Criticality Analysis - Revision history

Lisa Hacker at 05:19, 23 August 2012

2012-08-23T05:19:01Z

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			{{Banner Xfmea Articles}}{{Navigation box}}
	=== Introduction ===		=== Introduction ===

Richard House: /* Task 102 MIL-STD-1629A */

2012-08-21T00:22:02Z

Task 102 MIL-STD-1629A

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	where:		where:
	* <math>{{C}_{m}}=\,\!</math>Criticality number for a failure mode.		* <math>{{C}_{m}}=\,\!</math> Criticality number for a failure mode.
	* <math>\beta =\,\!</math>Conditional probability of mission loss.		* <math>\beta =\,\!</math> Conditional probability of mission loss.
	* <math>\alpha =\,\!</math>Failure mode ratio.		* <math>\alpha =\,\!</math> Failure mode ratio.
	* <math>{{\lambda }_{P}}=\,\!</math>Part failure rate.		* <math>{{\lambda }_{P}}=\,\!</math> Part failure rate.
	* <math>t=\,\!</math>Mission Duration.		* <math>t=\,\!</math> Mission Duration.

	In addition to the constant failure rate assumption, it is important to note that this formulation assumes that the part failure rate values for a mode are relatively small and thus the following simplification is made:		In addition to the constant failure rate assumption, it is important to note that this formulation assumes that the part failure rate values for a mode are relatively small and thus the following simplification is made:

Richard House: /* Derivation */

2012-08-21T00:21:21Z

Derivation

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	Given that the component failed before time T, the probability that it is caused by mode B is:		Given that the component failed before time T, the probability that it is caused by mode B is:




	::<math>\begin{align}		::<math>\begin{align}
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	& =\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{1-{{R}_{A}}(T){{R}_{B}}(T)}=\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{{{Q}_{component}}}		& =\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{1-{{R}_{A}}(T){{R}_{B}}(T)}=\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{{{Q}_{component}}}
	\end{align}</math>		\end{align}</math>




	The upper bound of P1 is:		The upper bound of P1 is:

	::<math>{{P}_{1}}=\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{{{Q}_{component}}}<\frac{\int_{0}^{T}{{{f}_{B}}(t)dt}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{AB}}}</math>.		::<math>{{P}_{1}}=\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{{{Q}_{component}}}<\frac{\int_{0}^{T}{{{f}_{B}}(t)dt}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{AB}}}</math>.


	The lower bound of P1 is:		The lower bound of P1 is:
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	Thus, the probability that the failure is caused by mode B is:		Thus, the probability that the failure is caused by mode B is:
	::<math>\frac{{{Q}_{B}}-{{Q}_{B}}{{Q}_{A}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{A}}{{Q}_{B}}}<{{P}_{1}}<\frac{{{Q}_{B}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{AB}}}</math>		::<math>\frac{{{Q}_{B}}-{{Q}_{B}}{{Q}_{A}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{A}}{{Q}_{B}}}<{{P}_{1}}<\frac{{{Q}_{B}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{AB}}}</math>


	Thus the approximation:		Thus the approximation:
	::<math>{{\tilde{P}}_{1}}=\frac{{{Q}_{B}}}{{{Q}_{B}}+{{Q}_{A}}}</math>		::<math>{{\tilde{P}}_{1}}=\frac{{{Q}_{B}}}{{{Q}_{B}}+{{Q}_{A}}}</math>
	is within the bounds.		is within the bounds.


	Similarly, given that the component failed before time T, the probability that it is caused by mode A is:		Similarly, given that the component failed before time T, the probability that it is caused by mode A is:


	::<math>\begin{align}		::<math>\begin{align}
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	& =\frac{\int_{0}^{T}{{{f}_{A}}(t){{R}_{B}}(t)dt}}{1-{{R}_{A}}(T){{R}_{B}}(T)}		& =\frac{\int_{0}^{T}{{{f}_{A}}(t){{R}_{B}}(t)dt}}{1-{{R}_{A}}(T){{R}_{B}}(T)}
	\end{align}</math>		\end{align}</math>


	Then the sum of P1 and P2 is:		Then the sum of P1 and P2 is:

