787 lines
26 KiB
TeX
787 lines
26 KiB
TeX
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\ifthenelse {\boolean{paper}}
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{
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\abstract{
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%% What I have done
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%%
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This paper presents a simple two level Failure Mode Modular De-Composition (FMMD)
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model of a theoretical system.
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Firstly a UML model is presented and the class relationships described.
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Secondly the theoretical model is developed and analysed.
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This model is then represented as a Directed Acyclic Graph (DAG),
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showing the data relationships between the {\fg}s
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components and failure modes.
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% What I have found
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%%
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From traversing the DAG, minimal cut sets (component level combinations
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that cause system level failures) are revealed.
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Common mode failure modes and same component dependencies
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can also be automatically determined.
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%% Sell it
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%%
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By having an FMMD data model, we can derive failure mode models
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for the traditional methodologies (such as FMEA, FMECA, FMEDA and FTA).
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Also, with statistical data, we can use the minimal cut set results
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to determine the likelihood of particular system failures, even
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if they have multiple causes.
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%
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} % abstract
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} % ifthenelse
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{
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%%% CHAPTER INTO NEARLT THE SAME AS ABSTRACT
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\section{Introduction}
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This chapter
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presents a simple two stage FMMD % Failure Mode Modular De-Composition (FMMD)
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model of a theoretical system.
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The Analysis model is then represented as a Directed Acyclic Graph (DAG), of the {\fg}s
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components and failure modes represented in it.
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% What I have found
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%%
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From traversing the DAG, minimal cut sets (component level combinations
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that cause system level failures) are revealed.
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Common mode failure modes and same component dependencies
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can also be automatically determined.
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%% Sell it
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%%
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By having an FMMD data model, we can derive failure mode models
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for the traditional methodologies (such as FMEA, FMECA, FMEDA and FTA).
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Also, with statistical data, we can use the minimal cut set results
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to determine the likelihood of particular system failures, even
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if they have multiple causes.
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}
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%{ \huge This might become a chapter in its own right after fmmdset }
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\section{From UML Model to Object Model}
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Let us consider a theoretical FMMD model. For the sake of simplicity
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consider that all base~components have %only
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two failure modes that
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we will label $a$ and $b$.
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We can start with some base components, of types C and K say, $\{ C_1, C_2, C_3, K_4, C_5, C_6, K_7 \}$.
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\input{./shortfm}
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\paragraph{Determining Failure Mode collections.}
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Thus applying the function $fm$ to any of the components
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gives error modes identified by a or b.
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As each component has two failure
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modes $a$ and $b$. So the function $fm$ applied to
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$C_1$ yields $C_{1 a}$ and $C_{1 b}$:
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i.e. $fm(C_1) = \{ C_{1 a}, C_{1 b} \}$.
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%HOW UML OBJECT MODEL OF COMPONENT AND ITS ERROR MODES
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\ifthenelse {\boolean{paper}}
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{
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We can organise these into functional groups (where the superscript
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represents the FMMD hierarchy level, or $\alpha$ value, thus:
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}
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{
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We can organise these into functional groups (where the superscript
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represents the $\alpha$ value, or FMMD hierarchy level, see section \ref{alpha}), thus:
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}
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$$ FG^0_1 = \{C_1, C_2\},$$
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$$ FG^0_2 = \{C_1, C_3, K_4\},$$
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$$ FG^0_3 = \{C_5, C_6, K_7\}.$$
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Note that in this model the base~component $C_1$ has been used in two separate functional groups.
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This could be a component that they
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both commonly use. A real world example of a component included in
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more than one {\fg} could
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be a power-supply or DCDC\footnote{A DCDC (direct current to direct current)
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converter, is a common feature in modern PCBs, used to provide isolation
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and/or voltage supplies at a different EMF from the source of power.}
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converter shared to power
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the functional groups $FG^0_1$ and $FG^1_1$.
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Also note that the component type $K$ has been used by
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two different functional groups.
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For the sake of example, let our temperature environment
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for the SYSTEM be ${{0}\oc}$ to ${{125}\oc}$, but let the component
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type `K' have a de-graded performance
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\footnote{ A real world example of
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degraded performace with temperature is the isolating opto coupler.
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These can typically only cope with lower baud rate ranges
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at high temperatures \cite{tlp181}.}
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failure mode between
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${{80}\oc}$ and ${{125}\oc}$.
