First passage probability for two-mode systems
Lutes, Loren D.
Master of Science
This study deals with the first-passage time and maximum response statistics of linear systems with one and two degrees of freedom (SDOF and 2-DOF). Emphasis is placed on empirical studies of the first-passage time of 2-DOF systems. The empirical results were obtained from a digital computer simulation, with the Gaussian white noise excitation obtained from a random number generator subroutine. Maximum response distribution is compared with the Gumbel type 1 distribution as well as Poisson crossings, Vanraarcke's and Mark's analytical approximate results. A simple empirical approximation equation is determined for SDOF systems from the study of the probability of first-passage time distribution. This equation is shown to be in good agreement in general, with the empirical data and with numerical results obtained by Roberts. The Gumbel type 1 distribution is compared with the empirical data. It appears to fit quite well for moderate failure levels and diverge for very large failure levels for both SDOF and 2-DOF systems. The results of Poisson crossings, Vanmarcke's and Mark's analytical theories do not give good approximations to the maximum response distribution of 2-DOF systems, particularly for small damping values. A considerable amount of empirical data is obtained for first-passage probability of 2-DOF systems. Emphasis is placed on the limiting decay rate (a) of survival probability, since this has been shown to be a useful parameter to characterize first-passage probability in SDOF systems. In particular, the dependence of a on the barrier level (b) is studied, and similarities with and differences from SDOF results are considered. No general formula is obtained for predicting a from knowledge of b, and system parameters.