3.2 Theoretical Overview of Reliability and Performance in Engineering Design 93 The last part of the curve, the increasing hazard rate region, is designated the ‘wear-out phase’ of the equipment. It starts when the equipment has passed its use- ful life and begins to wear out. During this phase, the number of failures begin to increase exponentially, and are known as ‘wear-out failures’. b) Component Reliability and Failure Distributions In the calculations for reliability, it is important to note that reliability is an indirect function of the probability of the occurrence of failure. The probability of the occurrence of failure is given by the failure distribution, or failure probability (FP) statistic. Thus, the probability of no failures occurring over a specific period of time is a measure of the component’s or equipment’s reliability and is given by the reliability probability (RP) statistic. Furthermore, if FP is the probability of failure occurring, and RP is the probabil- ity of no failure occurring, then FP = 1−RP or RP = 1−FP . (3.34) Reliability of components can thus be determined through the establishment of var- ious failure distributions, o riginating from their failure density functions. Reliability evaluation in designing for reliability assumes th at component reli- ability is known, and we are only interested in using this component reliability to compute system reliability. However, it is essential to understand how component reliability is determined, specifically from two important failure distributions, namely: • Exponential failure distribution. • Weibull failure distribution. 3.2.3.2 The Exponential Failure Distribution When a component is subject only to functional failures that occur at random in- tervals, and the expected number of failures is the same for equally long periods of time, its probability density function and its reliability can be defined by the expo- nential equation: Probability density function : f(t, θ )= 1 θ e −t/ θ . (3.35) Reliability: R(t, θ )=e −t/ θ (3.36) 94 3 Reliability and Performance in Engineering Design or, if it is expressed in terms of the failure rate, λ f(t, λ )= λ e − λ t , (3.37) and the reliability function is R(t, λ )=e − λ t , (3.38) where: f(t, λ )= probability density function of the Poisson process in terms of time t and failure rate λ . R(t, λ )= reliability of the Poisson process. t = operating time in the ‘useful life period’. θ = mean time between failures (MTBF). λ = 1/ θ , the failure rate for the component. This equation is applicable for determining component reliability, as long as the component is in its ‘useful life period’. This is the period during which the failure rate is constant,andfailure occurrences are predo minantly chance or random fail- ures. The ‘useful life period’ is considered to be the time after which ‘early failures’ no longer exist and ‘wear-out’ failures have not begun. Note that λ is the distribution scale parameter because it scales the exponential function. In reliability terms, λ is the failure r ate, which is the reciprocal of the mean time between failure. Because λ is constant for a Poisson process (exponential distribution function), the probability of failure at any time t depends only upon the elapsed time in the component’s ‘useful life period’. In complex electro-mech anical systems, the system failure rate is effectively con- stant over the ‘useful life period’, regardless of the failure patterns of individual components. An important point to note about Eqs. (3.37) and (3.38), with respect to designing for reliability, is that reliability in this case is a fu nction of operat- ing time (t) for the component, as well as the measure of mean time to failure (MTTF). a) Statistical Properties of the Exponential Failure Distribution The mean or MTTF The mean, or mean time to fail (MTTF) of the one-parameter exponential distribution is given by the following expression, where ¯ U is the MTTF ¯ U = ∞  0 tf(t)dt . (3.39) 3.2 Theoretical Overview of Reliability and Performance in Engineering Design 95 Relating f (t) to the exponential function gives the relationship ¯ U = ∞  0 t λ e − λ t dt ¯ U = 1 λ . (3.40) The median The median, ¯u, of the one-parameter exponential distribution is the value ¯u = 1 λ 0.693 ¯u = 0.693 ¯ U . The mode The mode, ˚u, of the one-parameter exponential distribution is given by ˚u = 0 . (3.41) For a continuousdistribution, the mode is the value of the variate that correspondsto the maximumprobability density function (p.d.f.). The modal life,˚u, is the maximum value of t that satisfies the expression d[ f(t)] dt = 0 . The standard deviation The standard deviation σ T of the one-parameter exponen- tial distribution is given by σ T = 1 λ = m . (3.42) The reliability function The one-parameter exponential reliability function is given by R(T)=e − λ T R(T)=e −T/m . This is the complement of the exponential cumulative distribution function where R(T)=1 − T  0 f(T)dT R(T)=1 − T  0 λ e − λ T dT R(T)= e − λ T . (3.43) 96 3 Reliability and Performance in Engineering Design Conditional reliability Conditional reliability calculates the probab ility of f urther successful functional duration, given that an item has already successfully func- tioned for a certain time. In this respect, conditional reliability could be considered to be the reliability of ‘used items or components’. This implies that the reliability for an added duration (mission) of t undertaken after the equipment