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===Confidence Bounds for Competing Failure Modes===
#REDIRECT [[Competing Failure Modes Analysis]]
 
The method available in Weibull++ for estimating the different types of confidence bounds, for competing failure modes analysis, is the Fisher matrix method, and is presented in this section.
 
[edit]Variance/Covariance Matrix
 
The variances and covariances of the parameters are estimated from the inverse local Fisher matrix, as follows:
 
 
 
 
where  is the log-likelihood function of the failure distribution, described in Chapter 5.
 
[edit]Bounds on Reliability
 
The competing failure modes reliability function is given by:
 
 
where:
•   is the reliability of the  mode,
•   is the number of failure modes.
The upper and lower bounds on reliability are estimated using the logit transformation:
 
 
where  is calculated using Eqn. (CFMReliability).  is defined by:
 
 
(If  is the confidence level, then  for the two-sided bounds, and  for the one-sided bounds.)
 
The variance of  is estimated by:
 
 
 
Thus:
 
 
where  is an element of the model parameter vector.
 
Therefore, the value of  is dependent on the underlying distribution.
 
For the Weibull distribution:
 
 
where:
 
and  is given in Chapter 6.
 
For the exponential distribution:
 
 
where  is given in Chapter 7.
 
For the normal distribution:
 
 
 
where  is given in Chapter 8.
 
For the lognormal distribution:
 
 
where  is given in Chapter 9.
 
[edit]Bounds on Time
 
The bounds on time are estimate by solving the reliability equation with respect to time. From Eqn. (CFMReliability) we have that:
 
 
 
where:
•   is inverse function for Eqn. (CFMReliability)
• for the Weibull distribution  is  , and  is 
• for the exponential distribution  is  , and  =0
• for the normal distribution  is  , and  is  , and
• for the lognormal distribution  is  , and  is 
Set:
 
The bounds on  are estimated from:
 
 
and:
 
Then the upper and lower bounds on time are found by using the equations
 
 
and:
 
is calculated using Eqn. (ka) and  is computed as:
 
 
[edit]Complex Competing Failure Modes
 
In addition to being viewed as a series system, the relationship between the different competing failures modes can be more complex. After performing separate analysis for each failure mode, a diagram that describes how each failure mode can result in a product failure can be used to perform analysis for the item in question. Such diagrams are usually referred to as Reliability Block Diagrams (RBD) (for more on RBDs see ReliaSoft's System Analysis Reference and ReliaSoft's BlockSim software).
 
A reliability block diagram is made of blocks that represent the failure modes and arrows and connects the blocks in different configurations. Note that the blocks can also be used to represent different components or subsystems that make up the product. Weibull ++ provides the capability to use a diagram to model, series, parallel, k-out-of-n configurations in addition to any complex combinations of these configurations.
 
In this analysis, the failure modes are assumed to be statistically independent. (Note: In the context of this reference, statistically independent implies that failure information for one failure mode provides no information about, i.e. does not affect, other failure mode). Analysis of dependent modes is more complex. Advanced RBD software such as ReliaSoft's BlockSim can handle and analyze such dependencies, as well as provide more advanced constructs and analyses (see http://www.reliasoft.com/BlockSim).

Latest revision as of 07:37, 29 June 2012

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