concepts and indicators of reliability theory are considered. The mathematical foundations of the theory of reliability and the foundations of the reliability of complex systems are outlined. The basic theoretical principles of technical diagnostics of machines are given.

The course of lectures is intended for full-time, full-time accelerated, part-time and distance learning students of specialties 190601 “Automobiles and Automotive Industry”, 190700 “Organization and Traffic Safety”, areas of training 190600 “Operation of Transport and Technological Machines and Complexes”.

Table 4. Il. 25. Bibliography: 12 titles.

© FSBEI “SibADI”, Contents Introduction………………………………………….…………...……. 1. Basic concepts and indicators of reliability theory…….. 1.1. Reliability as a science…………………..……….………..… 1.2. History of the development of reliability theory……………..………… 1.3. Basic concepts of reliability……………...………..……… 1.4. Life cycle of an object……………………………...……… 1.5. Maintaining the reliability of the facility during operation......... 1.6. Main reliability indicators………………………..….. 1.6.1. Indicators for assessing reliability…………...…….

.….. 1.6.2.Indicators for assessing durability…………..……...….. 1.6.3.Indicators for assessing preservation…………..……...….. 1.6. 4. Indicators for assessing maintainability……..…..…… 1.6.5. Comprehensive reliability indicators………………….….. 1.7. Obtaining information about the reliability of machines……….......….. 1.8. Standardization of reliability indicators………..………....…. Questions for self-test…………………………….……......…. 2. Mathematical foundations of reliability………….……….….... 2.1. Mathematical apparatus for processing random variables…………………………………………………….. 2.2. Some laws of distribution of a random variable...... 2.2.1. Normal distribution…………………...…….……..... 2.2.2. Exponential distribution……………………..…... 2.2.3. Weibull distribution…………………………………..... Self-test questions………………………………………………………..…. 3. Fundamentals of reliability of complex systems…………….……..…... 3.1. Features of complex systems…………………………..……. 3.2. Structure of complex systems……………………………..……. 3.3. Features of calculating the reliability of complex systems……..….. 3.3.1. Calculation of system reliability when connecting its elements in series…………………………….………… 3.3.2. Calculation of system reliability when connecting its elements in parallel……………………………..….… 3.4. Reservation…...………………….…………………....…… Self-test questions…………………….………………..…. 4. Wear…………………………………….....……… 4.1. Types of friction………………………………………………………..……... 4.2. Types of wear……………………………………..……… 4.3. Wear characteristics……………………………………. 4.4. Methods for determining wear……………………………..……Self-test questions………………………………………………………...…. 5. Corrosion damage……………………………..…….. 5.1. Types of corrosion………………………………………….……… 5.2. Methods of combating corrosion…………………………………….. Questions for self-test…………………………………….…..…. 6. Technical diagnostics…………………………………..…. 6.1. Basic concepts of technical diagnostics……………..… 6.2. Tasks of technical diagnostics…………………………..… 6.3. Selection of diagnostic parameters……………………..….. 6.4. Patterns of changes in state parameters during operation of machines……………………….………….. 6.5. Methods and types of diagnosis……………………….…... 6.6. Diagnostic tools………………...……………..….... 6.7. Classification of sensors………………………..……….….… 6.8. Computer diagnostics of a car…………………….. 6.9. Standards in automotive diagnostics………………..….. 6.10. General requirements for technical diagnostic tools……………………………….……. Self-test questions…………………………..…….………. Bibliography………………………..……………. The purpose of teaching the discipline “Fundamentals of Reliability Theory and Diagnostics” is to develop in students a system of scientific knowledge and professional skills in using the fundamentals of reliability theory and diagnostics in relation to solving problems of technical operation of vehicles at all stages of their life cycle:

design, production, control, storage and operation.

The main objectives of the discipline “Fundamentals of Reliability Theory and Diagnostics” are:

– study of the basic definitions of the structure and content of the concepts of reliability and diagnostics;

– mastering methods for collecting and processing information about the reliability of vehicles in operation, methods for assessing the results obtained and their systematization;

– study of patterns of changes in the technical condition of products and the occurrence of failures, as well as factors affecting the reliability and physical processes of product failures;

– obtaining reliability indicators of the main systems and components of vehicles under real operating conditions and determining the optimal service life of rolling stock;

– mastering diagnostic methods and calculating diagnostic parameters;

– study of product quality management methods using international standards of the ISO 9000 series.

1. BASIC CONCEPTS AND INDICATORS OF THE THEORY

RELIABILITY

Reliability characterizes the quality of a technical product.

Quality is a set of properties that determine the suitability of a product for its intended use and its consumer properties.

Reliability is a complex property of a technical object, which consists in its ability to perform specified functions while maintaining its basic characteristics within established limits.

The concept of reliability includes reliability, durability, maintainability and safety.

The subject of reliability is the study of the reasons that cause failures of objects, the determination of the laws to which they obey, the development of methods for quantitative measurement of reliability, methods of calculation and testing, the development of ways and means of increasing reliability.

The object of research into reliability as a science is one or another technical means: a separate part, a machine unit, an assembly, a machine as a whole, a product, etc.

There are general reliability theory and applied reliability theory. The general theory of reliability has three components:

1. Mathematical theory of reliability. Defines mathematical laws that govern failures and methods for quantitatively measuring reliability, as well as engineering calculations of reliability indicators.

2. Statistical theory of reliability. Processing of statistical information about reliability. Statistical characteristics of reliability and failure patterns.

3. Physical theory of reliability. Study of physicochemical processes, physical causes of failures, the influence of aging and strength of materials on reliability.

Applied reliability theories are developed in a specific field of technology in relation to objects in this field. For example, there is a theory of reliability of control systems, a theory of reliability of electronic devices, a theory of machine reliability, etc.

Reliability is related to the efficiency (e.g., cost-effectiveness) of the technology. Insufficient reliability of a technical device results in:

– decreased productivity due to downtime due to breakdowns;

– reduction in the quality of the results of using a technical device due to deterioration of its technical characteristics due to malfunctions;

– costs for repairs of technical equipment;

– loss of regularity in obtaining results (for example, decreased regularity of transportation for vehicles);

– reduction in the level of safety of using a technical device.

1.2. History of the development of reliability theory Stage I. First stage.

It begins with the beginning of the appearance of the first technical devices (this is the end of the 19th century (approximately 1880)) and ends with the advent of electronics and automation, aviation and rocket and space technology (mid-20th century).

Already at the beginning of the century, scientists began to think about how to make any machine unbreakable. There was such a thing as a “margin” of safety. But by increasing the safety margin, the weight of the product also increases, which is not always acceptable. Experts began to look for ways to solve this problem.

The basis for solving such problems was the theory of probability and mathematical statistics. Based on these theories, already in the 30s.

The concept of failure was formulated as an excess of load over strength.

With the beginning of the development of aviation and the use of electronics and automation in it, the theory of reliability begins to develop rapidly.

Stage II. The stage of formation of reliability theory (1950 – 1960).

In 1950, the US Air Force organized the first group to study problems of electronic equipment reliability. The group found that the main reason for the failure of electronic equipment was the low reliability of its elements. We began to understand this, to study the influence of various operational factors on the proper operation of the elements. We collected rich statistical material, which became the basis of the theory of reliability.

Stage III. Stage of classical reliability theory (1960 – 1970).

In the 60-70s. space technology is emerging that requires increased reliability. In order to ensure the reliability of these products, they begin to analyze the product design, production technology and operating conditions.

At this stage, it was established that the causes of machine breakdowns can be detected and eliminated. The theory of diagnostics of complex systems begins to develop. New standards for machine reliability are emerging.

Stage IV. Stage of system reliability methods (from 1970 to the present).

At this stage, new reliability requirements were developed, laying the foundation for modern reliability systems and programs. Standard methods for carrying out activities related to ensuring reliability have been developed.

These techniques are divided into two main areas:

the first direction relates to potential reliability, which takes into account design (choice of material, safety factor, etc.) and technological (tightening tolerances, increasing surface cleanliness, etc.) methods of ensuring reliability;

the second direction is operational, which is aimed at ensuring operational reliability (stabilizing operating conditions, improving maintenance and repair methods, etc.).

Reliability uses the concept of an object. An object is characterized by quality. Reliability is a component indicator of the quality of an object. The higher the reliability of an object, the higher its quality.

During operation, an object may be in one of the following states (Fig. 1.1):

1) Serviceable condition - the condition of the object in which it meets all the requirements of regulatory, technical and (or) design documentation.

2) Faulty state - a state of an object in which it does not meet at least one of the requirements of regulatory and technical and (or) design documentation.

3) Operable state - the state of the object in which the values of all parameters characterizing the ability to perform specified functions comply with the requirements of regulatory technical and (or) design documentation.

4) Inoperative state - a state of an object in which the value of at least one parameter characterizing the ability to perform specified functions does not meet the requirements of regulatory, technical and (or) design documentation.

There are malfunctions, coverings, and wear on the tread that lead to a failure (crack in the metal structure of the frame, bending of the fan blade - Inoperable torus of the engine cooling system).

A special case of an inoperative state is Fig. 1.1. The basic technical diagram shows the limit state. states: 1 – damage; 2 – refusal;

Limit state – 3 – repair; 4 – transition to a limiting state, in which the further operation of the object is unacceptable or impractical due to the presence of a critical state; III – a minor defect is different, or restoration of an operational state is impossible or impractical.

The transition of an object to a limiting state entails a temporary or permanent cessation of operation of the object, that is, the object must be taken out of service, sent for repairs or decommissioned. The limit state criteria are established in the regulatory and technical documentation.

Damage is an event consisting in a violation of the serviceable state of an object while maintaining a serviceable state.

Failure is an event consisting of a violation of the operational state of an object.

Restoration (repair) – returning an object to a working condition.

Damage and failure criteria are established in regulatory technical and (or) design documentation.

The classification of failures is given in table. 1.1.

II. Dependency III. Nature of occurrence IV. Nature of detection V. Cause of occurrence Dependent failure is a failure caused by other failures.

Sudden failure – characterized by a sharp change in one or more specified parameters of an object. An example of a sudden failure is a malfunction of the ignition system or engine power system.

Gradual failure – characterized by a gradual change in one or more specified parameters of the object. A typical example of a gradual failure is the malfunction of the brakes as a result of wear of the friction elements.

