# Reliability of Machine Components

## Mean Time Between Failure - *MTB* and reliability of machine components and systems

Reliability characterizes components or system of components by the probability they will perform the desired functions for a given time.

In general -

- more components and/or more complicated systems reduces reliability
- simpler systems with few components increases reliability

Reliability equations:

### Reliability

Reliability at a given time:

R = e^{-}^{λt}(1)

where

R = reliability. Values between 0 - 1 where value 1 indicates 100% live components and value 0 indicates 0% live components.

λ = proportional failure rate - a failure rate expressed as a proportion of initial number of live components - N_{o}

t = time

### Unreliability

The connection between reliability and unreliability:

R + Q = 1 (2)

where

Q = unreliability. Values between 0 - 1 where value 1 indicates 0% live components and value 0 indicates 100% live components.

(1) and (2) can be used to express unreliability

Q = 1 -e^{-}^{λt}(3)

### Number of Live Components

The number of live surviving components in a system at a given time:

N_{s}= N_{o}e^{-}^{λt}(4)

where

N_{s}= number of live surviving components at time t

N_{o}= initial number of live surviving components at time zero

### Number of Failure Components

The number of failure dead components in a system at a given time:

N_{s}= N_{o}(1 - e^{-}^{λt}) (5)

where

N_{s}= number of live surviving components at time t

N_{o}= initial number of live surviving components at time zero

### Mean Time Between Failures - *MTBF*

Mean time between failures - MTBF:

m = 1 / λ (6)

where

m = MTBF - Mean Time Between Failure

- MTTF - Mean Time To Failure is also used and equal to MTBF

Mean Time Between Failure (MTBF) can be determined by rating Total Surviving Hours against Number of Failures as

m = t_{s}/ n_{f}(7)

where

t_{s}= total surviving hours

n_{f}= number of failures

Combining (5) with the formulas for reliability and more

R = e^{-}^{t/m}(1b)

Q = 1 -e^{-}^{t/m}(3b)

N_{s}= N_{o}e^{-}^{t/m}(4b)

N_{s}= N_{o}(1 - e^{-}^{t/m}) (5b)