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+\begin{document} + +\section{F.M.E.A.} + +\begin{frame} +\begin{itemize} + \pause \item Failure + \pause \item Mode + \pause \item Effects + \pause \item Analysis +\end{itemize} + +\end{frame} + +% % \begin{itemize} +% \item Failure +% \item Mode +% \item Effects +% \item Analysis +% \end{itemize} + + +\subsection{FMEA basic concept} + +\begin{frame} +\begin{itemize} + \pause \item \textbf{F - Failures of given component} Consider a component in a system + \pause \item \textbf{M - Failure Mode} Look at one of the ways in which it can fail (i.e. determine a component `failure~mode') + \pause \item \textbf{E - Effects} Determine the effects this failure mode will cause to the system we are examining + \pause \item \textbf{A - Analysis} Analyse how much impact this symptom will have on the environment/people/the system its-self +\end{itemize} +\end{frame} + + +\begin{frame} + +Example: Let us consider a system, in this case a milli-volt reader, consisting +of instrumentation amplifiers connected to a micro-processor +that reports its readings via RS-232. +Let us perform an FMEA and consider how one of its resistors failing could affect +it. +For the sake of example let us choose a resistor in an OP-AMP +reading the milli-volt source and that if it were to go open, we would have a gain +of 1 from the amplifier. + +\begin{itemize} + \pause \item \textbf{F - Failures of given component} The resistor could fail by going OPEN or SHORT (EN298 definition). + \pause \item \textbf{M - Failure Mode} Consider the component failure mode OPEN + \pause \item \textbf{E - Effects} This will disconnect the feedback loop in the amplifier causing a LOW READING + \pause \item \textbf{A - Analysis} The reading will be out of normal range, and we will have an erroneous milli-volt reading +\end{itemize} +\end{frame} + + + +\begin{frame} +Note here that we have had to look at the failure~mode +in relation to the entire circuit. +We have used intuition to determine the probable +effect of this failure mode. +We have not examined this failure mode +against every other component in the system. +Perhaps we should.... this would be a more rigorous and complete +approach in looking for system failures. + +\end{frame} + + +\begin{frame} +Consider the analysis +where we look at all the failure modes in a system, and then +see how they can affect all other components within it. + + +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 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. + +\begin{equation} + \label{eqn:fmea_single} + N.(N-1).cfm % \\ + %(N^2 - N).cfm +\end{equation} + +This would mean an order of $N^2$ number of checks to perform +to perform `rigorous~FMEA'. Even small systems have typically +100 components, and they typically have 3 or more failure modes each. +$100*99*3=29,700$. +\end{frame} + + + + +\begin{frame} + +For looking at potential double failure scenarios (two components +failing within a given time frame) and the order becomes +$N^3$. + +\begin{equation} + \label{eqn:fmea_double} + N.(N-1).(N-2).cfm % \\ + %(N^2 - N).cfm +\end{equation} + +$100*99*98*3=2,910,600$. + +The European Gas burner standard (EN298:2003), demands the checking of +double failure scenarios (for burner lock-out scenarios). + +\end{frame} + +\section{Production FMEA : 1940's to present} + +\begin{frame} + + +Describe process, the probability times the cost +\end{frame} + + + + +\subsection{Production FMEA : Example Ford Pinto : 1975} +\begin{frame} + +\end{frame} + + + +\subsection{FMEA and complexity of each failure scenario analysis} +\begin{frame} + +Consider the FMEA type methodologies +where we look at all the failure modes in a system, and then +see how they can affect all other components within it, +to determine its system level symptom or failure mode. +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 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. + +\begin{equation} + \label{eqn:fmea_state_exp} + N.(N-1).cfm % \\ + %(N^2 - N).cfm +\end{equation} + + +The FMMD methodology breaks the analysis down into small stages, +by making the analyst choose functional groups, and then when analysed the groups +are treated as components to be used for a higher stage. +This is designed to address the state explosion (where $O$ is order +of complexity) $O=N^2$ inherent in equation~\ref{eqn:fmea_state_exp}. +\end{frame} + + +We can view the functional groups in FMMD as forming a hierarchy. +If for the sake of example we consider each functional group to +be three components, figure~\ref{fig:three_tree} shows +how the levels work and converge to a top or system level. + +\begin{figure} + \centering + \includegraphics[width=300pt]{./three_tree.png} + % three_tree.png: 780x226 pixel, 72dpi, 27.52x7.97 cm, bb=0 0 780 226 + \caption{Functional Group Tree example} + \label{fig:three_tree} +\end{figure} + +\clearpage +We can represent the number of failure scenarios to check in an FMMD hierarchy +with equation~\ref{eqn:anscen}. + +\begin{equation} + \label{eqn:anscen} + \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-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 is show in equation~\ref{eqn:anscen}. + + +So for a very simple analysis with three components forming a functional group where +each component has three failure modes, we have only one level (zero'th). +So to check every failure modes against the other components in the functional group +requires 18 checks. + +\begin{equation} + \label{eqn:anscen2} + \sum_{n=0}^{0} {3}^{0}.3.3.(3-1) = 18 +\end{equation} +\clearpage + + + +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). + +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_exp} 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. + + + +Where $N=3$ and $cfm=3$ we can see that the number of checks for this simple functional +group is the same for equation~\ref{eqn:fmea_state_exp} +and equation~\ref{eqn:anscen}. +\clearpage + +\section{Example} + +To see the effects of reducing `state~explosion' we need to look at a larger system. +Let us take a system with 3 levels and apply these formulae. +Having three levels (in addition to the top zero'th level) +will require 81 base level components. + +$$ +%\begin{equation} + \label{eqn:fmea_state_exp} + 81.(81-1).3 = 19440 % \\ + %(N^2 - N).cfm +%\end{equation} +$$ + +$$ +%\begin{equation} + % \label{eqn:anscen} + \sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720 +%\end{equation} +$$ + +Thus for FMMD we needed to examine 720 failure mode scenarios, and for traditional FMEA +type analysis methods 19440. +% In practical example followed through, no more than 9 components have ever been required for a functional +% group and the largest known number of failure modes has been 6. +% If we take these numbers and double them (18 components per functional group +% and 12 failure modes per component) and apply the formulas for a 4 level analysis +% (i.e. + +\clearpage +Note that for all possible double simultaneous failures the equation~\ref{eqn:fmea_state_exp} becomes +equation~\ref{eqn:fmea_state_exp2} essentially making the order $N^3$. +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:fmea_state_exp2} + N.(N-1).(N-2).cfm % \\ + %(N^2 - N).cfm +\end{equation} + +\begin{equation} + \label{eqn:anscen2} + \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-1).(fgn-2) +\end{equation} + + +\end{document}