Template:Cd power weibull - Revision history

Lisa Hacker: Redirected page to Time-Varying Stress Models#Cumulative Damage Power Relationship

2012-08-15T23:42:05Z

Redirected page to Time-Varying Stress Models#Cumulative Damage Power Relationship

← Older revision		Revision as of 23:42, 15 August 2012
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	~~==Cumulative Damage Power - Weibull==~~		#REDIRECT [[Time-Varying_Stress_Models#Cumulative_Damage_Power_Relationship]]
	~~<br>~~
	~~Given a time-varying stress <math>x(t)</math> and assuming the power law relationship, the characteristic life is given by:~~

	~~<br>~~
	~~::<math>\frac{1}{\eta (t,x)}=s(t,x)={{\left( \frac{x(t)}{a} \right)}^{n}}</math>~~

	~~<br>~~
	~~The reliability function of the unit under a single stress is given by:~~

	~~<br>~~
	~~::<math>R(t,x(t))={{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}</math>~~

	~~<br>~~
	~~where:~~

	~~<br>~~
	~~::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du</math>~~

	~~<br>~~
	~~Therefore, the <math>pdf</math> is:~~

	~~<br>~~
	~~::<math>f(t,x)=\beta s(t,x){{\left( I(t,x) \right)}^{\beta -1}}{{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}</math>~~

	~~<br>~~
	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 can be obtained utilizing the statistical properties definitions (e.g. mean life, failure rate, etc.) presented in previous chapters. The log-likelihood equation is as follows:

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln~~ [~~\beta s({{T}_{i}},{{x}_{i}}){{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta~~ -~~1}}~~]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta }} -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( I(T_{i}^{\prime },x_{i}^{\prime }) \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>\begin{align}~~
	~~& R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime }) \right)}^{\beta }}}} \\~~
	~~& R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime }) \right)}^{\beta }}}}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~and:~~
	~~<br>~~
	~~• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.~~

	~~• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data 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 <math>{{i}^{th}}</math> group of data intervals.~~

	~~• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.~~

	~~• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.~~
	~~<br>~~
	~~<br>~~

Athanasios Gerokostopoulos: /* Cumulative Damage Power - Weibull */

2012-02-22T17:20:02Z

Cumulative Damage Power - Weibull

← Older revision		Revision as of 17:20, 22 February 2012
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	===Cumulative Damage Power - Weibull===		==Cumulative Damage Power - Weibull==
	<br>		<br>
	Given a time-varying stress <math>x(t)</math> and assuming the power law relationship, the characteristic life is given by:		Given a time-varying stress <math>x(t)</math> and assuming the power law relationship, the characteristic life is given by:

Athanasios Gerokostopoulos: /* Cumulative Damage Power - Weibull */

2012-02-21T20:48:07Z

Cumulative Damage Power - Weibull

← Older revision		Revision as of 20:48, 21 February 2012
Line 13:		Line 13:

	<br>		<br>
	:where:		where:

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

	<br>		<br>
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\beta s({{T}_{i}},{{x}_{i}}){{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta -1}}]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta }} \\		& \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\beta s({{T}_{i}},{{x}_{i}}){{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta -1}}]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta }} -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( I(T_{i}^{\prime },x_{i}^{\prime }) \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 }]
	~~& &~~ -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( I(T_{i}^{\prime },x_{i}^{\prime }) \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>		\end{align}</math>

	<br>		<br>
	:where:		where:

	<br>		<br>
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	\end{align}</math>		\end{align}</math>

	~~:and:~~
	<br>		<br>
	:~~:<math>{{F}_{e}}</math>~~		and:
	<br>		<br>
	• is the number of groups of exact times-to-failure data points.		• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.

	• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.		• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.

Nicolette Young: Created page with '===Cumulative Damage Power - Weibull===
Given a time-varying stress $x(t)$ and assuming the power law relationship, the characteristic life is given by:
:…'

2012-01-16T21:25:48Z

Created page with '===Cumulative Damage Power - Weibull=== Given a time-varying stress <math>x(t)</math> and assuming the power law relationship, the characteristic life is given by: :…'

New page

===Cumulative Damage Power - Weibull===
 
Given a time-varying stress <math>x(t)</math> and assuming the power law relationship, the characteristic life is given by:

 
::<math>\frac{1}{\eta (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>R(t,x(t))={{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}</math>

 
:where:

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

 
Therefore, the <math>pdf</math> is:

 
::<math>f(t,x)=\beta s(t,x){{\left( I(t,x) \right)}^{\beta -1}}{{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}</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 can be obtained utilizing the statistical properties definitions (e.g. mean life, failure rate, etc.) 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 [\beta s({{T}_{i}},{{x}_{i}}){{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta -1}}]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta }} \\
& & -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( I(T_{i}^{\prime },x_{i}^{\prime }) \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>

 
:where:

 
::<math>\begin{align}
& R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime }) \right)}^{\beta }}}} \\
& R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime }) \right)}^{\beta }}}}
\end{align}</math>

:and:
 
::<math>{{F}_{e}}</math>
 
• is the number of groups of exact times-to-failure data points.

• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data 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 <math>{{i}^{th}}</math> group of data intervals.

• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.

• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.

Template:Cd power weibull - Revision history

Lisa Hacker: Redirected page to Time-Varying Stress Models#Cumulative Damage Power Relationship

Athanasios Gerokostopoulos: /* Cumulative Damage Power - Weibull */

Athanasios Gerokostopoulos: /* Cumulative Damage Power - Weibull */

Nicolette Young: Created page with '===Cumulative Damage Power - Weibull=== Given a time-varying stress x(t) and assuming the power law relationship, the characteristic life is given by: :…'

Nicolette Young: Created page with '===Cumulative Damage Power - Weibull===
Given a time-varying stress $x(t)$ and assuming the power law relationship, the characteristic life is given by:
:…'