《电子工程师手册》学习资料（英文版）ch110 Reliability engineering

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 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 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 oper￾ations 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 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. 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

110.4 Mean Time To Failure(MTTF The mean or expected value of the continuous random variable"time-to-failure"is the MTTE This is a very useful parameter and is often enough to assess the suitability of components. It can be obtained using either the failure density function f(r) or the reliability function R(t) as follows: MTTF=tf( dt or r(tdt (110.6) In the case of repairable components, the repair time can also be considered as a continuous random variable with an expected value of MTTR The mean time between failures, MTBE is the sum of MTTF and MTTR. Since for well-designed components MTTR<<MTTE MTBF and MTTF are often used interchangably. 110.5 Average Failure Rate The average failure rate over the time interval o to t is defined as In rlt AFR(O, T)= AFR(T) 110.6 A Posteriori Failure Probability When components are subjected to a burn-in(or wearin)period of duration T, and if the component survives during(0, T), the probability of failure during(T, T+n) is called the a posteriori failure probability Q(n. It can be found using f(E)dE (110.8) f(sds The probability of survival during(T, T+t)is R(n)=1-Q()= f(5R(T+) R(T) -4((109 f(s)ds 110.7 Units for Failure rates Several units are used to express failure rates. In addition to i(r) which is usually in number per hour, %/K is used to denote failure rate in percent per thousand hours and PPM/K is used to express failure rate in parts per million per thousand hours. The last unit is also known as FIT"fails in time". The relatio these units are given in Table 110.2 e 2000 by CRC Press LLC

© 2000 by CRC Press LLC 110.4 Mean Time To Failure (MTTF) The mean or expected value of the continuous random variable “time-to-failure” is the MTTF. This is a very useful parameter and is often enough to assess the suitability of components. It can be obtained using either the failure density function f(t) or the reliability function R(t) as follows: (110.6) In the case of repairable components, the repair time can also be considered as a continuous random variable with an expected value of MTTR. The mean time between failures, MTBF, is the sum of MTTF and MTTR. Since for well-designed components MTTR<<MTTF, MTBF and MTTF are often used interchangably. 110.5 Average Failure Rate The average failure rate over the time interval 0 to T is defined as (110.7) 110.6 A Posteriori Failure Probability When components are subjected to a burn-in (or wearin) period of duration T, and if the component survives during (0, T), the probability of failure during (T, T+t) is called the a posteriori failure probability Qc(t). It can be found using (110.8) The probability of survival during (T, T+t) is (110.9) 110.7 Units for Failure Rates Several units are used to express failure rates. In addition to l(t) which is usually in number per hour, %/K is used to denote failure rate in percent per thousand hours and PPM/K is used to express failure rate in parts per million per thousand hours. The last unit is also known as FIT for “fails in time”. The relationships between these units are given in Table 110.2. MTTF = t f(t)dt R(t)dt • • Ú Ú 0 0 or AFR T AFR T R T T 0, ln ( ) º ( ) = - ( ) Q t f d f d c T T t T ( ) = ( ) ( ) + • Ú Ú x x x x R tT Q t f d f d R T t R T d c T t T T T t ( ) = - ( ) = ( ) ( ) = ( + ) ( ) = - ( ) È Î Í ˘ ˚ ˙ + • • + Ú Ú Ú 1 x x x x exp l x x

TABLE 110.2 Relationships Between Different Failure Rate Units PPM/K(FIT) 105(%/K) %/K=105λ 0-(PPM/K) PPMK(FIT)=109λ 104(%/K PPM/K 110.8 Application of the Binomial Distribution In an experiment consisting of n identical independent trials, with each trial resulting in success or failure with probabilities of p and g, the probability P, of r successes and(n-r) failures is P,=Cp(1-p) (110.10) If X denotes the number of successes in n trials, then it is a discrete random variable with a mean value of (np) and a variance of (npg) In a system consisting of a collection of n identical components with a probability p that a component is defective, the probability of finding r defects out of n is given by the P, in Eq (110.10). If p is the probability of success of one component and if at least r of them must be good for system success, then the system reliability (probability of system success)is given by R=∑(-p)y For systems with redundancy, r0 (110.13) c 2000 by CRC Press LLC

© 2000 by CRC Press LLC 110.8 Application of the Binomial Distribution In an experiment consisting of n identical independent trials, with each trial resulting in success or failure with probabilities of p and q, the probability Pr of r successes and (n-r) failures is (110.10) If X denotes the number of successes in n trials, then it is a discrete random variable with a mean value of (np) and a variance of (npq). In a system consisting of a collection of n identical components with a probability p that a component is defective, the probability of finding r defects out of n is given by the Pr in Eq. (110.10). If p is the probability of success of one component and if at least r of them must be good for system success, then the system reliability (probability of system success) is given by (110.11) For systems with redundancy, r - l l , 0

