Assignment on the basics of the theory of reliability and diagnostics. Fundamentals of the theory of reliability and technical diagnostics
Ministry of Education and Science of the Russian Federation
State educational institution
higher professional education
"Omsk State Technical University"
A. V. Fedotov, N. G. Skabkin
Fundamentals of the theory of reliability and technical diagnostics
Lecture notes
Publishing house OmSTU
UDC 62-192 + 681.518.54
BBK 30.14 + 30.82
Reviewers: n. S. Galdin, Dr. Tech. Sciences, prof, department. PttMiG SibAdi; Yu. P. Kotelevsky, Cand. Tech. Sciences, gen. Director of LLC "adl-Omsk"
Fedotov, A.V.
Form 34 Fundamentals of the theory of reliability and technical diagnostics: lecture notes / A. V. Fedotov, N. G. Skabkin. - Omsk: Publishing house of OmSTU, 2010 .-- 64 p.
The basic concepts of the theory of reliability, qualitative and quantitative characteristics of reliability are considered. The mathematical foundations of the theory of reliability, calculations of reliability indicators, basic concepts, definitions and tasks of technical diagnostics are considered.
The abstract can be used both for the practical consolidation of theoretical material on the course "Diagnostics and reliability of automated systems" for full-time students, and for self-training of students of correspondence and distance learning forms.
Reprinted by the decision of the Editorial and Publishing Council
Omsk State Technical University
UDC 62-192 + 681.518.54
BBK 30.14 + 30.82
© GOU VPO "Omsk State
technical university ", 2010
General characteristics of reliability as a science
The emergence of technology and its widespread use in production processes has made the question of its effectiveness urgent. The efficiency of using machines is associated with their ability to continuously and efficiently perform the functions assigned to them. However, due to breakdowns or malfunctions, the quality of the machines' operation decreases, forced downtime occurs in their operation, there is a need for repairs to restore the operability and the required technical characteristics of the machines.
These circumstances led to the emergence of the concept of reliability of machines and other technical means. The concept of reliability is associated with the ability of a technical tool to perform the functions assigned to it for the required time and with the required quality. From the first steps in the development of technology, the task was to make a technical device so that it would work reliably. With the development and complication of technology, the problem of its reliability became more complicated and developed. To solve it, it was necessary to develop the scientific foundations of a new scientific direction - the science of reliability.
Reliability characterizes the quality of a technical device. 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 study of reliability as a qualitative indicator characterizing a technical device led to the emergence of the science "Reliability". The subject of scientific research is the study of the causes of failures of objects, the determination of the regularities 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.
Distinguish between general theory of reliability and applied theory of reliability. The general theory of reliability has three components:
1. Mathematical theory of reliability. Determines the mathematical laws that obey failures and methods for quantitative measurement of 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. Investigation of physical and chemical processes, physical causes of failures, the effect 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 area. For example, there is the theory of reliability of control systems, the theory of reliability of electronic devices, the theory of reliability of machines, etc.
Reliability is related to the efficiency (eg cost-effectiveness) of the technique. Insufficient reliability of a technical device results in:
reduced productivity due to downtime due to breakdowns;
a decrease in the quality of the results of using a technical device due to a deterioration in its technical characteristics due to malfunctions;
the cost of repairing a technical facility;
loss of regularity of obtaining the result (for example, a decrease in the regularity of transportation for vehicles);
decrease in the level of safety of using a technical device.
Diagnostics is directly related to reliability. Diagnostics - teaching about the methods and principles of disease recognition and diagnosis. Technical diagnostics considers issues related to the assessment of the actual state of technical systems. The task of diagnostics is to identify and prevent emerging failures of technical means in order to increase their overall reliability.
The process of technical diagnostics provides for the presence of a diagnostic object, diagnostic tools and a human operator. In the process of diagnostics, measuring, control and logical operations are performed. These operations are performed by the operator using diagnostic tools in order to determine the actual state of the technical device. The results of the assessment are used to make a decision on the further use of the technical means.
The assessment of the reliability indicator is the numerical values of indicators determined by the results of observations of objects in operation or special tests for reliability. When determining reliability indicators, two options are possible: the form of the operating time distribution law is known ...
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PAGE 2
TEST
"Fundamentals of the theory of reliability and diagnostics"
- The task
According to the results of testing products for reliability according to the plan [ N v z ] the following initial data were obtained for assessing reliability indicators:
- 5 sample values of mean time to failure (unit of measurement: thousand hours): 4.5; 5.1; 6.3; 7.5; 9.7.
- 5 sample values of the operating time before censoring (ie, 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 the mean time to failure;
- with a confidence level, the lower confidence limits and;
- build the following graphs to scale:
distribution function;
the likelihood of failure-free operation;
upper confidence limit;
lower confidence limit.
- Introduction
The computational part of the practical work contains an assessment of the reliability indicators for the given statistical data.
The assessment of the reliability indicator is the numerical values of indicators determined by the results of observations of objects in operation or special tests for reliability.
When determining reliability indicators, two options are possible:
The form of the distribution law of the operating time is known;
The form of the distribution law of the operating time 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 estimated, 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
The quantitative indicators of the reliability of the rolling stock can be determined by representative statistical data on failures obtained during operation or as a result of special tests, set taking into account the peculiarities of the structure, the presence or absence of repairs and other factors.
The initial set of objects of observation is called the general population. By the coverage of the population, there are 2 types of statistical observations: continuous and selective. Continuous observation, when each element of the population is studied, is associated with a significant investment of money and time, and sometimes it 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 some of its representative part - a sample population, which is also called a sample. Based on the results of studying the attribute in the sample population, a conclusion is made about the properties of the attribute in the general population.
The sampling method can be used in two ways:
Simple random selection;
Random selection by typical groups.
Dividing the sample into typical groups (for example, by models of open wagons, by years of construction, etc.) gives a gain in accuracy when assessing the characteristics of the entire general population.
No matter how thoroughly the sample observation is set, the number of objects is always finite, and therefore the amount 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, it is usually sought to choose a method that is consistent, unbiased, and efficient. A consistent estimate is an estimate that, with an increase in the number of objects of observation, converges in probability to the true value of the indicator (conv. 1).
An unbiased estimate is an estimate whose mathematical expectation is equal to the true value of the reliability indicator (conv. 2).
An estimate is called effective if its variance is the smallest in comparison with the variances of all other estimates (condition 3).
If conditions (2) and (3) are satisfied only for N tending to zero, then such estimates are called asymptotically unbiased and asymptotically effective, 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 assessment of the reliability indicator â ( N ), calculated from a sample set of objects of volume N is used as an approximate value of the reliability indicator for the entire general population. Such an estimate is called a point estimate.
Taking into account the probabilistic nature of reliability indicators and a significant scatter of statistical data on failures, when using point estimates of indicators instead of their true values, it is important to know what are the limits of possible error and what is its probability, that is, it is important to determine the accuracy and reliability of the estimates used. It is known that the quality of the point estimate is the higher, the more statistical material it is obtained. Meanwhile, the point estimate by itself does not carry any information about the amount of data on which it was obtained. This determines the need for interval estimates of reliability indicators.
The initial data for assessing the reliability indicators are determined by the observation plan. The initial data for the plan ( N V Z) are:
Selected values of MTBF;
Selected values of the operating time of machines that remained operational during the observation period.
The operating time of machines (products) that remained operational during the test is called the operating time before censoring.
Right censoring (clipping) is an event that results in the termination of testing or in-service observations of an object prior to the occurrence of a failure (limit state).
