AND Gate - Revision history

Lisa Hacker: Redirected page to Fault Tree Diagrams and System Analysis

2015-06-25T16:59:05Z

Redirected page to Fault Tree Diagrams and System Analysis

← Older revision		Revision as of 16:59, 25 June 2015
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	~~This page describes the AND gate and it's applications in Fault Tree diagrams used in Reliability and Safety Engineering.~~		#REDIRECT [[Fault_Tree_Diagrams_and_System_Analysis]]

	In an AND gate, the output event occurs if all input events occur (or event A AND event B AND event C). When used in reliability and safety engineering an event is usualy a failure, a failure mode, or other undesirable event.

	If one is looking at system reliability with each event representing a component of the system then this this implies that all components must fail (input) in order for the system to fail (output). When using RBDs, the equivalent is a simple parallel configuration.

	[[~~File:andgate.png\|center~~]]


	~~==System Reliability Example==~~

	Consider a system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail. Then the fault tree diagram is an AND gate (representing the system level) with two events ''A'' and ''B''. This is shown in the next figure.

	~~[[File:I10.2.gif\|center]]~~

	~~Assuming independence then the reliability equation for either configuration is:~~


	~~::<math>{{R}_{System}}={{R}_{A}}+{{R}_{B}}-{{R}_{A}}\cdot {{R}_{B}}</math>~~

	~~If we assume that the Reliability of A is 0.90 and of B 0.85 then~~

	~~::<math>{{R}_{System}}=0.90+0.85-0.90\cdot 0.85</math>~~
	~~::<math>{{R}_{System}}=0.985</math>~~


	Reliability deals with the probability of an event not occurring (the event being probability of failure of A, or Q(A)=P(A)), then if we are dealing directly with probabilities then P(A and B) = P(A ∩ B) = P(A) P(B).


	~~::<math>{{Q}_{System}}={{Q}_{A}}\cdot {{Q}_{B}}</math>~~

	or

	~~::<math>{{Q}_{System}}=(1-0.90)\cdot (1-0.85)</math>~~
	~~::<math>{{Q}_{System}}=(0.015</math>.~~

	~~Recall that~~

	~~::<math>{{R}_{System}}=1-{{Q}_{System}}</math>~~
	~~::<math>{{R}_{System}}=1-{0.015}=0.985.</math>.~~

	~~==Failure Modes==~~

	The same can be expanded to failure modes. Consider a component that can fail due to failure modes, A and B and that the component will only fail if both modes occur. Then the probability of failure is again

	~~::<math>{{Q}_{System}}={{Q}_{A}}\cdot {{Q}_{B}}</math>~~

	~~=Using Time Varying Formulations=~~

	Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to probability distributions. Consider again the prior system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail. Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then

	~~::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R(t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.~~

	~~If each of the components followed a e-parameter Weibull distribution or~~
	~~::<math> R(t)_A=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }} </math>~~
	~~::<math> R(t)_B=e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>~~
	~~then~~

	~~::<math>{{R(t)}_{System}}=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}~~
	~~+e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }}~~
	~~-e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}\cdot e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>~~

Pantelis: /* Using Time Varying Formulations */

2011-07-16T01:12:42Z

Using Time Varying Formulations

← Older revision		Revision as of 01:12, 16 July 2011
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	=Using Time Varying Formulations=		=Using Time Varying Formulations=

	Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to probability distributions.		Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to probability distributions. Consider again the prior system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail. Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then


	Consider again the prior system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail.

	~~Assuming independence then the reliability equation for either configuration is:~~

	Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then

	::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R(t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.		::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R(t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.

Pantelis at 01:11, 16 July 2011

2011-07-16T01:11:46Z

← Older revision		Revision as of 01:11, 16 July 2011
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	In an AND gate, the output event occurs if all input events occur (or event A AND event B AND event C). When used in reliability and safety engineering an event is usualy a failure, a failure mode, or other undesirable event.		In an AND gate, the output event occurs if all input events occur (or event A AND event B AND event C). When used in reliability and safety engineering an event is usualy a failure, a failure mode, or other undesirable event.

