Cumulative Damage-Power-Weibull Example: Difference between revisions

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Using the simple step-stress data given [[Time-Varying Stress Models#Model Formulation|here]], one would define  <math>x(t)</math>  as:
 
<center><math>\begin{align}
x(t)=\ & 2,\text{    }0<t\le 250 \\
=\ & 3,\text{    }250<t\le 350 \\
=\ & 4,\text{    }350<t\le 370 \\
=\ & 5,\text{    }370<t\le 380 \\
=\ & 6,\text{    }380<t\le 390 \\
=\ & 7,\text{    }390<t\le +\infty 
\end{align}</math></center>
 
 
Assuming a power relation as the underlying life-stress relationship and the Weibull distribution as the underlying life distribution, one can then formulate the log-likelihood function for the above data set as,
 
 
::<math>\begin{align}
  & \ln (L)= & \Lambda =\overset{F}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,\ln \left\{ \beta {{\left[ \frac{x(t)}{a} \right]}^{n}}{{\left[ \mathop{}_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta -1}} \right\} -\overset{F}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,\left\{ {{\left[ \mathop{}_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta }} \right\} 
\end{align}</math>
 
<br>
where:
<br>
• <math>F</math>  is the number of exact time-to-failure data points.
<br>
• <math>\beta </math>  is the Weibull shape parameter.
<br>
• <math>a</math>  and  <math>n</math>  are the IPL parameters.
<br>
• <math>x(t)</math>  is the stress profile function.
<br>
• <math>{{t}_{i}}</math>  is the  <math>{{i}^{th}}</math>  time to failure.
 
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The parameter estimates for  <math>\hat{\beta }</math> ,  <math>\hat{a}</math>  and  <math>\hat{n}</math>  can be obtained by simultaneously solving, <math>\tfrac{\partial \Lambda }{\partial a}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial n}=0</math> . Using ALTA, the parameter estimates for this data set are:
 
 
::<math>\begin{align}
\widehat{\beta }=\ & 2.67829 \\
  \widehat{\alpha }=\ & 9.842122 \\
  \widehat{n}=\ & -3.998466 
\end{align}</math>
 
<br>
Once the parameters are obtained, one can now determine the reliability for these units at any time  <math>t</math>  and stress  <math>x(t)</math>  from:
 
 
::<math>R\left( t,x\left( t \right) \right)={{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}</math>
 
 
or at a fixed stress level  <math>x(t)=2V</math>  and  <math>t=300</math> ,
 
 
::<math>R\left( t=300,x(t)=2 \right)={{e}^{-{{\left[ \mathop{}_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}=97.5%</math>
 
 
The mean time to failure (MTTF) at any stress  <math>x(t)</math>  can be determined by:
 
 
::<math>MTTF\left( x\left( t \right) \right)=\int_{0}^{\infty }t\left[ \left\{ \beta {{\left[ \frac{x\left( t \right)}{a} \right]}^{n}}{{\left[ \int_{0}^{t}{{\left[ \frac{x\left( u \right)}{a} \right]}^{n}}du \right]}^{\beta -1}} \right\}{{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}} \right]dt</math>
 
 
or at a fixed stress level  <math>x\left( t \right)=2V</math> ,
 
 
::<math>MTTF\left( x\left( t \right) \right)=1046.3hrs</math>
 
<br>
Any other metric of interest (e.g. failure rate, conditional reliability etc.) can also be determined using the basic definitions given in [[Appendix A: Brief Statistical Background|Appendix A]] and calculated automatically with ALTA.
 
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Latest revision as of 22:27, 15 September 2023

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