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We can term this
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degraded performance of component `K' as failure mode `d'.
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\paragraph{Symptom Extraction.}
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A process of symptom extraction is now applied to the functional groups.
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Again for the sake of example, let us say that each functional
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group has one or two symptoms again subscripted by $a$ and $b$.
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%Applying symptom abstraction to $FG^0_1$ i.e. $\bowtie fm ( FG^0_1 ) = \{ FG^0_{1 a}, FG^0_{1 b} \} $
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%We can now create a new derived component, $C^1_1$, whose failure
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%modes are the symptoms of $FG^0_1 $ thus $ fm ( {C}^1_1 ) = \{ FG^0_{1 a}, FG^0_{1 b} \} $.
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\paragraph{Building the Object Model}
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Using the UML model in figure \ref{fig:cfg2fmmd_data}, we apply FMMD analysis stages
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to build a hierarchy representing the whole system.
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We shall begin with the $FG^0$ level functional groups $ FG^0_1, FG^0_2 $ and $FG^0_3$ defined above.
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\begin{figure}[h]
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\centering
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\includegraphics[width=400pt,bb=0 0 702 464,keepaspectratio=true]{./fmmd_data_model/cfg2.jpg}
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% cfg2.jpg: 702x464 pixel, 72dpi, 24.76x16.37 cm, bb=0 0 702 464
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\caption{UML Class model for FMMD}
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\label{fig:cfg2fmmd_data}
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\end{figure}
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The UML model shows the relationships between data types (or classes) that
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are used in the FMMD process.
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The purpose of failure mode analysis, is to tie SYSTEM level failures
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to their possible causes in the base components.
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By doing this accurate statistics can be obtained for SYSTEM level
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failures, and an insight into how we can make the system safer
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can be determined.
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In order to do this, we need to be able to trace the component
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failure modes from the functional groups, to the symptoms
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they cause, and to the failure modes in the {\dcs}.
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We can use graph theory to represent this.
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As it would make no sense for a derived component to
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derive failure modes form itsself, we can apply an acyclic constraint
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to the graph. This means the graph must be a Directed Acylic
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Graph (DAG).
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% %\begin{figure}[h]
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% \centering
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% \includegraphics[width=400pt,keepaspectratio=true]{./fmmd_data_model/cfg2.jpg}
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% % cfg2.jpg: 702x464 pixel, 72dpi, 24.76x16.37 cm, bb=0 0 702 464
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% \caption{Complete UML diagram}
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% \label{fig:cfg2fmmd_data}
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% \end{figure}
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\pagebreak[4]
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\subsection{Find Failure Modes}
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Consider the SYSTEM environment with its temperature range of ${{0}\oc}$ to ${{125}\oc}$.
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We must check this against all components used.
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For our example, we component `K' which has an extra
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failure mode for degraded performance `d'. Thus applying the function $fm$
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to component type `K' under these temperature range conditions
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gives the following failure modes, $fm{K} =\{ K^0_a, K^0_b, K^0_d \}$.
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Were our system specified for a ${{0}\oc}$ to ${{80}\oc}$ range
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we could say $fm{K} =\{ K^0_a, K^0_b \}$.
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\pagebreak[3]
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\paragraph{Get the failure modes from the functional groups.}
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Applying the function $fm$ to our functional groups, with the SYSTEM environmental
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constraint applied to component type `K', yields
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%%//$$ FG^0_1 = \{C_1, C_2\},$$
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%%$$ FG^0_2 = \{C_1, C_3, K_4\},$$
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%%$$ FG^0_3 = \{C_5, C_6, K_7\}.$$
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$$ fm(FG^0_1) = \{C^0_{1 a}, C^0_{1 b}, C^0_{2 a}, C^0_{2 b}\},$$
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$$ fm(FG^0_2) = \{C^0_{1 a}, C^0_{1 b}, C^0_{3 a}, C^0_{3 b}, K^0_{4 a}, K^0_{4 b}, K^0_{4 d}\},$$
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$$ fm(FG^0_3) = \{C^0_{5 a}, C^0_{5 b}, C^0_{6 a}, C^0_{6 b}, K^0_{7 a}, K^0_{7 b}, K^0_{7 d}\}.$$
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The next stage is to look at the failure modes from the perspective of
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the functional groups, rather than the components.
