Normal Distribution equation

Analytieal solutions to equation 4.32 for a single load applieation are available for eertain eombinations of distributions. These coupling equations (so ealled beeause they eouple the distributional terms for both loading stress and material strength) apply to two eommon eases. First, when both the stress and strength follow the Normal distribution (equation 4.38), and seeondly when stress and strength ean be eharaeterized by the Lognormal distribution (equation 4.39). 

A size distribution that fits the normal distribution equation can be represented by two parameters, the arithmetic mean size, x, and the standard deviation, cr. The mean size, x, is the size at 50% of the distribution, also written as jcgo. The standard deviation is easily obtained from the cumulative distribution as... 

The normal probability function table given in the appendix d this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/cy/(In o- )] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, dg, and the geometric standard deviation,. The geometric mean size is the size at 50% of the distribution, d. The geometric standard deviation is easily obtained finm the following ratios ... 

The Central-Limit Theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend toward the normal distribution, equation (3.17), as the sample size becomes large. ... 

Figure 3 Elution Profile for Base Case Parameters assuming a normal distribution (equation 17).

The theory of coiled structures in polymer networks has also been applied to swelling In many cases the cross-linking, which has formed the network, took place in the dry polymer. It is then reasonable to assume that in this dry condition the normal distribution (equation 6, p. 98) of chain-endpoints prevails ... 

Gaussian or normal distribution. Equation (11) (see chapter 2, Molecular Mass of Macromolecules and its Distribution). We formulate it here in three dimensions ... 

The log-normal distribution is probably the most widely used type of function it is again a two-parameter function, it is skewed to the right and it gives equal probability to ratios of sizes rather than to size differences, as in the normal distribution. The log-normal distribution equation is obtained from the normal distribution in equation 2.20 by substitution of In x for x, Inxg for Xa and In Cg for a, i.e. 

Fig. 1.12 Three normal distributions with different values of a (Equation (1.55)). The functions are normalised, so the area under each curve is the same.

To obtain the expression for the log-normal distribution it is only necessary to substitute for I and a in Equation (1.52) the logarithms of these quantities. One thus obtains... 

The second complication is that the values of z shown in Table 4.11 are derived for a normal distribution curve that is a function of O, not s. Although is an unbiased estimator of O, the value of for any randomly selected sample may differ significantly from O. To account for the uncertainty in estimating O, the term z in equation 4.11 is replaced with the variable f, where f is defined such that f > z at all confidence levels. Thus, equation 4.11 becomes... 

In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error... 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

The particle size distribution of ball-milled metals and minerals, and atomized metals, follows approximately the Gaussian or normal distribution, in most cases when the logarithn of die diameter is used rather dran the simple diameter. The normal Gaussian distribution equation is... 

We need this speeial algebra to operate on the engineering equations as part of probabilistie design, for example the bending stress equation, beeause the parameters are random variables of a distributional nature rather than unique values. When these random variables are mathematieally manipulated, the result of the operation is another random variable. The algebra has been almost entirely developed with the applieation of the Normal distribution, beeause numerous funetions of random variables are normally distributed or are approximately normally distributed in engineering (Haugen, 1980). 

The varianee for any set of data ean be ealeulated without referenee to the prior distribution as diseussed in Appendix I. It follows that the varianee equation is also independent of a prior distribution. Here it is assumed that in all the eases the output funetion is adequately represented by the Normal distribution when the random variables involved are all represented by the Normal distribution. The assumption that the output funetion is robustly Normal in all eases does not strietly apply, partieularly when variables are in eertain eombination or when the Lognormal distribution is used. See Haugen (1980), Shigley and Misehke (1996) and Siddal (1983) for guidanee on using the varianee equation. 

Finally, it is worth investigating how deterministic values of material strength are calculated as commonly found in engineering data books. Equation 4.14 states that the minimum material strength, as used in deterministic calculations, equals the mean value determined from test, minus three standard deviations, calculated for the Normal distribution (Cable and Virene, 1967) ... 

The distributional parameters for Kt in the form of the Normal distribution can then be used as a random variable product with the loading stress to determine the final stress acting due to the stress concentration. Equations 4.23 and 4.24 show... 

Figure 4.28 Derivation of the coupling equation for the case when both loading stress and material strength are a Normal distribution...

This is essentially the eoupling equation for the ease when both stress and strength are a Normal distribution. A parameter to define the relative shapes of the stress and strength distributions is also presented, ealled the Loading Roughness, LR, given by ... 

Another consideration when using the approach is the assumption that stress and strength are statistically independent however, in practical applications it is to be expected that this is usually the case (Disney et al., 1968). The random variables in the design are assumed to be independent, linear and near-Normal to be used effectively in the variance equation. A high correlation of the random variables in some way, or the use of non-Normal distributions in the stress governing function are often sources of non-linearity and transformations methods should be considered. 

Assuming that all the variables follow a Normal distribution, a probabilistie model ean be ereated to determine the stress distribution for the first failure mode using the varianee equation and solving using the Finite Differenee Method (see Appendix XI). The funetion for the von Mises stress, L, on first assembly at the solenoid seetion is taken from equation 4.75 and is given by ... 

In the stress rupture ease, the interferenee of the stress, L, and strength, Sy, both following a Normal distribution ean be determined from the eoupling equation ... 

Equation 4.83 states that there are four variables involved. We have already determined the load variable, F, earlier. The load is applied at a mean distanee, /r, of 150 mm representing the eouple length, and is normally distributed about the width of the foot pad. The standard deviation of the eouple length, cr, ean be approximated by assuming that 6cr eovers the pad, therefore ... 

We ean use a Monte Carlo simulation of the random variables in equation 4.83 to determine the likely mean and standard deviation of the loading stress, assuming that this will be a Normal distribution too. Exeept for the load, F, whieh is modelled by a 2-parameter Weibull distribution, the remaining variables are eharaeterized by the Normal distribution. The 3-parameter Weibull distribution ean be used to model... 

Both of the torque eapaeities ealeulated, the holding torque of the hub and the shaft torque at yield, are represented by the Normal distribution, therefore we ean use the eoupling equation to determine the probability of interferenee, where ... 

Solving equation 4.92 using Monte Carlo simulation for the variables involved, the torque eapaeity of the shear pin is found to have a Normal distribution of Mwl A (4421.7,234.1)Nm. 

The allowable misalignment toleranee for the vertieal tie rod, tp = 1.5 , is also eonsidered to be normally distributed in praetiee. With the assumption that approximately 6 standard deviations are eovering this range, the standard deviation beeomes = 0.5 . The mean of the angle on whieh the prineipal plane lies is /i, and the loads must be resolved for this angle, but its standard deviation is the statistieal sum of cr and as given by equation 4.103 ... 

The shape of the Normal distribution is shown in Figure 3 for an arbitrary mean, /i= 150 and varying standard deviation, ct. Notice it is symmetrical about the mean and that the area under each curve is equal representing a probability of one. The equation which describes the shape of a Normal distribution is called the Probability Density Function (PDF) and is usually represented by the term f x), or the function of A , where A is the variable of interest or variate. 

Once the mean and standard deviation have been determined, the frequency distribution determined from the PDF can be compared to the original histogram, if one was constructed, by using a scaling factor in the PDF equation. For example, the expected frequency for the Normal distribution is given by ... 

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