Ramakumar, R. "Reliability Engineering The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Ramakumar, R. “Reliability Engineering” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
110 Reliability engineering 110.2 Catastrophic Failure Models 110.3 The Bathtub Curve 110.4 Mean Time To Failure(MTTF) 110.5 Average Failure Rate 110.6 A Posteriori Failure Probability 110.7 Units for Failure Rates 110.8 Application of the Binomial Distribution 110.9 Application of the Poisson Distribution 110.10TheE 110.11 The Weibull Distribution 110.12 Combinatorial Aspect 10.13 Modeling Maintenance 110.14 Markov models 110.15 Binary Model for a Repairable Component 110.16 Two Dissimilar Repairable Components 110.17 Two Identical Repairable Components 110.18 Frequency and Duration Techniques 110.19 Applications of Markov Process 110.20 Some Useful Approximations R. Ramakumar 110.21 Application Aspects Oklahoma State University 110.22 Reliability and Economics 110.1 Introduction Reliability engineering is a vast field and it has grown significantly during the past five decades(since World War II). The two major approaches to reliability assessment and prediction are (1)traditional methods based on probabilistic assessment of field data and(2)methods based on the analysis of failure mechanisms and physics of failure. The latter is more accurate, but is difficult and time consuming to implement. The first one, in spite of its many flaws, continues to be in use. Some of the many areas encompassing reliability engineering are reliability allocation and optimization, reliability growth and modeling, reliability testing including accel erated testing, data analysis and graphical techniques, quality control and acceptance sampling, maintenance engineering, repairable system modeling and analysis, software reliability, system safety analysis, Bayesian sis,reliability management, simulation and Monte Carlo techniques, Failure Modes, Effects and Criticality Analysis(FMECA), and economic aspects of reliability, to mention a few Application of reliability techniques is gaining importance in all branches of engineering because of its effectiveness in the detection, prevention, and correction of failures in the design, manufacturing, and operational ome of the material in this chapter was previously published by CRC Press in The Engineering Handbook, R C. Dorf, d,1996 c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 110 Reliability Engineering1 110.1 Introduction 110.2 Catastrophic Failure Models 110.3 The Bathtub Curve 110.4 Mean Time To Failure (MTTF) 110.5 Average Failure Rate 110.6 A Posteriori Failure Probability 110.7 Units for Failure Rates 110.8 Application of the Binomial Distribution 110.9 Application of the Poisson Distribution 110.10 The Exponential Distribution 110.11 The Weibull Distribution 110.12 Combinatorial Aspects 110.13 Modeling Maintenance 110.14 Markov Models 110.15 Binary Model for a Repairable Component 110.16 Two Dissimilar Repairable Components 110.17 Two Identical Repairable Components 110.18 Frequency and Duration Techniques 110.19 Applications of Markov Process 110.20 Some Useful Approximations 110.21 Application Aspects 110.22 Reliability and Economics 110.1 Introduction Reliability engineering is a vast field and it has grown significantly during the past five decades (since World War II). The two major approaches to reliability assessment and prediction are (1) traditional methods based on probabilistic assessment of field data and (2) methods based on the analysis of failure mechanisms and physics of failure. The latter is more accurate, but is difficult and time consuming to implement. The first one, in spite of its many flaws, continues to be in use. Some of the many areas encompassing reliability engineering are reliability allocation and optimization, reliability growth and modeling, reliability testing including accelerated testing, data analysis and graphical techniques, quality control and acceptance sampling, maintenance engineering, repairable system modeling and analysis, software reliability, system safety analysis, Bayesian analysis, reliability management, simulation and Monte Carlo techniques, Failure Modes, Effects and Criticality Analysis (FMECA), and economic aspects of reliability, to mention a few. Application of reliability techniques is gaining importance in all branches of engineering because of its effectiveness in the detection, prevention, and correction of failures in the design, manufacturing, and operational 1 Some of the material in this chapter was previously published by CRC Press in The Engineering Handbook, R. C. Dorf, Ed., 1996. R. Ramakumar Oklahoma State University