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We can call these failures modes `symptoms'.
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As this is a theoretical example, we shall have to skip this step\footnote{
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In a real analysis this would involve evaluating the effect of each components failure mode, (or combinations of)
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on the performance of the {\fg}.}.
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The next stage is to collect the common symptoms, or the symptoms that
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are the same {\em from the perspective of a user of the {\fg}}.
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We can define this stage as the function $\bowtie$ which has a set of failure modes as
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its range and {\dc} as its domain.
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For the sake of example let us determine some arbitary collections
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into symptoms. Let us group the symptoms from $ FG^0_1 $ as the following
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$ s1 = \{ C^0_{1 a}, C^0_{2 b} \}$ and $ s2 = \{ C^0_{1 b}, C^0_{2 a} \}$.
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We can represent the relationships between the failure modes, and desired failure modes or symptoms
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as a DAG (see figure \ref{fig:dag0}).
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\def\layersep{2.5cm}
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\begin{figure}
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\centering
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\begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep]
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\tikzstyle{every pin edge}=[<-,shorten <=1pt]
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\tikzstyle{fmmde}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt]
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\tikzstyle{component}=[fmmde, fill=green!50];
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\tikzstyle{failure}=[fmmde, fill=red!50];
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\tikzstyle{symptom}=[fmmde, fill=blue!50];
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\tikzstyle{annot} = [text width=4em, text centered]
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% Draw the input layer nodes
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%\foreach \name / \y in {1,...,4}
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% This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4}
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% \node[component, pin=left:Input \#\y] (I-\name) at (0,-\y) {};
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\node[component] (C-1) at (0,-1) {$C^0_1$};
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\node[component] (C-2) at (0,-3) {$C^0_2$};
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%\node[component] (C-3) at (0,-5) {$C^0_3$};
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%\node[component] (K-4) at (0,-8) {$K^0_4$};
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%\node[component] (C-5) at (0,-10) {$C^0_5$};
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%\node[component] (C-6) at (0,-12) {$C^0_6$};
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%\node[component] (K-7) at (0,-15) {$K^0_7$};
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% Draw the hidden layer nodes
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%\foreach \name / \y in {1,...,5}
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% \path[yshift=0.5cm]
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\node[failure] (C-1a) at (\layersep,-1) {a};
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\node[failure] (C-1b) at (\layersep,-2) {b};
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\node[failure] (C-2a) at (\layersep,-3) {a};
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\node[failure] (C-2b) at (\layersep,-4) {b};
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% Draw the output layer node
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% Connect every node in the input layer with every node in the
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% hidden layer.
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%\foreach \source in {1,...,4}
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% \foreach \dest in {1,...,5}
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\path (C-1) edge (C-1a);
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\path (C-1) edge (C-1b);
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\path (C-2) edge (C-2a);
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\path (C-2) edge (C-2b);
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%\node[symptom,pin={[pin edge={->}]right:Output}, right of=C-1a] (O) {};
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\node[symptom, right of=C-1a] (s1) {s1};
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\node[symptom, right of=C-2a] (s2) {s2};
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\path (C-2b) edge (s1);
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\path (C-1a) edge (s1);
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\path (C-2a) edge (s2);
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\path (C-1b) edge (s2);
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%\node[component, right of=s1] (DC) {$C^1_1$};
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%\path (s1) edge (DC);
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%\path (s2) edge (DC);
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% Connect every node in the hidden layer with the output layer
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%\foreach \source in {1,...,5}
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% \path (H-\source) edge (O);
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% Annotate the layers
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\node[annot,above of=C-1a, node distance=1cm] (hl) {Failure modes};
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\node[annot,left of=hl] {Base Components};
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\node[annot,right of=hl](s) {Symptoms};
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%\node[annot,right of=s](dcl) {Derived Component};
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\end{tikzpicture}
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% End of code
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\caption{DAG representing failure modes and symptoms of $FG^0_1$}
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\label{fig:dag0}
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\end{figure}
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%%%\begin{figure}[h]
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%%% \centering
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%%% \includegraphics[width=300pt,bb=0 0 466 270,keepaspectratio=true]{./fmmd_data_model/dag0.jpg}
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%%% % dag0.jpg: 466x270 pixel, 72dpi, 16.44x9.52 cm, bb=0 0 466 270
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%%% \caption{DAG reprsenting the failure modes from $FG^0_1$.}
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%%% \label{fig:dag0}
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%%%\end{figure}
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We can now create a new {\dc}. This will have an $\alpha$ value higher
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than the any of the components in the {\fg} that it was derived from.
