Template:Eyring-weibull stat prop sum - Revision history

Richard House: Redirected page to Eyring Relationship#Eyring-Weibull

2012-08-16T23:14:44Z

Redirected page to Eyring Relationship#Eyring-Weibull

← Older revision		Revision as of 23:14, 16 August 2012
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	~~===~~Eyring-Weibull ~~Statistical Properties Summary===~~		#REDIRECT [[Eyring_Relationship#Eyring-Weibull]]

	~~{{eyring-weib mean}}~~

	~~{{eyring-weib median}}~~

	~~{{eyring-weib mode}}~~

	~~{{eyring-weib sd}}~~

	~~{{eyring-weib rf}}~~

	~~{{eyring-weib cr}}~~

	~~{{eyring-weib rl}}~~

	~~{{eyring-weib frf}}~~

Nicolette Young: /* Eyring-Weibull Failure Rate Function */

2012-01-12T22:46:10Z

Eyring-Weibull Failure Rate Function

← Older revision		Revision as of 22:46, 12 January 2012
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	{{eyring-weib rl}}		{{eyring-weib rl}}

	~~====Eyring-Weibull Failure Rate Function====~~		{{eyring-weib frf}}
	~~<br>~~
	~~The Eyring-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)}=\beta~~ {{~~\left( T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta~~ -1}}~~</math>~~
	~~<br>~~

Nicolette Young: /* Reliable Life */

2012-01-12T22:45:39Z

Reliable Life

← Older revision		Revision as of 22:45, 12 January 2012
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	{{eyring-weib cr}}		{{eyring-weib cr}}

	~~====Reliable Life====~~		{{eyring-weib rl}}
	~~<br>~~

	~~For the Eyring-Weibull model, the reliable life, <math>{{t}_{R}}</math> , of a unit for a specified reliability and starting the mission at age zero is given by:~~

	~~<br>~~
	~~::<math>{{t}_{R}}=\frac{1}~~{~~V}{{e}^~~{-~~\left( A-\tfrac{B}{V} \right)}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }~~}}~~</math>~~


	====Eyring-Weibull Failure Rate Function====		====Eyring-Weibull Failure Rate Function====

Nicolette Young: /* Conditional Reliability Function */

2012-01-12T22:45:16Z

Conditional Reliability Function

← Older revision		Revision as of 22:45, 12 January 2012
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	{{eyring-weib rf}}		{{eyring-weib rf}}

	~~====Conditional Reliability Function====~~		{{eyring-weib cr}}
	~~<br>~~

	~~The Eyring-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( \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>
	~~<br>~~
	~~:or:~~

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

Nicolette Young: /* Eyring-Weibull Reliability Function */

2012-01-12T22:44:40Z

Eyring-Weibull Reliability Function

← Older revision		Revision as of 22:44, 12 January 2012
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	{{eyring-weib sd}}		{{eyring-weib sd}}

	~~====Eyring~~-~~Weibull Reliability Function====~~		{{eyring-weib rf}}

	~~<br>~~
	~~The Eyring-Weibull reliability function is given by:~~

	~~<br>~~
	~~::<math>R(T,V)={{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>~~
	~~<br>~~
	====Conditional Reliability Function====		====Conditional Reliability Function====
	<br>		<br>

Nicolette Young: /* Standard Deviation */

2012-01-12T22:44:16Z

Standard Deviation

← Older revision		Revision as of 22:44, 12 January 2012
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	{{eyring-weib mode}}		{{eyring-weib mode}}

	~~====Standard Deviation====~~		{{eyring-weib sd}}

	~~<br>~~
	~~The standard deviation, <math>{{\sigma }_{T}},</math>~~
	~~for the Eyring-Weibull model is given by:~~

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

	~~<br>~~
	====Eyring-Weibull Reliability Function====		====Eyring-Weibull Reliability Function====

Nicolette Young: /* Mode */

2012-01-12T22:43:54Z

Mode

← Older revision		Revision as of 22:43, 12 January 2012
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	{{eyring-weib median}}		{{eyring-weib median}}

	~~====Mode====~~		{{eyring-weib mode}}

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

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

	~~<br>~~

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

Nicolette Young: /* Median */

2012-01-12T22:43:35Z

Median

← Older revision		Revision as of 22:43, 12 January 2012
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	{{eyring-weib mean}}		{{eyring-weib mean}}

	~~====Median====~~		{{eyring-weib median}}

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

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

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

Nicolette Young: /* Mean or MTTF */

2012-01-12T22:42:21Z

Mean or MTTF

← Older revision		Revision as of 22:42, 12 January 2012
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	===Eyring-Weibull Statistical Properties Summary===		===Eyring-Weibull Statistical Properties Summary===

	~~====Mean or MTTF====~~		{{eyring-weib mean}}

	~~The mean, <math>\overline{T}</math>, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is given by:~~


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

	~~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>~~

	====Median====		====Median====

Nicolette Young: Created page with '===Eyring-Weibull Statistical Properties Summary=== ====Mean or MTTF==== The mean, $\overline{T}$ , or Mean Time To Failure (MTTF) for the Eyring-Weibull model is gi…'

2012-01-12T22:40:48Z

Created page with '===Eyring-Weibull Statistical Properties Summary=== ====Mean or MTTF==== The mean, <math>\overline{T}</math>, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is gi…'

New page

===Eyring-Weibull Statistical Properties Summary===

====Mean or MTTF====

The mean, <math>\overline{T}</math>, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is given by:

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

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> .

 

====Median====

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

 
::<math>\breve{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>

====Mode====

 
The mode, <math>\tilde{T},</math>
for the Eyring-Weibull model is given by:

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

 

====Standard Deviation====

 
The standard deviation, <math>{{\sigma }_{T}},</math>
for the Eyring-Weibull model is given by:

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

 
====Eyring-Weibull Reliability Function====

 
The Eyring-Weibull reliability function is given by:

 
::<math>R(T,V)={{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>
 
====Conditional Reliability Function====
 

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>
 
: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>
 
====Reliable Life====
 

For the Eyring-Weibull model, the reliable life, <math>{{t}_{R}}</math> , of a unit for a specified reliability and starting the mission at age zero is given by:

 
::<math>{{t}_{R}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>

====Eyring-Weibull Failure Rate Function====
 
The Eyring-Weibull failure rate function, <math>\lambda (T)</math> , is given by:
 

::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\beta {{\left( T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta -1}}</math>