Explicit failure is a failure detected visually or by standard methods and means of control and diagnostics when preparing an object for use or during its intended use.

Latent failure is a failure that is not detected visually or by standard methods and means of monitoring and diagnostics, but is detected during maintenance or special diagnostic methods.

Depending on the method of eliminating the failure, all objects are non-repairable (non-recoverable).

Repairable objects include objects that, when a failure occurs, are repaired and, after restoration of functionality, put back into operation.

Non-repairable objects (elements) are replaced after a failure occurs. Such elements include most asbestos and rubber products (brake linings, clutch disc linings, gaskets, cuffs), some electrical products (lamps, fuses, spark plugs), wear parts that ensure operational safety (liners and pins of steering rod joints, pivot bushings connections). Non-repairable machine elements also include rolling bearings, axles, pins, and fasteners.

Restoring the listed elements is not economically feasible, since repair costs are quite high, and the durability provided is significantly lower than that of new parts.

An object is characterized by a life cycle. The life cycle of an object consists of a number of stages: design of the object, manufacturing of the object, operation of the object. Each of these life cycle stages affects the reliability of the product.

At the design stage of an object, the foundations for its reliability are laid. The reliability of an object is affected by:

– selection of materials (strength of materials, wear resistance of materials);

– safety margins of parts and the structure as a whole;

– ease of assembly and disassembly (determines the complexity of subsequent repairs);

– mechanical and thermal stress of structural elements;

– redundancy of the most important or least reliable elements and other measures.

At the manufacturing stage, reliability is determined by the choice of production technology, compliance with technological tolerances, the quality of processing of mating surfaces, the quality of materials used, and the thoroughness of assembly and adjustment.

At the design and manufacturing stage, design and technological factors affecting the reliability of the object are determined. The effect of these factors is revealed at the stage of operation of the facility. In addition, at this stage of an object’s life cycle, operational factors also affect its reliability.

Operation has a decisive influence on the reliability of objects, especially complex ones. The reliability of the object during operation is ensured by:

– compliance with operating conditions and modes (lubrication, load conditions, temperature conditions, etc.);

– carrying out periodic maintenance in order to identify and eliminate emerging problems and maintain the facility in working condition;

– systematic diagnosis of the condition of the object, identification and prevention of failures, reduction of the harmful consequences of failures;

– carrying out preventive restoration repairs.

The main reason for the decrease in reliability during operation is wear and aging of object components. Wear leads to changes in size, malfunction (due to deterioration of lubrication conditions, for example), breakdowns, decreased strength, etc. Aging leads to changes in the physical and mechanical properties of materials, leading to breakdowns or failures.

Operating conditions are set in such a way as to minimize wear and aging: for example, wear increases under conditions of shortage or poor quality of lubricant. Aging increases when temperature conditions exceed acceptable limits (for example, sealing gaskets, valves, etc.).

The reliability of an object at the operating stage can be illustrated by a graph of the typical dependence of the failure rate of an object on the operating time, presented in Fig. 1.2.

Rice. 1.2. Dependence of failure rate on operating time: 1 – failure rate (t); 2 – aging curve; I – running-in period; II – period of normal operation; III – wear period; PS – limit state During the running-in period tп, reliability is, first of all, determined by design and technological factors, which leads to an increased failure rate. As these factors are identified and eliminated, the reliability of the object is brought to a nominal level, which is maintained over a long period of normal operation.

During operation, manifestations of wear and fatigue accumulate in an object, the intensity of which increases with increasing service life of the object (increasing curve 2 in Fig. 1.2). There begins a period of intense wear and tear of the object, which ends with its reaching a limiting state and decommissioning.

Annual operating costs are characterized by graphs (Fig. 1.3).

Rice. 1.3. Dependence of operating costs on operating time: 1 – operating costs; 2 – costs From the graphs it is clear that there is an optimal service life of the facility, at which the total operating costs are minimal. Long-term operation, significantly exceeding the optimal period, is economically unprofitable.

1.5. Maintaining the reliability of an object during operation Maintaining the required level of reliability of technical objects during operation is carried out through a set of organizational and technical measures. This includes periodic maintenance, preventative and remedial repairs. Periodic maintenance is aimed at timely adjustments, eliminating the causes of failures, and early detection of failures.

Periodic maintenance is carried out within the established time limits and to the established extent. The task of any maintenance is to check controlled parameters, adjust if necessary, identify and eliminate faults, and replace elements provided for in the operational documentation.

The procedure for performing simple work is determined by the maintenance instructions, and the procedure for performing complex work is determined by technological maps.

In the process of technical maintenance, diagnostics of the condition of the operated object (to one extent or another) is usually carried out.

Diagnostics consists of monitoring the condition of an object in order to identify and prevent failures. Diagnostics are carried out using diagnostic monitoring tools, which can be built-in or external. Built-in tools allow for continuous monitoring. Periodic monitoring is carried out using external means.

As a result of diagnostics, deviations in object parameters and the causes of these deviations are identified. The specific location of the malfunction is determined. The problem of predicting the state of the object is solved and a decision is made on its further operation.

An object is considered operational if its state allows it to perform the functions assigned to it. If during operation the characteristics of an object or its structure have changed unacceptably, then they say that a malfunction has occurred in the object. The occurrence of a malfunction cannot be identified with the loss of the object’s operability. However, a faulty object will always have a fault.

To restore the reliability indicators of an object when they decrease, preventive and restorative repairs are carried out.

Restorative repairs serve to restore the functionality of an object after a failure and maintain a given level of its reliability by replacing parts and assemblies that have lost their level of reliability or have failed.

The number of repairs is determined by economic feasibility. A typical dependence of the probability of failure-free operation of a repaired object on the operating time is shown in Fig. 1.4.

Rice. 1.4. Dependence of the probability of failure-free operation of a repaired object on the operating time:

P – probability of failure-free operation of the facility;

Pmin – minimum acceptable level of reliability;

N is the number of elements of the object that are replaced during repair. The next repair does not allow achieving the initial level of reliability of the object, and the service life of the object after this repair will be less than after the previous repair (t3 t2 t1). Thus, the effectiveness of each subsequent repair is reduced, which entails the need to limit the total number of repairs of the facility.

1.6. Main indicators of reliability In accordance with GOST 27.002, reliability is the property of an object to maintain over time, within established limits, the values of all parameters characterizing the ability to perform the required functions.

This standard specifies both single reliability indicators, each of which characterizes a separate aspect of reliability (failure-free operation, durability, storability or maintainability), and complex reliability indicators, which simultaneously characterize several reliability properties.

1.6.1. Indicators for assessing reliability Reliability is the property of an object to continuously maintain an operational state for some time or operating time.

Operating time means the duration of operation of the machine, expressed:

– for machines in general – in time (hours);

– for road transport – in kilometers of vehicle mileage;

– for aviation – in aircraft flight hours;

– for agricultural machinery – in hectares of conditional plowing;

– for engines – in engine hours, etc.

To assess reliability, the following indicators are used:

1. The probability of failure-free operation is the probability that, within a given operating time, an object failure does not occur.

The probability of failure-free operation varies from 0 to 1.

where is the number of objects operational at the initial time; n(t) – the number of objects that failed at time t from the start of testing or operation.

The probability of failure-free operation P of an object is related to the probability of failure F by the following relationship:

The probability of failure-free operation decreases with increasing operating time or operating time of the object. The dependences of the probability of failure-free operation P(t) and the probability of failure F(t) on the operating time t are presented in Fig. 1.5.

Rice. 1.5. Dependencies of the probability of failure-free At the initial moment of time for an operational object, the probability of its failure-free operation is equal to one (100%). As the object operates, this probability decreases and tends to zero. The probability of an object failure, on the contrary, increases with increasing service life or operating time.

2. Mean time to failure (mean time between failures) and mean time between failures.

Average time to failure is the mathematical expectation of the operating time of an object before the first failure. This metric is often referred to as mean time between failures.

where ti is the time to failure of the i-th object; N – number of objects.

Mean time between failures is the mathematical expectation of the time between adjacent failures of an object.

3. Failure probability density (failure frequency) - the ratio of the number of failed products per unit of time to the initial number under supervision, provided that the failed products are not restored or replaced with new ones.

where n(t) is the number of failures in the operating interval under consideration;

N is the total number of products under supervision; t is the value of the operating interval under consideration.

4. Failure rate is the conditional probability density of the occurrence of a failure of an object, determined provided that the failure did not occur before the considered point in time.

In other words, this is the ratio of the number of failed products per unit of time to the average number of fail-safe ones for a given period of time, provided that the failed products are not restored or replaced with new ones.

The failure rate is estimated using the following formula:

where f(t) – failure rate; P(t) – probability of failure-free operation;

n(t) – number of failed products during the time from t to t + t; t – operating interval under consideration; ср – average number of trouble-free working products:

where N(t) is the number of fail-safe products at the beginning of the operating interval under consideration; N(t + t) is the number of trouble-free products at the end of the operating interval.

1.6.2. Indicators for assessing durability Durability is the property of an object to maintain an operational state until a limit state occurs with an established maintenance and repair system.

The durability of machines is laid down during their design and construction, ensured during the production process and maintained during operation.

Resource – the operating time of a machine from the start of operation or its resumption after repair to the limit state.

Service life is the calendar duration of operation of the machine from the start of its operation or resumption after repair, until the onset of the limit state.

To assess durability, the following indicators are used:

1. Average resource – mathematical expectation of the resource where tpi – resource of the i-th object; N – number of objects.

2. Gamma-percentage resource - operating time during which the object will not reach the limit state with a given probability, expressed as a percentage.

To calculate the indicator, probability formula 3 is used. Average service life is the mathematical expectation of the service life where tслi is the service life of the i-th object.

4. Gamma-percentage service life is the calendar duration of operation during which the object does not reach the limiting state with probability expressed as a percentage.

1.6.3. Indicators for assessing storability Storability is the property of an object to retain, within specified limits, the values of parameters characterizing the ability of the object to perform the required functions during and after storage and (or) transportation.

To assess preservation, the following indicators are used:

1. Average shelf life is the mathematical expectation of the shelf life of an object.

2. Gamma-percentage shelf life - the calendar duration of storage and (or) transportation of an object, during and after which the indicators of reliability, durability and maintainability of the object will not go beyond the established limits with a probability expressed as a percentage.

Storability indicators essentially correspond to durability indicators and are determined using the same formulas.