The reasons for censoring are:
The timing of the beginning and (or) the 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 being investigated;
Transfer of products from one mode of use to another during testing or operation;
The need to assess the reliability before the onset of failures of all investigated products.
Working time before censoring is the working time of an object from the beginning of testing to the onset of censoring. A sample, the elements of which are MTBF and before censoring values, is called a censored sample.
A once-censored sample is a censored sample in which the values of all operating time before censoring are equal to each other and no less than the maximum operating time to failure. If the values of the operating time before censoring in the sample are not equal to each other, then such a sample is repeatedly censored.
- Evaluation of reliability indicators NON-PARAMETRIC METHOD
1 ... The operating time to failure and the operating time before censoring are arranged in a common variation series in the order of non-decreasing operating time (the operating times before censoring are marked *): 4,0*; 4,5; 5,0*; 5,1; 6,0*; 6,3; 7,5; 8,0*; 9,7; 10,0*.
2 ... We calculate the point estimates of the distribution function for the operating time using the formula:
where is the number of workable products j -th refusal in the variation series.
3. We calculate a point estimate of the mean time to failure using the formula:
Where;
Thousand. hour.
4. The point estimate of no-failure operation for the operating time of a thousand hours is determined by the formula:
Where;
5. We calculate point estimates by the formula:
6. Based on the calculated values, we plot the graphs of the operating time distribution functions and the reliability function.
7. The lower confidence limit for mean time to failure is calculated by the formula:
Where is the quantile of the normal distribution corresponding to the probability. It is taken according to the table depending on the confidence level.
According to the condition of the task, the confidence probability. We select the corresponding value from the table.
Thousand. hour.
8 The values of the upper confidence limit for the distribution function are calculated by the formula:
where is the quantile of the CHI-squared distribution with the number of degrees of freedom. Taken according to the table depending on the confidence level q.
The curly braces 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 no-failure operation are determined by the formula:
10. The lower confidence limit of the probability of no-failure operation at a given operating time thousand hours is determined by the formula:
Where; ...
Respectively
11. Using the calculated values and we plot the graphs of the functions of the upper confidence limit and the lower confidence limit as the previously constructed models of point estimates and
- CONCLUSION ON THE PERFORMED WORK
When examining the results of testing products for reliability according to the plan [ N v z ] obtained the values of the following reliability indicators:
Point estimate of the mean time to failure, thousand hours;
- point estimate of the probability of no-failure operation for operating time thousand hours;
- with a confidence level, the lower confidence limits, thousand hours and;
The found values of the distribution function, the probability of no-failure operation, the upper confidence limit and the lower confidence limit were plotted.
Based on the calculations performed, it is possible to solve similar problems that engineers face in production (for example, when operating wagons on a railway).
- Bibliography
- Chetyrkin E. M., Kalikhman I. L. Probability and statistics. Moscow: Finance and Statistics, 2012 .-- 320 p.
- Reliability of technical systems: Handbook / Ed. I.A.Ushakova. - M .: Radio and communication, 2005 .-- 608 p.
- Reliability of engineering products. A practical guide to rationing, validation and assurance. 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. S. 01.01.91, M .: Publishing house of standards, 2009 .-- 134 p. Group T51.
- Bolyshev L.N., Smirnov N.V. Tables of mathematical statistics. Moscow: 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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The foundations of the theory of reliability and diagnostics in relation to the most capacious component of the system man - car - road - environment are stated. The basic information about the quality and reliability of the car as a technical system is presented. Basic terms and definitions are given, indicators of reliability of complex and dismembered systems and methods of their calculation are given. Attention is paid to the physical foundations of vehicle reliability, methods of processing information about reliability and methods of testing for reliability. The place and role of diagnostics in the system of maintenance and repair of cars in modern conditions are shown.
For university students.
The concepts of "quality" and "reliability" of machines.
The life of modern society is inconceivable without the use of machines of a wide variety in design and purpose, which transform energy, materials, information, change the life of people and the environment.
Despite the huge variety of all machines, in the process of their development, uniform criteria are used to assess the degree of their perfection.
In the conditions of market relations, the creation of most new machines requires compliance with the most important condition of competitiveness, namely, giving them new functions and high technical and economic indicators of their use.
For effective use of machines, it is necessary that they have high quality and reliability indicators.
The international standard ISO 8402 - 86 (ISO - International Organization Standartization) gives the following definition: "Quality is a set of properties and characteristics of a product or service that give them the ability to satisfy conditioned or implied needs."
TABLE OF CONTENTS
Foreword
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 operation of enterprises in the transport and road complex
1.2. The concepts of "quality" and "reliability" of machines
1.3. Reliability and Human Issues
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. The main states of the object (technical system)
2.3. The transition of an object to 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 Recoverable Systems
2.8. Durability indicators
2.9. Persistence indicators
2.10. Repairability 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 the reliability of machines
3.2. Principles of collection and systematization of operational information on product reliability
3.3. Construction of empirical distribution and statistical estimation of its parameters
3.4. The distribution laws of the operating time to failure, most often used in the theory of reliability
3.5. Laplace transform
3.6. Confidence interval and confidence level
Chapter 4. Reliability of Complex Systems
4.1. Complex system and its characteristics
4.2. Reliability of Dismembered Systems
Chapter 5. Mathematical models of the reliability of the functioning of technical elements and systems
5.1. General reliability model of a technical element
5.2. General system reliability model in terms of integral equations
5.2.1. Basic notation and assumptions
5.2.2. State Matrix
5.2.3. Transition matrix
5.3. Reliability Models for Non-Recoverable Systems
Chapter 6. The life cycle of a technical system and the role of scientific and technical preparation of production to ensure the requirements of its quality
6.1. The structure of the life cycle of a technical system
6.2. Comprehensive product quality assurance system
6.3. Quality 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. Quality control methods, 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 presentation of data
6.5.4. Pareto chart
6.5.5. Causal diagram
6.5.6. Scatter plot
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 cars during their operation
7.1. Reasons for loss of performance and types of damage to machine elements
7.2. Physicochemical 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 in terms of strength
7.4. Tribological failures
7.5. Types of wear of car parts
7.6. Corrosion parameters failures
7.7. Wear chart and methods for measuring wear of car parts
7.8. Methods for determining the wear of machine parts
7.8.1. Periodic wear measurement
7.8.2. Continuous wear measurement
7.9. Influence of permanent deformations and aging of materials on wear of parts
7.10. Assessment of the reliability of elements and technical systems of vehicles during their design
7.11. The most common ways and methods of ensuring and predicting reliability used in the creation of machines
Chapter 8. System of maintenance and repair of machines
8.1. Maintenance and repair systems of machines, their essence, content and principles of construction
8.2. Requirements for the system of maintenance and repair, and methods for determining the frequency of their implementation
8.3. Operation of the machine in extreme situations
Chapter 9. Diagnostics as a method of monitoring and ensuring the reliability of the vehicle 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 limiting and permissible values of technical condition parameters
9.5. Principles of car diagnostics
9.6. Organization of car diagnostics in the maintenance and repair system
9.7. Types of car diagnostics
9.8. Diagnostics of car units during repair
9.9. Diagnostics of the state of the cylinder-piston group
9.10. The concept of diagnosing technology in modern conditions
9.11. Technical diagnostics is an important element of technological certification of services of service enterprises
9.12. Management of reliability, technical condition of machines based on the results of diagnostics
9.13. Diagnostics and vehicle safety
9.14. Brake system diagnostics
9.15. Headlight diagnostics
9.16. Suspension and steering diagnostics
Conclusion
Bibliography.
