Template:Confidence bounds for competing failure modes - Revision history

Kate Racaza: Redirected page to Competing Failure Modes Analysis

2012-06-29T07:37:41Z

Redirected page to Competing Failure Modes Analysis

Lisa Hacker: /* Variance/Covariance Matrix */

2012-03-30T01:15:36Z

Variance/Covariance Matrix

← Older revision		Revision as of 01:15, 30 March 2012
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	\end{align}</math>		\end{align}</math>

	where <math>\Lambda </math> is the log-likelihood function of the failure distribution, described in ~~Chapter~~ [[Parameter Estimation]].		where <math>\Lambda </math> is the log-likelihood function of the failure distribution, described in [[Parameter Estimation]].

	===Bounds on Reliability===		===Bounds on Reliability===

Lisa Hacker: /* Bounds on Reliability */

2012-03-30T01:15:17Z

Bounds on Reliability

← Older revision		Revision as of 01:15, 30 March 2012
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	::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>		::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>

	and <math>Var(\widehat{{{u}_{i}}})</math> is given in ~~Chapter~~ [[The Weibull Distribution]].		and <math>Var(\widehat{{{u}_{i}}})</math> is given in [[The Weibull Distribution]].


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	::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})</math>		::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})</math>

	where <math>Var(\widehat{{{\lambda }_{i}}})</math> is given in ~~Chapter~~ [[The Exponential Distribution]].		where <math>Var(\widehat{{{\lambda }_{i}}})</math> is given in [[The Exponential Distribution]].


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	::<math>{{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}}</math>		::<math>{{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}}</math>

	where <math>Var(\widehat{{{z}_{i}}})</math> is given in ~~Chapter~~ [[The Normal Distribution]].		where <math>Var(\widehat{{{z}_{i}}})</math> is given in [[The Normal Distribution]].


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	::<math>{{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}</math>		::<math>{{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}</math>

	where <math>Var(\widehat{{{z}_{i}}})</math> is given in ~~Chapter~~ [[The Lognormal Distribution]].		where <math>Var(\widehat{{{z}_{i}}})</math> is given in [[The Lognormal Distribution]].

	===Bounds on Time===		===Bounds on Time===

Harry Guo: /* Bounds on Time */

2012-02-21T17:41:41Z

Bounds on Time

← Older revision		Revision as of 17:41, 21 February 2012
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	where:		where:
	:• <math>\varphi </math> is inverse function for ~~Eqn~~. ~~(CFMReliability)~~		:• <math>\varphi </math> is inverse function for the reliabilty equation for competing faiure modes.
	:• for the Weibull distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\beta }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> is <math>{{\hat{\eta }}_{i}}</math>		:• for the Weibull distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\beta }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> is <math>{{\hat{\eta }}_{i}}</math>
	:• for the exponential distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\lambda }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> =0		:• for the exponential distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\lambda }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> =0
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	::<math>{{t}_{L}}={{e}^{{{u}_{L}}}}</math>		::<math>{{t}_{L}}={{e}^{{{u}_{L}}}}</math>

	<math>{{K}_{\alpha }}</math> is calculated using ~~Eqn. (ka)~~ and <math>Var(\widehat{u})</math> is computed as:		<math>{{K}_{\alpha }}</math> is calculated using the inverse standard normal distribution and <math>Var(\widehat{u})</math> is computed as:

	::<math>Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat{{{a}_{i}}})+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat{{{b}_{i}}})+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat{{{a}_{i}}},\widehat{{{b}_{i}}}) \right)</math>		::<math>Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat{{{a}_{i}}})+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat{{{b}_{i}}})+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat{{{a}_{i}}},\widehat{{{b}_{i}}}) \right)</math>

Harry Guo at 17:40, 21 February 2012

2012-02-21T17:40:19Z

← Older revision		Revision as of 17:40, 21 February 2012
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	::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>		::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>

	and <math>Var(\widehat{{{u}_{i}}})</math> is given in Chapter 6.		and <math>Var(\widehat{{{u}_{i}}})</math> is given in Chapter [[The Weibull Distribution]].


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	::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})</math>		::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})</math>

	where <math>Var(\widehat{{{\lambda }_{i}}})</math> is given in Chapter 7.		where <math>Var(\widehat{{{\lambda }_{i}}})</math> is given in Chapter [[The Exponential Distribution]].


