Dynamics of randomly perturbed nonlinear structural and mechanical systems with resonances

We study the dynamics of randomly perturbed nonlinear structural and mechanical systems with resonances. In the first part, we investigate the asymptotic behaviors of shell type structures under random excitations. The objective of this investigation is achieved by formulating methods to analyze complex interactions between nonlinearities and random excitation. The normal form is employed not only to capture the essential dynamics of the system in which significant nonlinearities are considered but also to reduce the dimensionality. Due to the random nature of forcing characteristics, the theory of stochastic processes is used to explore responses of the structural systems in the presence of noise. With adequate scaling of parameters, the stochastic averaging is applied for the analytical study. It turns out that the solution of the reduced model is approximated by a Markov process which takes its value on a graph with gluing conditions which furnish the complete specification of the dynamics of the reduced model. The limiting generator of the Markov process on the graph enables us to examine the standard statistical measures of response and stability such as mean exit time, probability density and stochastic bifurcation.; In the second part, we investigate the dynamics of 1-DOF systems excited by both periodic and random perturbation. The near resonant motion of such systems is not well understood. We will study this problem in depth with the aim of discovering a common geometric structure in the phase space, and determine the effects of noisy perturbations on the passage of trajectories through the resonance zone. We consider the noisy, periodically driven Duffing equation as a prototypical 1-DOF system and achieve a model-reduction through stochastic averaging. The solution of the reduced model can be approximated by a Markov process. Depending on the strength of the noise, the reduced Markov process takes its values on a line or on a graph with certain gluing conditions at the vertices of the graph. The reduced model will provide a framework for computing standard statistical measures of dynamics and stability. This work will also explain a counter-intuitive phenomenon of stochastic resonance.