Exponential failure curve

Suppose 1000 identical items (each with a failure rate of 1 every 1000 hours) start to run simultaneously (Lloyd and Tye, 1995, p. 48). After 100 hours, the accumulated operating time will be 100,000 h. In this period it is predicted that 100 items will fail. The surviving 900 items then mn for another 100 h and 90 will fail, and so on. In short, as the number of surviving items diminishes, so also will the number of failures. If this is plotted against total hours, we obtain a curve similar to the one in Fig. 10.3. This is the exponential failure curve P = 1 - e, with e being the exponential number 2.718. On exactly the same reasoning, it can be shown than an individual item has a probability of failing at time t equal to 1 - e . ... 

The degree of life acceleration that this test represents is uncertain. Fitting data to a temperature dependent first-order exponential (Arrhenius) failure curve is often inaccurate due to initial model assumptions. Depending upon the estimate of activation energies used, and the criteria used to define failure, an... 

The constant failure rate curve shown in Figiue 16.9 is a straight line corresponding to the exponential failure rate aheady discussed. It represents random events that occur independently of time. For example, operator error that can take place at any time. 

A considerable assumption in the exponential distribution is the assumption of a constant failure rate. Real devices demonstrate a failure rate curve more like that shown in Eigure 9. Eor a new device, the failure rate is initially high owing to manufacturing defects, material defects, etc. This period is called infant mortaUty. EoUowing this is a period of relatively constant failure rate. This is the period during which the exponential distribution is most apphcable. EinaHy, as the device ages, the failure rate eventually increases. 

To illustrate probability calculations involving tlie exponential and Weibull distributions introduced in conjunction willi llie batlitub curve of failure rate, consider first llie case of a mansistor having a constant rate of failure of 0.01 per tliousand hours. To find the probability tliat llie transistor will operate for at least 25,000 hours, substitute tlie failure rate... 

Figures 62.8, 62.9, 62.10 show the data for generator fan failure plotted on exponential, normal and log normal hazard paper respectively. The exponential plot is a reasonably straight line which indicates that the failure rate is relatively constant over the range of the data. It should be noted that the reason the probability scale on the exponential hazard plot is crossed out is because that is not the proper way to plot data. (This will be discussed later.) The normal plot is curved concave upward which...

The behavior of the failure rate as a function of time can be gaged from a hazard plot. If data are plotted on exponential hazard paper, the derivative of the cumulative hazard function at some time is the instantaneous failure rate at that time. Since time to failure is plotted as a function of the cumulative hazard, the instantaneous failure rate is actually the reciprocal of the slope of the plotted data, and the slope of the plotted data corresponds to the instantaneous mean time to failure. For the data that are plotted on one of the other hazard papers and that give a curved plot, one can determine from examining the changing slope of the plot whether the tme failure rate is increasing or decreasing relative to the failure rate of the theoretical distribution for the paper. Such information on the behavior of the failure rate cannot be obtained from probability plots. 

With shape parameter / < 1 a decreasing failure rate can be simulated, insuch as on the left side of the bathtub curve (Fig. 5.9.6). For / =1 the Weibull distribution reduces to a pure exponential with a constant failure rate that is given by the reciprocal of the characteristic time T ... 

Note that the simplest and most obvious initial assumption of a constant failure rate has actually led to an exponential curve. That is because the exponential curve is simply a succession of evenly spaced data points (very close to each other), that are in simple geometric progression, that is, the ratio of any point to its preceding point is a constant. Most natural processes behave similarly, and so e is encountered very frequently. 

Figure 9. Population growth curve. No physical phenomena, except perhaps the growth of the universe, follow the exponential law shown by the dashed line. The vertical line denotes the half-life x of the growth. Coatings failure is manifested as a curve that peaks below the saturation limit and levels off at a lower value.

In practice, initial failures and wear-out may play a role in the breakdown of equipment, giving a typical bathtub failure rate curve. Be that as it may, for the purpose of inventory control, we feel that the assumption of exponential running times between failures will in most cases give good resvilts in return for the limited amount of effort required. In cases where wear-out or initial failures play a significant role, we expect manual deviation from the recommended stock quantities. 

