3.3 Analytic Development of Reliability and Performance in Engineering Design 203 Fig. 3.36 Example exponential probability graph c) Determining the Maximum Likelihood Estimation Parameter The parameter of the exponential distribution can also be estimated using the maxi- mum likelihood estimation (MLE) method. This function is log-likelihood and com- posed of two summation portions Λ = ln(L)= F ∑ i=1 N i ln λ e − λ T i − S ∑ i=1 ˇ N i λ ˇ T i , (3.167) where: F is the number of groups of times-to-failure data points. N i is the number of tim es to failure in the ith time-to-failure data group. λ is the failure rate parameter (unknown a priori, only one to be found). T i is the time of the ith group of time-to-failure data. S is the number of groups of suspension data points. ˇ N i is the number of suspensions in the ith group of data points. ˇ T i is the time of the ith suspension data group. The solution will be found by solving for a parameter λ ,sothat ∂ ( Λ ) ∂λ = 0and ∂ ( Λ ) ∂λ = F ∑ i=1 N i 1 λ −T i − S ∑ i=1 ˇ N i ˇ T i , (3.168) 204 3 Reliability and Performance in Engineering Design where also: F is the number of groups of times-to-failure data points. N i is the number of times to failure in the ith time-to-failure data group. λ is the failure rate parameter (unknown a priori, only one to be found). T i is the time of the ith group of time-to-failure data. S is the number of groups of suspension data points. ˇ N i is the number of suspensions in the ith group of data points. ˇ T i is the time of the ith suspension data group. 3.3.3.3 Expansion of the Weibull Distribution Model a) Characteristics of the Two-Parameter Weibull Distribution The characteristics of the two-parameter Weibull distribution can be exemplified by examining the two parameters β and μ , and the effect they have on the Weibull probability density function, reliability function and failure rate function. Changing the value of β , the shape parameter or slope of the Weibull distribution changes the shape of the probability density function (p.d.f.), as shown in Tables 3.15 to 3.19. In addition, when the cumulative distribution function (c.d.f.) is plotted, as shown in Tables 3.20 and 3.21, a change in β results in a change in the slope of the distri- bution. Effects of β on the Weibull p.d.f. The parameter β is dimensionless, with the following effects on the Weibull p.d.f. • For 0 < β < 1, the failure rate decreases with time and: As T → 0, f(T) → ∞ . As T → ∞ , f(T) → 0. f(T) decreases monotonically and is convex as T increases. The mode ˚u is non-existent. • For β = 1, it becomes the exponential distribution, as a special case, with: f(T)=1/ μ e −T/ μ for μ > 0, T ≥ 0 1/ μ = λ the chance, useful life, or failure rate. • For β > 1, f(T) assumes wear-out type shapes, i.e. the failure rate increases with time: f(T)=0atT = 0 . f(T) increases as T → ˚u (mode) and decreases thereafter. • For β = 2, the Weibull p.d.f. becomes the Rayleigh distribution. • For β < 2.6, the Weibull p.d.f. is positively skewed. • For 2.6 < β < 3.7, its coefficient of skewness approaches zero (no tail), and approximates the normal p .d.f. 3.3 Analytic Development of Reliability and Performance in Engineering Design 205 Fig. 3.37 Weibull p.d.f. with 0 < β < 1, β = 1, β > 1andafixed μ (ReliaSoft Corp.) • For β > 3.7, the Weibull p.d.f. is negatively skewed. From Fig. 3.37: • For 0 < β < 1: T → 0, f(T) → ∞. T → ∞, f(T) → 0. • For β = 1: f(T)=1/ μ e −T/ μ . T →∞, f(T) → 0. • For β > 1: f(T)=0atT = 0. T → ˚u, f(T) > 0. Effects of β on the Weibull reliability function and the c.d.f. Considering first the Weibull unreliability function (Fig. 3.38), or cumulative distribution function, F(t), the following effects of β are observed: • For 0 < β < 1 and constant μ , F(T) is linear with minimum slop e and values of F(T) ranging from 5 to below 90.00. • For β = 1 and constant μ , F(T) is linear with a steeper slope and values of F(T) ranging from less than 1 to above 90.00. • For β > 1 and constant μ , F(T) is linear with maximum slope and values of F(T) ranging from well below 1 to well above 99.90. Considering the Weibull reliability function (Fig. 3.39), or one minus the cumu- lative distribution function, 1 −F(t), the following effects of β are observed: • For 0 < β < 1 and constant μ , R(T) is convex, and decreases sharply and mono- tonically. • For β = 1 and constant μ , R(T) is convex, and decreases monotonically but less sharply. 206 3 Reliability and Performance in Engineering Design Fig. 3.38 Weibull c.d.f. or unreliability vs. time (ReliaSoft Corp.) Fig. 3.39 Weibull 1–c.d.f. or reliability vs. time (ReliaSoft Corp.) • For β > 1 and constant μ , R(T) decreases as T increases but less sharply than before and, as wear-out sets in, it decreasessharply and goesthroughan inflection point. 