Parametrically Excited Nonlinear Gyroscopic Systems

题目：参数激励非线性陀螺仪系统我们提出的研究的总体目标是制定和发展方法来研究非线性陀螺仪系统的长期影响的小耗散，对称破坏和时间相关的扰动，特别是旋转轴和管道输送流体的动力学。本文提出了一个统一的框架来研究具有周期性或随机扰动的非线性系统。要理解参数激励陀螺仪系统的动力学，就必须研究随时间输入、对称性和非线性之间的复杂相互作用。我们的方法将包括一些最新的确定性和随机降维理论应用于相关的非线性陀螺仪模型。提出的工作主要由四个部分组成：适当的建模，一些理论考虑的进一步发展，数值算法和实验验证。结果将极大地增强对耗散和时变扰动下陀螺仪系统的稳定性和全局动力学的理解。在确定性的背景下，我们将能够预测全局动力学和在陀螺系统中引起全局分岔的机制。当激励频率略高于某一共振频率时，我们还将研究周期激励对陀螺仪系统的稳定化作用。在随机环境中，我们将能够以理论上正确且计算有效的方式计算许多标准稳定性指标（例如，平稳度量和退出时间）。在本研究的最后一部分，将对一个旋转轴进行动态实验，以验证所获得的理论结果。这些动力学实验将确定稳定性边界并检验非线性响应的性质。反过来，数值和实验结果将指导理论纳入观察到的任何新现象。

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.