1.6.4. Indicators for assessing maintainability Maintainability is a property of an object, which consists in its adaptability to maintaining and restoring an operational state through maintenance and repair.

Recovery time is the duration of restoration of an object's operational state.

Recovery time is equal to the sum of the time spent on finding and eliminating the failure, as well as on carrying out the necessary debugging and checks to ensure that the object is restored to operability.

To assess maintainability, the following indicators are used:

1. Average recovery time is the mathematical expectation of the object’s recovery time where tвi is the recovery time of the i-th failure of the object; N is the number of failures over a given period of testing or operation.

2. Probability of restoration of a working state – the probability that the time to restore the working state of an object will not exceed a specified value. For most mechanical engineering objects, the probability of recovery obeys an exponential distribution law where is the failure rate (assumed constant).

1.6.5. Complex indicators of reliability Each of the indicators described above allows us to evaluate only one of the aspects of reliability - one of the properties of the reliability of an object.

For a more complete assessment of reliability, complex indicators are used that allow simultaneous assessment of several important properties of an object.

1. Availability coefficient Kg – the probability that an object will be operational at any point in time, except for planned periods during which the object is not intended to be used for its intended purpose.

where To is the average mean time between failures; TV is the average recovery time of an object after a failure.

2. Technical utilization coefficient - the ratio of the mathematical expectation of the total time an object remains in working condition for a certain period of operation to the mathematical expectation of the total time the object remains in working condition and downtime due to maintenance and repair for the same period of operation.

where TR, TTO is the total duration of machine downtime for repairs and maintenance.

For cars, the main indicators of durability are the service life before replacement (before a certain type of repair) or write-off, gamma-percentage service life; the main indicator of failure-free operation is the time between failures of a certain complexity group (mean time between failures); the main indicators of maintainability are the specific labor intensity of maintenance, the specific labor intensity of current repairs and the specific total labor intensity of maintenance and routine repairs.

1.7. Obtaining information about the reliability of machines In order to determine the reliability of any machine, it is necessary to have information about the failures of its parts, assemblies, assemblies and the machine itself as a whole.

The collection of information about machine failures is carried out by:

– machine development organizations;

– machine manufacturers;

– operating and repair enterprises.

Development organizations (design institutes) collect and process information about the reliability of prototype machines by conducting special tests.

Manufacturing enterprises (machine-building plants) collect and process primary information about the reliability of mass-produced products and analyze the causes of machine failures. They collect information based on special factory and operational tests.

Operation and repair organizations collect primary information about the reliability of machines in operation.

The main source of information about reliability, especially of transport vehicles, is testing.

In road transport, the following types of tests are distinguished (Fig. 1.6):

1. Factory (resource) tests – tests of prototypes or first production samples. These tests are:

a) finishing;

b) suitability for mass production;

c) control;

d) acceptance documents;

e) research.

The purpose of development tests is to evaluate the impact on reliability of changes made during development of the design and production technology.

Tests for suitability for mass production determine the admissibility of vehicles for mass production based on their reliability.

Control tests are used to check whether mass-produced vehicles meet the established reliability standards.

Acceptance tests determine the compliance of a given batch of cars with the requirements of technical specifications and the possibility of its acceptance.

The purpose of research tests is to determine the endurance limit of cars, establish the law of resource distribution, study the dynamics of the wear process, and compare the resources of cars.

Based on the nature of the factory tests, they are divided into:

– for benches;

– polygon;

– road.

Bench tests are carried out on special stands that allow simulating various test conditions.

Test sites are tests of vehicles at special testing sites with roads with different characteristics.

Road tests are usually carried out under real operating conditions, but in different climatic zones.

In the Russian Federation, the main field tests are carried out at the NAMI Central Research Site. The landfill facilities include:

– ring express concrete road;

– straight road for dynamometer tests;

– ring dirt road;

– cobblestone road;

– special test roads.

2. Operational tests – tests of production vehicles under real operating conditions. This is basically a road test. Their goal is to obtain reliable data on the operational reliability of cars based on systematic observations.

Most operational tests are carried out at special motor transport enterprises located in various climatic zones. These tests provide the most objective information about the reliability of the car.

Finishing For suitability Testing for production Control Acceptance Research Fig. 1.6. Classification of test types Information is collected on controlled batches of cars. In this case, not only failures and malfunctions are recorded, but also various types of impacts on the vehicle (maintenance, routine repairs); vehicle operating conditions (cargo transported, length of travel, percentage of traffic on various types of roads). The information collected in this way is directly processed at the enterprise or sent to manufacturing plants in the form of special inquiry certificates, which are analyzed, systematized and statistically processed.

All types of tests are divided according to duration:

– to normal (full);

– accelerated;

– abbreviated (incomplete).

Normal (full) tests are carried out until failure of all tested vehicles (components, assemblies) placed for testing. These tests represent the full sample.

Accelerated - carried out until each of the N cars put for testing reaches a predetermined operating time or until a certain number of n cars (n N) fails.

Abbreviated (incomplete) tests are tests when, by the time the observations were stopped, n out of N vehicles delivered for testing had failed, and the rest were operational and had different operating hours.

The collection of information on machine reliability is carried out in accordance with the requirements of industry standard and technical documentation.

Information on machine reliability must meet the following requirements:

1) completeness of information, which means the availability of all information necessary for conducting reliability assessment and analysis;

2) reliability of information, i.e. all failure reports must be accurate;

3) timeliness of information allows you to quickly eliminate the causes of failures and take measures to eliminate identified deficiencies;

4) continuity of information allows you to compare the results of calculations obtained in the first and subsequent periods of operation and eliminates errors.

1.8. Standardization of reliability indicators In order to create highly reliable objects, it is necessary to standardize reliability - establish the nomenclature and quantitative values of the main reliability indicators of the elements of the object.

The range of reliability indicators is selected depending on the class of products, operating modes, nature of failures and their consequences. The choice of reliability indicators can be determined by the customer.

All products are divided into the following classes:

– non-repairable and non-restorable general purpose products. Components of products that cannot be restored on site and cannot be repaired (for example, bearings, hoses, toners, fasteners, radio components, etc.), as well as non-repairable products for independent functional purposes (for example, electric lamps, control devices, etc.);

– refurbished products undergoing scheduled maintenance, routine and medium repairs, as well as products undergoing major repairs;

– products designed to perform short-term one-time or periodic tasks.

Product operating modes can be as follows:

– continuous, when the product operates continuously for a certain time;

– cyclic, when the product operates at a specified frequency for a certain time;

– operational, when an indefinite period of downtime is replaced by a period of work of a given duration.

Usually the probability of failure-free operation P(t) is normalized with an estimate of the resource Tp during which it is regulated. The value of Tr must be consistent with the structure and frequency of repair work and maintenance, and the permissible probability of failure-free operation is a measure of the danger of the consequences of failure.

The gradation of products by reliability classes is presented in Table. 1.2.

The P(t) values are specified for a certain period of operation of the Tr, subject to strict regulation and compliance with operating modes and operating conditions.

Class zero includes low-critical parts and assemblies, the failure of which remains with virtually no consequences. For them, a good indicator of reliability may be the average service life, time between failures or a failure flow parameter.

Classes from the first to the fourth are characterized by increased requirements for trouble-free operation (the class number corresponds to the number of nines after the decimal point). The fifth class includes highly reliable products, the failure of which within a given period is unacceptable.

In the automotive industry, the values of the availability coefficient Kg, the average time in working condition Tr, the time to first failure and the average time between failures are usually set.

For transport vehicles, it is very important to identify and quantify failures that affect the safety of their operation. According to the American FMECA methodology, system safety is assessed by the probability of failure-free operation, taking into account two parallel indicators: the category of consequences and the level of danger.

Class I – failure does not lead to injury to personnel;

Class II – failure leads to injury to personnel;

Class III – failure results in serious injury or death;

Class IV – Failure results in serious injury or death to a group of people.

1. Explain the concepts of quality, reliability, subject, object of reliability, general theory of reliability, applied theory of reliability.

2. Stages of development of reliability theory.

3. Define the main states and events in reliability.

4. Give a classification of failures.

5. What is the difference between refurbished and non-refurbished products?

6. What is the curve of changes in failure rate over time and the curve of changes in operating costs from product operating time over time?

9. Define the main indicators of reliability, non-failure operation, durability, maintainability and storability.

11. Give definitions of indicators for assessing failure-free operation - probability of failure-free operation and probability of failure, failure flow parameter, average time between failures, average time to failure, gamma-percentage time to failure, failure rate. What are their units of measurement?

12. Define indicators for assessing durability - technical resource, service life, gamma-percentage resource and service life. What are their units of measurement?

13. What is the difference between technical resource and product service life?

14. Define indicators for assessing persistence - average and gamma-percentage shelf life.

15. Define indicators for assessing maintainability - recovery time and average time to restore functionality, the probability of restoring functionality within a given time frame, the intensity of recovery.

16. Give definitions of complex reliability indicators - technical utilization coefficient, availability coefficient.

17. List the main types of testing of technical objects.

18. Basic requirements for information about machine reliability.

19. List the main methods for normalizing reliability indicators.

20. Explain the gradation of products according to reliability classes.

22. What is the failure hazard level?

2. MATHEMATICAL FOUNDATIONS OF RELIABILITY

2.1. Mathematical apparatus for processing random variables The reliability of objects is violated by emerging failures. Failures are treated as random events. To quantify reliability, methods of probability theory and mathematical statistics are used.

Reliability indicators can be determined:

– analytically based on a mathematical model – mathematical determination of reliability;

– as a result of processing experimental data – statistical determination of the reliability indicator.

The moment of failure occurrence and the frequency of failure occurrence are random values. Therefore, the basic methods for reliability theory are the methods of probability theory and mathematical statistics.

A random variable is a quantity that, as a result of experiment, takes on one, unknown value in advance, depending on random reasons. Random variables can be discrete or continuous.

As is known from probability theory and mathematical statistics, the general characteristics of random variables are:

1. Arithmetic mean.

where xi is the realization of a random variable in each observation; n – number of observations.

2. Scope. The concept of range in statistics theory is used as a measure of the dispersion of a random variable.

where xmax is the maximum value of the random variable; xmin – minimum value of the random variable.