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In this case all components were base components and therefore have an $\alpha$ value of zero.
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Our derived component can thus take an $\alpha$ value of one.
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Our newly derived component can be
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$$ C^1_1 = \bowtie fm(FG^0_1) .$$
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Applying $fm$ to our new derived component will give us our symptoms from functional group $ FG^0_1 $
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thus
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$$ fm(C^1_1) = \{s1, s2 \}.$$
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We can represent $ C^1_1 $ as an addition to the DAG (see figure \ref{fig:dag1}).
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%%%\begin{figure}[h]
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%%% \centering
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%%% \includegraphics[width=300pt,bb=0 0 466 270,keepaspectratio=true]{./fmmd_data_model/dag1.jpg}
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%%% % dag0.jpg: 466x270 pixel, 72dpi, 16.44x9.52 cm, bb=0 0 466 270
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%%% \caption{DAG reprsenting the failure modes from $FG^0_1$ and $ DC^1_0 $.}
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%%% \label{fig:dag1}
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%%%\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep]
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\tikzstyle{every pin edge}=[<-,shorten <=1pt]
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\tikzstyle{fmmde}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt]
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\tikzstyle{component}=[fmmde, fill=green!50];
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\tikzstyle{failure}=[fmmde, fill=red!50];
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\tikzstyle{symptom}=[fmmde, fill=blue!50];
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\tikzstyle{annot} = [text width=4em, text centered]
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% Draw the input layer nodes
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%\foreach \name / \y in {1,...,4}
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% This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4}
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% \node[component, pin=left:Input \#\y] (I-\name) at (0,-\y) {};
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\node[component] (C-1) at (0,-1) {$C^0_1$};
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\node[component] (C-2) at (0,-3) {$C^0_2$};
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%\node[component] (C-3) at (0,-5) {$C^0_3$};
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%\node[component] (K-4) at (0,-8) {$K^0_4$};
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%\node[component] (C-5) at (0,-10) {$C^0_5$};
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%\node[component] (C-6) at (0,-12) {$C^0_6$};
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%\node[component] (K-7) at (0,-15) {$K^0_7$};
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% Draw the hidden layer nodes
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%\foreach \name / \y in {1,...,5}
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% \path[yshift=0.5cm]
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\node[failure] (C-1a) at (\layersep,-1) {a};
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\node[failure] (C-1b) at (\layersep,-2) {b};
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\node[failure] (C-2a) at (\layersep,-3) {a};
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\node[failure] (C-2b) at (\layersep,-4) {b};
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% Draw the output layer node
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% Connect every node in the input layer with every node in the
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% hidden layer.
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%\foreach \source in {1,...,4}
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% \foreach \dest in {1,...,5}
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\path (C-1) edge (C-1a);
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\path (C-1) edge (C-1b);
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\path (C-2) edge (C-2a);
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\path (C-2) edge (C-2b);
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%\node[symptom,pin={[pin edge={->}]right:Output}, right of=C-1a] (O) {};
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\node[symptom, right of=C-1a] (s1) {s1};
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\node[symptom, right of=C-2a] (s2) {s2};
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\path (C-2b) edge (s1);
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\path (C-1a) edge (s1);
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\path (C-2a) edge (s2);
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\path (C-1b) edge (s2);
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\node[component, right of=s1] (DC) {$C^1_1$};
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\path (s1) edge (DC);
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\path (s2) edge (DC);
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% Connect every node in the hidden layer with the output layer
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%\foreach \source in {1,...,5}
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% \path (H-\source) edge (O);
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% Annotate the layers
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\node[annot,above of=C-1a, node distance=1cm] (hl) {Failure modes};
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\node[annot,left of=hl] {Base Components};
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\node[annot,right of=hl](s) {Symptoms};
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\node[annot,right of=s](dcl) {Derived Component};
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\end{tikzpicture}
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% End of code
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\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow C^1_1$}
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\label{fig:dag1}
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\end{figure}
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\clearpage
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\subsection{ Creating Derived components from $FG^0_2$ and $FG^0_3$ }
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Applying the FMMD process for $FG^0_2$ and $FG^0_3$.