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Federal State Autonomous
educational institution
higher professional education
"SIBERIAN FEDERAL UNIVERSITY"
Department of Transport
Course work
In the discipline "Fundamentals of the theory of reliability and diagnostics"
Completed by a student, groups FT 10-06 V.V. Korolenko
Checked by V.V. Kovalenko
Received by Doctor of Technical Sciences, prof. N.F. Bulgakov
Krasnoyarsk 2012
INTRODUCTION
1 Analysis of research works on reliability and diagnostics
2 Assessment of vehicle reliability indicators
2.2 Point estimation
2.3 Interval assessment
2.5 Testing the null hypothesis
4 Second variation row
5 Assessment of indicators of the recovery process
CONCLUSION
LIST OF SOURCES USED
INTRODUCTION
reliability trouble-free operation recovery
The theory and practice of reliability studies the processes of failure and how to deal with them in the component parts of objects of any complexity - from large complexes to elementary details.
Reliability - the property of an object to keep in time within the established limits the value of all parameters characterizing the ability to perform the required functions in the specified modes and conditions of use, maintenance, repairs, storage and transportation.
Reliability is a complex property, which, depending on the purpose of the object and the conditions of its use, consists of combinations of properties: reliability, durability, maintainability and preservation.
There is an extensive system of state standards "Reliability in technology" described by GOST 27.001 - 81.
The main ones are:
GOST 27.002 - 83. Reliability in technology. Terms and Definitions.
GOST 27.003 - 83. Selection and standardization of reliability indicators. Basic provisions.
GOST 27.103 - 83. Criteria for failures and limit states. Basic provisions.
GOST 27.301-83. Predicting the reliability of products during design. General requirements.
GOST 27.410 - 83. Methods and plans for statistical control of reliability indicators on an alternative basis.
1 Analysis of research papers
The article tells about the outstanding engineer and entrepreneur A.E. Struve, who was the founder of the famous Kolomna machine-building plant (now OJSC “Kolomna plant). He was engaged in the construction of 400 railway platforms for the Moscow-Kursk railway. Under his leadership, the largest railway bridge in Europe across the Dnieper was built. Along with the freight yards, platforms and bridge structures, the Struve plant mastered the production of steam locomotives and passenger cars of all classes, service cars and tanks.
The article describes the activities of E.A. and M.E. Cherepanovs, who built the first steam locomotive in Russia. The steam locomotive, using a steam engine as a power plant, has long been the dominant type of locomotive and played a huge role in the development of railway communication.
The article describes the activities of V. Kh. Balashenko, a well-known creator of track technology, honored inventor, three times "Honorary Railwayman", laureate of the USSR State Prize. He has designed a snow-removing machine. At the same time, he manufactured a mobile conveyor for loading open wagons and a press for stamping anti-theft protectors from old-year rails. Developed 103 track lining machines, which replaced over 20 thousand track fitters.
The article tells about S.M.Serdinov, who was engaged in the feasibility study and preparation of the first projects of electrified sections, developed samples of electric rolling stock and equipment for power supply devices and subsequently put into operation the first electrified sections and their subsequent operation. Later S.M. Serdinov supported proposals to improve the energy efficiency of the 25 kV alternating current system, developed and implemented a 2x25 kV system, first on the Vyazma - Orsha section, and then on a number of other roads (more than 3 thousand km).
The article tells about B.S. Jacobi, who was one of the first in the world, used the electric motor he created for transport purposes - the movement of a boat (bot) with passengers along the Neva. He created a model of an electric motor consisting of eight electromagnets arranged in pairs on a movable and stationary wooden drum. For the first time, I used a commutator with rotating metal discs and copper levers in my electric motor, which, when sliding along the discs, provided current collection
The article describes the work of I.P. Prokofiev, who developed a number of original projects, including the arched floors of railway workshops at Perovo and Murom stations (the first three-span frame structures in Russia), the overlap of the landing stage (canopy in the train arrival and departure area) of the Kazan station in Moscow. He also developed a project for a railway bridge across the river. Kazanka and a number of typical designs of retaining walls of variable height.
The article describes the activities of V.G. Inozemtsev, Honored Scientist of the Russian Federation, inventor of brake technology, which is still used today. Created at VNIIZhT a unique laboratory base for studying the brakes of trains of large mass and length.
The article tells about F.P. Kochnev, doctor of technical sciences, professor. He developed the scientific principles of the organization of passenger transportation concerning the choice of a rational speed of movement of passenger trains and their weight. The solution of the problem of the rational organization of passenger traffic, the development of a system of technical and economic calculations for passenger traffic were of great importance.
The article tells about I. L. Perist, who established the technology of driving heavy freight trains, and improved the work of the passenger infrastructure and the formation of the largest networks of marshalling complexes. He was the main initiator of the unprecedented reconstruction of Moscow railway stations.
The article describes P. P. Melnikov, an outstanding Russian engineer, scientist and organizer in the field of transport, building the first long-distance railroad in Russia. The construction took almost 8 years.
The article describes the activities of I. I. Rerberg. He is a Russian engineer, architect, author of projects for the Kievsky railway station, he organized the protection of the line from snow drifts with the help of afforestation. On his initiative, the first in Russia impregnation plant was opened. He created mechanical workshops, which began the production of the first domestic cars. He worked to improve the working and living conditions of railway workers.
The article tells about the Russian engineer and scientist in the field of structural mechanics and bridge building N. A. Belyumbsky, who developed more than 100 projects of large bridges. The total length of bridges built according to his designs exceeds 17 km. These include bridges across the Volga, Dnieper, Ob, Kama, Oka, Neva, Irtysh, Belaya, Ufa, Volkhov, Neman, Selena, Ingulets, Chu sova yu, Berezina, etc.
The article describes the activities of S.P.Syromyatnikov, a Soviet scientist in the field of steam locomotive construction and heat engineering, who developed the design, modernization and thermal calculation of steam locomotives. The founder of the scientific design of steam locomotives; developed the theory and calculation of thermal processes, and also created the theory of the combustion process of steam locomotive boilers.
The article describes the work of V.N. Obraztsov, who proposed ways to solve the problems associated with the design of railway stations and junctions, organized the planning of sorting work on the railway network, as well as issues of interaction between railway services and various types of transport. He is the founder of the science of designing stations and junctions of a railway junction.
The article describes the activities of P.P. Roterte, the head of the metro construction, who organized the construction of the first stage of the Moscow metro. The following sections were approved for the first stage of construction: Sokolniki - Okhotny Ryad, Okhotny Ryad - Krymskaya Ploshchad and Okhotny Ryad - Smolenskaya Ploshchad. They provided for the construction of 13 stations and 17 ground lobbies.
2 Assessment of reliability indicators of railway facilities
78 35 39 46 58 114 137 145 119 64 106 77 108 112 159 160 161 101 166 179 189 93 199 200 81 215 78 80 91 98 216 224
2.1 Estimation of mean time between failures
As a result of statistical processing of the variational series, sample characteristics are obtained, which are necessary for further calculations.
2.2 Point estimation
A point estimate of the mean time to failure of an ATS element between replacements is the sample mean, thousand km:
where Li is the i-th member of the variation series, thousand km;
N - Sample size.
The number of members of the variation series is N = 32.
