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 exemp
Trang 1Fig 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 iln
λe−λ T i
−∑S
i=1
ˇ
N iλTˇi , (3.167)
where:
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)
Ti 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 parameterλ, so that
∂(Λ)
∂λ = 0 and ∂∂λ(Λ)=
F
∑
i=1
N i
1
λ − Ti
−∑S
i=1 ˇ
N i Tˇi , (3.168)
Trang 2204 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)
Ti 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 parametersβ and μ, and the effect they have on the Weibull probability density function, reliability function and failure rate function Changing the value ofβ, 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 inβ 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.
• Forβ = 1, it becomes the exponential distribution, as a special case, with:
f (T ) = 1/μe−T/μ for μ> 0, T ≥ 0
1/μ=λ the chance, useful life, or failure rate
• Forβ> 1, f (T ) assumes wear-out type shapes, i.e the failure rate increases with
time:
f (T ) = 0 at T = 0
f (T ) increases as T → ˚u (mode) and decreases thereafter.
• Forβ = 2, the Weibull p.d.f becomes the Rayleigh distribution.
• Forβ < 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
Trang 3Fig 3.37 Weibull p.d.f with 0<β< 1,β = 1, β> 1 and a fixedμ (ReliaSoft Corp.)
• Forβ> 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.
• Forβ= 1: f (T ) = 1/μe−T/μ T →∞, f(T ) → 0.
• Forβ> 1: f (T ) = 0 at T = 0 T → ˚u, f (T ) > 0.
Effects ofβ 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 ofβ are observed:
• For 0 <β < 1 and constantμ, F (T ) is linear with minimum slope and values of
F (T) ranging from 5 to below 90.00.
• Forβ= 1 and constantμ, F (T ) is linear with a steeper slope and values of F(T )
ranging from less than 1 to above 90.00
• Forβ > 1 and constantμ, 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 ofβ are observed:
• For 0 <β< 1 and constantμ, R (T) is convex, and decreases sharply and
mono-tonically
• Forβ= 1 and constantμ, R (T ) is convex, and decreases monotonically but less
sharply
Trang 4206 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.)
• Forβ > 1 and constantμ, R (T) decreases as T increases but less sharply than
before and, as wear-out sets in, it decreases sharply and goes through an inflection point
Trang 5Fig 3.40 Weibull failure rate vs time (ReliaSoft Corp.)
Effects ofβ on the Weibull failure rate function The Weibull failure rate for
0<β< 1 is unbounded at T = 0 The failure rateλ(T) decreases thereafter mono-tonically and is convex, approaching the value of zero as T → 0 orλ(∞) = 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 encountered in pilot tests, the following conclusions may
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 ofβ on the Weibull failure rate function and derived failure charac-teristics The effects ofβ on the hazard or failure rate function of the Weibull
dis-tribution result in several observations and conclusions about the characteristics of failure:
• Whenβ= 1, the hazard rateλ(T ) yields a constant value of 1/μwhere: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 useful life period of the compo-nent
Trang 6208 3 Reliability and Performance in Engineering Design
• Whenβ> 1, the hazard rateλ(T) increases as T increases, and becomes suitable
for representing the failure rate of components with wear-out type failures
• For 1 <β< 2, theλ(T) curve is concave Consequently, the failure rate increases
at a decreasing rate as T increases.
• Forβ = 2, theλ(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 ofλ(T ) = 0 at T = 0, and increasing thereafter with a slope
of 2/μ2 Thus, the failure rate increases at a constant rate as T increases.
• Whenβ> 2, theλ(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 parameterμ A change in the Weibull scale parameterμhas the same effect on the distribution (Fig 3.41) as a change of the abscissa scale:
• Ifμis increased whileβ is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location
• Ifμis decreased whileβis 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 withμ = 50, μ = 100, μ = 200 (ReliaSoft Corp.)
Trang 7b) 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 areβ andμ The mathematical model for reliability of the Weibull
distri-bution can also be determined from a three-parameter Weibull distridistri-bution 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 mathematical 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 no appreciable or noticeable degradation
or wear is evident beforeγhours 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 parameterμ 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 forγ = 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
parameterμrespectively
The shape or failure pattern parameterβ As its name implies,β determines the contour of the Weibull p.d.f By finding the value ofβ for a given set of data, the
particular phase of an equipment’s characteristic life may be determined:
Trang 8210 3 Reliability and Performance in Engineering Design
• Whenβ< 1, the equipment is in a wear-in or infant mortality phase of its char-acteristic life, with a resulting decreasing rate of failure.
• Whenβ = 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 increasresult-ing 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, furthermore, relatively few failures are needed 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 Parametersβ,μandγ
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
%
Origin
Weibull plot
Principal
ordinate
Principal
abscissa
p
p
q
q
σ n
Fig 3.42 Plot of the Weibull density function, F (t), for different values ofβ
Trang 9• If the plot is linear, thenγ= 0 If the plot is non-linear, thenγ= 0, and the
proce-dure to make it linear by calculation is to add a constant value to the parameterγ
in the event the plot is convex relative to the origin on the Weibull graph, or to subtract a constant value from the parameterγ 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 plot (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 forβ is then determined from theβ-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 until it intersects the principal abscissa The value forμis 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 theβ-scale on the Weibull graph:
μ/n and σ/n ,
where:
μ = characteristic life,
σ = standard deviation,
n = number of data points
The value on theμ/n scale, adjacent to the previously determined value ofβ, is determined This value is, in effect, the mean time between failures (MTBF), as
a ratio to the number of data points, or the percentage failures that were plotted on
the y-axis against the age at failure.
Thus, MTBF= scale value ofμ/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 parameterγ
to obtain the true MTBF, as shown below in Fig 3.43.
e) Procedure to Obtain the Standard Deviationσ
The standard deviation is the value on theσ/n scale, adjacent to the determined
value ofβ
σ= n × scale value ofσ/n
Trang 10212 3 Reliability and Performance in Engineering Design
True MTBF
MTBF μ γ
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 parametersβ andμ, whereβ is the shape
pa-rameter or failure pattern, andμis the scale parameter or characteristic life, have
an effect on the probability density function, reliability function and failure rate function (cf Fig 3.44)
The effect ofβ on the Weibull p.d.f is that whenβ > 1, the probability density function, f (T ), assumes a wear-out type shape, i.e the failure rate increases with
time
The effect ofβ on the Weibull reliability function, or one minus the cumulative distribution function c.d.f., 1− F(t), is that whenβ > 1 and 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 ofβ on the Weibull hazard or failure rate function is that whenβ > 1,
the hazard rateλ(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 parameterμhas the effect that whenμ, the char-acteristic life, is increased whileβ, the failure pattern, is constant, the distribution
f (T ) is spread out with a greater variance about the mean and, whenμis decreased
whileβ 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 beforeγhours 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