Made all function names lower case
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@ -61,36 +61,36 @@ verification checks in the process are stated formally.
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The first stage is to find the failure modes to consider for
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analysis.
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From the earlier definition of the function `FM':
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From the earlier definition of the function `fm':
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The function $FM$ applied to a component returns the failure modes for that component.
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The function $fm$ applied to a component returns the failure modes for that component.
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The function $FM$ takes a flat set components $\mathcal{FG}$ and returns a set of failure modes $\mathcal{F}$.
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The function $fm$ takes a flat set components $\mathcal{FG}$ and returns a set of failure modes $\mathcal{F}$.
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$$ FM: \mathcal{FG} \rightarrow \mathcal{F}$$
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$$ fm: \mathcal{FG} \rightarrow \mathcal{F}$$
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%Let $FG$ be the set of components in the functional group under analysis, and $c$
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%be components that are members of it. This function returns a flat set of failure modes $F$.
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given by
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$$FM(FG) = F$$
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$$fm(FG) = F$$
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%%
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%% Algorithm 1
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%%
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\begin{algorithm}[h+]
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~\label{alg1}
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\caption{Determine Failure Modes: FM( $FG$ )} \label{alg11}
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\caption{Determine Failure Modes: fm( $FG$ )} \label{alg11}
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\begin{algorithmic}[1]
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\REQUIRE {FG is a set of components (a functional~group)}
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\STATE { Let $FG$ be a set of components } \COMMENT{The functional group should be chosen to be minimally sized collections of components that perform a specific function}
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\FORALL { $c \in FG $ }
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\REQUIRE{ Each component $c \in FG $ has a known set of failure modes i.e. $ \forall c \in FG \; such \; that\; FM(c) \neq \emptyset$ }
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\REQUIRE{ Each component $c \in FG $ has a known set of failure modes i.e. $ \forall c \in FG \; such \; that\; fm(c) \neq \emptyset$ }
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\ENDFOR
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\STATE {let $F=FM(FG)$ be a set of all failure modes to consider for the functional~group $FG$}
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\STATE {let $F=fm(FG)$ be a set of all failure modes to consider for the functional~group $FG$}
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\RETURN { $F$ }
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@ -100,7 +100,7 @@ $$FM(FG) = F$$
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\end{algorithmic}
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\end{algorithm}
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Algorthim \ref{alg11} has taken a functional~group $FG$ and returned a set of failure~modes $F=FM(FG)$
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Algorthim \ref{alg11} has taken a functional~group $FG$ and returned a set of failure~modes $F=fm(FG)$
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(given that each component has a known set of failure~modes).
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The next task is to formulate `test cases'. These are a collection of combinations of these failure~modes and will be used
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in the analysis stages.
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@ -118,11 +118,11 @@ The test cases are collections of failure modes.
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These could be formed from single failure modes or failure modes in combination.
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Let $TC$ be the set of test cases associated with the functional group $FG$.
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$$ DTC: \mathcal{F} \rightarrow \mathcal{TC} $$
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$$ dtc: \mathcal{F} \rightarrow \mathcal{TC} $$
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given by
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$$ DTC(F) = TC $$
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$$ dtc(F) = TC $$
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%%
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%% Algorithm 2
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@ -131,7 +131,7 @@ $$ DTC(F) = TC $$
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\begin{algorithm}[h+]
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~\label{alg2}
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\caption{Determine Test Cases: DTC: (F) } \label{alg22}
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\caption{Determine Test Cases: dtc: (F) } \label{alg22}
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\begin{algorithmic}[1]
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\REQUIRE {F is a flat set of failure modes }
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@ -184,7 +184,7 @@ $$ DTC(F) = TC $$
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\end{algorithmic}
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\end{algorithm}
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Algorithm \ref{alg22} has taken the set of failure modes $ F=FM(FG) $ and returned a set of test cases $TC$.
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Algorithm \ref{alg22} has taken the set of failure modes $ F=fm(FG) $ and returned a set of test cases $TC$.
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The next stage is to analyse the effect of each test case on the functional group.
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@ -202,13 +202,13 @@ The next stage is to analyse the effect of each test case on the functional grou
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The test cases are now analysed for their impact on the behaviour of the functional~group.
