In the first half of the thesis, a novel approach is developed for determining the response of a lightly damped nonlinear single-degree of freedom system to a random excitation with an evolutionary broad-band power spectrum. The new approach is based on the coupling of the concepts of stochastic averaging and equivalent linearization. The nonlinearities can be either of the hysteretic or of the 'zero-memory' kind. Moreover, approximate analytical relationships for evaluating the response variance are derived for a number of oscillators. The efficiency and accuracy of the approach is demonstrated by pertinent digital simulations.;In the second half of the thesis, an approximate analytical approach is presented for examining the first-passage problem in context with the response of a class of lightly damped nonlinear oscillators to broad-band random excitations. A Markovian approximation both of the response amplitude envelope and of the response energy envelope is considered. This modeling leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. The Kolmogorov equation is solved approximately by employing a Galerkin approach. A set of confluent hypergeometric functions is used as an orthogonal basis for the expansions which are involved in the application of the Galerkin approach. The reliability of the derived analytical solution is demonstrated by comparisons of digital data derived by Monte Carlo simulation.
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