	If one is looking at system reliability with each event ~~represnting~~ a component of the system then this this implies that all components must fail (input) in order for the system to fail (output). When using RBDs, the equivalent is a simple parallel configuration.		If one is looking at system reliability with each event representing a component of the system then this this implies that all components must fail (input) in order for the system to fail (output). When using RBDs, the equivalent is a simple parallel configuration.

	[[File:andgate.png\|center]]		[[File:andgate.png\|center]]
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	==System Reliability Example==		==System Reliability Example==

	Consider a system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail. Then the fault tree diagram is an AND gate (~~represinting~~ the system level) with two events ''A'' and ''B''. This is shown in the next figure.		Consider a system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail. Then the fault tree diagram is an AND gate (representing the system level) with two events ''A'' and ''B''. This is shown in the next figure.

	[[File:I10.2.gif\|center]]		[[File:I10.2.gif\|center]]
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	::<math>{{R}_{System}}={{R}_{A}}+{{R}_{B}}-{{R}_{A}}\cdot {{R}_{B}}</math>		::<math>{{R}_{System}}={{R}_{A}}+{{R}_{B}}-{{R}_{A}}\cdot {{R}_{B}}</math>

	If we assume that the ~~Relaibility~~ of A is 0.90 and of B 0.85 then		If we assume that the Reliability of A is 0.90 and of B 0.85 then

	::<math>{{R}_{System}}=0.90+0.85-0.90\cdot 0.85</math>		::<math>{{R}_{System}}=0.90+0.85-0.90\cdot 0.85</math>
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	Reliability deals with the probability of an event not ~~occuring~~ (the event being probability of failure of A, or Q(A)=P(A)), then if we are dealing directly with probabilities then P(A and B) = P(A ∩ B) = P(A) P(B).		Reliability deals with the probability of an event not occurring (the event being probability of failure of A, or Q(A)=P(A)), then if we are dealing directly with probabilities then P(A and B) = P(A ∩ B) = P(A) P(B).


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	==Failure Modes==		==Failure Modes==

	The same can be expanded to failure modes. Consider a component that can fail due to failure modes, A and B and that the component will only fail if both modes occur. Then the ~~probaility~~ of failure is again		The same can be expanded to failure modes. Consider a component that can fail due to failure modes, A and B and that the component will only fail if both modes occur. Then the probability of failure is again

	::<math>{{Q}_{System}}={{Q}_{A}}\cdot {{Q}_{B}}</math>		::<math>{{Q}_{System}}={{Q}_{A}}\cdot {{Q}_{B}}</math>
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	=Using Time Varying Formulations=		=Using Time Varying Formulations=

	Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to ~~probility~~ distributions.		Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to probability distributions.

Pantelis: /* Using Time Varying Formulations */

2011-07-16T01:10:13Z

Using Time Varying Formulations

← Older revision		Revision as of 01:10, 16 July 2011
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	then		then

	::<math>{{R}_{System}}=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}		::<math>{{R(t)}_{System}}=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}
	+e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }}		+e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }}
	-e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}\cdot e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>		-e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}\cdot e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>

Pantelis: /* Using Time Varying Formulations */

2011-07-16T01:09:51Z

Using Time Varying Formulations

← Older revision		Revision as of 01:09, 16 July 2011
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	Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then		Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then

	::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R*t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.		::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R(t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.

	If each of the components followed a e-parameter Weibull distribution or		If each of the components followed a e-parameter Weibull distribution or
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	::<math>{{R}_{System}}=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}		::<math>{{R}_{System}}=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}
	+~~R(t)_B=~~e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }}		+e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }}
	-e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}\cdot ~~R(t)_B=~~e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>		-e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}\cdot e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>

Pantelis: /* Using Time Varying Formulations */

2011-07-16T01:09:10Z

Using Time Varying Formulations

← Older revision		Revision as of 01:09, 16 July 2011
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	::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R*t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.		::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R*t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.