or component has already accumulatedT hours of operation from age zero is a function only of the added time duration, and not a function of the age at the beginning of the mission. The conditional r eliability function for the one-parameter exponential distribu- tion is given by the following expression R(T,t)= R(T +t) R(T) R(T,t)= e − λ (T+t) e − λ T R(T,t)= e − λ t . (3.44) Reliable life The reliable life, or the mission duration for a desired reliability goal for the one-parameter exponential distribution is given by R(t R )=e − λ t R ln{R(t R )} = − λ t R t R = −ln{R(t R )} λ . (3.45) Residual life Let T denote the time to failure for an item. The conditional survival function can then be expressed as R(t)=P(T > t) . The conditional survival function is the probability that the item will survive for period t given that it has survived without failure for period T.Theresidual life is thus the extended duration or operational life t where the component has already accumulated T hours of operation from age zero, subject to the conditional survival function. The conditional survival function of an item that has survived (without failure) up to time x is R(t|x)=P(t > t + x|T > x) = P(T > t + x) P(T > t) = R(t + x) R(x) . (3.46) R(t|x) denotes the probability that a used item of age x will survive an extra time t. 3.2 Theoretical Overview of Reliability and Performance in Engineering Design 97 The mean residual life (MRL) of a used item of age x can thus be expressed as MRL(x)= ∞  0 R(t|x)dt . (3.47) When x = 0, the initial age is zero, imp lying a new item and, consequently MRL(0)=MTTF . In considering the reliable life for the one-parameter exponential distribution com- pared to the residual life, it is of interest to study the function h(x)= MRL(x) MTTF . (3.48) There are certain characteristics of comparison, when the initial age is zero (i.e. x = 0), between the mean residual life MRL (x) and the mean or the mean time to fail (MTTF). Characteristics of comparison between the mean residual life MRL (x)andthe mean, or mean time to fail (MTTF), are the following: • When the time to failure for an item, T, has an exponential distribution, then h(x)=1forallx. • When T has a Weibull distribution with shape parameter β < 1 (i.e. decreasing failure rate), then h(x) is an increasing function. • When T has a Weibull d istribution w ith shape parameter β > 1 (i.e. increasing failure rate), then h(x) is a decreasing function. Failure rate function The exponential failure rate function is given by λ t = f(T) R(T) = λ e − λ T e − λ T = λ (3.49) f(T) R(T) = hazard rate h(t),and λ (t) is constant λ . The hazard rate is a constant with respect to time for the exponential failure dis- tribution function. For other distributions, such as the Weibull distribution or the log-normal distribution, the hazard rate is not constant with respect to time. 3.2.3.3 The Weibull Failure Distribution Although the determination of equipment reliability and corresponding system reliability during the period of the equipment’s useful life period is based on the exponential failure distribution,thefailure rate of the equipment may not be con- stant throughoutthe period of its use or operation. In most engineering installations, particularly with the integration of complex systems, the purpose of determining 98 3 Reliability and Performance in Engineering Design equipment criticality, or combinations of critical equipment, is predominantly to assess the times to wear-out failures, rather than to assess the times to chance or random failures. In such cases, the exponential failure distribution does not apply, and it becomes necessary to substitute a general failure distribution, such as the Weibull distribution. The Weibull distribution is particularly useful because it can be applied to all three of the phases of the hazard rate curve, which is also called the equipment ‘life characteristic curve’. The equation for the two-parameter Weibull cumulative distribution function (c.d.f.) is given by F(t)= 1  0 f(t| βμ )dt . (3.50) The equation for the two-parameter Weibull probability density function (p.d.f.) is given by f(t)= β t ( β −1) e −t/ μ β μ β , (3.51) where: t = the operating time for which the reliability R(t) of the component must be determined. β = parameter of the Weibull distribution referred to as the shape parameter. μ = parameter of the Weibull distribution referred to as the scale parameter. a) Statistical Properties o f the Weibull Distribution The mean or MTTF The mean, ¯ U, of the two-parameter Weibull probability den- sity function (p.d.f.) is given by ¯ U = μΓ (1/ β + 1) , (3.52) where Γ (1/ β + 1) is the g amma function, evaluated at (1/ β + 1). The median The median, ¯u, of the two-parameter Weibull distribution is given by ¯u = μ (ln2) 1/ β . (3.53) The mode The mode or value with maximum probability, ˚u, of the two-parameter Weibull distribution is given by ˚u = μ  1− 1 β  1/ β . (3.54) 3.2 Theoretical Overview of Reliability and Performance in Engineering Design 99 The standard deviation The standard deviation, σ T , of the two-parameter Weibull is given by σ T = μ  Γ  2 β + 1  − Γ  1 β +  2 . (3.55) The cumulative distribution function (c.d.f.) The c.d.f. of the two-parameter Weibull distribution is given by F(T)=1 − e −(T/ μ ) β . (3.56) Reliability function The Weibull reliability function is given by R(T)=1 −F(t)= e −(T/ μ ) β . (3.57) The conditional reliability function Equation (3.58) gives the