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	::<math>{{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}}</math>		::<math>{{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}}</math>

	where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter 8.		where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter [[The Normal Distribution]].


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	::<math>{{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}</math>		::<math>{{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}</math>

	where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter 9.		where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter [[The Lognormal Distribution]].

	===Bounds on Time===		===Bounds on Time===
	The bounds on time are estimate by solving the reliability equation with respect to time. From ~~Eqn. (CFMReliability)~~ we have that:		The bounds on time are estimate by solving the reliability equation with respect to time. From the reliabilty equation for competing faiure modes, we have that:

	::<math>\hat{t}=\varphi (R,{{\hat{a}}_{i}},{{\hat{b}}_{i}})</math>		::<math>\hat{t}=\varphi (R,{{\hat{a}}_{i}},{{\hat{b}}_{i}})</math>
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	::<math>i=1,...,n</math>		::<math>i=1,...,n</math>

	:where:		where:
	:• <math>\varphi </math> is inverse function for Eqn. (CFMReliability)		:• <math>\varphi </math> is inverse function for Eqn. (CFMReliability)
	:• for the Weibull distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\beta }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> is <math>{{\hat{\eta }}_{i}}</math>		:• for the Weibull distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\beta }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> is <math>{{\hat{\eta }}_{i}}</math>
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	:• for the lognormal distribution <math>{{\hat{a}}_{i}}</math> is <math>\hat{\mu }_{i}^{\prime }</math> , and <math>{{\hat{b}}_{i}}</math> is <math>\hat{\sigma }_{i}^{\prime }</math>		:• for the lognormal distribution <math>{{\hat{a}}_{i}}</math> is <math>\hat{\mu }_{i}^{\prime }</math> , and <math>{{\hat{b}}_{i}}</math> is <math>\hat{\sigma }_{i}^{\prime }</math>

	:Set:		Set:

	::<math>u=\ln (t)</math>		::<math>u=\ln (t)</math>
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	::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>		::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>

	:and:		and:

	::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>		::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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	::<math>{{t}_{U}}={{e}^{{{u}_{U}}}}</math>		::<math>{{t}_{U}}={{e}^{{{u}_{U}}}}</math>

	:and:		and:

	::<math>{{t}_{L}}={{e}^{{{u}_{L}}}}</math>		::<math>{{t}_{L}}={{e}^{{{u}_{L}}}}</math>

Harry Guo: /* Bounds on Reliability */

2012-02-21T17:37:50Z

Bounds on Reliability

← Older revision		Revision as of 17:37, 21 February 2012
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	Therefore, the value of <math>Var({{\hat{R}}_{i}})</math> is dependent on the underlying distribution.		Therefore, the value of <math>Var({{\hat{R}}_{i}})</math> is dependent on the underlying distribution.


	For the Weibull distribution:		For the Weibull distribution:
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	and <math>Var(\widehat{{{u}_{i}}})</math> is given in Chapter 6.		and <math>Var(\widehat{{{u}_{i}}})</math> is given in Chapter 6.


	For the exponential distribution:		For the exponential distribution:
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	where <math>Var(\widehat{{{\lambda }_{i}}})</math> is given in Chapter 7.		where <math>Var(\widehat{{{\lambda }_{i}}})</math> is given in Chapter 7.


	For the normal distribution:		For the normal distribution:
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	where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter 8.		where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter 8.


	For the lognormal distribution:		For the lognormal distribution:

Harry Guo: /* Bounds on Reliability */

2012-02-21T17:37:10Z

Bounds on Reliability

← Older revision		Revision as of 17:37, 21 February 2012
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	::<math>\widehat{R}=\underset{i=1}{\overset{n}{\mathop \prod }}\,{{\hat{R}}_{i}}</math>		::<math>\widehat{R}=\underset{i=1}{\overset{n}{\mathop \prod }}\,{{\hat{R}}_{i}}</math>

