In certain cases when one or more of the characteristics of the distribution change based on an outside factor, one may be interested in formulating a model that includes both the life distribution and a model that describes how a characteristic of the distribution changes. In reliability, the most common "outside factor" is the stress applied to the component. In system analysis, stress comes into play when dealing with units in a load sharing configuration. When components of a system operate in a load sharing configuration, each component supports a portion of the total load for that aspect of the system. When one or more load sharing components fail, the operating components must take on an increased portion of the load in order to compensate for the failure(s). Therefore, the reliability of each component is dependent upon the performance of the other components in the load sharing configuration.
Figure 3.7: Single pdf
Figure 3.8: pdf and life-stress relationship.
Traditionally in a reliability block diagram, one assumes independence and thus an item's failure characteristics can be fully described by its failure distribution. However, if the configuration includes load sharing redundancy, then a single failure distribution is no longer sufficient to describe an item's failure characteristics. Instead, the item will fail differently when operating under different loads and the load applied to the component will vary depending on the performance of the other component(s) in the configuration. Therefore, a more complex model is needed to fully describe the failure characteristics of such blocks. This model must describe both the effect of the load (or stress) on the life of the product and the probability of failure of the item at the specified load. The models, theory and methodology used in Quantitative Accelerated Life Testing (QALT) data analysis can be used to obtain the desired model for this situation. The objective of QALT analysis is to relate the applied stress to life (or a life distribution). Identically in the load sharing case, one again wants to relate the applied stress (or load) to life. Figure 3.7 graphically illustrates the probability density function (pdf) for a standard item, where only a single distribution is required. Figure 3.8 represents a load sharing item by using a 3-D surface that illustrates the pdf, load and time. Figure 3.9 shows the reliability curve for a load sharing item vs. the applied load.
Figure 3.9: Reliability and life-stress relationship.
To formulate the model, a life distribution is combined with a life-stress relationship. The distribution choice is based on the product's failure characteristics while the life-stress relationship is based on how the stress affects the life characteristics. Figure 3.10 graphically shows these elements of the formulation and Figure 3.11 shows the combination of both an underlying distribution and a life-stress model, by plotting a pdf against both time and stress.
Figure 3.10: A life distribution and a life-stress relationship.
Figure 3.11: pdf vs. time and stress
The assumed underlying life distribution can be any life distribution. The most commonly used life distributions include the Weibull, the exponential and the lognormal. The life-stress relationship describes how a specific life characteristic changes with the application of stress. The life characteristic can be any life measure such as the mean, median, R(x), F(x), etc. It is expressed as a function of stress. Depending on the assumed underlying life distribution, different life characteristics are considered. Typical life characteristics for some distributions are shown in the next table.
|
Distribution |
Parameters |
Life Characteristic |
|
Weibull |
β*, η |
Scale parameter, η |
|
Exponential |
λ |
Mean Life, (1/λ) |
|
Lognormal |
|
Median, |
*Usually assumed constant
For example, when considering the Weibull distribution, the scale parameter, η, is chosen to be the life characteristic that is stress-dependent while β is assumed to remain constant across different stress levels. A life-stress relationship is then assigned to η. The three life-stress models supported by BlockSim are presented next.
The Inverse Power Law (IPL) model is given by:
(19)
Where:
L represents a quantifiable life measure, such as mean life, characteristic life, median life, B(x) life, etc.
V represents the stress level.
K is one of the model parameters to be determined, (K > 0).
n is another model parameter to be determined.
The Arrhenius model is given by:
(20)
Where:
L represents a quantifiable life measure, such as mean life, characteristic life, median life or B(x) life, etc.
V represents the stress level (in absolute units if it is temperature).
C is one of the model parameters to be determined, (C > 0).
B is another model parameter to be determined.
The Eyring model is given by:
(21)
Where:
L represents a quantifiable life measure, such as mean life, characteristic life, median life, B(x) life, etc.
V represents the stress level.
A is one of the model parameters to be determined.
B is another model parameter to be determined.
We will illustrate the use of the life distributions and life-stress relationships by combining the Weibull distribution and the IPL model.
The IPL-Weibull model can be derived by setting η = L(V), yielding the following IPL-Weibull pdf:
(22)
The IPL-Weibull model yields the IPL-exponential model for 
= 1.
The mean, 
(also called MTTF),
of the IPL-Weibull relationship is given by:
(23)
Where
is the Gamma function evaluated at the value of 
.
The IPL-Weibull reliability function is given by:
The IPL-Weibull conditional reliability function at a specified stress level is given by:
(24)
Or:
For the IPL-Weibull relationship, the reliable life, TR, of a unit for a specified reliability and starting the mission at age zero is given by:
(25)
The IPL-Weibull failure rate function, λ(T), is given by:
(26)
See Also:
Statistical Background
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