Template:Cd exponential - Revision history

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

2012-08-16T00:40:42Z

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

← Older revision		Revision as of 00:40, 16 August 2012
Line 1:		Line 1:
	~~=Cumulative Damage Exponential Relationship=~~		#REDIRECT [[Time-Varying_Stress_Models#Cumulative_Damage_Exponential_Relationship]]
	~~<br>~~
	~~This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life~~-~~stress relationship is based on the exponential relationship. Given a time-varying stress <math>x(t)</math> and assuming the exponential relationship, the life-stress relationship is given by:~~


	~~::<math>L(x(t))=C{{e}^{bx(t)}}</math>~~


	~~In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the exponential relationship:~~


	~~Therefore, instead of dis<math>C</math>playing and <math>b</math> as the calculated parameters, the following reparameterization is used:~~


	~~::<math>\begin{align}~~
	~~{{\alpha }_{0}}=\ & \ln (C) \\~~
	~~{{\alpha }_{1}}=\ & b~~
	~~\end{align}</math>~~

Richard House at 00:38, 16 August 2012

2012-08-16T00:38:52Z

← Older revision		Revision as of 00:38, 16 August 2012
Line 17:		Line 17:
	{{\alpha }_{1}}=\ & b		{{\alpha }_{1}}=\ & b
	\end{align}</math>		\end{align}</math>


	~~{{cd exponential exponential}}~~

	~~{{cd exponential weibull}}~~

	~~{{cd exponential lognormal}}~~

Athanasios Gerokostopoulos: /* Cumulative Damage Exponential Relationship */

2012-02-22T17:22:08Z

Cumulative Damage Exponential Relationship

← Older revision		Revision as of 17:22, 22 February 2012
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	==Cumulative Damage Exponential Relationship==		=Cumulative Damage Exponential Relationship=
	<br>		<br>
	This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life-stress relationship is based on the exponential relationship. Given a time-varying stress <math>x(t)</math> and assuming the exponential relationship, the life-stress relationship is given by:		This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life-stress relationship is based on the exponential relationship. Given a time-varying stress <math>x(t)</math> and assuming the exponential relationship, the life-stress relationship is given by:

Athanasios Gerokostopoulos: /* Cumulative Damage Exponential Relationship */

2012-02-21T22:46:42Z

Cumulative Damage Exponential Relationship

← Older revision		Revision as of 22:46, 21 February 2012
Line 7:		Line 7:


	In ALTA ~~PRO~~, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the exponential relationship:		In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the exponential relationship:


Line 14:		Line 14:

	::<math>\begin{align}		::<math>\begin{align}
	& {{\alpha }_{0}}= & \ln (C) \\		{{\alpha }_{0}}=\ & \ln (C) \\
	& {{\alpha }_{1}}= & b		{{\alpha }_{1}}=\ & b
	\end{align}</math>		\end{align}</math>

Nicolette Young: /* Cumulative Damage Exponential - Lognormal */

2012-01-16T21:23:16Z

Cumulative Damage Exponential - Lognormal

← Older revision		Revision as of 21:23, 16 January 2012
Line 23:		Line 23:
	{{cd exponential weibull}}		{{cd exponential weibull}}

	~~===Cumulative Damage Exponential - Lognormal===~~		{{cd exponential lognormal}}
	~~<br>~~
	~~Given a time-varying stress <math>x(t)</math> and assuming the exponential life-stress relationship, the median life is:~~


	~~::<math>\frac~~{1}{~~\breve{T}(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))=1-\Phi (z)</math>~~

	~~:where:~~


	~~::<math>z(t,x)=\frac{\ln I(t,x)}{\sigma _{T}^{\prime }}</math>~~

	~~:and:~~


	~~::<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)=\frac{s(t,x)\varphi (z(t,x))}{\sigma _{T}^{\prime }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 [\frac{s({{T}_{i}},{{x}_{i}})\varphi (z({{T}_{i}},{{x}_{i}}))}{\sigma _{T}^{\prime }I({{T}_{i}},{{x}_{i}})}] \\
	& & \overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(T_{i}^{\prime },x_{i}^{\prime })) \right)+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
	~~\end{align}</math>~~