R()=c t (110.14) Qt)=Q()=1-c (110.15) The a posteriori failure probability Q(n) is independent of the prior operating time T, indicating that the component does not degrade no matter how long it operates. Obviously, such a scenario is valid only during the useful lifetime(horizontal portion of the bathtub curve)of the component. The mean and standard deviation of the random variable "lifetime"are μ≡MTTF 1 1 (110.16) 110.11 The Weibull Distribution The Weibull distribution has two parameters, a scale parameter a and a shape parameter B By adjusting these two parameters, a wide range of experimental data can be modeled in system reliability studies. Th 入() ab5a>0,B>0,t≥0 (110.17) R C With B=l, the Weibull distribution reduces to the constant hazard model with n=(1/a). with B= 2, the Weibull distribution reduces to the Rayleigh distribution. The associated MTTF is MTTF=u=a∏1+ 110.20) where I denotes the gamma function. 110.12 Combinatorial Aspects Analysis of complex systems is facilitated by decomposition into functional entities consisting of subsystems or units and by the application of combinatorial considerations and network modeling techniques A series or chain structure consisting of n units is shown in Fig. 110.2. From the reliability point of view, the system will succeed only if all the units succeed. The units may or may not be physically in series. If R, is the probability of success of the ith unit, then the series system reliability R, is given as (110.21) e 2000 by CRC Press LLC

© 2000 by CRC Press LLC (110.14) (110.15) The a posteriori failure probability Qc(t) is independent of the prior operating time T, indicating that the component does not degrade no matter how long it operates. Obviously, such a scenario is valid only during the useful lifetime (horizontal portion of the bathtub curve) of the component. The mean and standard deviation of the random variable “lifetime” are (110.16) 110.11 The Weibull Distribution The Weibull distribution has two parameters, a scale parameter a and a shape parameter b. By adjusting these two parameters, a wide range of experimental data can be modeled in system reliability studies. The associated functions are (110.17) (110.18) (110.19) With b = 1, the Weibull distribution reduces to the constant hazard model with l = (1/a). With b = 2, the Weibull distribution reduces to the Rayleigh distribution. The associated MTTF is (110.20) where G denotes the gamma function. 110.12 Combinatorial Aspects Analysis of complex systems is facilitated by decomposition into functional entities consisting of subsystems or units and by the application of combinatorial considerations and network modeling techniques. A series or chain structure consisting of n units is shown in Fig. 110.2. From the reliability point of view, the system will succeed only if all the units succeed. The units may or may not be physically in series. If Ri is the probability of success of the ith unit, then the series system reliability Rs is given as (110.21) R t e t ( ) = - l Q t Q t e c t ( ) = ( ) = - - 1 l m l s l º = MTTF = 1 1 and l b a a b b b t t ( ) = > > t ³ -1 ; 0, 0, 0 f t t t ( ) = -Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ - b a a b b b 1 exp R t t ( ) = -Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ exp a b MTTF = = + Ê Ë Á ˆ ¯ m a ˜ b G 1 1 R R s i i n = ’=1

Cause Unit 1 nit 2 Effect FIGURE 110.2 Series or chain structure Caus Effect FIGURE 110.3 Parallel structure if the units do not interact with each other. If they do, then the conditional probabilities must be carefully If each of the units has a constant hazard then )=∏pλ) (11022) where n; is the constant failure rate for the ith unit or component. This enables us to replace the n components in series by an equivalent component with a constant hazard A, where ∑ If the components are identical, then As =nm and the MTTF for the equivalent component is(1/n)of the MTTF of one component. A parallel structure consisting of n units is shown in Fig. 110.3. From the reliability point of view, the system will succeed if any one of the n units succeeds. Once again, the units may or may not be physically or topologically in parallel. If Q is the probability of failure of the ith unit, then the parallel system reliability R, is given as if the units do not interact with each other(meaning independent) c 2000 by CRC Press LLC

© 2000 by CRC Press LLC if the units do not interact with each other. If they do, then the conditional probabilities must be carefully evaluated. If each of the units has a constant hazard, then (110.22) where li is the constant failure rate for the ith unit or component. This enables us to replace the n components in series by an equivalent component with a constant hazard ls where (110.23) If the components are identical, then ls = nl and the MTTF for the equivalent component is (1/n) of the MTTF of one component. A parallel structure consisting of n units is shown in Fig. 110.3. From the reliability point of view, the system will succeed if any one of the n units succeeds. Once again, the units may or may not be physically or topologically in parallel. If Qi is the probability of failure of the ith unit, then the parallel system reliability Rp is given as (110.24) if the units do not interact with each other (meaning independent). FIGURE 110.2 Series or chain structure. FIGURE 110.3 Parallel structure. Cause Unit 1 Unit 2 Unit n Effect Unit 1 Cause Effect Unit 2 Unit n R t t s i i n ( ) = -( ) ’= exp l 1 l l s i i n = = Â1 R Q p i i n = - ’= 1 1

