Template:Cd gll exponential - Revision history

Richard House: Redirected page to Time-Varying Stress Models#Cumulative Damage General Log-Linear Relationship

2012-08-16T00:47:13Z

Redirected page to Time-Varying Stress Models#Cumulative Damage General Log-Linear Relationship

← Older revision		Revision as of 00:47, 16 August 2012
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	~~==Cumulative Damage General Log-Linear - Exponential==~~		#REDIRECT [[Time-Varying_Stress_Models#Cumulative_Damage_General_Log-Linear_Relationship]]
	~~<br>~~
	~~Given <math>n</math> time~~-~~varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))</math> , the life~~-~~stress relationship is:~~


	::<math>\frac{1}{m\left( t,\overset{\_}{\mathop{x}}\, \right)}=s(t,\overset{\_}{\mathop{x}}\,)={{e}^{-{{a}_{0}}-\underset{j=1}{\mathop{\overset{n}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{a}_{j}}{{x}_{j}}(t)}}</math>


	~~where <math>{{\alpha }_{0}}</math> and <math>{{\alpha }_{j}}</math> are model parameters.~~
	~~This relationship can be further modified through the use of transformations and can be reduced to the relationships discussed previously (power, Arrhenius and exponential), if so desired.~~
	~~The exponential reliability function of the unit under multiple stresses is given by:~~


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


	~~where:~~


	::<math>I(t,\overset{\_}{\mathop{x}}\,)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{du}{{{e}^{^{^{{{\alpha }_{0}}+\overset{n}{\mathop{\underset{j=1}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}(t)}}}}}</math>


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


	~~::<math>f(t,\overset{\_}{\mathop{x}}\,)=s(t,\overset{\_}{\mathop{x}}\,){{e}^{-I(t,\overset{\_}{\mathop{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}},{{\overset{\_}{\mathop{x}}\,}_{i}})~~]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}) \right) -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( I(T_{i}^{\prime },\overset{\_}{\mathop{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 },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })= & {{e}^{-I(T_{Li}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })}} \\~~
	~~& R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })= & {{e}^{-I(T_{Ri}^{\prime \prime },\overset{\_}{\mathop{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.~~
	~~<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>~~

Athanasios Gerokostopoulos: /* Cumulative Damage General Log-Linear - Exponential */

2012-02-22T17:24:15Z

Cumulative Damage General Log-Linear - Exponential

← Older revision		Revision as of 17:24, 22 February 2012
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	===Cumulative Damage General Log-Linear - Exponential===		==Cumulative Damage General Log-Linear - Exponential==
	<br>		<br>
	Given <math>n</math> time-varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))</math> , the life-stress relationship is:		Given <math>n</math> time-varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))</math> , the life-stress relationship is:

Athanasios Gerokostopoulos: /* Cumulative Damage General Log-Linear - Exponential */

2012-02-21T23:30:58Z

Cumulative Damage General Log-Linear - Exponential

← Older revision		Revision as of 23:30, 21 February 2012
Line 4:		Line 4:


	::<math>\frac{1}{m\left( t,\overset{\_}{\mathop{x}}\, \right)}=s(t,\overset{\_}{\mathop{x}}\,)={{e}^{-{{a}_{0}}-\underset{j=1}{\mathop{\overset{n}{\mathop{\mathop{}_{}^{}}}\,}}\,{{a}_{j}}{{x}_{j}}(t)}}</math>		::<math>\frac{1}{m\left( t,\overset{\_}{\mathop{x}}\, \right)}=s(t,\overset{\_}{\mathop{x}}\,)={{e}^{-{{a}_{0}}-\underset{j=1}{\mathop{\overset{n}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{a}_{j}}{{x}_{j}}(t)}}</math>


Line 14:		Line 14:
	::<math>R(t,\overset{\_}{\mathop{x}}\,)={{e}^{-I(t,\overset{\_}{\mathop{x}}\,)}}</math>		::<math>R(t,\overset{\_}{\mathop{x}}\,)={{e}^{-I(t,\overset{\_}{\mathop{x}}\,)}}</math>

	~~:where:~~

			where:


