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(Created page with '===Arrhenius-Weibull Statistical Properties Summary=== <br> ====Mean or MTTF==== <br> The mean, <math>\overline{T}</math> (also called <math>MTTF</math> by some authors), o…')
 
 
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===Arrhenius-Weibull Statistical Properties Summary===
#REDIRECT [[Arrhenius_Relationship#Arrhenius-Weibull_Statistical_Properties_Summary]]
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====Mean or MTTF====
 
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The mean,  <math>\overline{T}</math>  (also called  <math>MTTF</math>  by some authors), of the Arrhenius-Weibull relationship is given by:
 
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::<math>\overline{T}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>
 
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where  <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math>  is the gamma function evaluated at the value of  <math>\left( \tfrac{1}{\beta }+1 \right)</math> .
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====Median====
 
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The median, <math>\breve{T},</math> 
for the Arrhenius-Weibull model is given by:
 
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::<math>\breve{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
 
====Mode====
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The mode,  <math>\tilde{T},</math> 
for the Arrhenius-Weibull model is given by:
 
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::<math>\tilde{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
 
====Standard Deviation====
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The standard deviation,  <math>{{\sigma }_{T}},</math>  for the Arrhenius-Weibull model is given by:
 
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::<math>{{\sigma }_{T}}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>
 
====Arrhenius-Weibull Reliability Function====
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The Arrhenius-Weibull reliability function is given by:
 
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::<math>R(T,V)={{e}^{-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}</math>
 
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If the parameter  <math>B</math>  is positive, then the reliability increases as stress decreases.
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[[Image:ALTA6.7.gif|thumb|center|300px|Behavior of the reliability function at different stress and constant parameter values.]]
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The behavior of the reliability function of the Weibull distribution for different values of  <math>\beta </math>  was illustrated in Chapter 5. In the case of the Arrhenius-Weibull model, however, the reliability is a function of stress also. A 3D plot such as the ones shown in Fig. 8 is now needed to illustrate the effects of both the stress and  <math>\beta .</math>
 
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::<math></math>
 
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[[Image:ALTA6.8.gif|thumb|center|300px|Reliability function for <math>\Beta<1 </math>, <math>\Beta=1 </math>, and <math>\Beta>1 </math>.]]
 
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====Conditional Reliability Function====
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The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:
 
 
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::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>
 
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or:
 
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::<math>R(T,t,V)={{e}^{-\left[ {{\left( \tfrac{T+t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }} \right]}}</math>
 
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====Reliable Life====
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For the Arrhenius-Weibull relationship, the reliable life,  <math>{{t}_{R}}</math> , of a unit for a specified reliability and starting the mission at age zero is given by:
 
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::<math>{{t}_{R}}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left\{ -\ln \left[ R\left( {{t}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
 
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This is the life for which the unit will function successfully with a reliability of  <math>R({{t}_{R}})</math> . If  <math>R({{t}_{R}})=0.50</math>  then  <math>{\breve{T}</math>,
the median life, or the life by which half of the units will survive.
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====Arrhenius-Weibull Failure Rate Function====
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The Arrhenius-Weibull failure rate function,  <math>\lambda (T)</math> , is given by:
 
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::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}</math>
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[[Image:ALTA6.9.gif|thumb|center|300px|Failure rate function for <math>\Beta<1 </math>, <math>\Beta=1 </math>, and <math>\Beta>1 </math>.]]
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Latest revision as of 04:38, 16 August 2012

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