Parametrically Excited Nonlinear Gyroscopic Systems

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PI: N. Sri NamachchivayaUniversity of Illinois @ Urbana-ChampaignProposal (# 0084944) Title: Parametrically Excited Nonlinear Gyroscopic SystemsThe overall goal of our proposed investigation is to formulate and develop methods to study the long term effects of small dissipative, symmetry-breaking and time-dependent perturbations on nonlinear gyroscopic systems, particularly the dynamics of rotating shafts and pipes conveying fluid. This proposal outlines a unified framework to study nonlinear systems with either periodic} or stochastic perturbations. An understanding of the dynamics of parametrically excited gyroscopic systems necessitates a study of the complex interactions between time-dependent inputs, symmetries, and nonlinearities. Our approach will consist of the application of some recent theories of deterministic and stochastic dimensional reduction to relevant nonlinear gyroscopic models. The proposed work consists of essentially four components: appropriate modelling, further development of some theoretical considerations, numerical algorithms, and experimental verification. The outcome will be a greatly-enhanced understanding of the stability and global dynamics of gyroscopic systems under dissipation and time-dependent perturbations. In the deterministic context, we will be able to predict global dynamics and the mechanisms which give rise to global bifurcations in gyroscopic systems. We shall also examine stabilization of gyroscopic systems by periodic excitations when the excitation frequency is slightly above a certain resonance frequency.In the stochastic context, we will be able to compute a number of standard stability indices (e.g., stationary measures and exit times) in a theoretically correct and computationally efficient way. In the final part of this research, dynamic experimentswill be conducted on a rotating shaft to verify the theoretical results obtained. These dynamics experiments will locate the stability boundaries and examine the nature of the nonlinear response. The numerical and experimental results will, in turn, guide the theories to incorporate any new phenomena observed.