Richard House: /* Example */

2012-08-21T00:20:46Z

Example

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	To illustrate this assume that a component has three failure modes (FM1, FM2 and FM3) where any mode can cause system failure (i.e., a series system).		To illustrate this assume that a component has three failure modes (FM1, FM2 and FM3) where any mode can cause system failure (i.e., a series system).
	The unreliability is defined by the failure model of each mode at t=1,000 hours. (In this case FM1 is Weibull		The unreliability is defined by the failure model of each mode at t=1,000 hours. (In this case FM1 is Weibull
	(<math>\beta=3</math>,<math>\eta=3000</math> hrs) and FM2 and 3 are exponential with MTTF=10,000 hours.)		(<math>\beta=3\,\!</math>,<math>\eta=3000\,\!</math> hrs) and FM2 and 3 are exponential with MTTF=10,000 hours.)

Richard House: /* Task 102 MIL-STD-1629A */

2012-08-21T00:19:54Z

Task 102 MIL-STD-1629A

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	::<math>{{C}_{m}}=\beta \alpha {{\lambda }_{P}}t\,\!</math>		::<math>{{C}_{m}}=\beta \alpha {{\lambda }_{P}}t\,\!</math>


	where:		where:
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	In addition to the constant failure rate assumption, it is important to note that this formulation assumes that the part failure rate values for a mode are relatively small and thus the following simplification is made:		In addition to the constant failure rate assumption, it is important to note that this formulation assumes that the part failure rate values for a mode are relatively small and thus the following simplification is made:


	::<math>Q(t)=1-{{e}^{-{{\lambda }_{P}}\cdot t}}\approx {{\lambda }_{P}}\cdot t \,\!</math>		::<math>Q(t)=1-{{e}^{-{{\lambda }_{P}}\cdot t}}\approx {{\lambda }_{P}}\cdot t \,\!</math>


	This approximation holds true when using an exponential distribution and small values of lambda (<1E-4), but not for all values.		This approximation holds true when using an exponential distribution and small values of lambda (<1E-4), but not for all values.

Richard House: /* General Criticality Number Computation */

2012-08-21T00:19:23Z

General Criticality Number Computation

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	* <math>{{C}_{mode}}= \,\!</math> Criticality number for the failure mode.		* <math>{{C}_{mode}}= \,\!</math> Criticality number for the failure mode.
	* <math>{{\beta }_{mode}}= \,\!</math> Conditional probability of mission loss.		* <math>{{\beta }_{mode}}= \,\!</math> Conditional probability of mission loss.
	* <math>{{\alpha }_{mode}}=</math>Failure mode ratio.		* <math>{{\alpha }_{mode}}= \,\!</math> Failure mode ratio.
	* <math>Q{{(t)}_{mode}}= \,\!</math> Unreliability or probability of failure due to that mode at time t.		* <math>Q{{(t)}_{mode}}= \,\!</math> Unreliability or probability of failure due to that mode at time t.
	* <math>Q{{(t)}_{component}}= \,\!</math> Unreliability or probability of failure of the component at time t.		* <math>Q{{(t)}_{component}}= \,\!</math> Unreliability or probability of failure of the component at time t.
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	Note that this differs from the specialized definition given in Task 102 of MIL-STD-1629A, Section 3.2.1.6 (described next).		Note that this differs from the specialized definition given in Task 102 of MIL-STD-1629A, Section 3.2.1.6 (described next).

	=== Task 102 MIL-STD-1629A ===		=== Task 102 MIL-STD-1629A ===

Lisa Hacker: /* General Criticality Number Computation */

2012-04-03T17:07:24Z

General Criticality Number Computation

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	Note that this differs from the specialized definition given in Task 102 of MIL-STD-1629A, Section 3.2.1.6 (described next).		Note that this differs from the specialized definition given in Task 102 of MIL-STD-1629A, Section 3.2.1.6 (described next).

