Failure bathtub curve

FIGURE 11.14 Failure bathtub curve statistic. The early failure drop is generally canceled with design, type, and routine tests. 

Monitoring the trends of a machine-train or process system will provide the ability to prevent most catastrophic failures. The trend is similar to the bathtub curve used to schedule preventive maintenance. The difference between the preventive and predictive bathtub curve is that the latter is based on the actual condition of the equipment, not a statistical average. 

Figure 11-2 A typical bathtub failure rate curve for process hardware. The failure rate is approximately constant over the midlife of the component.

Early failures and wear out failures are reflected in the curve known as the bathtub curve (see Figure 11.14). 

The Weibull distribution provides a inatlieiiiatical model of all tluee stages of the batlitub curve. Tliis is now discussed. An assumption about failure rate tliat reflects all tluee stages of tlie bathtub curve is... 

To illustrate probability calculations involving Uie exponenUal and Weibull distributions introduced in conjunction with Uie bathtub curve of failure rate, consider first Uie case of a transistor having a constant rate of failure of... 

As described in Problem HZA.7, the failure rate of equipment frequently exhibits three stages a break-in stage with a declining failure rate, a useful life stage characterized by a fairly constant failure rate, and a wearout period characterized by an increasing failure rate. Many industrial parts and components follow this path. A failure rate curve exhibiting these three phases is called a bathtub curve. 

The LINUX cluster had been managed under continually increasing load for a period of months. Systems administrators had responded by incremental addition of PCs, but the performance had shown continued decline. The system was loaded near capacity, which is often far below theoretical, 70% of maximum being a common benchmark. Another feature of the system, which is especially relevant to clusters, was the impact of component failure. For computers, this is likely the disk drive. Drives have a mean time between failure of one to two years and follow a bathtub curve of burn-in failure, followed by reliable performance and then high failure near the end of the lifecycle. Cluster design must take into account the need for graceful degradation. 

Fig. 5.9.6 Schematic bathtub curve of failure rates over time with increasing reliability, the curves are lower and less steep at the end of a product s life...

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 ... 

Develop bathtub curve based on the failure rate data and the test data. 

Pattern A. Referred to as a bathtub curve with three identifiable regions, namely (i) the initial period of high probability of failure (ii) region of constant and low probability of failure (iii) a wear-out region of high probability of failure. 

The lifetime of a population of units at the component, board, box, or system level can be divided into three distinct periods. This is most often defined by the so-called reliability bathtub curve (Fig. 6.16). The bathtub curve describes the cradle-to-grave failure rates or frequency of failures as a function of time. The curve is divided into three distinct areas early failure rate (also known as infant mortality), the useful life period, and the wearout failure period. The infant-mortality portion of the curve, also known as the early life period, is the initial steep slope from the start to... 

The bathtub curve represents the curve of the failure rate of a product, which is denoted by X(t) (L. Scheidt, et al. 1994. Niu Peng-zhi, et al. 2007. Wang Rong-hua, et al. 2002) In terms of the maintainability of a product, the failure rate points to the malfunction rate. The rate of failure is also called the rate of damage failure. The rate of safety damage shows the damage probability of system, which is the same as the rate of failure. RD (t) is thus equal to X(t). 

An item s failure rate is generally not a single value— it will vary with time and the age of the item. The bathtub curve, shown in Figure 16.9, illustrates this phenomenon (the term bathtub comes from the rather fanciful resemblance of the shape of the overall failure rate to that of a bathtub). 

This was based on the bathtub curve (Fig. 1) which depicts the increase in the failure rate of items in time. Time-based maintenance does not account for the fact that the condition of the items depends not only on elapsed time but also on operational and environmental conditions. Based on advances in diagnostic techniques in the 1970s, condition-based maintenance (or predictive maintenance) was introduced. Here, maintenance activities occur when symptoms of wear or failure are determined either by monitoring or diagnosis, i.e., maintenance effectiveness depends on the existence of appropriate diagnostic techniques. 

The shape of the plot in Figure 3-2 is characteristic of many components and well known to reliability engineers. The shape is called the "bathtub curve." Three regions are distinct. In the early portion of the plot, failure rates are higher. This area is called "infant mortality." The middle portion of the curve is known as "useful Ufe." The final portion of the curve is called "end of tife" or "wearout region."... 

Failures occur when some "stress" exceeds the corresponding "susceptibility" in any component. As a concept it makes perfect sense in the context of mechanical engineering. The stress is usually a force, and the susceptibility is the point where the mechanical component can no longer resist that force. An analogous concept has been developed for electronic components (Ref. 3). Simulations using the stress-susceptibility concept generate failure rate curves similar to the bathtub curve (Ref. 4 and 5). 

Field data may include failures occurring during the wearout portion of the bathtub curve... 

As can be seen in Eq. (9.23) the survival probability of a component is completely determined by its failure rate )t(t). Its general shape is known as the bathtub curve , which is similar to the curve of human mortality. The bathtub curve is shown in Fig. 9.12. 

The shape factor a, which is the density, distribution and failure rate function, and represents specific regularity failure occurrence, in function with the Peta parameter and acquires three different shapes P <1, P = 1, P> 1, which defines the "bathtub curve" as shown in figure 4. 

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 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. 

In principle, it is possible to differentiate the damage behavior regarding the product life cycle into three superior damage phases Early failure phase, coincidence failure phase and wearout failure phase. The serial combination of these phases ( bathtube curve ) is shown in Fig. 1, focused on the failure rate 7.(t). Furthermore, Fig. 1 includes the range of the WeibuU form parameter b in reference to the failure phase. 

Statistical values can also be used to determine expected periods of optimum performance in the life cycle of products, systems, hardware, or equipment. For example, if the life cycle of humans were plotted on a curve, the period of their lives that may be considered most useful, in terms of productivity and success, could be represented as shown in Figure 5.4. This plotted curve is often referred to as the bathtub curve because of its obvious shape. A similar curve can be used to determine the most productive period of a product s life cycle according to the five known phases of that life cycle, as discussed in Chapter 3. The resultant curve, known as a product s reliability curvey would resemble the curve that appears in Figure 5.5. During the breakin period, failures in the system may occur more frequently, but decreasingly less frequently as the curve begins to level toward the useful life period. Then, as the system reaches the end of its useful life and approaches wearout, more frequent failure experience is likely until disposal. 

Hence, statistical evaluation of failures that occur during a product s life cycle help develop a failure curve that, because of its shape, is referred to as a bathtub curve. When considering the usefulness of a product, the curve becomes a reliability curve for that product. 

Installed safety equipment is subject to the same operational stresses as control equipment. However, when control equipment fails, the failure can be detected because the process behaves abnormally. In contrast, safety equipment typically operates on demand only, i.e., when an abnormal condition occurs, so failure may not be detected until it is required to act. Equipment often demonstrates a failure rate over time that follows a so-called bathtub curve. 

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