FORM, SORM and simulation techniques for nonlinear random vibrations

Approximate solution methods for nonlinear random vibration problems are developed using computational tools of the time-invariant structural reliability theory. The basic framework of the approach is composed of: (1) representation of the input stochastic excitation in terms of a finite number of random variables, (2) formulation of each response statistic of interest in terms of one or more limit-state functions of the random variables, and (3) estimation of the response statistic using computational reliability tools.; An important step in the proposed approach is the finding of the design point, which is the point on the limit-state surface that is nearest to the origin in a transformed standard normal space. This point is usually found by solving a constrained optimization problem, requiring repeated computations of the limit-state function and its gradient. In this thesis, a new method for determining the excitation corresponding to the design point is presented. The basic idea starts from the finding that, for a linear oscillator subjected to a stationary Gaussian white-noise excitation, the "design-point excitation" is a linear function of the unit-impulse response function of the oscillator. Inspired by this idea, we investigate the dynamic characteristics of nonlinear oscillators by observing their free vibration motion and its mirror image. It is shown that for a nonlinear elastic single-degree-of-freedom (SDOF) oscillator subjected to a stationary Gaussian white noise, the design-point excitation is identical to the excitation that generates the mirror image of the free vibration response, when the oscillator is released from the target threshold. This idea is extended to general nonlinear systems, including systems having hysteretic behavior and multi-degree-of-freedom systems subjected to non-white and non-stationary excitation, for an approximate solution of the design-point excitation. The design-point excitation is the most likely realization of the stochastic excitation to produce a target response threshold.; The accuracy and effectiveness of solution tools such as the first-order reliability method (FORM), the second-order reliability method (SORM), and various sampling techniques are investigated in the context of nonlinear random vibration analysis.