Validation of Results

Abstract

Given the cumulative distribution function of oscillator reliability R, the computation of the ordinary moments of time to first passage failure can be easily calculated by Equation 2.30. The inverse problem, calculation of the cumulative probability distribution function given the ordinary moments of the distribution, is a much more formidable task and will be addressed in Chapter V. In this section, however, the comparison of the ordinary moments of first passage time found in Chapter III by directly solving the steady state Pontriagin-Vitt equation will be compared with those found by using Equation 2.29 in conjunction with the transient solution of the first passage problem. Equation 2.29 can be approximated by

$${T^{(r)}}({_o}) \simeq \,\sum\limits_{i = 0}^\infty {{{[({\tau _{i\,}} + \,{\tau _{i + l}})/2]}^r}\,[R({\tau _i}|{{}_o}) - R({\tau _{i + l}}|} {_o})]$$

(4.1) where R is the cummulative distribution function of oscillator reliability.