3. The standard deviation is also a measure of the dispersion of a random variable.

4. The coefficient of variation also characterizes the dispersion of a random variable taking into account the average value. The coefficient of variation is determined by the formula. There are random variables with small variation (V0.1), medium variation (0.1V0.33) and large variation (V0.33). If the coefficient of variation is V0.33, then the random variable obeys the normal distribution law. If the coefficient of variation is 0.33V1, then it follows the Weibull distribution. If the coefficient of variation V=1, then – to an equiprobable distribution.

In the theory and practice of reliability, the following distribution laws are most often used: normal, logarithmically normal, Weibull, exponential.

The distribution law of a random variable is a relationship that establishes a connection between the possible values of a random variable and their corresponding probabilities.

To characterize the distribution law of a random variable, the following functions are used.

1. The distribution function of a random variable is a function F(x), which determines the probability that the random variable X will take a value less than or equal to x as a result of testing:

The distribution function of a random variable can be represented by a graph (Fig. 2.1).

Rice. 2.1. Distribution function of a random variable 2. Probability density of a random variable The probability density characterizes the probability that a random variable will take a specific value x (Fig. 2.2).

Rice. 2.2. Probability distribution density An experimental estimate of the probability density of a random variable is the histogram of the distribution of the random variable (Fig. 2.3).

Rice. 2.3. Histogram of distribution of a random variable A histogram shows the dependence of the number of observed values of a random variable in a certain interval of values on the boundaries of these intervals. Using the histogram, you can approximately judge the distribution density of a random variable.

When constructing a histogram in a sample of a random variable x from n values, the largest xmax and smallest xmin values are determined.

The range of changes in the value of R is divided into m equal intervals. Then the number of observed values of the random variable ni falling into each i-th interval is counted.

2.2. Some laws of distribution of a random variable The law of normal distribution is fundamental in mathematical statistics. It is formed when, during the process under study, its result is influenced by a relatively large number of independent factors, each of which, individually, has only a minor effect compared to the total influence of all the others.

The distribution density (failure rate) under the normal law is determined by the formula The distribution function (failure probability) of this law is found by the formula The reliability function (probability of failure-free operation) is opposite to the distribution function The failure rate is calculated by the formula Graphs of the main reliability characteristics under the normal law are shown in Fig. 2.4.

Rice. 2.4. The reliability characteristics of cars under more than 40% of various random phenomena associated with the operation of cars are described by the normal law:

– clearances in bearings due to wear;

– gaps in the main gear engagement;

– gaps between the brake drum and pads;

– frequency of first failures of springs and engine;

– frequency of TO-1 and TO-2, as well as the time for performing various operations.

2.2.2. Exponential distribution The law of exponential distribution has found wide application, especially in technology. The main distinguishing feature of this law is that the probability of failure-free operation does not depend on how long the product has worked since the beginning of operation. The law does not take into account gradual changes in technical condition parameters, but considers the so-called “ageless” elements and their failures. As a rule, this law describes the reliability of a product during its normal operation, when gradual failures do not yet appear and reliability is characterized only by sudden failures. These failures are caused by an unfavorable combination of various factors and therefore have a constant intensity. The exponential distribution is often called the fundamental law of reliability.

The distribution density (failure rate) under an exponential law is determined by the formula The probability of failure-free operation under an exponential law is expressed by where is the failure rate.

The failure rate for the exponential distribution is a constant value.

MTBF is found using the formula: With the exponential law, the standard deviation and the coefficient of variation are calculated as follows:

Graphs of the main reliability characteristics under the exponential law are shown in Fig. 2.5.

Rice. 2.5. Characteristics of machine reliability at The exponential law describes the failure of the following parameters quite well:

– operating time to failure of many non-repairable elements of radio-electronic equipment;

– operating time between adjacent failures with the simplest flow of failures (after the end of the running-in period);

– recovery time after failures, etc.

The Weibull distribution is universal, since when the parameters change, it can describe almost any process: normal distribution, lognormal, exponential.

The distribution density (failure rate) under the Weibull distribution is determined by the formula where is the scale parameter; – form parameter.

The probability of failure-free operation under the Weibull distribution law is expressed by the Failure rate is determined by the formula In Fig. Figure 2.6 shows reliability graphs for the Weibull distribution.

Rice. 2.6. Characteristics of vehicle reliability under the Weibull distribution law describes the failures of many components and parts of vehicles:

– rolling bearings;

– steering joints, cardan transmission;

– destruction of axle shafts.

1. Define the scattering characteristics of random distributions - mean value, standard deviation and coefficient of variation.

2. Give the concept and explain the purpose of the laws of distribution of random variables.

3. In what cases in practice is it advisable to use the normal distribution, what is the form of its density curves and distribution function?

4. In what cases in practice is it advisable to use exponential distribution, what is the form of its density curves and distribution function?

5. In what cases in practice is it advisable to use the Weibull distribution, what is the form of its density curves and distribution function?

6. What is the concept and methodology for constructing a histogram and an empirical distribution curve?

3. FUNDAMENTALS OF RELIABILITY OF COMPLEX SYSTEMS

A complex system is understood as an object designed to perform specified functions, which can be divided into elements, each of which also performs certain functions and interacts with other elements of the system.

The concept of a complex system is relative. It can be applied both to individual components and mechanisms (engine, fuel supply system to the engine), and to the machine itself (machine tool, tractor, car, airplane).

1. A complex machine consists of a large number of elements, each of which has its own reliability characteristics.

Example: a car consists of 15–18 thousand parts, each of which has its own reliability characteristics.

2. Not all elements have the same effect on the reliability of the machine.

Many of them affect only the effectiveness of its work, and not its failure. The degree of influence of each element on the reliability of the machine depends on many factors, such as: the purpose of the element, the nature of the interaction of the element with other elements of the machine, the structure of the machine, the type of connections between the elements.

For example: a malfunction of the car's power system can cause excessive fuel consumption, i.e. malfunction, and failure of the ignition system can lead to failure of the entire vehicle.

3. Each instance of a complex machine has individual features, because slight variations in the properties of individual machine elements affect the output parameters of the machine itself. The more complex the machine, the more individual features it has.

When analyzing the reliability of complex machines, they are divided into elements (links) in order to first consider the parameters and characteristics of the elements, and then evaluate the performance of the entire machine.

Theoretically, any complex machine can be conditionally divided into a large number of elements, understanding an element as a unit, assembly or part.

By element we mean an integral part of a complex machine, which can be characterized by independent input and output parameters.

When analyzing the reliability of a complex product, it is advisable to divide all its elements and parts into the following groups:

1. Elements whose performance remains virtually unchanged over their service life. For a car, this is its frame, body parts, lightly loaded elements with a large margin of safety.

2. Elements whose performance changes during the service life of the machine. These elements, in turn, are divided into:

2.1. Not limiting the reliability of the machine. The service life of such elements is comparable to the service life of the machine itself.

2.2. Limiting machine reliability. The service life of such elements is less than the service life of the machine.

2.3. Reliability critical. The service life of such elements is not very long, from 1 to 20% of the service life of the machine itself.

In relation to a car, the number of these elements is distributed as follows (Table 3.1).

Element number From the standpoint of reliability theory, the following structures of complex machines can be (Fig. 3.1):

1) dismembered - in which the reliability of individual elements can be determined in advance, since the failure of an element can be considered as an independent event;

2) related - in which the failure of elements is a dependent event associated with a change in the output parameters of the entire machine;

3) combined – consisting of subsystems with a related structure and with independent formation of reliability indicators for each of the subsystems.

A transport vehicle as a complex system is characterized by a combined structure, when the reliability of individual subsystems (units, components) can be considered independently.

The connection of elements in a complex machine can be serial, parallel and mixed (combined).

In the design of a car there are all types of connections, examples of which are shown in Fig. 3.2.

Rice. 3.2. Types of connections of elements in a car structure:

a) sequential; b) parallel; c) combined 3.3. Features of calculating the reliability of complex systems 3.3.1. Calculation of system reliability with sequential The most typical case is when the failure of one element disables the entire system, as is the case with a sequential connection of elements (Fig. 3.2, a).

For example, most machine drives and transmission mechanisms obey this condition. So, if any gear, bearing, coupling, etc. in a machine drive fails, then the entire drive will stop functioning. In this case, the individual elements do not necessarily have to be connected in series. For example, bearings on a gearbox shaft work structurally in parallel with each other, but failure of any one of them leads to system failure.

Probability of failure-free operation of a system with a series connection of elements. The formula shows that even if a complex machine consists of elements of high reliability, then in general it has low reliability due to the presence of a large number of elements in its design connected in series.

In the design of a car, elements are mainly connected in series. In this case, the failure of any element causes the failure of the car itself.

An example of a calculation from the field of automobile transport: for a car unit consisting of four series-connected elements, the probability of failure-free operation of the elements for a certain operating time is P1 = 0.98; P2 = 0.65; P3 = 0.88 and P4 = 0.57. In this case, the probability of failure-free operation for the same operating time of the entire unit is equal to Рс = 0.98·0.65·0.88·0.57 = 0.32, i.e. very, very low.

In other words, the reliability of a car with elements connected in series is lower than the reliability of its weakest link.

Therefore, as the design of a car, its units and systems becomes more complex, one of the manifestations of which is an increase in the number of elements in the system, the requirements for the reliability of each element and their uniform strength increase sharply.

3.3.2. Calculation of system reliability with parallel connection When connecting elements in parallel, the probability of failure-free operation of the system For example: if the probability of failure-free operation of each element is P = 0.9, and the number of elements is three (n = 3), then P(t) = 1-(0, 1)3 = 0.999. Thus, the probability of failure-free operation of the system increases sharply and it becomes possible to create reliable systems from unreliable elements.

Parallel connection of elements in complex systems increases its reliability.

To increase the reliability of complex systems, structural redundancy is often used, that is, the introduction into the structure of an object of additional elements that perform the functions of the main elements in the event of their failure.

The classification of various reservation methods is carried out according to the following criteria:

1. According to the reserve switching scheme:

1.1. General reservation, in which the object as a whole is reserved.

1.2. Separate reservation, in which individual elements or their groups are reserved.

1.3. Mixed reservation, in which different types of reservation are combined in one object.

2. According to the method of switching on the reserve:

2.1. Permanent redundancy – without rebuilding the structure of an object when a failure of its element occurs.

2.2. Dynamic redundancy, in which when an element fails, the circuit structure is rebuilt. In turn, it is divided:

– for redundancy by replacement, in which the functions of the main element are transferred to the backup one only after the failure of the main one;

– sliding reservation, in which several main elements are reserved by one or more reserve ones, each of which can replace any main one (i.e. the groups of main and reserve elements are identical).

