Reliability of Machine Components

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 - at time t

t = time (hours) - Note! other units can be used as long as the use is consistent through the calculations.

The failure rate at time t can be expressed as

λ = N_F / (N_o t) 

    = (N_o - N_s) / (N_o t)                 (2)

where

N_F = N_o - N_s = number of failing components at time t

N_s = number of live surviving components at time t

N_o = initial number of live components at time zero

Example - Failure Rate and Reliability

A car manufacturer sells 400000 cars of a certain model in one year. During the the first tree years the owners of 50000 of these cars experience major failures. The failure rate can be calculated as

λ = (50000 cars)/ ((400000 cars) (3 years))

  = 0.042 (per year)

The reliability of a model of this car within three years can be calculated as

 R = e^{-(0.042 1/year) (3 year)}      

    = 0.88

    = 88 %

Unreliability - the Probability for a Device to Fail

The connection between reliability and unreliability:

R + Q = 1                              (3)

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                               (4)

Example - Unreliability

The unreliability of the car model in the example above within three years can be calculated as

 Q = 1 - e^{-(0.042 1/year) (3 year)}      

    = 0.12

    = 12 %

Number of Live Components

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

N_s = N_o e^-λt                            (5)

Number of Failure Components

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

N_s = N_o (1 - e^-λt)                        (6)

Mean Time Between Failures - MTBF

Mean time between failures - MTBF:

MTBF = 1 / λ                         (7)

where

MTBF = Mean Time Between Failure (hours)

• MTTF - Mean Time To Failure is an alternative to MTBF

Example - Mean Time Between Failure

From the example above the failure rate is 0.42 per year. MTBF can be calculated as

MTBF = 1 / (0.042 1/year)

        = 23.8 years

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

MTBF = t_s / N_F                      (8)

where

t_s = total surviving hours (hours)

Combining (5) with the formulas for reliability and more

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

Q = 1 -e^-t/m                       (4b)

N_s = N_o e^-t/m                      (5b)

N_f = N_o (1 - e^-t/m)                     (6b)

MTBF vs. MTTF vs. MTTR

• MTBF - Mean Time Between Failures > commonly used to determine average time between failures
• MTTF - Mean Time To Failure > commonly used for replaceable products that cannot be repaired
• MTTR - Mean Time To Repair > commonly used to determine how long it will take to get a failed product running again

MTBF and MTTF are often used interchangeably.

Systems Reliability

There are two basic types of reliability systems - series and parallel - and combinations of them.

• in a series system - all devices in the system must work for the system to work
• in a parallel system - the system works if at least one device in the system works

Reliability of Systems in Series

Reliability of systems in series can be expressed as

R_s = R₁ R₂                        (9)

where

R_s = system reliability

R_1,2 = subsystem reliability

Example - Reliability of Systems in Series

From the example above the reliability of a car over three year is 0.88. If we depends on two cars for a mission - whatever it is - the reliability for our mission will be

R_s = 0.88 0.88

    = 0.77

    = 77 %

Reliability of Systems in Parallel

Reliability of systems in parallel can be expressed as

R_s = R₁ Q₂ + R₂ Q₁ + R₁ R₂                       (10)

where

Q_1,2 = (1 - R_1,2) subsystem unreliability

Example - Reliability of Systems in Parallel

From the example above the reliability of a car over three year is 0.88. If it is enough with one of our two cars for our mission - the reliability for our mission can be calculated as

R_s = 0.88 (1- 0.88) + 0.88 (1- 0.88) + 0.88 0.88

    = 0.99

    = 99 %

Note! - two systems in parallel with 99 % reliability vs. one system alone with reliability 88 % shows the power of redundancy.