# Weibull Distribution

Last edit: 28/02/2024

It is now clear that, in Functional Safety, the failure rate of any component has to be constant: the issue are components subject to wear, like contactors and solenoid valves, since their failure rates are usually not constant. Therefore, the exponential curve is not helpful to model their life distribution: that is where the Weibull distribution comes in.

The Weibull distribution is one of the most widely used Life Distributions in Reliability analysis. The distribution is named after the Swedish professor Waloddi Weibull (1887-1979), who developed the distribution for modelling the strength of materials.

The Weibull distribution is very flexible and can, through an appropriate choice of parameters, model many types of failure rate behaviours. It is therefore used to model the failure behaviour of electromechanical components.

## The Probability Density Function

The Weibull Probability Density Function is the following:

f(t)=β∙(t^(β-1))/(η^β)∙e^(-(t/η)^β)

• η is called the Life Characteristics
• β is referred as the shape parameter

In figure the Weibull distribution is plotted for η=1 and for some values of β.

## The Cumulative Density Function

The Weibull Cumulative Density Function is the following:

F(t)=1- e^(-(t/η)^β )

Please notice that when t = η

F(η)=1-e^(-(η/η)^β )=1-e^(-(1)^β )=1-e^(-1)=0,63

Therefore, regardless of the distribution shape parameter β, when t = η, the Probability of unavailability F(t) of the component = 63%.

The parameter η is defined as the characteristic lifetime of the distribution.

## The Instantaneous Failure Rate

Finally, the Instantaneous Failure Rate is the following:

λ(t)=f(t)/(1-F(t))=β∙(t^(β-1))/(η^β)

When β =1, the failure rate is constant and equal to:

λ= 1/η

In this case, the Weibull distribution is identical to the exponential one.

When β < 1 the failure rate decreases with time. Both electronic and mechanical systems may initially have high failure rates. Manufacturers conduct production process control, production acceptance tests, “burn-in,” or reliability stress screening (RSS), to prevent early failures before delivery to customers. Therefore, shape parameters of less than one indicates the following:

• lack of adequate process control;
• inadequate burn-in or stress screening;
• production problems, mis-assembly, poor quality control;
• overhaul problems;
• mixture of populations;
• run-in or wear-in.

Many electronic components during their useful life show a decreasing instantaneous failure rate, thus featuring shape parameters less than 1. Preventive maintenance on such a component is not appropriate, as old parts are better than new.

When β > 1 the failure rate increases with time. That behaviour is attributed, first of all, to components in the wear-out, or end of life, phase. Some typical examples of these cases are:

• wear;
• corrosion;
• crack propagation;
• fatigue;
• moisture absorption;
• diffusion;
• evaporation (weight loss);
• damage accumulation.

Design measures have to ensure that those phenomena do not significantly contribute to the probability of product failure during the expected operational life, however that is typically the behaviour of Contactors and Solenoid valves during their entire life.

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