Template:Aaw: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
(Created page with '==Arrhenius-Weibull== <br> The <math>pdf</math> for 2-parameter Weibull distribution is given by: <br> ::{{weibull2pdf}} <br> The scale parameter (or characteristic life) of …')
 
(Redirected page to Arrhenius Relationship)
 
(6 intermediate revisions by 2 users not shown)
Line 1: Line 1:
==Arrhenius-Weibull==
#REDIRECT [[Arrhenius_Relationship]]
<br>
The  <math>pdf</math>  for 2-parameter Weibull distribution is given by:
 
<br>
::{{weibull2pdf}}
 
<br>
The scale parameter (or characteristic life) of the Weibull distribution is  <math>\eta </math> .
 
<br>
The Arrhenius-Weibull model pdf can then be obtained by setting  <math>\eta =L(V)</math>  in Eqn. (arrhenius):
 
<br>
::<math>\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}</math>
 
 
<br>
and substituting for  <math>\eta </math>  in Eqn. (Weibullpdf):
 
<br>
::<math>f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}</math>
 
<br>
An illustration of the  <math>pdf</math>  for different stresses is shown in Fig. 6.  As expected, the  <math>pdf</math>  at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3 in Fig. 6). This behavior is observed when the parameter  <math>B</math>  of the Arrhenius model is positive.
 
<br>
[[Image:ALTA6.6.gif|thumb|center|300px|Behavior of the probability density function at different stresses and with the parameters held constant.]]
 
<br>
The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in Chapter 5.
 
===Arrhenius-Weibull Statistical Properties Summary===
<br>
 
====Mean or MTTF====
 
<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====
 
<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====
<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====
<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====
<br>
The Arrhenius-Weibull reliability function is given by:
 
<br>
::<math>R(T,V)={{e}^{-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}</math>
 
<br>
If the parameter  <math>B</math>  is positive, then the reliability increases as stress decreases.
<br>
[[Image:ALTA6.7.gif|thumb|center|300px|Behavior of the reliability function at different stress and constant parameter values.]]
<br>
 
<br>
The behavior of the reliability function of the Weibull distribution for different values of  <math>\beta </math>  was illustrated in Chapter 5. In the case of the Arrhenius-Weibull model, however, the reliability is a function of stress also. A 3D plot such as the ones shown in Fig. 8 is now needed to illustrate the effects of both the stress and  <math>\beta .</math>
 
<br>
::<math></math>
 
<br>
[[Image:ALTA6.8.gif|thumb|center|300px|Reliability function for <math>\Beta<1 </math>, <math>\Beta=1 </math>, and <math>\Beta>1 </math>.]]
 
<br>
====Conditional Reliability Function====
<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====
<br>
 
For the Arrhenius-Weibull relationship, 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}}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left\{ -\ln \left[ R\left( {{t}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
 
<br>
This is the life for which the unit will function successfully with a reliability of  <math>R({{t}_{R}})</math> . If  <math>R({{t}_{R}})=0.50</math>  then  <math>{\breve{T}</math>,
the median life, or the life by which half of the units will survive.
<br>
<br>
 
====Arrhenius-Weibull Failure Rate Function====
<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>
 
===Parameter Estimation===
<br>
====Maximum Likelihood Estimation Method====
<br>
The Arrhenius-Weibull log-likelihood function is as follows:
 
<br>
::<math>\begin{align}
  & \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}} \right] \\
&  & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }] 
\end{align}</math>
 
<br>
where:
 
<br>
::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>
 
 
<br>
::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>
 
<br>
and:
<br>
• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
 
• ..  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
 
• <math>\beta </math>  is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
 
• <math>B</math>  is the Arrhenius parameter (unknown, the second of three parameters to be estimated).
 
• <math>C</math>  is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).
 
• <math>{{V}_{i}}</math>  is the stress level of the  <math>{{i}^{th}}</math>  group.
 
• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
 
• <math>S</math>  is the number of groups of suspension data points.
 
• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
 
• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
 
• <math>FI</math>  is the number of interval data groups.
 
• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the i <math>^{th}</math>  group of data intervals.
 
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the i <math>^{th}</math>  interval.
 
• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the i <math>^{th}</math>  interval.
 
The solution (parameter estimates) will be found by solving for  <math>\widehat{\beta },</math>  <math>\widehat{B},</math>  <math>\widehat{C}</math>  so that  <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math>  <math>\tfrac{\partial \Lambda }{\partial B}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
&  & -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
&  & -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} 
\end{align}</math>
 
 
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial B}= & -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}+\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }} \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} 
\end{align}</math>
 
 
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial C}= & -\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} 
\end{align}</math>

Latest revision as of 07:26, 8 August 2012

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Aaw&oldid=30398"