Lav = 1/32 3928 = 122.75
Dispersion (unbiased) of the point estimate of mean time to failure, (thousand km) 2:
D (L) = 1/31 (577288 - 482162) = 3068.5745
Mean square deviation, thousand km,
S (L) = = 55.39471
Coefficient of variation of a point estimate of the mean time to failure
The Weibull - Gnedenko form parameter in is determined according to Table 11, depending on the obtained coefficient of variation V.
If it is difficult to determine the shape in by the coefficient of variation, then we calculate the shape in according to the following algorithm:
1. Divide the obtained coefficient of variation into the sum of two numbers, and one of them determines the value of the form in from the table
V = 0.4512 = 0.44 + 0.0112
2. We find from table 11 the value of the form in for the coefficient of variation, decomposed in the sum and the next value of the form in
for V1 = 0.44 B1 = 2.4234
for V2 = 0.46 V2 = 2.3061
3. Find the difference? V and? In for the values we found
V = 0.46 - 0.44 = 0.02
B = 2.4234 - 2.3061 = 0.1173
4. Composing the proportion
5. Find the value of the form in for the coefficient of variation V = 0.45128
in = in (0.44) - in = 2, 4234 - 0, 06568 = 2, 35772
Let us determine q at b = 0.90, for which we calculate the significance level e and select the value (64) from Table 12:
Quantile of distribution:
Required accuracy of MTBF estimation:
e = (1-0.9) / 2 = 0.05
The calculated value of the marginal relative error:
q = ((2 * 32 / 46.595) ^ (1 / 2.3577)) - 1 = 0.1441
2.3 Interval assessment
With probability b, it can be argued that the mean time to failure of the L-13U pantograph is in the interval, which is the interval estimate.
The lower and upper boundaries of this interval are as follows:
Lsrn = 122.75 * (1-0.1441) = 105.0617
Lav = 122.75 * (1 + 0.1441) = 140.4382
As a result, we obtain point and interval estimates of the mean time to failure of the L-13U pantograph - one of the quantitative safety indicators. For non-renewable elements, it is at the same time an indicator of durability - an average resource.
2.4 Estimation of the scale parameter of the Weibull - Gnedenko law
The point estimate of the scale parameter a of the Weibull - Gnedenko law, we calculate by the formula, thousand km:
where Г (1 + 1 / в) is the gamma function for the argument x = 1 + 1 / в, which is taken from Table 12 depending on the coefficient of variation V. To find the gamma function Г (1 + 1 / в), we use by the same algorithm, similarly to the estimation of the shape parameter in the Weibull - Gnedenko law.
G (1 = 1 / c) = 0.8862
We obtain, respectively, the lower bound of the scale parameter
Upper bound
2.5 Testing the null hypothesis
The correspondence of the Weibull-Gnedenko law to the experimental distribution is checked by X2 - Pearson's goodness-of-fit criterion. There is no reason to reject the null hypothesis if the condition
X2calculation< Х2табл(,к), (2.9)
where is the criterion value calculated from experimental data;
The critical point (tabular value) of the criterion at the level of significance and the number of degrees of freedom (see Table 12 Appendix 1).
The significance level is usually taken equal to one of the values of the series: 0.1, 0.05, 0.025, 0.02, 0.01.
Number of degrees of freedom
k = S - 1 - r, (2.10)
where S is the number of partial sampling intervals;
r is the number of parameters of the assumed distribution.
With the two-parameter Weibull-Gnedenko law, k = S-3.
The null hypothesis is tested using the following algorithm:
S = 1 + 3.32 * lnN (2.11)
Divide the range of the variation series into S intervals, i.e. the difference between the largest and smallest numbers. The boundaries of the intervals are found by the formula
where j - 1,2,…., S.
Determine empirical frequencies, i.e. nj - the number of members of the variation series that fell into the j -th interval. When a zero interval occurs (nj = 0), this interval is divided into two parts and added to the neighboring ones with a recalculation of their boundaries and the total number of intervals.
where j = 1,2,…, S.
The failure distribution function included in formula (14) is determined by the formula (for the Weibull-Gnedenko law).
3) Determine the calculated value of the criterion
Hrasch2 = (2.15)
We will consider the assessment of the X2 criterion using the previously given example of a variation series.
1) Number of intervals S = 1 + 3.332 * ln316. The number of degrees of freedom k = 6 - 3 = 3. The level of significance is assumed to be 0.1. The tabular value of the X2tabl criterion (0.1; 3) = 6.251 (see Table 12). The range of the variation series 224-35 = 189 thousand km is divided into 6 intervals: 189/6 = 31.5 thousand km. It should be noted that the first interval starts at zero, and the last one ends at infinity.
Table 1 - Calculation of empirical frequencies
2) We calculate the theoretical frequencies by the formula (2.13) and determine the calculated value of the criterion X2calculated by the formula (2.15). For clarity, the calculation is summarized in Table 2.
Table 2 - Calculation of X2 - Pearson's goodness-of-fit test
3) As a result, we get that the calculated value of the criterion:
X2calculated = 33.968 - 32 = 1.968
X2calculated = 1.968 X2tabl = 6.251
The null hypothesis is accepted.
3 Assessment of quantitative characteristics of reliability and durability
3.1 Evaluation of the probability of failure-free operation
We calculate the quantitative characteristics of reliability using the example of the brake system. The probability of failure-free operation of the L-13U pantograph is estimated according to the Weibull-Gnedenko law, using the formula:
P (L) = exp [- (L / a)]. (3.1)
The interval estimate is determined by substituting the values of a and a in the formula (3.1), respectively, instead of a.
Table 3 - Point estimate of the probability of failure-free operation of the brake system before the first failure
|
L, thousand km |
||||
Figure 1 - Graph of the probability of failure-free operation of the pantograph L-13U
3.2 Estimation of gamma percent mean time to failure
According to GOST 27.002 - 83 gamma-percent operating time to failure Lj, thousand km, is the operating time during which a failure of an ATS element does not occur with probability j. For non-recoverable elements, it is at the same time an indicator of durability - a gamma - percentage resource (operating time during which an ATS element will not reach the limit state with a given probability j). For the Weibull - Gnedenko law, its point estimate, thousand km,
Lj = a * (- ln (j / 100)) 1 / b. (3.2)
We take the probability j to be equal to 90%, respectively. Then we get:
3.3 Assessment of failure rate
Failure rate (L), thousand km-1, is the conditional density of the probability of failure of the pantograph L-13U, determined for the considered moment of time, provided that no failure has occurred before this moment.
For the Weibull - Gnedenko law, its point estimate, refusal, thousand km,
(L) = in / av * (L) in-1. (3.3)
b = 2.3577; a = 138.1853
The interval estimate is determined by substituting into formula (3.3) instead of a the values of an and a.
Table 4 - Point estimate of the failure rate of the pantograph L-13U
|
L, thousand km |
||||
Figure 2 - Graph of the failure rate of the pantograph L-13U
3.4 Estimation of the density of distribution of failures
The failure distribution density f (L), thousand km-1, is the probability density that the operating time of the L-13U pantograph to failure will be less than L. For the Weibull - Gnedenko law:
f (L) = b / a * (L / a) b-1 * (3.4)
f (10) = 2.357 / 138.185 * (10 / 138.185) 2.3577-1 * 0.00048
Table 5 - Density of distribution of operating time to failure of pantograph L-13U
Figure 3 - Graph of the density distribution of failures of the pantograph L-13U
4 To simplify the problem, we calculate the second variational series using a computer program.
Variational range:
54 67 119 14 31 41 68 90 94 112 80 130 146 71 45 148 88 99 113
As a result of the calculation, we obtain the following tables and graphs.
