A system is designed to operate for 100 years. The system consists of 3 componen

Question

A system is designed to operate for 100 years. The system consists of 3 components in series. Their failure distributions are Weibull with shape parameter 12 and scale parameter 840 days Lognormal with shape parameter (s) 0-7 and medium 435 days; constant failure rate of 0.0001 compute the system reliability If 2 units of components 1 and 2 are available, determine the high-level redundancy reliability. Assume that components 1 and 2 can be configured as a sub-assembly. If two units of components 1 and 2 available, determine the low-level redundancy reliability.

Explanation / Answer

Reliability is defined as the probability that an element (that is, a component, subsystem or full system) will accomplish its assigned task within a specified time, which is designated as the interval t = [0, tM]. This book deals only with systems consisting of elements that can take on one of two states: either the element is operational (designated as the 1 state) or the element has failed (designated as the 0 state). Furthermore, the book considers only coherent systems, which have the following characteristics: (a) the reliability of the system increases if the reliability of its components increases, and (b) the system has no irrelevant components. The failure of any component or set of components in a coherent system cannot cause an increase in reliability, and every component has some effect, however small, on the overall reliability. If the reliability of the ith component of a system is pi and if this component has an unreliability qi, then pi = 1 qi (2.1) 7 8 2 Basic Elements of System Reliability and qi = 1 pi . (2.2) Also, since the component is always in one of the two possible states (operational or failed), qi + pi = 1 . (2.3) To perform quantitative system reliability analysis, it is necessary to ascribe a probability that the individual components pi either are operational or have failed. A reliability function, also called a survivor function, defines the probability that the component will perform its intended task (usually subject to some stated set of environmental conditions, such as vibration and temperature) for some specified performance period. The performance period may be a function of cycles, distance or time. Although the techniques presented here can employ a reliability function that depends on any of these three parameters, the focus is on determining the probability of system failure as a function of time. Additionally, although the estimation of system element reliabilities is outside the scope of this book, it is essential that the failure characteristics of the elements be determined using an approach that appropriately accounts for the environment in which they operate. It is also critical that the reliabilities are estimated in a legitimate and appropriately conservative fashion. Several different functions have been used to characterize the probability distribution of failures as a function of time. Some of the more common reliability functions include the exponential, normal, log-normal and Weibull distributions. In this book, however, the exponential probability distribution is used almost exclusively.1 The exponential distribution is appropriate for components with a failure rate that is time independent. Most electronic devices demonstrate such a constant failure rate during their useful lifetime, which is the time following a “burn in” that eliminates any weak or faulty components. The reliability function for a single-component system associated with the exponential distribution is r(, t) = et , (2.4) where r(, t) is the probability that a component with failure rate will be operational at time t. The Mathematica function implementing Equation is r[ , t ]:=et ; This book typically depicts system reliability graphically using log-log plots of the probability of failure as a function of time. Figure 2.1 shows the probability of failure for components with failure rates = 100, 200 and 400 failures per million hours (fpmh). Figure 2.1 illustrates several noteworthy points. Obviously, the probability of component failure increases with time and with . Note that the curves, when shown on a log-log plot, are nearly straight lines in the time range of interest. Also, the 1 The techniques illustrated in this book, however, are not limited to use of the exponential distribution; substitution of another distribution in lieu of the exponential is perfectly valid. (2.4) 2.2 Reliability Functional Block Diagrams 9 1 2 5 10 time hrs 0.0001 0.0002 0.0005 0.001 0.002 0.005 Probability of failure 200 100 400 fpmh Fig. 2.1 Probability of component failure for various failure rates, probability of failure curves increase uniformly by equal amounts for each doubling of the failure rate, . The slope of these curves is approximately one decade of unreliability per decade of time, which is a general feature of all simplex systems with a high degree of reliability over the time span of interest.2 It is shown later that the slope of a system unreliability curve is related to the level of system redundancy, with the slope increasing as the level of redundancy increases. 2.2 Reliability Functional Block Diagrams To support the analysis of system reliability, the analyst should first, after careful study of the entire system, depict the overall system design in the form of a reliability functional block diagram.3 The purpose of the block diagram is to describe the system at its simplest level, while still retaining all of the significant subsystem or component failure information, and to describe the effect that these failures have on the overall reliability of the system. Generally, this means that the functional block diagram represents the system as a collection of “black boxes,” each of which is subject to independent failure with respect to the other system elements. Note that 2 Obviously, since p = et and q = 1 p, q 1 as t , and therefore, all of the curves in Figure 2.1 asymptotically approach unity after a sufficiently long period of time. 3 This book uses functional block diagrams to describe the functional relationships between system elements that are capable of independent failure. Functional block diagrams are similar to reliability block diagrams but do not strictly adhere to the same conventions. Also, functional block diagrams often require an accompanying explanation to unambiguously describe the characteristics of the system. 10 2 Basic Elements of System Reliability a given element may be inoperative owing to the failure of other elements on which it depends, but it still may be capable of failure independently of the other elements in the system. In the following discussion, each element of the overall system is represented as a block, with the appropriate inputs and outputs representing its relationship to the remainder of the system. Each block contains an element pi that is assumed to have a failure rate i. Figure 2.2 represents a single component, p1, of a system that in turn could be part of another block diagram depicting a larger system. This component block has a single input and a single output, but in the more general case, the block might have multiple inputs and multiple outputs. By convention, inputs are generally assumed to enter either from the left side or from the top side of the box, and outputs exit either from the right side or from the bottom side.

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