diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex index ffb7d42..c18c9a8 100644 --- a/component_failure_modes_definition/component_failure_modes_definition.tex +++ b/component_failure_modes_definition/component_failure_modes_definition.tex @@ -339,23 +339,6 @@ Thus if the failure modes of a component $F$ are unitary~state, we can say $F \ An example of a component with an obvious set of ``unitary~state'' failure modes is the electrical resistor. Electrical resistors can fail by going OPEN or SHORTED. -%% CUNT -%% CUNT For a given resistor R we can apply the -%% CUNT the function $fm$ to find its set of failure modes thus $ fm(R) = \{R_{SHORTED}, R_{OPEN}\} $. -%% CUNT A resistor cannot fail with both conditions open and short active at the same time! The conditions -%% CUNT OPEN and SHORT are thus mutually exclusive. -%% CUNT Because of this, the failure mode set $F=fm(R)$ is `unitary~state'. -%% CUNT -%% CUNT -%% CUNT Thus because both fault modes cannot be active at the same time, the intersection of $ R_{SHORTED} $ and $ R_{OPEN} $ cannot exist. -%% CUNT -%% CUNT The intersection of these is therefore the empty set, $$ R_{SHORTED} \cap R_{OPEN} \eq \emptyset $$, -%% CUNT therefore -%% CUNT $ fm(R) \in \mathcal{U} $. -%% CUNT -%% CUNT - - For a given resistor R we can apply the the function $fm$ to find its set of failure modes thus $ fm(R) = \{R_{SHORTED}, R_{OPEN}\} $. @@ -366,7 +349,6 @@ Because of this, the failure mode set $F=fm(R)$ is `unitary~state'. Thus because both fault modes cannot be active at the same time, the intersection of $ R_{SHORTED} $ and $ R_{OPEN} $ cannot exist. -%%CUNT The intersection of these is therefore the empty set, $$ R_{SHORTED} \cap R_{OPEN} \eq \emptyset $$, The intersection of these is therefore the empty set, $$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$, therefore $ fm(R) \in \mathcal{U} $. @@ -578,12 +560,6 @@ components $C_j$ are in \end{itemize} %} -%% CUNT -%% CUNT \begin{equation} -%% CUNT |{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} -%% CUNT - {\sum{j \in J} {|FM({C_{j})}| \choose 2}}} -%% CUNT \label{eqn:correctedccps} -%% CUNT \end{equation} \begin{equation} |{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} - {\sum_{j \in J} {|FM({C_{j})}| \choose 2}} @@ -667,7 +643,7 @@ Thus the statistical sample space $\Omega$ for a component or derived~component $$ \Omega(C) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$ The failure mode set $F$ for a given component or derived~component $C$ is therefore -$$ F = \Omega(C) \backslash OK $$ +$$ F = \Omega(C) \backslash \{OK\} $$ The $OK$ statistical case is the largest in probability, and is therefore of interest when analysing systems from a statistical perspective.