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(Created page with '===Generalized Eyring-Lognormal=== <br> By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by: <br> ::<m…')
 
 
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===Generalized Eyring-Lognormal===
#REDIRECT [[Eyring_Relationship#Generalized_Eyring-Lognormal]]
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By setting  <math>\sigma _{T}^{\prime }=L(V,U)</math>  from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by:
 
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::<math>f(t,V,U)=\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}</math>
 
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:where:
 
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::<math></math>
 
====Generalized Eyring-Lognormal Reliability Function====
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The generalized Erying lognormal reliability function is given by:
 
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::<math>R(T,V,U)=1-\Phi (z)</math>
 
====Parameter Estimation====
 
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Substituting the generalized Eyring model into the lognormal log-likelihood equation yields:
 
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::<math>\begin{align}
  & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}]\overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(t_{i}^{\prime })) \right) \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}</math>
 
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:where:
 
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::<math>z_{Ri}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}</math>
 
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::<math>z_{Li}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}</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>A,B,C,D</math>  are 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.
• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
• <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}}</math>  group of data intervals.
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  <math>{{i}^{th}}</math>  interval.
• <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>A,</math>  <math>B,</math>  <math>C,</math> and  <math>D</math>  so that  <math>\tfrac{\partial \Lambda }{\partial A}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial B}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial D}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial D}=0</math> .
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Latest revision as of 00:48, 17 August 2012

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