Template:Cd exponential exponential - Revision history

Richard House: Redirected page to Time-Varying Stress Models#Cumulative Damage Exponential Relationship

2012-08-16T00:39:37Z

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

← Older revision		Revision as of 00:39, 16 August 2012
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	~~==Cumulative Damage Exponential - Exponential==~~		#REDIRECT [[Time-Varying_Stress_Models#Cumulative_Damage_Exponential_Relationship]]
	~~<br>~~
	~~Given a time-varying stress <math>x(t)</math> and assuming the exponential life-stress relationship, the mean life is given by:~~


	~~::<math>\frac{1}{m(t,x)}=s(t,x)=\frac{{{e}^{-bx(t)}}}{C}</math>~~


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


	~~::<math>R(t,x(t))={{e}^{-I(t,x)}}</math>~~

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


	~~::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du</math>~~

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

	~~<br>~~
	~~::<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 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~~ [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>\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:~~
	~~<br>~~
	~~• <math>{{F}_{e}}</math> is the number of groups of exact time-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 Exponential - Exponential */

2012-02-22T17:22:18Z

Cumulative Damage Exponential - Exponential

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

Athanasios Gerokostopoulos: /* Cumulative Damage Exponential - Exponential */

2012-02-21T22:48:13Z

Cumulative Damage Exponential - Exponential

← Older revision		Revision as of 22:48, 21 February 2012
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	::<math>R(t,x(t))={{e}^{-I(t,x)}}</math>		::<math>R(t,x(t))={{e}^{-I(t,x)}}</math>

	:where:		<br>
			where:


	<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\~~mathop{}_~~{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du</math>		::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du</math>

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

			<br>
	::<math>f(t,x)=s(t,x){{e}^{-I(t,x)}}</math>		::<math>f(t,x)=s(t,x){{e}^{-I(t,x)}}</math>

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	::<math>\begin{align}		::<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) \\		& \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 }]
	~~& &~~ -\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>		\end{align}</math>


	:where:		where:


Line 40:		Line 42:


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

Nicolette Young: Created page with '===Cumulative Damage Exponential - Exponential===
Given a time-varying stress $x(t)$ and assuming the exponential life-stress relationship, the mean life is giv…'

2012-01-16T21:22:02Z

Created page with '===Cumulative Damage Exponential - Exponential=== Given a time-varying stress <math>x(t)</math> and assuming the exponential life-stress relationship, the mean life is giv…'

New page

===Cumulative Damage Exponential - Exponential===
 
Given a time-varying stress <math>x(t)</math> and assuming the exponential life-stress relationship, the mean life is given by:

::<math>\frac{1}{m(t,x)}=s(t,x)=\frac{{{e}^{-bx(t)}}}{C}</math>

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

::<math>R(t,x(t))={{e}^{-I(t,x)}}</math>

:where:

<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\mathop{}_{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du</math>

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

::<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 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 [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>\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>{{F}_{e}}</math> is the number of groups of exact time-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 exponential exponential - Revision history

Richard House: Redirected page to Time-Varying Stress Models#Cumulative Damage Exponential Relationship

Athanasios Gerokostopoulos: /* Cumulative Damage Exponential - Exponential */

Athanasios Gerokostopoulos: /* Cumulative Damage Exponential - Exponential */

Nicolette Young: Created page with '===Cumulative Damage Exponential - Exponential=== Given a time-varying stress x(t) and assuming the exponential life-stress relationship, the mean life is giv…'

Nicolette Young: Created page with '===Cumulative Damage Exponential - Exponential===
Given a time-varying stress $x(t)$ and assuming the exponential life-stress relationship, the mean life is giv…'