Product Maturity 1. Franck Bayle

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Figure 1.3. Failure instants for β = 3 and β = 10

It can be noted that failure instants are more dispersed for β = 3 (on the left) than for β = 10 (on the right). On the other hand, for β = 1, equation [1.2] is written as:

      [1.3]

This represents the exponential distribution law modeling catastrophic failures. The failure rate for this category of failures is constant, which means that failure instants do not depend on the elapsed time. This specificity of the exponential law is known as the “memoryless property” (it is the only continuous law with this property). Indeed, returning to the analogy with human beings, a catastrophic failure is, for example, a car accident occurring when a driver cuts off another driver. This “failure” does not depend on the distance traveled, but is due solely to the recklessness of another person. This is entirely different from an aging failure, for which the failure instant directly depends on the distance traveled, because this relates to driver fatigue.

      It is important to note that the concept of maturity has no qualitative meaning for non-maintained products. Indeed, the objective of reliability is a probability of success; the mission is achieved by the survival function, which for a Weibull distribution is defined as:

      [1.4]

      This survival function – and this is the case regardless of the law used – is a strictly decreasing function of time. Therefore, the concept of constant reliability is not applicable. For most non-maintained industrial applications, exponential distribution is preferred to Weibull distribution; this is because the reliability objective is a probability of achieving the mission, whose value is obviously high (generally such that R ∈[90% ; 99%].

      In this case, we can return to an exponential distribution because, for these values of the survival function, it is conservative, with respect to a Weibull distribution, whose shape parameter is greater than 1. Indeed, from a mathematical perspective, the ratio of the two survival functions can be calculated as follows:

      [1.5]

      with β > 1 and Tm = mission duration.

      To obtain a sufficiently high probability of success in the mission (survival function) requires Tm/η ≪ 1. Consequently, β being greater than 1,

      [1.6]

      Since Tm/η is greater than

      Another, more physical way to view this result is to remember that the shape parameter β represents the dispersion of time until failure. The greater β is, the less dispersed the time until failure. Since the Weibull shape parameter is > 1, the corresponding failure instants are less dispersed around the scale parameter η.

      Consequently, failures following an exponential distribution with an identical scale parameter occur earlier than those following the Weibull distribution. Therefore, the survival function of the exponential distribution at any instant “t” is weaker than that of the Weibull distribution, which proves the conservative character of this approach.

1.2. Temperature effect

      Temperature is systematically involved in component failure mechanisms. The Arrhenius law is generally used in order to model its effect on the reliability of components. Based on an empirical research method, that is, studied through a number of experiments, the Arrhenius law is used to model the variation in the speed of certain chemical reactions under the influence of temperature. With respect to the previously described Weibull law, the following formulation is obtained:

      [1.7]

with
	Ea is the activation energy.
	Kb is the Boltzmann constant.

1.3. Effect of maintenance

In most industrial applications, the focus is on the reliability observed on the ground, which must take into account the maintenance actions carried out. Maintenance can take several forms, depending on the level at which it is being performed (components, products, etc.). At the component level, maintenance is generally referred to as “perfect”, also known as “corrective maintenance”, since defective components are replaced with new ones.

      At the product level, maintenance may be referred to as “preventive”. This is the case with cars, for example, where engine oil, various filters, etc., are changed on a regular basis without any failures having been observed. СКАЧАТЬ