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

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The  <math>pdf</math>  for 2-parameter Weibull distribution is given by:
The  <math>pdf</math>  for 2-parameter Weibull distribution is given by:
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::<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>
 
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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):
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>\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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:or:
or:
 
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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>
 
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<br>
Substituting for  <math>\eta </math>  into Eqn. (Eyrpdf):
Substituting for  <math>\eta </math>  into Eqn. (Eyrpdf):
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<br>
::<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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Revision as of 21:49, 10 February 2012

Webnotes-alta.png

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]

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