3. According to the reserve status:

3.1. Loaded (hot) backup, in which backup elements (or one of them) are constantly connected to the main ones and are in the same operating mode as them; it is used when it is not allowed to interrupt the functioning of the system while switching a failed element to a backup one.

3.2. Lightweight redundancy, in which the backup elements (at least one of them) are in a less loaded mode compared to the main ones, and the probability of their failure during this period is low.

3.3. Unloaded (cold) redundancy, in which the backup elements are in an unloaded mode before they begin to perform functions. In this case, an appropriate device is required to enable the reserve. Failure of unloaded backup elements before switching on in place of the main element is impossible.

1. Explain the concept of a complex system and its features from the standpoint of reliability.

2. List four groups of elements of complex systems.

3. Explain the differences between the main types of structures of complex systems - dissected, connected and combined.

4. Explain the calculation of circuit reliability of complex systems when connecting elements in series.

5. Explain the calculation of circuit reliability of complex systems with parallel connection of elements.

6. Explain the term structural redundancy.

7. List the types of redundancy depending on the scheme for switching on the reserve.

8. List the types of reservation depending on the method of inclusion of the reserve.

9. List the types of reservation depending on the state of the reserve.

From 80 to 90% of moving machine interfaces fail due to wear. At the same time, the efficiency, accuracy, efficiency, reliability and durability of machines are reduced. The process of interaction of surfaces during their relative motion is studied by such a scientific and technical discipline as tribology, which combines the problems of friction, wear and lubrication.

There are four types of friction:

1. Dry friction occurs in the absence of lubrication and contamination between the rubbing surfaces. Typically, dry friction is accompanied by abrupt movement of surfaces.

2. Boundary friction is observed in the case when the surfaces of the rubbing bodies are separated by a layer of lubricant with a thickness of 0.1 microns to the thickness of one molecule, which is called boundary. Its presence reduces friction forces from two to ten times compared to dry friction and reduces wear of mating surfaces by hundreds of times.

3. Semi-dry friction is mixed friction, when on the contact area of the bodies the friction is boundary in places, and dry in the rest of the area.

4. Fluid friction is characterized by the fact that the rubbing surfaces are completely separated by a thick layer of lubricant. Lubricant layers located at a distance of more than 0.5 microns from the surface are able to move freely one relative to the other.

In liquid friction, the resistance to movement consists of the resistance to sliding of the lubricant layers relative to each other along the thickness of the lubricating layer and depends on the viscosity of the lubricating fluid.

This mode is characterized by a very low friction coefficient and is optimal for the friction unit in terms of its wear resistance.

It should be noted that sometimes different types of friction are observed in the same mechanism. For example, in an internal combustion engine, the cylinder walls in the lower part are abundantly lubricated, as a result of which, when the piston moves mid-stroke, the friction of the rings and the piston on the cylinder wall approaches liquid friction.

When the piston moves near top dead center (especially during the intake stroke), the lubrication conditions for the rings and piston deteriorate sharply, since the oil film remaining on the cylinder walls undergoes changes under the influence of the high temperature of the combustion products. The upper part of the cylinder is especially poorly lubricated. After starting a cold engine, boundary and even dry friction of the compression rings against the cylinder walls is possible, which is one of the reasons for increased wear of the cylinders in the upper part.

Wear is the process of destruction and separation of material from the surface of a solid body and (or) accumulation of its residual deformation during friction, manifested in a gradual change in the size and (or) shape of the body.

Wear is usually divided into two groups:

1. Mechanical - occurs as a result of the cutting or scratching action of solid particles located between the friction surfaces:

1) abrasive - wear of the surface of a part, which occurs as a result of the cutting or scratching action of solid bodies or particles;

2) erosive (water-abrasive, gas-abrasive, electro-erosive) - wear occurs as a result of the impact on the surface of the part of a flow of liquid, gas, solid particles moving at high speed, as a result of the impact of discharges during the passage of electric current;

3) cavitation - wear occurs during the relative movement of a solid and liquid under conditions of cavitation. Cavitation is observed in a liquid when the pressure in it drops to the saturated vapor pressure, when the continuity of the liquid flow is disrupted and cavitation bubbles form. When the maximum size is reached, they begin to slam shut at high speed, which leads to a hydraulic shock on the metal surface;

4) fatigue – wear under the influence of alternating stresses. It affects gears, rolling and sliding bearings;

5) adhesive - wear (wear due to seizing) occurs when metals set during friction with the formation of strong metal bonds in areas of direct contact of surfaces;

6) wear during fretting is mechanical wear of slipping areas of tightly contacting surfaces under load during oscillatory, cyclic, reciprocating relative movements with small amplitudes.

2. Corrosion-mechanical – occurs during friction of materials that enter into chemical interaction with the environment:

1) oxidative wear - occurs when oxygen contained in the air or lubricant interacts with the metal and forms an oxide film on it, which, during friction, abrades or comes off the metal and is removed with the lubricant, and then forms again ( An example of oxidative wear is the wear of the upper part of the cylinders of an internal combustion engine under the action of acid corrosion, which occurs at low wall temperatures, especially when the engine is running cold);

2) wear during fretting corrosion consists of the formation of ulcers and corrosion products in the form of powder or plaque on the surfaces of mutual contact of parts. Wear in this case depends on the simultaneous processes of microsetting, fatigue, corrosion-mechanical and abrasive effects.

The main quantitative characteristics of wear are wear, wear rate, wear intensity.

Wear is the result of wear and tear, defined in established units. Wear (absolute or relative) characterizes the change in geometric dimensions (linear wear), mass (weight wear) or volume (volumetric wear) of a part due to wear and is measured in appropriate units.

Wear rate Vi (m/h, g/h, m3/h) – the ratio of wear U to the time interval during which it occurred:

Wear rate J is the ratio of wear to the determined path L along which wear occurred, or the amount of work done:

With linear wear, the wear intensity is a dimensionless quantity, and with weight wear, it is measured in units of mass per unit friction path.

The property of a material to resist wear under certain friction conditions is characterized by wear resistance - the reciprocal value of wear rate or intensity, in appropriate units.

During machine operation, wear indicators of parts and joints do not maintain constant values. Changes in wear of parts over time can generally be represented in the form of a model proposed by V.F. Lorenz. During the initial period of operation, called the running-in period, fairly rapid wear of parts is observed (Fig. 4.1, section I). The duration of this period is determined by the quality of the surfaces and the operating mode of the mechanism and is usually 1.5-2% of the life of the friction unit. After running-in, a period of steady-state wear begins (Figure 4.1, section II), which determines the durability of the joints. The third period - the period of catastrophic wear (Fig. 4.1, section III) - characterizes the limiting state of the mechanism and limits the resource. As can be seen from the graphs above, the wear process has a direct, determining effect on the occurrence of failures and malfunctions of machine friction units. The change in reliability indicators over time is identical to the change in wear indicators.

The higher steepness of the m = () curve in section II is explained by the fact that with operating time, failures arise, caused, in addition to wear, by fatigue, corrosion failure or plastic deformation.

Running-in is the process of changing the geometry of friction surfaces and the physico-chemical properties of the surface layers of the material in the initial period of friction, usually manifested under constant external conditions in a decrease in the friction force, temperature and wear rate. The running-in process is characterized by intensive separation of wear products from friction surfaces, increased heat generation and changes in the microgeometry of surfaces.

Rice. 4.1 – Changing pairing parameters during operation:

1 – wear U; 2 – wear rate V; 3 – failure rates m;

With the correct choice of the ratio of the hardness of the parts and the running-in modes, the period of so-called normal, or steady wear begins quite quickly (Fig. 4.1, section II). This period is characterized by a small, approximately constant wear rate and continues until changes in the size or shape of parts affect their operating conditions, or until the material reaches its fatigue limit.

The accumulation of changes in the geometric dimensions and physical and mechanical properties of parts leads to a deterioration in the operating conditions of the interface. The main factor in this case is an increase in dynamic loads due to an increase in gaps in the rubbing pairs. As a result, a period of catastrophic or progressive wear begins (Fig. 4.1, section III). The described pattern is conditional and serves only to illustrate the process of wear of machine elements.

1) Micrometering method. The method is based on measurement using a micrometer or a measuring device with an indicator of parameters before and after wear.

Disadvantages of the method:

– inevitable disassembly and assembly of the product before and after work in order to measure the part;

– the detected change in size may be a consequence not only of surface wear, but also the result of part deformation;

– disassembly and assembly of products during operation sharply reduces the performance of the machines.

2) Method of artificial bases. It consists of extruding or cutting out depressions of a given shape (pyramid or cone) and depth on the surface. By observing the change in the size of the print, the relationship of which with depth is known in advance, local linear wear can be determined. Special instruments are used that make it possible to determine with an accuracy of 1.5 to 2 microns for the holes of engine cylinders, shafts, and also flat surfaces.

The disadvantage of the method is that in most cases it also requires preliminary disassembly of products and therefore has the same disadvantages as the micrometering method.

3) Method of measuring wear by weight reduction. Based on weighing the part before and after wear. Typically used when testing light weight parts.

The disadvantage of the method is that it may be unacceptable when wear occurs due not only to particle separation, but also to plastic deformation.

4) Method for analyzing the iron content in oil. Based on chemical analysis of ash obtained by burning an oil sample. During the period between two consecutive samplings, the total amount of oil in the crankcase, its loss and the amount of oil added are taken into account.

This analysis is integral, since wear products are usually separated simultaneously from several rubbing parts.

Accurate determination of the amount of iron is complicated by the fact that large particles of wear products can settle on the crankcase walls.

5) Method of radioactive isotopes. It consists of introducing a radioactive isotope into the material of the part being studied. In this case, along with wear products, a proportional amount of radioactive isotope atoms will enter the oil. By the intensity of their radiation in an oil sample, one can judge the amount of metal that entered the oil over the period of time under consideration.

Advantages of the method:

– the wear of a specific part is determined, and not the total for several parts;

– sensitivity increases hundreds of times;

– the research process is accelerated.

Disadvantages of the method:

– special preparation of samples of test parts is required;

– availability of special equipment for measuring radiation intensity and taking precautions to protect human health.

