function definitions domain and range and specific

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -243,10 +243,20 @@ This corresponds to the `mutually exclusive' definition in
 probability theory\cite{probandstat}.
 \end{definition}
 
-We can define a function $FM$ to 
+Let the set of all possible tomponents to be $\mathcal{C}$
-take a given component $C$ and return its set of failure modes $F$.
+and let the set of all possible failure modes be $\mathcal{F}$.
 
-$$  FM : C \mapsto F $$
+We can define a function $FM$  
+
+\begin{equation}
+FM : \mathcal{C} \mapsto \mathcal{F}
+\end{equation}
+
+defined by
+
+$$  FM ( C ) = F $$
+
+i.e. take a given component $C$ and return its set of failure modes $F$.
 
 \begin{definition}
 We can define a set $\mathcal{U}$ which is a set of sets of failure modes, where

pt100/pt100.tex

@@ -83,6 +83,23 @@ Thus a little loss of supply current due to resistance in the cables
 does not impinge on accuracy.
 The resistance to temperature conversion is achieved 
 through the published PT100 tables\cite{eurothermtables}.
+The standard voltage divider equations (see figure \ref{fig:vd} and 
+equation \ref{eqn:vd} can be used to  calculate
+expected voltages for failure mode and temperature reading purposes.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./pt100/voltage_divider.png}
+ % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
+ \caption{Voltage Divider}
+ \label{fig:vd}
+\end{figure}
+%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
+
+\begin{equation}
+\label{eqn:vd}
+ V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
+\end{equation}
 
 \section{Safety case for 4 wire circuit}
 
@@ -235,21 +252,21 @@ valid temperature range for this example ({0\oc} .. {300\oc}) we can calculate
 valid voltage reading ranges by using the standard voltage divider equation \ref{eqn:vd}
 for the circuit shown in figure \ref{fig:vd}.
 
-
+%
-\begin{figure}[h]
+%\begin{figure}[h]
- \centering
+% \centering
- \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./pt100/voltage_divider.png}
+% \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./pt100/voltage_divider.png}
- % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
+% % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
- \caption{Voltage Divider}
+% \caption{Voltage Divider}
- \label{fig:vd}
+% \label{fig:vd}
-\end{figure}
+%\end{figure}
-%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
+%%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
-
+%
-\begin{equation}
+%\begin{equation}
-\label{eqn:vd}
+%\label{eqn:vd}
- V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
+% V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
-\end{equation}
+%\end{equation}
-
+%
 
 
 \subsection{Proof of Out of Range \\ Values for Failures}
@@ -286,8 +303,7 @@ Since the lowreading or sense- is directly connected to the 0V rail,
 both temperature readings will be 0V.
 $$ lowreading = 5V.\frac{100\Omega}{2k2+100\Omega} = 0.218V$$
 With pt100 at the high end of the temperature range 300\oc.
-$$ highreading = 5V $$
+$$ highreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
-$$ lowreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
 
 Thus with $R_2$ shorted both readings are outside the 
 proscribed range in table \ref{ptbounds}.
@@ -433,7 +449,7 @@ This figure is referred to as a FIT\footnote{FIT values are measured as the numb
 failures per Billion (${10}^9$) hours of operation, (roughly 114,000 years). The smaller the 
 FIT number the more reliable the fault~mode} Failure in time.
 
-The formula given for a thermistor in  MIL-HDBK-217F\cite{mil1992}[9.8] is reporoduced in
+The formula given for a thermistor in  MIL-HDBK-217F\cite{mil1992}[9.8] is reproduced in
 equation \ref{thermistorfit}. The variable meanings and values are described in table \ref{tab:thermistor}.
 
 \begin{equation}
@@ -525,5 +541,8 @@ conditions.
 \clearpage
 \section{ PT100 Double Simultaneous \\ Fault Analysis}
 
+
+
+DO THE DOUBLE
 % typeset in {\Huge \LaTeX} \today
 

symptom_ex_process/symptom_ex_process.tex

@@ -3,15 +3,14 @@
 In failure mode analysis, it is essential to
 know the failure modes of the sub-systems and components used.
 This paper outlines a technique for determining the failure modes of a sub-system given 
-its component parts.
+its components.
-%, and the failure modes of those parts.
 
 This chapter describes a process for taking a functional group of components, applying FMEA analysis and then determining how that functional group can fail.
-With this information, we can trat the functional group
+With this information, we can treat the functional group
 as a component in its own right.
 