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Let $R$ be a set of test case analysis results, indexed by $j$ (the same index used to identify the test cases $tc_{j}$).
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$$ ATC: \mathcal{TC} \rightarrow \mathcal{R} $$A
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$$ atc: \mathcal{TC} \rightarrow \mathcal{R} $$A
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given by
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$$ ATC(TC) = R $$
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$$ atc(TC) = R $$
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\begin{algorithm}[h+]
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~\label{alg3}
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\caption{Analyse Test Cases: ATC(TC) } \label{alg33}
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\caption{Analyse Test Cases: atc(TC) } \label{alg33}
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\begin{algorithmic}[1]
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\STATE { let r be a `test case result'}
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\STATE { Let the function $Analyse : tc \mapsto r $ } \COMMENT { This analysis is a human activity, examining the failure~modes in the test case and determining how the functional~group will fail under those conditions}
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@ -268,19 +268,19 @@ That is to say, each result in a symptom set, from the perspective of the functi
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has the same failure symptom.
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Let set $SP$ be the family of symptom sets for the functional group $FG$.
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$$FCS: \mathcal{R} \rightarrow \mathcal{SP} $$
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$$fcs: \mathcal{R} \rightarrow \mathcal{SP} $$
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given by
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$$ FCS(R) = SP $$
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$$ fcs(R) = SP $$
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\begin{algorithm}[h+]
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~\label{alg4}
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\caption{Find Common Symptoms: FCS($R$)} \label{alg44}
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\caption{Find Common Symptoms: fcs($R$)} \label{alg44}
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\begin{algorithmic}[1]
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%\REQUIRE {All failure modes for the components in $fm_i = FM(fg_i)$}
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%\REQUIRE {All failure modes for the components in $fm_i = fm(fg_i)$}
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\STATE {Let $sp_l$ be a set of `test cases results' where $l$ is an index set $L$}
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\STATE {Let $SP$ be a set whose members are the indexed `symptoms' $sp_l$}
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\COMMENT{ $SP$ is the set of `fault symptoms' for the sub-system}
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@ -357,16 +357,16 @@ This derived component may now be used to build
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new functional groups at higher levels of fault abstraction.
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Let $DC$ be a derived component with its own set of failure~modes.
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$$ CDC: \mathcal{SP} \rightarrow \mathcal{DC} $$
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$$ cdc: \mathcal{SP} \rightarrow \mathcal{DC} $$
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given by
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$$ CDC(SP) = DC $$
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$$ cdc(SP) = DC $$
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\begin{algorithm}[h+]
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~\label{alg5}
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\caption{Create Derived Component: CDC(SP) } \label{alg55}
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\caption{Create Derived Component: cdc(SP) } \label{alg55}
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\begin{algorithmic}[1]
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@ -377,7 +377,7 @@ $$ CDC(SP) = DC $$
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\STATE { $DC := DC \cup f_l$ } \COMMENT{ this is saying place $f_l$ into $DC$'s collection of failure modes}
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\ENDFOR
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\ENSURE { $FM(DC) \neq \emptyset$ } \COMMENT{Ensure that DC has a known set of failure modes}
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\ENSURE { $fm(DC) \neq \emptyset$ } \COMMENT{Ensure that DC has a known set of failure modes}
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\RETURN DC
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%\hline
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@ -400,11 +400,11 @@ $$ \bowtie: \mathcal{FG} \mapsto \mathcal{DC} $$
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\begin{algorithmic}[1]
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\STATE {F = FM (FG)} \COMMENT{ collect all component failure modes }%from the from the components in the functional~group }
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\STATE {TC = DTC (F)} \COMMENT{ determine all test cases } %to apply to the functional group }
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\STATE {R = ATC (TC)} \COMMENT{ analyse the test cases }%, for failure mode behaviour of the functional~group }
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\STATE {SP = FCS (R)} \COMMENT{ find common symptoms }%of failure for the functional group }
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\STATE {DC = CDC (SP)} \COMMENT{ create a derived component }
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\STATE {F = fm (FG)} \COMMENT{ collect all component failure modes }%from the from the components in the functional~group }
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\STATE {TC = dtc (F)} \COMMENT{ determine all test cases } %to apply to the functional group }
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\STATE {R = atc (TC)} \COMMENT{ analyse the test cases }%, for failure mode behaviour of the functional~group }
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\STATE {SP = fcs (R)} \COMMENT{ find common symptoms }%of failure for the functional group }
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\STATE {DC = cdc (SP)} \COMMENT{ create a derived component }
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\RETURN $DC$
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@ -21,7 +21,8 @@ of a system can be built from the bottom~up. This process can continue
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until there is a complete hierarchy representing the failure mode
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behaviour of the entire system under analysis.