INFORMATION MANAGEMENT SYSTEM FOR MANUFACTURING EFFICIENCY A t current schedules, each of NASA's four Space Shuttle Orbiters must fly two or three times a year. Preparing an orbiter for its next mission is an incredibly complex process and much of the work is accomplished in the Orbiter Processing Facility(OPF)at Kennedy Space Center The average "flow-the complete cycle of refurbishing an orbiter- requires the integration of approximately 10,000 work events, takes 65 days, and some 40,000 technician labor hours. Under the best conditions, scheduling each of the 10,000 work events in a single flow would be a task of monumental proportions. But the job is further complicated by the fact that only half the work is standard and predictable; the other half is composed of problem-generated tasks and jobs specific to the next mission, which creates a highly dynamic processing environment and requires frequent rescheduling For all the difficulties, Kennedy Space Center and its prime contractor for shuttle processing Lockheed Space Operations Company(LSOC)-are doing an outstanding job of managing OPF oper ations with the help of a number of processing innovations in recent years. One of the most important the Ground Processing Scheduling System, or GPSS. The GPSS is a software system for enhancing efficiency by providing an automated scheduling tool that predicts conflicts between scheduled tasks, helps human schedulers resolve those conflicts, and searches for near-optimal schedules GPSS is a cooperative development of Ames Research Center, Kennedy Space Center, LSOC, and a lated company, Lockheed Missiles and Space Company. It originated at Ames, where a group of computer scientists conducted basic research on the use of artificial intelligence techniques to automate the scheduling process. A product of the work was a software system for complex, multifaceted operations known as the Gerry scheduling engine Kennedy Space Center brought Ames and Lockheed together and the group formed an inter-center/NASA ntractor partnership to transfer the technology of the Gerry scheduling engine to the Space Shuttle program. The transfer was successfully accomplished and gPSs has become the accepted general purpose scheduling tool for OPF operations. Courtesy of National Aeronautics and Space Administration. Pace Center technicians are preparing a space Shuttle orbiter for its next mission,①i时t extremely complex scheduling job. ( Photo courtesy of National Aeronautics and Space Administration. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC INFORMATION MANAGEMENT SYSTEM FOR MANUFACTURING EFFICIENCY t current schedules, each of NASA’s four Space Shuttle Orbiters must fly two or three times a year. Preparing an orbiter for its next mission is an incredibly complex process and much of the work is accomplished in the Orbiter Processing Facility (OPF) at Kennedy Space Center. The average “flow” — the complete cycle of refurbishing an orbiter — requires the integration of approximately 10,000 work events, takes 65 days, and some 40,000 technician labor hours. Under the best conditions, scheduling each of the 10,000 work events in a single flow would be a task of monumental proportions. But the job is further complicated by the fact that only half the work is standard and predictable; the other half is composed of problem-generated tasks and jobs specific to the next mission, which creates a highly dynamic processing environment and requires frequent rescheduling. For all the difficulties, Kennedy Space Center and its prime contractor for shuttle processing — Lockheed Space Operations Company (LSOC) — are doing an outstanding job of managing OPF operations with the help of a number of processing innovations in recent years. One of the most important is the Ground Processing Scheduling System, or GPSS. The GPSS is a software system for enhancing efficiency by providing an automated scheduling tool that predicts conflicts between scheduled tasks, helps human schedulers resolve those conflicts, and searches for near-optimal schedules. GPSS is a cooperative development of Ames Research Center, Kennedy Space Center, LSOC, and a related company, Lockheed Missiles and Space Company. It originated at Ames, where a group of computer scientists conducted basic research on the use of artificial intelligence techniques to automate the scheduling process. A product of the work was a software system for complex, multifaceted operations