Template:Cd power exponential: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
Line 1: Line 1:
===Cumulative Damage Power Relationship===
===Cumulative Damage Power - Exponential===
<br>
Given a time-varying stress <math>x(t)</math> and assuming the power law relationship, the mean life is given by:
 
 
::<math>\frac{1}{m(t,\,x)}=s(t,\,x)={{\left( \frac{x(t)}{a} \right)}^{n}}</math>
 
   
The reliability function of the unit under a single stress is given by:


This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life-stress relationship is based on the power relationship. Given a time-varying stress  <math>x(t)</math>  and assuming the power law relationship,  the life-stress relationship is given by:


::<math>R(t,\,x(t))={{e}^{-I(t,\,x)}}</math>
<br>
where:
<br>
<br>
::<math>L(x(t))={{\left( \frac{a}{x(t)} \right)}^{n}}</math>
::<math>I(t,\,x)=\underset{0}{\mathop{\overset{t}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du</math>


<br>
<br>
In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the power law relationship:
Therefore, the <math>pdf</math> is:


<br>
<br>
::<math>L(x(t))={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}\ln \left( x(t) \right)}}</math>
::<math>f(t,\,x)=s(t,\,x){{e}^{-I(t,\,x)}}</math>
 
 
Parameter estimation can be accomplished via maximum likelihood estimation methods, and confidence intervals can be approximated using the Fisher matrix approach. Once the parameters are determined, all other characteristics of interest (e.g. mean life, failure rate, etc.) can be obtained utilizing the statistical properties definitions presented in previous chapters. The log-likelihood equation is as follows:
 
 
::<math>\begin{align}
  & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [s({{T}_{i}},\,{{x}_{i}})]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},\,{{x}_{i}}) \right) -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( I(T_{i}^{\prime },\,x_{i}^{\prime }) \right)+\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>
 
 
where:


<br>
Therefore, instead of displaying  <math>a</math>  and  <math>n</math>  as the calculated parameters, the following reparameterization is used:


<br>
::<math>\begin{align}
::<math>\begin{align}
{{\alpha }_{0}}=\ & \ln ({{a}^{n}}) \\  
  & R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },\,x_{i}^{\prime \prime })= & {{e}^{-I(T_{Li}^{\prime \prime },\,x_{i}^{\prime \prime })}} \\  
  {{\alpha }_{1}}=\ & -n  
  & R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },\,x_{i}^{\prime \prime })= & {{e}^{-I(T_{Ri}^{\prime \prime },\,x_{i}^{\prime \prime })}}  
\end{align}</math>
\end{align}</math>


and:
<br>
• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
<br>
• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.
<br>
• <math>{{T}_{i}}</math>  is the exact failure time of the <math>{{i}^{th}}</math> group.
<br>
• <math>S</math> is the number of groups of suspension data points.
<br>
• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
<br>
• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
<br>
• <math>FI</math> is the number of interval data groups.
<br>
• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math>  group of data intervals.
<br>
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.
<br>
• <math>T_{Ri}^{\prime \prime }</math> is the ending of the  <math>{{i}^{th}}</math>  interval.
<br>
<br>

Revision as of 22:33, 21 February 2012

Cumulative Damage Power - Exponential


Given a time-varying stress [math]\displaystyle{ x(t) }[/math] and assuming the power law relationship, the mean life is given by:


[math]\displaystyle{ \frac{1}{m(t,\,x)}=s(t,\,x)={{\left( \frac{x(t)}{a} \right)}^{n}} }[/math]


The reliability function of the unit under a single stress is given by:


[math]\displaystyle{ R(t,\,x(t))={{e}^{-I(t,\,x)}} }[/math]


where:

[math]\displaystyle{ I(t,\,x)=\underset{0}{\mathop{\overset{t}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du }[/math]


Therefore, the [math]\displaystyle{ pdf }[/math] is:


[math]\displaystyle{ f(t,\,x)=s(t,\,x){{e}^{-I(t,\,x)}} }[/math]


Parameter estimation can be accomplished via maximum likelihood estimation methods, and confidence intervals can be approximated using the Fisher matrix approach. Once the parameters are determined, all other characteristics of interest (e.g. mean life, failure rate, etc.) can be obtained utilizing the statistical properties definitions presented in previous chapters. The log-likelihood equation is as follows:


[math]\displaystyle{ \begin{align} & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [s({{T}_{i}},\,{{x}_{i}})]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},\,{{x}_{i}}) \right) -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( I(T_{i}^{\prime },\,x_{i}^{\prime }) \right)+\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]


where:


[math]\displaystyle{ \begin{align} & R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },\,x_{i}^{\prime \prime })= & {{e}^{-I(T_{Li}^{\prime \prime },\,x_{i}^{\prime \prime })}} \\ & R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },\,x_{i}^{\prime \prime })= & {{e}^{-I(T_{Ri}^{\prime \prime },\,x_{i}^{\prime \prime })}} \end{align} }[/math]


and:
• [math]\displaystyle{ {{F}_{e}} }[/math] is the number of groups of exact times-to-failure data points.
• [math]\displaystyle{ {{N}_{i}} }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}} }[/math] time-to-failure data group.
• [math]\displaystyle{ {{T}_{i}} }[/math] is the exact failure time of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ S }[/math] is the number of groups of suspension data points.
• [math]\displaystyle{ N_{i}^{\prime } }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}} }[/math] group of suspension data points.
• [math]\displaystyle{ T_{i}^{\prime } }[/math] is the running time of the [math]\displaystyle{ {{i}^{th}} }[/math] suspension data group.
• [math]\displaystyle{ FI }[/math] is the number of interval data groups.
• [math]\displaystyle{ N_{i}^{\prime \prime } }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}} }[/math] group of data intervals.
• [math]\displaystyle{ T_{Li}^{\prime \prime } }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.
• [math]\displaystyle{ T_{Ri}^{\prime \prime } }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Cd_power_exponential&oldid=17271"