ADS-79D-HDBK Downloaded from https://www.everyspec.com parameter, β , indicates whether the failure rate is decreasing ( β β = 1), or increasing ( β > 1). Applied to CBM, if the assumption is made that an increase in CI value (magnitude) corresponds to an increase in the probability of the presence of a fault ( β > 1), then Weibull analysis should provide a reasonably accurate assessment of CI data and component health. I.4.3.2 Minimum Sample Size Derivation The cumulative distribution function for the two-parameter Weibull distribution is given by: ( Equation 5) Where:
x = the CI value to assess β = the Weibull slope (or shape) parameter η = the Weibull scale parameter Applied to CBM, F(x) is the probability of encountering a fault up to a CI value of x . The complement of F(x) is reliability, R(x) , the probability of not encountering a fault up to a CI value of x : (Equation 6) If the natural logarithm is taken of both sides of Equation 6 , then we can solve for x : (Equation 7) Using Equation 7 , we know β and η from the Weibull distribution, and we are interested in knowing the threshold CI value, X 0 , where R(X 0 ) = 90%, the required reliability at X 0 . So the threshold CI value is given by:
(Equation 8) Or, rearranging the terms of Equation 8 , we are able to express η in terms of the threshold CI value and its associated reliability: (Equation 9) 167