From bfabf24ddfe0050fd43d9d8ab6f251e3757ebc33 Mon Sep 17 00:00:00 2001 From: Robin Date: Mon, 5 Apr 2010 11:55:15 +0100 Subject: [PATCH] skeleton paragraphs for bayes and hyp-sig-P --- .../component_failure_modes_definition.tex | 32 ++ .../component_failure_modes_definition.tex~ | 108 +++++ ...mponent_failure_modes_definition_paper.tex | 79 ++++ component_failure_modes_definition/paper.aux | 8 + component_failure_modes_definition/paper.log | 447 ++++++++++++++++++ component_failure_modes_definition/paper.pdf | Bin 0 -> 68316 bytes 6 files changed, 674 insertions(+) create mode 100644 component_failure_modes_definition/component_failure_modes_definition.tex~ create mode 100644 component_failure_modes_definition/component_failure_modes_definition_paper.tex create mode 100644 component_failure_modes_definition/paper.aux create mode 100644 component_failure_modes_definition/paper.log create mode 100644 component_failure_modes_definition/paper.pdf diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex index 660885f..b4f25fd 100644 --- a/component_failure_modes_definition/component_failure_modes_definition.tex +++ b/component_failure_modes_definition/component_failure_modes_definition.tex @@ -1,6 +1,8 @@ \abstract{ This chapter defines what is meant by the terms components, component fault modes and `unitary~state' component fault modes. +The application of Bayes theorem in current methodologies, and +the unsuitability of the `null hypothesis' or p value statistical approach. Mathematical constraints and definitions are made using set theory. } @@ -12,6 +14,9 @@ For most common electrical and mechanical components, the failure modes for a given type of part can be obtained from standard literature\cite{mil1991} \cite{mech}. %The failure modes for a given component $K$ form a set $F$. + +\subsection{Unitary State Component Failure Mode sets} + An important factor in defining a set of failure modes is that they should be as clearly defined as possible. % @@ -73,7 +78,34 @@ the failure mode set is not unitary~state and does not exist in the family of se Note where that are more than two failure~modes, by banning pairs from happening at the same time we have banned larger combinations as well + + +\subsection{Bayes Theorem} + +Describe application - likely hood of faults being the cause of symptoms - +probablistic approach - no direct causation paths to the higher~abstraction fault mode. +Often for instance a component in a module within a module within a module etc +that has a probability of causing a SYSTEM level fault. + +Used in FTA\cite{NASA}\cite{NUK}. Problems, difficult to get reliable stats +for probability to cause because of small sample numbers... + +FMMD approach can by traversing down the tree use known component failure figures +to %$$ c1 \cap c2 \eq \emptyset | c1 \neq c2 \wedge c1,c2 \in C \wedge C \in U $$ %Thus if the failure~modes are pairwaise mutually exclusive they qualify for inclusion into the %unitary~state set family. + + + +\subsection{Tests of Hypotheses and Significance} + +In high reliability systems the fauls are often logged - strange occurances - +processors resetting - what are the common factors - P values - +for instance very high voltage spikes can reset micro controllers - +but how do you corrollate that with unshielded suppressed contactors... + +Maybe looking at the equipment and seeing if there is a 5\% +level of the error being caused ? +i.e. using it to search for these conditions ? diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex~ b/component_failure_modes_definition/component_failure_modes_definition.tex~ new file mode 100644 index 0000000..18c3cd1 --- /dev/null +++ b/component_failure_modes_definition/component_failure_modes_definition.tex~ @@ -0,0 +1,108 @@ + +\abstract{ This chapter defines what is meant by the terms +components, component fault modess and `unitary~state' component fault modes. +Mathematical constraints and definitions are made using set theory. +} + + +\section{Introduction} +When building a system from components, +we should be able to find all known failure modes for each component. +For most common electrical and mechanical components, the failure modes +for a given type of part can be obtained from standard literature\cite{mil1991} +\cite{mech}. %The failure modes for a given component $K$ form a set $F$. + +An important factor in defining a set of failure modes is that they +should be as clearly defined as possible. +% +It should not be possible for instance for +a component to have two or more failure modes active at once. + +Having a set of failure modes whhere $N$ modes could be active simultaneously +would mean having to consider $2^N$ failure mode scenarios. +% +Should a component be analysed and simultaneous failure mode cases exit, +the combinations could be represented by a new failure modes, or +the component should be considered from a fresh perspective, +perhaps considering it as several smaller components +within one package. + +\begin{definition} +A set of failure modes where only one fault mode +can be active at a time is termed a `unitary~state' failure mode set. +\end{definition} + +We can define a function $FM()$ to +take a given component $K$ and return its set of failure modes $F$. + +$$ FM : K \mapsto F $$ + +We can further define a set $U$ which is a set of sets of failure modes, where +the component failure modes in each of its members are unitary~state. +Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$. + + +\subsection{Component failure modes : Unitary State example} + +A component with simple ``unitary~state'' failure modes is the electrical resistor. + +Electrical resistors can fail by going OPEN or SHORTED. +However they cannot fail with both conditions active. The conditions +OPEN and SHORT are mutually exlusive. +Because of this the failure mode set $F=FM(R)$ is `unitary~state'. +%A more complex component, say a micro controller could have several +%faults active. It could for instance have a broken I/O output +%and an unstable ADC input. Here the faults cannot be considered `unitary~state'. + +% A set of failure