|
|
| (16 intermediate revisions by 5 users not shown) |
| Line 1: |
Line 1: |
| ===Characteristics===
| | #REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]] |
| The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta, <math>\beta ,</math> and eta, <math>\eta ,</math> and the effect they have on the <math>pdf,</math> reliability and failure rate functions.
| |
| | |
| ====Looking at <math>\beta </math>====
| |
| Beta, <math>\beta ,</math> is called the shape parameter or slope of the Weibull distribution. Changing the value of <math>\beta </math> forces a change in the shape of the <math>pdf</math> as shown in Fig. 6. In addition, when the <math>cdf</math> is plotted on Weibull probability paper, as shown in Fig. 7, a change in beta is a change in the slope of the distribution on Weibull probability paper.
| |
| <br>
| |
| '''Effects of <math>\beta </math> on the ''pdf'''''
| |
| <br>
| |
| [[Image:ALTA4.3.gif|thumb|center|400px|Weibull ''pdf'' with <math>0<\beta<1</math>, <math>\Beta=1 </math> and a fixed <math>\eta </math>.]] | |
| <br>
| |
| <br>
| |
| :• For <math>0<\beta <1</math> , the failure rate decreases with time and:
| |
| ::o As <math>T\to 0,</math> <math>f(T)\to \infty .</math>
| |
| ::o As <math>T\to \infty </math> , <math>f(T)\to 0</math> .
| |
| ::o <math>f(T)</math> decreases monotonically and is convex as <math>T</math> increases.
| |
| ::o The mode is non-existent.
| |
| :• For <math>\beta =1,</math> it becomes the exponential distribution, as a special case, <br>
| |
| <br>
| |
| :or:
| |
| <br>
| |
| ::<math>f(T)=\frac{1}{\eta }{{e}^{-\tfrac{T}{\eta }}};\text{ }\eta >0,T\ge 0</math>
| |
| <br>
| |
| :where <math>\tfrac{1}{\eta }=\lambda =</math> chance, useful life, or failure rate.
| |
| <br>
| |
| :• For <math>\beta >1</math> , <math>f(T),</math> the Weibull assumes wear-out type shapes (i.e. the failure rate increases with time) and:
| |
| ::o <math>f(T)=0</math> at <math>T=0</math> .
| |
| ::o <math>f(T)</math> increases as <math>T\to \tilde{T}</math> (mode) and decreases thereafter.
| |
| ::o For <math>\beta =2</math> it becomes the Rayleigh distribution as a special case. For <math>\beta <2.6</math> the Weibull <math>pdf</math> is positively skewed (has a right tail), for <math>2.6<\beta <3.7</math> its coefficient of skewness approaches zero (no tail); consequently, it may approximate the normal <math>pdf</math> , and for <math>\beta >3.7</math> it is negatively skewed (left tail).
| |
| :• The parameter <math>\beta </math> is a pure number, i.e. it is dimensionless.
| |
| | |
| <br>
| |
| '''Effects of <math>\beta </math> on the Reliability Function and the ''cdf'''''
| |
| <br>
| |
| [[Image:ALTA4.4.gif|thumb|center|400px|Weibull ''cdf'', or Unreliability vs. Time, on Weibull probability plotting paper with <math>0<\Beta<1</math>, <math>\Beta=1 </math> and a fixed <math>\eta </math>.]]
| |
| | |
| <br>
| |
| [[Image:ALTA4.5.gif|thumb|center|400px|Weibull 1-''cdf'', or Reliability vs. Time, on linear scales with <math>0<\Beta<1</math>, <math>\Beta=1 </math> and a fixed <math>\eta </math>.]]
| |
| | |
| <br>
| |
| :• <math>R(T)</math> decreases sharply and monotonically for <math>0<\beta <1</math> , it is convex, and decreases less sharply for the same <math>\beta </math> .
| |
| :• For <math>\beta =1</math> and the same <math>\eta </math> , <math>R(T)</math> decreases monotonically but less sharply than for <math>0<\beta <1</math> , and is convex.
| |
| :• For <math>\beta >1</math> , <math>R(T)</math> decreases as <math>T</math> increases but less sharply than before, and as wear-out sets in, it decreases sharply and goes through an inflection point.
| |
| <br>
| |
|
| |
| '''Effects of <math>\beta </math> on the Failure Rate Function'''
| |
| | |
| [[Image:ALTA4.6.gif|thumb|center|400px|Weibull Failure Rate vs. Time with <math>0<\Beta<1 </math>, <math>\Beta=1 </math>, <math>\Beta>1 </math>.]]
| |
| <br>
| |
| :• The Weibull failure rate for <math>0<\beta <1</math> is unbounded at <math>T=0</math> . The failure rate, <math>\lambda (T),</math> decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \infty </math> or <math>\lambda (\infty )=0</math> . This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When such behavior is encountered, one or more of the following conclusions can be drawn:
| |
| ::o Burn-in testing and/or environmental stress screening are not well implemented.
| |
| ::o There are problems in the production line.
| |
| ::o Inadequate quality control.
| |
| ::o Packaging and transit problems.
| |
| :• For <math>\beta =1</math> , <math>\lambda (T)</math> yields a constant value of <math>\tfrac{1}{\eta }</math> , or:
| |
| ::<math>\lambda (T)=\lambda =\frac{1}{\eta }</math>
| |
| This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.
| |
| :• For <math>\beta >1</math> , <math>\lambda (T)</math> increases as <math>T</math> increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For <math>1<\beta <2</math> the <math>\lambda (T)</math> curve is concave, consequently the failure rate increases at a decreasing rate as <math>T</math> increases.
| |
| :• For <math>\beta =2</math> , or for the Rayleigh distribution case, the failure rate function is given by:
| |
| ::<math>\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)</math>
| |
| <br>
| |
| hence there emerges a straight line relationship between <math>\lambda (T)</math> and <math>T</math> , starting at a value of <math>\lambda (T)=0</math> at <math>T=0</math> , and increasing thereafter with a slope of <math>\tfrac{2}{{{\eta }^{2}}}</math> . Consequently, the failure rate increases at a constant rate as <math>T</math> increases. Furthermore, if <math>\eta =1</math> the slope becomes equal to 2, and <math>\lambda (T)</math> becomes a straight line which passes through the origin with a slope of 2.
| |
| :• When <math>\beta >2</math> the <math>\lambda (T)</math> curve is convex, with its slope increasing as <math>T</math> increases. Consequently, the failure rate increases at an increasing rate as <math>T</math> increases, indicating wear-out life.
| |
| <br>
| |
| | |
| ====Looking at <math>\eta </math>====
| |
| Eta, <math>\eta ,</math> is called the scale parameter of the Weibull distribution. The parameter <math>\eta </math> has the same units as <math>T</math> , such as hours, miles, cycles, actuations, etc.
| |
| <br>
| |
| [[Image:ALTA4.7.gif|thumb|center|300px| ]]
| |
| <br>
| |
| :• A change in the scale parameter <math>\eta </math> has the same effect on the distribution as a change of the abscissa scale.
| |
| ::o If <math>\eta </math> is increased while <math>\beta </math> is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
| |
| ::o If <math>\eta </math> is decreased while <math>\beta </math> is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.
| |
| <br>
| |