Dynamics of Noisy Nonlinear Mechanical Systems

这项研究的总体目标是建立和发展方法来分析旋转和陀螺系统中固有的噪声、对称性和非线性之间的复杂相互作用，特别是旋转柔性盘和旋转轴的动力学。该方法包括合理地构建低维随机模型，系统地捕捉噪声机械系统的基本特征。所提出的工作的基础是随机过程的极限理论的集合，其建模一类具有主导哈密顿动力学的工程问题。对于简化的模型，人们寻找响应和稳定性的标准统计度量；例如，平均退出时间、李亚普诺夫指数和平稳度量，以便有可能设计可靠的工程系统。这项拟议的工作将是一项跨学科的工作，从建模和新的分析技术到实验验证。我们计划将拟议计划中开发的创新理论和模拟转化为行业创新理论和模拟，这将使各种旋转系统的设计得到改进，并提高各种旋转系统的性能。拟议工作的最终结果之一是基于网络的模拟器，将用于高年级本科生和研究生课程，并为工业设计师提供避免不必要行为的预测工具。这将增强对噪音对实际工程问题影响的直观理解。

The overall goal of this investigation is to formulate and develop methods to analyze the complex interactions between noise, symmetries, and nonlinearities inherent in rotating and gyroscopic systems, particularly the dynamics of spinning flexible discs and rotating shafts.The approach consists of a rational construction of low-dimensional stochastic models that systematically capture essential features of noisy mechanical systems. The basis of the proposed work is a collection of limit theories for stochastic processes which model a class of engineering problems with dominant Hamiltonian dynamics. For the reduced model, one seeks standard statistical measures of response and stability; e.g., mean exit times, Lyapunov exponents and stationary measures, so that it may be possible to design reliable engineering systems. The proposed work will be a cross-disciplinary one, spanning the spectrum from modeling and novel analytical techniques to experimental verification.We plan to transfer to industry innovative theory and simulations developed in the proposed program which will allow improved design and greater performance of various rotating systems.An integral part of the final outcome of the proposed work will be web-based simulators to be used in senior undergraduate and graduate courses and to provide industrial designers with a predictive tool to avoid unwanted behavior. This will enhance visual understanding of the effect of noise on real engineering problems.