known as the Gerry scheduling engine. Kennedy Space Center brought Ames and Lockheed together and the group formed an inter-center/NASA contractor partnership to transfer the technology of the Gerry scheduling engine to the Space Shuttle program. The transfer was successfully accomplished and GPSS has become the accepted general purpose scheduling tool for OPF operations. (Courtesy of National Aeronautics and Space Administration.) Kennedy Space Center technicians are preparing a Space Shuttle Orbiter for its next mission, an intricate task that requires scheduling 10,000 separate events over 65 days. A NASA-developed computer program automated this extremely complex scheduling job. (Photo courtesy of National Aeronautics and Space Administration.) A
phases of products and systems. Increasing emphasis being placed on quality of components and systems, pled with pressures to minimize cost and increase value, further emphasize the need to study, understand, quantify, and predict reliability and arrive at innovative designs and operational and maintenance procedure From the electrical engineering point of view, two(among several)areas that have received significant attention are electronic equipment(including computer hardware)and electric power systems. Other major areas include communication systems and software engineering. As the complexity of electronic equipment grew during and after World War II and as the consequences of failures in the field became more and more apparent, the U.S. military became seriously involved, promoted the formation of groups, and became instru- mental in the development of the earliest handbooks and specifications. The great northeast blackout in the U.S. in November 1965 triggered the serious application of reliability concepts in the power systems area. he objectives of this chapter are to introduce the reader to the fundamentals and applications of classical reliability concepts and bring out the important benefits of reliability considerations. Brief summaries of application aspects of reliability for electronic systems and power systems are also included 110.2 Catastrophic Failure Models Catastrophic failure refers to the case in which repair of the component is either not possible or available or of no value to the successful completion of the mission originally planned. Modeling such failures is typicall based on life test results We nsider the "lifetime or "time to failure as a continuous random variable Then P(survival up to time t)=P(T>I=R(O) where R(r) is the reliability function. Obviously, as t -oo, R(t)0 since the probability of failure increases with time of operation. Moreover, P(failure at t)=P(T <)=Q(0 where Q(r) is the unreliability function. From the definition of the distribution function of a continuous random variable, it is clear that Q(r) is indeed the distribution function for T. Therefore, the failure density function f(r) can be obtained d dt The hazard rate function (t) is defined as probability of failure in(t, t+Ar). given survival up to (110.4) It can be shown that f(t (110.5) R(t) The four functions, f(t), Q(n), R(t), and n(r) constitute the set of functions used in basic reliability analysis. The relationships between these functions are given in Table 110.1 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC phases of products and systems. Increasing emphasis being placed on quality of components and systems, coupled with pressures to minimize cost and increase value, further emphasize the need to study, understand, quantify, and predict reliability and arrive at innovative designs and operational and maintenance procedures. From the electrical engineering point of view, two (among several) areas that have received significant attention are electronic equipment (including computer hardware) and electric power systems. Other major areas include communication systems and software engineering. As the complexity of electronic equipment grew during and after World War II and as the consequences of failures in the field became more and more apparent, the U.S. military became seriously involved, promoted the formation of groups, and became instrumental in the development of the earliest handbooks and specifications. The great northeast blackout in the U.S. in November 1965 triggered the serious application of reliability concepts in the power systems area. The objectives of this chapter are to introduce the reader to the fundamentals and applications of classical reliability concepts and bring out the important benefits of reliability considerations. Brief summaries of application aspects of reliability for electronic systems and power systems are also included. 