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Richard House: Redirected page to Arrhenius Relationship#Arrhenius-Weibull Statistical Properties Summary

2012-08-16T04:38:48Z

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	~~===~~Arrhenius-~~Weibull Statistical Properties Summary===~~		#REDIRECT [[Arrhenius_Relationship#Arrhenius-Weibull_Statistical_Properties_Summary]]
	~~<br>~~

	~~{{aaw mean}}~~

	~~{{aaw median}}~~

	~~{{aaw mode}}~~

	~~{{aaw sd}}~~

	~~{{aaw rf}}~~

	~~{{aaw crf}}~~

	~~{{aaw rl}}~~

	~~{{aaw frf}}~~

2012-01-12T19:16:49Z

Arrhenius-Weibull Failure Rate Function

← Older revision		Revision as of 19:16, 12 January 2012
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	{{aaw rl}}		{{aaw rl}}

	~~====Arrhenius-Weibull Failure Rate Function====~~		{{aaw frf}}
	~~<br>~~
	~~The Arrhenius-Weibull failure rate function, <math>\lambda (T)</math> , is given by:~~

	~~<br>~~
	~~::<math>\lambda \left( T,V \right)=\frac~~{~~f\left( T,V \right)}{R\left( T,V \right)}=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^~~{~~\beta -1~~}}~~</math>~~
	~~<br>~~
	~~[[Image:ALTA6.9.gif\|thumb\|center\|300px\|Failure rate function for <math>\Beta<1 </math>, <math>\Beta=1 </math>, and <math>\Beta>1 </math>.]]~~
	~~<br>~~

2012-01-12T19:16:28Z

Reliable Life

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 {{aaw crf}}
-====Reliable Life====
+{{aaw rl}}
 ====Arrhenius-Weibull Failure Rate Function====

2012-01-12T19:15:34Z

Conditional Reliability Function

2012-01-12T19:15:17Z

Conditional Reliability Function

← Older revision		Revision as of 19:15, 12 January 2012
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	~~====Conditional Reliability Function====~~		{{aaw crf
	~~<br>~~
	~~The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:~~


	~~<br>~~
	~~::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>~~

	~~<br>~~
	~~or:~~

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

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

2012-01-12T19:13:20Z

Arrhenius-Weibull Reliability Function

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 {{aaw sd}}
-====Arrhenius-Weibull Reliability Function====
+{{aaw rf}}
 ====Conditional Reliability Function====
 <br>

2012-01-12T19:13:03Z

Standard Deviation

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	{{aaw mode}}		{{aaw mode}}

	~~====Standard Deviation====~~		{{aaw sd}}
	~~<br>~~
	~~The standard deviation, <math>~~{{~~\sigma }_{T}},</math> for the Arrhenius-Weibull model is given by:~~

	~~<br>~~
	~~::<math>{{\sigma }_{T}}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}~~}}~~</math>~~

	====Arrhenius-Weibull Reliability Function====		====Arrhenius-Weibull Reliability Function====

2012-01-12T19:12:41Z

Mode

← Older revision		Revision as of 19:12, 12 January 2012
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	{{aaw median}}		{{aaw median}}

	~~====Mode====~~		{{aaw mode}}
	~~<br>~~
	~~The mode, <math>\tilde~~{~~T},</math>~~
	~~for the Arrhenius-Weibull model is given by:~~

	~~<br>~~
	~~::<math>\tilde~~{~~T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }~~}}~~</math>~~

	====Standard Deviation====		====Standard Deviation====

2012-01-12T19:12:19Z

Median

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	{{aaw mean}}		{{aaw mean}}

	~~====Median====~~		{{aaw median}}

	~~<br>~~
	~~The median, <math>\breve~~{~~T},</math>~~
	~~for the Arrhenius-Weibull model is given by:~~

	~~<br>~~
	~~::<math>\breve~~{~~T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }~~}}~~</math>~~

	====Mode====		====Mode====

2012-01-12T19:11:52Z

Mean or MTTF

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	<br>		<br>

	~~====Mean or MTTF====~~		{{aaw mean}}

	~~<br>~~
	~~The mean, <math>\overline{T}</math> (also called <math>MTTF</math> by some authors), of the Arrhenius-Weibull relationship is given by:~~

	~~<br>~~
	~~::<math>\overline{T}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>~~

	~~<br>~~
	~~where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta }+1 \right)</math> .~~
	~~<br>~~
	~~<br>~~
	====Median====		====Median====