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%FMMD hierarchy
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Using the FMMD technique the hierarchy is built from the bottom up to ensure complete failure mode coverage.
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Using the FMMD technique the hierarchy is built from the bottom up to
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ensure complete failure mode coverage.
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Because the process is bottom-up, syntax checking and tracking can ensure that
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no component failure mode can be overlooked.
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Once a hierarchy is in place, it can be converted into a fault data model.
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@ -4,14 +4,13 @@
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% TO DO: separate these two:
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\paragraph{Symptom Extraction Objective}
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The objective of `symptom abstraction' is to analyse the functional~group and find
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how it can fail
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when specified components within it fail.
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Once we know how a functional~group can fail, we can treat it as a component or sub-system
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with its own set of failure modes.
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\paragraph{FMEA applied to the Functional Group}
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\paragraph{fmEA applied to the Functional Group}
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As the functional~group is a set of components, the failure~modes
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that we have to consider are all the failure modes of its components.
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Each failure mode (or combination of) investigated is termed a `test case'.
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@ -62,14 +61,16 @@ To sumarise:
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\item Choose a set of components to form a functional group.
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% \item Obtain the list of components in the functional group
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\item Collect the failure modes of each component into a flat set.
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\item Choose all single instances and selected combinations of the failure modes to
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\item Choose all single instances (and optional selected combinations\footnote{
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Some specific combinations of failure modes might be included, or with EN298 \cite{en298}
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all possible double failure mode conditions} of the failure modes to
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form `test cases'.
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% \item Draw these as contours on a diagram
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% \item Where si,ultaneous failures are examined use overlapping contours
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% \item For each region on the diagram, make a test case
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\item Using the `test cases' determine their effects on the failure~mode behaviour of the functional group.
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\item Using the `test cases' determine their effects on the failure~mode behaviour of the functional group. This is a human process involving detailed analysis of the failure modes oin the test case on the operation of the {\fg}.
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\item Collect common~symptoms. i.e. determine which test cases produce the same fault symptoms {\em from the perspective of the functional~group}.
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\item The common~symptoms are now the fault mode behaviour of the functional~group.
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\item The common~symptoms are now the fault mode behaviour of the {\fg}. i.e. given the {\fg} as a `black box' the symptoms are the ways in which it can fail.
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\item A new `derived component' can now be created where each common~symptom, or lone test case is a failure~mode of this new component.
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\end{itemize}
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@ -83,7 +84,7 @@ Consider a functional group $FG$ with components $C_1$, $C_2$ and $C_3$.
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$$ FG = \{ C_1 , C_2 , C_3 \} $$
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Each component has a set of related fault modes (i.e. ways in which it can fail to operate correctly).
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Let us define the following failure modes for each component, defining a function $FM()$
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Let us define the following failure modes for each component, defining a function $fm()$
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that is passed a component and returns the set of failure modes associated with it
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\footnote{Base component failure modes are defined, often with
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statistics and evironmental factors in a variety of sources. \cite{mil1991}
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@ -92,7 +93,7 @@ statistics and evironmental factors in a variety of sources. \cite{mil1991}
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\ifthenelse {\boolean{paper}}
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{
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\subsection{Define Failure mode function FM}
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\subsection{Define Failure mode function fm}
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}
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{
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To re-cap from the definitions chapter \ref{chap:definitions}.
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@ -101,44 +102,44 @@ To re-cap from the definitions chapter \ref{chap:definitions}.
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Let the set of all possible components be $\mathcal{C}$
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and let the set of all possible failure modes be $\mathcal{F}$.