 This new component is a derived component.
-For a top down technique this would correspond to a sub-system.
+For a top down technique this would correspond to a low~level sub-system.
 %The technique uses a graphical notation, based on Euler\cite{eulerviz} and Constraint diagrams\cite{constraint} to model failure modes and failure mode common symptom collection.  The technique is designed for making building blocks for a hierarchical fault model.
 
 Once the failure modes have been determined for a sub-system/derived~component,
@@ -30,10 +29,10 @@ Once a hierarchy is in place it can be converted into a fault data model.
 %
 From the fault data model, automatic generation
 of FTA\cite{nasafta} (Fault Tree Analysis) and mimimal cuts sets\cite{nucfta} are possible. 
-Also statistical reliability\cite{en61508} and MTTF (Mean Time to Failure) calculations can be produced 
+Also statistical reliability/probability of failure~on~demand\cite{en61508} and MTTF (Mean Time to Failure) calculations can be produced 
 automatically, where component failure mode statistics are available\cite{mil1991}.
 %
-This paper focuses on the process of building the blocks that are used in the hierarchy.
+This paper focuses on the process of building the blocks, that are key to creating an FMMD hierarchy.
 
 \end{abstract}
 
@@ -55,7 +54,8 @@ and checks will be made, and finally a component or a low level sub-system
 will be found to be faulty.
 A natural fault finding process is thus top~down.
 \subsection{FMMD - Bottom~up Analysis}
-The FMMD technique described here works the other way. It works from the bottom up.
+The FMMD technique does not follow the `natural fault finding' or top down approach,
+it instead works from the bottom up.
 Starting with a collection of components that form
 a simple functional group, the effect of all component error modes are
 examined, as to their effect on the functional group.
@@ -102,7 +102,7 @@ The aims are
 It is helpful here to define some terms, `system', `functional~group', `component', `base~component' and `derived~component/sub-system'.
 These are listed in table~\ref{tab:symexdef}.
 
-A System, is really any coherent entity that would be sold as a safety critical product.
+A System, is really any coherent entity that would be sold as a product. % safety critical product.
 A sub-system is a system that is part of some larger system.
 For instance a stereo amplifier separate is a sub-system. The 
 whole Sound System, consists perhaps of the following `sub-systems': 
@@ -112,8 +112,8 @@ CD-player, tuner, amplifier~separate, loudspeakers and ipod~interface.
 %and is the way in which FTA\cite{nucfta} analyses a System
 %and breaks it down.
 
-A sub-system will be composed of component parts, which
+A sub-system will be composed of components, which
-may themselves be sub-systems. However each `component part'
+may themselves be sub-systems. However each `component'
 will have a fault/failure behaviour and it should
 always be possible to obtain a set of failure modes
 for each `component'.  In FMMD terms a sub-system is a derived component.
@@ -166,6 +166,7 @@ Currently this sort of information is generally only  available for generic comp
 
 %\vspace{0.3cm}
 \begin{table}[h]
+\center
 \begin{tabular}{||l|l||} \hline \hline
   {\em Definition } & {\em Description}    \\ \hline
 System & A product designed to  \\
@@ -205,8 +206,9 @@ that we have to consider are all the failure modes of its components.
 Each failure mode (or combination of) investigated is termed a `test case'.
 Each `test case' is analysed.
 The component failure modes are examined with respect to their effect on the functional~group.
-The philosophy behind this analysis is, how will the functional~group react
+The aim of this analysis is to find out how the functional~group react
-to each of the test case conditions. The aim is to produce a set of failure modes from the perspective of the functional~group.
+to each of the test case conditions.
+The goal of the process is to produce a set of failure modes from the perspective of the functional~group.
 \paragraph{Symptom Identification}
 When all `test~cases' have been analysed a second phase is applied.
 %
@@ -222,20 +224,12 @@ will both cause the same failure; $no\_sound$ !
 \paragraph{Collection of Symptoms}
 The common symptoms of failure and lone~component failure~modes are identified and collected.
 We can now consider the functional~group as a component and the common symptoms as its failure modes.
-Note  that here because this is bottom up, we can ensure that all failure modes
+Note  that here because the process is bottom up, we can ensure that all failure modes
-associated with a functional~group have been handled. 
+associated with a functional~group have been handled.
+Were failure~modes missed any failure mode model could be dangerously incomplete. 
 It is possible here for an automated system to flag unhandled failure modes.
 \ref{requirement at the start}
 