	~~:where:~~

	~~::<math>\begin{align}~~
	~~& z_{Ri}^{\prime \prime }= & \frac{\ln I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} \\~~
	~~& z_{Li}^{\prime \prime }= & \frac{\ln I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}~~
	~~\end{align}</math>~~

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

Nicolette Young: /* Cumulative Damage Exponential - Weibull */

2012-01-16T21:22:43Z

Cumulative Damage Exponential - Weibull

@@ Line 21: / Line 21: @@
 {{cd exponential exponential}}
-===Cumulative Damage Exponential - Weibull===
+{{cd exponential weibull}}
 ===Cumulative Damage Exponential - Lognormal===

Nicolette Young: /* Cumulative Damage Exponential - Exponential */

2012-01-16T21:21:54Z

Cumulative Damage Exponential - Exponential

← Older revision		Revision as of 21:21, 16 January 2012
Line 19:		Line 19:


	~~===Cumulative Damage Exponential - Exponential===~~		{{cd exponential exponential}}
	~~<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>~~

	~~: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:~~
	~~<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>~~

	===Cumulative Damage Exponential - Weibull===		===Cumulative Damage Exponential - Weibull===

Nicolette Young: Created page with '==Cumulative Damage Exponential Relationship==
This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and th…'

2012-01-16T21:14:59Z

Created page with '==Cumulative Damage Exponential Relationship== This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and th…'

New page

==Cumulative Damage Exponential Relationship==
 
This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life-stress relationship is based on the exponential relationship. Given a time-varying stress <math>x(t)</math> and assuming the exponential relationship, the life-stress relationship is given by:

::<math>L(x(t))=C{{e}^{bx(t)}}</math>

In ALTA PRO, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the exponential relationship:

Therefore, instead of dis<math>C</math>playing and <math>b</math> as the calculated parameters, the following reparameterization is used:

::<math>\begin{align}
& {{\alpha }_{0}}= & \ln (C) \\
& {{\alpha }_{1}}= & b
\end{align}</math>

===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.
 
 

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

::<math>\frac{1}{\eta (t,x)}=s(t,x)=\frac{{{e}^{-b\cdot x(t)}}}{C}</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{}_{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}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 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.
 
 

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

::<math>\frac{1}{\breve{T}(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))=1-\Phi (z)</math>

:where:

::<math>z(t,x)=\frac{\ln I(t,x)}{\sigma _{T}^{\prime }}</math>

:and:

::<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)=\frac{s(t,x)\varphi (z(t,x))}{\sigma _{T}^{\prime }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 [\frac{s({{T}_{i}},{{x}_{i}})\varphi (z({{T}_{i}},{{x}_{i}}))}{\sigma _{T}^{\prime }I({{T}_{i}},{{x}_{i}})}] \\
& & \overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(T_{i}^{\prime },x_{i}^{\prime })) \right)+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}</math>

:where:

::<math>\begin{align}
& z_{Ri}^{\prime \prime }= & \frac{\ln I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} \\
& z_{Li}^{\prime \prime }= & \frac{\ln I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}
\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 exponential - Revision history

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

Richard House at 00:38, 16 August 2012

Athanasios Gerokostopoulos: /* Cumulative Damage Exponential Relationship */

Athanasios Gerokostopoulos: /* Cumulative Damage Exponential Relationship */

Nicolette Young: /* Cumulative Damage Exponential - Lognormal */

Nicolette Young: /* Cumulative Damage Exponential - Weibull */

Nicolette Young: /* Cumulative Damage Exponential - Exponential */

Nicolette Young: Created page with '==Cumulative Damage Exponential Relationship== This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and th…'

Nicolette Young: Created page with '==Cumulative Damage Exponential Relationship==
This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and th…'