If each of the units has a constant hazard then R()=1 (110.25) and we do not have the luxury of being able to replace the parallel system by an equivalent component with a constant hazard. The parallel system does not exhibit constant-hazard behavior even though each of the units has constant-hazard The MTTF of the parallel system can be obtained by using Eq (110.25)in(110.6). The results for the case of components with identical hazards n are:(1.5/7),(1.833/), and (2.083/)for n=2, 3, and 4 respectively. The largest gain in MTTF is obtained by going from one component to two components in parallel. It is uncommon to have more than two or three components in a truly parallel configuration because of the cost involved. For two non-identical components in parallel with hazard rates of M, and n, the MTTF is given as MTTF I (110.26) An r-out-of-n structure, also known as a partially redundant system, can be evaluated using Eq (110.11) all the components are identical, independent, and have a constant hazard of n, then the system reliability R()=∑cc(-c-)y (110.27) For r=l, the structure becomes a parallel system and for r=n, it becomes a series system Series-Parallel systems are evaluated by repeated application of the expressions derived for series and parallel configurations by employing the well-known network reduction techniques. Several general techniques are available for evaluating the reliability of complex structures that do not come under purely series or parallel or series parallel. They range from inspection to cutset and tieset methods and connection matrix techniques that are amenable for computer programming 110.13 Modeling maintenance Maintenance of a component could be a scheduled (or preventive)one or a forced(corrective)one. The latter follows in-service failures and can be handled using Markov models discussed later. Scheduled maintenance is conducted at fixed intervals of time, irrespective of the system continuing to operate satisfactor Scheduled maintenance, under ideal conditions, takes very little time(compared to the time between main makes sense only for those components with increasing hazard rates. Most mechanica swis mi zirable, tenances)and the component is restored to an"as new"condition. Even if the component is not re scheduled maintenance postpones failure and prolongs the life of the component. Schedule ms come under this category. It can be shown that the density function f (o) with scheduled maintenance included can be 所()=∑(一kT,)() (110.28) e 2000 by CRC Press LLC

© 2000 by CRC Press LLC If each of the units has a constant hazard, then (110.25) and we do not have the luxury of being able to replace the parallel system by an equivalent component with a constant hazard. The parallel system does not exhibit constant-hazard behavior even though each of the units has constant-hazard. The MTTF of the parallel system can be obtained by using Eq. (110.25) in (110.6). The results for the case of components with identical hazards l are: (1.5/l), (1.833/l), and (2.083/l) for n = 2, 3, and 4 respectively. The largest gain in MTTF is obtained by going from one component to two components in parallel. It is uncommon to have more than two or three components in a truly parallel configuration because of the cost involved. For two non-identical components in parallel with hazard rates of l1 and l2, the MTTF is given as (110.26) An r-out-of-n structure, also known as a partially redundant system, can be evaluated using Eq. (110.11). If all the components are identical, independent, and have a constant hazard of l, then the system reliability can be expressed as (110.27) For r = 1, the structure becomes a parallel system and for r = n, it becomes a series system. Series-parallel systems are evaluated by repeated application of the expressions derived for series and parallel configurations by employing the well-known network reduction techniques. Several general techniques are available for evaluating the reliability of complex structures that do not come under purely series or parallel or series parallel. They range from inspection to cutset and tieset methods and connection matrix techniques that are amenable for computer programming. 110.13 Modeling Maintenance Maintenance of a component could be a scheduled (or preventive) one or a forced (corrective) one. The latter follows in-service failures and can be handled using Markov models discussed later. Scheduled maintenance is conducted at fixed intervals of time, irrespective of the system continuing to operate satisfactorily. Scheduled maintenance, under ideal conditions, takes very little time (compared to the time between main￾tenances) and the component is restored to an “as new” condition. Even if the component is not repairable, scheduled maintenance postpones failure and prolongs the life of the component. Scheduled maintenance makes sense only for those components with increasing hazard rates. Most mechanical systems come under this category. It can be shown that the density function fT *(t) with scheduled maintenance included can be expressed as (110.28) Rt t p i i n ( ) =- - - [ ] ( ) ’= 1 1 1 exp l MTTF =+- + 11 1 1 2 12 l l ll Rt C e e n k kt t n k k r n ( ) = - ( ) - - - = Â l l 1 f t f t kT R T T M k M k * ( ) = - ( ) ( ) = • Â 1 0