Failure distribution

The failure distribution describes the probability distribution over time with which materials, electrical components or mechanical components fail.

Typical failure distributions are:

the general distribution of failure
the bathtub curve
the Weibull distribution
the logarithmic normal distribution

General failure distribution

The general failure distribution is the exponential failure distribution, which is based on the exponential density function . This leads to a constant risk rate.

{\ displaystyle F (t) = \ int _ {0} ^ {t} \ lambda e ^ {- \ lambda x} \, dx = 1-e ^ {- \ lambda t} \!}

Bathtub effect

The bathtub effect (also known as bathtub curve ) is used to describe reliability in technology, although the general concept is also applicable to humans. The bathtub curve describes the graph that resembles a cross section of a bathtub , a specific function of the mean operating time between failures , which contains three parts:

The first part is a decreasing failure rate known as early failures or infant mortality.
The second part is a constant failure rate known as random failures.
The third part is an increasing disorder rate, known as aging.

Or, as a ( non-differentiable ) function of time t :

{\ displaystyle y (t) = {\ begin {cases} c_ {0} -c_ {1} t + \ lambda, & 0 \ leq t \ leq c_ {0} / c_ {1} \\\ lambda, & c_ {0 } / c_ {1} <t \ leq t_ {0} \\ c_ {2} (t-t_ {0}) + \ lambda, & t_ {0} <t \ end {cases}} \!}

However, not all products follow this function.

Device life

The device service life can typically be shown in a diagram of a bathtub curve as the failure rate of technical devices or systems depending on the service life.

Bathtub curve

The X-axis shows the time span from the beginning of the commissioning of the device, while the Y-axis shows the failure rate per unit of time. Reliable data are only possible if a statistically sufficiently large number of devices is used. Over the life cycle of a product , the failure rate curve often takes the form of a bathtub : The failure rate is particularly high at the beginning (I) and at the end (III) of the service life , while it is lower and fairly constant in the period in between (II). Phase I is typically shorter than the following two, whereby phase II can have very different durations, depending on the design of the product. ${\ displaystyle t}$ ${\ displaystyle \ lambda}$

The reasons for this effect are that design, production or material defects are often noticed right at the start of operation or lead to damage. The devices not affected by such defects function reliably, which is why the curve flattens out in the middle. At the end of the service life, gradually increasing failures or damage occur. These are mainly due to aging and wear and tear . When this area is reached, either major repairs or replacements are necessary. End of life tests can be used as a basis to determine the service life , e.g. B. the Highly Accelerated Life Test .

In order to reduce the number of early failures for the user, devices can be subjected to artificial aging before production is complete. The devices are z. B. operated with increased ambient temperatures, increased operating voltage (" burn-in ") or a vibration test.

Devices are developed to achieve a predetermined service life . The end of the estimated useful life is ideally at the end of Phase II, but it can be shifted forwards or backwards due to fluctuations in equipment and production quality.

The device failure in phase III occurs due to the failure of various parts. To increase the service life, it is important to know which parts fail first in order to specifically improve these parts. An evaluation of any repairs is necessary. Information about errors that have occurred must systematically go from customer service (repairs) to the development department so that they do not incorporate the same weak points when developing new devices. If this is not done intensively enough, weak points can persist across product generations.

Parameters of the failure distribution

swell

↑ Larry George: The Bathtub Curve Doesn't Always Hold Water. (Registration required)

[1] Larry George: The Bathtub Curve Doesn't Always Hold Water. (Registration required)