			=== Task 102 MIL-STD-1629A ===

	==== Task 102 MIL-STD-1629A ====

	Specifically, a procedure is given for computing the “Failure mode criticality number” when an assumption of a constant failure mode (i.e., an exponential distribution) is used and for small values of lambda:		Specifically, a procedure is given for computing the “Failure mode criticality number” when an assumption of a constant failure mode (i.e., an exponential distribution) is used and for small values of lambda:

Lisa Hacker at 17:06, 3 April 2012

2012-04-03T17:06:52Z

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			=== Introduction ===

	Failure mode criticality, as used in FMECA, can be generically described as the joint probability of three events:		Failure mode criticality, as used in FMECA, can be generically described as the joint probability of three events:

Lisa Hacker at 17:06, 3 April 2012

2012-04-03T17:06:11Z

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	Failure mode criticality, as used in FMECA, can be generically described as the joint probability of three events:		Failure mode criticality, as used in FMECA, can be generically described as the joint probability of three events:

	#The probability of failure of the component;		#The probability of failure of the component.
	#The probability that the mode under consideration was the culprit~~; and~~		#The probability that the mode under consideration was the culprit.
	#The probability that this mode would result in a system failure (which for the case of non-redundancy or if in series =1).		#The probability that this mode would result in a system failure (which for the case of non-redundancy or if in series =1).

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	==== Task 102 MIL-STD-1629A ====		==== Task 102 MIL-STD-1629A ====

	Specifically a procedure is given for computing the “Failure mode criticality number” when an assumption of a constant failure mode (i.e., an exponential distribution) is used and for small values of lambda:		Specifically, a procedure is given for computing the “Failure mode criticality number” when an assumption of a constant failure mode (i.e., an exponential distribution) is used and for small values of lambda:

	This is given by:		This is given by:
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	===Example===		===Example===

	To illustrate this assume that a component has three failure modes, FM1, FM2 and FM3 ~~and~~ where any mode can cause system failure (i.e., a series system).		To illustrate this assume that a component has three failure modes (FM1, FM2 and FM3) where any mode can cause system failure (i.e., a series system).
	The unreliability is defined by the failure model of each mode at t=1,000 ~~hrs~~. (In this case FM1 is Weibull		The unreliability is defined by the failure model of each mode at t=1,000 hours. (In this case FM1 is Weibull
	(<math>\beta=3</math>,<math>\eta=3000</math> hrs) and FM2 and 3 are exponential with MTTF=10,000 ~~hrs~~.)		(<math>\beta=3</math>,<math>\eta=3000</math> hrs) and FM2 and 3 are exponential with MTTF=10,000 hours.)


	Then		Then:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	and		and:

	::<math>\begin{align}		::<math>\begin{align}
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	The upper bound on P1 is:		The upper bound of P1 is:

	::<math>{{P}_{1}}=\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{{{Q}_{component}}}<\frac{\int_{0}^{T}{{{f}_{B}}(t)dt}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{AB}}}</math>.		::<math>{{P}_{1}}=\frac{\int_{0}^{T}{{{f}_{B}}(t){{R}_{A}}(t)dt}}{{{Q}_{component}}}<\frac{\int_{0}^{T}{{{f}_{B}}(t)dt}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{component}}}=\frac{{{Q}_{B}}}{{{Q}_{A}}+{{Q}_{B}}-{{Q}_{AB}}}</math>.
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	Thus the approximation		Thus the approximation:
	::<math>{{\tilde{P}}_{1}}=\frac{{{Q}_{B}}}{{{Q}_{B}}+{{Q}_{A}}}</math>		::<math>{{\tilde{P}}_{1}}=\frac{{{Q}_{B}}}{{{Q}_{B}}+{{Q}_{A}}}</math>
	is within the bounds.		is within the bounds.
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	\end{align}</math>		\end{align}</math>

	Thus the approximation		Thus the approximation:
	::<math>{{\tilde{P}}_{2}}=\frac{{{Q}_{A}}}{{{Q}_{B}}+{{Q}_{A}}}</math>		::<math>{{\tilde{P}}_{2}}=\frac{{{Q}_{A}}}{{{Q}_{B}}+{{Q}_{A}}}</math>
	is within these bounds.		is within these bounds.

	Also note		Also note:
	::<math>{{\tilde{P}}_{1}}+{{\tilde{P}}_{2}}=1</math>.		::<math>{{\tilde{P}}_{1}}+{{\tilde{P}}_{2}}=1</math>

Richard House at 23:15, 21 March 2012

2012-03-21T23:15:06Z

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	where		where:
	* <math>{{C}_{m}}=\,\!</math>Criticality number for a failure mode.		* <math>{{C}_{m}}=\,\!</math>Criticality number for a failure mode.
	* <math>\beta =\,\!</math>Conditional probability of mission loss.		* <math>\beta =\,\!</math>Conditional probability of mission loss.