Table 6 - initial data for estimating the mean time to failure
Table 7 - Calculation of X2 - Pearson's agreement criterion
X2calculated = 1.6105 X2tabl = 11.345
The null hypothesis is accepted.
Table 8 - Point estimate of the probability of failure-free operation of the pantograph L-13U
|
L, thousand km |
||||
Figure 4 - Graph of the probability of failure-free operation of the pantograph L-13U
Table 9 - Point estimate of the failure rate of the pantograph L-13U
|
L, thousand km |
||||
Figure 5 - Graph of the intensity of the first failures of the pantograph L-13U
Table 10 - Density of distribution of operating time to failure of pantograph L-13U
Figure 5 - Graph of the density distribution of failures of the pantograph L-12U
Table 11 - The results of calculating the main parameters of the 1st, 2nd variation series
|
Indicator |
First row |
Second row |
|
5 Assessment of indicators of the recovery process (graphic-analytical method)
Let's calculate an estimate of the average operating time before the first, second recovery:
Let's calculate the estimate of the standard deviation before the first, second recovery:
Let's calculate the distribution composition function before the first, second, third recovery, and enter the calculated data into the table.
The calculation of the functions of the distribution composition of the operating time before replacing the elements of the L-13U pantograph will be performed according to the formula:
where lcp is the mean time between failures;
Up - distribution quantile;
K - standard deviation
Table 12 - Calculation of the composition function of the distribution of operating time before replacements
|
l№ср ± Uр? у№к |
lІср ± Uр? уІк |
|||
Let's make a graphical construction of the distribution composition functions. Let's calculate the values of the leading function and the parameter of the failure flow at the intervals chosen by us. Let's enter the calculated data into tables and make a graphical construction (see Figure 6).
The calculation is carried out by the graphic-analytical method, the indicators are taken from the resulting graph and entered into the table.
Table 13 - Leading function definitions
The failure flow parameter is determined by the formula:
substitute values for
Let's calculate the parameter of the flow of failures for other values of the mileage, and enter the result into the table.
Table 13 - Determination of the parameter of the recovery flow
Figure 6 - Graphoanalytical method for calculating the characteristics of the recovery process,? (L) and u (L) pantograph L-13U
CONCLUSION
In the course of the course work, the theoretical knowledge in the discipline "Fundamentals of the theory of reliability and diagnostics", "Fundamentals of the operability of technical systems" was consolidated. For the first sample, the following were made: an estimate of the average technical resource before the replacement of vehicle elements (point estimate); calculation of the confidence interval for the average technical resource of the vehicle; estimation of the scale parameter of the Weibull-Gnedenko law; evaluation of the parameters of the null hypothesis, evaluation of the characteristics of probability theory: probability density and failure distribution function f (L), F (L); estimation of the probability of failure-free operation; determination of the need for spare parts; assessment of gamma - percentage time to failure; failure rate assessment; assessment of indicators of the recovery process (by graphic-analytical method); calculation of the leading recovery function; calculation of the parameter of the recovery flow; graphic-analytical method for calculating the leading function and the parameter of the restoration flow. The second variational series was calculated in a computer program developed especially for students “Model of statistical evaluation of the characteristics of reliability and efficiency of technology”.
The reliability assessment system allows not only to constantly monitor the technical condition of the rolling stock, but also to manage their performance. Operational planning of production, quality management of maintenance and repair of railway vehicles is facilitated.
LIST OF USED SOURCES
1 Bulgakov N.F., Burkhiev Ts. Ts. Quality management of vehicle prevention. Modeling and optimization: Textbook. allowance. Krasnoyarsk: IPC KSTU, 2004.184 p.
2 GOST 27.002-89 Reliability in technology. Basic concepts. Terms and Definitions.
3 Kasatkin G.S. Journal"Railway transport" No. 10, 2010.
4 Kasatkin GS Journal "Railway transport" No. 4, 2010.
5 Sadchikov P.I., Zaitseva T.N. Journal "Railway Transport" No. 12, 2009.
6 Prilepko A. I. Journal "Railway transport" No. 5, 2009.
7 Shilkin P.M. Journal "Railway Transport" No. 4, 2009.
8 Kasatkin G.S. Journal "Railway Transport" No. 12, 2008.
9 Balabanov V.I. Journal "Railway Transport" No. 3, 2008.
10 Anisimov P.S. Journal "Railway Transport" No. 6, 2006.
11 Levin B.A. Railway transport "No. 3, 2006.
12 X Abstract. The builder of the first railway in Russia. http://xreferat.ru.
13 GZD News. Bronze bust of Ivan Rerberg. http://gzd.rzd.ru.
14 Websib. Nikolay Apollonovich Belelyubsky. http://www.websib.ru.
15 Syromyatnikov S. P. Bibliography of scientists of the USSR. "Izvestia of the Academy of Sciences of the USSR. Department of technical sciences", 1951, No. 5.64s.
16 Wikipedia. Free encyclopedia. V.N. Obraztsov. http://ru.wikipedia.org.
17 Kasatkin G.S. Kasatkin "Railway Transport" No. 5 2010.
18 GZD News. An outstanding figure in the railway industry. http://www.rzdtv.ru.
19 Methodical manual "Fundamentals of the theory of reliability and diagnostics." 2012
Posted on Allbest.ru
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Professor T.P. Voskresenskaya
INTRODUCTION The importance of reliability theory
in modern technology.
The modern period of development of technology is characterized by the development and implementation of complex technical systems and complexes.
The main concepts that are used in this discipline are the concepts of a complex dynamic system and a technical device (TC) or an element that is part of the system. Complexity usually means complexity systems of individual elements, while not just the sum of the elements is considered, but their interaction. The interaction of elements and their properties change over time. The complexity of the interaction of elements and their number are two aspects of the concept of a complex dynamic system. The complexity of a system is determined not so much by the number of elements as by the number of connections between the elements themselves and between the system and the environment.
Complex dynamic systems are systems oversaturated with internal connections of elements and external connections with the environment.
Let us define a complex dynamic system as the formation of elements of various natures that have some functions and properties that are absent in each of the elements, and are able to function, statically correlating in a certain range with the environment, and due to this, preserve their structure during the continuous change of interacting elements along complex dynamic laws.
Complex dynamical systems are essentially nonlinear systems, the mathematical description of which is not always possible at the present stage.
Any complex dynamic system is created to solve a specific theoretical or production problem. In connection with the deterioration of the properties of the system during operation, there is a need for periodic maintenance, the purpose of which is to preserve the ability of the system to perform its functions. Therefore, information processes are of fundamental importance for complex dynamic systems. The cyclical nature of information processes is provided by a feedback mechanism. Based on information about the behavior of the system, management of its state is organized, taking into account the results of which the subsequent management of the system is adjusted.
When designing technical systems, it is necessary to provide for maintenance issues during the intended operation. Among other problems of design and creation of the complex:
Compliance with the specified technical requirements;
The cost-effectiveness of the complex, taking into account the tests and conditions of the intended operation;
Development of technical means of maintenance of the complex and software for them;
Ensure the suitability of the complex for work in the link "man - machine", etc.
Thus, even when designing a complex, attention should be focused on all the noted, interconnected issues as a whole, and not on each individual of them.