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\paragraph{Applying FMMD $ \bowtie fm(FG^0_2) $:}
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The failure modes $fm(FG^0_2) = \{C^0_{1 a}, C^0_{1 b}, C^0_{3 a}, C^0_{3 b}, K^0_{4 a}, K^0_{4 b}, K^0_{4 d}\}.$
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Let us say new symptom s3 can be caused by failure modes $\{C^0_{1 a}, C^0_{3 b}, K^0_{4 b} \}$
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, let us say new symptom s4 can be caused by failure modes $\{C^0_{1 b}, C^0_{3 a}, K^0_{4 d} \}$
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and let us say new symptom s5 can be caused by failure mode $\{K^0_{4 a} \}$.
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We can create a derived component $C^1_2$ using
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$\bowtie fm(FG^0_2) = C^1_2$.
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Applying $fm$ to our {\dcs} gives $fm(C^1_2) = \{ s3,s4,s5 \}$.
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We can respresent this in the DAG in figure \ref{fig:dag2}.
|
|
|
|
|
|
%
|
|
% DAG INCLUDING DC^1_2
|
|
%
|
|
|
|
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep]
|
|
\tikzstyle{every pin edge}=[<-,shorten <=1pt]
|
|
\tikzstyle{fmmde}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt]
|
|
\tikzstyle{component}=[fmmde, fill=green!50];
|
|
\tikzstyle{failure}=[fmmde, fill=red!50];
|
|
\tikzstyle{symptom}=[fmmde, fill=blue!50];
|
|
\tikzstyle{annot} = [text width=4em, text centered]
|
|
|
|
% Draw the input layer nodes
|
|
%\foreach \name / \y in {1,...,4}
|
|
% This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4}
|
|
% \node[component, pin=left:Input \#\y] (I-\name) at (0,-\y) {};
|
|
|
|
\node[component] (C-1) at (0,-1) {$C^0_1$};
|
|
\node[component] (C-2) at (0,-3) {$C^0_2$};
|
|
\node[component] (C-3) at (0,-5) {$C^0_3$};
|
|
\node[component] (K-4) at (0,-8) {$K^0_4$};
|
|
%\node[component] (C-5) at (0,-10) {$C^0_5$};
|
|
%\node[component] (C-6) at (0,-12) {$C^0_6$};
|
|
%\node[component] (K-7) at (0,-15) {$K^0_7$};
|
|
|
|
% Draw the hidden layer nodes
|
|
%\foreach \name / \y in {1,...,5}
|
|
% \path[yshift=0.5cm]
|
|
\node[failure] (C-1a) at (\layersep,-1) {a};
|
|
\node[failure] (C-1b) at (\layersep,-2) {b};
|
|
\node[failure] (C-2a) at (\layersep,-3) {a};
|
|
\node[failure] (C-2b) at (\layersep,-4) {b};
|
|
\node[failure] (C-3a) at (\layersep,-5) {a};
|
|
\node[failure] (C-3b) at (\layersep,-6) {b};
|
|
\node[failure] (K-4a) at (\layersep,-7) {a};
|
|
\node[failure] (K-4b) at (\layersep,-8) {b};
|
|
\node[failure] (K-4d) at (\layersep,-9) {d};
|
|
|
|
% Draw the output layer node
|
|
|
|
% Connect every node in the input layer with every node in the
|
|
% hidden layer.
