Failure rate bathtub curve

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

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. 

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. 

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). 

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. 

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. 

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. 

As proof-test intervals are increased, it is also important to recognize that the useful life of the equipment as it relates to the random failure rate portion of the bathtub curve should not be exceeded. 

Figure 4.6 Bathtub curve-time-dependent failure rate.

The lifetime of an entire population of products often is graphically represented by a set of curves collectively called the bathtub curve. Bathtub curve has been depicted in Fig. VII/1.2.2-1. The bathtub curve consists of three periods. First is an infant mortality or burnt-in period with a decreasing failure rate showing early-life failure. These... 

A plot of the failure rate of a product as a function of time typically takes the shape of a bathtub curve (see Fig. 57.2). This curve illustrates the three phases that occur during the lifespan of a product from a reliability perspective. In the first, infant mortahty phase, there is an initially high but rapidly declining failure rate caused by infant mortahty. Infant mortality is typically caused by manufacturing defects that went undetected during inspection and testing and lead to rapid failure in service. Burn-in can be used to remove these units before shipment. The second phase, the normal operating life of the product, is characterized by a period of stable, relatively low failure rates. 

In the case of the bathtub curve, failure rate during useful life is constant. Letting this constant be a leads to ... 

In doing so, the failure rate X plays an important role, several roles respectively, and it a priori assumes pure random failures. For the scope of the standard, which is safety related parts of control systems , this is a very simple comparison standard. It fits quite well, if a constant failure rate X can be suggested, e g. for electronical products, which can be connected to the horizontal part of the bathtub curve . In the VDW s study on operational dependability, by contrast, the Weibull distribution is used, because at machine tools the failures are not purely random, but the failure rate rises as the operating hours increase (as a consequence of wear and tear, ageing). 

The bathtub curve shows three areas over time. The early failure phase describes the time frame in which the failure behavior is not sufficiently developed through unknown influences, environment parameters, correct materials and bias points. This should be investigated for the development of components within the context of the design verification so that phase 2, the usage phase, can be entered at the beginning of series production. The usage phase should be designed in a way that the failure rate only starts after the expiration of the statistical life expectation of the components. In reality the failure rate is placed below the bathtub curve, as far as necessary so that an age induced increase can be seen and a sufficient robustness level ensures that the statistical life expectation is achieved. ISO 26262 does not mention any requirements, for example in order to prevent early failure behavior. 

In the automobile industry, AEC (Q) 100 is used for complex components. It is a standard for the qualification of electric components. Simple components as resistors or capacitors are not covered in this standard. Since these simple components would often push aU statistic boimdaries through their variety of elements, such statistic observations are often insufficient for safety engineering. The risk for such simple components is that harmful components can be delivered to the production undetected. This is why the eligibility and whether the components are actually sufficiently dimensioned for their case of application are tested in the context of the qualification of the entire electric assembly group. The value for failiue rates is taken from the reliability handbooks. However, for the correct qualification including the proof of lifetime efficiency of the entire electronic assembly group it is assumed that the simple components is within the constant phase of failure rates of the bathtub curve. 

There are two chapters in ISO 2jS2iS2, Part 5, which cover the topic quantified safely analyses. The main referenced based on reUabilily analysis and failure rates for electrical parts, which addresses only random hardware faults. Therefore, the interpretation of the bathtub curve, the sufficient bust, or confidence in the used data and the significance of the determined results are always a question of how the analyses have been performed with which aim. Both metrics of ISO 26262 have different objectives. Part 5, Chap. 8 describes the following objective ... 

Generally, bathtub hazard rate curve is used to describe the failure rate of engineering items/systems and is shown in Figure 3.5. 

A typical failure rate curve for process equipment and other hardware has the bathtub shape shown in Fig. 10.9. For much of its lifetime, the failure rate is approximately constant, with a value denoted by jx. In the subsequent analysis, a constant failure rate is assumed for each component however, extensions can be made to include time-varying failure rates. 

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