Time-Varying Stress Models - Revision history

Lisa Hacker at 22:02, 20 April 2023

2023-04-20T22:02:24Z

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Lisa Hacker: Reverted edits by Lisa Hacker (talk) to last revision by Sharon Honecker

2023-04-20T21:59:00Z

Reverted edits by Lisa Hacker (talk) to last revision by Sharon Honecker

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Lisa Hacker at 21:57, 20 April 2023

2023-04-20T21:57:23Z

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Lisa Hacker at 21:46, 20 April 2023

2023-04-20T21:46:50Z

← Older revision		Revision as of 21:46, 20 April 2023
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	*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending 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 Arrhenius Relationship=
			See [[Cumulative Damage Arrhenius]]
			=Cumulative Damage Exponential=
			See [[Cumulative Damage Exponential]]
			=Cumulative Damage General Log-Linear Relationship=
			See [[Cumulative Damage General Loglinear]]

	=Confidence Intervals=		=Confidence Intervals=

Lisa Hacker at 21:43, 20 April 2023

2023-04-20T21:43:26Z

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Sharon Honecker: /* Mathematical Formulation for a Step-Stress Model */

2017-02-01T23:58:47Z

Mathematical Formulation for a Step-Stress Model

← Older revision		Revision as of 23:58, 1 February 2017
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	::<math>\begin{align}		::<math>\begin{align}
	{{F}_{1}}({{t}_{1}};{{S}_{1}})=\ & {{F}_{2}}({{\varepsilon }_{1}},{{S}_{2}}) \\		{{F}_{1}}({{t}_{1}};{{S}_{1}})=\ & {{F}_{2}}({{\varepsilon }_{1}};{{S}_{2}}) \\
	1-{{e}^{-{{(KS_{1}^{n}{{t}_{1}})}^{\beta }}}}=\ & 1-{{e}^{-{{(KS_{2}^{n}{{\varepsilon }_{1}})}^{\beta }}}} \\		1-{{e}^{-{{(KS_{1}^{n}{{t}_{1}})}^{\beta }}}}=\ & 1-{{e}^{-{{(KS_{2}^{n}{{\varepsilon }_{1}})}^{\beta }}}} \\
	S_{1}^{n}{{t}_{1}}=\ & S_{2}^{n}{{\varepsilon }_{1}} \\		S_{1}^{n}{{t}_{1}}=\ & S_{2}^{n}{{\varepsilon }_{1}} \\

Sharon Honecker: /* Cumulative Damage Power - Weibull */

2017-02-01T23:57:36Z

Cumulative Damage Power - Weibull

← Older revision		Revision as of 23:57, 1 February 2017
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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 [\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 }]		& \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>		\end{align}\,\!</math>

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	::<math>\begin{align}		::<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[ \int_{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[ \int_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta }} \right\}		& \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[ \int_{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[ \int_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta }} \right\}
	\end{align}\,\!</math>		\end{align}\,\!</math>

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	::<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>		::<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>,		or at a fixed stress level <math>x(t)=2 \text{ V}\,\!</math> and <math>t=300 \text{ hours}\,\!</math>,

	::<math>R\left( t=300,x(t)=2 \right)={{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}=97.5%\,\!</math>		::<math>R\left( t=300,x(t)=2 \right)={{e}^{-{{\left[ \int_{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:		The mean time to failure <math>(MTTF)\,\!</math> 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>		::<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>,		or at a fixed stress level <math>x\left( t \right)=2 \text{ V}\,\!</math>,

	::<math>MTTF\left( x\left( t \right) \right)=1046.~~3hrs~~\,\!</math>		::<math>MTTF\left( x\left( t \right) \right)=1046.3 \text{ hours}\,\!</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.		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.

Sharon Honecker: /* Mathematical Formulation for a Step-Stress Model */

2017-02-01T23:37:19Z

Mathematical Formulation for a Step-Stress Model

← Older revision		Revision as of 23:37, 1 February 2017
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	::<math>{{f}_{i}}(t,{{S}_{i}})=-\frac{d}{dt}\left[ {{F}_{i}}(t,{{S}_{i}}) \right]\,\!</math>		::<math>{{f}_{i}}(t,{{S}_{i}})=-\frac{d}{dt}\left[ {{F}_{i}}(t,{{S}_{i}}) \right]\,\!</math>

	Once the model has been formulated, model parameters (i.e., <math>K\,\!</math>, <math>n\,\!</math> and <math>\beta \,\!</math> ) can be computed utilizing maximum likelihood estimation methods.		Once the model has been formulated, model parameters (i.e., <math>K\,\!</math>, <math>n\,\!</math> and <math>\beta \,\!</math>) can be computed utilizing maximum likelihood estimation methods.

	The previous example can be expanded for any time-varying stress. ALTA allows you to define any stress profile. For example, the stress can be a ramp stress, a monotonically increasing stress, sinusoidal, etc. This section presents a generalized formulation of the cumulative damage model, where stress can be any function of time.		The previous example can be expanded for any time-varying stress. ALTA allows you to define any stress profile. For example, the stress can be a ramp stress, a monotonically increasing stress, sinusoidal, etc. This section presents a generalized formulation of the cumulative damage model, where stress can be any function of time.

Kate Racaza: /* Model Formulation */

2015-12-01T18:23:17Z

Model Formulation

← Older revision		Revision as of 18:23, 1 December 2015
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	The previous example can be expanded for any time-varying stress. ALTA allows you to define any stress profile. For example, the stress can be a ramp stress, a monotonically increasing stress, sinusoidal, etc. This section presents a generalized formulation of the cumulative damage model, where stress can be any function of time.		The previous example can be expanded for any time-varying stress. ALTA allows you to define any stress profile. For example, the stress can be a ramp stress, a monotonically increasing stress, sinusoidal, etc. This section presents a generalized formulation of the cumulative damage model, where stress can be any function of time.

			<div class="noprint">
	{{Example:CD-GLL_Weibull}}		{{Example:CD-GLL_Weibull}}
			</div>

	=Cumulative Damage Power Relationship=		=Cumulative Damage Power Relationship=

David J. Groebel: /* Confidence Intervals */

2015-08-12T15:03:55Z

Confidence Intervals

← Older revision		Revision as of 15:03, 12 August 2015
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	=Confidence Intervals=		=Confidence Intervals=
	Using the same methodology as in previous sections, approximate confidence intervals can be derived and applied to all results of interest using the Fisher Matrix approach discussed in [[Appendix A: Brief Statistical Background\|Appendix A]]. ALTA utilizes such intervals on all results~~. The formulas for such intervals are beyond the scope of this reference and are thus omitted. Interested readers can contact ReliaSoft for internal document ALTA-CBCD, detailing these derivations~~.		Using the same methodology as in previous sections, approximate confidence intervals can be derived and applied to all results of interest using the Fisher Matrix approach discussed in [[Appendix A: Brief Statistical Background\|Appendix A]]. ALTA utilizes such intervals on all results.

	=Notes on Trigonometric Functions=		=Notes on Trigonometric Functions=
	Trigonometric functions sometime are used in accelerated life tests. However ALTA does not include them. In fact, a trigonometric function can be defined by its frequency and magnitude. Frequency and magnitude then can be treated as two constant stresses. The GLL model discussed in [[General Log-Linear Relationship]] then can be applied for modeling.		Trigonometric functions sometime are used in accelerated life tests. However ALTA does not include them. In fact, a trigonometric function can be defined by its frequency and magnitude. Frequency and magnitude then can be treated as two constant stresses. The GLL model discussed in [[General Log-Linear Relationship]] then can be applied for modeling.