|
|
%\foreach \source in {1,...,4}
|
|
% \foreach \dest in {1,...,5}
|
|
\path (C-1) edge (C-1a);
|
|
\path (C-1) edge (C-1b);
|
|
\path (C-2) edge (C-2a);
|
|
\path (C-2) edge (C-2b);
|
|
\path (C-3) edge (C-3a);
|
|
\path (C-3) edge (C-3b);
|
|
\path (K-4) edge (K-4a);
|
|
\path (K-4) edge (K-4b);
|
|
\path (K-4) edge (K-4d);
|
|
|
|
%\node[symptom,pin={[pin edge={->}]right:Output}, right of=C-1a] (O) {};
|
|
\node[symptom, right of=C-1a] (s1) {s1};
|
|
\node[symptom, right of=C-2a] (s2) {s2};
|
|
\node[symptom, right of=C-3a] (s3) {s3};
|
|
\node[symptom, right of=C-3b] (s4) {s4};
|
|
\node[symptom, right of=K-4b] (s5) {s5};
|
|
|
|
|
|
|
|
\path (C-2b) edge (s1);
|
|
\path (C-1a) edge (s1);
|
|
|
|
\path (C-2a) edge (s2);
|
|
\path (C-1b) edge (s2);
|
|
|
|
\path (C-1a) edge (s3);
|
|
\path (C-3b) edge (s3);
|
|
\path (K-4b) edge (s3);
|
|
|
|
\path (C-1b) edge (s4);
|
|
\path (C-3a) edge (s4);
|
|
\path (K-4d) edge (s4);
|
|
|
|
\path (K-4a) edge (s5);
|
|
|
|
\node[component, right of=s1] (DC-1) {$C^1_1$};
|
|
\node[component, right of=s4] (DC-2) {$C^1_2$};
|
|
|
|
\path (s1) edge (DC-1);
|
|
\path (s2) edge (DC-1);
|
|
|
|
\path (s3) edge (DC-2);
|
|
\path (s4) edge (DC-2);
|
|
\path (s5) edge (DC-2);
|
|
|
|
|
|
|
|
|
|
|
|
% Connect every node in the hidden layer with the output layer
|
|
%\foreach \source in {1,...,5}
|
|
% \path (H-\source) edge (O);
|
|
|
|
% Annotate the layers
|
|
\node[annot,above of=C-1a, node distance=1cm] (hl) {Failure modes};
|
|
\node[annot,left of=hl] {Base Components};
|
|
\node[annot,right of=hl](s) {Symptoms};
|
|
\node[annot,right of=s](dcl) {Derived Component};
|
|
\end{tikzpicture}
|
|
% End of code
|
|
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow C^1_1$ and $FG^0_2 \rightarrow C^1_2$}
|
|
\label{fig:dag2}
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
|
|
%/\clearpage
|
|
|
|
\paragraph{Applying FMMD $\bowtie fm(FG^0_3) $ :}
|
|
Let us say new symptom s6 can be caused by failure modes $\{C_{5 a}, C_{6 b}, K_{4 b} \}$
|
|
, let us say new symptom s7 can be caused by failure modes $\{C_{5 b}, C_{6 a}, K_{7 d} \}$
|
|
and let us say new symptom s8 can be caused by failure mode $\{K_{7 a} \}$.
|
|
|
|
We can create a derived component $C^1_3$ using
|
|
$\bowtie fm(FG^0_3) = C^1_3$
|
|
where $fm(C^1_3) = \{ s6,s7,s8 \}$.
|
|
|
|
We can now represent the first stage of FMMD, all base component
|
|
failure modes analysed and our first set of derived components determined.
|
|
This is shown in the DAG in figure \ref{fig:dag3}.
|
|
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep]
|
|
\tikzstyle{every pin edge}=[<-,shorten <=1pt]
|
|
\tikzstyle{fmmde}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt]
|
|
\tikzstyle{component}=[fmmde, fill=green!50];
|
|
\tikzstyle{failure}=[fmmde, fill=red!50];
|
|
\tikzstyle{symptom}=[fmmde, fill=blue!50];
|
|
\tikzstyle{annot} = [text width=4em, text centered]
|
|
|
|
% Draw the input layer nodes
|
|
%\foreach \name / \y in {1,...,4}
|
|
% This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4}
|
|
% \node[component, pin=left:Input \#\y] (I-\name) at (0,-\y) {};
|
|
|
|
\node[component] (C-1) at (0,-1) {$C^0_1$};
|
|
