Template:TNT Lognormal - Revision history

Richard House: Redirected page to Temperature-NonThermal Relationship#T-NT Lognormal

2012-08-15T06:16:48Z

Redirected page to Temperature-NonThermal Relationship#T-NT Lognormal

Chris Kahn: /* The Mode */

2012-03-09T18:41:37Z

The Mode

← Older revision		Revision as of 18:41, 9 March 2012
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	====The Mode====		====The Mode====
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	• The mode of the T-NT lognormal model is given by:		The mode of the T-NT lognormal model is given by:

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Chris Kahn: /* The Standard Deviation */

2012-03-09T18:41:25Z

The Standard Deviation

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	====The Standard Deviation====		====The Standard Deviation====
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	• The standard deviation of the T-NT lognormal model (standard deviation of the times-to-failure), <math>{{\sigma }_{T}}</math> , is given by:		The standard deviation of the T-NT lognormal model (standard deviation of the times-to-failure), <math>{{\sigma }_{T}}</math> , is given by:

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	• The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:		The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:

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Chris Kahn: /* The Mean */

2012-03-09T18:41:14Z

The Mean

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	====The Mean====		====The Mean====
	• The mean life of the T-NT lognormal model (mean of the times-to-failure), <math>\bar{T}</math> , is given by:		The mean life of the T-NT lognormal model (mean of the times-to-failure), <math>\bar{T}</math> , is given by:

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	• The mean of the natural logarithms of the times-to-failure, <math>{{\bar{T}}^{^{\prime }}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:		The mean of the natural logarithms of the times-to-failure, <math>{{\bar{T}}^{^{\prime }}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:

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Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-17T01:30:03Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 01:30, 17 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}{{\phi }_{pdf}}\left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)+n\ln ({{U}_{i}})-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}{{\phi }_{pdf}}\left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)+n\ln ({{U}_{i}})-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (C)+n\ln ({{U}_{i}})-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \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 })]
	~~& &~~ \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (C)+n\ln ({{U}_{i}})-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \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>		\end{align}</math>

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	:where:		where:

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	::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>		::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

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	:and:		and:
	<br>		<br>
	• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.		• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.

Athanasios Gerokostopoulos: /* Lognormal Failure Rate */

2012-02-17T01:28:34Z

Lognormal Failure Rate

← Older revision		Revision as of 01:28, 17 February 2012
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	::<math>\lambda (T,U,V)=\frac{f(T,U,V)}{R(T,U,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\~~mathop{}_~~{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>		::<math>\lambda (T,U,V)=\frac{f(T,U,V)}{R(T,U,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>

	<br>		<br>

	===Parameter Estimation===		===Parameter Estimation===
	====Maximum Likelihood Estimation Method====		====Maximum Likelihood Estimation Method====

Athanasios Gerokostopoulos: /* Reliable Life */

2012-02-17T01:28:13Z

Reliable Life

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	:where:		where:

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	:and:		and:

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	::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{-\infty }^{z({T}',U,V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>		::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',U,V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

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Athanasios Gerokostopoulos: /* T-NT Lognormal Reliability */

2012-02-17T01:27:12Z

T-NT Lognormal Reliability

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	<br>		<br>
	:or:		or:

	<br>		<br>

Athanasios Gerokostopoulos: /* T-NT Lognormal Reliability */

2012-02-17T01:26:50Z

T-NT Lognormal Reliability

← Older revision		Revision as of 01:26, 17 February 2012
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	::<math>R(T,U,V)=\~~mathop{}_~~{T}^{\infty }f(t,U,V)dt</math>		::<math>R(T,U,V)=\int_{T}^{\infty }f(t,U,V)dt</math>

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	::<math>R(T,U,V)=\~~mathop{}_~~{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>		::<math>R(T,U,V)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>
	<br>		<br>

	====Reliable Life====		====Reliable Life====
	<br>		<br>

Athanasios Gerokostopoulos: /* The Mode */

2012-02-17T01:26:17Z

The Mode

← Older revision		Revision as of 01:26, 17 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\		& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} = & {{e}^{\ln (C)-n\ln (U)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
	& = & {{e}^{\ln (C)-n\ln (U)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
	\end{align}</math>		\end{align}</math>