	:where:		where:
	::• <math>{{R}_{i}}</math> is the reliability of the <math>{{i}^{th}}</math> mode,		::• <math>{{R}_{i}}</math> is the reliability of the <math>{{i}^{th}}</math> mode,
	::• <math>n</math> is the number of failure modes.		::• <math>n</math> is the number of failure modes.
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	\end{align}</math>		\end{align}</math>

	where <math>\widehat{R}</math> is calculated using ~~Eqn. (CFMReliability)~~.		where <math>\widehat{R}</math> is calculated using the reliability equation for competing failure modes.
	<math>{{K}_{\alpha }}</math> is defined by:		<math>{{K}_{\alpha }}</math> is defined by:

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	::<math>\frac{\partial R}{\partial {{R}_{i}}}=\underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{{{R}_{j}}}</math>		::<math>\frac{\partial R}{\partial {{R}_{i}}}=\underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{{{R}_{j}}}</math>

	:Thus:		Thus:

	::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( \underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{R}_{j}^{2} \right)Var({{\hat{R}}_{i}})</math>		::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( \underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{R}_{j}^{2} \right)Var({{\hat{R}}_{i}})</math>
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	::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}{{e}^{{{{\hat{u}}}_{i}}}} \right)}^{2}}Var({{\hat{u}}_{i}})</math>		::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}{{e}^{{{{\hat{u}}}_{i}}}} \right)}^{2}}Var({{\hat{u}}_{i}})</math>

	:where:		where:

	::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>		::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>
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	where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter 8.		where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter 8.

	:For the lognormal distribution:		For the lognormal distribution:

	::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\cdot {{{\hat{\sigma }}}^{\prime }} \right)}^{2}}Var({{\hat{z}}_{i}})</math>		::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\cdot {{{\hat{\sigma }}}^{\prime }} \right)}^{2}}Var({{\hat{z}}_{i}})</math>

Harry Guo: /* Variance/Covariance Matrix */

2012-02-21T17:36:07Z

Variance/Covariance Matrix

← Older revision		Revision as of 17:36, 21 February 2012
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	\end{align}</math>		\end{align}</math>

	where <math>\Lambda </math> is the log-likelihood function of the failure distribution, described in Chapter 5.		where <math>\Lambda </math> is the log-likelihood function of the failure distribution, described in Chapter [[Parameter Estimation]].

	===Bounds on Reliability===		===Bounds on Reliability===

Harry Guo: /* Variance/Covariance Matrix */

2012-02-21T17:35:43Z

Variance/Covariance Matrix

@@ Line 7: / Line 7: @@
 The variances and covariances of the parameters are estimated from the inverse local Fisher matrix, as follows:
+where <math>\Lambda </math> is the log-likelihood function of the failure distribution, described in Chapter 5.
 ===Bounds on Reliability===

Harry Guo at 16:59, 21 February 2012

2012-02-21T16:59:11Z

← Older revision		Revision as of 16:59, 21 February 2012
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	::<math>Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat{{{a}_{i}}})+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat{{{b}_{i}}})+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat{{{a}_{i}}},\widehat{{{b}_{i}}}) \right)</math>		::<math>Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat{{{a}_{i}}})+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat{{{b}_{i}}})+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat{{{a}_{i}}},\widehat{{{b}_{i}}}) \right)</math>

	~~===Complex Failure Modes Analysis===~~
	In addition to being viewed as a series system, the relationship between the different competing failures modes can be more complex. After performing separate analysis for each failure mode, a diagram that describes how each failure mode can result in a product failure can be used to perform analysis for the item in question. Such diagrams are usually referred to as Reliability Block Diagrams (RBD) (for more on RBDs see ReliaSoft's System Analysis Reference and ReliaSoft's BlockSim software).

	A reliability block diagram is made of blocks that represent the failure modes and arrows and connects the blocks in different configurations. Note that the blocks can also be used to represent different components or subsystems that make up the product. Weibull ++ provides the capability to use a diagram to model, series, parallel, k-out-of-n configurations in addition to any complex combinations of these configurations.

	In this analysis, the failure modes are assumed to be statistically independent. (Note: In the context of this reference, statistically independent implies that failure information for one failure mode provides no information about, i.e. does not affect, other failure mode). Analysis of dependent modes is more complex. Advanced RBD software such as ReliaSoft's BlockSim can handle and analyze such dependencies, as well as provide more advanced constructs and analyses (see http://www.ReliaSoft.com/BlockSim).