Template:Introduction of Mixed Weibull Distribution

Introduction
Besides the Weibull, exponential, normal and lognormal, there are other distributions that are used to model reliability and life data. However, these four represent the most prominent distributions in Weibull++. In this chapter, we will discuss other distributions that are used under special circumstances: the mixed Weibull, the generalized gamma, the Gumbel, the logistic and the loglogistic distributions. In this section, we will introduce mixed Weibull distribution. Other distributions will be discussed in later Chapters.

The mixed Weibull distribution (also known as a multimodal Weibull) is used to model data that do not fall on a straight line on a Weibull probability plot. Data of this type, particularly if the data points follow an S-shape on the probability plot, may be indicative of more than one failure mode at work in the population of failure times. Field data from a given mixed population may frequently represent multiple failure modes. The necessity of determining the life regions where these failure modes occur is apparent when it is realized that the times-to-failure for each mode may follow a distinct Weibull distribution, thus requiring individual mathematical treatment. Another reason is that each failure mode may require a different design change to improve the component's reliability [19].

A decreasing failure rate is usually encountered during the early life period of components when the substandard components fail and are removed from the population. The failure rate continues to decrease until all such substandard components fail and are removed. This corresponds to a decreasing failure rate. The Weibull distribution having $$\beta <1$$  is often used to depict this life characteristic.

A second type of failure prevails when the components fail by chance alone and their failure rate is nearly constant. This can be caused by sudden, unpredictable stress applications that have a stress level above those to which the product is designed. Such failures tend to occur throughout the life of a component. The distributions most often used to describe this failure rate characteristic are the exponential distribution and the Weibull distribution with $$\beta \approx 1$$.

A third type of failure is characterized by a failure rate that increases as operating hours are accumulated. Usually, wear has started to set in and this brings the component's performance out of specification. As age increases further, this wear-out process removes more and more components until all components fail. The normal distribution and the Weibull distribution with a $$\beta >1$$  have been successfully used to model the times-to-failure distribution during the wear-out period.

Several different failure modes may occur during the various life periods. A methodology is needed to identify these failure modes and determine their failure distributions and reliabilities. This section presents a procedure whereby the proportion of units failing in each mode is determined and their contribution to the reliability of the component is quantified. From this reliability expression, the remaining major reliability functions, the probability density, the failure rate and the conditional-reliability functions are calculated to complete the reliability analysis of such mixed populations.