3.3 Analytic Development of Reliability and Performance in Engineering Design 207 Fig. 3.40 Weibull failure rate vs. time (ReliaSoft Corp.) Effects of β on the Weibull failure rate function The Weibull failure r ate for 0 < β < 1 is unbounded at T = 0. The failure rate λ (T) decreases thereafter mono- tonically and is convex, approaching the value of zero as T → 0or λ (∞)=0. This behaviour makes it suitable for representing the failure rates of components that exhibit early-type failures, for which the failure rate decreases with age (Fig. 3.40). When such behaviour is encounteredin pilot tests, the followingconclusionsmay be drawn: • Burn-in testing and/or environmental stress screening are not well implemented. • There are problems in the process line, affecting the expected life of the compo- nent. • Inadequate quality control of component manufacture is bringing about early failure. Effects of β on the Weibull failure rate function and derived failure charac- teristics The effects of β on the hazard or failure rate function of the Weibull dis- tribution result in several observations and conclusions about the characteristics of failure: • When β = 1, the hazard rate λ (T) yields a constant value of 1/ μ where: λ (T)= λ = 1/ μ . This parameter becomes suitable for representing the hazard or failure rate of chance-type or random failures, as well as the u seful life period of the compo- nent. 208 3 Reliability and Performance in Engineering Design • When β > 1, the hazard rate λ (T) increases as T increases, and becomessuitable for representing the failure rate of components with wear-out type failures. • For 1 < β < 2, the λ (T) curveis concave.Consequently, the failure rate increases at a decreasing rate as T increases. • For β = 2, the λ (T) curve represents the Rayleigh distribution where: λ (T)= 2/ μ (T/ μ ). There emerges a straight-line relationship between λ (T) and T, starting with a failure rate value of λ (T)=0atT = 0, and increasing thereafter with a slope of 2/ μ 2 . Thus, the failure rate increases at a constant rate as T increases. • When β > 2, the λ (T) curve is convex, with its slope increasing as T increases. Consequently, the failure rate increases at an increasing rate as T increases, indicating component wear-out. The scale parameter μ A change in the Weibull scale parameter μ has the same effect on the distribution (Fig. 3.41) as a change of the abscissa scale: • If μ is increased while β is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location. • If μ is decreased while β is kept the same, the distribution gets pushed in towards the left ( i.e. towards 0) and its height increases. Fig. 3.41 Weibull p.d.f. with μ = 50, μ = 100, μ = 200 (ReliaSoft Corp.) 3.3 Analytic Development of Reliability and Performance in Engineering Design 209 b) The Three-Parameter Weibull Model The mathematical model for reliability of the Weibull distribution has so far been determined from a two-parameter Weibull distribution formula, where the two pa- rameters are β and μ . The mathematical model fo r reliability o f the Weibull distri- bution can also be determined from a three-parameter Weibull distribution formula, where the three parameters are: β = shape parameter or failure pattern μ = scale parameter or characteristic life γ = location, position or minimum life parameter. This reliability model is given as R(t)=e −[(t− γ )/ μ ] β . (3.169) The three-parameter Weibull distribution has wide applicability. The math ematical model for the cumulative probability, or the cumulative distribution function (c.d.f.) of the three-parameter Weibull distribution is F(t)=1− e −[(t− γ )/ μ ] β , (3.170) where: F(t)=cumulative probability of failure, γ = location or position parameter, μ = scale parameter, β = shape parameter. The location, position, or minimum life parameter γ This parameter can be thought of as a guarantee period within which no failures occur, and a guaranteed minimum life could exist. This means that n o appreciable or noticeable degradation or wear is evident before γ hours of operation. However, when a component is sub- ject to failure immediately after being placed in service, no guarantee or failure-free period is apparent; then, γ = 0. The scale or characteristic life parameter μ This parameter is a constant and, by definition, is the mean operating period or, in terms of system unreliability, the operating period during which at least 63% of the system’s equipment is expected to fail. This ‘unreliability’ value of 63%, which is obtained from the previous formula Q = 1 −R = 100 −37%, can readily be determined from the reliability model by substituting specific values for γ = 0, and t = μ in the case of the Weibull graph being a straight line, and the period t being equal to the characteristic life or scale parameter μ respectively. The shape or failure pattern parameter β As its name implies, β determines the contour of the Weibull p.d.f. By finding the value of β for a given set of data, the particular phase of an equipment’s characteristic life may be determined: 210 3 Reliability and Performance in Engineering Design • When β < 1, the equipment is in a wear-in or infant mortality phase of its char- acteristic life, with a resulting decreasing rate of failure. • When β = 1, the equipment is in the steady operational period or service life phase of its characteristic life, with a resulting constant rate of failure. • When β > 1, the equipment begins to fail due to aging and/or degradation through use, and is in a wear-out phase of its characteristic life, with a result- ing increasing rate of failure. Since the probability of survival p(s), or the reliability for the Weibull distribution, is the unity complement of the probability of failure p(f ), or failure distribution F(t), the following mathematical model for reliability will plot a straight line on logarithmic scales R(t)=p(s)=e −[(t− γ )/ μ ] β . (3.171) To facilitate calculations for the Weibull parameters, a Weibull graph has been de- veloped. The principal advantage of this method of the Weibull analysis of failure is that it gives a complete picture of the type of distribution that is represented by the failure data and, furthermo re, relatively few failures are need ed to be able to make a satisfactory evaluation of the