modes, where only one or no failure modes +% are active is termed an `unitary~state' failure mode set. This +% will be donoted as set $A$. +% +To define `unitary~state' using set theory we can define a function +`active'. +The function $active(f)$ deontes that the failure mode $f$ (where $f$ is an element of $F$) is currently active. + +Thus for the set $F$ to exist in $U$ the following condition must be true. + +\begin{equation} +\label{unitarystate_def} + F \in U | f \in F \wedge active(f) \wedge f1 \in F \wedge f1 \neq f \wedge \neg active(f1) +\end{equation} + +As an example the resistor $R$ +has two failure modes $R_{open}$ and $R_{shorted}$. + +$$ FM(R) = F = \{ R_{open}, R_{shorted} \} $$ + +Applying equation \ref{`unitarystate'_definition} to a resistor +for both fault modes + + $$ active(R_{short}) | R_{short} \in F \wedge R_{open} \in F \wedge R_{open} \neq R_{short} \wedge \neg active(R_{open}) $$ + $$ active(R_{open}) | R_{open} \in F \wedge R_{short} \in F \wedge R_{short} \neq R_{open} \wedge \neg active(R_{short}) $$ + +For the case of the resistor with only two failure modes the results above, being true, +show that the failure modes for a resistor of $ F = \{ R_{open}, R_{shorted} \} $ are `unitary~state' +component failure modes. + +Thus + $$ FM(R) = \{ R_{open}, R_{shorted} \} \in U $$ + + +A general case can be stated by taking equation \ref{unitary_state_def} and making it a function thus. + + +\begin{equation} +\label{`unitarystate'_def} + UnitaryState(F) = \forall f \in F | active(f) \wedge f1 \in F \wedge f1 \neq f \wedge \neg active(f1) +\end{equation} + +%Which can be written + +%$$ UnitaryState(FM(K)) $$ + + + +% should this be a paragraph in Symptom Abstraction ???? + + + diff --git a/component_failure_modes_definition/component_failure_modes_definition_paper.tex b/component_failure_modes_definition/component_failure_modes_definition_paper.tex new file mode 100644 index 0000000..660885f --- /dev/null +++ b/component_failure_modes_definition/component_failure_modes_definition_paper.tex @@ -0,0 +1,79 @@ + +\abstract{ This chapter defines what is meant by the terms +components, component fault modes and `unitary~state' component fault modes. +Mathematical constraints and definitions are made using set theory. +} + + +\section{Introduction} +When building a system from components, +we should be able to find all known failure modes for each component. +For most common electrical and mechanical components, the failure modes +for a given type of part can be obtained from standard literature\cite{mil1991} +\cite{mech}. %The failure modes for a given component $K$ form a set $F$. + +An important factor in defining a set of failure modes is that they +should be as clearly defined as possible. +% +It should not be possible for instance for +a component to have two or more failure modes active at once. + +Having a set of failure modes where $N$ modes could be active simultaneously +would mean having to consider $2^N$ failure mode scenarios. +% +Should a component be analysed and simultaneous failure mode cases exit, +the combinations could be represented by a new failure modes, or +the component should be considered from a fresh perspective, +perhaps considering it as several smaller components +within one package. + +\begin{definition} +A set of failure modes where only one fault mode +can be active at a time is termed a `unitary~state' failure mode set. +This is termed the $U$ set thoughout this study. +This corresponds to the `mutually exclusive' definition in +probability theory\cite{probandstat}. +\end{definition} + +We can define a function $FM()$ to +take a given component $K$ and return its set of failure modes $F$. + +$$ FM : K \mapsto F $$ + +We can further define a set $U$ which is a set of sets of failure modes, where +the component failure modes in each of its members are unitary~state. +Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$. + + +\subsection{Component failure modes : Unitary State example} + +A component with simple ``unitary~state'' failure modes is the electrical resistor. + +Electrical resistors can fail by going OPEN or SHORTED. +However they cannot fail with both conditions active. The conditions +OPEN and SHORT are mutually exclusive. +Because of this the failure mode set $F=FM(R)$ is `unitary~state'. + + +Thus + +$$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$ + + +We can make this a general case by taking a set $C$ representing a collection +of component failure modes, +We can now state that + + +$$ c1 \cap c2 \neq \emptyset | c1 \neq c2 \wedge c1,c2 \in C \wedge C \not\in U $$ + +That is to say that if it is impossible that any pair of failure modes can be active at the same time +the failure mode set is not unitary~state and does not exist in the family of sets $U$ + + Note where that are more than two failure~modes, by banning pairs from happening at the same time + we have banned larger combinations as well + +%$$ c1 \cap c2 \eq \emptyset | c1 \neq c2 \wedge c1,c2 \in C \wedge C \in U $$ + +%Thus if the failure~modes are pairwaise mutually exclusive they qualify for inclusion into the +%unitary~state set family. diff --git a/component_failure_modes_definition/paper.aux b/component_failure_modes_definition/paper.aux new file mode 100644 index 0000000..3f5afa9 --- /dev/null +++ b/component_failure_modes_definition/paper.aux @@ -0,0 +1,8 @@ +\relax +\citation{mil1991} +\citation{mech} +\citation{probandstat} +\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Component failure modes : Unitary State example}{1}} +\bibstyle{plain} +\bibdata{vmgbibliography,mybib} diff --git a/component_failure_modes_definition/paper.log b/component_failure_modes_definition/paper.log new file mode 100644 index 0000000..9f916b8 --- /dev/null +++ b/component_failure_modes_definition/paper.log @@ -0,0 +1,447 @@ +This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2010.2.1) 2 APR 2010 16:35 +entering extended mode + %&-line parsing enabled. +**paper.tex +(./paper.tex +LaTeX2e <2005/12/01> +Babel and hyphenation patterns for 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