1. What is wear and tear?

2. Name the differences and give examples of dry, boundary, semi-dry and liquid friction.

3. Give a general classification of wear.

4. Give a classification of mechanical wear.

5. Give the classification of corrosion-mechanical wear.

6. Define wear characteristics - wear (linear, volumetric, mass), wear rate and intensity, wear resistance and relative wear resistance.

7. Explain the methods of the following experimental methods for determining wear: micrometering, the artificial base method, the method of measuring wear by mass reduction, the method of analyzing the iron content in oil, the method of radioactive isotopes.

What are the advantages and disadvantages of the listed methods?

9. Name the main methods for reducing wear rates.

5. CORROSION DAMAGE

Corrosion of metals and alloys is their spontaneous destruction as a result of chemical, electrochemical interaction with the external environment, as a result of which they pass into an oxidized state and change their physical and mechanical properties.

Cars used in conditions of dust, high humidity, and temperatures are pronounced objects susceptible to corrosion damage. In this case, the most characteristic elements are parts made of thin-sheet steel of the body, frame and suspension, threaded and welded connections, parts of fuel equipment (exhaust valves, the upper part of cylinder liners and piston heads), gas pipelines.

Corrosion processes, depending on the mechanism of interaction of the metal with the environment, are divided into two types - chemical and electrochemical corrosion, and 36 types, the most common of which are:

a) depending on the nature of the corrosive environment:

– atmospheric, – gas, – liquid, – underground (soil), – biological;

b) depending on the conditions of the corrosion process:

– structural, – subsurface, – intergranular, – contact, – crevice, – stress corrosion, – corrosion cavitation, – fretting corrosion;

c) depending on the type of corrosion destruction:

– continuous, – local (local).

Chemical corrosion is the process of destruction of a material as a result of direct interaction at high temperatures with atmospheric oxygen, hydrogen sulfide, and water vapor.

The main condition for the occurrence of chemical corrosion is the absence of an electrically conductive medium, which is not typical for vehicle parts. However, this corrosion can be observed in some body elements. This is how exhaust pipes and mufflers are destroyed (burnt out), and body elements directly adjacent to the engine exhaust pipe or intake pipe (for example, a bus body skirt, rear buffer of passenger cars) are destroyed.

Electrochemical corrosion occurs as a result of exposure of the metal to the environment (electrolyte). It is associated with the emergence and flow of electric current from one surface to another.

The intensity of the electrochemical corrosion process depends on the access of oxygen to the metal surface, the chemical composition of the alloy, the density of corrosion products, which can sharply slow down the electrochemical process of structural heterogeneity of the metal, the presence and distribution of internal stresses.

Gas corrosion occurs at high temperatures in an environment of aggressive gases in the absence of moisture.

Intergranular corrosion. Invisible to the naked eye, it represents the destruction of metal between crystals under the action of alternating loads.

Contact corrosion occurs when two metals of different potentials are joined and an electrolyte is present.

Stress corrosion occurs when a part is corroded by dynamic or static stress.

Crevice corrosion is especially common in bodies due to the large number of cracks and gaps in them. Crevice corrosion develops in places where bolts, rivets, and spot welding are installed.

Corrosive cavitation is typical for those body parts that are exposed to water, such as the underbody. Drops of moisture falling on the bottom create a closure of cavitation bubbles and hydraulic shocks.

Complete corrosion occurs when vehicles are operated in a polluted atmosphere, starting on the lower surface of the bottom, from the inside of the wings, and in the internal cavities of doors and power elements (thresholds, cross members, reinforcements). Inside the cabin, it usually occurs under the floor mats.

Local corrosion can be intercrystalline and in the form of ulcers, spots, threads. Corrosion in the form of ulcers leaves individual centers of destruction on the metal, and in the case of thin sheet metal – through ones. Pitting corrosion occurs on parts that have passivating films and has the form of dots; its products fall out in the form of columns. Filament corrosion is close in nature to intercrystalline corrosion and occurs under a layer of paint or other protective coating in the form of a winding thread that deeply affects the metal.

Corrosion protection methods are conventionally divided into three groups:

a) methods for increasing the corrosion resistance of metals:

– application of paint and varnish, galvanic (chrome plating, nickel plating, galvanizing), chemical (oxidation, phosphating) or plastic (flame, vortex and other spraying methods) protective coatings;

– the use of alloys that are homogeneous in composition or with alloying additives, for example, chromium, aluminum, silicon;

b) methods of influencing the environment - sealing joints, eliminating gaps, introducing anti-corrosion additives into the environment of operating materials;

c) combined methods.

1. Explain the concept and importance of the problem of corrosion for road transport.

2. List the types of corrosion depending on the nature of the corrosive environment, the conditions for the occurrence of corrosion destruction, and the type of corrosion destruction.

3. What are the mechanisms of chemical and electrochemical corrosion?

4. List and explain with specific examples the main methods of combating corrosion.

6. TECHNICAL DIAGNOSTICS

6.1. Basic concepts of technical diagnostics Diagnostics is a branch of science that studies the various states of a technical object, has methods for determining the state of a technical object at the present time, and assessing the state in the past and future.

The technical condition of a machine (component, unit) is assessed by parameters, which are divided into structural and diagnostic.

A structural parameter is a physical quantity that directly characterizes the technical condition (operability) of a machine (for example, the dimensions of mating parts and the gaps between them); it is determined by direct measurements.

A diagnostic parameter is a physical quantity that indirectly characterizes the condition of the machine (for example, the amount of gases breaking into the crankcase, engine power, oil waste, knocking, etc.); it is monitored using diagnostic tools. Diagnostic parameters reflect changes in structural parameters.

There is a certain quantitative relationship between the structural and corresponding diagnostic parameters. For example, the size of the gaps in the interfaces of cylinder-piston groups (CPG) is diagnosed by the amount of gases breaking into the crankcase and the waste of crankcase oil; the size of the gaps in the crankshaft bearings - according to the pressure in the oil line; the degree of rarefaction of the battery - according to the density of the electrolyte.

A quantitative measure of state parameters (structural and diagnostic) are their values, which can be nominal, acceptable, limit and current (Fig. 6.1).

The nominal value of the parameter corresponds to the value established by calculation and is guaranteed by the manufacturer in accordance with the specifications. The nominal value is observed for new and overhauled components.

The permissible value (deviation) of a parameter is its limit value at which a component of the machine, after control, is allowed to operate without maintenance or repair operations. This value is given in the technical documentation for machine maintenance and repair. If the parameter value is acceptable, the component part of the machine operates reliably until the next scheduled inspection.

The limit value of a parameter is the largest or smallest value of a parameter that an operational component can have. At the same time, further operation of the component or the machine as a whole without repair is unacceptable due to a sharp increase in the wear rate of the joints, an excessive decrease in the efficiency of the machine, or a violation of safety requirements.

Figure 6.1. Definition of the concepts nominal, permissible, limit value of a parameter: I – operational and serviceable condition;

II – pre-failure (workable, but faulty) state;

III – inoperable (respectively faulty) state The current value of the parameter is the value of the parameter at each specific moment in time.

Limit values of state parameters, depending on what criteria (signs) they are established on the basis of, are divided into three groups:

– technical;

– technical and economic;

– technological (quality).

Technical criteria (signs) characterize the limiting state of the components when they can no longer perform their functions for technical reasons (for example, a maximum increase in the chain pitch above 40% of the nominal value leads to its slipping on the sprockets and falling off) or when further operation of the facility will lead to emergency failure (for example, operation at maximum oil pressure in the line leads to failure of the diesel engine).

Technical and economic criteria characterizing the limit state indicate a decrease in the efficiency of using the object due to a change in the technical condition (for example, with extreme wear of the CPG, crankcase oil burn increases by more than 3.5%, which indicates the inappropriateness of working on such an engine).

Technological criteria characterize a sharp deterioration in the quality of work due to the limiting state of the working parts of machines.

Based on the volume and nature of information, diagnostic parameters are divided into:

a) to general (integral);

b) element-by-element.

General parameters are parameters that characterize the technical condition of the object as a whole. In most cases, they do not provide information about a specific malfunction of the machine.

In relation to road transport, these include:

power at the drive wheels, engine power, fuel consumption, braking distance, vibration, noise, etc.

Element-by-element parameters are parameters that indicate a very specific malfunction of a machine unit or mechanism.

6.2. Tasks of technical diagnostics The main tasks of technical diagnostics are:

– establishing the type and scope of maintenance work on the machine after it has completed a certain amount of operating time;

– determination of the residual life of the machine and the degree of its readiness to perform mechanized work;

– implementation of quality control of preventive operations during maintenance;

– identifying the causes and nature of malfunctions that arise during the use of the machine.

The main task of technical diagnostics is to determine the technical condition of an object (machine) at the required point in time. When solving this problem, depending on the point in time at which it is necessary to determine the technical condition of the machine, three interrelated and complementary directions are distinguished:

– technical diagnostics, i.e. determining the technical condition of the machine in which it is currently located;

– technical forecasting, i.e. scientific prediction of the technical state of a machine in which it will find itself at some future moment;

– technical genetics, i.e. determination of the technical state of the machine in which it was at some point in time in the past (in technical literature the term “retrospection” is often used instead of the term “technical genetics”).

The introduction of technical diagnostics allows:

– reduce downtime of cars and other machines due to technical faults by 2...2.5 times by preventing failures; increase the time between repairs of assembly units and machine assemblies by 1.3...1.5 times;

– eliminate premature disassembly of units and components and thereby reduce the wear rate of parts and connections;

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Reliability indicator assessment is the numerical values of indicators determined based on the results of observations of objects under operating conditions or special reliability tests. When determining reliability indicators, two options are possible: the type of operating time distribution law is known...

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PAGE 2

TEST

“Fundamentals of the theory of reliability and diagnostics”

Exercise

Based on the results of testing products for reliability according to plan [ N v z ] the following initial data were obtained for assessing reliability indicators:
- 5 sample values of time to failure (unit: thousand hours): 4.5; 5.1; 6.3; 7.5; 9.7.
- 5 sample values of operating time before censoring (i.e., 5 products remained in working condition by the time the tests were completed): 4.0; 5.0; 6.0; 8.0; 10.0.

Define:

- point estimate of average time to failure;

- with confidence probability lower confidence limits and;
- draw the following graphs to scale:

distribution function;

probability of failure-free operation;

upper confidence limit;

lower confidence limit.

