Design Decisions under Uncertainty with Limited Information

An Efficient Method to Calculate the Failure Rate of Dynamic Systems with Random Parameters Using the Total Probability Theorem

“…Because both S YY (ω) and R YY (τ) are real and even functions, we have [19] (8) Note that for a wide-sense stationary process, (9) where −1 ≤ ρ≤+1 is the correlation coefficient and…”

“…(17) yields, (20) Because the integral in Eq. (20) is non-negative, f(t) has the following lower bound (21) If we continue the process of using Equations (17) and (19) to derive Eq. (20), the following Rice's inclusion-exclusion series [4,19] can be obtained (22) where (23) The PDF f(t) is then used in Eq.…”

“…In this case, for each realization of the input random variables X, the output process Y(t) is also wide-sense stationary and Gaussian with zero mean, a constant standard deviation and a PSD of (7) where H(ω) is the frequency response function of the system between the i th (input) and j th (output) degrees of freedom (DOF) [19].…”

“…The joint up-crossing rate v ++ (t,τ 1 ) indicates the probability that an up-crossing occurs at both t and τ 1 < t. It can be expressed as (19) since the up-crossing at τ 1 is the first or the first up-crossing has occurred at some previous time τ 2 < τ 1 . Integration of Eq.…”

“…(2) where the failure rate λ(t) is approximated by the up-crossing rate v + (t) (i.e., λ(t) = v + (t)). Because the excitation process is Gaussian and wide-sense stationary and the dynamic system is linear with constant parameters, the up-crossing rate of threshold α is constant given by [19] (12) where (13) is the up-crossing rate for α = 0 and (14) is the standard deviation of the derivative process . …”