Failure estimate plots

On occasion it is necessary to produce failure estimate plots instead of survival estimates plots. Fortunately, this requires only a simple modification to the preceding Kaplan-Meier survival estimates program. The only changes necessary to this program to get a failure plot are to alter the title and axis labels, and to change the survival variable reference to failure because the failure variable is also present in the ProductLimitEstimates data set. The resulting failure estimate plot looks like the following ... 

The failure points plotted and a straight line is fitted to estimate the WeibuU population. 

Plot the actual test data to establish or update previous estimates. Plot failure intensity execution time the y intercept of the straight line fit is an estimate of A,q. Plot failure intensity failure number the X intercept of the straight line fits is an estimate of vq. 

The data from the entitlement experiment is plotted in Fig. 7.9. The truncation of both datasets at about 10 years is clearly visible, particularly for the clear dataset, for which the majority of the samples had still not reached a crack severity rating of 3. There are three points in this graph, two in the clear dataset and one in the black dataset, for which the aacking time to failure is shown as exceeding 10 years. These points are the cracking time to failure estimations for the three samples that had only reached a crack severity of 1 as of the last read date. Since cracking time to failure is determined at a severity rating of 1.5, these points resulted in predicted exposure times beyond that of the last read. 

In this paper, we have introduced an approach to compute the cumulative number of failures at fleet level when the recurrent failure events of units within the fleet are chronologically ordered. The proposed model is examined with three random samples. It is observed that in all three samples, the actual total number of failures (estimated by the proposed model) are slightly higher than the expected total number of failures (using the MCF). It is also observed that when the data set is non-censored, the ATNF for all three cases is slightly lower than the ETNF. In addition, the plots show that until around 125,000 fleet-wise operation hours, the ETNF and ATNF approach generates the same results for both censored and non-censored data sets. Therefore, the results obtained by the ATNF of a mature fleet can be used to estimate the number of failures for a new fleet, when the fleet is operated under approximately the same conditions. 

From the quality-eost arguments made in Seetion 1.2, it is possible to plot points on the graph of Oeeurrenee versus Severity and eonstruet lines of equal failure eost (% isoeosts). Figure 2.22 shows this graph, ealled a Conformability Map. Beeause of uneertainty in the estimates, only a broad band has been defined. 

The line the data supports on a hazard plot determines engineering information relating to the distribution of time to failure. Fan failure data and simulated data are illustrated here to explain how the information is obtained. The methods provide estimates of distribution parameters, percentiles, and probabilities of failure. The methods that give estimates of distribution parameters differ slightly from the hazard paper of one theoretical distribution to another and are given separately for each distribution. The methods that give estimates of distribution percentiles and failure probabilities are the same for all papers and are given first. 

Suppose, for example, that an estimate based on a Wei-bull fit to the fan data is desired of the fifth percentile of the distribution of time to fan failure. Enter the Weibull plot. Figure 62.6, on the probability scale at the chosen percentage point, 5 per cent. Go vertically down to the fitted line and then horizontally to the time scale where the estimate of the percentile is read and is 14,000 hours. 

An estimate of the probability of failure before some chosen specific time is obtained by the following. Suppose that an estimate is desired of the probability of fan failure before 100,000 hours, based on a Weibull fit to the fan data. Enter the Weibull plot on the vertical time scale at the chosen time, 100,000 hours. Go horizontally to the fitted line and then up to the probability scale where the estimate of the probability of failure is read and is 38 per cent. In other words, an estimated 38 per cent of the fans will fail before they run for 100,000 hours. 

For the purpose of showing how to obtain from an exponential hazard plot an estimate of the exponential mean time to failure, assume that the straight line on Figure 62.9 is the one fitted to the data. Enter the plot at the 100 per cent point on the horizontal cumulative hazard scale at the bottom of the paper. Go up to the fitted line and then across horizontally to the vertical time scale where the estimate of the mean time to failure is read and is 1000 hours. The corresponding estimate of the failure rate is the reciprocal of the mean time to failure and is 1/100 = 0.001 failures per hour. 

When a set of data does not plot as a straight line on any of the available papers, then one may wish to draw a smooth curve through the data points on one of the plotting papers, and use the curve to obtain estimates of distribution percentiles and probabilities of failure for various given times. With such a nonparametric fit to the data, it is usually unsatisfactory to extrapolate beyond the data because it is difficult to determine how to extend how to extend the curve. Nonparametric fitting is best used only if the data contain a reasonably large number of failures. 

If estimated of distribution parameters are desired from data plotted on a hazard paper, then the straight line drawn through the data should be based primarily on a fit to the data points near the center of the distribution the sample is from and not be influenced overly by data points in the tails of the distribution. This is suggested because the smallest and largest times to failure in a sample tend to vary considerably from the true cumulative hazard function, and the middle times tend to lie close to it. Similar comments apply to the probability plotting. 

This equation may be used to create a plot, similar to those in Figs. 4.27 and 4.28, in which the logarithm of the estimated lifetime is plotted against the reciprocal of the failure temperature. From plots of this type, the dramatic increases in estimated lifetimes for small decreases in temperature can be more easily visualized. 

Droplet growth rates and viscosity decline rates both are exponential processes, following a straight line on a semi-log plot (log x or log vs. time), where is the mean droplet diameter. Emulsion failure is also associated with a certain minimum viscosity, depending on water content, crude-oil content, temperature, etc. Viscosity and mean droplet size may be projected to estimate the time remaining before emulsion failure. The ultimate droplet size and viscosity should be determined experimentally for the same formulation in a pilot-plant pipe loop. 

For pure liquids, the Debye equation suggests that the molar polarization should be a linear function of the reciprocal temperature. Furthermore, one should be able to analyze relative permittivity data for a polar liquid like water as a function of temperature to obtain the dipole moment and polarizability from the slope and intercept, respectively. In fact, if one constructs such a plot using data for a polar solvent, one obtains results which are unreasonable on the basis of known values of p and ocp from gas phase measurements. The reason for the failure of the Debye model in liquids is the fact that it neglects the field due to dipoles in the immediate vicinity of a given molecule. However, it provides a reasonable description of the dielectric properties of dilute polar gases. In liquids, relatively strong forces, both electrostatic and chemical, determine the relative orientation of the molecules in the system, and lead to an error in the estimation of the orientational component of the molar polarization. 

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