It is possible to design a complex that meets the given technical requirements, but does not meet the economic requirements, the requirements for maintenance and the functioning of the complex in the "man-machine" link. Consequently, the problem of creating a complex must be solved from the standpoint of a systems approach. The essence of this approach can be demonstrated with a simple example. Suppose that we have selected one car for each of the brands available for sale. Then we ask a group of experts to study them and choose the best carburetor, then choose the best engine, distributor, transmission, etc., until we collect all the car parts from different cars. We are unlikely to be able to assemble a car from these parts, and if we can, it will hardly work well. The reason is that the individual parts will not fit together. Hence the conclusion: it is better when the parts of the system fit together well, even if they work separately and do not perfectly, than when the perfectly working parts do not fit together. This is the essence of the systems approach.
Sometimes the improvement of one part of the complex leads to a deterioration in the technical characteristics of the other, so that the improvement becomes meaningless. A systematic approach to the analysis of the considered phenomena involves the use of a set of various mathematical methods, modeling methods and experiments.
In the proposed course, the solution of particular problems of servicing complex systems and their elements by the analytical method is considered and the features of solving more complex problems of operation by the method of statistical modeling are noted. In practice, the implementation of the obtained methods will lead to the analysis of the complex from the standpoint of a systematic approach.
The main features of a complex system or technical device (TU) are as follows:
Possessing a certain unity of purpose and promoting the development of optimal outputs from the available set of inputs; the optimality of the outputs should be assessed according to a previously developed criterion of optimality;
Performing a large number of different functions, which are carried out by many parts of the system;
Complexity of functioning, i.e. a change in one variable entails a change in many variables and, as a rule, in a non-linear manner;
High degree of automation;
Ability to quantitatively describe the disturbance entering the system.
The operation of a complex technical specification is a continuous process that includes a number of activities that require a planned, continuous impact on the technical specification to maintain it in working order. These activities include: routine maintenance, recovery after failure, storage, preparation for work, etc. The above definition of operation does not cover all those activities that make up the process of operating complex systems. Therefore, operation in a broad sense should be understood as the process of using technical specifications for the intended purpose and maintaining it in a technically sound condition.
The state of technical specifications is determined by a set of values of its technical characteristics. During operation, the technical characteristics of the device change continuously. For the organization of operation, it is important to distinguish between the states of technical specifications that meet the extreme or permissible (boundary) values of the technical characteristics, which correspond to the operating state, failure, the state of maintenance, storage, recovery, etc. For example, an engine is in working order if it provides the required thrust, provided that the values of all other characteristics are within the limits established in the technical documentation. The engine must be in a maintenance condition if its specification values have reached the appropriate limits. In this case, its immediate use for its intended purpose is impossible.
The main task of the theory of operation is to scientifically predict the states of complex systems or technical specifications and to develop, using special models and mathematical methods for the analysis and synthesis of these models, recommendations for organizing their operation. When solving the main problem of operation, a probabilistic-statistical approach is used to predict and control the states of complex systems and to model operational processes.
Some issues of the theory of operation, such as predicting the reliability of technical units under operating conditions, organizing the recovery of technical units during the execution of a task, diagnosing failures in complex systems, determining the required number of spare elements, etc., have received sufficient development in the theory of reliability, the theory of recovery and the theory of queuing. , in technical diagnostics and the theory of inventory management.
1. Basic concepts and definitions
reliability theory.
Reliability theory - the science of methods of ensuring and maintaining reliability in the design, manufacture and operation of systems.
The ability of any product or system to maintain its original technical characteristics during operation is determined by its reliability. The physical meaning of reliability is the ability of a technical specification to maintain its characteristics over time.
Operational characteristics are also readiness for use, recoverability, maintenance parameters. Reliability can be determined as an independent operational characteristic of the technical unit, and serve as a component of other operational characteristics.
Under reliability the property of technical specifications is understood to perform the specified functions, maintaining its performance indicators within the specified limits for the required period of time or the required operating time under certain operating conditions.
As follows from the definition, reliability depends on what functions the product performs in the time during which the performance of these functions must be ensured, and on the operating conditions.
Any product has many performance indicators and it is necessary to strictly stipulate in each case when the technical parameters or the technical specification property should be taken into account when determining its reliability.
In this regard, the concept is introduced operability , which is defined as the state of technical specifications, in which it is able to perform the specified functions with the parameters established by the requirements of the technical documentation. The introduction of the concept of operability is necessary to determine the technical parameters and properties of technical specifications that determine the performance of the specified functions and the permissible limits of their change.
It also follows from the definition of reliability that reliability consists in the ability of a technical specification to maintain its initial technical characteristics over time. However, even the most reliable technical specifications cannot maintain their initial technical characteristics for an unlimited time. Therefore, it makes no sense to talk about reliability without defining a specific period of time during which these characteristics should be provided. In addition, the actual reliability of each TU largely depends on the operating conditions. Any predetermined value of reliability is valid only for specific operating conditions, including modes of use of technical specifications.
In the theory of reliability, the concepts of an element and a system are introduced. The difference between them is purely conditional and consists in the fact that when determining the reliability, the element is considered indivisible, and the system is presented as a set of separate parts, the reliability of each of which is determined separately.
The concepts of element and system are relative. For example, one cannot assume that an airplane is always a system, and one of its engines is an element. An engine can be considered an element if, when determining reliability, it is considered as a whole. If it is divided into its constituent parts (combustion chamber, turbine, compressor, etc.), each of which has its own reliability value, then the engine is a system.
It is much more difficult to quantify or measure the reliability of a DUT than to measure any of its technical characteristics. As a rule, only the reliability of elements is measured, for which special, sometimes rather complex and lengthy tests are carried out or the results of observations of their behavior in operation are used.
The reliability of the systems is calculated based on the data on the reliability of the elements. As a starting point in determining the quantitative values of reliability, events are used, consisting in a malfunction of the TC and called failures.
Under rejection an event is understood after which the technical unit ceases to perform (partially or completely) its functions. The concept of failure is fundamental in the theory of reliability and the correct understanding of its physical essence is the most important condition for the successful solution of reliability issues.
In some cases, the system continues to perform the specified functions, but with some elements violations of technical characteristics appear. This state of the element is called a malfunction.
Malfunction - the state of the element, in which it currently does not meet at least one of the requirements established both in relation to the main and secondary parameters.
Let's consider some other concepts that characterize the operational qualities of technical specifications. In some cases, it is required that the technical device not only works reliably for a certain period of time, but, in spite of the presence of failures during interruptions in operation, would generally retain the ability to perform the specified functions for a long time.
The property of technical specifications to maintain operability with the necessary interruptions for maintenance and repairs up to the limit state defined in the technical documentation is called durability ... The limiting states of technical conditions can be: breakdown, limit wear, drop in power or productivity, decrease in accuracy, etc.
Tu can lose efficiency not only during operation, but also during long-term storage, as a result of aging. To emphasize the property of technical specifications to remain operational during storage, the concept of persistence was introduced, which makes sense of the reliability of technical specifications in storage conditions.
Persistence the property of technical specifications is called to have conditioned performance indicators during and after the storage and transportation period established in the technical documentation.
The concepts of service life, operating time and resource are of great importance in determining the operational characteristics of technical devices.
Service life is called the calendar duration of operation of the TU until the occurrence of the limiting state specified in the technical documentation. Under operating time means the duration (in hours or cycles) or the volume of work of the technical specification (in liters, kilograms, t-km, etc.) until a failure occurs ... Resource is called the total operating time of technical specifications to the limiting state specified in the technical documentation.
