The dynamics of two three-degree-of-freedom systems with quadratic nonlinearities are studied. The first system has two simultaneous two-to-one internal resonances. The second has a combination internal resonance. In both cases the response to a primary resonant excitation of the third mode is studied. The method of multiple time scales is used to obtain the equations that govern the amplitudes and phases of the first system. Then the fixed points of these equations are obtained and their stability is determined. The fixed points undergo Hopf bifurcations, and the overall system response can be periodic or periodically, quasiperiodically, or chaotically modulated. The method of the time-averaged Lagrangian is used to obtain the equations that govern the amplitudes and phases of the second system. The fixed points of these equations are obtained and their stability is determined. These fixed points undergo Hopf bifurcations, and the overall system response can be periodic or a two- or three-torus. / Master of Science

Three methods are developed to study the response of nonlinear systems to combined deterministic and stochastic excitations. They are a second-order closure method, a generalized method of equivalent linearization, and a reduced non-Gaussian closure scheme. Primary resonances of single- and two-degree-of-freedom systems in the presence of different internal (autoparametric) resonances are investigated. We propose these methods to overcome some of the limitations of the existing methods of solution and explain some observed experimental results in the context of the response of randomly excited nonlinear systems. These include non-Gaussian responses, broadening effects, and shift in the resonant frequency. When available, the results of these methods are compared with those obtained by using the exact stationary solutions of the Fokker-Planck-Kolmogorov equation. It is found that the presence of the nonlinearities bend the frequency-response curves and this causes multi-valued regions for the mean-square responses. The multi-valuedness is responsible for a jump phenomenon. The results show that for some range of parameters, noise can expand the stability region of the mean response. As applications, we study the response of a shallow arch, a string, and a hinged-clamped beam to random excitations. / Ph. D.