Cumulative Damage-Power-Weibull Example - Revision history

Lisa Hacker: Changed redirect target from Accelerated Life Testing Data Analysis Reference to Main Page

2023-09-15T22:27:06Z

Changed redirect target from Accelerated Life Testing Data Analysis Reference to Main Page

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	#REDIRECT [[~~Accelerated_Life_Testing_Data_Analysis_Reference~~]]		#REDIRECT [[Main Page]]

Lisa Hacker: Redirected page to Accelerated Life Testing Data Analysis Reference

2015-06-30T00:28:53Z

Redirected page to Accelerated Life Testing Data Analysis Reference

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Kate Racaza: duplicate of a page that was merged with the page: Time-Varying_Stress_Models

2012-08-01T09:19:05Z

duplicate of a page that was merged with the page: Time-Varying_Stress_Models

← Older revision		Revision as of 09:19, 1 August 2012
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	~~{\| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"~~		[[Category: For Deletion]]
	\|-
	~~\| valign="middle" align="left" bgcolor=EEEDF7\|[[Image: ALTA-Examples-banner.png\|400px\|center]]~~
	\|}
	~~<br>~~
	~~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.~~

	~~<br>~~
	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.~~

	~~<br>~~
	~~<br>~~

Nikki Helms: Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image: ALTA-Exa…'

2012-03-21T22:53:02Z

Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image: ALTA-Exa…'

New page

{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"
|-
| valign="middle" align="left" bgcolor=EEEDF7|[[Image: ALTA-Examples-banner.png|400px|center]]
|}
 
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>

 
where:
 
• <math>F</math> is the number of exact time-to-failure data points.
 
• <math>\beta </math> is the Weibull shape parameter.
 
• <math>a</math> and <math>n</math> are the IPL parameters.
 
• <math>x(t)</math> is the stress profile function.
 
• <math>{{t}_{i}}</math> is the <math>{{i}^{th}}</math> time to failure.

 
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>

 
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>

 
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.