2. A quantitative measure of the reliability of complex systems
To select rational measures aimed at ensuring reliability, it is very important to know the quantitative indicators of the reliability of elements and systems. A feature of the quantitative characteristics of reliability is their probabilistic and statistical nature. Hence, the peculiarities of their definition and use follow. As practice shows, entering into operation similar technical specifications, for example, cars, even being manufactured at the same plant, show different ability to maintain their performance. In the process of operation, technical specifications failures occur at the most unexpected, unforeseen moments. The question arises, are there any patterns in the appearance of failures? Exist. Only to establish them, it is necessary to conduct observations not for one, but for many technical devices in operation, and for processing the results of observations, apply the methods of mathematical statistics and the theory of probability.
The use of quantitative assessments of reliability is necessary when solving the following problems:
Scientific substantiation of requirements for newly created systems and products;
Improving the quality of design;
Creation of scientific methods of testing and control of the level of reliability;
Justification of ways to reduce economic costs and reduce the time for product development;
Improving the quality and stability of production;
Development of the most effective operating methods;
Objective assessment of the technical condition of the equipment in operation;
Currently, in the development of the theory of reliability are distinguished two main directions :
Progress in technology and improvement of technology for the manufacture of elements and systems;
Rational use of elements in the design of systems - the synthesis of systems in terms of reliability.
3. Quantitative indicators of reliability
elements and systems.
The quantitative indicators of the reliability of elements and systems include:
Reliability factor R r ;
Probability of failure-free operation for a certain period of time P ( t ) ;
Mean time to first failure T cf for non-recoverable systems;
MTBF t Wed for recoverable systems:
Failure rate λ( t ) ;
Average recovery time τ Wed ;
μ( t ) ;
Reliability function R r ( t ).
Definitions of the named quantities:
R r – the likelihood of finding the product in working order.
P ( t ) - the probability that for a given period of time ( t ) the system will not fail.
T cf Is the mathematical expectation of the system operation time until the first failure.
t Wed is the mathematical expectation of the operating time of the system between successive failures.
λ( t ) - mathematical expectation of the number of failures per unit of time; for a simple stream of failures:
λ( t )= 1/ t Wed .
τ Wed Is the mathematical expectation of the system recovery time.
μ( t ) - mathematical expectation of the number of restorations per unit of time:
μ( t ) = 1 / τ av.
R r ( t ) - change in the reliability of the system over time.
4. Classification of systems for the purpose of calculating reliability.
Systems for the purpose of calculating reliability are classified according to several criteria.
1. By the peculiarities of functioning during the period of application:
Disposable systems; these are systems the reuse of which is impossible or impractical for any reason;
Reusable systems; these are systems that can be reused and can be carried out after the system has performed the functions assigned to it for the previous cycle of use.
2. By adaptability to recovery after failures:
Recoverable, if their performance, lost during failure, can be restored during operation;
Non-recoverable, if their performance, lost upon failure, cannot be recovered.
3. For the implementation of maintenance:
Out of service - systems, the technical condition of which is not monitored during operation and measures are not taken to ensure their reliability;
Serviced - systems, the technical condition of which is monitored during operation and appropriate measures are taken to ensure their reliability.
4. By the type of implemented maintenance:
With periodic maintenance - systems in which measures to ensure reliability are implemented only when carrying out scheduled maintenance work at predetermined intervals T about ;
With a random maintenance period - systems in which measures to ensure reliability are implemented at random intervals corresponding to the appearance of failures or the system reaching its maximum operational state;
Combined maintenance - systems in which, in the presence of scheduled maintenance and repair work, there are maintenance items with a random period.
5. Classification of systems by structure.
The indicators of the reliability of systems depend not only on the indicators of the reliability of the elements, but also on the methods of “connecting” the elements into the system. Depending on the method of "connecting" the elements into the system, block diagrams are distinguished: a. sequential (main connection); b. parallel (redundant connection); in. combined (in the block diagram, there is both the main and the redundant connection of the elements); see fig. one.
Fig. 1. Structures of systems for the purpose of calculating reliability.
The assignment of the system structure to the main or redundant does not depend on the physical relative arrangement of the elements in the system; it only depends on the effect of element failures on the reliability of the entire system.
The main structures of the system are characterized by the fact that the failure of one element causes the failure of the entire system.
Redundant system structures are those in which a failure occurs when all or a certain number of elements that make up the system fail.
Redundant structures can be with general redundancy, redundancy by groups of elements and with element-by-element redundancy (see Fig. 2, a., B., C.).
Figure 2. Options for system redundancy.
The classification of the system in terms of structure is not constant, but depends on the purpose of the calculation. The same system can be primary and redundant; for example, what "connection" do the engines of a four-engined airplane have? The answer is twofold.
If we consider the system from the point of view of a technician servicing the aircraft, then the engines are "connected" in series, since the plane cannot be released on a flight if at least one engine is faulty; thus, failure of one element (motor) means failure of the entire system.
If we consider the same system in flight, then from the point of view of the pilots, it will be redundant, because the system will fail completely if all engines fail.
6. Classification of failures and malfunctions of systems and elements.
Failures are of a different nature and are classified according to several criteria. The main ones are as follows:
- the impact of failure on work safety : dangerous, safe;
- the impact of failure on the operation of the main mechanism : leading to downtime; reducing the performance of the main mechanism; not leading to downtime of the main mechanism;
- nature of failure elimination : urgent; not urgent; compatible with the operation of the main mechanism; incompatible with the operation of the main mechanism;
- outward manifestation of failure : explicit (obvious); implicit (hidden);
- duration of failure elimination : short-term; long;
- nature of failure : sudden; gradual; dependent; independent;
- cause of failure : constructional; manufacturing; operational; erroneous; natural;
- time of failure : during storage and transportation; during the start-up period; before the first major overhaul; after a major overhaul.
All of the above types of failures are of a physical nature and are considered technical.
In addition to them, technological failures can occur in systems consisting of autonomous elements (machines, mechanisms, devices).
Technological - these are failures associated with the performance of individual elements of auxiliary operations that require stopping the operation of the main mechanism of the system.
Technological failures occur in the following cases:
Performing operations preceding the cycle of the main system mechanism;
Execution of operations following the cycle of the main mechanism, but not compatible with the execution of a new cycle;
The cycle of development of the main mechanism of the system is less than the cycle of development of an auxiliary element in the technological process;
A technological operation performed by any element is incompatible with the operation of the main mechanism of the system;
System transition to a new state;
Inconsistency of the operating conditions of the system with the conditions specified in the passport characteristics of the system mechanisms.
7. The main quantitative dependencies in the calculation of systems for reliability.
7.1. Statistical analysis of the operation of elements and systems.
The qualitative and quantitative characteristics of the reliability of the system are obtained as a result of the analysis of statistical data on the operation of elements and systems.
When determining the type of the distribution law of a random variable, which includes the intervals of no-failure operation and recovery time, the calculations are performed in the sequence:
Preparation of experimental data; this operation consists in the fact that primary sources about the operation of systems and elements are analyzed to identify clearly erroneous data; the statistical rad is represented in the form of a variational, i.e. placed as the random value increases or decreases;
Building a histogram of a random variable;
Approximation of experimental distribution by theoretical dependence; verification of the correctness of the approximation of the experimental distribution by the theoretical one using the goodness-of-fit criteria (Kolmogorov, Pearson, omega-square, etc.).
As observations carried out in various fields of technology show, the flow of failures and restorations is the simplest, i.e. possesses ordinariness, stationarity and absence of aftereffect.