			::<math>I(t,\overset{\_}{\mathop{x}}\,)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{du}{{{e}^{^{^{{{\alpha }_{0}}+\overset{n}{\mathop{\underset{j=1}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}(t)}}}}}</math>

	::<math>I(t,\overset{\_}{\mathop{x}}\,)=\underset{0}{\mathop{\overset{t}{\mathop{\mathop{}_{}^{}}}\,}}\,\frac{du}{{{e}^{^{^{{{\alpha }_{0}}+\overset{n}{\mathop{\underset{j=1}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}(t)}}}}}</math>

	Therefore, the <math>pdf</math> is:		Therefore, the <math>pdf</math> is:
Line 29:		Line 31:

	::<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}},{{\overset{\_}{\mathop{x}}\,}_{i}})]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}) \right) \\		& \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [s({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}})]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}) \right) -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( I(T_{i}^{\prime },\overset{\_}{\mathop{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 },\overset{\_}{\mathop{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 43:		Line 44:


	: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.
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	• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.		• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
	<br>		<br>
	• .. is the number of groups of suspension data points.		• <math>S</math> is the number of groups of suspension data points.
	<br>		<br>
	• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group 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.
Line 59:		Line 60:
	• <math>FI</math> is the number of interval data groups.		• <math>FI</math> is the number of interval data groups.
	<br>		<br>
	• .. is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.		• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.
	<br>		<br>
	• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.		• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.

Nicolette Young: Created page with '===Cumulative Damage General Log-Linear - Exponential===
Given $n$ time-varying stresses $\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))$ …'

2012-01-16T21:31:03Z

Created page with '===Cumulative Damage General Log-Linear - Exponential=== Given <math>n</math> time-varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))</math> …'

New page

===Cumulative Damage General Log-Linear - Exponential===
 
Given <math>n</math> time-varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))</math> , the life-stress relationship is:

::<math>\frac{1}{m\left( t,\overset{\_}{\mathop{x}}\, \right)}=s(t,\overset{\_}{\mathop{x}}\,)={{e}^{-{{a}_{0}}-\underset{j=1}{\mathop{\overset{n}{\mathop{\mathop{}_{}^{}}}\,}}\,{{a}_{j}}{{x}_{j}}(t)}}</math>

where <math>{{\alpha }_{0}}</math> and <math>{{\alpha }_{j}}</math> are model parameters.
This relationship can be further modified through the use of transformations and can be reduced to the relationships discussed previously (power, Arrhenius and exponential), if so desired.
The exponential reliability function of the unit under multiple stresses is given by:

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

:where:

::<math>I(t,\overset{\_}{\mathop{x}}\,)=\underset{0}{\mathop{\overset{t}{\mathop{\mathop{}_{}^{}}}\,}}\,\frac{du}{{{e}^{^{^{{{\alpha }_{0}}+\overset{n}{\mathop{\underset{j=1}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}(t)}}}}}</math>

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

::<math>f(t,\overset{\_}{\mathop{x}}\,)=s(t,\overset{\_}{\mathop{x}}\,){{e}^{-I(t,\overset{\_}{\mathop{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}},{{\overset{\_}{\mathop{x}}\,}_{i}})]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}) \right) \\
& & -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( I(T_{i}^{\prime },\overset{\_}{\mathop{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 },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })= & {{e}^{-I(T_{Li}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })}} \\
& R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })= & {{e}^{-I(T_{Ri}^{\prime \prime },\overset{\_}{\mathop{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.
 
• .. 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.
 
• .. 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 gll exponential - Revision history

Richard House: Redirected page to Time-Varying Stress Models#Cumulative Damage General Log-Linear Relationship

Athanasios Gerokostopoulos: /* Cumulative Damage General Log-Linear - Exponential */

Athanasios Gerokostopoulos: /* Cumulative Damage General Log-Linear - Exponential */

Nicolette Young: Created page with '===Cumulative Damage General Log-Linear - Exponential=== Given n time-varying stresses \underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t)) …'

Nicolette Young: Created page with '===Cumulative Damage General Log-Linear - Exponential===
Given $n$ time-varying stresses $\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))$ …'