Systemfunktion (Strukturformel)
$$
S = B \wedge H \wedge R \wedge S \wedge (K_1 \vee K_2 \vee K_3)
$$
Fehlerbaum: Strukturbaum der Negation der Systemfunktion
$$
\begin{array}{ll}
\neg S &=\neg\left(B \wedge H \wedge R \wedge S \wedge\left(K_{1} \vee K_{2} \vee K_{3}\right)\right) \\
&=\neg B \vee \neg H \vee \neg R \vee \neg S \vee \neg\left(K_{1} \vee K_{2} \vee K_{3}\right) \\
&=\neg B \vee \neg H \vee \neg R \vee \neg S \vee\left(\neg K_{1} \wedge \neg K_{2} \wedge \neg K_{3}\right)
\end{array}
$$
Funktionswahrscheinlichkeit
1-aus-n Systeme (Bsp: n =3)
Das System fällt aus $\Leftrightarrow$ Alle 3 Komponenten $K_1$ bis $K_3$ fallen aus
System funktionfähig $\Leftrightarrow$ Mindeste $n$ aus $m$ Komponente funktionfähig
Bsp: 2-aus-3 System
System funktionsfähig, wenn die Komponenten 1&2, 1&3, 2&3 oder 1&2&3 funktionsfähig sind
System NICHT funktionsfähig, wenn nur Komponente 1, 2, oder 3 funktionsfähig ist
Funktionsfähigkeit
$$
\varphi_{m}^{n}=\sum_{k=n}^{m}\left(\begin{array}{c}
m \\
k
\end{array}\right) * \varphi(K)^{k} *(1-\varphi(K))^{(m-k)}
$$
$\varphi(K)$: Funktionswahrscheinlichkeit der Komponenten
Bsp: Aufg1 2(a)
Systeme mit Mehrheitsentscheider
$$
\varphi_{m}^{n}= \varphi(V) \sum_{k=n}^{m}\left(\begin{array}{c}
m \
k
\end{array}\right) * \varphi(K)^{k} *(1-\varphi(K))^{(m-k)}
$$
$\varphi(V)$: Funktionswahrscheinlichkeit des Entscheiders (Voter)
Failure metrics
MTTF (Mean Time to Failure)
mittlere Funktionszeit
Einheit, die nicht instand gesetzt wird (Ersatz)
MTBF (Mean Time between Failures)
mittlere Zeit zwischen Ausfällen
Einheit, die wieder instand gesetzt wird
Betriebszeit zwischen zwei aufeinanderfolgenden Ausfällen
MTTR (Mean Time to Repair/Recover)
mittlere Reparaturzeit
Für über die Zeit konstante Ausfallraten gilt außerdem:
refers to the amount of time required to repair a system and restore it to full functionality.
MTTR clock starts ticking when the repairs start and it goes on until operations are restored. This includes repair time, testing period, and return to the normal operating condition
Calculation:
$$
\mathrm{MTTR}=\frac{\text { total maintenance time }}{\text { total number of repairs }}
$$
E.g.: a pump that fails three times over the span of a workday. The time spent repairing each of those breakdowns totals one hour. In that case, MTTR would be 1 hour / 3 = 20 minutes.
Note: MTTR can also refer to Mean Time To Recovery
Mean Time To Recovery is a measure of the time between the point at which the failure is first discovered until the point at which the equipment returns to operation. So, in addition to repair time, testing period, and return to normal operating condition, it captures failure notification time and diagnosis.
Mean Time Between Failures (MTBF)
measures the predicted time that passes between one previous failure of a mechanical/electrical system to the next failure during normal operation.
helps to predict how long an asset can run before the next unplanned breakdwon happens.
Note: the repair time is not included in the calculation of MTBF.
Calculation:
$$
\mathrm{MTBF}=\frac{\text { total operational time }}{\text { total number of failures }}
$$
Mean Time To Failure
basic measure of reliability used for non-repairable systems.
represents the length of time that an item is expected to last in operation until it fails (commonly refer to as the lifetime of any product or a device).
MTBF Vs. MTTF:
MTBF is used only when referring to repairable items, MTTF is used to refer to non-repairable items.
Calculation
$$
\mathrm{MTTF}=\frac{\text { total hours of operational }}{\text { total number of units }}
$$
E.g.: we tested three identical pumps until all of them failed. The first pump system failed after eight hours, the second one failed at ten hours, and the third failed at twelve hours. MTTF in this instance would be (8 + 10 + 12) / 3 = 10 hours.
Alterung, Ausfall
Alterungseffekte
Vereinfachtes Modell: Konstante Ausfallrate
Reale Systeme: ariable Ausfallwahrscheinlichkeit über Zeit