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We can define a function $FM$
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We can define a function $fm$
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\begin{equation}
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{FM} : \mathcal{C} \mapsto \mathcal{P}\mathcal{F}
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{fm} : \mathcal{C} \mapsto \mathcal{P}\mathcal{F}
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\end{equation}
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defined by (where $C$ is a component and $F$ is a set of failure modes):
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$$ FM ( C ) = F $$
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$$ fm ( C ) = F $$
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%\\
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e.g.
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%And for this example:
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$$ FM(C_1) = \{ a_1, a_2, a_3 \} $$
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$$ FM(C_2) = \{ b_1, b_2 \} $$
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$$ FM(C_3) = \{ c_1, c_2 \} $$
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$$ fm(C_1) = \{ a_1, a_2, a_3 \} $$
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$$ fm(C_2) = \{ b_1, b_2 \} $$
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$$ fm(C_3) = \{ c_1, c_2 \} $$
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\paragraph{Finding all failure modes within the functional group}
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For FMMD failure mode analysis, we need to consider the failure modes
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For fmMD failure mode analysis, we need to consider the failure modes
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from all the components in the functional group as a flat set.
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This can be found by applying function $FM$ to all the components
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This can be found by applying function $fm$ to all the components
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in the functional~group and taking the union of them thus:
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$$ FunctionalGroupAllFailureModes = \bigcup_{j \in \{1...n\}} FM(C_j) $$
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$$ FunctionalGroupAllFailureModes = \bigcup_{j \in \{1...n\}} fm(C_j) $$
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We can actually overload the notation for the function FM
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We can actually overload the notation for the function fm
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and define it for the set components within a functional group $FG$ (i.e. where $FG \subset \mathcal{C} $) thus:
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\begin{equation}
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FM : FG \mapsto \mathcal{F}
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fm : FG \mapsto \mathcal{F}
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\end{equation}
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Applied to the functional~group $FG$ in the example above:
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\begin{equation}
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FM(FG) = \{a_1, a_2, a_3, b_1, b_2, c_1, c_2 \}
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fm(FG) = \{a_1, a_2, a_3, b_1, b_2, c_1, c_2 \}
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\end{equation}
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This can be seen as all the failure modes that can affect the failure mode group $FG$.
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@ -146,7 +147,7 @@ This can be seen as all the failure modes that can affect the failure mode group
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\subsection{Analysis of the functional group failure modes}
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For this example we shall consider single failure modes.
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%For each of the failure modes from $FM(FG)$ we shall
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%For each of the failure modes from $fm(FG)$ we shall
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%create a test case ($g_i$). Next each test case is examined/analysed
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%and its effect on the functional group determined.
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@ -213,13 +214,13 @@ The next stage of the process could be applied automatically.
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Each common symptom becomes a failure mode of
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a newly created derived component. Let $DC$ be the newly derived component.
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This is assigned the failure modes that were derived from the functional~group.
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We can thus apply the function $FM$ on this newly derived component thus:
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We can thus apply the function $fm$ on this newly derived component thus:
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$$ FM(DC) = \{ SP1, SP2, SP3 \} $$
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$$ fm(DC) = \{ SP1, SP2, SP3 \} $$
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Note that $g_6$ has \textbf{not dissappeared from the analysis process}.
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Were the designer to have overlooked this test case, it would appear as a failure mode of the derived component.
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i.e. were it not to have been grouped in $SP3$, $ FM(DC)$ would have been $ \{ SP1, SP2, g_6 \}$.
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i.e. were it not to have been grouped in $SP3$, $ fm(DC)$ would have been $ \{ SP1, SP2, g_6 \}$.
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This is rather like a child not eating his lunch and being served it cold for dinner\footnote{Although I was only ever threatened with a cold dinner once, my advice to all nine year olds faced with this dilemma, it is best to throw the brussel sprouts out of the dining~room window while the adults are not watching!}!