characteristics of component failure. Figure 3.42 shows the basic features of the Weibull graph. c) Procedure to Calculate the Weibull Parameters β , μ and γ The procedure to calculate the Weibull parameters using the Weibull graph illus- trated in Fig. 3.42 is given as follows: • The percentage failure is plotted on the y-axis against the age at failure on the x-axis (q −q). Failure age 4.0 3.0 2.0 1.0 0.0 % Fail. 0.0 1.0 Origin Weibull plot Principal ordinate Principal abscissa β μ n p p q q σ n Fig. 3.42 Plot of the Weibull density function, F(t), for diff erent values of β 3.3 Analytic Development of Reliability and Performance in Engineering Design 211 • If the plot is linear, then γ = 0. If the plot is non-linear, then γ = 0, and the proce- dure to make it linear by calculation is to add a constant value to the parameter γ in the event the plot is convex relative to the origin on the Weibull graph, or to subtract a constant value from the parameter γ in the event the plot is concave. A best fit straight line through the original plot would suffice. • A line (pp) is drawn through the origin of the chart, parallel to the calculated linear Weibull plo t (qq), or estimated straight line fit. • The line pp is extended until it intersects the principal ordinate, (point i in Fig. 3.37). The value for β is then determined from the β -scale at a point hori- zontally opposite the line pp intersection with the principal ordinate. • The linear Weibull plot (qq), or the graphically estimated straight line fit, is ex- tended u ntil it intersects the principal abscissa. The value for μ is then found at the bottom of the graph, vertically opposite the linear principal abscissa intersec- tion. d) Procedure to Derive the Mean Time Between Failures (MTBF) Once the Weibull parameters have been determined, the mean time between failures (MTBF) may be evaluated. There are two other scales parallel to the β -scale on the Weibull graph: μ /n and σ /n , where: μ = characteristic life, σ = standard deviation, n = number of d ata points. The value on the μ /n scale, adjacent to the previously determined value of β ,is determined. This value is, in effect, the mean time b etween failures (MTBF), as a ratio to the number of data points, or the percentage failures that wer e plotted on the y-axis against the age at failure. Thus, MTBF = scale value of μ /n . It is important to note that this mean value is referenced from the beginning of the Weibull distribution and should therefore be added to the minimum life parameter γ to obtain the true MTBF, as shown below in Fig. 3.43. e) Procedure to Obtain the Standard Deviation σ The standard deviation is the value on the σ /n scale, adjacent to the determined value of β . σ = n ×scale value of σ /n . 212 3 Reliability and Performance in Engineering Design Start Commence Weibull Time True MTBF MTBFμγ True MTBF = from Start to Commence Weibull to Time True MTBF = γ + μ Fig. 3.43 Minimum life parameter and true MTBF The standard deviation value of the Weibull distribution is used in the conventional manner and can be applied to obtain a general idea of the shape of the distribution. Summary of Quantitative Analysis of the Weibull Distribution Model In the two-parameter Weibull, the parameters β and μ ,where β is the shape pa- rameter or failure pattern,and μ is the scale parameter or characteristic life,have an effect on the probability density function, reliability fu nction and failure rate function (cf. Fig. 3.44). The effect of β on the Weibull p. d.f. is that when β > 1, the probability density function, f(T), assumes a wear-out type shape, i.e. the failure rate increases with time. The effect of β on the Weibull reliability function, or one minus the cumulative distribution function c.d.f ., 1 −F(t), is that when β > 1and μ is constant, R(T) decreases as T increases until wear-out sets in, when it decreases sharply and goes through an inflection point. The effect of β on the Weibull hazard o r failure rate function is that when β > 1, the hazard rate λ (T) increases as T increases, and becomes suitable for representing the failure rate of components with wear-out type failures. A change in the Weibull scale parameter μ has the effect that when μ ,thechar- acteristic life,isincreased while β ,thefailure pattern, is constant, the distribution f(T) is spread out with a greater variance about the mean and, when μ is decreased while β is constant, the distribution is peaked. With the inclusion of γ , the location or minimum life parameter in a three- parameter Weibull distribution, no appreciable or noticeable degradation or wear is evident before γ hours of operation. 3.3.3.4 Qualitative Analysis of the Weibull Distribution Model It was stated earlier that the principal advantage of Weibull analysis is that it gives a complete picture of the type of distribution that is represented by the failure data, and that relatively few failures are needed to be able to make a satisfactory assess- . time of the ith group of time-to-failure data. S is the number of groups of suspension data points. ˇ N i is the number of suspensions in the ith group of data points. ˇ T i is the time of the. time of the ith group of time-to-failure data. S is the number of groups of suspension data points. ˇ N i is the number of suspensions in the ith group of data points. ˇ T i is the time of the. sets in, it decreasessharply and goesthroughan in ection point. 3.3 Analytic Development of Reliability and Performance in Engineering Design 207 Fig. 3.40 Weibull failure rate vs. time (ReliaSoft