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

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The  <math>pdf</math> of the lognormal distribution is given by:
| [http://help.synthesis8.com/weibull_alta8/alta_standard_folios.htm the help files...]
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| [[Image:Book blue.png]]
| [http://reliawiki.com/index.php/Template:Erying-log#Eyring-Lognormal the theory textbook...]  
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| [[Image:Articleblue.png]]
| [http://www.weibull.com/hotwire/issue74/relbasics74.htm related article(s)...]
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| [[Image:Bulbblue.png]]
| [http://reliawiki.com/index.php/Template:Example:Eyring#Eyring_Example use example(s)...]
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<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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where:
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<math>{T}'=\ln (T)</math>
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<math>T=\text{times-to-failure}</math>
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and:
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• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.
• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.
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The Eyring-lognormal model can be obtained first by setting  <math>\breve{T}=L(V)</math> in Eqn. (eyring). Therefore:
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<math>\breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>
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or:
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<math>{{e}^{{{\overline{T}}^{\prime }}}}=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>
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Thus:
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<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math>
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Substituting Eqn. (eyr-logn-mean) into Eqn. (Eyr-logn-pdf) yields the Eyring-lognormal model  <math>pdf</math> 
or:
<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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| valign="middle" | [http://reliawiki.com/index.php/Template:Erying-log#Eyring-Lognormal Get More Details...]
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| valign="middle" | [http://reliawiki.com/index.php/Template:Example:Eyring#Eyring_Example See an example]
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Revision as of 17:14, 7 March 2012

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Standard Folio Data Eyring-Lognormal

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