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%
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\ifthenelse {\boolean{paper}}
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@ -252,12 +253,12 @@ $$
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% \bowtie(FG_{cfm}) = DC
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%\end{equation}
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%
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%or applying the function $FM$ to obtain the $FG_{cfm}$ set
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%or applying the function $fm$ to obtain the $FG_{cfm}$ set
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%
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Where DC is a derived component, and FG is a functional group:
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\begin{equation}
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\bowtie(FM(FG)) = DC
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\bowtie(fm(FG)) = DC
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\end{equation}
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@ -59,9 +59,10 @@ Top down fault isolation/finding techniques are described in \ref{NETWORKDECOMPO
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%% FMEA and FTA and safety engineering people used the term SUB_SYSTEM ALOT
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%% this study needs to use this term to keep the interested/in context.
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The term `sub-system' is typically used in top down methodologies.
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It has two equivalents in FMMD. The initial phase, where it is called
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a functional~group, and the analysed phase where it is called a derived~component.
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The term sub-system will be used alongside both functional~group and derived~component where necessary.
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It has two equivalents in FMMD.
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Both {\fg} and {\dc} correspond to the top doiwn concept of a `sub-system'.
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In FMMD a {\fg} becomes a {\dc} after analysis.
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The term sub-system will be used alongside both {\fg} and {\dc} where necessary.
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\subsection{Top-Down System De-Composition}
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@ -104,10 +105,10 @@ of the components.
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\subsection{Systems, functional groups, sub-systems and failure modes}
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It is helpful here to define some terms, `system', `functional~group', `component', `base~component', `symptom' and `derived~component/sub-system'.
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It is helpful here to define the terms, `system', `functional~group', `component', `base~component', `symptom' and `derived~component/sub-system'.
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These are listed in table~\ref{tab:symexdef}.
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A System, is really any coherent entity that would be sold as a product. % safety critical product.
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A System, is any coherent entity that would be sold as a product. % safety critical product.
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A sub-system is a system that is part of some larger system.
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For instance a stereo amplifier separate is a sub-system. The
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whole Sound System, consists perhaps of the following `sub-systems':
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@ -116,7 +117,7 @@ CD-player, tuner, amplifier~separate, loudspeakers and ipod~interface.
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%Thinking like this is a top~down analysis approach
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%and is the way in which FTA\cite{nucfta} analyses a System
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%and breaks it down.
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\paragraph{Sub-systems, {\fgs} and components}
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A sub-system will be composed of components, which
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may themselves be sub-systems. However each `component'
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will have a fault/failure behaviour and it should
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@ -132,10 +133,10 @@ component failure modes.
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Using the reasoning that working from the bottom up forces the consideration of all possible
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component failures (which can be missed in a top~down approach)
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component failures (which can be missed in a top~down approach \cite{faa}[Ch.9])
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we are presented with a problem. Which initial collections of base components should we choose?
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For instance in the CD~player example; to start at the bottom; we are presented with
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For instance in the CD~player example; if we start at the bottom, we are presented with
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a massive list of base~components, resistors, motors, user~switches, laser~diodes, all sorts!
|
||||
Clearly, working from the bottom~up, we need to pick small
|
||||
collections of components that work together in some way.
|
||||
@ -143,7 +144,7 @@ These are termed `functional~groups'. For instance the circuitry that powers th
|
||||
to illuminate the CD might contain a handful of components, and as such would make a good candidate
|
||||
to be one of the base level functional~groups.
|
||||
|
||||
|
||||
\paragraph{{\fg} to {\dc} process outline}
|
||||
In choosing the lowest level (base component) sub-systems we would look
|
||||
for the smallest `functional~groups' of components within a system.
|
||||
We can define a functional~group as a set of components that interact
|
||||
@ -172,7 +173,7 @@ The symptoms are the failure modes of this new `derived component'.
|
||||
|
||||
Electrical components have detailed datasheets associated with them. A useful extension of this could
|
||||
be failure modes of the component, with environmental factors and MTTF statistics.
|
||||
Currently this sort of failure mode information is generally only available for generic component types\cite{mil1991}.
|
||||
Currently this sort of failure mode information is generally only available for generic component types \cite{mil1991}.
|
||||
|
||||
%\vspace{0.3cm}
|
||||
\begin{table}[h]
|
||||
|
||||
Loading…
Reference in New Issue
Block a user