The reliability of complex systems, as a rule, obeys an exponential law, which is characterized by dependencies:
Uptime probability:
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Uptime distribution function:
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Uptime distribution density:
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f (t)These dependencies correspond to the simplest flow of failures and are characterized by constants:
Failure rate λ( t ) = const ;
Recovery rate μ( t ) = const ;
MTBF t Wed = 1 / λ ( t ) = const ;
Recovery time τ cf = 1 / μ ( t ) = const .
Parameters λ( t ), t Wed ; μ( t ) and τ Wed - are obtained as a result of processing the variation series by timing observation of the operation of elements and systems.
7.2. Calculation of the coefficient of reliability of elements.
The element reliability coefficient is determined according to the data of statistical processing of the variation series according to the formulas:
or 
(1)
as well as in terms of failure rate and recovery λ( t ) and μ( t ) :
. (2)
In industrial transport systems, a distinction should be made between technical and technological failures. Accordingly, the characteristics of the reliability of elements in technical and technological terms are the coefficients of technical r t i and technological r ci reliability of elements. The reliability of the element as a whole is determined by the dependence:
r r i = r t i · r ci . (3)
7.3. Calculation of the technical reliability of the system.
The reliability of the main system (system of series-connected elements) is determined in the presence of only technical failures by the dependence:
with equally reliable elements:
Where n - the number of elements connected in series in the system;
When calculating quantitative indicators of redundant and combined structures of systems, it is necessary to know not only their reliability, but also the unreliability of the element; since reliability r i and unreliability q i element make up the total sum of probabilities equal to one, then:
q i =(1 - r i ) . (6)
The unreliability of a redundant system (with parallel connection of elements) is defined as the probability that all elements of the system have failed, i.e.:
(7)
Reliability, respectively, is determined by the dependence:
(8)
Or, with equally reliable elements
, (9)
Where m - the number of spare elements.
Power ( m + 1) when calculating the reliability of the system, it is explained by the fact that in the system one element is required, and the number of reserve elements can vary from 1 to m .
As already noted, redundancy in combined systems can be element-by-element, group of elements, and element-by-element. System reliability indicators depend on the type of redundancy in the combined system. Consider these options for different ways of developing the system.
The reliability of combined redundant systems with general redundancy (system redundancy) is determined by the dependence:
(10)
with equally reliable elements (hence, subsystems):
(11)
The reliability of combined systems with redundancy by groups of elements is determined sequentially; first, the reliability of the redundant subsystems is determined, then the reliability of the system of serially connected subsystems.
The reliability of combined systems with element-by-element (separate) redundancy is determined sequentially; firstly, the reliability of block elements is determined (an element reserved by one, two, etc. to m elements), then - the reliability of the system of series-connected block elements.
The reliability of the block element is equal to:
; (12)
R to j for item-by-item reservation is equal to:
; (13)
or with equally reliable elements:
(14)
Consider example calculation of system reliability without redundancy and with various forms of its development (redundancy).
A system consisting of four elements is given (see Fig. 1.):
| r 1 = 0,95 | r 2 = 0,82 | r 3 = 0,91 | r 4 = 0,79 | |||||||
Figure 1. Block diagram of the (main) system.
Main system reliability:
0.95 0.82 0.91 0.79 = 0.560.
The reliability of the combined system with general (system) redundancy will be (see Fig. 2):
| r 1 = 0,95 | r 2 = 0,82 | r 3 = 0,91 | r 4 = 0,79 | |||||||||||||||||
| r 1 = 0,95 | r 2 = 0,82 | r 3 = 0,91 | r 4 = 0,79 | |||||||||||||||||
Figure 2. Block diagram of a combined system with system redundancy.
1- (1- 0,560) 2 = 1 – 0,194 = 0,806.
The reliability of a combined system when backing up with groups of elements will depend on how the elements are grouped; in our example, we group the elements as follows (see Fig. 3):
| r 1 = 0,95 | r 2 = 0,82 | r 3 = 0,91 | r 4 = 0,79 | |||||||||||||||||
| r 1 = 0,95 | r 2 = 0,82 | r 3 = 0,91 | r 4 = 0,79 | |||||||||||||||||
Figure 3. Block diagram of a combined system with redundancy by groups of elements.
Reliability of the first subgroup R o1 of the 1st and 2nd series connected elements will be equal to:
0.95 0.82 = 0.779;
Reliability of a block element of the first subgroup:
= 1- (1- 0,779) 2 = 0,951.
Reliability of the second subgroup R OP of the 3rd and 4th elements connected in series will be equal to:
0.91 0.79 = 0.719.
Reliability of a block element of the second subgroup:
= 1 – (1 – 0,719) 2 = 0,921.
System reliability R cop of two subsystems connected in series will be equal to:
0.951 0.921 = 0.876.
Combined system reliability R to j with element-by-element redundancy, it is equal to the product of the reliability of block elements, each consisting of one element of the system (see Fig. 4)
| r 1 = 0,95 | r 2 = 0,82 | r 3 = 0,91 | r 4 = 0,79 | ||||||||||||||||||||||
| r 1 = 0,95 | r 2 = 0,82 | r 3 = 0,91 | r 4 = 0,79 | ||||||||||||||||||||||
Figure 4. Block diagram of a combined system with element-by-element redundancy.
The reliability of a block element is determined by the formula:
;
For the first element: r j 1 = 1 – (1 – 0,95) 2 = 0,997;
For the second element: r j 2 = 1 – (1 – 0,82) 2 = 0,968;
For the third element: r j 3 = 1 – (1 – 0,91) 2 = 0, 992;
For the fourth element: r j 4 = 1 – (1 – 0,79) 2 = 0,956.
For a system of series-connected block elements:
0.997 0.968 0.992 0.956 = 0.915.
As the calculation example shows, the more connections between the elements of the system, the higher its reliability.
7.4. Calculation of the technical readiness of the system.
The system readiness parameters in the presence of technical and technological failures are determined by the formula:
.
Where r r i - technical reliability of the element;
r ci - technological reliability of the element;
r r i - generalized reliability of the element.
When reserving elements, the change in technical and technological reliability occurs in different ways: technical - according to a multiplicative scheme, technological - according to an additive scheme, while the maximum technological reliability can be equal to one.
Hence, with double redundancy of the element, we obtain its reliability of the block element:
With an arbitrary number of spare elements m:
where m is the number of spare elements.
The availability of combined systems is determined similarly to the definition of reliability in the presence of only technical failures, i.e. the readiness of the block elements is determined, and according to their indicators, the readiness of the entire system.
7. Formation of the optimal structure of the system.
As the results of calculations show, with the development of the structure of the system, its reliability asymptotically approaches one, while the cost of forming the system increases linearly. Since the operational productivity of the system is the product of its reliability by the nominal (passport) productivity, the outrunning increase in costs in the formation of the system with a slowing growth of its reliability will lead to the fact that the costs per unit of productivity will increase and further development of the structure of the system will become economically inexpedient. Thus, the solution to the question of the expedient reliability of the system is an optimization problem.
The objective function of the system optimization is as follows:
where is the total cost of the system; - the availability factor of the combined system achieved on the basis of these costs.
EXAMPLE Initial conditions: the main system of the form is set (see figure):
Figure 5. Structure of the main system, reliability indicators
elements and notional values of elements.
It is required to determine the optimal redundancy ratio of the third element of the system (the rest of the elements are not redundant).
Decision:
1. Determine the reliability of the main system:
0.80 · 0.70 · 0.65 · 0.90 = 0.328.
2. Determine the cost of the main system:
С о == 20 + 30 + 12 + 50 = 112 USD
3. Determine the unit costs to achieve a given availability factor of the main system:
