From 4059c7f79c41751586cbfee4da61a8a4c8e18c93 Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Thu, 27 Oct 2011 17:14:15 +0100 Subject: [PATCH] Working on presentation while ill (have not eaten for 54+ hours) --- presentations/fmea/Makefile | 2 +- presentations/fmea/fmea_pres.tex | 102 ++++++++++++++++++++---------- presentations/fmea/fmmd_exm_h.dia | Bin 1451 -> 1971 bytes presentations/fmea/three_tree.dia | Bin 0 -> 6455 bytes presentations/fmea/three_tree.png | Bin 13487 -> 0 bytes 5 files changed, 68 insertions(+), 36 deletions(-) create mode 100644 presentations/fmea/three_tree.dia delete mode 100644 presentations/fmea/three_tree.png diff --git a/presentations/fmea/Makefile b/presentations/fmea/Makefile index b9c6128..6d0c6b8 100644 --- a/presentations/fmea/Makefile +++ b/presentations/fmea/Makefile @@ -1,5 +1,5 @@ -DIAPNG= component.png fmmd_env_op_uml.png fmmd_exm_h.png master_uml.png mvampcircuit.png mvamp.png n_inv_dc.png pd.png pd_euler2.png pd_euler.png +DIAPNG= three_tree.png component.png fmmd_env_op_uml.png fmmd_exm_h.png master_uml.png mvampcircuit.png mvamp.png n_inv_dc.png pd.png pd_euler2.png pd_euler.png %.png:%.dia dia -t png $< diff --git a/presentations/fmea/fmea_pres.tex b/presentations/fmea/fmea_pres.tex index 39084fc..d4b33e0 100644 --- a/presentations/fmea/fmea_pres.tex +++ b/presentations/fmea/fmea_pres.tex @@ -22,7 +22,9 @@ \frametitle{FMEA} %\tableofcontents[currentsection] \end{frame} - +This talk introduces Failure Mode Effects Analysis, and the different ways it is applied. +These techniques are discussed, and then +a refinement is proposed, which is essentially a modularisation of the FMEA process. \begin{frame} \frametitle{FMEA} \begin{itemize} @@ -129,13 +131,13 @@ We need to look at a large number of failure scenarios to do this completely (all failure modes against all components). This is represented in the equation below. %~\ref{eqn:fmea_state_exp}, where $N$ is the total number of components in the system, and -$cfm$ is the number of failure modes per component. +$f$ is the number of failure modes per component. \begin{equation} \label{eqn:fmea_single} - N.(N-1).cfm % \\ - %(N^2 - N).cfm + N.(N-1).f % \\ + %(N^2 - N).f \end{equation} \end{frame} @@ -159,11 +161,14 @@ $N^3$. \begin{equation} \label{eqn:fmea_double} - N.(N-1).(N-2).cfm % \\ - %(N^2 - N).cfm + N.(N-1).(N-2).f % \\ + %(N^2 - N).f \end{equation} $100*99*98*3=2,910,600$. +\pause + +.\\ The European Gas burner standard (EN298:2003), demands the checking of double failure scenarios (for burner lock-out scenarios). @@ -438,6 +443,7 @@ against all safe and dangerous failure probabilities. Again this is usually expressed as a percentage. $$ SFF = \big( \Sigma\lambda_S + \Sigma\lambda_{DD} \big) / \big( \Sigma\lambda_S + \Sigma\lambda_D \big) $$ +\pause SFF determines how proportionately fail-safe a system is, not how reliable it is ! \pause Weakness in this philosophy; \pause adding extra safe failures (even unused ones) improves the SFF. @@ -577,12 +583,12 @@ judged to be in critical sections of the product. % to do this completely (all failure modes against all components). % This is represented in equation~\ref{eqn:fmea_state_exp}, % where $N$ is the total number of components in the system, and -% $cfm$ is the number of failure modes per component. +% $f$ is the number of failure modes per component. % % \begin{equation} % \label{eqn:fmea_state_exp} -% N.(N-1).cfm % \\ -% %(N^2 - N).cfm +% N.(N-1).f % \\ +% %(N^2 - N).f % \end{equation} @@ -606,6 +612,8 @@ This creates an analysis hierarchy. \pause \item Collect Symptoms. \pause \item Create a '{\dc}', where its failure modes are the symptoms of the {\fg} from which it was derived. \pause \item The {\dc} is now available to be used in higher level {\fgs}. + \pause \item We can represent this process as a function which converts a {\fg} into a {\dc} and use the symbol $ \bowtie $ to represet it. + \pause \item i.e. $ \bowtie ( FunctionalGroup ) \rightarrow {DerivedComponent} $ \end{itemize} \end{frame} @@ -623,7 +631,7 @@ This creates an analysis hierarchy. We can return to the milli-volt amplifier as an example to analyse. \pause We can begin by looking for functional groups.\pause -The resistors would together to perform a fairly common function in electronics, that of the potential divider. +The resistors perform a fairly common function in electronics, that of the potential divider. So our first functional group is $\{ R1, R2 \}$.\pause We can now take the failure modes for the resistors (OPEN and SHORT EN298) and see what effect each of these failures will have on the {\fg} (the potential divider). @@ -652,6 +660,7 @@ Resistor and its failure modes represented as a directed graph. \begin{table} \begin{tabular}{|| l | l | c | c | l ||} \hline \textbf{Failure Scenario} & & \textbf{Pot Div Effect} & & \textbf{Symptom} \\ +\textbf{ / test case } & & \textbf{ } & & \textbf{ } \\ \hline FS1: R1 SHORT & & $LOW$ & & $PDLow$ \\ \hline FS2: R1 OPEN & & $HIGH$ & & $PDHigh$ \\ \hline @@ -782,36 +791,50 @@ how the levels work and converge to a top or system level. \caption{Functional Group Tree example} \label{fig:three_tree} \end{figure} + \end{frame} -\begin{frame} -\frametitle{FMMD - Failure Mode Modular De-Composition} + \begin{frame} + \frametitle{FMMD - Failure Mode Modular De-Composition} The fact FMMD analyses small groups of components at a time, and organises them into a hierarchy addresses the state explosion problem. \pause -Where $O$ is order -of complexity $O(N^2)$ in the equation below. +For FMEA where we check every component failure mode rigorously +against all the other components (we could call this \textbf{RFMEA}) +Where $N$ is the number of components, we can determine the order +of complexity $ O(N^2) $ thus. +% % \begin{equation} \label{eqn:fmea_single2} - N.