Bertholon (Ziani, 2008) model combines an exponential and a WeibuU for the overaU second and third parts of the bathmb curve (occasional failures and wear). This model seems hkely to be used in the future to characterize the reliabihty of electronic components whose integration should lead to more and more severe life limitations. The model consists of two blocks in series, one corresponding to an exponential law and the second to a WeibuU law. Its reUabiUty is expressed by equation 1. 

Bertholon model can be generalized into a model to 7 parameters characterizing the three phases of the bathtub curve a first Weibull law with P < I for the phase of youth failure, an exponential law for the phase of occasional failures and a second Weibull with P > for the wear phase. It corresponds to three blocks in series, the first is a Weibull, initiated at t = 0 (y = 0) and limited to duration T, and the other two corresponding to the Bertholon model. The occurrence of failure can be simulated by the formula 4 under Excel. 

They most often lead to exponential, eventually WeibuU probabilities distribution. The time curves of fault rate and reparations, eventually other stochastic influences during the reliabUily of complicated systems ensuring in real operation, are not taken into account in these models (CHOVANEC, A.). In the real case the operation reliability, respectively its partial properties are coimected with processes, which are necessary for the failure removal (control process, supplying system, repairing process, etc.). That s why also the model may have several states and distrihutions of random variables. 

Using the indicator variable Ki an expectation value E Ki) is established which is equal to the component reliability Ri, where the component reliability can be described by arbitrary distributions, e.g. with a constant failure rate or using a Weibull distribution. In the case of aircraft systems, component failures are generally assumed to be independent of age and arise randomly based on the bath-tub curve, so that an exponential distribution with a constant failure rate X,- is applied (19). 

To improve the performance of SiC and to increase its resistance against creep failure, generally various constituents are added to monolithic SiC ceramics. Additives in various shapes and sizes are usually added to SiC to achieve a better material for structural use and to extend its service lifetime. An evaluation of creep failure, commonly referred to as creep rupture or stress rupture , is a critical step in evaluating the suitability of a certain ceramic for use in the desired application. The stress rupture and creep properties of a SiC matrix reinforced with SiC fiber (i.e., a SiC/SiC composite) has been evaluated by tests conducted in order to assess the propensity of SiC/SiC for high-temperature appfications over an extended lifetime. In Fig. 6.105, plots of stress versus time-to-rupture are shown for several temperatures. As commonly done, these plots are on a log-log scale. Each curve can be fitted by means of an empirical relation, similar to the earlier exponential equation expressing the time-to-rupture, h, to a stress exponent for stress rupture as ... 

For reliability calculation of electronic components, handbooks are typically used (Roller et al. 2011). The handbooks calculate constant failure rates based on exponential distribution. The exponential distribution corresponds to a Weibull distribution with shape parameter b =. Therefore only random failures are described in section II of the bath tub curve, which are characterized by a constant failure rate. 

Figure 1 makes clear, that the empirical failure behaviour of the analysed safety-related function can approximately be illustrated with a Weibull distribution, because the theoretical failure probability of the Weibull distribution has a similar curve progression with the empirical failure probability. The coefficient of determination with i 0.954 is explicitly better than the coefficient of determination of the exponential distribution with R 0.845. The results of the other examined machines, which are not visualised here, also confirm the first outcomes of the first VDW-study that the exponential distribution in not adequate for field data analysis of systems with an ageing process. 

A risk assessment mainly concentrates on the useful life in the bathtub curve in Figure 3.2. In the useful life region, the failure rate is constant over the period of time. In other words, a failure could occur randomly regardless of when a previous failure occurred. This results in a negative exponential distribution for the failure frequency. The failure density function of an exponential distribution is as follows ... 

The results obtained using an exponential distribution is not very useful as it does not reflect a curve that increases in D(T) as the inspection period increases. From these results, the most suited distribution was found to be the Weibull and the truncated standard normal distributions. These two distributions gave clear indications of the optimum inspection period. The values of a and P in the Weibull distribution can be estimated by a collection of test data or by using available failure data of the equipment, and since the failure data available is associated with a high degree of uncertainty, this distribution is not used here. As such for the... 

---