Weibull three-parameter

Alloy Number Tested Number Failed Two-Parameter Weibull Three-Parameter Weibull ... 

Weibull distribution This distribution has been useful in a variety of reliability applications. The Weibull distribution is described by three parameters, and it can assume many shapes depending upon the values of the parameters. It can be used to model decreasing, increasing, and constant failure rates. 

The Weibull distribution, illustrated in Figure 2, is most attractive, as it permits characterization of all typical cases of a PDF and CDF with only three parameters (2-4) ... 

Moore and Caux (1997), in the same paper examining the relationships between hypothesis testing and effects levels, also characterized some important properties of the regression approach. One of the critical questions is which model to use and how much of a difference it makes. Logistic, probit, Weibull, and three parameter logistic models incorporating a slope parameter were compared in these data sets. The differences in using these models for extrapolation depend upon the structure of the data set. 

The Weibull distribution function contains three parameters ... 

In 1951, Weibull [5] proposed the use of a mathematical function that, by changing the value of its three parameters, a, and x> can cover many shapes and can approximate to a normal distribution under certain conditions. The Weibull distribution is defined by ... 

Due to its versatility, the Weibull distribution is very well suited for many data evaluation applications. However, the exact determination of the three parameters which are its determinants sometimes creates considerable difficulties, in particular in the case of insufficiently large data... 

Where /(t) stands for probability density function, /(t) >0. From this it is clear that it is a case of three-parameter model. Shape and locations can be varied in accordance with selection of range of parameter for various variables viz. rj (>0), defines the bulk of the distribution parameter (scale), 8 (>0), determines shape (also often called Weibull slope) and y, defines the location of the distribution in time and has wide variation. 

This parameter allows for a lower limit stress a below which the probabihty of failure is zero. The underlying distribution is called three-parameter Weibull distribution. In the following, we will usually assume <7i = 0, however. 

However, to determine the field operating life, one needs to know the life of the parts in lab testing. The raw data obtained from the lab test could be fitted with a failure distribution to determine the mean life of the parts. Typical failure distributions include Weibull, Normal, Lognormal, and Exponential. For wear-out type of failures, the Weibull and Lognormal distributions are usually used, with Weibull being the most common. Weibull distributions are lowest value distributions derived from the weakest-link theory. Solder joint interconnects can be considered as connected in series. Usually, the failure of one joint at a critical location could cause the entire device to fail. The joints that fail early are usually located at the highest stress locations in the package. Devices with more resilient joints would not fail early. A Weibull distribution captures the minimum solder joint life, and the shape parameter captures the quality of the joints as a function of their construction and the applied stress. There are different types of Weibull distributions one-parameter, two-parameter, and three-parameter. The three-parameter Weibull Probability Distribution Function (PDF) is as shown in Eq.59.1. 

For the CARES/Lz/e reliability analysis the Weibull parameters obtained from the three-point flexure bars were used to predict the strength response of the disks. Utilizing Eq. (2) with maximum likelihood analysis and assuming volume flaws a Weibull modulus my = 11.96, a characteristic strength uej/= 612.7 MPa, and a Weibull scale parameter uok = 453.8 MPa mm / was obtained for the flexure... 

The usual description of the failure behaviour of mechanical or automotive components is provided by the three-parameter Weibull distribution (Bertsche 2008), with the pdf... 

Still, the short-term distribution of the peaks is found to be well described by a three-parameter exponential distribution. This distribution was found to provide the best fit as compared to the Weibull and the Lognormal models. These latter distributions are frequently being applied in order to represent ice loading magnitudes. The conditional three-parameter model with the ice thickness (i.e. the constant h) as an indexing parameter is expressed as follows ... 

Figure 17. Variation in shape of three-parameter Weibull distribution for a proportionality factor of 0.5 Inverse intensity parameter of first exponential distribution varies between 0.2 and 1.2. Inverse intensity parameter of second exponential distribution varies between 0.02 and 0.12.

The three parameters m, a, and (Tq are constants for the material m is the Weibull modulus and (T is the maximum value of stress under which the specimen will not fail, which is usually set to zero. Rearranging, setting <7equal to the largest value of stress in the... 

Equations 7.3 and 7.4 are called the three-parameter Weibull functions. The three parameters are o, Oq, and m. For ceramics, a two-parameter form is used. In this, the a is considered to be zero. Then the function can be written as ... 

The failure data of area array solder joints (e.g., flip chips, ceramic ball grid arrays) are often fitted to a lognormal distribution. The choice of the distribution selected may depend on the confidence level chosen for the fit. A lognormal distribution may satisfactorily represent failure data at a lower confidence level in some cases. There are two variations for WeibuU distributions the two-parameter distribution and the three-parameter distribution. The third parameter, called the location parameter, represents the minimum time-to-failure. Sometimes the failure data exhibit a slight curvature at a lower failure probability deviating from a two-parameter Weibull distribution. A three-parameter Weibull distribution can be utilized to better fit the data. 

Although the three-parameter Weibull distributions enable the incorporation of early fatigue failures into the analysis, it is crucial to ensure that the same failure mechanism is operative to cause failures represented later in the distribution. A detailed failure analysis of early failures as well as later failures enables this determination. Otherwise, the distribution may be purely a statistical artifact without a common physics-of-failure basis. Any reliability evaluation and inferences based upon it are incomplete without supporting evidence from failure analysis. Although a three-parameter Weibull distribution may provide a more accurate estimate of the time for early failures, a larger sample size for evaluation may also be required. Hence, an investigator must strike a balance by weighing all the aspects in the choice of a distribution to be used. 

Some of the three-parameter distribution functions can fit distillation data with good accuracy Weibull and Gamma distribution have standard deviations of 0.86% and 1.07% and are ranked within the best five. 

The three-parameter Weibull cumulative distribution fimction, F t), that predicts the cumulative probability of failure up to a specific time, t, is mathematically expressed by Equation 6.11. The probability density fimction,/(t), which is a derivative of the cumulative distribution fimction, is expressed by Equation 6.12 ... 

Three statistieal distributions that are eommonly used in engineering are the Normal (see Figure 4.3(a)), Lognormal (see Figure 4.3(b)) and Weibull, both 2-and 3-parameter (see Figure 4.3(e) for a representation of the 2-parameter type). 

Three ranges of values of n were considered, >1, 0.7—1.0 and <0.7. When n> 1, and particularly when 3 < n < 4, the Weibull distribution readily reduces to a normal distribution if the Erofe ev function is symmetrical about a = 0.5. [The Weibull distribution is symmetrical for n = 3.26, i.e. (1 — In 2)-1, and the inflection point varies only slowly with n.] Thus, under these conditions (3 < n < 4 and symmetry about a = 0.5), we may derive the parameters of the corresponding normal distribution (where p defines the half-life of the reaction and the dispersion parameter, a, is a measure of the lack of homogeneity of the surface centres), viz. 

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