-% \paragraph{symptom abstraction represented on the diagram} This process can be applied using a diagram.  From the collection of parts for the sub-system under analysis, a set of failure modes for each component is obtained. A diagram is then drawn with each component failure mode represented by a contour.  Component failure mode combinations are chosen for `test cases'.\footnote{Combinations of component failure modes can be represented by overlapping contours} A `test case' is represented on the diagram as a point or asterisk, in a region enclosed by the contours representing the failure modes it investigates.  The effect on the sub-system of each test case is analysed.  %It is then represented on the diagram by an asterisk on the contour representing the failure mode.  The `test~case~results' are archived.  When all test cases have been analysed, we switch our attention to a higher abstraction level.  % We treat the sub-system as a black box, or as a component part itsself.  % We can now look at the test case results from the perspective of a `user' % of this sub-system.  % % 
-% We treat the sub-system as a `black box' and view the effects of the component failure
-% at the sub-system level. This mean we are not interested so much in what the compoent does,
-% but how the sub-system reacts when it fails in a certain way.
-% 
-% Each `test case' is labelled from the perspective of the failure as seen at sub-system level.
-
-% We can now try to simplfy by determining common symptoms.  A common symptom, in this context, is defined as faults caused by different  component failure modes that have the same effect from the perspective of a `user' of the sub-system.  Test case results can now viewed as failure modes of the sub-sytem or `black box', and grouped together where there are common symptoms.  These are grouped together by joining them with lines. These lines form  collected groups (or `spiders').  See figure \ref{fig:gensubsys3}.
-% It can be seen now that each {\em lone test case} and {\em spider} on the diagram is a distinct failure mode of the sub-system.  This means that these failure modes represent the fault behaviour of the sub-system.  We can now treat this sub-system as a component in its own right, or in other words, we have derived a failure mode model at a higher level of abstraction.  We can now draw a new diagram to represent the failure modes of the sub-system.  Each spider or lone test case, becomes a contour representing a failure mode of the sub-system in this new diagram (see figure \ref{fig:gensubsys4}.  
 
 \section{The Process : To analyse a base level Derived~Component/sub-system}
 
@@ -260,39 +254,70 @@ Determine which test cases produce the same fault symptoms {\em from the perspec
 
 \section{A general derived Component/Sub-System example}
 
-Consider a functional group $FG$ with component parts $A$,$B$ and $C$.
+Consider a functional group $FG$ with components $C_1$, $C_2$ and $C_3$.
 
-$$ FG = \{ A, B , C \} $$
+$$ FG = \{ C_1 , C_2 , C_3 \} $$
 
-Each part has a set of related fault modes (i.e. ways in which it can fail to operate correctly).
+Each component has a set of related fault modes (i.e. ways in which it can fail to operate correctly).
-Let us define the following failure modes for each component part, defining a function $FM()$ where $K$
+Let us define the following failure modes for each component, defining a function $FM()$ 
-is a component part and $F$ is its set of failure modes\footnote{Base component failure modes are defined, often with 
+that is passed a component and returns the set of failure modes associated with it
+\footnote{Base component failure modes are defined, often with 
 statistics and evironmental factors in  a variety of sources. \cite{mil1991}
 }.
 
-$$ 
+To re-cap from the definitions chapter \ref{chap:definitions}.
-FM : K  \mapsto F
-$$
-\\
-For our example above
-\\
-$$ FM(A) =  \{ a_1, a_2, a_3 \} $$
-$$ FM(B) =  \{ b_1, b_2 \} $$ 
-$$ FM(C) =  \{ c_1, c_2 \} $$
 
-\paragraph{NOTE TO ANDREW : SHOULD I DEFINE A FUNCTION HERE THAT CONVERTS A FUNCTIONAL GROUP
+Let the set of all possible components  be $\mathcal{C}$
-TO the set of failure modes in all its component parts ??? Am I being lazy here ???}
+and let the set of all possible failure modes be $\mathcal{F}$.
 
