ALTA ALTA Standard Folio Data Eyring-Weibull: Difference between revisions

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The <math>pdf</math> of the 1-parameter exponential distribution is given by:
The <math>pdf</math> for 2-parameter Weibull distribution is given by:


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


<br>
<br>
It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:
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>  in Eqn. (eyring):


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


<br>
<br>
:thus:
:or:


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


<br>
<br>
The Eyring-exponential model <math>pdf</math>  can then be obtained by setting  <math>m=L(V)</math>  in Eqn. (eyring):  
Substituting for <math>\eta </math>  into Eqn. (Eyrpdf):


<br>
<br>
::<math>m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</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>
 
<br>
and substituting for  <math>m</math>  in Eqn. (pdfexpm2):
 
<br>
::<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>
<br>


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| align="center" valign="middle" | [http://reliawiki.com/index.php/Template:Alta_a-e.e-e#Eyring-Exponential Get More Details...]
| align="center" valign="middle" | [http://reliawiki.com/index.php/Template:Alta_eyring-weibull#Eyring-Weibull Get More Details...]
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| align="center" valign="middle" | [Link2 See Examples...]
| align="center" valign="middle" | [Link2 See Examples...]

Revision as of 22:17, 16 January 2012

Reliability Web Notes

Standard Folio Data Eyring-Weibull
ALTA

The [math]\displaystyle{ pdf }[/math] for 2-parameter Weibull distribution is given by:


[math]\displaystyle{ 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]\displaystyle{ \eta }[/math] . The Eyring-Weibull model [math]\displaystyle{ pdf }[/math] can then be obtained by setting [math]\displaystyle{ \eta =L(V) }[/math] in Eqn. (eyring):


[math]\displaystyle{ \eta =L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}} }[/math]


or:


[math]\displaystyle{ \frac{1}{\eta }=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} }[/math]


Substituting for [math]\displaystyle{ \eta }[/math] into Eqn. (Eyrpdf):


[math]\displaystyle{ 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]
Get More Details...
[Link2 See Examples...]



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