Introduction

The calculation part of the practical work contains an assessment of reliability indicators based on given statistical data.

Reliability indicator assessment these are numerical values of indicators determined based on the results of observations of objects under operating conditions or special reliability tests.

When determining reliability indicators, two options are possible:

The type of operating time distribution law is known;

The type of operating time distribution law is not known.

In the first case, parametric assessment methods are used, in which the parameters of the distribution law included in the calculation formula of the indicator are first assessed, and then the reliability indicator is determined as a function of the estimated parameters of the distribution law.

In the second case, nonparametric methods are used, in which reliability indicators are assessed directly from experimental data.

BRIEF THEORETICAL INFORMATION

Quantitative indicators of the reliability of rolling stock can be determined from representative statistical data on failures obtained during operation or as a result of special tests carried out taking into account the operating characteristics of the structure, the presence or absence of repairs and other factors.

The sampling method can be used in two ways:

Simple random selection;

Random selection according to typical groups.

Dividing the sample population into typical groups (for example, by gondola car models, by years of construction, etc.) gives an increase in accuracy when estimating the characteristics of the entire population.

When calculating an estimate, one usually tries to choose a method so that it is consistent, unbiased, and efficient. A consistent estimate is one that, with an increase in the number of observation objects, converges in probability to the true value of the indicator (condition 1).

An estimate is called unbiased, the mathematical expectation of which is equal to the true value of the reliability indicator (condition 2).

An estimate is called effective, the variance of which, compared to the dispersions of all other estimates, is the smallest (condition 3).

If conditions (2) and (3) are satisfied only when N tending to zero, then such estimates are called asymptotically unbiased and asymptotically efficient, respectively.

Consistency, unbiasedness and efficiency are qualitative characteristics of assessments. Conditions (1)-(3) allow for a finite number of objects N observations, write down only an approximate equality

a~â(N)

Thus, the estimate of the reliability indicator â( N ), calculated from a sample set of volume objects N is used as an approximate value of the reliability indicator for the entire population. This estimate is called a point estimate.

The initial data for assessing reliability indicators are determined by the observation plan. The initial data for the plan ( N V Z ) are:

Selected time-to-failure values;

Selected operating hours of machines that remained operational during the observation period.

The operating time of machines (products) that remained operational during testing is called the operating time before censoring.

Censoring (cut-off) on the right is an event leading to the termination of testing or operational observations of an object before the onset of failure (limit state).

Reasons for censoring are:

Different times of the beginning and (or) end of testing or operation of products;

Removal from testing or operation of some products for organizational reasons or due to failures of components whose reliability has not been studied;

Transfer of products from one application mode to another during testing or operation;

The need to assess the reliability before failures of all tested products.

Operating time before censoring is the operating time of the object from the start of testing to the onset of censoring. A sample whose elements are the values of time to failure and before censoring is called a censored sample.

A once-censored sample is a censored sample in which the values of all times before censoring are equal to each other and are not less than the longest time before failure. If the values of the operating time before censoring in the sample are not equal, then such a sample is repeatedly censored.

Evaluation of reliability indicators USING NON-PARAMETRIC METHOD

1 . We arrange the time to failure and the time to censoring into a general variation series in non-decreasing order of the time (the time before censoring is marked *): 4,0*; 4,5; 5,0*; 5,1; 6,0*; 6,3; 7,5; 8,0*; 9,7; 10,0*.

2 . We calculate point estimates of the distribution function for operating time using the formula:

where is the number of functional products j -th failure in the variation series.

3. We calculate the point estimate of the average time to failure using the formula:

Where;

Thousand hour.

4. The point estimate of failure-free operation per thousand hours is determined using the formula:

Where;

5. We calculate point estimates using the formula:

6. Based on the calculated values, we construct graphs of the operating time distribution functions and reliability functions.

7. The lower confidence limit for the average time to failure is calculated using the formula:

Where is the quantile of the normal distribution corresponding to the probability. Accepted according to the table depending on the confidence level.

According to the conditions of the task, confidence probability. We select the corresponding value from the table.

Thousand hour.

8 We calculate the values of the upper confidence limit for the distribution function using the formula:

where is the quantile of the chi-squared distribution with the number of degrees of freedom. Accepted according to the table depending on the confidence level q.

The curly brackets in the last formula mean taking the integer part of the number enclosed in these brackets.

For;
For;
For;
For;
For.

9. The values of the lower confidence limit of the probability of failure-free operation are determined by the formula:

10. The lower confidence limit of the probability of failure-free operation at a given operating time, thousand hours, is determined by the formula:

Where; .

Respectively

11. Based on the calculated values, we construct graphs of the functions of the upper confidence limit and lower confidence limit as previously constructed models of point estimates and

CONCLUSION ON THE WORK DONE

When studying the results of testing products for reliability according to plan [ N v z ] the following reliability indicators were obtained:

Point estimate of mean time to failure thousand hours;
- point estimate of the probability of failure-free operation per thousand hours of operation;
- with confidence probability lower confidence limits thousand hours and;

Using the found values of the distribution function, the probability of failure-free operation, the upper confidence limit and the lower confidence limit, graphs were constructed.

Based on the calculations performed, it is possible to solve similar problems that engineers face in production (for example, when operating cars on the railway).

Bibliography
Chetyrkin E. M., Kalikhman I. L. Probability and statistics. M.: Finance and Statistics, 2012. 320 p.
Reliability of technical systems: Handbook / Ed. I. A. Ushakova. M.: Radio and Communications, 2005. 608 p.
Reliability of engineering products. A practical guide to standardization, confirmation and provision. M.: Publishing house of standards, 2012. 328 p.
Methodical instructions. Reliability in technology. Methods for assessing reliability indicators based on experimental data. RD 50-690-89. Enter. P. 01.01.91, M.: Standards Publishing House, 2009. 134 p. Group T51.
Bolyshev L. N., Smirnov N. V. Tables of mathematical statistics. M.: Nauka, 1983. 416 p.
Kiselev S.N., Savoskin A.N., Ustich P.A., Zainetdinov R.I., Burchak G.P. Reliability of mechanical systems of railway transport. Tutorial. M.: MIIT, 2008 -119 p.

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	Linear automatic systems. Modern control systems R. Control systems with feedback. Nyquist proposed a stability criterion based on the frequency characteristics of a system in an open state and in 1936.

The fundamentals of the theory of reliability and diagnostics are outlined in relation to the most capacious component of the person - car - road - environment system. Basic information about the quality and reliability of the car as a technical system is presented. Basic terms and definitions are given, reliability indicators of complex and dissected systems and methods for their calculation are given. Attention is paid to the physical foundations of vehicle reliability, methods of processing information about reliability and reliability testing methods. The place and role of diagnostics in the system of vehicle maintenance and repair in modern conditions is shown.
For university students.

The concepts of “quality” and “reliability” of machines.
The life of modern society is unthinkable without the use of machines of a wide variety of designs and purposes that transform energy, materials, information, and change people’s lives and the environment.
Despite the enormous diversity of all machines, in the process of their development, uniform criteria are used to assess the degree of their perfection.

In market conditions, the creation of most new machines requires compliance with the most important condition for competitiveness, namely giving them new functions and high technical and economic indicators of their use.
For efficient use of machines, it is necessary that they have high levels of quality and reliability.

The international standard ISO 8402 - 86 (ISO - International Organization Standardization) gives the following definition: “Quality is the set of properties and characteristics of a product or service that give it the ability to satisfy stated or anticipated needs.”

TABLE OF CONTENTS
Preface
Introduction
Chapter 1. Reliability is the most important property of product quality
1.1. The quality of products and services is the most important indicator of the successful activities of enterprises in the transport and road complex
1.2. The concepts of “quality” and “reliability” of machines
1.3. Reliability and universal problems
Chapter 2. Basic concepts, terms and definitions adopted in the field of reliability
2.1. Objects considered in the field of reliability
2.1.1. General concepts
2.1.2. Classification of technical systems
2.2. Basic states of an object (technical system)
2.3. Transition of an object into various states. Types and characteristics of failures of technical systems
2.4. Basic concepts, terms and definitions in the field of reliability
2.5. Reliability indicators
2.6. Reliability criteria for non-recoverable systems
2.7. Reliability criteria for restored systems
2.8. Durability indicators
2.9. Storability indicators
2.10. Maintainability indicators
2.11. Comprehensive reliability indicators
Chapter 3. Collection, analysis and processing of operational data on product reliability
3.1. Goals and objectives of collecting information and assessing machine reliability
3.2. Principles of collecting and systematizing operational information on product reliability
3.3. Construction of an empirical distribution and statistical assessment of its parameters
3.4. Time-to-failure distribution laws, most often used in reliability theory
3.5. Laplace transform
3.6. Confidence interval and confidence probability
Chapter 4. Reliability of complex systems
4.1. Complex system and its characteristics
4.2. Reliability of dismembered systems
Chapter 5. Mathematical models of reliable functioning of technical elements and systems
5.1. General reliability model of a technical element
5.2. General model of system reliability in terms of integral equations
5.2.1. Basic notations and assumptions
5.2.2. State Matrix
5.2.3. Transition Matrix
5.3. Reliability models for non-recoverable systems
Chapter 6. Life cycle of a technical system and the role of scientific and technical preparation of production to ensure its quality requirements
6.1. Life cycle structure of a technical system
6.2. Comprehensive product quality assurance system
6.3. Quality level assessment and reliability management
6.3.1. International quality standards ISO 9000-2000 series
6.3.2. Quality control and its methods
6.3.3. Methods of quality control, analysis of defects and their causes
6.4. Technical and economic management of product reliability
6.5. Seven simple statistical methods for assessing quality used in ISO 9000 standards
6.5.1. Classification of statistical quality control methods
6.5.2. Data layering
6.5.3. Graphical representation of data
6.5.4. Pareto chart
6.5.5. Cause and effect diagram
6.5.6. Scatter diagram
6.5.7. Checklist
6.5.8. Control card
Chapter 7. The physical essence of the processes of changing the reliability of structural elements of automobiles during their operation
7.1. Causes of loss of performance and types of damage to machine elements
7.2. Physico-chemical processes of destruction of materials
7.2.1. Classification of physical and chemical processes
7.2.2. Processes of mechanical destruction of solids
7.2.3. Aging of materials
7.3. Failures based on strength parameters
7.4. Tribological failures
7.5. Types of wear of car parts
7.6. Failures due to corrosion parameters
7.7. Wear chart and methods for measuring wear of car parts
7.8. Methods for determining wear of machine parts
7.8.1. Periodic wear measurement
7.8.2. Continuous wear measurement
7.9. The influence of residual deformations and aging of materials on the wear of parts
7.10. Assessing the reliability of vehicle elements and technical systems during their design
7.11. The most common methods and techniques for ensuring and predicting reliability used in creating machines
Chapter 8. Machine maintenance and repair system
8.1. Machine maintenance and repair systems, their essence, content and principles of construction
8.2. Requirements for the maintenance and repair system, and methods for determining the frequency of their implementation
8.3. Machine operation in extreme situations
Chapter 9. Diagnostics as a method of monitoring and ensuring vehicle reliability during operation
9.1. General information about diagnostics
9.2. Basic concepts and terminology of technical diagnostics
9.3. Diagnostic value
9.4. Diagnostic parameters, determination of limit and permissible values of technical condition parameters
9.5. Principles of car diagnostics
9.6. Organization of vehicle diagnostics in the maintenance and repair system
9.7. Types of car diagnostics
9.8. Diagnosis of vehicle components during repairs
9.9. Diagnosing the condition of the cylinder-piston group
9.10. The concept of diagnosing equipment in modern conditions
9.11. Technical diagnostics is an important element of technological certification of services of service enterprises
9.12. Management of reliability and technical condition of machines based on diagnostic results
9.13. Vehicle diagnostics and safety
9.14. Brake system diagnostics
9.15. Diagnostics of headlights
9.16. Diagnostics of suspension and steering
Conclusion
Bibliography.

