Eyring Relationship - Revision history

Sharon Honecker: /* Conditional Reliability Function */ changed R(t,T,V) to R((t|T),V)

2017-02-08T21:07:06Z

Conditional Reliability Function: changed R(t,T,V) to R((t|T),V)

← Older revision		Revision as of 21:07, 8 February 2017
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	The Eyring-Weibull conditional reliability function at a specified stress level is given by:		The Eyring-Weibull conditional reliability function at a specified stress level is given by:

	::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}{{{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}\,\!</math>		::<math>R((t\|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}{{{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}\,\!</math>

	or:		or:

	::<math>R(T,t,V)={{e}^{-\left[ {{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }} \right]}}\,\!</math>		::<math>R((t\|T),V)={{e}^{-\left[ {{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }} \right]}}\,\!</math>

	====Reliable Life====		====Reliable Life====

Sharon Honecker: /* Conditional Reliability */ changed R(t,T,V) to R((t|T),V)

2017-02-08T21:06:20Z

Conditional Reliability: changed R(t,T,V) to R((t|T),V)

← Older revision		Revision as of 21:06, 8 February 2017
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	The conditional reliability function for the Eyring-exponential model is given by:		The conditional reliability function for the Eyring-exponential model is given by:

	::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}\,\!</math>		::<math>R((t\|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}\,\!</math>

	====Reliable Life====		====Reliable Life====

Chris Kahn at 22:18, 25 January 2013

2013-01-25T22:18:59Z

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Richard House at 22:18, 21 September 2012

2012-09-21T22:18:07Z

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Richard House at 20:40, 4 September 2012

2012-09-04T20:40:09Z

← Older revision		Revision as of 20:40, 4 September 2012
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	{{template:ALTABOOK\|5}}		{{template:ALTABOOK\|5}}

	The Eyring relationship was formulated from quantum mechanics principles [[~~Reference Appendix D~~: ~~References~~\|[9]]] and is most often used when thermal stress (temperature) is the acceleration variable. However, the Eyring relationship is also often used for stress variables other than temperature, such as humidity. The relationship is given by:		The Eyring relationship was formulated from quantum mechanics principles, as discussed in Glasstone et al. [[Appendix_E:_References\|[9]]], and is most often used when thermal stress (temperature) is the acceleration variable. However, the Eyring relationship is also often used for stress variables other than temperature, such as humidity. The relationship is given by:

	::<math>L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>		::<math>L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>

Richard House at 03:58, 29 August 2012

2012-08-29T03:58:29Z

Richard House at 00:02, 23 August 2012

2012-08-23T00:02:08Z

← Older revision		Revision as of 00:02, 23 August 2012
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	::<math>L(V)=C\cdot {{e}^{\tfrac{B}{V}}}</math>		::<math>L(V)=C\cdot {{e}^{\tfrac{B}{V}}}</math>

	Comparing the above equation to the Arrhenius relationship, it can be seen that the only difference between the two relationships is the <math>\tfrac{1}{V}</math> term above. In general, both relationships yield very similar results. Like the Arrhenius, the Eyring relationship is plotted on a log-reciprocal paper.		Comparing the above equation to the Arrhenius relationship, it can be seen that the only difference between the two relationships is the <math>\tfrac{1}{V}\,\!</math> term above. In general, both relationships yield very similar results. Like the Arrhenius, the Eyring relationship is plotted on a log-reciprocal paper.

	[[Image:ALTA7.2.png\|center\|400px\|Eyring relationship plotted on Arrhenius paper.]]		[[Image:ALTA7.2.png\|center\|400px\|Eyring relationship plotted on Arrhenius paper.]]

Richard House: /* Maximum Likelihood Estimation Method */

2012-08-23T00:00:43Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 00:00, 23 August 2012
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	*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending 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>\beta ,</math> <math>A</math> and <math>B</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <math>\tfrac{\partial \Lambda }{\partial A}=0</math> and <math>\tfrac{\partial \Lambda }{\partial B}=0</math>		The solution (parameter estimates) will be found by solving for the parameters <math>\beta ,\,\!</math> <math>A\,\!</math> and <math>B\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,\,\!</math> <math>\tfrac{\partial \Lambda }{\partial A}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial B}=0\,\!</math>

	where:		where:

Richard House: /* Maximum Likelihood Estimation Method */

2012-08-22T23:58:55Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 23:58, 22 August 2012
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	*<math>FI\,\!</math> is the number of interval data groups.		*<math>FI\,\!</math> is the number of interval data groups.

	*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the i <math>^{th}\,\!</math> group of data intervals.		*<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_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.