\node[component] (C-2) at (0,-3) {$C^0_2$};
|
|
\node[component] (C-3) at (0,-5) {$C^0_3$};
|
|
\node[component] (K-4) at (0,-8) {$K^0_4$};
|
|
\node[component] (C-5) at (0,-10) {$C^0_5$};
|
|
\node[component] (C-6) at (0,-12) {$C^0_6$};
|
|
\node[component] (K-7) at (0,-15) {$K^0_7$};
|
|
|
|
% Draw the hidden layer nodes
|
|
%\foreach \name / \y in {1,...,5}
|
|
% \path[yshift=0.5cm]
|
|
\node[failure] (C-1a) at (\layersep,-1) {a};
|
|
\node[failure] (C-1b) at (\layersep,-2) {b};
|
|
\node[failure] (C-2a) at (\layersep,-3) {a};
|
|
\node[failure] (C-2b) at (\layersep,-4) {b};
|
|
\node[failure] (C-3a) at (\layersep,-5) {a};
|
|
\node[failure] (C-3b) at (\layersep,-6) {b};
|
|
\node[failure] (K-4a) at (\layersep,-7) {a};
|
|
\node[failure] (K-4b) at (\layersep,-8) {b};
|
|
\node[failure] (K-4d) at (\layersep,-9) {d};
|
|
|
|
|
|
\node[failure] (C-5a) at (\layersep,-10) {a};
|
|
\node[failure] (C-5b) at (\layersep,-11) {b};
|
|
\node[failure] (C-6a) at (\layersep,-12) {a};
|
|
\node[failure] (C-6b) at (\layersep,-13) {b};
|
|
\node[failure] (K-7a) at (\layersep,-14) {a};
|
|
\node[failure] (K-7b) at (\layersep,-15) {b};
|
|
\node[failure] (K-7d) at (\layersep,-16) {d};
|
|
|
|
% Draw the output layer node
|
|
|
|
% Connect every node in the input layer with every node in the
|
|
% hidden layer.
|
|
%\foreach \source in {1,...,4}
|
|
% \foreach \dest in {1,...,5}
|
|
\path (C-1) edge (C-1a);
|
|
\path (C-1) edge (C-1b);
|
|
\path (C-2) edge (C-2a);
|
|
\path (C-2) edge (C-2b);
|
|
\path (C-3) edge (C-3a);
|
|
\path (C-3) edge (C-3b);
|
|
\path (K-4) edge (K-4a);
|
|
\path (K-4) edge (K-4b);
|
|
\path (K-4) edge (K-4d);
|
|
|
|
\path (C-5) edge (C-5a);
|
|
\path (C-5) edge (C-5b);
|
|
\path (C-6) edge (C-6a);
|
|
\path (C-6) edge (C-6b);
|
|
\path (K-7) edge (K-7a);
|
|
\path (K-7) edge (K-7b);
|
|
\path (K-7) edge (K-7d);
|
|
|
|
%\node[symptom,pin={[pin edge={->}]right:Output}, right of=C-1a] (O) {};
|
|
\node[symptom, right of=C-1a] (s1) {s1};
|
|
\node[symptom, right of=C-2a] (s2) {s2};
|
|
|
|
\node[symptom, right of=C-3a] (s3) {s3};
|
|
\node[symptom, right of=C-3b] (s4) {s4};
|
|
\node[symptom, right of=K-4b] (s5) {s5};
|
|
|
|
|
|
\node[symptom, right of=C-5a] (s6) {s6};
|
|
\node[symptom, right of=C-6b] (s7) {s7};
|
|
\node[symptom, right of=K-7b] (s8) {s8};
|
|
|
|
\path (C-2b) edge (s1);
|
|
\path (C-1a) edge (s1);
|
|
|
|
\path (C-2a) edge (s2);
|
|
\path (C-1b) edge (s2);
|
|
|
|
\path (C-1a) edge (s3);
|
|
\path (C-3b) edge (s3);
|
|
\path (K-4b) edge (s3);
|
|
|
|
\path (C-1b) edge (s4);
|
|
\path (C-3a) edge (s4);
|
|
\path (K-4d) edge (s4);
|
|
|
|
\path (K-4a) edge (s5);
|
|
|
|
|
|
|
|
\path (C-5a) edge (s6);
|
|
\path (C-6b) edge (s6);
|
|
\path (K-7b) edge (s6);
|
|
|
|
\path (C-5b) edge (s7);
|
|
\path (C-6a) edge (s7);
|
|
\path (K-7d) edge (s7);
|
|
|
|
\path (K-7a) edge (s8);
|
|
|
|
|
|
\node[component, right of=s1] (DC-1) {$C^1_1$};
|
|
\node[component, right of=s4] (DC-2) {$C^1_2$};
|
|
\node[component, right of=s7] (DC-3) {$C^1_3$};
|
|
|
|
\path (s1) edge (DC-1);
|
|
\path (s2) edge (DC-1);
|
|
|
|
\path (s3) edge (DC-2);
|
|
\path (s4) edge (DC-2);
|
|
\path (s5) edge (DC-2);
|
|
|
|