-We can now represent the functional~group $FG$ as a set of component faulure modes $FG_{cfm}$,
+We can define a function $FM$  
-thus 
 
 \begin{equation}
- FG_{cfm} = \{a_1, a_2, a_3, b_1, b_2, c_1, c_2 \}
+FM : \mathcal{C} \mapsto \mathcal{P}\mathcal{F}
+\end{equation}
+
+defined by
+
+$$  FM ( C ) = F $$
+
+%\\
+
+And for this example:
+
+$$ FM(C_1) =  \{ a_1, a_2, a_3 \} $$
+$$ FM(C_2) =  \{ b_1, b_2 \} $$ 
+$$ FM(C_3) =  \{ c_1, c_2 \} $$
+
+
+\paragraph{Finding all failure modes within the functional group}
+
+For FMMD failure mode analysis we need to consider the failure modes
+from all the components in the functional group as a flat set.
+This can be found by applying function $FM$ to all the components
+in the functional~group and taking the union of them thus:
+
+$$ FunctionalGroupAllFailureModes = \bigcup_{j \in \{1...n\}} FM(C_j) $$
+
+We can actually overload the notation for the function FM
+and define it for the set components within a functional group $FG$ (i.e. where $FG \subset \mathcal{C} $) thus:
+
+\begin{equation}
+FM : FG \mapsto \mathcal{F}
+\end{equation}
+
+Applied to the functional~group $FG$ in the example above:
+\begin{equation}
+ FM(FG) = \{a_1, a_2, a_3, b_1, b_2, c_1, c_2 \}
 \end{equation} 
 
+This can be seen as all the failure modes that can affect the failure mode group $FG$.
 
+\subsection{Analysis of the functional group failure modes}
 
-% The failure modes of the components can be represented as contours on on the diagram in \ref{fig:gensubsys1}.  \begin{figure} \centering \includegraphics[width=3in,height=3in,bb=0 0 513 541]{symptom_abstraction/synmptom_abstraction.jpg} % synmptom_abstraction.jpg: 570x601 pixel, 80dpi, 18.10x19.08 cm, bb=0 0 513 541 \label{fig:gensubsys1} \caption{$FG_{cfm}$ Component Failure modes represented as contours} \end{figure} % % DIAGRAM WITH SPIDER % \begin{figure} %  \centering %  \includegraphics[scale=20]{./synmptom_abstraction.jpg} %  % synmptom_abstraction.jpg: 570x601 pixel, 80dpi, 18.10x19.08 cm, bb=0 0 513 541 %  \label{fig:gensubsys2} %  \caption{$SS_{cfm}$ Component Failure modes represented as contours} % \end{figure} We can now look at the effects that  component failure modes have on the sub-system.  This process involves examining `test cases'. Each `test case' represents the fault behaviour of the sub-system due to particular combinations of component fault modes.  Each test case can be represented on the diagram as a labeled point.  The labeled point will reside in a region on the diagram enclosed by the contours representing particular component fault modes.  The label will indicate the fault symptom from the perspective of the sub-system.  For the sake of example, only single component failure modes are considered.  We can now assign a test~case to each contour, and mark it on the diagram.  % \begin{figure}[h+] %  \centering %  \includegraphics[scale=20]{./symptom_abstraction2.jpg} %  % synmptom_abstraction.jpg: 570x601 pixel, 80dpi, 18.10x19.08 cm, bb=0 0 513 541 %  \label{fig:gensubsys2} %  \caption{Component Failure modes with analysed test cases} % \end{figure} \begin{figure} \centering \includegraphics[width=3in,height=3in,bb=0 0 513 541]{symptom_abstraction/symptom_abstraction2.jpg} % symptom_abstraction2.jpg: 570x601 pixel, 80dpi, 18.10x19.08 cm, bb=0 0 513 541 \label{fig:gensubsys2} \caption{Component Failure modes with analysed test cases} \end{figure}
+For this example we shall consider single failure modes.
+%For each of the failure modes from $FM(FG)$ we shall
+%create a test case ($fgfm_i$). Next each test case is examined/analysed
+%and its effect on the functional group determined.
 
 \par
 %\vspace{0.3cm}
@@ -300,6 +325,7 @@ thus
 \begin{tabular}{||c|c|c|c||} \hline \hline
   {\em Component Failure Mode } & {\em test case} & {\em Functional Group} & {\em Functional Group}    \\ 
   {\em                        } & {\em          } & {\em failure mode}     & {\em Symptom}                \\ \hline
+%
 $a\_1$ & $fs\_1$ & $fgfm_{1}$ & SP2     \\      \hline
 $a\_2$ & $fs\_2$ & $fgfm_{2}$ & SP1 \\    \hline 
 $a\_3$ & $fs\_3$ & $fgfm_{3}$ & SP2\\    \hline  
@@ -307,6 +333,7 @@ $b\_1$ & $fs\_4$ & $fgfm_{4}$ & SP1 \\    \hline
 $b\_2$ & $fs\_5$ & $fgfm_{5}$ & SP1 \\     \hline  
 $c\_1$ & $fs\_6$ & $fgfm_{6}$ & \\    \hline  
 $c\_2$ & $fs\_7$ & $fgfm_{7}$ & SP2\\    \hline
+%
  \hline
 \end{tabular}
 \caption{Component to functional group to failure symptoms example}
@@ -315,13 +342,13 @@ $c\_2$ & $fs\_7$ & $fgfm_{7}$ & SP2\\    \hline
 %\vspace{0.3cm}
 
