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==Generalized Eyring Relationship==
#REDIRECT [[Eyring_Relationship#Generalized_Eyring_Relationship]]
 
===Introduction===
 
<br>
The generalized Eyring relationship is used when temperature and a second non-thermal stress (e.g. voltage) are the accelerated stresses of a test and their interaction is also of interest. This relationship is given by:
 
<br>
::<math>L(V,U)=\frac{1}{V}{{e}^{A+\tfrac{B}{V}+CU+D\tfrac{U}{V}}}</math>
 
<br>
:where:
 
<br>
•     is the temperature (in absolute units ).
 
• <math>U</math>  is the non-thermal stress (i.e. voltage, vibration, etc.).
<math>A,B,C,D</math> are the parameters to be determined.
 
<br>
The Eyring relationship is a simple case of the generalized Eyring relationship where  <math>C=D=0</math>  and  <math>{{A}_{Eyr}}=-{{A}_{GEyr}}.</math>
Note that the generalized Eyring relationship includes the interaction term of  <math>U</math>  and  <math>V</math>  as described by the  <math>D\tfrac{U}{V}</math>  term. In other words, this model can estimate the effect of changing one of the factors depending on the level of the other factor.
 
{{gen-eyring acceleration factor}}
 
===Generalized Eyring-Exponential===
<br>
By setting  <math>m=L(V,U)</math>  as given in Eqn. (Gen-Eyr), the exponential  <math>pdf</math>  becomes:
 
<br>
::<math>f(t,V,U)=\left( V{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right){{e}^{-tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}}}}</math>
 
====Generalized Eyring-Exponential Reliability Function====
 
The generalized Eyring exponential model reliability function is given by:
 
<br>
::<math>R(T,U,V)={{e}^{-tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}}}}</math>
 
====Parameter Estimation====
 
<br>
Substituting the generalized Eyring relationship into the exponential log-likelihood equation yields:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( {{V}_{i}}{{e}^{-A-\tfrac{B}{{{V}_{i}}}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{{{V}_{i}}}}} \right) \\
&  & \overset{Fe}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( {{T}_{i}}{{V}_{i}}{{e}^{-A-\tfrac{B}{{{V}_{i}}}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{{{V}_{i}}}}} \right) \\
&  & -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( T_{i}^{\prime }V_{i}^{\prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime }}-CU_{i}^{\prime }-D\tfrac{U_{i}^{\prime }}{V_{i}^{\prime }}}} \right) \\
&  & +\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>
 
<br>
:where:
 
<br>
::<math>R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime })={{e}^{-T_{Li}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}}}}</math>
 
<br>
::<math>R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime })={{e}^{-T_{Ri}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}}}}</math>
 
<br>
:and:
<br>
• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
<br>
• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
<br>
• <math>A,B,C,D</math>  are parameters to be estimated.
<br>
• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>{{U}_{i}}</math>  is the non-thermal stress level of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>S</math>  is the number of groups of suspension data points.
<br>
• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
<br>
• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
<br>
• <math>FI</math>  is the number of interval data groups.
<br>
• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the  <math>{{i}^{th}}</math>  group of data intervals.
<br>
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  <math>{{i}^{th}}</math>  interval.
<br>
• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the  <math>{{i}^{th}}</math>  interval.
<br>
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> .
<br>
 
===Generalized Eyring-Weibull===
<br>
By setting  <math>\eta =L(V,U)</math>  from Eqn. (Gen-Eyr), the generalized Eyring Weibull model is given by:
 
<br>
::<math>\begin{align}
  & f(t,V,U)= & \beta \left( V{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right){{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta -1}} \\
&  & .{{e}^{-{{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta }}}} 
\end{align}</math>
 
 
====Generalized Eyring-Weibull Reliability Function====
 
The generalized Eyring Weibull reliability function is given by:
 
 
<br>
::<math>R(T,V,U)={{e}^{-{{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta }}}}</math>
 
 
====Parameter Estimation====
 
Substituting the generalized Eyring model into the Weibull log-likelihood equation yields:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\beta \left( V{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right) \\
&  & {{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta -1}}] \\
&  & -\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( {{t}_{i}}{{V}_{i}}{{e}^{-A-\tfrac{B}{{{V}_{i}}}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{{{V}_{i}}}}} \right)}^{\beta }} \\
&  & -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( t_{i}^{\prime }V_{i}^{\prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime }}-CU_{i}^{\prime }-D\tfrac{U_{i}^{\prime }}{V_{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>
 
<br>
:where:
<br>
<br>
::<math>R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime })={{e}^{-{{\left( T_{Li}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}} \right)}^{\beta }}}}</math>
 
<br>
::<math>R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime })={{e}^{-{{\left( T_{Ri}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}} \right)}^{\beta }}}}</math>
 
<br>
:and:
<br>
• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
 
• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
 
• <math>A,B,C,D</math>  are parameters to be estimated.
 
• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
 
• <math>{{U}_{i}}</math>  is the non-thermal stress level of the  <math>{{i}^{th}}</math>  group.
 
• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
 
• <math>S</math>  is the number of groups of suspension data points.
 
• <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.
 
• <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.
<br>
<br>
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> .
<br>
 
===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>
::<math>f(t,V,U)=\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}</math>
 
<br>
:where:
 
<br>
::<math></math>
 
====Generalized Eyring-Lognormal Reliability Function====
<br>
 
The generalized Erying lognormal reliability function is given by:
 
<br>
 
::<math>R(T,V,U)=1-\Phi (z)</math>
 
====Parameter Estimation====
 
<br>
Substituting the generalized Eyring model into the lognormal log-likelihood equation yields:
 
<br>
::<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>
 
<br>
:where:
 
<br>
::<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>
 
<br>
::<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>
 
<br>
:and:
<br>
• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
<br>
• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
<br>
• <math>A,B,C,D</math>  are parameters to be estimated.
<br>
• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>{{U}_{i}}</math>  is the non-thermal stress level of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>S</math>  is the number of groups of suspension data points.
<br>
• <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.
<br>
• <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.
<br>
<br>
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> .
<br>
 
===Example===
<br>
The following data set represents failure times (in hours) obtained from an electronics epoxy packaging accelerated life test performed to understand the synergy between temperature and humidity and estimate the  <math>B10</math>  life at the use conditions of  <math>T=350K</math>  and  <math>H=0.3</math> . The data set is modeled using the lognormal distribution and the generalized Eyring model.
<br>
 
[[Image:altaproject.png|thumb|center|400px| ]]
 
[[Image:altaaccelerated.png|thumb|center|400px| ]]
 
[[Image:altatimes2fail.png|thumb|center|400px|]]
 
[[Image:tempNhumin.png|thumb|center|400px|]]
 
[[Image:406-1-2.png|thumb|center|400px|]]
<math></math>
 
The probability plot at the use conditions is shown next.
 
<br>
[[Image:plotfolio426.png|thumb|center|400px|]]
<br>
The  <math>B10</math>  information is estimated to be 3004.63 hours, as shown next.
<br>
[[Image:tempBX.png|thumb|center|400px|]]
 
<br>

Latest revision as of 00:30, 17 August 2012

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