	If each of the components followed a Weibull distribution ~~then~~		If each of the components followed a e-parameter Weibull distribution or
	:::<math> R(T)_A=e^{-\left( { \frac{T-\gamma_A }{\eta_A }}\right) ^{\beta_A }} </math>		::<math> R(t)_A=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }} </math>
			::<math> R(t)_B=e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>
			then

			::<math>{{R}_{System}}=e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}
	::<math>{{R}_{System}}=~~0.90~~+~~0.85~~-~~0.90~~\~~cdot 0.85</math>~~		+R(t)_B=e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }}
	~~::<math>~~{{R}_{~~System~~}}~~=0.985~~</math>		-e^{-\left( { \frac{t-\gamma_A }{\eta_A }}\right) ^{\beta_A }}\cdot R(t)_B=e^{-\left( { \frac{t-\gamma_B }{\eta_B }}\right) ^{\beta_B }} </math>

Pantelis: /* Using Time Varying Formulations */

2011-07-16T01:06:29Z

Using Time Varying Formulations

← Older revision		Revision as of 01:06, 16 July 2011
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	Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then		Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then

	::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R*t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>		::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R*t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>.

			If each of the components followed a Weibull distribution then
			:::<math> R(T)_A=e^{-\left( { \frac{T-\gamma_A }{\eta_A }}\right) ^{\beta_A }} </math>


	::<math>{{R}_{System}}=0.90+0.85-0.90\cdot 0.85</math>		::<math>{{R}_{System}}=0.90+0.85-0.90\cdot 0.85</math>
	::<math>{{R}_{System}}=0.985</math>		::<math>{{R}_{System}}=0.985</math>

Pantelis at 01:04, 16 July 2011

2011-07-16T01:04:15Z

← Older revision		Revision as of 01:04, 16 July 2011
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	Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to probility distributions.		Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to probility distributions.


			Consider again the prior system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail.

			Assuming independence then the reliability equation for either configuration is:

			Furthermore assume that the reliability of each component is given by a reliability distribution R(t). Then

			::<math>{{R(t)}_{System}}={{R(t)}_{A}}+{{R*t)}_{B}}-{{R(t)}_{A}}\cdot {{R(t)}_{B}}</math>

			::<math>{{R}_{System}}=0.90+0.85-0.90\cdot 0.85</math>
			::<math>{{R}_{System}}=0.985</math>

Pantelis: /* Failure Modes */

2011-07-16T01:01:43Z

Failure Modes

← Older revision		Revision as of 01:01, 16 July 2011
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	::<math>{{Q}_{System}}={{Q}_{A}}\cdot {{Q}_{B}}</math>		::<math>{{Q}_{System}}={{Q}_{A}}\cdot {{Q}_{B}}</math>

	=Using Time Varying Formulations"		=Using Time Varying Formulations=

	Traditional fault trees defined events using a single constant probability. This concept can be expanded ~~via~~		Traditional fault trees defined events using a single constant probability. This concept can be easily expanded to probility distributions.

Pantelis: /* System Reliability Example */

2011-07-16T01:00:34Z

System Reliability Example

← Older revision		Revision as of 01:00, 16 July 2011
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	=System Reliability Example=		==System Reliability Example==

	Consider a system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail. Then the fault tree diagram is an AND gate (represinting the system level) with two events ''A'' and ''B''. This is shown in the next figure.		Consider a system with two components, ''A'' and ''B''. The system fails if both ''A'' and ''B'' fail. Then the fault tree diagram is an AND gate (represinting the system level) with two events ''A'' and ''B''. This is shown in the next figure.
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	::<math>{{R}_{System}}=1-{{Q}_{System}}</math>		::<math>{{R}_{System}}=1-{{Q}_{System}}</math>
	::<math>{{R}_{System}}=1-{0.015}=0.985.</math>.		::<math>{{R}_{System}}=1-{0.015}=0.985.</math>.

			==Failure Modes==

			The same can be expanded to failure modes. Consider a component that can fail due to failure modes, A and B and that the component will only fail if both modes occur. Then the probaility of failure is again

			::<math>{{Q}_{System}}={{Q}_{A}}\cdot {{Q}_{B}}</math>

			=Using Time Varying Formulations"

			Traditional fault trees defined events using a single constant probability. This concept can be expanded via