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==T-NT Weibull==
#REDIRECT [[Temperature-NonThermal_Relationship#T-NT_Weibull]]
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By setting  <math>\eta =L(U,V)</math>, the T-NT Weibull model is given by:
 
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::<math>f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}</math>
 
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===T-NT Weibull Statistical Properties Summary===
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====Mean or MTTF====
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The mean,  <math>\overline{T}</math> , for the T-NT Weibull model is given by:
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::<math>\overline{T}=\frac{C}{{{U}^{n}}{{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> .
 
====Median====
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The median,  <math>\breve{T},</math>
for the T-NT Weibull model is given by:
 
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::<math>\breve{T}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
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====Mode====
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The mode,  <math>\tilde{T},</math>  for the T-NT Weibull model is given by:
 
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::<math>\tilde{T}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
 
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====Standard Deviation====
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The standard deviation,  <math>{{\sigma }_{T}},</math>  for the T-NT Weibull model is given by:
 
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::<math>{{\sigma }_{T}}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>
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====T-NT Weibull Reliability Function====
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The T-NT Weibull reliability function is given by:
 
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::<math>R(T,U,V)={{e}^{-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}</math>
 
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====Conditional Reliability Function====
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The T-NT Weibull conditional reliability function at a specified stress level is given by:
 
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::<math>R(T,t,U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac{{{e}^{-{{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}</math>
 
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:or:
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::<math>R(T,t,U,V)={{e}^{-\left[ {{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }} \right]}}</math>
 
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====Reliable Life====
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For the T-NT Weibull model, 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}}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},U,V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
 
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====T-NT Weibull Failure Rate Function====
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The T-NT Weibull failure rate function,  <math>\lambda (T)</math> , is given by:
 
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::<math>\lambda \left( T,U,V \right)=\frac{f\left( T,U,V \right)}{R\left( T,U,V \right)}=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}</math>
 
===Parameter Estimation===
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====Maximum Likelihood Estimation Method====
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Substituting the T-NT relationship into the Weibull log-likelihood function yields:
 
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::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta }}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}T_{i}^{\prime } \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
\end{align}</math>
 
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where:
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::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}}</math>
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::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}}</math>
 
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and:
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• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
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• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
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• <math>\beta </math>  is the Weibull shape parameter (unknown, the first of four parameters to be estimated).
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• <math>B</math>  is the first T-NT parameter (unknown, the second of four parameters to be estimated).
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• <math>C</math>  is the second T-NT parameter (unknown, the third of four parameters to be estimated).
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• <math>n</math>  is the third T-NT parameter (unknown, the fourth of four parameters to be estimated).
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• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{U}_{i}}</math>  is the non-thermal stress level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
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• <math>S</math>  is the number of groups of suspension data points.
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• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
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• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
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• <math>FI</math>  is the number of interval data groups.
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• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the  <math>{{i}^{th}}
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• </math>  group of data intervals.
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• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  <math>{{i}^{th}}</math>  interval.
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• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the  <math>{{i}^{th}}</math>  interval.
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The solution (parameter estimates) will be found by solving for the parameters  <math>B,</math> <math>C,</math> <math>n</math>  and  <math>\beta </math>  so that  <math>\tfrac{\partial \Lambda }{\partial B}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial C}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial n}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial \beta }=0</math> .
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Latest revision as of 06:15, 15 August 2012

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