 Table~\ref{tab:fexsymptoms} shows the analysis process.
-In this example we are only looking at single fault possibilities.
+As we are only looking at single fault possibilities for this example each failure mode
+is represented by a test~case.
 The Component failure modes become test cases\footnote{The test case stage is necessary because for more complex analysis we have to consider the effects of combinations of component failure modes}.
 The test cases are analysed w.r.t. the functional~group.
 These become functional~group~failure~modes ($fgfm$'s).
 The functional~group~failure~modes are how the functional group fails for the test~case, rather than how the components failed.
 
-% The sub-system fault symptoms are now represented on the diagram as in figure \ref{fig:gensubsys2}.  A second stage of analysis is now applied.  Empirically, it is often noticed that a sub-system will fail in the same way due to a variety of reasons.  To the `user' of the sub-system, it does not matter which component or combination of components has failed.  The sub-system can thus be considered to have its own set of failure modes.  This stage of the analysis is to determine these, to collect `like symptoms'.  This is performed on the diagram by linking the test cases with lines to form `spiders' 
 For the sake of example let us consider the fault symptoms of $\{fgfm_2, fgfm_4, fgfm_5\}$  be 
 identical from the perspective of the functional~group.
 That is to say, that the way in which functional~group fails if $fgfm_2$, $fgfm_4$ or $fgfm_5$ % failure modes
@@ -339,7 +366,6 @@ Likewise
 let $SP2 = \{fgfm_1, fgfm_3, fgfm_7\}$  be  an identical failure mode {\em from the perspective of the functional~group}.
 Let $\{fgfm_6\}$ be a distinct failure mode {\em from the perspective of the functional~group i.e. it cannot be grouped as a common symptom}.
 
-% The diagram can now be drawn as in figure \ref{fig:gensubsys3}.  % \begin{figure}[h+] %  \centering %  \includegraphics[scale=20]{./symptom_abstraction3.jpg} %  % synmptom_abstraction.jpg: 570x601 pixel, 80dpi, 18.10x19.08 cm, bb=0 0 513 541 %  \label{fig:gensubsys3} %  \caption{Common failure modes collected as `Spiders'} % \end{figure} \begin{figure}[h+] \centering \includegraphics[width=3in,height=3in,bb=0 0 513 541]{symptom_abstraction/symptom_abstraction3.jpg} % symptom_abstraction3.jpg: 570x601 pixel, 80dpi, 18.10x19.08 cm, bb=0 0 513 541 \label{fig:gensubsys3} \caption{Common failure modes collected as `Spiders'} \end{figure}
 
 We have now in $SP1$, $SP2$ and $fgfm_6$ as the three ways in which this functional~group can fail.
 In other words we have derived failure modes for this functional~group.
@@ -420,14 +446,27 @@ this section
 using set theory.  
 
 The {\em symptom abstraction process} (given the symbol `$\bowtie$') takes a functional group $FG$
-and converts it to a sub-system $DC$.
+and converts it to a derived~component/sub-system $DC$.
 %The sub-system $SS$ is a collection
 %of failure~modes of the sub-system.
+Note that
 $DC$ is a derived component at a higher level of fault analysis abstraction,
-it may be thus be treated  
+than the functional~group it was derived from.
+However, it can still be treated  
 as a component with a known set of failure modes.
-Thus $DC$ can be used as a system building block at a higher
+\paragraph{enumerating abstraction levels}
-level of fault abstraction.
+If $DC$ is included in a functional~group
+that functional~group must be considered to be a a higher level of
+abstraction than a base level functional~group.
+%
+In fact if the abstraction level is enumerated 
+the functional~group must take the abstraction level
+of the highest assigned to any of its components.
+%
+$DC$ can be used as a system building block at a higher
+level of fault abstraction. Because the derived components
+merge to form functional  groups, a converging hierarchy is
+naturally formed with the abstraction level increasing with each tier.
 
 
 The algorithm, representing the function $\bowtie$, has been broken down into five stages, each following on from the other.