Tolerance distribution models

Tolerance Distribution Model. This model assumes that each member of the population at risk has an individual tolerance below which no response will be produced and that these tolerances vary in the members of the population according to some probability distribution (F). The probability distribution is also assumed to involve parameters of location (a) and scale (0 > 0) and can be generally denoted by F (a + 0 log z), where z is the tolerance level to a particular toxic agent. The probability, P(d), that a random individual will suffer a response from a dose, d, is P(d) = F (a + 0 log d) = dF(x). 

The bilateral tolerance stack model including a factor for shifted component distributions is given below. It is derived by substituting equations 3.11 and 3.18 into equation 3.2. This equation is similar to that derived in Harry and Stewart (1988), but using the estimates for Cp and a target Cp for the assembly tolerance... 

The inadequacy of the worst case model is evident and the statistical nature of the tolerance stack is more realistic, especially when including the effects of shifted distributions. This has also been the conclusion of some of the literature discussing tolerance stack models (Chase and Parkinson, 1991 Harry and Stewart, 1988 Wu et al., 1988). Shifting and drifting of component distributions has been said to be the chief reason for the apparent disenchantment with statistical tolerancing in manufacturing (Evans, 1975). Modern equipment is frequently composed of thousands of components, all of which interact within various tolerances. Failures often arise from a combination of drift conditions rather than the failure of a specific component. These are more difficult to predict and are therefore less likely to be foreseen by the designer (Smith, 1993). 

Other mathematical models of tolerance distributions which produce a sigmoid appearance of their corresponding dose-response functions have been suggested. The most commonly used is the log logistic function. 

The shown characteristics are based on a real dental drill For this paper the data set has to be synthesized. The tolerance areas of the drill characteristics as well as their expected normal distributions (step 2, Fig. 1) parameters regarding the manufacturing process are shown in Table 1. The assumption of the expected normal distribution model was verified and confirmed by a Kolmogorov-Smimov conformance test (Sachs 2002). 

The continuously analysis and control of the manufacturing process is the precondition for the obtaining of excellent products. The two established methods are quality control charts and the relayed process capability indices. Quality Control Charts (QCC) can be used without an exact knowledge of the characteristic distribution model and are useful in the analysis of manufacturing risks. QCCs show the mean and scattering of each product characteristic in relation to the given tolerance areas. Furthermore, alert and intervention limits can be adapted inside QCC, if needed. This method is an industrial standard, further explanations can be found in (Timischl 2002). It is possible... 

The visualisation in Eigure 3 shows, that the distribution model of Cl is not completely inside the tolerance area, therefore the related manufacturing process step has to be improved. 

Sufficient data on process distributions and costs must be collated to characterize manufacturing processes for advanced tolerance models. 

As described above, microchannel reactor scale-up requires integrated models, which include the reaction chemistry with heat transfer, pressure drop, flow distribution, and manufacturing tolerances. The culmination of scale-up models is their successful demonstration. 

Outliers may heavily influence the result of PCA. Diagnostic plots help to find outliers (leverage points and orthogonal outliers) falling outside the hyper-ellipsoid which defines the PCA model. Essential is the use of robust methods that are tolerant against deviations from multivariate normal distributions. 

The basis of all performance criteria are prediction errors (residuals), yt - yh obtained from an independent test set, or by CV or bootstrap, or sometimes by less reliable methods. It is crucial to document from which data set and by which strategy the prediction errors have been obtained furthermore, a large number of prediction errors is desirable. Various measures can be derived from the residuals to characterize the prediction performance of a single model or a model type. If enough values are available, visualization of the error distribution gives a comprehensive picture. In many cases, the distribution is similar to a normal distribution and has a mean of approximately zero. Such distribution can well be described by a single parameter that measures the spread. Other distributions of the errors, for instance a bimodal distribution or a skewed distribution, may occur and can for instance be characterized by a tolerance interval. 

Note that z can be larger than the number of objects, n, if for instance repeated CV or bootstrap has been applied. The bias is the arithmetic mean of the prediction errors and should be near zero however, a systematic error (a nonzero bias) may appear if, for instance, a calibration model is applied to data that have been produced by another instrument. In the case of a normal distribution, about 95% of the prediction errors are within the tolerance interval 2 SEP. The measure SEP and the tolerance interval are given in the units of v, and are therefore most useful for model applications. 

The uniform distribution specifies that the part is equally likely to have a value anywhere in the specified tolerance range. The first example we will look at is the 5% resistor model included in class.lib. If you look at Appendix E on page 620, you will see the following line in the file class.lib ... 

The model name is R5pcnt. It is a resistor model because the model type is RES. The nominal value of the model is R=1 and it has a 5% tolerance. The distribution is uniform. A uniform distribution means that the model parameter R is equally likely to have a value of 1.05, 1,0.95, or any other value between 1.05 and 0.95. When you use this model, the actual value of the resistor is the value specified in the schematic times the parameter R. Thus, if the value of a 5% resistor is specified as lk in the schematic, it may have a value anywhere between 950 and 1050 when used with the Monte Carlo analyses. An equivalent model is ... 

In amplifier design it is important to know how your bias will change with device tolerances. In this section we will find the minimum and maximum collector current of a BJT when we include variations in the transistor current gain, 0F, and resistor tolerances. The circuit above was previously simulated in the Transient Analysis and AC Sweep parts. We will use the same resistor values as before, but we will change the resistor models to include tolerance. The BJT is also changed to the model QBf. This model allows 0F to have a uniform distribution between 50 and 350. 

Probit model. This assumes a lognormal distribution for tolerance in the exposed population. 

Logistic Models. This model is based upon the assumption of a logistic distribution of the logarithms of the individual tolerances. 

The technique involves superimposing the tolerance limits on the graphical representation (i.e., distribution curve) of the process capability curve. (See Fig. 1.) If the curve fits well within the tolerance limits, the inherent reproducibility of the process is considered adequate. If the width of the curve straddles the tolerance limits, however, the inherent reproducibility is considered inadequate. Finally, if the curve is skewed near the right or left limit, the model will predict that defects should occur. 

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