FIRST-PASSAGE PROBABILITY FOR STRUCTURES WITH STOCHASTIC EXCITATIONS
Doctor of Philosophy
First-passage probabilities are investigated for the absolute value of the zero-start response of simple oscillators excited by stationary white noise with a normal probability distribution. Both small-time (non-stationary) and large-time (stationary) behavior are included, but more emphasis is given to the latter. The oscillator responses studied included the linear SDF system, linear 2DF systems, and yielding SDF systems. The basic purposes of this study are, for each oscillator, to investigate the behavior of the first-passage probability by use of Monte Carlo simulation data and theoretical approaches, and to seek empirical formulas which approximate the results adequately for design purposes. Simple procedures are presented for the linear SDF system which allow approximate prediction of the probability of first-passage for both long-time and short-time situations. This includes both the zero-start situation and the stationary-start situation. For a linear 2DF system with nearly equal modal frequencies, an equivalent SDF model is developed by consideration of response correlation coefficients. The model is shown to agree with the 2DF first-passage behavior when modal frequencies differ by less than 20 percent. A diffusion model is developed to approximate the first-passage probability of linear 2DF systems. The procedure is basically parallel to the ideas of Roberts for SDF response. The approximate formulation uses the amplitudes of two response modes as components of a Markov vector to establish a two-dimensional diffusion model. Within any short-time increment the two amplitudes are assumed to evolve independently of each other. The model is simplified to a one-dimensional diffusion when the second frequency is very large. Based on simulation and diffusion results, relatively simple empirical formulas are presented to approximate the long-time first-passage probabilities for 2DF systems. The yielding SDF system studied is the bilinear hysteretic oscillator which has previously been approximated by third-order linear systems and two-mode linear systems in studies of power spectral density and mean-squared response. Here the first-passage statistics of the yielding system are compared with results for the "equivalent" two-mode and third-order linear models. Diffusion results for the two-mode linear system are feasible only when both modes are under-damped, and this excludes approximation of several yielding situations. Diffusion results for the third-order linear model seem to better fit the existing simulation data for the yielding system than do the results for two-mode models.