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| ===Arrhenius-Weibull Statistical Properties Summary===
| | #REDIRECT [[Arrhenius_Relationship#Arrhenius-Weibull_Statistical_Properties_Summary]] |
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| {{aaw mode}}
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| ====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>,
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| 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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