Richard House: /* Confidence Bounds on Time */

2012-08-22T23:56:24Z

Confidence Bounds on Time

← Older revision		Revision as of 23:56, 22 August 2012
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	\end{align}</math>		\end{align}</math>

	Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:		Solving for <math>{{T}_{U}}\,\!</math> and <math>{{T}_{L}}\,\!</math> yields:

	::<math>\begin{align}		::<math>\begin{align}

← Older revision		Revision as of 03:58, 29 August 2012
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	=Eyring-Exponential=		=Eyring-Exponential=
	The ~~<math>~~pdf~~</math>~~ of the 1-parameter exponential distribution is given by:		The ''pdf'' of the 1-parameter exponential distribution is given by:

	::<math>f(t)=\lambda \cdot {{e}^{-\lambda \cdot t}}</math>		::<math>f(t)=\lambda \cdot {{e}^{-\lambda \cdot t}}</math>
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	::<math>f(t)=\frac{1}{m}\cdot {{e}^{-\tfrac{t}{m}}}</math>		::<math>f(t)=\frac{1}{m}\cdot {{e}^{-\tfrac{t}{m}}}</math>

	The Eyring-exponential model ~~<math>~~pdf~~</math>~~ can then be obtained by setting <math>m=L(V)\,\!</math>:		The Eyring-exponential model ''pdf'' can then be obtained by setting <math>m=L(V)\,\!</math>:

	::<math>m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>		::<math>m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>

	and substituting for <math>m\,\!</math> in the exponential ~~<math>~~pdf~~</math>~~ equation:		and substituting for <math>m\,\!</math> in the exponential ''pdf'' equation:

	::<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>		::<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>
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	=Eyring-Weibull=<!-- THIS SECTION HEADER IS LINKED TO: Eyring Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->		=Eyring-Weibull=<!-- THIS SECTION HEADER IS LINKED TO: Eyring Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
	The ~~<math>~~pdf~~</math>~~ for 2-parameter Weibull distribution is given by:		The ''pdf'' for 2-parameter Weibull distribution is given by:

	::<math>f(t)=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{\eta } \right)}^{\beta }}}}</math>		::<math>f(t)=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{\eta } \right)}^{\beta }}}}</math>

	The scale parameter (or characteristic life) of the Weibull distribution is <math>\eta \,\!</math> . The Eyring-Weibull model ~~<math>~~pdf~~</math>~~ can then be obtained by setting <math>\eta =L(V)\,\!</math>:		The scale parameter (or characteristic life) of the Weibull distribution is <math>\eta \,\!</math> . The Eyring-Weibull model ''pdf'' can then be obtained by setting <math>\eta =L(V)\,\!</math>:

	::<math>\eta =L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>		::<math>\eta =L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
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	::<math>\frac{1}{\eta }=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>		::<math>\frac{1}{\eta }=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>

	Substituting for <math>\eta \,\!</math> into the Weibull ~~<math>~~pdf~~</math>~~ yields:		Substituting for <math>\eta \,\!</math> into the Weibull ''pdf'' yields:

	::<math>f(t,V)=\beta \cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{\left( t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>		::<math>f(t,V)=\beta \cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{\left( t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>
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	=Eyring-Lognormal=		=Eyring-Lognormal=
	The ~~<math>~~pdf~~</math>~~ of the lognormal distribution is given by:		The ''pdf'' of the lognormal distribution is given by:

	::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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	::<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math>		::<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math>

	Substituting this into the lognormal ~~<math>~~pdf~~</math>~~ yields the Eyring-lognormal model ~~<math>~~pdf~~</math>~~:		Substituting this into the lognormal ''pdf'' yields the Eyring-lognormal model ''pdf'':

	::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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	==Generalized Eyring-Exponential==		==Generalized Eyring-Exponential==
	By setting <math>m=L(V,U)\,\!</math>, the exponential ~~<math>~~pdf~~</math>~~ becomes:		By setting <math>m=L(V,U)\,\!</math>, the exponential ''pdf'' 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>		::<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>
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	==Generalized Eyring-Weibull==		==Generalized Eyring-Weibull==
	By setting <math>\eta =L(V,U)\,\!</math> to the Weibull ~~<math>~~pdf~~</math>~~, the generalized Eyring Weibull model is given by:		By setting <math>\eta =L(V,U)\,\!</math> to the Weibull ''pdf'', the generalized Eyring Weibull model is given by:

	::<math>\begin{align}		::<math>\begin{align}
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	==Generalized Eyring-Lognormal==		==Generalized Eyring-Lognormal==
	By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> to the lognormal ~~<math>~~pdf~~</math>~~, the generalized Erying lognormal model is given by:		By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> to the lognormal ''pdf'', the generalized Erying lognormal model is given by:

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