reliability for an extended operational period, or mission duration of t, having already accumulated T hours of operation up to the start of this mission duration, and estimates whether the component will begin the next mission successfully. It is termed conditionalbecause th e reliability of the following operational period or new mission can be estimated, based on the fact that the component has already successfully accumulated T hours of operation. The Weibull conditional reliability function is given by R(T,t)= R(T +t) R(T) = e −(T+t/ μ ) β e −(T/ μ ) β = e − [ (T+t/ μ ) β −(T/ μ ) β ] , (3.58) The reliable life For the two-parameter Weibull distribution, the reliable life, T R , of a component for a specified reliability, starting at age zero, is given by T R = μ {−ln[R(T R ]} 1/ β (3.59) b) The Weibull Shape Parameter The range of shapes that the Weibull density function can take is very broad, de- pending on the value of the shape parameter β . This value is usually indicated as β < 1, β = 1and β > 1. Figure 3.20 illustrates the shape of the Weibull c.d.f. F(t) for different values of β . The amount the curve is spread out along the abscissa or x-axis depends on the parameter μ , thus being called the Weibull scale parameter. 100 3 Reliability and Performance in Engineering Design Fig. 3.20 Shape of the Weibull density function, F(t), for different values of β For β < 1, the Weibull curve is asymptotic to both the x-axis and the y-axis, and is skewed. For β = 1, the Weibull curve is identical to the exponential density function. For β > 1, the Weibull curve is ‘bell shaped’ but skewed. c) The Weibull Distribution Function, Reliability and Hazard Integrating out the Weibull cumulative distribution function (c.d.f.)given in Eq. (3.50) gives the following F(t)= 1  0 f(t| βμ )dt F(t)=1−e −t/ μ β . (3.60) The mathematical model of reliability for the Weibull density function is R(t)=1 −F(t) R = e −t/ μ β , (3.61) where: R is the ‘probability of success’ or reliability. t is the equipment age. μ is the characteristic lif e or scale parameter. β is the slope or shape parameter. 3.2 Theoretical Overview of Reliability and Performance in Engineering Design 101 The Weibull hazard rate function, λ (t), is derived from a ratio between the Weibull probability density function (p.d.f.) and the Weibull reliability function λ (t)= f(t) R(t) λ (t)= β (t) β −1 μ β , (3.62) where: μ = the scale parameter, β = the shape parameter. To use this model, one must estimate the values of μ and β . Estimates of these pa- rameters fro m the Weibull pr obability density function are computationally difficult to obtain. There are analytical methods for estimating these parameters but they in- volve the solution o f a system of transcendental equations. An easier and commonly used method is based on a graphical technique that makes use of the Weibull graph chart. d) The Weibull Graph Chart The values of the failure distribution, expressed as percentage values of failure oc- currences, are plotted against the y-axis of the chart displayed in Fig. 3.21, and the correspondingtime between failures plotted against the x-axis.If the plot is a straight Fig. 3.21 The Weibull graph chart for different percentage values of the failure distribution 102 3 Reliability and Performance in Engineering Design line, then the Weibull distribution is applicable an d the relevantparameters are deter- mined. If the plot is not a straight line, then the two-parameter Weibull distribution is not applicable and more detailed analysis is required . Such d etailed analysis is presented in Section 3.3.3. To explain the format of the chart in Fig. 3.21, each axis of the chart is considered. • The scale of the x-axis is given as a log scale. • The description given along the y-axis is: • ‘cumulative percent’for‘cumulative distribution function (%)’ • The scale of the y-axis is given as a log–log scale. 3.2.3.4 Reliability Evaluation of Two-State Device Networks The f ollowing models present reliability evaluation of series and parallel two-state device networks (Dhillon 1983): a) Series Network This network denotes an assembly of which the components are connected in series. If any one of the components malfunctions, it will cause the assembly to fail. For the k non-identical and independent component series, which are time t-dependent, the formula for R S (t), the network reliability, is given in R S (t)={1 −F 1 (t)}·{1−F 2 (t)}·{1−F 3 (t)}· ·{1−F k (t)} And: {1−F i (t)}≈R i (t) . (3.63) The ith component cumulative distribution function (failure probability) is defined by F i (t)= t  0 f i (t)dt , (3.64) where: F i (t) is the ith component failure probability for i = 1,2,3, ,k. R i (t) is the ith component reliability, for i = 1,2,3, ,k. By definition: f i (t)= lim Δt→0 α S (t) − α S (t + Δt) α 0 Δt f i (t)= dF i (t) dt , where: Δt = the time interval, . or operation. In most engineering installations, particularly with the integration of complex systems, the purpose of determining 98 3 Reliability and Performance in Engineering Design equipment. Theoretical Overview of Reliability and Performance in Engineering Design 93 The last part of the curve, the increasing hazard rate region, is designated the ‘wear-out phase’ of the equipment. It. = μ  1− 1 β  1/ β . (3.54) 3.2 Theoretical Overview of Reliability and Performance in Engineering Design 99 The standard deviation The standard deviation, σ T , of the two-parameter Weibull is given by σ T = μ  Γ  2 β +