110.2 Catastrophic Failure Models Catastrophic failure refers to the case in which repair of the component is either not possible or available or of no value to the successful completion of the mission originally planned. Modeling such failures is typically based on life test results. We can consider the “lifetime” or “time to failure” T as a continuous random variable. Then, (110.1) where R(t) is the reliability function. Obviously, as t Æ •, R(t) Æ 0 since the probability of failure increases with time of operation. Moreover, (110.2) where Q(t) is the unreliability function. From the definition of the distribution function of a continuous random variable, it is clear that Q(t) is indeed the distribution function for T. Therefore, the failure density function f(t) can be obtained as (110.3) The hazard rate function l(t) is defined as (110.4) It can be shown that (110.5) The four functions, f(t), Q(t), R(t), and l(t) constitute the set of functions used in basic reliability analysis. The relationships between these functions are given in Table 110.1. P(survival up to time t) = > P(T t) º R(t) P(failure at t) = £ P(T t) º Q(t) f t d dt ( ) = Q t( ) l t t t t t t ( ) º Æ [ ] + D D D 0 1 lim ( , ), given survival up to t probability of failure in l t f t R t ( ) = ( ) ( )
TABLE 110.1 Relationships Between Different Reliability Functie f(r) (t) R(o f(r)=f(r) M(r)exp1-m(E 入(r)= I-Q0) dr(Q(r) In r(r) Q(t)=f()d 1-R(r) R()=1-|f(k 1-Q) R() () FIGURE 110.1 Bathtub-shaped hazard function 110.3 The Bathtub Curve Of the four functions discussed, the hazard rate function i(r)displays the different stages during the lifetime of a component most clearly. In fact, typical(o) plots have the general shape of a bathtub as shown in Fig. 110.1 The first region corresponds to wearin(infant mortality)or early failures during debugging. The hazard rate goes down as debugging continues. The second region corresponds to an essentially constant and low failure rate and failures can be considered to be nearly random. This is the useful lifetime of the component. The third region corresponds to wearout or fatigue phase with a sharply increasing hazard rate. Burn-in"refers to the practice of subjecting components to an initial operating period of t(see Fig 110.1) before delivering them to the customer. This eliminates all the initial failures from occurring after delivery to customers requiring high-reliability components. Moreover, it is prudent to replace a component as it approaches the wearout region, i.e., after an operating period of (t-4). Electronic components tend to have a long useful life(constant hazard) period. Wearout region tends to dominate in the case of mechanical components. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 110.3 The Bathtub Curve Of the four functions discussed, the hazard rate function l(t) displays the different stages during the lifetime of a component most clearly. In fact, typical l(t) plots have the general shape of a bathtub as shown in Fig. 110.1. The first region corresponds to wearin (infant mortality) or early failures during debugging. The hazard rate goes down as debugging continues. The second region corresponds to an essentially constant and low failure rate and failures can be considered to be nearly random. This is the useful lifetime of the component. The third region corresponds to wearout or fatigue phase with a sharply increasing hazard rate. “Burn-in” refers to the practice of subjecting components to an initial operating period of t1 (see Fig. 110.1) before delivering them to the customer. This eliminates all the initial failures from occurring after delivery to customers requiring high-reliability components. Moreover, it is prudent to replace a component as it approaches the wearout region, i.e., after an operating period of (t2-t1). Electronic components tend to have a long useful life (constant hazard) period.Wearout region tends to dominate in the case of mechanical components. TABLE 110.1 Relationships Between Different Reliability Functions f(t) l(t) Q(t) R(t) f(t) = f(t) l(t) Q(t) 1 – R(t) 1 – Q(t) R(t) FIGURE 110.1 Bathtub-shaped hazard function l(t d ) exp l(x) x t - È Î Í ˘ ˚ ˙ Ú0 d dt Q( )t - d dt R( )t l x x ( ) ( ) ( ) t f t f d t = - Ú 1 0 1 1 - Q t d dt Q t ( ) ( ( )) - d dt [ln R(t)] Q t f d t ( ) = ( ) Ú x x 0 1 0 - -È Î Í ˘ ˚ ˙ Ú exp l(x)dx t R t f d t ( ) = - ( ) Ú 1 0 x x exp - ( ) È Î Í ˘ ˚ ˙ Ú l x dx t 0 l (t) t1 t2 t I II III