1.1. Fundamentals of reliability theory

a) Reliability and solving problems of accelerating scientific and technological progress.

As technology becomes more complex, the areas of its use expand, the level of automation increases, and loads and speeds increase, the role of reliability issues increases. Their solution is one of the main sources of increasing the efficiency of equipment, saving material, labor and energy costs.

Example 1. The cost of a 10% increase in the service life of car tires is 0.2% of their cost. Increased tire reliability leads to a corresponding reduction in the need for them. As a result, the cost of producing tires that provide a solution to a specific transport problem is 0.898 of their original cost.

Due to the increasing complexity of equipment, the cost of malfunctions arising during its operation has increased significantly.

Example 2. The E-652 excavator replaces the work of 150 excavators. One hour of its downtime leads to significant material losses.

Insufficiently, a high level of reliability is one of the main reasons for unreasonably high costs for maintenance, repair of equipment and production of spare parts.

Example 3. To maintain tractors in working condition, twice as much money is spent on repairs and maintenance during their service life as on purchasing a new one.

b) Basic concepts of reliability.

Reliability is a property of the system preserve in time within established limits, the values of all parameters characterizing the ability to perform the required functions in given modes of use, maintenance, repair, storage and transportation.

Reliability is a complex, but nevertheless clearly (at the GOST level) regulated property of the system.

Let us consider sequentially, in accordance with cause-and-effect relationships, the basic concepts used in describing reliability.

Reliability as a complex property of a system is determined by a combination of four simpler properties, namely: reliability, durability, maintainability and storability. Moreover, depending on the design and operation features of the system, one or another property (or properties) may not be included in the reliability. For example, if a rolling bearing cannot be repaired, then repairability is not included in the reliability property. The classification of reliability properties is shown in Fig. 1.1.

Reliability is a property of the system continuously maintain an operational state when operating for a period of time some(specified) time or some(given) operating time.

Durability is the property of a system to function until ultimate condition under the established procedure for maintenance and repair.

Maintainability is a property of a system consisting in adaptability to warning and detection pre-failure conditions, failures and damage, maintaining and restoring an operational state through maintenance and repair.

Storability is the property of a system to retain the values of indicators of reliability, durability and maintainability during and after storage and (or) transportation.

When determining the reliability properties, concepts were used that define various states of the system. Their classification is shown in Fig. 1.2.

Serviceable – the state of the system in which it currently corresponds to all requirements, established as in relation main parameters, characterizing the functioning of the system, and in relation to minor parameters, characterizing ease of use, appearance, etc.

Faulty - the state of the system in which it is currently from the requirements established both in relation to main, so secondary parameters.

Operable – the state of the system in which it currently corresponds to all requirements established in relation to main parameters.

Inoperative - the state of the system in which it is currently does not match at least one from the requirements established for main parameters.

Limit – a state of a system in which it temporarily or permanently cannot be operated. The limit state criteria for different systems are different and are established in the regulatory and technical design or operational documentation.

From the above definitions it follows that a faulty system can be operational (for example, a car with damaged body paint), and an inoperative system can also be faulty.

The transition of a system from one state to another occurs as a result of an event. The classification of events is shown in Fig. 1.3., and the graph explaining it in Fig. 1.4.

Damage is an event as a result of which the system ceases to meet the requirements for minor parameters.

Failure is an event as a result of which the system ceases to meet the requirements in relation to the main and primary and secondary parameters, i.e. complete or partial loss of performance.

Failure – failure with self-healing.

Resource exhaustion is an event as a result of which the system goes into a limit state. Of the listed events, the most important is failure, which is classified:

A. By significance (critical, essential, insignificant).

B. By the nature of occurrence (sudden, gradual).

B. By the nature of detectability (explicit, hidden).

D. Due to its occurrence (structural, production, operational, degradation).

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Introduction

The calculation part of the practical work contains an assessment of reliability indicators based on given statistical data.

Reliability indicator assessments are numerical values ​​of indicators determined based on the results of observations of objects under operating conditions or special reliability tests.

When determining reliability indicators, two options are possible:

- the type of operating time distribution law is known;

- the type of operating time distribution law is not known.

In the first case, parametric assessment methods are used, in which the parameters of the distribution law included in the calculation formula of the indicator are first assessed, and then the reliability indicator is determined as a function of the estimated parameters of the distribution law.

In the second case, nonparametric methods are used, in which reliability indicators are assessed directly from experimental data.

1. Brief theoretical information

failsafe trust distribution point

The sampling method can be used in two ways:

- simple random selection;

- random selection according to typical groups.

Dividing the sample population into typical groups (for example, by gondola car models, by years of construction, etc.) gives an increase in accuracy when estimating the characteristics of the entire population.

When calculating an estimate, one usually tries to choose a method so that it is consistent, unbiased, and efficient. A consistent estimate is one that, with an increase in the number of observation objects, converges in probability to the true value of the indicator (condition 1).

An unbiased estimate is one whose mathematical expectation is equal to the true value of the reliability indicator (condition 2).

An estimate is called effective, the variance of which, compared to the dispersions of all other estimates, is the smallest (condition 3).

If conditions (2) and (3) are satisfied only when N tends to zero, then such estimates are called asymptotically unbiased and asymptotically efficient, respectively.

Consistency, unbiasedness and efficiency are qualitative characteristics of assessments. Conditions (1) - (3) allow us to write down only an approximate equality for a finite number of observation objects N

a~b(N)

Thus, the estimate of the reliability indicator in (N), calculated from a sample population of objects of volume N, is used as an approximate value of the reliability indicator for the entire population. This estimate is called a point estimate.

The initial data for assessing reliability indicators are determined by the observation plan. The initial data for the plan (N V Z) are:

- sample values ​​of time to failure;

- sample values ​​of operating time of machines that remained operational during the observation period.

The operating time of machines (products) that remained operational during testing is called the operating time before censoring.

Censoring (cutting off) on the right is an event leading to the termination of testing or operational observations of an object before the onset of failure (limit state).

Reasons for censoring are:

- different times of the beginning and (or) end of testing or operation of products;

- removal from testing or operation of some products for organizational reasons or due to failures of components, the reliability of which is not investigated;

- transfer of products from one mode of use to another during testing or operation;

- the need to assess the reliability before failure of all tested products.

Operating time before censoring is the operating time of the object from the start of testing to the onset of censoring. A sample whose elements are the values ​​of time to failure and before censoring is called a censored sample.

2. Estimation of reliability indicators using a nonparametric method

1 . We arrange the time to failure and the time to censoring in a general variation series in the order of non-decreasing operating time (the time before censoring is marked *): 4.0*; 4.5; 5.0*; 5.1; 6.0*; 6.3; 7.5; 8.0*; 9.7; 10.0*.

2 . We calculate point estimates of the distribution function for operating time using the formula:

; ,

where is the number of serviceable products of the j-th failure in the variation series.

;

;

;

;

3. We calculate the point estimate of the average time to failure using the formula:

,

Where;

;

.

;

thousand hours

4. The point estimate of failure-free operation per thousand hours is determined using the formula:

,

Where;

.

;

5. We calculate point estimates using the formula:

.

;

;

;

.

6. Based on the calculated values, we construct graphs of the operating time distribution functions and reliability functions.

7. The lower confidence limit for the average time to failure is calculated using the formula:

,

where is the quantile of the normal distribution corresponding to the probability. Accepted according to the table depending on the confidence level.

According to the conditions of the task, confidence probability. We select the corresponding value from the table.

thousand hours

8 . We calculate the values ​​of the upper confidence limit for the distribution function using the formula:

,

where is the quantile of the chi-squared distribution with the number of degrees of freedom. Accepted according to the table depending on the confidence level q.

.

The curly brackets in the last formula mean taking the integer part of the number enclosed in these brackets.

For;

For;

For;

For;

For.

;

;

;

;

Reliability indicator assessments are numerical values of indicators determined based on the results of observations of objects under operating conditions or special reliability tests.

- sample values of time to failure;

- sample values of operating time of machines that remained operational during the observation period.

Operating time before censoring is the operating time of the object from the start of testing to the onset of censoring. A sample whose elements are the values of time to failure and before censoring is called a censored sample.

1 . We arrange the time to failure and the time to censoring in a general variation series in the order of non-decreasing operating time (the time before censoring is marked ): 4.0; 4.5; 5.0; 5.1; 6.0; 6.3; 7.5; 8.0; 9.7; 10.0.

8 . We calculate the values of the upper confidence limit for the distribution function using the formula:

9. The values of the lower confidence limit of the probability of failure-free operation are determined by the formula:

Using the found values of the distribution function, the probability of failure-free operation, the upper confidence limit and the lower confidence limit, graphs were constructed.