\path (s6) edge (DC-3);
|
|
\path (s7) edge (DC-3);
|
|
\path (s8) edge (DC-3);
|
|
|
|
|
|
% Connect every node in the hidden layer with the output layer
|
|
%\foreach \source in {1,...,5}
|
|
% \path (H-\source) edge (O);
|
|
|
|
% Annotate the layers
|
|
\node[annot,above of=C-1a, node distance=1cm] (hl) {Failure modes};
|
|
\node[annot,left of=hl] {Base Components};
|
|
\node[annot,right of=hl](s) {Symptoms};
|
|
\node[annot,right of=s](dcl) {Derived Component};
|
|
\end{tikzpicture}
|
|
% End of code
|
|
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow C^1_1$ and $FG^0_2 \rightarrow C^1_2$}
|
|
\label{fig:dag3}
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
%\clearpage
|
|
%\pagebreak[4]
|
|
\subsection{Using Derived Components in Functional Groups}
|
|
|
|
The DAG we have in figure \ref{fig:dag3} does not yet give us SYSTEM or `top~level'
|
|
failure modes.
|
|
We can apply $fm$ to the derived components and
|
|
this returns the failure modes. We can notate
|
|
these with $a$ and $b$ etc as before, but can give them
|
|
a subscript representing the symptom they were sourced from thus:
|
|
$$ fm(C^1_1) = \{ a_{s1}, b_{s2} \}, $$
|
|
$$ fm(C^1_2) = \{ a_{s3}, b_{s4}, c_{s5} \}, $$
|
|
$$ fm(C^1_3) = \{ a_{s6}, b_{s7}, c_{s8} \}. $$
|
|
|
|
In order to determine SYSTEM level symptoms, we need to
|
|
use the derived components to form a higher level functional
|
|
group and analyse that.
|
|
|
|
For the sake of example, let us assume that we
|
|
can use all three derived components to
|
|
create a top~level functional group.
|
|
|
|
Let
|
|
$ FG^1_1 = \{ C^1_1, C^1_1, C^1_1 \} $.
|
|
|
|
Applying $fm(FG^1_1) = \{ a_{s1}, b_{s2}, a_{s3}, b_{s4}, c_{s5}, a_{s6}, b_{s7}, c_{s8} \}$.
|
|
To get our system level derived component we can apply $ \bowtie fm(FG^1_1) = C^2_1 $.
|
|
|
|
NOW THINK ABOUT THIS
|
|
|
|
NEED INTERESTING FAULTS
|
|
|
|
TO RACE BACK DOWN THE DAG
|
|
|
|
|
|
|
|
\section{Directed Acyclic Graph}
|
|
|
|
Show how the hierarchy can be represented as a DAG
|
|
|
|
draw a dag
|
|
|
|
\subsection{Traversing the datamodel}
|
|
|
|
Show how we can find multiple causes for a SYSTEM level error
|
|
|
|
\subsubsection{Common mode failure detection}
|
|
|
|
Describe what a common mode failure is.
|
|
|
|
show how common mode failures can be detected by using the parts list (same components can all have their
|
|
error modes turned on, and the effect can be seen on the system, automatically tracing
|
|
common mode failures.
|
|
|
|
|
|
\subsubsection{Common dependency detection}
|
|
|
|
The same component can be relied on by different functional groups within a system
|
|
For instance a power supply spur (i.e. supplying a particular isolated voltage say)
|
|
could have many functional groups depending or linked to its failure modes.
|
|
|
|
Show how FMMD makes this tracable
|
|
|
|
|
|
% clear the page if its a paper to keep the diagram out of the references
|
|
\ifthenelse {\boolean{paper}}
|
|
{
|
|
\clearpage
|
|
}
|
|
{
|
|
}
|
|
|
|
|
|
\section{Current Static Failure Mode Methodologies}
|
|
\ifthenelse {\boolean{paper}}
|
|
{
|
|
paper
|
|
}
|
|
{
|
|
chapter
|
|
}
|
|
|
|
|
|
|
|
\vspace{60pt}
|
|
\today
|