(N-1).cfm % \\ - %(N^2 - N).cfm + N.(N-1).f \end{equation} +% +% %\end{frame} + \end{frame} + +\begin{frame} +\frametitle{FMMD - comparing number of checks RFMEA $\ldots$ FMMD} +%\end{frame} +If we consider $c$ to be the number of components in a {\fg}, $f$ is the number of failure modes per component, and +$L$ to be the number of levels in the hierarchy of FMMD analysis. + +%\begin{frame} We can represent the number of failure scenarios to check in an FMMD hierarchy with equation~\ref{eqn:anscen}. - +\pause \begin{equation} \label{eqn:anscen} - \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-1) + \sum_{n=0}^{L} {c}^{n}.c.f.(c-1) \end{equation} -Where $fgn$ is the number of components in each functional group, -and $cfm$ is the number of failure modes per component -and L is the number of levels, the number of -analysis scenarios to consider. +% Where $c$ is the number of components in each functional group, +% and $f$ is the number of failure modes per component +% and L is the number of levels, the number of +% analysis scenarios to consider. -~\ref{eqn:fmea_state_exp}. +%%~\ref{eqn:fmea_state_exp}. \end{frame} @@ -833,18 +856,18 @@ analysis scenarios to consider. % In other words, we have three components in our functional group, % and nine failure modes to consider. % So taking each failure mode and looking at how that could affect the functional group, -% we must compare each failure mode against the two other components (the `$fgn-1$' term). +% we must compare each failure mode against the two other components (the `$c-1$' term). % % For the one `zero' level FMMD case we are doing the same thing as FMEA type analysis % (but on a very simple small sub-system). % We are looking at how each failure~mode can effect the system/top level. % We can use equation~\ref{eqn:fmea_state_exp44} to represent % the number of checks to rigorously perform FMEA, where $N$ is the total -% number of components in the system, and $cfm$ is the number of failures per component. +% number of components in the system, and $f$ is the number of failures per component. % -% Where $N=3$ and $cfm=3$ we can see that the number of checks for this simple functional +% Where $N=3$ and $f=3$ we can see that the number of checks for this simple functional % group is the same for equation~\ref{eqn:fmea_state_exp22} % and equation~\ref{eqn:anscen}. % \clearpage @@ -855,7 +878,7 @@ analysis scenarios to consider. To see the effects of reducing `state~explosion' we can use an example. % with fixed numbers %for components in a functional group, and failure modes per component. -Let us take a system with 4 levels (with a top/system 0 level), +Let us take a system with 3 levels of FMMD analysis, with three components per functional group and three failure modes per component, and apply these formulae. Having 4 levels (in addition to the top zeroth level) @@ -865,14 +888,14 @@ $$ %\begin{equation} \label{eqn:fmea_state_exp22} 81.(81-1).3 = 19440 % \\ - %(N^2 - N).cfm + %(N^2 - N).f %\end{equation} $$ $$ %\begin{equation} % \label{eqn:anscen} - \sum_{n=0}^{4} {3}^{n}.3.3.(2) = 2178 + \sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720 %\end{equation} $$ \end{frame} @@ -884,10 +907,10 @@ $$ \begin{itemize} - \pause \item Thus for FMMD we needed to examine 2178 failure~modes against functionally adjacent components, and for traditional FMEA + \pause \item Thus for FMMD we needed to examine 720 failure~modes against functionally adjacent components, and for traditional FMEA type analysis methods, the number rises to 19440. \pause \item 19440 `checks' is not practical - \pause \item 2178 checks is alot, but... + \pause \item 720 checks is quite alot, but... \pause \item Modules in FMMD can be re-used... \end{itemize} % In practical example followed through, no more than 9 components have ever been required for a functional @@ -907,22 +930,31 @@ To determine all possible double simultaneous failures for rigorous FMEA \begin{equation} \label{eqn:fmea_state_exp2} - N.(N-1).(N-2).cfm % \\ - %(N^2 - N).cfm + N.(N-1).(N-2).f % \\ + %(N^2 - N).f \end{equation} + +Or express in terms of the level + +\begin{equation} + \label{eqn:fmea_state_exp2} + c^{L+1}.(c^{L+1}-1).(c^{L+1}-2).f % \\ + %(N^2 - N).f +\end{equation} + \pause The FMMD case (equation~\ref{eqn:anscen2}), is cubic within the functional groups only, not all the components in the system. \begin{equation} \label{eqn:anscen2} - \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-1).(fgn-2) + \sum_{n=0}^{L} {c}^{n}.c.f.(c-1).(c-2) \end{equation} \end{frame} \begin{frame} \frametitle{FMMD - Failure Mode Modular De-Composition} \textbf{Traceability} -Because each reasoning stage contains associations ($FailureMode \mapsto Symptom$) +Because each reasoning stage contains associations ($FailureMode \rightarrow Symptom$) we can trace the `reasoning' from base level component failure mode to top level/system failure, by traversing the tree/hierarchy. 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