This work focuses on mechanical systems in which one or more of the frequencies of oscillation are dynamical variables, whose evolutions are dependent on the state of the system. As time evolves these systems can slowly move in and out of a state of resonance. This proposal investigates the dynamical behavior of these slowly-varying systems near resonance, accounting for the full complexity of the coupling between components that exists in these systems. Two primary research topics are to be addressed in this work, i) the effects of resonances on the dynamical behavior of directly coupled, multi-degree-of-freedom systems, and ii) resonant effects in nonlinear systems in the presence of period-amplitude dependence of one or more of the components. These include mistuned systems where the components with dynamical frequencies are coupled to all the remaining resonant modes as well as examples in the presence of additional internal resonances. The multi-degree-of-freedom systems will be reduced using singular perturbation methods and the transport of phase space through the resonance region will be analyzed through analytical techniques and numerical investigations. Finally, the analytical predictions will be verified on an existing experimental apparatus that will be used to more fully account for the complexities of physical systems. Resonances in dynamical systems provide a mechanism by which complex coupled systems can undergo unexpected qualitative changes in their dynamical behavior, and this work will lead to a more complete understanding of this behavior. These topics represent a significant class of problems with relevance to complex, interconnected mechanical systems and will identify parameters and the characteristics of coupling which play an important role in determining the impact of resonances in interconnected mechanical systems.
