Template:Generalized eyring - Revision history

Richard House: Redirected page to Eyring Relationship#Generalized Eyring Relationship

2012-08-17T00:30:03Z

Redirected page to Eyring Relationship#Generalized Eyring Relationship

← Older revision		Revision as of 00:30, 17 August 2012
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	~~{{generalized eyring relationship}}~~		#REDIRECT [[Eyring_Relationship#Generalized_Eyring_Relationship]]

Nicolette Young: /* Generalized Eyring Relationship */

2012-01-18T16:49:28Z

Generalized Eyring Relationship

← Older revision		Revision as of 16:49, 18 January 2012
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	~~==Generalized 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}}~~

	~~{{generalized eyring-weibull}}~~

	~~{{generalized eyring-log}}~~

	~~===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>~~

Nicolette Young: /* Generalized Eyring-Lognormal */

2012-01-12T23:11:59Z

Generalized Eyring-Lognormal

@@ Line 29: / Line 29: @@
 {{generalized eyring-weibull}}
-===Generalized Eyring-Lognormal===
+{{generalized eyring-log}}
 ===Example===

Nicolette Young: /* Generalized Eyring-Weibull */

2012-01-12T23:11:36Z

Generalized Eyring-Weibull

@@ Line 27: / Line 27: @@
 {{generalized eyring-exponential}}
-===Generalized Eyring-Weibull===
+{{generalized eyring-weibull}}
 ===Generalized Eyring-Lognormal===

Nicolette Young: /* Generalized Eyring-Exponential */

2012-01-12T23:11:02Z

Generalized Eyring-Exponential

@@ Line 25: / Line 25: @@
 {{gen-eyring acceleration factor}}
-===Generalized Eyring-Exponential===
+{{generalized eyring-exponential}}
 ===Generalized Eyring-Weibull===

Nicolette Young: /* Acceleration Factor */

2012-01-12T23:10:22Z

Acceleration Factor

@@ Line 23: / Line 23: @@
 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.
-===Acceleration Factor===
+{{gen-eyring acceleration factor}}
 ===Generalized Eyring-Exponential===

Nicolette Young: Created page with '==Generalized Eyring Relationship== ===Introduction===
The generalized Eyring relationship is used when temperature and a second non-thermal stress (e.g. voltage) are the …'

2012-01-12T23:09:34Z

Created page with '==Generalized Eyring Relationship== ===Introduction=== The generalized Eyring relationship is used when temperature and a second non-thermal stress (e.g. voltage) are the …'

New page

==Generalized Eyring Relationship==

===Introduction===

 
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:

 
::<math>L(V,U)=\frac{1}{V}{{e}^{A+\tfrac{B}{V}+CU+D\tfrac{U}{V}}}</math>

 
:where:

 
• 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.

 
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.

===Acceleration Factor===

 
Most models in actual use do not include any interaction terms, therefore, the acceleration factor can be computed by multiplying the acceleration factors obtained by changing each factor while keeping the other factors constant. In the case of the generalized Eyring relationship, the acceleration factor is derived differently.

 
The acceleration factor for the generalized Eyring relationship is given by:

 
::<math>\begin{align}
& {{A}_{F}}= & \frac{{{L}_{USE}}}{{{L}_{Accelerated}}}=\frac{\tfrac{1}{{{V}_{U}}}{{e}^{A+\tfrac{B}{{{V}_{U}}}+C{{U}_{U}}+D\tfrac{{{U}_{U}}}{{{V}_{U}}}}}}{\tfrac{1}{{{T}_{A}}}{{e}^{A+\tfrac{B}{{{V}_{A}}}+C{{U}_{A}}+D\tfrac{{{U}_{A}}}{{{V}_{A}}}}}} \\
& = & \frac{\tfrac{1}{{{V}_{U}}}{{e}^{A+\tfrac{B}{{{V}_{U}}}+C{{U}_{U}}+D\tfrac{{{U}_{U}}}{{{V}_{U}}}}}}{\tfrac{1}{{{V}_{A}}}{{e}^{A+\tfrac{B}{{{V}_{A}}}+C{{U}_{A}}+D\tfrac{{{U}_{A}}}{{{V}_{A}}}}}}
\end{align}</math>

 
where:
 
• <math>{{L}_{USE}}</math> is the life at use stress level.

• <math>{{L}_{Accelerated}}</math> is the life at the accelerated stress level.

• <math>{{V}_{u}}</math> is the use temperature level.

• <math>{{V}_{A}}</math> is the accelerated temperature level.

• <math>{{U}_{A}}</math> is the accelerated non-thermal level.

• <math>{{U}_{u}}</math> is the use non-thermal level.
 

===Generalized Eyring-Exponential===
 
By setting <math>m=L(V,U)</math> as given in Eqn. (Gen-Eyr), the exponential <math>pdf</math> becomes:

 
::<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:

 
::<math>R(T,U,V)={{e}^{-tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}}}}</math>

====Parameter Estimation====

 
Substituting the generalized Eyring relationship into the exponential log-likelihood equation yields:

 
::<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>

 
:where:

 
::<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>

 
::<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>

 
:and:
 
• <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.
 
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> .
 

===Generalized Eyring-Weibull===
 
By setting <math>\eta =L(V,U)</math> from Eqn. (Gen-Eyr), the generalized Eyring Weibull model is given by:

 
::<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:

 
::<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:

 
::<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>

 
:where:
 
 
::<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>

 
::<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>

 
:and:
 
• <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.
 
 
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> .
 

===Generalized Eyring-Lognormal===
 
By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by:

 
::<math>f(t,V,U)=\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}</math>

 
:where:

 
::<math></math>

====Generalized Eyring-Lognormal Reliability Function====
 

The generalized Erying lognormal reliability function is given by:

 

::<math>R(T,V,U)=1-\Phi (z)</math>

====Parameter Estimation====

 
Substituting the generalized Eyring model into the lognormal log-likelihood equation yields:

 
::<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>

 
:where:

 
::<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>

 
::<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>

 
:and:
 
• <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.
 
 
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> .
 

===Example===
 
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.
 

[[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.

 
[[Image:plotfolio426.png|thumb|center|400px|]]
 

The <math>B10</math> information is estimated to be 3004.63 hours, as shown next.
 
[[Image:tempBX.png|thumb|center|400px|]]